ABSTRACT
Title of dissertation: Access Scheduling and Controller Design
in Networked Control Systems
Lei Zhang, Doctor of Philosophy, 2005
Dissertation directed by: Assistant Professor Dimitrios Hristu-Varsakelis
Department of Mechanical Engineering
A Networked Control System (NCS) is a control system in which the sensors
and actuators are connected to a feedback controller via a shared communication
medium. In an NCS, the shared medium can only provide a limited number of
simultaneous connections for the sensors and actuators to communicate with the
controller. As a consequence, the design of an NCS involves not only the specifica-
tion of a feedback controller but also that of a communication policy that schedules
access to the shared communication medium. Up to now, this task has posed a sig-
nificant challenge, due in large part to the modeling complexity of existing NCS ar-
chitectures, under which the control and communication design problems are tightly
intertwined.
This thesis proposes an alternative NCS architecture, whereby the plant and
controller choose to ?ignore? the actuators and sensors that are not actively com-
municating. This new architecture leads to simpler NCS models in which the design
of feedback controller and communication polices can be effectively decoupled. In
that setting, we propose a set of medium access scheduling strategies and accom-
panying controller design methods that address a broad range of stabilization, esti-
mation, and optimization problems for a general class of NCSs. The performance
of the proposed methods is illustrated through a set of simulations and hardware
experiments.
ACCESS SCHEDULING AND CONTROLLER DESIGN
IN NETWORKED CONTROL SYSTEMS
by
Lei Zhang
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2005
Advisory Committee:
Assistant Professor Dimitrios Hristu-Varsakelis, Chair/Advisor
Professor William S. Levine, Dean?s Representative
Professor Amr Baz
Professor Balakumar Balachandran
Associate Professor Jeffrey Herrmann
c? Copyright by
Lei Zhang
2005
To my mother, Shulan An, and my father, Chongxi Zhang.
ii
ACKNOWLEDGMENTS
I would like to express my sincere appreciation to my advisor, Prof. Dimitrios
Hristu-Varsakelis, who has guided me through my four-year Ph.D. research with his
patience, vision, and wisdom. Dr. Hristu is never satisfied by mediocre research, he
has always encouraged me to challenge myself with perfectionism and persistence. I
thank him for helping me understand the essence of scientific research and find the
real potential of myself.
My thanks also go to my thesis committee members, Prof. William S. Levine,
Prof. Balakumar Balachandran, Prof. Amr Baz, and Prof. Jeffrey Herrmann, for
their invaluable advice on how to improve my thesis.
I must acknowledge the help and support of the following professors, fellow
students, and friends:
? Prof. P. S. Krishnaprasad, who taught me geometric control theory in his
class ENEE769Q, and inspired many of my research ideas during the spring
semester of 2003, while he was serving as my temporary advisor.
? Prof. Stuart. S. Antman, who led me into the magnificent palace of classical
mechanics during his class AMSC698P in fall 2002.
? Mr. Cheng Shao who has been my labmate, roommate, and always a friend
iii
in need.
? My labmates, Mr. Kevin Roy Kefauver, Mr. Philip Yip, and Mr. Harry
Kurnianto Himawan for their help and friendship.
? Cheng Shao, Gengsheng Lv, Guojing Tang, Peng Lin, Wei Xi, Fang Rong,
Xue Mei, and Wenjing Weng, my eight closest friends in Maryland. I am glad
to be part of this fantastic ?gang of nine?.
My gratitude to my mother, Shulan An, my father, Chongxi Zhang, and my
little sister, Xin Zhang, can not be expressed in words. Their love and support has
always been the greatest source of comfort in the thirty year of my life.
Last, but not the least, I want to thank my girlfriend, Sha Li, who has accom-
panied me through this journey, helped me to have rediscovered myself, and given
me the most wonderful time in my life.
iv
Contents
1 Introduction 1
1.1 Fundamental Issues in NCSs . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Scope and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Networked Control Systems 10
2.1 A Brief History on NCSs . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Data Rate Constraints . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Quantized Feedback . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Data Packet Dropout . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.5 Medium Access Constraints . . . . . . . . . . . . . . . . . . . 21
3 Medium Access Constraints in NCSs 29
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Communication Sequence . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Effects of Medium Access Constraints . . . . . . . . . . . . . . . . . . 32
v
3.4 The Extended Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.2 Switched System Expression . . . . . . . . . . . . . . . . . . . 35
3.5 A Discrete-time Formulation . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 NCSs with Nonlinear Plants . . . . . . . . . . . . . . . . . . . . . . . 37
4 Static Medium Access Scheduling 39
4.1 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Communication Sequences that Preserve
Controllability and Observability . . . . . . . . . . . . . . . . 42
4.1.2 Output Feedback Stabilization . . . . . . . . . . . . . . . . . . 46
4.1.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 Communication Sequences that Preserve
Reachability and Observability . . . . . . . . . . . . . . . . . 51
4.2.2 Choosing Communication Sequences . . . . . . . . . . . . . . 56
4.2.3 Output Feedback Stabilization . . . . . . . . . . . . . . . . . . 58
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Example 1: Continuous-time . . . . . . . . . . . . . . . . . . . 61
4.3.2 Example 2: Discrete-time . . . . . . . . . . . . . . . . . . . . 65
5 Dynamic Medium Access Scheduling 68
5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 An Equivalent Switched System . . . . . . . . . . . . . . . . . . . . . 70
5.3 Stable Convex Combinations . . . . . . . . . . . . . . . . . . . . . . . 73
vi
5.4 Dynamic Medium Access Scheduling Policies . . . . . . . . . . . . . . 76
5.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 LQG Control in NCSs 89
6.1 An Equivalent NCS Model . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1.1 Effects of Medium Access Constraints . . . . . . . . . . . . . . 90
6.1.2 The Extended Plant . . . . . . . . . . . . . . . . . . . . . . . 92
6.2 LQG Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Communication Sequences that Preserve
Stabilizability and Detectability . . . . . . . . . . . . . . . . . . . . . 95
6.3.1 Controllability and Reconstructibility . . . . . . . . . . . . . . 96
6.3.2 Stabilizability and Detectability . . . . . . . . . . . . . . . . . 99
6.4 The LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.1 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.2 LQ Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4.3 Periodic Riccati Equations . . . . . . . . . . . . . . . . . . . . 104
6.4.4 Convergence of the LQG Controller . . . . . . . . . . . . . . . 106
6.4.5 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 NCS Experiments 112
7.1 Experiment Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.1.1 Hardware Configuration . . . . . . . . . . . . . . . . . . . . . 112
7.1.2 Software Configuration . . . . . . . . . . . . . . . . . . . . . . 114
vii
7.2 Experiment 1: Stabilization under Dynamic
Medium Access Scheduling . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2.1 Inverted Pendulum Model . . . . . . . . . . . . . . . . . . . . 118
7.2.2 NCS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2.3 Simulink Implementation . . . . . . . . . . . . . . . . . . . . . 125
7.2.4 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Experiment 2: LQG Control under
Static Access Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3.1 NCS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.3.2 Simulink Implementation . . . . . . . . . . . . . . . . . . . . . 141
7.3.3 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . 143
8 Conclusions and Future Work 145
8.1 Opportunities of Future Work . . . . . . . . . . . . . . . . . . . . . . 147
A Calculating the Time-varying Feedback and Observer Gains 151
A.1 Solving for the Gramian Wa(t,t+T) . . . . . . . . . . . . . . . . . . 152
A.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A.1.2 The Gramian ODEs . . . . . . . . . . . . . . . . . . . . . . . 154
A.2 Solving for the Gramian [e?ATTMa(t?T,t)e?AT] . . . . . . . . . . . 156
A.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.2.2 The Gramian ODEs . . . . . . . . . . . . . . . . . . . . . . . 157
B The Dynamics of a Rotary Inverted Pendulum 159
viii
List of Tables
7.1 Parameters of the two pendulum rods. . . . . . . . . . . . . . . . . . 120
7.2 Electro-mechanical parameters of the servo motor the gear set, and
the rotary arm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.3 Average medium access percentage of two pendulums under the WFD
rule. The average shown are taken over time and over three runs of
the experiments (5000 data points in each run). . . . . . . . . . . . . 128
7.4 Average dwell-times under the WFD rule and GD rule. The average
shown are taken over time and over three runs of the experiments
(5000 data points in each run). . . . . . . . . . . . . . . . . . . . . . 137
ix
List of Figures
1.1 A control system with a point-to-point wiring configuration. . . . . . 2
1.2 A control system with a decentralized control configuration. . . . . . 3
1.3 A control system with a hierarchical control configuration. . . . . . . 4
1.4 A control system with an NCS configuration: sensors and actuators
are connected to the controller via a shared communication medium. 4
2.1 Controller Area Networks (CAN) in automobiles. . . . . . . . . . . . 12
2.2 Generic configuration of an NCS: sensors and actuators of an MIMO
plant are connected to a centralized feedback controller via a shared
communication medium. . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 A control system with time delays in its feedback loop. . . . . . . . . 14
2.4 An NCS having data-rate constraints in the communication medium. 17
2.5 An MIMO plant with medium access constraints existing only on the
output side. A ZOH is used at the receiving end of the communication
medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 A ?block-diagonal? NCS: a collection of uncoupled continuous linear
systems share a communication medium to close their feedback loops. 24
3.1 An NCS having m actuators and p sensors, with medium access con-
straints existing on both sensor and actuator sides. . . . . . . . . . . 30
3.2 The extended plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
x
4.1 Output feedback stabilization of an NCS: The controller consists of
an observer followed by a time-varying feedback gain K(t). . . . . . . 46
4.2 Output feedback stabilization of an NCS: The controller consists of
a discrete-time observer and a time-varying feedback gain K(k). . . . 59
4.3 Simulation results: Stabilization of an NCS using an observer-based
continuous-time controller. The plant was an LTI system with two
inputs and two outputs. The communication medium provided one
input channel and one output channel. The communication sequence
?(t) assigned 30% of the input channel access time to u1 and 70%
to u2; while the communication sequence ?(t) assigned 70% of the
output channel access time to y1 and 30% to y2. . . . . . . . . . . . . 63
4.4 Simulation results: Stabilization of the same NCS under the same
communication sequences and controller as in Fig. 4.3, while a white
noise of 0.01W as well as a 0.02s fixed transmission delay was added
at each input and output of the plant. . . . . . . . . . . . . . . . . . 64
4.5 Simulation results: Stabilization of an NCS using an observer-based
discrete-time controller. The plant was a discrete-time LTI system
with two inputs and two outputs. The communication medium pro-
vided one input channel and one output channel. . . . . . . . . . . . 67
5.1 An NCS in which the plant?s state information is available, and the
controller is constant feedback. . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Simulation results: Stabilization of an NCS using the FD rule. The
plant was an LTI system with 2 inputs and 4 outputs. The commu-
nication medium provided one input channel and one output channel. 84
5.3 Simulation results: Stabilization of an NCS using the GD rule (?0 =
0.1, ?? = 1). The plant was an LTI system with 2 inputs and 4
outputs. The communication medium provided one input channel
and one output channel. . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Simulation results: Stabilization of an NCS using the WFD rule.
The plant was an LTI system with 2 inputs and 4 outputs. The
communication medium provided one input channel and one output
channel. The input u1 was given higher access priority. . . . . . . . . 87
xi
5.5 Simulation results: Stabilization of an NCS using the WFD rule.
The plant was an LTI system with two inputs and two outputs. The
communication medium provided one input channel and one output
channel. The input u2 was given higher access priority. . . . . . . . . 88
6.1 The extended plant under the new NCS formulation. . . . . . . . . . 92
6.2 An NCS stabilized by the LQG controller. . . . . . . . . . . . . . . . 101
6.3 Evolution of tr(P(k)) and tr(K(k)). The P(k) satisfy the Riccati
equation (6.17), while K(k) satisfy the backwards Riccati equation
(6.21). Both P(k) and K(k) converge to SPPS solutions. . . . . . . . 110
6.4 Simulation Results: Stabilization of an NCS under the LQG con-
troller. The plant was a stochastic LTI system with 2 inputs and 2
outputs. The communication medium provided 1 input channel and
1 output channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1 Hardware configuration of the NCS experiment platform. . . . . . . . 113
7.2 Flow of data and controls in RTLinux [1]. . . . . . . . . . . . . . . . 115
7.3 The SimuLinux-RT real-time control system prototyping environment
[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4 Experiment 1: Stabilizing two rotary inverted pendulums at their
unstable equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.5 Configuration of a rotary inverted pendulum: a pendulum rod con-
nects with a rotary arm via a hinge, and the rotary arm is driven by
a servo motor. Two encoders output the angles of the rotary arm and
the pendulum rod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.6 NCS configuration of Experiment 1: the plant consisted of two un-
coupled rotary inverted pendulums. The two inverted pendulums
communicated with a remote controller via a shared RS232 medium. 119
7.7 A simplified model of rotary inverted pendulum. . . . . . . . . . . . . 120
7.8 Simulink implementation of Experiment 1. . . . . . . . . . . . . . . . 125
7.9 The WFD rule switching logic. . . . . . . . . . . . . . . . . . . . . . . 126
xii
7.10 The GD rule switching logic. . . . . . . . . . . . . . . . . . . . . . . . 127
7.11 Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.1, w2 = 0.9. . . . . . . . . . . 129
7.12 Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 =
0.1, w2 = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.13 Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.5, w2 = 0.5. . . . . . . . . . . 131
7.14 Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 =
0.5, w2 = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.15 Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.9, w2 = 0.1. . . . . . . . . . . 133
7.16 Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 =
0.9, w2 = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.17 Experiment data: Evolution of the states under the GD rule with
?0 = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.18 Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the GD rule with ?0 = 0.05. . . . . . . . . . . . . . . . . . 136
7.19 NCS configuration of Experiment 2: the plant is a simulated LTI
system with 4 inputs and 4 outputs; the plant communicates with
an LQG controller via the shared RS232 communication channel. A
global clock is used to synchronize the dynamics of the plant and the
controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.20 Simulink implementation of Experiment 2. . . . . . . . . . . . . . . . 142
7.21 Experiment data: Stabilization of an NCS using LQG controller. The
plant was an LTI system with 2 inputs and 2 outputs. The commu-
nication medium provided 1 input channel and 1 output channel. . . 144
xiii
Chapter 1
Introduction
In the last two decades, advances in communication, control, and computation tech-
nologies have fueled the rise of a modern control system architecture in which
sensors and actuators exchange information with a feedback controller through a
shared communication medium. Control systems having this configuration have
been termed Networked Control Systems (NCSs). Compared to conventional sys-
tem architectures, the advantages of NCSs include reduced system wiring, ease of
diagnosis and maintenance, low cost, and increased system flexibility.
NCSs are rapidly becoming ubiquitous, and have made inroads in a broad range
of applications. Most modern aircrafts are now built with the ?fly-by-wire? [3] tech-
nology, which relies on a databus (e.g., MIL-STD-1553 and ARINC-629) to connect
various sensors, actuators, and controllers within an airplane. The Controller Area
Networks (CAN) are widely used to close a series of important feedback loops in
the control systems of modern automobiles. NCSs can also be found in modern
factories, where Fieldbuses [4] play an important role to connect physically sep-
arated devices in machinery and instruments; in modern office buildings, where
special-purpose networks are used to control and monitor various security and en-
1
vironmental functions. More recently, NCSs have become the enabling architecture
for many emerging applications which require coordination of significant numbers of
spatially distributed sensors and actuators. These include tele-surgery, satellite ar-
rays, sensor networks [5], swarms of unmanned aerial and underwater vehicles [6] [7],
arrays of micro-valves [8], optical switches [9], RF switches [10], and MEMS-based
spatial light modulators [11], to name only a few.
The rise of NCSs stems in part from necessity. Traditionally, control systems
have been implemented using point-to-point wiring, i.e., each sensor and actuator is
connected to a centralized controller (often a micro-processor) via a designated wire
(Fig.1.1). This configuration ensures real-time communication between components
of a control system. However, as the complexity and scale of a control system
increase, point-to-point wiring becomes cumbersome or impractical: the increased
wiring burden brings problems involving weight, cost, maintenance, and reliability.
At the same time, the microprocessor in which the controller is implemented provides
a limited number of input/output (I/O) ports and limited computing capability, so
that point-to-point wiring becomes impossible when the number of sensors and
actuators is greater than the number of I/O ports or when the computation load
needed exceeds the capability of the processor.
....Actuator Actuator
Sensor
Actuator Actuator
Sensor SensorSensor
Actuator
Sensor
Actuator
Sensor
Plant
Controller
....
.... ....
Figure 1.1: A control system with a point-to-point wiring configuration.
2
An alternative to point-to-point wiring is to introduce additional controllers and
allow decisions to be made ?locally?. This idea gave rise to the ?decentralized?
(Fig. 1.2) and the ?hierarchical?(Fig. 1.3) control system configurations. Decentral-
ized control is well-suited in applications where the plant consists of multiple sub-
systems which have little or no coupling with each other. In the cases where control
decisions must be made by considering information that is not locally available, one
can add controllers to the decentralized configuration, resulting in a hierarchical
structure where proper actuation of the plant relies on the intelligence of both local
and high-level decision makers. Both the decentralized and hierarchical configura-
tions require a certain degree of ?decoupling? in the dynamics of the plant, so that
some decisions can in fact be made locally. This is often not the case in modern com-
plex systems. Moreover, the introduction of decentralized or hierarchical controllers
increases the costs of such systems and creates other design difficulties.
Controller
Actuator
Sensor
ActuatorActuator
Sensor Sensor
Actuator Actuator
Sensor Sensor
Actuator
Sensor
Plant
.... ....
....
....
Controller Controller Controller
Figure 1.2: A control system with a decentralized control configuration.
The introduction of NCSs (Fig. 1.4) in the 1970s provided a neat solution to the
problems highlighted above. In an NCS, the controller is connected to a communi-
cation medium that provides access to all the sensors and actuators; the medium
is shared by all its users, so that only a limited number of connections can be sup-
ported simultaneously. The NCS configuration has proved remarkably flexible, en-
abling many novel features in feedback control systems. For example, using wireless
3
Controller
Actuator
Sensor
Actuator
Sensor Sensor
Actuator
Sensor
Actuator
Sensor
Actuator Actuator
Sensor
Plant
.... ....
....
....
Controller Controller Controller
Controller Controller
Controller
Figure 1.3: A control system with a hierarchical control configuration.
communication, sensors and actuators in an NCS can easily change their locations
to form different ad hoc groups that are customized for different tasks.
Controller
Actuator Actuator
Sensor Sensor
Actuator Actuator
Sensor SensorSensor
Actuator
Sensor
Actuator.... ....
Plant
Shared Communication Medium
Figure 1.4: A control system with an NCS configuration: sensors and actuators are
connected to the controller via a shared communication medium.
1.1 Fundamental Issues in NCSs
Despite its flexibility and effectiveness, the NCS architecture also introduces new
problems which have until recently been beyond the scope of traditional control
4
systems theory. Classical systems theory was developed based largely on the as-
sumptions of real-time, unlimited exchanges of data between sensors, actuators, and
the feedback controller. However, in an NCS, data exchanges are always subject to
various communication constraints. These include [12]:
? Communication delays that arise in the data exchanges between components
(e.g., from a controller to an actuator). Depending on the protocol that is used
in the communication medium, these delays can be constant, time varying, or
randomly distributed.
? Bandwidth limitations that depend on the communication protocol used, the
network equipment, andthephysical properties ofthecommunication medium.
? The possibility of data packet loss. Data packet loss can be caused by hardware
failure or by the communication protocol (TCP/IP is an well-known example).
? The maximum number of simultaneous medium access channels allowed by
the medium. This number could be much smaller than the number of sensors
and actuators in the system. This medium access constraint will be the focus
of this thesis.
These constraints all present potential problems whose effects on closed-loop per-
formance and controller design must be understood and dealt with. In Chapter 2,
we will give a detailed review of recent theoretical advances in the analysis and de-
sign of NCS under the above listed constraints. Furthermore, we remark that, aside
from communication medium related constraints, similar limitations on information
exchanges within a control system may also arise because of the limited computing
capability of the controller or the performance of the sensors and actuators. Thus,
the fundamental issues addressed in the study of NCSs in fact concern Feedback
5
Control using Limited Information. This is a relatively new aspect of systems the-
ory, which had received little attention before NCSs became a prominent topic of
research. In today?s information-centric world, the efforts made in studying the in-
terplays between communication and control are likely to continue to contribute in
a broad range of problems involving technological, scientific, and social-economical
systems.
1.2 Scope and Contributions
The aim of this thesis is to explore the design of NCS control and communication
policies under the presence of medium access constraints. Specifically, we study
NCSs in which the communication medium can only provide a limited number of
simultaneous medium access channels. As a consequence, at any one time, only
a limited number of actuators and sensors are able to access the communication
medium to exchange information with the controller, while others must wait. We
address the following basic problems:
1. Schedule the access to a shared communication medium among multiple sensor
and actuators.
2. Design feedback controllers that achieve various control objectives (e.g., stabi-
lization, estimation) with limited access to the system?s sensors and actuators.
In previously proposed NCS architectures, the interaction between control and
communication gave rise to models whose complexity made it difficult to design
controllers and communication policies. Existing methods only address the design
of a controller for a fixed communication sequence or the selection of a stabilizing
6
medium access policy given a controller that was designed in advance without con-
sidering communication constraints. The co-design of the controller and the medium
access scheduling strategy remains an open problem. Moreover, most previous re-
sults only address simple NCS configurations where the plant is block-diagonal [13]
or the access constraints only exist on the output side of the plant [14, 15].
This thesis will show that the difficulties encountered in previous works were
partly due to the use of a zero order hold (ZOH) at the ?receiving end? of a com-
munication medium (i.e., at the plant?s and controller?s input stages). The use of
ZOH elements greatly increases the complexity of the closed-loop NCS because it
introduces time-varying delays and leads to closed-loop dynamics in which commu-
nication and control are tightly coupled (see, for example, the ?extensive form? in
[14]).
The main contribution of this thesis is to propose an alternative NCS architec-
ture, whereby the plant and controller forgo the use of a ZOH and instead choose
to ?ignore? (in a manner to be made precise) the actuators and sensors that are not
actively communicating. We will show that the new architecture leads to simpler
NCS models and elucidates the interaction between medium access scheduling and
the dynamics of the underlying plant and controller. Moreover, the new architecture
allows one to concurrently design the access scheduling strategy and the feedback
controller, and can be used to analyze and design for a more general class of NCSs
in which the plant has fully coupled dynamics, and medium access constraints ex-
ist in both the input and output stages of the plant. For this new configuration,
we will present a set of medium access scheduling policies and accompanying con-
troller design methods that address the fundamentals of stabilization, estimation,
and optimization in NCSs.
7
1.3 Thesis Outline
Chapter 2: We give a brief history of NCSs, and present a detailed literature
review of the state of the art in the design and analysis of NCSs under various
communication constraints.
Chapter 3: We introduce a new NCS protocol, in which the plant and controller
forgo the use of ZOHs, and instead choose to ?ignore? the actuators and sensors that
are not actively communicating. A pair of ?communication sequences? are used to
describe the instantaneous medium access status of the sensors and actuators. We
show that, under the new protocol, the effect of medium access constraints can be
modeled by cascading the underlying plant with a pair of ?communication sequence
matrices?.
Chapter 4: We study the stabilization of NCSs under periodic medium ac-
cess scheduling. We show that, for continuous-time systems, the controllability and
observability of a linear plant can be preserved as long as the communication se-
quences grant each input and output a finite interval of medium access during every
period; in the discrete-time case, one can always find (via a simple algorithm) peri-
odic communication sequences that preserve the reachability and observability of a
reversible linear plant. Using these effective communication sequences, an NCS can
be exponentially stabilized by an observer-based controller.
Chapter 5: We study the stabilization of NCSs under dynamic medium access
scheduling strategies. In this case, the communication sequences are determined on-
line, based on instantaneous state information of the plant. We present an algorithm
that provides a straightforward method for simultaneously determining stabilizing
gains and the scheduling policy. We introduce several feedback-based scheduling
policies that quadratically stabilize an NCS while achieving various objectives re-
8
lated to the system?s rate of convergence, the priorities of different sensors and
actuators, and the avoidance of chattering.
Chapter 6: We discuss Kalman filtering and LQ optimal control of a discrete-
time NCS under periodic medium access scheduling. We show that these problems
can be formulated as a standard LQG problem for an equivalent periodic system.
Moreover, there always exist periodic communication sequences that preserve the
detectability and stabilizability of the NCS, and thus make it possible to guarantee
the existence of a stabilizing LQG controller.
Chapter 7: We describe two NCS experiments which were designed to illustrate
the design techniques introduced in Chapter 5 and Chapter 6. The communication
medium we chose in these experiments was a RS232 serial channel. The medium
access scheduling strategies and the accompanying feedback controllers in the exper-
iments were designed within the SimuLinux-RT environment and were implemented
in the real-time operating system RT-Linux.
Chapter 8: We summarize the contributions of the thesis and discuss the pos-
sibilities of future research.
9
Chapter 2
Networked Control Systems
In this chapter, we first give a brief history of the use of communication networks
as part of a control system. We then continue with a literature review of recent
theoretical developments concerning the analysis and design of NCSs.
2.1 A Brief History on NCSs
The beginnings of NCSs can be traced at least as far back as the 1970s, when the
nuclear science community developed a parallel bus protocol called CAMAC [16].
CAMAC allowed a wide range of modular instruments to be interfaced to a stan-
dardized backplane called a DATAWAY. All modules could then be controlled by a
centralized controller by interfacing the DATAWAY to a computer. In 1973, another
bus protocol MIL-STD-1553 was released to meet the needs of aviation electronics
(referred to as avionics) applications. MIL-STD-1553 used ?Time Division Multi-
plexing (TDM)? to allow data transfers between multiple avionics units over a single
communication medium. Using MIL-STD-1553, navigation, weapon control, flight
control, and various other avionics subsystems in an airplane were able to share
10
their information without increasing the complexity of wiring. MIL-STD-1553 was
used in the U.S. Air Force?s F-16 and the Army?s attack helicopter AH-64A Apache
[17].
By the 1980s, engineers had realized that in the light of numerous new func-
tionalities added to a control system by the use of shared networks, the reduction
or elimination of wiring harnesses was just a convenient by-product. Since then, a
number of successful industrial control network protocols have been developed and
widely implemented in different industries. The Controller Area Network (CAN)
[18] was first developed by Robert Bosch GmbH, Germany in 1986 to connect ECUs
(electronic control units) in automobiles. The use of CAN networks in an automobile
enabled information exchange between power train and body subsystems such as en-
gine, transmission, brake, air-conditioning and lighting (Fig. 2.1). The earliest CAN
was based on a non-destructive arbitration mechanism, referred to as CSMA/CR
(Carrier Sense, Multiple Access, with Collision Resolution), which would grant bus
access to the message with highest priority without delay; there was no central bus
master. Today?s CAN can be implemented in topologies which are far more complex
than a single data bus. The latest version of CAN (Ver 2.0) uses a multi-master
bus configuration, and runs at a baud-rate of up to 1Mbit/s. The first CAN chip
was fabricated in 1987 by Intel, and more than 100 million CAN devices were sold
in the year 2000. Today, almost every new passenger car manufactured in Europe
is equipped with at least one CAN network. CAN is now an international standard
(ISO11898), its application is growing from vehicles to factory automation, building
automation, and beyond. New generations in the CAN family include DeviceNet
and CANopen.
Another family of industrial control network is Fieldbus. Fieldbus is a digital,
bi-directional, multidrop, serial communication network used to connect isolated
11
Transmission
Control
Cruise 
Control
Brake
Lock
Anti? Active
Control
Suspension
Control
Climate
HVAC
Power
Window
Power
Door
Lock
Dash Board Body BusBusDrive Train
Seat
Power
Control
Lighting
Control
Engine 
Figure 2.1: Controller Area Networks (CAN) in automobiles.
field devices, such as controllers, transducers, actuators and sensors in a machine or
instrument. Fieldbuses were first installed in early 1970s; by late 1980s, the most
popular field buses in Europe were French FIP (Factory Instrumentation Protocol)
and German ProfiBus (Process Field Bus). ProfiBus, for example, is a token-passing
network, it allows transfer speed of up to 500Kbit/s and the maximum length of
the bus is 1200m [4]. In 1995, after a long wait for the European Fieldbus protocols
to be standardized, the American companies decided to define their own field-bus,
named the Foundation Fieldbus (FF), which was optimized for the process control
industry. In 2000, a comprehensive standard, IEC61158, was finally released to
accommodate all field bus systems. Newer members in the Fieldbus family include
ControlNet, WorldFIP, P-Net, InterBus, and the family is still growing.
Building automation is also an important application area of control networks
and has given rise to communication protocols tailored to buildings and large struc-
tures. Released in 1995, BACnet (ANSI/ASHRAE135-1995) is ?a data communi-
cation protocol for Building Automation and Control Networks. It was designed to
control and monitor various equipments such as air-conditioning, lighting, alarm-
ing, and elevators in a modern office building. Being a new generation of control
network, BACnet is object-oriented. It also provides 35 BACnet services which are
standard messages that can be sent across a computer network for monitoring and
12
control of standard objects which specify the functionality of the various hardware
connected to the network. A BACnet can be implemented on common local area
networks (LAN) such as Ethernet, ARCNET and LonTalk [19] and hence can have
rather complex topologies.
As a result of the accelerating technological convergence of communications,
control, and computing, more recently, standard communication network protocols
such as daisy-chained RS232 [20], Ethernet [21], and IEEE802.11 wireless Ethernet
[22] are also making their way into the area of control systems. See [22] for a
detailed comparison of these network protocols from an NCS perspective and see
[21] for performance evaluations of these protocols.
2.2 Literature Review
Shared Communication Medium
sensor 2
sensor 1
...
sensor p
actuator 1
actuator 2
...
Plant
actuator m
Controller
Figure 2.2: Generic configuration of an NCS: sensors and actuators of an MIMO
plant are connected to a centralized feedback controller via a shared communication
medium.
The generic configuration of an NCS is illustrated in Fig. 2.2. As mentioned in
Chapter 1, data exchanges between sensors, actuators, and the feedback controller
of an NCS are always subject to various constraints that are imposed by the com-
13
munication medium. These constraints often manifest themselves in the forms of:
i) time delays, ii) data-rate limit, iii) data-packet dropout, and iv) medium access
constraints. The field of NCSs began to draw the attention of academic researchers
in the 1990s. Since then, a considerable volume of theoretical tools and results have
been produced. In this section, we give a literature review of some of the works
most relevant to this thesis. For organizational convenience, these works have been
categorized according to the communication constraints they address.
2.2.1 Time Delays
An obvious constraint likely to be imposed by the presence of a communication
medium is the time delay that occurs between the transmission and reception of a
data packet over that medium. Important components that contribute to the time
delays in an NCS include preprocessing time, waiting time, transmission time, post-
processing time, and et. al. [23]. The effects of all these delay components can be
typically captured by the sensor-to-controller delay ?sc and the controller-to-actuator
delay ?ca (Fig. 2.3).
CommuncationMediumCommuncation
Medium
Plant sensoractuator
Controller ?
sc?ca
Figure 2.3: A control system with time delays in its feedback loop.
In state-space models, time delays in the feedback loop of a control system can
be effectively captured by introducing additional states to keep track of the delayed
information. This technique is often termed ?state augmentation?(also known as
14
?state lifting? or ?state extensification?). For example, it is shown [24] that , for
a sampled-data control systems with constant feedback and a sample period h, a
constant delay ? of (r?1)h < ? < rh (where r ? N) will increase the system order
by a factor of r. For scalar linear systems, the relationship between the sampling
period and allowable time delay can be illustrated by a stability region plot [25],
which can be obtained via analytical or numerical methods.
Bounded time-varying delays in a control system?s feedback loop can also be
modeled byproper state augmentationto include theplant?s state andallthe delayed
control information; the details of this technique is illustrated in [25] and [26]. Also
using the augmented state model, some stability and performance analysis tools are
given in [27] for MIMO systems having multiple time delays in different feedback
loops. In [28], an LQR optimal control problem is formulated and studied based on
the same model.
A particularly thorough investigation of NCSs with random time delays can
be found in [4], where two classes of random delays are identified: independently
random delay and Markovian delay. For both cases, it is assumed that a time-stamp
is used to mark all transfered data with the time when they were generated, so that
the delay is always known to the controller. If the delays are independently random,
it is shown that the stability of the closed loop system under linear controller can
be checked with a criterion based on Kronecker product; if the random delays are
Markovian, then the state of the NCS can be modeled as a Markov chain and the
stability of the closed loop system can be checked with a similar criterion. Also in [4],
LQG optimal controllers are developed for systems having independently random or
Markovian delays, respectively. It is shown that, for both cases, the LQG optimal
controller is the combination of an optimal state feedback gain and a Kalman filter,
i.e., the separation principle applies.
15
It is possible to compensate known time delays in the feedback loop: if the
delays ?sc and ?ca are always known to the controller, the controller can propagate
its estimations of the plant?s states (based on a good model of the plant?s dynamics)
to the time when the control signals arrive at the actuators. The control signals
can then be calculated based on the adjusted state estimation. An example of such
a delay compensator is given in [25] for NCSs having fixed transmission delays.
Important results on time delays in dynamic systems are recently collected in [29],
another good survey on this topic can be found in [30].
2.2.2 Data Rate Constraints
Communication constraints also manifest themselves in the form of data rate limits
on the communication medium. The effects of data rate limits on networked control
systems have typically been studied from information theoretic perspectives, and
the communication medium is often modeled as a coded channel with a bandwidth
limit.
The work in [31] investigates state estimation in NCSs where the observations are
transmitted to an estimator with a finite data rate. The authors introduce a recur-
sive coder-estimator scheme, in which the coding decision can be dependent on the
whole past history of the observation process, and the estimator can be dependent
on the whole sequence of past codewords. Necessary and sufficient conditions are es-
tablished for the existence of stable and asymptotically convergent coder-estimator
schemes. Under the coder-estimator framework, feedback stabilization under data
rate constraint is investigated in [32], where the NCS configuration studied is illus-
trated Fig. 2.4. It is shown that, if the plant is a continuous-time LTI system, then
memoryless coding and control suffice to ensure the containability of the system,
16
meaning that given small enough initial conditions, the trajectory of the system will
lie in an n-dimensional sphere of an arbitrary size.
Communication medium with data?rate limit
actuator Plant sensor
Control
codeword
decoder
Quantizer/
Encoder
Controller
Codeword?based
Figure 2.4: An NCS having data-rate constraints in the communication medium.
The work in [33] investigates the stabilizability of infinite-dimensional linear
discrete-time plant when the controller receives observation data at a known rate.
It is shown that, under a finite horizon cost equal to the m-th output moment, the
problem reduces to quantizing the initial output. As the horizon approaches infinity,
asymptotic quantization theory can be applied to directly obtain the limiting coding
and control scheme. Necessary and sufficient conditions can then be derived for the
system to be asymptotically stabilizable in the m-th moment at a given data rate.
Under some restrictions on the initial condition distribution, a coding-estimation
scheme is presented in [34]; this scheme works for finite dimensional, time-varying
nonlinear system that satisfy a Lipschitz-type condition.
In [35], a sequential rate distortion function is used to study the achievable per-
formance of a discrete-time LQG system having a noisy observation channel with
limited bandwidth. By assuming equimemory on the encoder and decoder, it is
shown that the separation principle holds (i.e., the encoder-estimator and the con-
troller can be designed independently) and the optimal control law can be obtained
17
by solving the Riccati equation for a deterministic counterpart of the original prob-
lem. Based on this frame work, one can obtain the minimum data rate that ensures
the stability of the closed-loop system.
2.2.3 Quantized Feedback
An analog signal must be quantized before being transmitted via a digital commu-
nication medium. A quantizer acts as a functional that maps a real-valued function
into a piecewise constant function taking on a finite set of values [36]. Effective
quantization of sensor and actuator signals can help reduce the sizes of data packets
and thus reduce the data-rate in the communication medium.
A recursive quantized feedback stabilization policy is presented in [36], in which
the sensitivity of a quantizer increases as the system state approaches to zero. It
is shown that if a linear system can be stabilized by linear time-invariant feedback,
then it can be stabilized by quantized feedback with the presented policy. It is also
shown that choices of sampling period are decoupled from the issues regarding the
implementation of the quantized feedback control policy. This idea is independently
explored in [37].
In [38], it is shown that the coarsest quantizer that quadratically stabilize a SISO
discrete-time linear time-invariant system is a logarithmic quantizer, which can be
computed by solving an expensive control LQR problem. Moreover, a closed form
expression of the smallest logarithmic base can be derived exclusively in terms of
unstable eigenvalues of the system. The minimum densities for both sampling and
quantization that ensures stability were also discussed.
The work in [16] analyzes the effects of a general class of quantizers in the
sampled-data system setting, where a weaker notion of stability (similar to ?con-
18
tainability?) of the system is studied. The authors present analytical methods for
designing an effective quantizer, a sampled data controller, and to find the bound
on the sampling period that guarantee quadratic stability (the weaker notion) of the
closed-loop control system at a given decay rate. Related work can also be found in
[39], where the worst-case analysis and design of sampled-data control systems was
investigated.
The problem of jointly optimization of quantization, estimation and control is
studied in [40], where the plant to be controlled is modeled as a Hidden Markov
chain. Dynamic programming method is used to find the optimal quantization and
control scheme that minimize a cost function related to the estimation error, running
cost, the communication cost, and delay.
The work in [41] shows that, for the scalar case, binary control (a quantization
with only two admissible control values) is the most robust control strategy under a
time-varying data-rate constraint and asynchronism of sampling and control actu-
ation. A control synthesis approach based on binary control is also explored. This
approach uses a side channel (whose bandwidth can be arbitrarily low) to adjust
the magnitude of the binary control.
2.2.4 Data Packet Dropout
In real-world NCSs, data packets transmission between the plant and controller
may be ?dropped out? due to network congestion, unreliable hardware, or because
of the transmission protocol used (the transmission control protocol (TCP) and the
user datagram protocol (UDP) are two well-known examples). Also, sometimes it
is desirable to drop ?old? data-packets that fail to arrive in time and use newly-
generated data instead.
19
In [25], it is shown that an NCS with data packet dropout can be modeled as an
asynchronous dynamical system (ADS) with rate constraints on events [42]. Using
that model, one can calculate the minimum transmission rate that guarantees the
stability of an NCS whose closed-loop dynamics are stable without the presence of
packet dropout. Another possibility for addressing dropped data packets is to model
the arrival of data as a random process. For example, the work in [43] studies state
estimation of SISO NCSs, in which scalar observations arrive according to a Poisson
process; the work in [44] presents a Kalman Filter scheme for MIMO systems in
which the arrival of output information (all outputs as a packets) is modeled as a
Bernoulli process. These works are generalized in [45] where outputs are divided into
two parts each of which can be received or lost by the Kalman Filter independently.
In the case when packet dropout also exists in the data transmission from the
controller to the actuators, the design of estimator and controller depends strongly
on whether the communication protocol includes ?acknowledgment? (ACK) packets
that allow the controller to know whether the data was received by the actuator or
not. It can be shown that [46], if all successful data transmission is acknowledged,
the controller and the estimation can be designed separately, i.e., the separation
principle holds. In this case, if the data arrival from the controller to the plant and
from the plant to the controller are modeled as i.i.d. Bernoulli, then the relationship
between the maximum data packet dropout rate and the stability of a linear NCS
under a LQR controller can be related by a condition provided in [46]. The problem
of LQG optimal control with a similar NCS setup is discussed in [47].
20
2.2.5 Medium Access Constraints
One of the fundamental communication constraints in a communication network-
and indeed the one this thesis focuses on - is that of medium access. It comes about
because a communication medium can only provide limited number of simultaneous
medium access channels for its users. As a consequence, in an NCS, only limited
number of sensors and actuators are allowed to communicate with the controller at
any one time.
In modern communication networks, medium access constraints are often re-
solved via various Medium Access Control (MAC) protocols which define the access
scheduling and collision arbitration policies in the network. MAC protocols can be
roughly divided into two categories, namely sequential MAC protocols and random
MAC protocols. Under sequential MAC protocols, each user of the network accesses
the shared medium according to a pre-configured sequence. Under random MAC
protocols, every user attempts to access the media whenever it has a packet to trans-
mit; if there are other users wanting to access the medium at the same moment, an
arbitration policy is used to resolve the packet collision.
The advantages ofsequential MAC protocolsarelow latency, bounded time delay,
and guaranteed data rate. They are well-suited for those applications where every
user in the network has a fixed bit-rate and small data packets. Examples of sequen-
tial MAC protocols includes polling, token passing (used in ProfiBus), TDMA(Time
Division Multiplexing Access, used in T1 telephone systems and GSM cellular phone
systems). On the other hand, random MAC protocols allow real-time medium ac-
cess arbitrations based on the priority (importance) of the data packages. These
protocols suit well in the applications where users in a network have burst trans-
mission and relatively large data packet. The first random MAC protocol, ALOHA,
21
was developed by the University of Hawaii for wireless communication. Successors
to the ALOHA protocol are many variations of the Carrier Sense Multiple Access
(CSMA) protocol. For example, Carrier Sense Multiple Access/Collision Detec-
tion (CSMA/CD) is used in Ethernet and Carrier Sense Multiple Access/Collision
Avoidance (CSMA/CA) is used in IEEE802.11 Wireless LAN.
NCSs with medium access constraints
Due to access constraints in the communication medium, the design of an NCS in-
volves not only a feedback controller but also a medium access scheduling strategy,
which defines the to the MAC protocol implemented in the underlying communica-
tion medium. Another related problem is to determine what information the plant
or controller should assume when an actuator or sensor losses access to the commu-
nication medium. Most previous works (detailed in the next paragraphs) assume
that there is a zero order hold (ZOH) implemented at the ?receiving end? of the
communication medium (i.e., at the plant?s and controller?s input stages), if a sensor
or actuator fails to access the medium, the most recently updated values stored in
the ZOH will be used by the plant or the controller. In this thesis, we will intro-
duce an alternative way of handling medium access disruptions so that if a senor
or actuator is not actively accessing the communication medium, their input to the
controller or plant will be ignored.
Static medium access scheduling
One possible scheduling strategy is to let the medium access of different sensors
and actuators follow a pre-defined sequence, termed communication sequence [48].
Medium access scheduling strategies of this kind are often called static scheduling.
The work in [48] studies an NCS in which a collection of uncoupled discrete-time
22
linear plants send their sensors? information to a controller via a shared serial com-
munication medium. The medium access status of the sensors at each discrete time
instance is described by a fixed periodic communication sequence. Assuming ZOH
at the receiving side of the communication medium, a state augmentation technique
called extensification is used to model the closed loop NCS dynamics under a con-
stant feedback gain matrix. It is shown that any periodic communication sequence
defines an affine subspace of matrices in an augmented matrix space. Each matrix
in this space corresponds to a specific choice of feedback gains. In general, given
a periodic communication sequence, a stabilizing feedback gain can not be found
analytically and in fact the question of its existence has been proved to be NP-hard
[49].
The work in [48] is generalized in [14] where the plant studied is a coupled MIMO
system (Fig.2.5). Using the state extensification technique, the closed-loop system
is shown to define an affine matrix space of higher dimension. A more general NCS
configuration is studied in [50] where the medium access constraints are extended
to both sensors and actuators. To account for the effects of ZOHs at both the
plant and the controller?s input stages, a higher dimensional state extensification is
needed to model the closed-loop dynamics, which again define an affine matrix space.
Under this formulation, it is shown that [14, 50], given a periodic communication
sequence, the solution fora stabilizing feedback gainmay be obtainedvia a simulated
annealing algorithm that seeks to minimize the spectral radius of the closed-loop
systems, however the computation cost related to this algorithm is very high, and
the existence of a solution is not guaranteed.
If the feedback controller of an NCS is given in advance, the problem of design-
ing a stabilizing communication sequence is also challenging. Previously proposed
methods only handle NCS whose dynamics are ?block-diagonal?, in the sense that
23
Plant
Controller ZOH
Communication
medium
... ...
Figure 2.5: An MIMO plant with medium access constraints existing only on the
output side. A ZOH is used at the receiving end of the communication medium.
the sensors and actuators are connected to uncoupled sub-plants. The work in [13]
studies a block-diagonal NCS configuration where a collection of uncoupled contin-
uous linear systems share a communication medium to close their feedback loops
(shown in Fig. 2.6). It is shown that the existence of a stabilizing communication
sequence can be linked a condition that is related to the decay/growth rate of the
Lyapunov functions of the closed/open-loop dynamics of these subsystems.
...
actuator Plant 1 sensor
Controller 1
...
Communication
Medium
actuator sensor
Controller 2
actuator sensor
Controller N
Plant 2
Plant N
...
Figure 2.6: A ?block-diagonal? NCS: a collection of uncoupled continuous linear
systems share a communication medium to close their feedback loops.
24
A similar NCS configuration is discussed in [51, 15], it is shown that, depending
on the parameters of their dynamics, each subsystem needs to close its feedback loop
at a specific rate in order to achieve stability. Based on these rate requirements,
the Rate Monotonic (RM) [52] algorithm is then used to schedule the access to the
shared communication medium. The idea of RM scheduling is that the subsystem
that needs to close its loop at a higher rate obtains higher medium access priority.
Under the RM scheduling rule, the schedulability of a set of non-preemptive periodic
tasks can be checked by a sufficient condition given in [53].
An effective communication sequence can also be obtained by solving an opti-
mization problem that searches over all possible choices of communication sequence
over a finite horizon. It is shown [54] that a finite horizon optimal communica-
tion sequence can be found by performing backwards recursion of an LQ (Linear
Quadratic) cost function combined with tree pruning. The work in [55] uses a sim-
ilar optimization setup but the optimal communication sequence is obtained via
exhaustive search.
Dynamic medium access scheduling
Under static scheduling, the controller may not be able to respond quickly to a
sensor or actuator that requires immediate attention, and thus an NCS may be
less robust when the plant is subject to unpredictable disturbances. Moreover,
when implementing a static communication sequence, a global timer is needed to
synchronize all the sensors, actuators, and the controller. These restrictions have
given rises to research on dynamic medium access scheduling strategies in which the
access to the medium is determined ?on-line? based on a feedback-based arbitration
policy. Dynamic medium access scheduling policies can be implemented via various
random MAC protocols.
25
The ?CLS-?? dynamic access scheduling policy is introduced in [56] for NCSs
having block-diagonal configurations. This policy, which seeks to ?Contain Largest
State?, is inspired by the ?Clear Largest Buffer? policy originally introduced in [57]
for distributed manufacturing system. According to the CLS-? policy, the subsystem
whose state is farthest from the origin wins medium medium access; the medium
access is re-determined when the norm of the state of that subsystem is driven to a
small positive real number ?. It is shown that, by gradually decreasing the value of
?, the NCS can be asymptotically stabilized by the CLS-? policy.
The works in [58, 59] proposed the MEF-TOD (Maximum Error First, Try Once
Discard) policy which can be used to stabilize a general NCS in which the plant is a
coupled MIMO system. According to the MEF-TOD policy, i) at any time, the input
or output with the greatest weighted error from its most recently transmitted value
win the access of the medium; ii) if an input or output fails to win the competition
for the network access, it discards its current value, real-time access arbitration of
medium access is always based on real-time value of the inputs and outputs. An
important assumption made in the MEF-TOD policy is that, for each sensor and
actuator, the maximum interval between its two successful transmission is bounded
by a small number, ?m, referred to as the ?Maximum Allowable Transfer Interval?
(MATI). Treating the network induced error as a vanishing perturbation, the work
in [58] shows that an NCS is asymptotically stable under the MEF-TOD policy if
the MATI is less than a given bound, which can be obtained by applying Bellman-
Gronwall Lemma [60] to the closed-loop system. This result is extended to nonlinear
NCSs in [61]. It is shown [62, 13] that, due to the conservativeness of Bellman-
Gronwall Lemma, the bound obtained in [58, 59] on MATI is very conservative.
Neither the ?CLS-?? nor the ?MEF-TOD? policy guarantees an upper-bound for
the switching rate of medium access. High-speed switching is often impractical in
26
communication networks and may bring undesirable high frequency disturbances in
the feedback loop. An effective way to bound the switching rate is to introduce a
minimum dwell time ? > 0 and restrict the time interval between any two consec-
utive access switches to be no smaller than ? [63]. In Chapter 5 of this thesis, we
will introduce a dynamic medium access scheduling policy that ensures a dwell-time
between consecutive switches.
Relationships with this thesis
For NCSs under static access scheduling, previously proposed NCS synthesis meth-
ods only allow one to design a controller for a given communication sequence, or to
find a stabilizing communication sequence while assuming that the controller was
designed without considering communication constraints; the co-design of communi-
cation sequence and the feedback controller in NCSs has thus far been a challenging
task. When designing a stabilizing communication sequence, it is also often assumed
that the plant is block-diagonal, or that the medium access constraints exist only
on the output side of the plant, in order to simplify the analysis. When it comes to
controller design, the methods introduced in [14] and [48] used simulated annealing
algorithm which is complicated and does not guarantee the existence of a solution.
In dynamic scheduling of NCSs, most previous results only address simple con-
figurations where the plant is block-diagonal [13]; the work in [58] discusses NCSs
with coupled plant dynamics, but it is attached to very conservative assumptions
on the MATI.
In the following chapters, we describe a new model for NCS under medium
access constraints. We will show that this new model avoids the complexities in
previously proposed NCS models and elucidates the interaction between medium
access scheduling and the underlying dynamics of the control system. Based on this
27
new model, we will present a set of theoretical tools that allow one to concurrently
design the medium access scheduling strategy (static or dynamic) and the feedback
controller that stabilize a general class of NCSs in which the plant has?fully coupled?
dynamics and the medium access constraints exist on both input and output side
of the plant.
28
Chapter 3
Medium Access Constraints in
NCSs
The focus of this thesis is NCS design under medium access constraints. As we
have already mentioned, previously proposed synthesis methods do not provide a
complete solution to the NCS stabilization problem. The difficulties encountered in
previous works are partly due to the complexities introduced by assuming zero order
holds (ZOHs) at the ?receiving end? of a communication medium. In this chapter,
we introduce a new NCS architecture, in which the plant and controller forgo the
use of a ZOH, and instead choose to ?ignore? the actuators and sensors that are not
actively communicating. This new architecture leads to simpler NCS models and
elucidates the interaction between medium access scheduling and the dynamics of
the underlying plant and the feedback controller.
29
3.1 Problem Formulation
Consider the NCS configuration shown in Fig. 3.1, in which the plant is a linear
time-invariant (LTI) system and communication constraints exist on both input
and output side of the plant. Let the dynamics of the plant be given by:
?x(t) = Ax(t) +Bu(t), x ? Rn,u? Rm, (3.1)
y(t) = Cx(t), y ? Rp,
where the state of the plant x = [x1,??? ,xn]T ? Rn consists of n scalar components;
the input u = [u1,??? ,um]T ? Rm consist of m scalar components; and the output
y = [y1,??? ,yp]T ? Rp consists of p scalar components. Each of the plant?s m scalar
inputs receives its control signal ui from a designated actuator; each output yi is
measured by a designated sensor.
actuator 1
actuator 2
...
actuator m
sensor 2
sensor 1
...
sensor p
Controller
Input Channels Output Channels
Shared Communication Medium
Plant
yp
u2
um
u1 y1
y2
Figure 3.1: An NCS having m actuators and p sensors, with medium access con-
straints existing on both sensor and actuator sides.
In the sequel, we will use the terms input channel and output channel to refer
to communication links that enable data transmission from controller to actuators
30
and from sensors to controller, respectively, through the communication medium.
We assume that the shared communication medium can simultaneously provide w?
(1 ? w? < p) output channels and w? (1 ? w? < m) input channels. In such an
NCS, only w? of the p sensors can access these channels to communicate with the
controller at any one time, while others must wait. Similarly, at the input side of
the plant, the controller can only communicate with w? of the m actuators at any
one time to update their control signals.
It is important to note that the proposed NCS model does not capture settings
where sensors and actuators share a single communication channel. However, the
results presented in Chapter 4 and Chapter 6 can be modified to apply to those
settings.
We also remark that the central issue addressed in this thesis is medium access
constraints in an NCS. The effects of any other communication constraints, such as
data-rate limits, transmission delays, and possible data-packet have been neglected
for simplicity. However, the NCS model proposed in this chapter can be augmented
to include the effects of other communication constraints.
3.2 Communication Sequence
An effective way to model medium access constraints in NCSs is to use the notion
of a communication sequence [48, 14], which specifies the order in which the sensors
and actuators are to access the communication medium. Let us first consider the
output side of the plant. For i = 1,??? ,p, let the binary-valued function ?i(t) denote
the medium access status of sensor i at time t, i.e. ?i(t) : R mapsto?{0,1}, where 1 means
?accessing? and 0 means ?not accessing?. The medium access status of the p sensors
31
over time can then be represented by the p-to-w? communication sequence
?(t) =
?
??
??
??
??
?1(t)
?2(t)
...
?p(t)
?
??
??
??
??
.
Definition 3.1. An M-to-N (N < M) communication sequence ? is a map, ?(t) :
R mapsto?{0,1}M, satisfying bardbl?(t)bardbl2 = N, ?t.
Similarly, at the plant?s input side, we use a binary-valued function ?i(t) to
denote the medium access status of actuator i at time t. The medium access status
of the plant?s m actuators can then be described by an m-to-w? communication
sequence ?(t) with
?(t) =
?
??
??
??
??
?1(t)
?2(t)
...
?m(t)
?
??
??
??
??
.
In the sequel, we will refer to the ? and ? as the ?input? and ?output? commu-
nication sequences, respectively.
3.3 Effects of Medium Access Constraints
We now go on to investigate the effects of medium access constraints on the dynamics
of an NCS. A sensor?s output, say yi(t), is available to the controller only when the
sensor i is accessing the communication medium, i.e., ?i(t) = 1. When that is not
32
the case (?i(t) = 0), we let the controller ignore that sensor by assuming a zero
output value for its output. Let ?yi(t) denote the output signal that is assumed by
the controller for sensor i at time t. The above protocol leads to:
?yi(t) = ?i(t)?yi(t), ?i.
To simplify the notation, it is also convenient to define the matrix form of a
communication sequence (or the communication sequence matrix).
Definition 3.2. Let ?(t) be an M-to-N communication sequence, the ?matrix form?
of ?(t), M?(t) , is defined as
M?(t) definesdiag(?(t)).
Now let ?y(t) denote the output vector that is assumed by the controller at time
t, based on the above communication protocol. We thus have
?y(t) = M?(t)?y(t), (3.2)
where ?y = [?y1, ?y2,??? , ?yp]T.
Similarly, at the plant?s input side, when actuator j loses its access to the com-
munication medium, the control signal generated by the controller for that actuator
will be unavailable to and hence ignored by that actuator. Instead, the actuator
sets uj = 0 until it obtains medium access again. Let ?u = [?u1,????um]T denote the
control signals generated by the controller and u(t) denote the input signals actually
used by the actuator, we thus have
u(t) = M?(t)?u(t). (3.3)
33
3.4 The Extended Plant
Combining (3.1), (3.3), and (3.3), we obtain a linear time-varying (LTV) system
with ?u as its inputs and ?y as its outputs:
?x(t) = Ax(t) +BM?(t)?u(t), (3.4)
?y(t) = M?(t)Cx(t).
The last equations describe the plant from the controller?s point of view. We call
(3.4) the extended plant, and note that it incorporates the dynamics of the plant
together with the two communication sequences. The block diagram of the extended
plant is shown in Fig. 3.2.
Plant
Controller
Extended plant
y(t)
?u(t) ?y(t)
M?(t)M?(t)
u(t)
Figure 3.2: The extended plant
The properties of the extended plant can be summarized as follows:
1. The extended plant (3.4) is a linear time varying system with a constant A
matrix.
2. The extended plant (3.4) has fewer ?effective? inputs and outputs than the
original plant (3.1).
3. The state of the extended plant (3.4) coincides with that of the original plant
(3.1). Hence the NCS can be stabilized by designing a feedback controller that
34
stabilizes the extended plant (3.4).
3.4.1 Remark
Previous works have often assumed that, when a sensor or actuator loses medium
access, the latest available values stored in the ZOHs will be fed into the controller
or plant. Aside from increasing the system?s complexity by introducing time-varying
delays (see, for example, the ?extensive form? in [14] and other similar construc-
tions), the use of ZOHs may not necessarily benefit system performance: Consider
the scalar system ?x = x+ u stabilized by the feedback law u = ?2x. Suppose that
the medium access of u is cut off when x = 1. With a ZOH, the system will then
evolve according to ?x = x ? 2 thereafter. When x becomes negative, the control
u = ?2 would result in a greater divergence rate than a zero control. In this work,
we have chosen to ?ignore? the sensors or actuators that are not actively communi-
cating. We have shown that, doing this, the effect of medium access constraints can
be captured by simply cascading the original plant with a pair of communication
sequence matrices.
3.4.2 Switched System Expression
Note that according to Definition 3.1, ?(?) and ?(?) can only take on parenleftbigmw?parenrightbig and
parenleftbig p
w?
parenrightbig possible values, respectively. This makes the extended plant (3.4) essentially a
switched system [63] switching between parenleftbigmw?parenrightbig?parenleftbig pw?parenrightbig LTI systems. Using the switched
system notation, the extended plant can also be expressed as:
?x(t) = Ax(t) +B?(t)?u(t), (3.5)
?y(t) = C?(t)x(t),
35
where the communication sequences ?(t), ?(t) are now serving as the switching
signals of the switched system. Also, (3.5) represents a special class of switched
systems because the dynamics A are always constant. In Chapter 5, this property
will allow us to develop a group of dynamic medium access scheduling strategies
that asymptotically stabilize NCSs under constant feedback gains.
3.5 A Discrete-time Formulation
Feedback controllers of modern control systems are often implemented via digital
computers, especially when it comes to NCSs. In that setting, the plant?s outputs
are sampled periodically by an Analog-to-Digital (A/D) converter; a Digital-to-
Analog (D/A) converter is used to convert the digital control signals generated at the
controller to analog ones for the actuators. A dynamical system with synchronized
D/A and A/D converters at its inputs and outputs is often referred to as a ?sampled-
data system?. It is well-known that a sampled-data system can be modeled as a
discrete-time dynamical system. Suppose therefore, that the plant in the NCS of
Fig. 3.1 is a sampled-data system, and let the dynamics of the plant be given by
the discrete-time LTI system
x(k + 1) = Ax(k) +Bu(k), x ? Rn,u? Rm, (3.6)
y(k) = Cx(k), y ? Rp.
Suppose also that data transmissions and receptions via the communication medium
only take place at discrete-time instances. Then, the medium access status of the p
sensors and m actuators over time can be represented by the discrete-time p-to-w?
36
and m-to-w? communication sequences
?(k) =
?
??
??
??
??
?1(k)
?2(k)
...
?p(k)
?
??
??
??
??
, and ?(k) =
?
??
??
??
??
?1(k)
?2(k)
...
?m(k)
?
??
??
??
??
, respectively.
Definition 3.3. A discrete-time M-to-N communication sequence ? is a map, ?(k) :
Z mapsto?{0,1}M, satisfying bardbl?(k)bardbl2 = N, ?k.
Based on the same communication protocols we introduced in Section 3.3, the
extended plant to be controlled by the controller is the discrete-time linear time-
varying (LTV) system:
x(k + 1) = Ax(k) +BM?(k)?u(k), (3.7)
?y(k) = M?(k)Cx(k),
where M?(k) defines diag(?(k)), and M?(k) defines diag(?(k)) are the communication se-
quence matrices.
3.6 NCSs with Nonlinear Plants
Although we will not discuss the design of NCSs with non-linear dynamics, the
modeling techniques introduced in this chapter easily extends to those systems. For
example, if the plant is described by
?x(t) = f(x(t),u(t),t), (3.8)
y(t) = g(x(t),u(t),t),
37
then, under the communication sequences ?(?) and ?(?), the extended plant can be
expressed as:
?x(t) = f(x(t),M?(t)?u(t),t), (3.9)
?y(t) = M?(t)g(x(t),M?(t)?u(t),t).
38
Chapter 4
Static Medium Access Scheduling
In this chapter, we study the stabilization of NCSs under static medium access
scheduling. We show that, in the continuous-time case, the controllability and
observability of a linear plant can be preserved under periodic communication se-
quences as long as each input and output are granted a finite interval of medium
access during every period; in the discrete-time case, there always exist periodic
communication sequences (can be found via a simple algorithm) that preserve the
reachability and observability of a reversible plant. Using these effective commu-
nication sequences, one can easily design observer-based feedback controllers that
exponentially stabilize an NCS at an arbitrary decay rate.
4.1 Continuous-time Case
We begin with studying static medium access scheduling in continuous-time. Let
the plant be the LTI system
39
?x(t) = Ax(t) +Bu(t), x ? Rn,u? Rm, (4.1)
y(t) = Cx(t); y ? Rp.
Suppose that the medium access of the actuators and sensors is governed by the m-
to-w? input communication sequence ?(t) and the p-to-w? output communication
sequence ?(t), respectively. Based on our NCS model from Chapter 3, the dynamics
of the extended plant are:
?x(t) = Ax(t) +BM?(t)?u(t), (4.2)
?y(t) = M?(t)Cx(t).
For convenience, we re-write the dynamics of the extended plant as
?x(t) = Ax(t) + ?B(t)?u(t), (4.3)
?y(t) = ?C(t)x(t).
where ?C(t) defines M?(t)C, ?B(t) defines BM?(t). Without loss of generality, we assume that
the plant (4.1) is controllable and observable, i.e.,
rankparenleftbig[B,AB,??? ,An?1B]parenrightbig = rank
?
??
??
??
??
?
??
??
??
??
C
CA
...
CAn?1
?
??
??
??
??
?
??
??
??
??
= n. (4.4)
40
Note that, in the presence of medium access constraints, the controllability and
observability of the plant may be ?lost? if the communication sequences are not
chosen carefully, as would be the case, for example, if one of the inputs never ob-
tained medium access. In the continuous-time case, this scenario will turn out to be
the only one that we must guard against; on the other hand, the discrete-time case
will require a more sophisticated choice of communication sequences, as we shall see
in the sequel. In order to stabilize the NCS under consideration, our strategy will
be to first find communication sequences which preserve the controllability and ob-
servability in the extended plant (4.2), and then design an observer-based feedback
controller that will stabilize the extended plant. Before proceeding with our analy-
sis, we review several definitions regarding the controllability and observability of
LTV systems, such as (4.2). These definitions can be found in most standard linear
system theory texts (e.g., [64]).
Definition 4.1. The extended plant (4.2) is controllable on [t0,tf] if starting from
any initial condition at t0, there exists an input ?u(t) that steers x(t) to zero at tf.
Definition 4.2. The extended plant (4.2) is controllable if there exists ? > 0 such
that (4.2) is controllable on [t,t+?] for all t.
Definition 4.3. The extended plant (4.2) is observable on [t0,tf] if any initial
condition at t0 can be uniquely determined from the output ?y(t) on [t0,tf].
Definition 4.4. The extended plant (4.2) is observable if there exist a ? > 0 such
that (4.2) is observable on [t,t+?] for all t.
41
4.1.1 Communication Sequences that Preserve
Controllability and Observability
To investigate the controllability and observability of the extended plant (4.2), we
will begin with the worst-case communication condition, where there is only one
input channel and one output channel available at the communication medium (i.e.,
w? = w? = 1). Later we will generalize those results to the general multiple in-
put/output channels cases.
Let us first study the controllability of the extended plant. Consider the case
where w? = 1. Now, ?(t) is an m-to-1 communication sequence. By definition, ?(t)
only takes values on the set of m-dimensional standard basis vectors,
Em = {e1m,e2m,??? ,emm},
where e1m = [1,0???0]T, e2m = [0,1,0???0]T, ??? emm = [0???0,1]T. Let B =
[b1,b1,??? ,bm], then for all t, ?B(t) takes values on a finite set,
?B(t) ?{B1,B2,...,Bm},
where Bi = B?diag(eim) = [0,??? ,0,bi,0,??? ,0], i.e., Bi only retains the ith column
of B, and sets all the other columns to zero.
Note that the LTV system (4.2) is controllable on [t0,tf] if and only if the
controllability Gramian
W(t0,tf) =
integraldisplay tf
t0
?(t0,t) ?B(t) ?B(t)T?T(t0,t)dt (4.5)
is invertible, where ?(t0,t) = eA(t0?t). Inspired by a similar problem studied in [65],
42
we have the following:
Theorem 4.1. Suppose that the plant (4.1) is controllable. Let t0 < t1 < ??? <
tm = tf, then the m-to-1 communication sequence
?(t) = eim; ti?1 ? t < ti (i = 1,2,???m), (4.6)
is such that the extended plant (4.2) is controllable on [t0,tf].
Proof. Under the communication sequence (4.6), the controllability Gramianof (4.2)
over [t0,tf] can then be written as
W(t0,tf) =
integraldisplay t1
t0
?(t0,t)B1BT1 ?T(t0,t)dt+
integraldisplay t2
t1
?(t0,t)B2BT2 ?T(t0,t)dt+ (4.7)
???+
integraldisplay tm
tm?1
?(t0,t)BmBTm?T(t0,t)dt.
We prove the controllability of the extended plant by contradiction. If the control-
lability Gramian (4.7) were not invertible, then there should exit a vector xa negationslash= 0
such that xTaW(t0,tm)xa = 0. This would imply for all i = 1,??? ,m
xTa?(t0,t)Bi = 0; ti?1 ? t < ti (4.8)
because each integrand in the RHS of (4.7) is symmetric and non-negative definite.
Now differentiate both sides of (4.8) for n?1 times with respect to t, and evaluate
the resulting equation at t = t0 to obtain
xTa[Bi,ABi,??? ,An?1Bi] = 0, ?i = 1,??? ,m.
This would imply xTa[B,AB,??? ,An?1B] = 0, meaning that [B,AB,??? ,An?1B]
43
has rank less than n; this contradicts the assumption that (4.1) is controllable.
The proof of Theorem 4.1 suggests that the extended plant is controllable on
the time interval [t0,tf] if every input obtains some finite amount of medium access
during that time interval. This is summarized in the following corollary, whose proof
follows easily from that of the previous theorem:
Corollary 4.1. Suppose that the plant (4.1) is controllable. If the m-to-1 commu-
nication sequence ?(t) satisfies:
? for all i = 1,??? ,m, there exist t0 ? ti1 < ti2 ? tf, such that ?i(t) = 1 for all
t ? [ti1,ti2], where ?i(t) is the i-th component of ?(t),
then the extended plant (4.2) is controllable on [t0,tf].
The above theorem illustrate the extended plant?s controllability during a time
interval [t0,tf]; we now extend it to its controllability for all time.
Corollary 4.2. Suppose that the plant (4.1) is controllable, and that T > 0 is a real
number. If the m-to-1 communication sequence ?(t) satisfies:
1. ?(t) = ?(t+T), for all t;
2. for all i = 1,??? ,m, there exist 0 ? ti1 < ti2 ? T, such that ?i(t) = 1 for all
t ? [ti1,ti2], where ?i(t) is the i-th component of ?(t);
then the extended plant (4.2) is controllable on [t,t + T] for all t, and is hence
controllable.
Proof. Under the given communication sequence ?(?), during each period [t,t+T],
each input obtains a chance of medium access. In the spirit of Corollary 4.1, the
extended plant is controllable at [t,t+T] for all t.
44
For an NCS with more than one input channels (w? > 1), each possible value
taken by ?B(t) will consist of w? columns from B. An argument similar to those in the
proof of Theorem 4.1 shows that, the controllability of the plant (4.1) is preserved
in the extended plant (4.2), if every input obtains a finite interval of medium access
periodically.
Lemma 4.1. Suppose that the plant (4.1) is controllable and that T > 0 is a real
number. If the m-to-w? (1 ? w? < m) communication sequence ?(?) is T-periodic
and gives each of the m actuators a chance to access the communication medium
during every period T, i.e.,
1. for all t, ?(t) = ?(t+T);
2. for all i = 1,??? ,m, there exist ti1, ti2, such that 0 ? ti1 < ti2 ? T, and
?i(t) = 1 for all t ? [ti1,ti2], where ?i(t) is the i-th component of ?(t);
then the extended plant (4.2) is controllable on [t,t + T], for all t, and is hence
controllable.
Because observability is a dual property of controllability, the results we just
proved for controllability can be extended to the following statements regarding the
observability of the extended plant. The proofs are straightforward (they follow the
?dual? arguments from the proof of Theorem 4.1) and will not be included here.
Lemma 4.2. Suppose that the plant (4.1) is observable and that T > 0 is a real
number. if, for all t, the p-to-w? (1 ? w? < p) communication sequence ?(?) is
T-periodic and gives each of the p sensors a chance to access the communication
medium during every period T, i.e.,
1. for all t, ?(t) = ?(t+T);
45
2. for all i = 1,??? ,p, there exist ti1, ti2, such that 0 ? ti1 < ti2 ? T, and
?i(t) = 1 for all t ? [ti1,ti2], where ?i(t) is the i-th component of ?(t);
then the extended plant (4.2) is observable on [t,t + T], for all t, and is hence
observable.
4.1.2 Output Feedback Stabilization
Under periodic communication, the extended plant (4.2) becomes a linear periodic
time-varying system. The stabilization of such systems has been extensively studied
in the controls literature, see, for example, [66], [67], and [68]. Here we use an
observer-based controller described in [64].
From the two lemmas stated in the previous section, we know that the the con-
trollability and observability of the plant can be preserved in the extended plant if
the communication sequences are chosen properly. Using those ?effective? commu-
nication sequences, the extended-plant can be stabilized at an arbitrary decay rate
via output feedback. The feedback controller consists of a state observer followed
by a time varying feedback gain K(t) (shown in Fig. 4.1).
Observer
Plant
?u(t)
K(t)
M?(t)M?(t) u(t) y(t)
?x(t) ?y(t)
Figure 4.1: Output feedback stabilization of an NCS: The controller consists of an
observer followed by a time-varying feedback gain K(t).
46
Define an observer for the extended plant (4.2) by
??x(t) = A?x(t) + ?B(t)?u(t) +H(t)[?y(t)? ?C(t)?x(t)], (4.9)
where ?x is the observer?s state, ?C(t) = M?(t)C, ?B(t) = BM?(t), and H(t) is the
time-varying observer gain. Suppose that the extended plant is controlled by the
time-varying feedback law:
?u(t) = K(t)?x(t).
We then have the following:
Theorem 4.2. Let the plant (4.1) be controllable and observable, and the commu-
nication sequences ?(t) and ?(t) satisfy the conditions stated in Lemma 4.1 and
Lemma 4.2. Then given ? > 0, for any ? > 0 the feedback and observer gains
K(t) = ??BT(t)W?1?+?(t,t+T), (4.10)
H(t) = [e?ATTM?+?(t?T,t)e?AT]?1 ?CT(t), (4.11)
are such that the closed-loop system is uniformly exponentially stable with rate ?,
where
W?+?(t0,tf) defines
integraldisplay tf
t0
2e4(?+?)(t0??)eA(t0??) ?B(?) ?BT(?)eAT(t0??)d?, (4.12)
M?+?(t0,tf) defines
integraldisplay tf
t0
2e4(?+?)(??tf)eAT(??t0) ?CT(?) ?C(?)eA(??t0)d?. (4.13)
Proof. As defined in [64], the controllability and observability Gramians of the ex-
47
tended plant (4.2) are
W(t0,tf) =
integraldisplay tf
t0
eA(t0??) ?B(?) ?BT(?)eAT(t0??)d?,
M(t0,tf) =
integraldisplay tf
t0
eAT(??t0) ?CT(?) ?C(?)eA(??t0)d?.
If the communication sequences ?(t) and ?(t) satisfy the conditions of Lemmas
4.1 and 4.2, the extended plant will be controllable and observable on [t,t +T) for
all t. Hence, the Gramians of the extended plant (4.2), W(t,t + T) and M(t ?
T,t), are positive definite for all t. Also it is easy to show that, under T-periodic
communication sequences, the Gramians W(t,t + T), and M(t?T,t) are both T-
periodic in t. Therefore, we can find positive constants ?1 and ?2 such that, for all
t,
?1I ?W(t,t +?) ? ?2I, (4.14)
?1I ? ?T(t?T,t)M(t?T,t)?(t?T,t) ? ?2I, (4.15)
because ?(t?T,t) = e?AT is constant for all t.
Moreover, since ?B(t) is piecewise constant, and it always takes value on a finite
set, there exist positive constants ?1, ?2 such that, for all t, ? with t ? ?,
integraldisplay t
?
bardbl?B(?)bardbl2d? ? ?1 +?2(t??). (4.16)
Given (4.14)-(4.16), we can apply Theorem 15.5 in [64] for the extended plant (4.2)
to obtain the desired result.
48
4.1.3 Remarks
We remark that under the T-periodic communication sequences, ?(t) and ?(t), the
feedback and observer gains K(t) and H(t) given in Theorem 4.2 are also T-periodic.
Hence, when implementing the proposed observer-based controller, one only needs
to calculate H(t) and K(t) for t0 ? t < T + t0, where t0 is the initial time. The
Gramians (4.12), (4.13), can be computed by integrating a pair of linear ordinary
differential equations (ODEs). In particular, one can show that W?+?(t,t + T)
satisfies a differential equation similar to that satisfied by (4.7):
d
dtW?+?(t,t+T) = (A+ 2(? +?)I)W?+?(t,t+T)+
W?+?(t,t+T)(A+ 2(?+?)I)T ?2 ?B(t) ?BT(t)+
2e?(A+2(?+?)I)T ?B(t) ?BT(t)e?(A+2(?+?)I)T T. (4.17)
The initial condition of W?+?(t,t + T) at t = 0, W?+?(0,T), is obtained by inte-
grating the differential equation:
d
dtW?+?(t,T) = (A+ 2(?+?)I)W?+?(t,T)?2
?B(t) ?BT(t) + (4.18)
W?+?(t,T)(A+ 2(?+?)I)T; (4.19)
W?+?(T,T) = 0. (4.20)
The matrix M?+?(t?T,t) satisfies a similar differential equation. The detailed
derivations of the ODEs satisfied by the Gramians, (4.12) and (4.13), are given in
Appendix A.
49
4.2 Discrete-time Case
We now revisit static medium access scheduling and the accompanied controller de-
sign problem in discrete-time. Recall the discrete time NCS formulation introduced
in Chapter 3, where the plant is the discrete-time LTI system
x(k + 1) = Ax(k) +Bu(k), x ? Rn,u? Rm, (4.21)
y(k) = Cx(k); y ? Rp.
Suppose that the medium access of the actuators and sensors is governed by the
m-to-w? input communication sequence ?(k) and p-to-w? output communication
sequence ?(k), respectively. Then the dynamics of the extended are
x(k + 1) = Ax(k) +BM?(k)?u(k), (4.22)
?y(k) = M?(k)Cx(k).
Without loss of generality, we assume that the plant (4.21) is reachable and observ-
able, i.e.,
rankparenleftbig[B,AB,??? ,An?1B]parenrightbig = rank
?
??
??
??
??
?
??
??
??
??
C
CA
...
CAn?1
?
??
??
??
??
?
??
??
??
??
= n. (4.23)
Following our analysis for the continuous-time case, in order to stabilize the
extended plant, our strategy will be to first find communication sequences which
preserve the reachability and observability in the extended plant (4.22), and then
50
design an observer-based feedback controller that stabilizes the extended plant. It
will turn out to be the case that designing effective communication sequences is
more difficult in discrete-time than in continuous-time; in particular, it will not be
sufficient to simply give every sensor and actuator some amount of medium access
periodically. Before proceeding, we review some relevant definitions that concern
the reachability and observability of discrete-time LTV systems [64].
Definition 4.5. The extended plant (4.22) is reachable on [k0,kf] if given any xf,
there exists an input ?u(k) that steers (4.22) from x(k0) = 0 to x(tf) = xf.
Definition 4.6. The extended plant (4.22) is l-step reachable if l is a positive integer
and (4.22) is reachable on [k,k +l] for any k.
Definition 4.7. The extended plant (4.22) is observable on [k0,kf] if any initial
condition at k0 can be uniquely determined by the corresponding response ?y(k) for
k ? [k0,kf].
Definition 4.8. The extended plant (4.22) is l-step observable if l is a positive
integer and (4.22) is observable on [k,k +l] for any k.
4.2.1 Communication Sequences that Preserve
Reachability and Observability
We continue by studying the reachability of the extended plant. Suppose that
x(0) = 0, and let the extended plant (4.22) evolve from k = 0 to k = kf. Then
x(kf) = R(0,kf)?
bracketleftbigg
?u(0) ?u(1) ??? ?u(kf ?1)
bracketrightbiggT
,
51
where
R(0,kf) = [Akf?1BM?(0),Akf?2BM?(1),??? ,BM?(kf ?1)]. (4.24)
The extended plant (4.22) is reachable on [0,kf] if rank(R(0,kf)) = n. Notice
that for each k, ?(k) is an m-dimensional vector consisting of w? ones and m?w?
zeros. Hence, at each step k, the communication sequence matrix M?(k) has the
effect of ?selecting? w? columns from the m columns of the term Akf?k?1B on the
RHS of (4.24). The matrix R will have full rank if the kf ?w? columns that M?(k)
selects for k = 0,??? ,kf ?1 contain n linearly independent columns.
We now prove that if the plant (4.21) is ?reversible? (i.e., A is an invertible
matrix), then it is always possible to design a communication sequence ?(?), such
that the extended plant is also reachable.
Theorem 4.3. Suppose that A is invertible and that the plant (4.21) is reachable.
For any integer 1 ? w? < m, there exists an m-to-w? communication sequence ?(?)
and an integer kf ?
ceilingleftBig
n
w?
ceilingrightBig
? n, such that the extended plant (4.22) is reachable on
[0,kf].
Proof. First, consider the worst-case scenario where w? = 1. The theorem holds if
it is always possible to design an m-to-1 communication sequence ?(k), such that
the sequence M?(k) selects n independent columns (RHS of (4.24)) by picking one
column from the term Akf?k?1B at each step k = 0,???kf ?1, where kf ? n2.
Let ?i =
bracketleftbigg
AniB,Ani+1B,??? ,Ani+n?1B
bracketrightbigg
. The matrix ?i has rank n for all
i = 0,???n?1 because A is invertible and the extended plant is reachable. Now let
?0i ,??? ,?n?1i be any n linearly independent columns from ?i and let Li be the set
{?0i ,??? ,?n?1i }. Consider the following algorithm
52
1. Let L = L0.
2. Replace ?10 in L by a column from L1, while keeping the rank of L equal to n.
Such a replacement can always be found because rank(L1) = n.
3. For i = 2,??? ,n?1, replace ?i0 in L by a column from Li while keeping the
rank of L fixed.
The resulting matrix L includes one column from each ?i (i = 0,??? ,n ? 1) and
has rank n. The above algorithm ensures that it is possible to select n linearly
independent columns as long as one can select one column from each ?i. However,
on the RHS of (4.24), M?(?) can actually select n columns from each ?i, hence there
always exists an M?(?) that selects n independent columns in at most n2 steps.
Now consider the less restrictive case w? > 1. The sequence M?(?) can select at
least w? independent columns from each ?i (i = 0,1,???). Using a similar algorithm
as in the single-channel case (This time, replace w? columns in L at each step
while keeping the rank of L equal to n. It is easy to prove that such replacements
can always be found because rank(Li) = n.), one can thus design an m-to-w?
communication sequence ? such that M?(?) selects n independent columns from the
?i?s for i = 0,1,??? ,
ceilingleftBig
n
w?
ceilingrightBig
?1 in at most
ceilingleftBig
n
w?
ceilingrightBig
?n steps.
Remark 4.1. The bound for kf in Theorem 4.3 is conservative. Also notice that
for each k, ?(k) can only have parenleftbigmw?parenrightbig possible values. Thus it is possible to find the
minimum kf by searching (off-line) over all possible communication sequences ?(?)
in the interval k =
bracketleftBig
0,
ceilingleftBig
n
w?
ceilingrightBig
?n
bracketrightBig
.
We now extend the result in Theorem 4.3 to the ?l-step reachability? of the
extended plant (4.22).
53
Definition 4.9. A discrete-time communication sequence ?(?) is called N-periodic
if ?(k) = ?(k +N) for all k.
Corollary 4.3. Suppose that A is invertible and that the plant (4.21) is reachable.
For any integer 1 ? w? < m, there exist integers l,N > 0 and an N-periodic
m-to-w? communication sequence ?(?) such that the extended plant (4.22) is l-step
reachable.
Proof. From Theorem 4.3, there exists an integer kf and a communication se-
quence ?(k) such that the extended plant is reachable on [0,kf]. Hence the se-
quence M?(k) can select n independent columns from the matrices Akf?k?1B dur-
ing k = 0,??? ,kf ? 1. Now let N = kf and extend ?(k) for k ? kf by setting
?(k) = ?(k+N), ?k. Because A is invertible, the N-periodic sequence M?(k) will se-
lect n independent columns in every interval k = [jN,(j+1)N?1] (j = 0,1,2???).
Now, let l be any integer greater than or equal 2N?1, then for all i ? 0 there always
exists an integer j ? 0, such that [jN,(j + 1)N ?1] ? [i,i +l]. Hence the periodic
sequence M?(k) will select n independent columns on [i,i+l] for all i. We conclude
that the extended plant is l-step reachable under the N-periodic communication
sequence ?(k).
We now address the observability of the extended plant (4.22). It is easy to show
that the extended plant (4.22) is observable on [k0,kf] if the matrix
O(k0,kf) =
?
??
??
??
??
M?(0)C
M?(1)CA
...
M?(kf ?1)CAkf
?
??
??
??
??
(4.25)
54
satisfies
rank(O(k0,kf)) = n. (4.26)
Notice that, on the RHS of (4.25), at each step k, M?(k) selects w? rows from the
term CAk. Then, the duality of reachability and observability leads to the following
results.
Theorem 4.4. Suppose that A is invertible and that the plant (4.21) is observable.
For any integer 1 ? w? < p, there exists a p-to-w? communication sequence ?(?)
and an integer kf ?
ceilingleftBig
n
w?
ceilingrightBig
?n, such that the extended plant (4.22) is observable on
[0,kf].
Corollary 4.4. Suppose that A is invertible and that the plant (4.21) is observable.
For any integer 1 ? w? < p, there exist integers l,N > 0 and an N-periodic p-to-w?
communication sequence ?(?) such that the extended plant (4.22) is l-step observable.
The proofs of Theorem 4.4 and Corollary 4.4 are similar to those in the proofs of
Theorem 4.3 and Corollary 4.3, one only need to switch from column manipulations
to row manipulations.
Remark 4.2. The invertibility of A is necessary if one wants to preserve the reach-
ability and observability in the extended plant. A counterexample is:
A =
?
??
??
?
1 1 1
0 0 0
0 0 0
?
??
??
?
, B =
?
??
??
?
1
1
1
?
??
??
?
.
Here, (A,B) is reachable but there is no 3-to-1 communication sequence ?(k) such
that the extended plant is reachable on [0,kf], for any kf. Of course, the invertibil-
ity of A is guaranteed if A is obtained by discretizing a continuous-time LTI plant.
55
In Chapter 6, we will show that if A is not invertible (thus reachability or observ-
ability may be lost), we can still find communication sequences that preserve the
stabilizability and detectability in the extended plant.
4.2.2 Choosing Communication Sequences
We have shown that, in continuous-time NCSs, an LTI plant?s controllability and
observability are preserved in the extended plant if each sensor and actuator obtains
some finite amount of medium access during every period of the communication se-
quence. However, this is not always sufficient in discrete-time NCSs, as the following
examples illustrate
Example 4.1. Consider the case when the plant (4.21) has 2 inputs, and the plant?s
dynamics are given by
A =
?
?? 1 0
0 1
?
??, B = [b
1,b2] =
?
?? 1 0
0 1
?
??.
Since A is the identity matrix, the reachability testing matrix (4.24) will have the
form
R(0,kf) = [[b1,b2]M?(0)??? ,[b1,b2]BM?(kf ?2),[b1,b2]M?(kf ?1)].
Then matrix R(0,kf) will have full rank if each of the two inputs, u1, u2, is granted
a chance of medium access during the period [0,kf ?1].
56
Example 4.2. Now consider the case when
A =
?
?? 0 1
1 0
?
??, B = [b
1,b2] =
?
?? 1 0
0 1
?
??.
For this choice of A, B, we have Ab1 = b2, Ab2 = b1. Hence the reachability testing
matrix (4.24) will have the form
R(0,kf) = [Akf?1[b1,b2]M?(0)??? ,[b2,b1]BM?(kf ?2),[b1,b2]M?(kf ?1)].
If the communication sequence is chosen as the 2-periodic sequence
{?(0),?(1),???} = {[1, 0]T,[0, 1]T,[1, 0]T,[0, 1]T,???}.
then the matrix R(0,kf) will only consist of b1 or b2 (depending on whether kf is an
odd or even number), and hence lose rank. However, if we choose the communication
sequence as
{?(0),?(1),???} = {[1, 0]T,[1, 0]T,[1, 0]T,[1, 0]T,???},
i.e., only u1 gets medium access, then the matrix R(0,kf) will be full rank.
Example 4.3. Now consider the plant whose dynamics are obtained by combining
57
the previous two cases:
A =
?
??
??
??
??
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
?
??
??
??
??
, B =
?
??
??
??
??
1 0
0 1
1 0
0 1
?
??
??
??
??
.
Now, neither the communication sequence
{?(0),?(1),???} = {[1, 0]T,[0, 1]T,[1, 0]T,[0, 1]T,???},
nor the sequence
{?(0),?(1),???} = {[1, 0]T,[1, 0]T,[1, 0]T,[1, 0]T,???},
yields full rank in R(0,kf). In order for R(0,kf) to obtain full rank, one have to
stick to one input for more than 1 step and grant both input a chance of medium
access. For example, the 3-periodic communication sequence,
{?(0),?(1),???} = {[1, 0]T,[1, 0]T,[0, 1]T,[1, 0]T,[1, 0]T,[0, 1]T,???},
will work.
4.2.3 Output Feedback Stabilization
It is known that a discrete-time LTV system can be stabilized via output feedback
if it is l-step reachable and l-step observable [64]. A controller that guarantees
exponential decay rate of the closed-loop system consists of a state observer and a
58
time varying feedback gain K(k) (shown in Fig. 4.2).
Observer
Plant
?u(k)
K(k)
M?(k)M?(k) u(k) y(k)
?x(k) ?y(k)
Figure 4.2: Output feedback stabilization of an NCS: The controller consists of a
discrete-time observer and a time-varying feedback gain K(k).
The observer of the extended plant (4.22) is given by the following dynamical
system:
?x(k + 1) = A?x(k) + ?B(k)?u(k) +H(k)[?y(k)? ?C(k)?x(k)], (4.27)
where ?C(k) defines M?(k)C, ?B(k) defines BM?(k), and ?x is the state estimation generated
by the observer. The observer-based feedback law is:
?u(k) = K(k)?x(k).
Theorem 4.5. Let the extended plant (4.22) be l-step reachable, l-step observable
under the periodic communication sequences ?(k) and ?(k), and suppose that A is
invertible. Then given a constant ? > 1 and ? > 1 the feedback and observer gain
K(k) = ??BT(k)(A?1)TW?1?? (k,k +l), (4.28)
H(k) = [(A?l)TM??(k?l + 1,k + 1)A?l]?1(A?1)T ?CT(k), (4.29)
are such that the closed-loop system is uniformly exponentially stable with rate ?,
59
where
W??(k0,kf) defines
kf?1summationdisplay
j=k0
(??)4(k0?j)Ak0?j?1 ?B(j) ?BT(j)(Ak0?j?1)T,
M??(k0,kf) defines
kf?1summationdisplay
j=k0
(??)4(j?kf+1)(Aj?k0)T ?CT(j) ?C(j)Aj?k0.
Proof. The reachability and observability Gramians of (4.22) are defined as
W(k0,kf) =
kf?1summationdisplay
j=k0
Akf?j?1 ?B(j) ?BT(j)(Akf?j?1)T,
M(k0,kf) =
kf?1summationdisplay
j=k0
(Aj?k0)T ?CT(j) ?C(j)Aj?k0.
If the extended plant (4.22) is l-step reachable and l-step observable under the
periodic communication sequences ?(k) and ?(k), then, for all k, the reachability
and observability Gramians of (4.22), W(k,k+l) andM(k?l+1,k+1), are positive
definite and periodic in k. Hence there exist positive constants ?1,?2 such that
?1I ? A?lW(k,k +l)(A?l)T ? ?2I, (4.30)
?1I ? (A?l)TM(k?l + 1,k + 1)A?l ? ?2I. (4.31)
Moreover, since ?B(k) only takes values on a finite set, there exist positive constants
?1, ?2 such that for all k, j, with k ? j + 1
k?1summationdisplay
i=j
bardbl?B(i)bardbl2bardblA?1(i)bardbl? ?1 +?2(k?j ?1). (4.32)
Given (4.30)-(4.32), apply Theorem 29.5 in [64] for the extended plant (4.22) to
obtain the desired result.
60
We remark that the feedback and observer gains given in Theorem 4.5 are in-
dependent of the system?s state and hence can be calculated off-line according to
their update equations. Under N-periodic communication sequences, the feedback
and observer gains K(k) and H(k) are also N-periodic. Hence, when implementing
the observer-based controller, one only needs to calculate these gains for 0 ? k < N
and store them in a time-indexed look-up table.
4.3 Examples
In this section, we give two examples that illustrate the medium access scheduling
and controller design methods introduced in this chapter.
4.3.1 Example 1: Continuous-time
Consider an NCS in which the plant is a 4-th order unstable batch reactor having
two inputs and two outputs [69]:
?x =
?
??
??
??
??
1.38 ?0.2077 6.715 ?5.676
?0.5814 ?4.29 0 0.675
1.067 4.273 ?6.654 5.893
0.048 4.273 1.343 ?2.104
?
??
??
??
??
x+
?
??
??
??
??
0 0
5.67 0
1.136 ?3.146
1.136 0
?
??
??
??
??
u,
y =
?
?? 1 0 1 ?1
0 1 0 0
?
??x.
The plant is being controlled by a controller via a shared communication medium
which has only one input channel and one output channel (i.e., w? = w? = 1).
We chose the communication period to be T = 0.5s, and the input and output
61
communication sequences to be
?(t) =
?
???
???
[1, 0]T : 0 ? t < .3T
[0, 1]T : .3T ? t < T
, (4.33)
and
?(t) =
?
???
???
[1, 0]T : 0 ? t < .7T
[0, 1]T : .7T ? t < T
. (4.34)
Under these sequences, 30% of the input medium access time was given to u1, 70%
was given to u2, the reverse held true for the outputs y1 and y2.
For the given ?(t) and ?(t), we calculated the observer and state feedback gains
from the formulas of Theorem 4.2 with ? = 1 and ? = .2. We then implemented the
observer-based controller to stabilize the batch reactor under limited communica-
tion. The plant?s and the observer?s initial conditions were set to x(0) = [10,5,2,1]T
and ?x(0) = [10,10,10,10]T, respectively. Fig. 4.3 illustrates the simulation results
of the state and error evolutions of the batch reactor under the above communication
sequences and controller.
To evaluate the robustness performance of this NCS design strategy, we per-
formed a second simulation in which we added a white noise of 0.01W and a fixed
transmission delay of 0.02 second at each input and output of the plant, and used
the same communication sequence and feedback controller as in the previous sim-
ulation. Fig. 4.4 illustrates the simulation results of the state and observer error
evolution of the closed-loop NCSs under noise and delays. The simulation results
showed that the system remained stable in the presence of unmodeled noise and
transmission delays.
62
0 1 2 3 4 5?15
?10
?5
0
5
10
15
Time (s)
x(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) State evolution.
0 1 2 3 4 5?12
?10
?8
?6
?4
?2
0
2
4
t (s)
e(t)
e1(t)
e2(t)
e3(t)
e4(t)
(b) Observer error evolution, e(t) = x(t)? ?x(t).
Figure 4.3: Simulation results: Stabilization of an NCS using an observer-based
continuous-time controller. The plant was an LTI system with two inputs and two
outputs. The communication medium provided one input channel and one output
channel. The communication sequence ?(t) assigned 30% of the input channel access
time to u1 and 70% to u2; while the communication sequence ?(t) assigned 70% of
the output channel access time to y1 and 30% to y2.
63
0 10 20 30 40 50?25
?20
?15
?10
?5
0
5
10
15
20
Time (s)
x(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) State evolution.
0 10 20 30 40 50
?30
?20
?10
0
10
20
30
t (s)
e(t)
e1(t)
e2(t)
e3(t)
e4(t)
(b) Observer error evolution, e(t) = x(t)? ?x(t).
Figure 4.4: Simulation results: Stabilization of the same NCS under the same com-
munication sequences and controller as in Fig. 4.3, while a white noise of 0.01W as
well as a 0.02s fixed transmission delay was added at each input and output of the
plant.
64
4.3.2 Example 2: Discrete-time
Consider the NCS in which the plant is a 4-th order unstable discrete-time LTI
system having 2 inputs and 2 outputs:
x(k + 1) =
?
??
??
??
??
1 1/5 0 0
0 11/4 0 1/5
1 1/5 1/3 3/4
0 ?1 0 1/4
?
??
??
??
??
x(k) +
?
??
??
??
??
0 0
1 1
0 0
1 1
?
??
??
??
??
u(k),
y(k) =
?
?? 1 1 0 0
0 0 1 0
?
??x(k).
Suppose that the communication medium connecting the plant and the controller
provides only one input channel and one output channel (i.e., w? = w? = 1). Using
the algorithm presented in Section 4.2.1, we found that, using the 2-periodic input
and output communication sequences
{?(0),?(1),???} = {[0, 1]T,[1, 0]T,???},
{?(0),?(1),???} = {[1, 0]T,[0, 1]T,???},
the extended plant is 7-step reachable and 7-step observable.
We then design the observer-based output feedback controller described in Sec-
tion 4.2.3. The observer gains H(k) and the feedback gains K(k) were calculated
from the formulas in Theorem 4.5, where ? and ? were chosen as ? = 2, ? = 1.2.
The resulting periodic feedback and observer gains are
65
K(2i) =
?
?? ?6.4042 ?3.9222 ?0.0036 ?0.2097
0 0 0 0
?
??,
K(2i + 1) =
?
?? 0 0 0 0
?6.4042 ?3.9222 ?0.0036 ?0.2097
?
??,
H(2i) =
?
??
??
??
??
0 0.1953
0 ?5.3283
0 1.1311
0 2.1968
?
??
??
??
??
, H(2i+ 1) =
?
??
??
??
??
0.1963 0
2.6818 0
?0.1400 0
?1.1059 0
?
??
??
??
??
, i ? Z.
In this example, the plant and the observer?s initial conditions were set to x(0) =
[3,5,7,6]T, ?x(0) = [1,1,3,4]T, respectively. The simulation results on the evolution
of the plant?s states and the observer error of the closed-loop NCS are shown in
Fig. 4.5, from which we can see that the NCS is effectively stabilized by using the
communication sequences and the output feedback controller.
66
0 5 10 15?600
?400
?200
0
200
400
600
800
Time (k)
X(k)
x1(k)
x2(k)
x3(k)
x4(k)
(a) State evolution.
0 2 4 6 8 10 12 14?15
?10
?5
0
5
10
15
20
25
30
35
Time (k)
e(k)
e1(k)
e2(k)
e3(k)
e4(k)
(b) Observer error evolution, e(k) = x(k)? ?x(k).
Figure 4.5: Simulation results: Stabilization of an NCS using an observer-based
discrete-time controller. The plant was a discrete-time LTI system with two inputs
and two outputs. The communication medium provided one input channel and one
output channel.
67
Chapter 5
Dynamic Medium Access
Scheduling
Static medium access scheduling is easy to implement but it may be less robust
when the plant is subject to unpredictable disturbances, because the controller may
not be able to respond quickly to a sensor or actuator that requires immediate
attention. Moreover, when implementing a static communication sequence, a global
timer is needed to synchronize all the sensors, actuators, and the controller. These
limitations give rise to the idea of dynamic medium access scheduling, where the
medium access of the sensors or actuators is determined on-line based on the real-
time information of the plant.
In this chapter, we take the advantage of the NCS model introduced in Chapter 3
to study dynamic access scheduling without any requirements on the structure of
the plant, other than it being LTI. We present an algorithm for simultaneously de-
signing stabilizing gains and communication policies and avoids the computational
complexity and limitations associated with previously proposed methods. We in-
troduce a set of dynamic access scheduling policies that quadratically stabilize the
68
closed-loop NCS while achieving various objectives related to the system?s rate of
convergence, the priorities of different sensors and actuators, and the avoidance of
chattering.
5.1 Problem Formulation
Plant
M?(t)
K
M?(t)
?u ?x
u1
u2
um
x1
x2
xn
Figure 5.1: An NCS in which the plant?s state information is available, and the
controller is constant feedback.
Consider the continuous-time NCS shown in Fig. 5.1, where the dynamics of the
plant are given by the MIMO LTI system
?x(t) = Ax(t) +Bu(t), x ? Rn,u? Rm. (5.1)
For now, we will assume state feedback, meaning that the state x = [x1,??? ,xn]T
are available (although not simultaneously) for measurement at the plant?s outputs.
Let the medium access status of the actuators and sensors be governed by the m-
to-w? input communication sequence ?(t) and the p-to-w? output communication
sequence ?(t), respectively. Let ?x = [?x1, ?x2,??? , ?xn]T, denote the output signals
that are available to the controller. Based on the NCS model we introduced in
69
Chapter 3, we have
?x(t) = M?(t)?x(t). (5.2)
Then the dynamics of the extended plant can be expressed as
?x(t) = Ax(t) +BM?(t)?u(t), (5.3)
?x(t) = M?(t)x(t).
We will adopt a constant-gain feedback controller
?u(t) = K ? ?x(t). (5.4)
From (5.1)-(5.4), we see that the closed-loop dynamics of the NCS are
?x(t) = (A+BM?(t)KM?(t))x(t). (5.5)
The remainder of this chapter will address the following problem:
Problem 5.1. Find a feedback gain K and a dynamic medium access scheduling
policy
[?(t), ?(t)] = [?(x(t)), ?(x(t))], (5.6)
such that the closed-loop system (5.5) is asymptotically stable.
5.2 An Equivalent Switched System
Under constant feedback, the closed loop NCS (5.5) is essentially a switched system
[70, 71, 63], as we will illustrate shortly. Recall [63], that a switched system can be
70
described by a differential equation of the form:
?x(t) = fs(t)(x), (5.7)
where {fp : p ?P} is a family of sufficiently regular functions from Rn to Rn that is
parametrized by some index set P, and s : R mapsto?P is a piecewise constant function of
time, called the switching signal. In the special case where all the fp(?)?s are linear,
we obtain a switched linear system:
?x = As(t)x. (5.8)
First, consider the general case where the communication medium provides w?
(1 < w? < m) input channels and w? (1 < w? < n) output channels. Now ?(t) and
?(t) arem-to-w? andn-to-w? communication sequences, respectively. Forsimplicity,
it will be helpful to introduce one additional piece of notation.
Let ?(t) be a p-to-w (1 < w < p) communication sequence. Then, ?(t) is an p-
dimensional vector that takes parenleftbigpwparenrightbig possible different values. Denote one permutation
of the parenleftbigpwparenrightbig different values by
braceleftBig
?1, ?2, ??? , ?(pw)
bracerightBig
.
Definition 5.1. The ?scalar form? of the p-to-w communication sequence ?(t) is
the map
??(t) : R mapsto?
braceleftbigg
1,2??? ,
parenleftbiggp
w
parenrightbiggbracerightbigg
,
such that ?(t) = ???(t),?t. In other words, ??(t) equals i if ?(t) = ?i.
Now let the scalar forms of the communication sequences ?(t) and ?(t) be ??(t)
71
and ??(t), respectively, and define the switching signal s(t) as:
s(t) = [??(t),??(t)].
Then the closed-loop dynamics of the NCS (5.5) can be described by the following
switched system
?x = As(t)x, (5.9)
where the switching signal
s(t) = [??(t),??(t)]
is defined as:
s(t) : R mapsto?
braceleftbigg
1,2,???
parenleftbiggm
w?
parenrightbiggbracerightbigg
?
braceleftbigg
1,2,???
parenleftbigg n
w?
parenrightbiggbracerightbigg
.
Governed by the signal s(t), the system (5.9) switches between parenleftbigmw?parenrightbig?parenleftbig nw?parenrightbigpossible
dynamics:
As(t) ?
braceleftbigg
Aij : i = 1,??? ,
parenleftbiggm
w?
parenrightbigg
; j = 1,??? ,
parenleftbigg n
w?
parenrightbiggbracerightbigg
,
with
Aij = A+BKij, (5.10)
where Kij = diag(?i)?K ?diag(?j).
In the special case where the communication medium only provides one input
channel and one output channel (i.e., w? = w? = 1), the scalar form communication
sequences can be defined such that [??,??] = [i,j] corresponds to ? = eim and ? = ejn,
where eim and ein denote the i-th standard basis vector in Rm and Rn, respectively.
Doing so, Aij denotes the linear system dynamics when actuator i and sensor j are
72
accessing the communication medium. From (5.5), it clear that
Aij = A+BKij, (5.11)
where Kij = diag(eim)?K ?diag(ejn).
Without loss of generality, we will assume w? = w? = 1 in the rest of this
chapter. All of the analysis and results that follow apply to the multiple channel
case with a few straightforward modifications.
5.3 Stable Convex Combinations
Definition 5.2. [72] The switched system (5.9) is said to be quadratically stable
if there exists a positive definite quadratic function V(x(t)) = xT(t)Px(t), a posi-
tive number ? and a switching rule s(t) such that ddtV(x(t)) < ??xT(t)x(t) for all
trajectories x(?) of the system (5.9).
Fact 5.1. [70] The switched system (5.9) can be quadratically stabilized under a
feedback-based switching rule s(t) if there exist positive real numbers ?ij, i = 1,??? ,m,
j = 1,??? ,n, satisfying
msummationdisplay
i=1
nsummationdisplay
j=1
?ij = 1, (5.12)
such that the convex combination of Aij?s,
Adefines
msummationdisplay
i=1
nsummationdisplay
j=1
?ijAij, (5.13)
is stable.
Proof. We give here a sketch of the proof, as it will be useful in proving subsequence
results. For additional details, see [70].
73
If A is stable, then there exist positive definite matrices P,Q such that
ATP +PA = ?Q. (5.14)
Hence, for all x(t) negationslash= 0,
xT(t)(ATP +PA)x(t) = ?xT(t)Qx(t) < 0.
The last equation can be rewritten as
summationdisplay
i,j
?ijxT(t)(ATijP +PAij)x(t) = ?xT(t)Qx(t) < 0,
for all x(t) negationslash= 0. Because ?ij > 0, it follows that, for all x(t) negationslash= 0, there always exist
indices i(x) ?{1,???m}, and j(x) ?{1,???n} such that
xT(t)(ATi(x)j(x)P +PAi(x)j(x))x(t) < 0.
Notice that the Lyapunov function V = xT(t)Px(t) is continuous and piecewise
differentiable along trajectories of (5.9). Then between any two consecutive switches
d
dtV = x
T(t)(AT
s(t)P +PAs(t))x(t). (5.15)
Hence if the switched system is switched according to s(t) = [i(x(t)),j(x(t))], the
Lyapunov function V(x(t)) = xT(t)Px(t) will always be decreasing.
From Fact 5.1, the stabilizability of the switched system (5.9) relies on the exis-
tence of a stable convex combination (5.13). However, if the Aij?s are given and the
number of possible dynamics (in our case, m?n) is greater than two, the question
74
of whether such a stable convex combination exists is NP-hard [72]. Fortunately,
in an NCS, we also have the freedom to choose the controller in addition to the
communication policy, in order to obtain a stable convex combination (5.13). From
(5.11), we see that the A matrix in (5.13) can be expressed as
Adefines
summationdisplay
i,j
?ijAij = A+BK, (5.16)
where
K =
?
??
??
??
??
?11k11 ?12k12 ??? ?1nk1n
?21k21 ?22k22 ??? ?2nk2n
??? ???
?m1km1 ?m2km2 ??? ?mnkmn
?
??
??
??
??
, (5.17)
and kij is the (i,j) entry of the feedback gain K. Now a feedback gain K that
guarantees a stable convex combination (5.13) can be found by the following
Algorithm:
1. Choose m?n positive real numbers ?ij?s such that (5.12) is satisfied.
2. Choose a set of desired (stable) eigenvalues for A.
3. Solve the pole-placement problem forK, such thatA = A+BK has the desired
eigenvalues, provided that (A,B) is controllable.
4. Solve for K = [kij]m?n from (5.17).
Notice that for the same choice of A, different choices of ?ij?s results in different
values of the feedback gain K. A larger ?ij leads to a smaller kij. This fact gives
us additional freedom in the design of K. By properly choosing the ?ij?s we can
75
make the controller K meet certain optimization or design criteria, for example,
maxi,j |kij| < km, where km is the highest gain a controller may provide.
5.4 Dynamic Medium Access Scheduling Policies
Assume now that the feedback gain K is designed such that the convex combination
A (5.16) is stable. In this section, we introduce a set of dynamic medium access
scheduling policies (equivalently, feedback-based switching signals for (5.9)) that
quadratically stabilizes the closed-loop NCS (5.5).
Definition 5.3. Weighted Fastest Decay* (WFD*) rule:
For all t, let the switching signal s(t) be determined by
s(t) = argmin
i,j
?ijxT(t)[ATijP +PAij]x(t), (5.18)
where Aij is defined in (5.11), and P satisfies the Lyapunov equation (5.14).
Theorem 5.1. If A is stable, the system (5.9) is quadratically stable under the
switching rule WFD*.
Proof. Because A is stable, there exist positive definite matrices P, Q such that
(5.14) holds. Then,
msummationdisplay
i=1
nsummationdisplay
j=1
?ijxT(t)[ATijP +PAij]x(t) = ?xT(t)Qx(t). (5.19)
Equations (5.21) and (5.19) imply that, for all t,
?s(t)xT(t)[ATs(t)P +PAs(t)]x(t) ??x
T(t)Qx(t)
m?n .
76
Hence,
?V(x(t)) = xT(t)(ATs(t)P +PAs(t))x(t)
??x
T(t)Qx(t)
m?n??s(t) ???
?xT(t)x(t),
where
?? defines ?min(Q)m?n??
max
, (5.20)
and ?min(Q) denotes the smallest eigenvalue of Q, and ?max defines maxi,j ?ij.
From (5.18), we see that each ?ij acts as a weight associated with the dynamics
Aij. A greater ?ij will result in a greater chance that (5.9) is switched to Aij, all
else being equal. Note that Aij corresponds to the NCS dynamics when ui and
xj are accessing the communication medium. Hence, the choice of ?ij?s, is akin to
assign ?priorities? to every input and output of the NCS. Furthermore, from the
gain design algorithm introduced in Section 5.3, we notice that the choice of ?ij?s
also affects the value of the feedback gain, K: a greater ?ij will result in a smaller
gain kij. In order to decouple the controller design and medium access weighting, we
can modify the WFD* rule by replacing ?ij in (5.21) with a different set of weights,
wij, (i = 1???m, j = 1???n), this leads to the ?WFD? switching rule.
Definition 5.4. Weighted Fastest Decay (WFD) rule:
For all t, let the switching signal s(t) be determined by
s(t) = argmin
i,j
wijxT(t)[ATijP +PAij]x(t), (5.21)
where Aij is defined in (5.11), P satisfies the Lyapunov equation (5.14), and wij > 0,
(i = 1???m, j = 1???n) are real numbers defining the relative weight assigned to
77
the Aij dynamics.
Theorem 5.2. If A is stable, the system (5.9) is quadratically stable under the
switching rule WFD.
Proof. Suppose that the switching signal s(t) of the switched system (5.9) is gov-
erned by the WFD rule (5.21). Let the switching signal s?(t) be determined by the
WFD* rule (5.18). Then, by the definition of the WFD rule, we have, for all t,
ws(t)xT(t)[ATs(t)P +PAs(t)]x(t) ? ws?(t)xT(t)[ATs?(t)P +PAs?(t)]x(t).
From the proofs of Theorem 5.1, we know that
xT(t)[ATs?(t)P +PAs?(t)]x(t) ????xT(t)x(t).
Hence, we have
ws(t)xT(t)[ATs(t)P +PAs(t)]x(t) ??ws?(t)??xT(t)x(t),
and
xT(t)[ATs(t)P +PAs(t)]x(t) ??ws?(t)w
s(t)
??xT(t)x(t) ??wminw
max
??xT(t)x(t),
where wmin = mini,j wij, and wmax = maxi,j wij.
As suggested in[70], the following switching ruleensures maximum instantaneous
decay of the Lyapunov function V:
Definition 5.5. Fastest Decay (FD) rule:
78
For all t, let the switching signal s(t) be determined by
s(t) = argmin
i,j
xT(t)[ATijP +PAij]x(t), (5.22)
where Aij is defined in (5.11), and P satisfies the Lyapunov equation (5.14).
Theorem 5.3. If A is stable, the system (5.9) is quadratically stable under the
switching rule FD.
Proof. Under the FD rule, at any time t, system (5.9) is switched to the set of
dynamics that gives the fastest decay of V(x(t)). The instantaneous value of ?V (see
(5.15)) is, by definition, less than or equal to that when (5.9) is under any other
switching rules, including WFD*. Hence, for all t, ?V(x(t)) ? ???xT(t)x(t), where
?? is defined in (5.20).
Although Theorems 5.3 and 5.2 provide switching rules (equivalently feedback-
based medium access scheduling policies) that guarantee quadratic stability, the
medium access switching rate under the FD or WFD rule is not bounded. Under
these policies, it is theoretically possible that the medium access is switched for
infinitely many times in a finite time interval (?chattering?). High-speed switching
is often impractical and may result in undesirable high frequency actuator inputs.
One way to bound the switching rate is to introduce a minimum dwell time [63],
? > 0, restricting the time interval between any two consecutive switches to be no
smaller than ?. The GD switching rule we will introduce in the next guarantees a
dwell time between switchings. The idea is to let the system evolve with one set
of dynamics until the decay rate of the Lyapunov function V is less than a certain
threshold.
Definition 5.6. Guaranteed Dwell-time (GD) rule:
79
Let ?0 be a number satisfying 0 < ?0 < ??, where ?? is defined in (5.20);
1. Denote the current switch time by t0, choose s(t0) according to (5.22);
2. Let s(t) = s(t0) for all t ? [t0,t1), where t1 is the next switch time determined
by
t1 = inft>t
0
xT(t)(ATs(t0)P +PAs(t0))x(t) ???0xT(t)x(t); (5.23)
3. Repeat from step 1 for t1.
Theorem 5.4. If A is stable, the system (5.9) is quadratically stable under the
switching rule GD. Moreover, there exists ? > 0, such that the time between any
consecutive switches is no less than ?.
Proof. The quadratic stability of (5.13) is immediate because, according to the GD
rule, ?V < ??0xT(t)x(t) for all t. We only need to prove boundedness of the dwell
time. Let t0 and t1 (t0 < t1) be any two consecutive switching times. For t ? [t0,t1),
define
?(t) = ?x
T(t)(AT
s(t0)P +PAs(t0))x(t)
xT(t)x(t) ,
where ?(?) is continuous and differentiable in [t0,t1). LetATs(t0)P+PAs(t0) = ?Qs(t0),
then for all t ? [t0,t1)
??(t) = xTRs(t0)x?xTx?xTQs(t0)x?xTSs(t0)x
(xTx)2 , (5.24)
80
where Rs(t0) definesATs(t0)Qs(t0) +Qs(t0)As(t0) and Ss(t0) definesATs(t0) +As(t0). Now let
?Q = max
i,j
r(Qij),
?R = max
i,j
r(ATijQij +QijAij), and
?S = max
i,j
r(ATij +Aij),
where Qij defines ?(ATijP + PAij), and r(B) denotes the spectral radius of the square
matrix B:
r(B) defines max
i
(|?i(B)|).
From (5.24), it is easy to verify that for all t ? [t0,t1),
|??(t)|? ?R +?Q ??S.
Hence,
|?(t?1 )??(t0)|? (?R +?Q ??S)(t1 ?t0),
where t?1 denotes the instant immediately before the switch taking place at t1.
Notice that ?(t0) ? ?? because, at the beginning of each switch, s(t) is determined
by (5.22). Also ?(t?1 ) = ?0 according to the GD policy. Hence
|?(t?1 )??(t0)|? ?? ??0,
and
t1 ?t0 ? ?
? ??0
?R +?Q ??S. (5.25)
81
5.5 An Example
We now present an example in which we stabilize an MIMO plant using the dynamic
medium access scheduling policies and the controller design algorithm introduced
in the previous sections. Consider the NCS in which the plant is the 2-input 4-th
order unstable batch reactor[69]:
?x =
?
??
??
??
??
1.38 ?0.2077 6.715 ?5.676
?0.5814 ?4.29 0 0.675
1.067 4.273 ?6.654 5.893
0.048 4.273 1.343 ?2.104
?
??
??
??
??
x+
?
??
??
??
??
0 0
5.67 0
1.136 ?3.146
1.136 0
?
??
??
??
??
u.
Suppose that state feedback is available, and that the plant is controlled by a con-
stant feedback gain K via a shared communication medium which has only one
input and one output channel. The closed-loop NCS is equivalent to a switched
system that transitions between eight possible linear dynamics
?x = Aijx, (i = 1,2, j = 1,2,3,4).
We chose the weights ?ij to be
[?ij] =
?
?? 1/8 1/8 1/8 1/8
1/8 1/8 1/8 1/8
?
??, (5.26)
and placed the eigenvalues of the convex combination A (see (5.13)) at [?5,?6,?4
,?3]. Solving the pole-placement problem (5.16) for K, and then solving for K from
(5.17), we obtained the feedback gain
82
K =
?
?? 0.7285 ?4.2600 ?1.1423 ?2.9334
15.3458 2.8926 6.9400 ?1.7744
?
??.
We then chose Q = I, and solved for P in the Lyapunov equation (5.14). The
resulting P matrix was
P =
?
??
??
??
??
0.4665 ?0.0871 0.2263 ?0.3208
?0.0871 0.1378 ?0.0348 0.1576
0.2263 ?0.0348 0.1967 ?0.1438
?0.3208 0.1576 ?0.1438 0.4805
?
??
??
??
??
.
Using the calculated feedback gain K and the matrix P, we first used the FD
rule as the medium access scheduling policy. Fig. 5.2 illustrates the evolution of the
plant?s state x(t) and the communication sequences under the FD scheduling rule.
With the same K and P, we next used the GD rule as the medium access
scheduling policy. From (5.20), we calculated ?? = 1. We then chose ?0 = 0.1
in the GD rule. Fig. 5.3 illustrates the plant?s state evolution and the resulting
communication sequences under the GD rule. Compared with the results in Fig. 5.2,
we see that the GD rule significantly reduced the switch rates in the communication
sequences ??(t) and ??(t). Not surprisingly, as a trade-off, the converging rate is slower
with the GD rule than with the FD rule.
Finally, we used the WFD rule to schedule medium access in the NCS, and
demonstrated how the wij?s act as medium access priorities. Initially, we chose
[wij] = W1 =
?
?? 1/6 1/6 1/6 1/6
1/12 1/12 1/12 1/12
?
??.
83
0 0.2 0.4 0.6 0.8 1?10
?5
0
5
10
15
Time (s)
X(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) States evolution.
0 0.2 0.4 0.6 0.8 1
1
2
Time (s)
??(t)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
Time (s)
??(t)
(b) Evolution of communication sequences ??(t) and ??(t).
Figure 5.2: Simulation results: Stabilization of an NCS using the FD rule. The
plant was an LTI system with 2 inputs and 4 outputs. The communication medium
provided one input channel and one output channel.
84
0 0.2 0.4 0.6 0.8 1?10
?5
0
5
10
15
Time (s)
X(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) States evolution.
0 0.2 0.4 0.6 0.8 1
1
2
Time (s)
??(t)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
Time (s)
??(t)
(b) Evolution of communication sequences ??(t) and ??(t).
Figure 5.3: Simulation results: Stabilization of an NCS using the GD rule (?0 =
0.1, ?? = 1). The plant was an LTI system with 2 inputs and 4 outputs. The
communication medium provided one input channel and one output channel.
85
Notice that the first row in [wij] corresponds to the priority of u1, while the second
row corresponds to the priority of u2. Thus, by using W1, we assigned to the input
u1 a higher access priority than u2. Fig. 5.4 illustrates the the plant?s state evolution
and associated communication sequences using the WFD rule with [wij] = W1. The
resulting communication sequence in Fig. 5.4 shows that 90.5% of medium access
time to the input channel was assigned to u1 under this choice of [wij].
Exchanging the rows of W1, we formed the weight matrix
[wij] = W2 =
?
?? 1/12 1/12 1/12 1/12
1/6 1/6 1/6 1/6
?
??,
which give u2 higher priority to access the input communication medium. Fig. 5.5
illustrates the state evolution and associated communication sequences when using
the WFD rule with [wij] = W2. This time, 96.5% of the medium access time to the
input channel was allocated to u2.
86
0 0.2 0.4 0.6 0.8 1?10
?5
0
5
10
15
Time (s)
X(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) States evolution.
0 0.2 0.4 0.6 0.8 1
1
2
Time (s)
??(t)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
Time (s)
??(t)
(b) Evolution of communication sequences ??(t) and ??(t).
Figure 5.4: Simulation results: Stabilization of an NCS using the WFD rule. The
plant was an LTI system with 2 inputs and 4 outputs. The communication medium
provided one input channel and one output channel. The input u1 was given higher
access priority.
87
0 0.2 0.4 0.6 0.8 1?10
?5
0
5
10
15
Time (s)
X(t)
x1(t)
x2(t)
x3(t)
x4(t)
(a) States evolution.
0 0.2 0.4 0.6 0.8 1
1
2
Time (s)
??(t)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
Time (s)
??(t)
(b) Evolution of communication sequences ??(t) and ??(t).
Figure 5.5: Simulation results: Stabilization of an NCS using the WFD rule. The
plant was an LTI system with two inputs and two outputs. The communication
medium provided one input channel and one output channel. The input u2 was
given higher access priority.
88
Chapter 6
LQG Control in NCSs
In this chapter, we study state estimation and optimal control of NCSs under the
classical LQG (linear quadratic Gaussian) framework. Using existing results on
periodic Riccati equations, together with properly chosen periodic communication
sequences, we show that the estimation error covariance and the Kalman gain as-
sociated with the LQG problem both converge to unique periodic solutions. Fur-
thermore, the resulting LQG controller renders both the estimation error dynamics
and the closed-loop NCS dynamics asymptotically stable. The LQG design method
presented in this chapter avoids the complexity associated with previously proposed
models and addresses MIMO NCSs whose dynamics are ?fully coupled?.
6.1 An Equivalent NCS Model
In the NCS model presented in Chapter 3, unavailable inputs and outputs are set to
zero and are thus effectively ignored by the plant and the controller. In this chapter,
we introduce a new NCS model where unavailable outputs and inputs are simply
removed from the output and input vectors of the extended plant. This new model
89
will turn out to be equivalent to the one discussed in Chapter 3. This modified
model will simplify the solution of the LQG problem, as we will illustrate in Section
6.4.3. A similar technique is mentioned in [73] for multi-rate systems.
We begin with a deterministic NCS setting, which will be generalized to the
stochastic setting in Section 6.4, whereupon the LQG optimal estimation and control
problem will be formulated. Consider the NCS in which the plant is the discrete-time
LTI system:
x(k + 1) = Ax(k) +Bu(k), (6.1)
y(k) = Cx(k),
where x = [x1,??? ,xn]T ? Rn, u = [u1,??? ,um]T ? Rm, and y = [y1,??? ,yp]T ? Rp
are the plant?s states, inputs, and outputs, respectively. Suppose that the communi-
cation medium connecting the plant and the can only accommodate w? (1 ? w? < p)
output channels and w? (1 ? w? < m) output channels. Also suppose that the
medium access of the actuators and sensors is governed by the m-to-w? input com-
munication sequence (see Definition 3.3) ?(k) and p-to-w? output communication
sequence ?(k), respectively.
6.1.1 Effects of Medium Access Constraints
At the output side of the plant, only w? of the p outputs, y1,y2??? ,yp are granted
medium access at any time k, and all other outputs are effectively ignored by the
controller. Let the output information received by the controller at time k be de-
noted by
?y(k) = [?y1(k), ?y2(k),??? , ?yw?(k)]T.
90
For all k, ?y(k) contains those elements from y(k) for which ?i(k) = 1. To establish
the relationship between y(k) and ?y(k), we will make use of the following definition:
Definition 6.1. Let ?(k) be an M-to-N communication sequence. Then, for all
k ? N, the N ?M matrix ??(k) is obtained by removing the M ?N all-zero rows
from the M ?M matrix diag(?(k)).
Example 6.1. Let ?(1) = [1,1,0,1]T, then
??(1) =
?
??
??
?
1 0 0 0
0 1 0 0
0 0 0 1
?
??
??
?
.
Using the last definition, we can express ?y(k) as
?y(k) = ??(k)y(k), (6.2)
where ?(k) is the output communication sequence.
Similarly, at the input side of the plant, only w? of the m inputs, u1,??? ,um,
can be updated by the controller at any time k. When an input uj loses its access
to the communication medium, the plant ignores that input until the corresponding
actuator regains medium access. This is equivalent to setting uj = 0 while ?j = 0.
Let
?u(k) = [?u1(k), ?u2(k)??? ,?uw?(k)]T
denote the w? update input values to be sent to the plant from the controller at
time k. Under the protocol outlined above, u(k) can be expressed as
u(k) = ??(k)T ?u(k). (6.3)
91
6.1.2 The Extended Plant
Combining (6.1),(6.2), and (6.3), we obtain a new expression of the extended plant:
x(k + 1) = Ax(k) +B??(k)T ?u(k), (6.4)
?y(k) = ??(k)Cx(k).
Notice that (6.4) is a time-varying system with w? inputs and w? outputs, and it
describes the dynamics of the plant from the controller?s point of view. A block
diagram of the extended plant is shown in Figure 6.1. This new extended plant
formula shouldbe comparedwith theformula (3.7)presented inChapter 3: Equation
(6.4) is obtained by removing from ?u and ?y (in (3.7)) the input and output elements
that are ignored by the plant and the controller due to their being unavailable. These
two extended formulas are essentially equivalent.
Extended Plant
Plant
Controller
??(k)?T? (k) u(k) y(k)
?y(k)?u(k)
Figure 6.1: The extended plant under the new NCS formulation.
92
6.2 LQG Problem Formulation
We now go on to formulate the main problem we will investigate in this Chapter.
Consider the NCS in which the plant is the discrete-time stochastic LTI system:
x(k + 1) = Ax(k) +Bu(k) +v(k) (6.5)
y(k) = Cx(k) +w(k), k = 0,1,???N ?1,
where x ? Rn, u ? Rm, y ? Rp, N ? Z+, and the process noise v(?) and the
measurement noise w(?) are each Gaussian i.i.d.. Without loss of generality, we
assume that x(0),v(0),???v(N ? 1),w(0),???w(N ? 1) are independent random
variables with
v(?) ?N(0,G), w(?) ?N(0,Ip?p),
where Ip?p is the p ?p identity matrix and G is a positive definite n ?n matrix.
We will also assume that the plant?s initial condition x(0) is Gaussian with
x(0) ?N(x0,?0).
As before, suppose that the communication medium connecting the plant and
the controller provides w? (1 ? w? < m) input channels, and w? (1 ? w? < p)
output channels. Using the model presented in the previous section with the pair
of communication sequences ?(?) and ?(?), we obtain the dynamics of the extended
plant:
93
x(k + 1) = Ax(k) +B?T? (k)?u(k) +v(k), (6.6)
?y(k) = ??(k)Cx(k) +??(k)w(k).
or, equivalently
x(k + 1) = Ax(k) + ?B(k)?u(k) +v(k), (6.7)
?y(k) = ?Cx(k) + ?w(k).
where ?B(k) = B?T? (k), ?C = ??(k)C, and ?w(k) = ??(k)w(k). The remainder of this
chapter will give solutions to the following problem:
Problem 6.1. Design an optimal controller for the extended plant (6.7), such that
the quadratic cost function
J = E
bracketleftBigg
xT(N)Qx(N) +
N?1summationdisplay
k=0
xT(k)Qx(k) + ?uT(k)?u(k)
bracketrightBigg
(6.8)
is minimized.
Note that, if ?(?) is determined off-line, ?w(0),??? ?w(N?1),x(0),v(0),???v(N?1)
are independent random variables because ?w(k) is a sub-vector of w(k) for all k.
Moreover, because of the linearity of ??(k), ?w(k) is also Gaussian with
?w(k) ?N(0,Iw??w?), ?k.
Thus Problem 6.1 is essentially a standard LQG problem for the stochastic discrete-
time extended plant (6.7).
94
6.3 Communication Sequences that Preserve
Stabilizability and Detectability
Under periodic communication sequences, ?B(k) and ?C(k) are both periodic as well.
The extended plant (6.7) is thus a stochastic periodic linear time-varying (LTV)
system. We know [74] that, the stability of such a system under an LQG controller
depends on the stabilizability and detectability of its deterministic counterpart,
which, in our case, is the extended plant (6.4). Before going further, we review the
following relevant concepts for discrete-time LTV systems.
Definition 6.2. The system (6.4) is controllable on [k0,kf] if, given any x0, there
exists an input ?u(?) that steers (6.4) from x(k0) = x0 to the origin at time kf. We
say that (6.4) is l-step controllable (or ?controllable?) if, for any k, there exists a
positive integer l such that (6.4) is controllable on [k,k +l].
In discrete-time systems, the dual concept of controllability is reconstructibility.
Definition 6.3. The system (6.4) is reconstructible on [k0,kf] if x(kf) can be
determined by the corresponding response ?y(k) for k ? [k0,kf]. We say that (6.4)
is l-step reconstructible (or ?reconstructible?) if, for any k, there exists a positive
integer l such that (6.4) is reconstructible on [k,k +l].
Recall that, controllability and reconstructibility are weaker notions than reach-
ability and observability. A linear system is controllable if it is reachable, and is
reconstructible if it is observable (see, for example, [64]). In the case where A is
invertible, the controllability and reachability of the extended plant (6.4) are equiv-
alent, as are observability and reconstructibility. Another pair of dual concepts are
stablizability and detectablility.
95
Definition 6.4. A discrete-time linear system is called stabilizable if its uncontrol-
lable subspace is asymptotically stable.
Definition 6.5. A discrete-time linear system is called detectable if its unrecon-
structible subspace is asymptotically stable.
6.3.1 Controllability and Reconstructibility
We have shown in Chapter 4 (for an equivalent extended plant expression (4.22))
that, if A is invertible, there always exist periodic communication sequences ?(?) and
?(?) that preserve the reachability and observability of the plant. We will show that
these results still hold for the controllability and reconstructibility of our modified
NCS model. Moreover, the requirement for A being invertible can be removed.
Case 1: ?A? Invertible
Let us begin with the case where the matrix A in (6.4) is invertible. Now, con-
trollability and reachability are equivalent notions, so are observability and recon-
structibility. Let
R(0,kf) = [Akf?1B?T? (0),Akf?2B?T? (1),??? ,B?T? (kf ?1)]. (6.9)
The system (6.4) is controllable on [0,kf] iff rank(R(0,kf)) = n. Notice that, at
each step k, ?T? (k) has the effect of ?selecting? w? columns from the m columns of
the term Akf?k?1B on the RHS of (6.9).
We are now encountering the same situation in the proof of Theorem 4.3, where
we analyzed the reachability of an equivalent extended plant expression (4.22). We
can follow the same arguments given in Theorem 4.3 and Corollary 4.3 to arrive at
the following:
96
Corollary 6.1. Let A be invertible and the pair (A,B) be controllable. For any
integer 1 ? w? < m, there exist integers l,N > 0 and an N-periodic m-to-w?
communication sequence ?(?) such that the extended plant (6.4) is l-step controllable.
By switching from column manipulations to row manipulations, we obtain the
dual result for reconstructibility.
Corollary 6.2. Let A be invertible and the pair (A,C) be reconstructible. For
any integer 1 ? w? < p, there exist integers l,N > 0 and an N-periodic p-to-w?
communication sequence ?(?) such that the system (6.4) is l-step reconstructible.
Case 2: ?A? Non-invertible
Suppose A contains q (1 ? q < n) zero eigen-values. Then, there exists an invertible
matrix L such that
L?1AL =
?
?? A1 0
0 A0
?
??, L?1B =
?
?? B1
B0
?
??,
where A1 is an (n?q)?(n?q) invertible matrix, A0 is a q?q square matrix with
only zero eigenvalues, B1 is a (n?q)?m matrix, and B0 is a q?m matrix. Define
a new state variable
x? = L?1x =
?
?? x1
x0
?
??,
where x1 denotes the (n ? q)-dimensional subspace corresponding to A1, and x0
is the q-dimensional subspace corresponding to A0. Then, under the similarity
97
transformation L, the dynamics of the extended plant (6.4) can be rewritten as
?
?? x1(k + 1)
x0(k + 1)
?
?? =
?
?? A1 0
0 A0
?
??
?
?? x1(k)
x0(k)
?
??+
?
?? ?B1(k)
?B0(k)
?
?? ?u(k), (6.10)
where ?B0 defines B0?T? (k) and ?B1 defines B1?T? (k). In (6.10), the extended plant (6.4) has
been decomposed into two uncoupled sub-plants: the (n?p)-dimensional sub-plant
with dynamics (A1, ?B1(?)) and state variable x1, and the p-dimensional sub-plant
with dynamics (A0, ?B0(?)) and state variable x0. The following is a well-know fact
from linear system theory:
Fact 6.1. The controllability (or reachability, stabilizability) and reconstructibility
(or observability, detectability) of a linear (time-varying) system do not change un-
der similarity transformations (i.e., replacing the state variable x with x? = L?1x,
where L is an invertible matrix).
It should therefore be clear that, in order to study the controllability and recon-
structibility of the extended plant (6.4), one only needs to study the controllability
and reconstructibility of (6.10). Suppose therefore, that the pair (A,B) is control-
lable; then, (A1,B1) must be controllable as well. Moreover, because A1 is invertible,
the results we derived for the A-invertible case imply that there exists an integer
k1 > 0 and a communication sequence ?(k), for k = [0,k1], such that the pair
(A1, ?B1(?)) is controllable in [0,k1]. Hence, for any initial condition, there exist a
control sequence ?u(k), for k = [0,k1], that drives the state x1(k) to zero at k = k1.
We use that particular communication sequence ?(k) and control sequence ?u(k),
for the time interval [0,k1]; after k1, we set ?u(k) to zero and let the communication
sequence ?(?) be arbitrary. Then, there must exist kf > k1, such that x0(kf) = 0
because A0 has all zero eigen-values. Also, notice that x1(kf) = 0 as well because
98
no control will be fed into the plant during [k1 + 1,kf]. We have thus constructed
a control sequence ?u(k) and a communication sequence ?(k) for the time interval
[0,kf], such that the new state variable x? is driven to zero at time kf regardless of
initial conditions. This is equivalent to stating that (6.10) is controllable in [0,kf]
under the communication sequence ?(k). The above discussion, combined with an
argument similar to that in Corollary 4.3, yields the following results, which do not
require the invertibility of A.
Theorem 6.1. Let the pair (A,B) be controllable. For any integer 1 ? w? < m,
there exist integers l,N > 0 and an N-periodic m-to-w? communication sequence
?(?) such that the extended plant (6.4)is l-step controllable.
Then the duality of controllability and reconstructibility gives the following
Theorem 6.2. Let the pair (A,C) be reconstructible. For any integer 1 ? w? < p,
there exist integers l,N > 0 and an N-periodic p-to-w? communication sequence
?(?) such that the system (6.4) is l-step reconstructible.
6.3.2 Stabilizability and Detectability
We now generalize the results developed in the previous section, to cover the notion
of stabilizability and detectability. Suppose that the pair (A,B) is stabilizable; then,
there exist an invertible matrix ?, which transforms the pair (A,B) into its Kalman
controllability canonical form:
??1A? =
?
?? Ac A?c
0 A?c
?
??, ??1B =
?
?? Bc
0
?
??,
99
where (Ac,Bc) and (A?c,0) corresponds to the controllable and uncontrollable sub-
spaces of the pair (A,B), respectively. We define the new state variable
x?? = ??1x =
?
?? xc
x?c
?
??,
so that the dynamics of the extended plant (6.4) can be rewritten as
?
?? xc(k + 1)
x?c(k + 1)
?
?? =
?
?? Ac A?c
0 A?c
?
??
?
?? xc(k)
x?c(k)
?
??+
?
?? ?Bc(k)
0
?
??u(k), (6.11)
where ?Bc(k) defines Bc?T? (k). Because (Ac,Bc) is controllable, we know from Theo-
rem 6.1 that there exists a periodic communication sequence ?(?) such that the pair
(Ac, ?Bc(?)) is also controllable. Moreover, the uncontrollable subsystem A?c is stable
because we have assumed the pair (A,B) to be stabilizable. We can thus obtain the
following:
Theorem 6.3. Suppose the pair (A,B) is stabilizable. For any integer 1 ? w? < m,
there exist an integer N > 0 and an N-periodic m-to-w? communication sequence
?(?) such that the pair (A, ?B(?)) is stabilizable.
A dual theorem for detectability is the following:
Theorem 6.4. Suppose the pair (A,C) is detectable. For any integer 1 ? w? < p,
there exist an integer N > 0 and an N-periodic p-to-w? communication sequence
?(?) such that the pair (A, ?C(?)) is detectable.
100
6.4 The LQG Controller
With the above results in hand, we can now give a complete solution to Problem 6.1.
Note that the extended plant (6.5) is an LTV system. It is well known that (e.g.,
[75]) the LQG optimal controller for an LTV system, such as the extended plant
(6.7), consists of two parts (Fig. 6.2):
1. A Kalman filter that provides the optimal state estimate ?x(k) based on the
outputs ?y(?).
2. The LQ gain L(k), obtained by solving an LQR (linear quadratic regulator)
problem for the the extended plant (6.7)?s deterministic counterpart, (6.4).
Kalman Filter
Plant
?u(k)
?L(k)
??(k)??(k) u(k) y(k)
?x(k) ?y(k)
Figure 6.2: An NCS stabilized by the LQG controller.
The separation principle ensures that the two sub-problems can be solved inde-
pendently. The resulting optimal control law ?u? is:
?u?(k) = ?L(k)?x(k).
6.4.1 Kalman Filtering
The optimal estimator for the extended plant (6.7) is the discrete-time Kalman
filter, which can be described with the following recursive algorithm, starting with
101
initial conditions ?x(0) = ?x0, ?(0) = ?0.
1. Time update
?x(k?) = A?x(k?1) + ?B(k?1)?u(k?1), (6.12)
P(k) = A?(k ?1)AT +G, (6.13)
where ?x(k?) is the prediction of the state variable x(k) before the measure-
ment of ?y(k), and P(k) is the variance of the prediction error:
?x(k?) defines E[x(k)|?y(0)????y(k?1)],
P(k) defines E[(x(k)? ?x(k?))(x(k)? ?x(k?))T].
2. Measurement update
H(k) = P(k)?CT(k)( ?C(k)P(k)?CT(k) +I)?1, (6.14)
?x(k) = ?x(k?) +H(k)(?y(k)? ?C(k)?x(k?)), (6.15)
?(k) = (I ?H(k) ?C(k))P(k), (6.16)
where ?x(k)is the estimation of state variable x(k) conditioned on the mea-
surements ?y(0)????y(k), ?(k) is the covariance of the estimation error:
?x(k) defines E[x(k)|?y(0)????y(k)],
?(k) defines E[(x(k)? ?x(k))(x(k)? ?x(k))T].
102
From (6.13), (6.16), and (6.14), it is easy to verify that the sequence P(?) satisfies
the time varying discrete time Riccati equation
P(k + 1) =AP(k)AT +G?
AP(k) ?CT(k)[I + ?C(k)P(k)?CT(k)]?1 ?C(k)P(k)AT. (6.17)
Let e(k) denote the ?one-step prediction? error, e(k) = x(k)? ?x(k?). We have
e(k + 1) = (A??(k) ?C(k))e(k) +v(k)??(k)w(k),
and
E[e(k + 1)] = (A??(k) ?C(k))E[e(k)], (6.18)
where ?(k) = AH(k) is termed the ?Kalman gain?. The error equation (6.18) is
stable if A??(k) ?C(k) is stable.
6.4.2 LQ Regulator
The LQ optimal gain L(k) is obtained by solving an LQR problem for the extended
plant (6.7)?s deterministic counterpart, (6.4). The gain matrix L(k) satisfies the
equation:
L(k) = ( ?BTK(k + 1) ?B(k) +I)?1 ?B(k)TK(k + 1)A, (6.19)
and the symmetric positive semidefinite matrix K(?) satisfies the backwards Riccati
equation
103
K(N) =Q, (6.20)
K(k) =ATK(k + 1)A+Q?
ATK(k + 1) ?B(k)( ?BT(k)K(k + 1) ?B(k) +I)?1 ?BT(k)K(k + 1)A. (6.21)
With state feedback, the closed loop dynamics of the system (6.4) under the LQR
optimal gain L(k) are:
x(k + 1) = (A? ?B(k)L(k))x(k). (6.22)
The closed loop system is stable if (A? ?B(k)L(k)) is stable.
6.4.3 Periodic Riccati Equations
Because of the periodicity imposed by the communication sequences ?(?) and ?(?),
?B(?), ?C(?) are periodic as well; hence, the Riccati equations, (6.17) and (6.21),
both become Discrete-time Periodic Riccati Equations (DPREs). DPREs have been
extensively studied in the 80s and early 90s (see [74, 73, 76] and the references
therein). We go on to review some fundamental results from [74].
Definition 6.6. [74] A Discrete-time Periodic Riccati Equation (DPRE) is a dif-
ference equation of the form
P(k + 1) = A(k)P(k)AT(k) +B(k)B(k)T? (6.23)
A(k)P(k)CT(k)[I +C(k)P(k)CT(k)]?1C(k)P(k)AT(k),
where A(k) : Z mapsto? Rn?n, B(k) : Z mapsto? Rn?m, C(k) : Z mapsto? Rp?n and A(?), B(?), and
104
C(?) are T-periodic.
Theorem 6.5. ([74], Theorem 5) Consider the Kalman gain,
K(k) = A(k)P(k)CT(k)(C(k)P(k)CT(k) +I)?1,
associated with any symmetric positive semidefinite solution P(?) of (6.23). If (A(?),
B(?)) is stabilizable and (A(?), C(?)) detectable, then the corresponding closed-loop
matrix ?A(?) = A(?)?K(?)C(?) is exponentially stable.
In the last theorem, the exponential stability of a time varying matrix A(?) is to
be understood as the exponential stability of the associated LTV system with the
dynamics x(k + 1) = A(k)x(k).
Theorem 6.6. ([74], Theorem 6) There exists a unique Symmetric Periodic Positive
Semidefinite (SPPS) solution ?P(?) of the DPRE (6.17) and ??A(?) = A(?)? ?K(?)C(?)
is asymptotically stable iff (A(?),B(?)) is stabilizable and (A(?),C(?)) is detectable,
where ?K(?) is the Kalman gain associated with ?P(?).
Theorem 6.7. ([74], Theorem 7) Suppose that (A(?), B(?)) is stabilizable and (A(?),
C(?)) detectable. Then, every symmetric and positive semidefinite solution of the
DPRE converges to the unique SPPS solution.
The above results give a necessary and sufficient condition (Theorem 6.6) for the
existence and uniqueness of an SPPS solution as well a stability condition (Theorem
6.5) for the closed-loop system. An asymptotic convergence theorem (Theorem 6.7)
is provided to guarantees the convergence of a solution to the unique SPPS solution.
It is also shown that the closed-loop system is asymptotically stable under the SPPS
solution.
105
6.4.4 Convergence of the LQG Controller
Theorems 6.3, 6.4 combined with the facts we reviewed in Section 6.4.3 and applied
to the DPREs (6.17) and (6.21), imply the following:
Theorem 6.8. Suppose that, for the extended plant (6.7),
1. the communication sequence ?(?) is N-periodic, and the pair (A, ?C(?)) is de-
tectable;
2. the pair (A,g) is stabilizable, where G = ggT is the variance of the noise term
v(k) in (6.5);
then, starting from any positive definite initial conditions, as k ? ?, the Riccati
equation associated with the Kalman filter (6.17) converges to a unique N-periodic
solution ??(k). Moreover, the Kalman filter?s one-step prediction error dynamics
(6.18) are exponentially stable.
Theorem 6.9. Suppose that, for the extended plant (6.7),
1. the communication sequence ?(?) is N-periodic, and the pair (A, ?B(?)) is sta-
bilizable;
2. the pair (A,qT) is detectable, where Q = qqT is the weight matrix in the
quadratic cost function (6.8);
Then, starting from any positive definite initial conditions, as k ? ?, the Riccati
equation associated with the LQ regulator (6.21) converges to a unique N-periodic
solution ?K(k). Moreover, the closed-loop dynamics (6.22) under the LQ regulator
are exponentially stable.
106
6.4.5 Remark
The matrices ?B(k) and ?C(k) are N-periodic under N-periodic communication se-
quences ?(?) and ?(?). Hence, if the extended plant (6.7) is stabilizable and de-
tectable, the optimal feedback gain L(k) in (6.19) and the optimal observer gain
H(k) in (6.14) will also converge to unique periodic solutions, for all initial condi-
tions. One can therefore construct a sub-optimal LQG controller by implementing
the two periodic solutions. By virtue of Theorem 6.6, we know that both the Kalman
filter?s one-step prediction error dynamics (6.18) and the closed-loop dynamics (6.22)
are asymptotically stable under this sub-optimal controller.
6.5 An Example
We simulated a 4-th order unstable stochastic LTI plant with 2 inputs and 2 outputs.
The dynamics of the plant were of the form (6.5), with
A =
?
??
??
??
??
1 1/5 0 0
0 11/4 0 1/5
1 1/5 1/3 3/4
0 ?1 0 1/4
?
??
??
??
??
,B =
?
??
??
??
??
0 0
1 0
0 0
0 1
?
??
??
??
??
,
C =
?
?? 1 1 0 0
0 0 1 0
?
??.
The process noise v(?) and the measurement noise w(?) satisfied v(?) ? N(0,4I4?4)
and w(?) ?N(0,I2?2). The plant was controlled remotely via a shared communica-
tion medium which had only one input and one output channel (i.e., w? = w? = 1).
Using the algorithm introduced in Section 4.2.1, we found that the extended plant
107
is stabilizable and detectable under the 2-periodic input and output communication
sequences
{?(0),?(1),???} = {[0, 1]T,[1, 0]T,???}, and
{?(0),?(1),???} = {[0, 1]T,[1, 0]T,???}.
We formulated the LQG problem described in Section 6.4, where Q = 25I4?4
and the initial conditions were x(0) = [100,50,7,6]T, ?x(0) = [1,1,3,4]T, and ?(0) =
4I4?4. The simulation results illustrate what was developed earlier in this chapter:
the solution of the periodic Riccati equation (6.17), which is associated with the
Kalman filter error covariance, converged to a 2-periodic SPPS solution ?P(?) in
5 steps; the solution of the periodic backwards Riccati equation (6.21), which is
associated with the LQ optimal gain, converged to a 2-periodic SPPS solution ?K(?)
in 15 steps. The evolutions of tr(P(k)) and tr(K(k)) are shown in Fig. 6.3. Finally,
the SPPS solutions to the Riccati equations (6.17) and (6.21) were
?P(2i) =
?
??
??
??
??
8.92 1.15 5.00 ?0.40
1.15 267.59 ?21.96 ?105.34
5.00 ?21.96 14.12 10.06
?0.40 ?105.34 10.06 46.61
?
??
??
??
??
,
?P(2i+ 1) =
?
??
??
??
??
9.53 ?17.70 9.01 7.29
?17.70 71.87 ?29.46 ?27.60
9.01 ?29.46 23.52 13.57
7.29 ?27.60 13.57 15.79
?
??
??
??
??
,
108
?K(2i) =
?
??
??
??
??
456.09 62.55 12.61 34.65
62.55 79.70 0.33 ?7.34
12.61 0.33 28.10 7.54
34.65 ?7.34 7.54 45.73
?
??
??
??
??
,
?K(2i+ 1) =
?
??
??
??
??
496.33 284.55 11.30 39.56
284.55 715.87 3.60 53.81
11.30 3.60 27.98 6.83
39.56 53.81 6.83 43.83
?
??
??
??
??
, for i ? Z+.
Using the solutions for ?P(?) and ?K(?), the Kalman filter and LQ regulator were
constructed based on the formulas in Sections 6.4.1, 6.4.2. The state evolution of
the closed-loop system is shown in Fig. 6.4(a). The evolution of the Kalman filter?s
one-step prediction error, e(k), is shown in Fig. 6.4(b).
109
0 10 20 30 40 500
200
400
600
800
1000
1200
1400
k
tr(P(k))
tr(K(k))
Figure 6.3: Evolution of tr(P(k)) and tr(K(k)). The P(k) satisfy the Riccati equa-
tion (6.17), while K(k) satisfy the backwards Riccati equation (6.21). Both P(k)
and K(k) converge to SPPS solutions.
110
0 10 20 30 40 50?600
?400
?200
0
200
400
600
k
X(k)
x1(k)
x2(k)
x3(k)
x4(k)
(a) State evolution.
0 10 20 30 40 50?200
?100
0
100
200
300
k
e(k)
e1(k)
e2(k)
e3(k)
e4(k)
(b) Evolutions of Kalman filter?s one-step prediction error e(k) = x(k)??x(k?).
Figure 6.4: Simulation Results: Stabilization of an NCS under the LQG controller.
The plant was a stochastic LTI system with 2 inputs and 2 outputs. The commu-
nication medium provided 1 input channel and 1 output channel.
111
Chapter 7
NCS Experiments
In this chapter, we describe two NCS experiments which were designed to illustrate
the design techniques introduced in this thesis. The first experiment exercised our
dynamic medium access scheduling policies with a constant feedback controller; the
second experiment used a static medium access strategy and an LQG controller.
7.1 Experiment Platform
We begin with a general description of the hardware and software configurations of
our experiment platform. This platform provided a friendly NCS design and rapid
prototyping interface, and can be used for implementing various medium access
scheduling strategies and controller design methods that are proposed in this thesis.
7.1.1 Hardware Configuration
The hardware configuration of the experiment platform is depicted in Fig. 7.1. It
consisted of two personal computers, henceforth referred to as PC1 and PC2, which
were connected to each other via a RS232 serial communication channel. The two
112
PCs were equipped with Intel Pentium-4 CPUs running at 1GHz, and had 1G byte
RAM each. PC1 was also equipped with a data acquisition (DAQ) card which could
be used to connect the sensors and actuators of a physical plant to the computer.
The DAQ card used in the experiment platform (a MultiQ-PCI by Quanser Consult-
Figure 7.1: Hardware configuration of the NCS experiment platform.
ing Inc.) provided sixteen 14-bit A/D input channels and four D/A output channels
with a conversion time of 17?s. It also had six 24-bit encoder input and forty-eight
digital I/O channels. The DAQ card could communicate with its host PC via the
PCI interface running at 33MHz.
In an NCS experiment involving a physical plant, the outputs of the plant?s
sensors were first sampled by PC1 via the DAQ card. Then, following a user-
specified medium access scheduling strategy, those outputs were sent to PC2, over
a RS232 serial channel. Using the output information it receives from PC1, the
controller running in PC2 calculated the control signals needed by the plant; then,
113
those control signals were sent to PC1, over the RS232 channel, again following a
user-specified medium access scheduling strategy. The control signals were finally
applied to the appropriate actuator of the plant via the DAQ card.
7.1.2 Software Configuration
RTLinux
The operating systems used in both PC1 and PC2 were Real-Time Linux (RTLinux).
It works by treating the ?usual? Linux kernel as a task executing under a small real-
time OS kernel. In RTLinux, all interrupts are initially handled by the real-time
kernel and passed to the Linux task only when there are no real-time tasks to run.
Thus, Linux is the idle task for the real-time kernel, executing only when there are
no real-time tasks to run. The Linux task can never block interrupts or prevent
itself from being preempted. The technical key to all this is a software emulation
of interrupt control hardware. When Linux tells the hardware to disable interrupts,
the real-time kernel intercepts the request, records it, and returns to Linux. Linux
is never allowed to really disable hardware interrupts. No matter what state Linux
is in, it cannot add latency to the real-time system interrupt response time [1]. A
diagram that illustrates the block level design of RTLinux is shown in Fig. 7.2. The
kernel version of the RTLinux OS that was used in the experiment platform we
developed was Linux2.2.14-rtl2.3.
SimuLinux-RT
The controller and medium access scheduler design interface used in the experiment
platform was the SimuLinux-RT environment, developed by Quanser Consulting
Inc. and Mathworks Inc.. SimuLinux-RT is a hard real-time control system rapid-
114
Figure 7.2: Flow of data and controls in RTLinux [1].
prototyping environment that runs ?on top of? RTLinux, Matlab-Simulink, and
Realtime-Workshop.
As illustrated in Fig. 7.3, when designing a control system in SimuLinux-RT, the
system is first modeled in Simulink. After the modeling is completed, the Realtime-
Workshop is used to first generate the necessary C codes based on the Simulink
model, and then compile that codes into executable binary programs that runs in
RTLinux [2]. The SimuLinux-RT environment allows the user to examine real-time
data in virtual scopes and to record the time histories of model variables. Moreover,
some design parameters, such as feedback gains and constants, can be adjusted
on-line in the Simulink model without interrupting the real-time process.
115
12345267
Generate C codes
Compile to 
real-time executable
891051112239
131415716171418
System Modeling
1914611159105111223920219156952022232642425 891051112239202191569520
8262326424
PC
Run
real-time executable
Interrupts Data Acquisition
Figure 7.3: The SimuLinux-RT real-time control system prototyping environment
[2].
7.2 Experiment 1: Stabilization under Dynamic
Medium Access Scheduling
Using the infrastructure described in Section 7.1, we first built an NCS in which
the plant consisted of two rotary inverted pendulums. The pendulums shared a
RS232 communication medium to communicate with a remote feedback controller.
The goal of this experiment was to stabilize the two pendulums at their unstable
equilibria using the dynamic medium access policies and controller design methods
discussed in Chapter 5. The experiment setup is shown in Fig. 7.4.
As shown in Fig. 7.5, the actuator in each inverted pendulum was a servo motor
116
Figure 7.4: Experiment 1: Stabilizing two rotary inverted pendulums at their un-
stable equilibria.
which drove the pendulum rod via a set of gears and a rotary arm. The input
voltages applied to the two servo motors were denoted u1 and u2. Each inverted
pendulum was equipped with two sensors which measured the angles of the rotary
arm ? and the pendulum rod ?, with ? = 0 corresponding to the upright position.
Fig. 7.6 depicts the NCS configuration of this experiment: the two inverted pen-
dulums shared a RS232 communication medium to communicate with the controller
running in PC2. The medium access constraints were such that, at any one time,
only one of the two actuators was able to receive control updates from the controller;
furthermore, only one of the two pendulums could transmit their states, x1 or x2
to the controller.
As will be shown shortly, the dynamics of each rotary inverted pendulum can be
117
(a) Assembly schematic. (b) Photo.
Figure 7.5: Configuration of a rotary inverted pendulum: a pendulum rod connects
with a rotary arm via a hinge, and the rotary arm is driven by a servo motor. Two
encoders output the angles of the rotary arm and the pendulum rod.
described by a 4-dimensional state-space model, with states (?,?, ??, ??). However,
the encoders of each inverted pendulum only provided the angles ? and ?. In order
to retrieve ?? and ?? from ? and ?, we implemented a series of software differentiators
in the (?state reconstruction? block of Fig. 7.6) to reconstruct the full states x1 and
x2 of the two pendulums.
7.2.1 Inverted Pendulum Model
Fig. 7.7 depicts a simplified model of the rotary inverted pendulum. The pendulum
rod is considered a point mass m, the length of the pendulum is denoted by l, and
the radius of the rotary arm is denoted by r. The moment of inertia of the rotary
118
DAQ
RS232 RS232PC1
RS232 RS232
Controller
Pendulum1 Pendulum2
PC2
State 
reconstruction
Figure7.6: NCS configuration of Experiment 1: the plant consisted of two uncoupled
rotary inverted pendulums. The two inverted pendulums communicated with a
remote controller via a shared RS232 medium.
arm, gears, and motor about the rotary axis of the arm are represented by a single
rotary disc with an equivalent moment of inertia J. The rotation of the rotary
arm is driven by a geared servo motor (Fig. 7.5), which provides the torque T. It
can be shown (see Appendix B) that the linearized dynamics of the rotary inverted
pendulum in Fig. 7.7 can be described by the following LTI system:
?
??
??
??
??
??
??
??
??
?
??
??
??
??
=
?
??
??
??
??
0 0 1 0
0 0 0 1
0 ?mrgJ ?K2gKmKbRmJ 0
0 gJ+mr2lJ K2gKmKbrJRml 0
?
??
??
??
??
?
??
??
??
??
?
?
??
??
?
??
??
??
??
+
?
??
??
??
??
0
0
KmKg
JRm
?KmKgrJRml
?
??
??
??
??
u (7.1)
119
Figure 7.7: A simplified model of rotary inverted pendulum.
Parameter Symbol Rod 1 Rod 2 Units
Pendulum Length lp = 2?l 0.399 0.267 m
Pendulum Weight m 0.1529 0.1064 kg
Table 7.1: Parameters of the two pendulum rods.
The parameters of the servo plant and the two pendulum rods are given in
Table 7.2 and Table 7.1. Using these parameters, we obtained the dynamics of the
two rotary inverted pendulums:
Parameter Symbol Value Units
Motor Torque Constant Km 0.00767 Nm/amp
Motor Back EMF Constant Kb 0.00767 V/(rad/sec)
Armature Resistance Rm 2.6 ?
Arm Length r 0.210 m
Internal Gear Ratio Kgi 14:1 -
External Gear Ratio Kge 5:1 -
Gear Ratio Kg = Kgi ?Kge 70:1 -
Servo Plant Equivalent Inertia J 0.004 kgm2
Table 7.2: Electro-mechanical parameters of the servo motor the gear set, and the
rotary arm.
120
Pendulum 1 (longer rod):
?x1 = A1x1 +B1u1
=
?
??
??
??
??
0 0 1 0
0 0 0 1
0 ?65.4 ?27.6 0
0 141.1 34.8 0
?
??
??
??
??
x1 +
?
??
??
??
??
0
0
51.5
?64.8
?
??
??
??
??
u1
Pendulum 2 (shorter rod):
?x2 = A2x2 +B2u2
=
?
??
??
??
??
0 0 1 0
0 0 0 1
0 ?54.6 ?27.6 0
0 159.3 43.5 0
?
??
??
??
??
x2 +
?
??
??
??
??
0
0
51.5
?81.0
?
??
??
??
??
u2
Hence, the dynamics of the plant of the NCS was described by the LTI system:
?x =
?
?? A1
A2
?
??x+
?
?? B1
B2
?
??u,
where x =
?
?? x1
x2
?
??, and u =
?
?? u1
u2
?
??.
121
7.2.2 NCS Design
We let the controller for the pair of pendulums be made of two uncoupled constant
feedback gains
?u =
?
?? K1
K2
?
?? ?x.
Using our results from Chapter 5, the closed-loop NCS can be modeled as a switched
system that switches between four possible dynamics:
?x = As(t)x; As(t) ?{Aij : i = 1,2; j = 1,2}. (7.2)
where
A12 = A21 =
?
?? A1
A2
?
??,
A11 =
?
?? A1 +B1K1
A
?
??, A
22 =
?
?? A2
A2 +B2K2
?
??.
However, observe that one should never switch to theA12 andA21 dynamics because,
in these two cases, the feedback loops of both pendulums are open. Since the open-
loop dynamics, A1 and A2, are both unstable, it is always better for one feedback
loop to be closed. Thus our switching policy only allowed (7.2) to switch between
A11 and A22. Note that, by ruling out A12 and A21, the input communication
sequence (see definition in Chapter 5) ?(t) and the output communication sequence
?(t) will be identical at all times, i.e.,
?(t) = ?(t), ?t.
To simplify the notation, we defineA1 definesA11 andA1 definesA22. Now the closed-loop
122
NCS is equivalent to
?x = As(t)x; As(t) ?{Ai : i = 1,2}. (7.3)
Wechose ?1 = ?2 = 0.5anddenoted theconvex combination ofthetwo dynamics
by
A = ?1A1 +?2A2 =
?
?? A1 + 0.5B1K1
A2 + 0.5B2K2
?
??.
Based on the controller design method from Chapter 5, we obtained the following
feedback gains,
K1 = [8.94, 66.83, 6.27, 9.64],
K2 = [8.94, 56.51, 5.94, 6.98],
which were designed such that the eigenvalues of the convex combination A were
placed at [?115.53, ?3.88?2.78i, ?4.3, ?144.88, ?4.08?3.03i, ?4.52].
In this experiment, we studied the stabilization of the system under both the
WFD and the GD rules. For the two-pendulum system, the WFD rule was:
Weighted Fastest Decay (WFD):
??(t) = ??(t) = arg min
i=1,2
wixT(t)[ATi P +PAi]x(t), ?t (7.4)
In the last expression, x(t) = [x1(t), x2(t)]T, wi > 0 is the medium access priority
of pendulum i, the ?bar? notation denotes the scalar form of a communication
sequence (see Definition 5.1). The matrix P is obtained by solving the Lyapunov
equation
ATP +PA = ?I.
123
The resulting P was the block diagonal matrix P =
?
?? P1
P2
?
?? with
P1 =
?
??
??
??
??
2.19 4.34 0.60 0.57
4.34 30.71 3.68 3.60
0.60 3.68 0.50 0.48
0.57 3.60 0.48 0.47
?
??
??
??
??
,
P2 =
?
??
??
??
??
2.17 3.34 0.53 0.33
3.34 21.71 2.75 1.80
0.53 2.75 0.42 0.27
0.33 1.80 0.27 0.18
?
??
??
??
??
.
The GD scheduling rule for this NCS can be expressed as follows:
Guaranteed Dwell-time (GD):
Let ?0 be a number satisfying 0 < ?0 < ??, where
?? defines ?min(Q)2??
max
= 12?0.5 = 1. (7.5)
1. Denote the current time by t0, choose ?(t0) and ??(t0) according to
??(t0) = ??(t0) = arg min
i=1,2
xT(t0)[ATi P +PAi]x(t0). (7.6)
2. Let ??(t) = ??(t0) and ??(t) = ??(t0) for all t ? [t0,t1), where t1 is the next switch
time determined by
t1 = inf
t>t0
xT(t)(ATs(t0)P +PAs(t0))x(t) ???0xT(t)x(t). (7.7)
124
3. Repeat from step 1 for t1.
7.2.3 Simulink Implementation
Figure 7.8: Simulink implementation of Experiment 1.
The Simulink implementation of this experiment is shown in Fig. 7.8. At PC1,
the outputs ?, ? of each pendulum were first measured by the DAQ card, then
angular velocities, ?? and ??, are obtained by differentiating the angles in software.
The transfer functions of the differentiators were chosen as s/(0.05s+1). The WFD
and GD access scheduling rules were implemented in the scheduler block, which
generated the communication sequences ? and ? based on the feedback from the
states, x1 and x2. According to the communication sequence ?(t), one of the state
variables, x1 and x2, was selected and sent to the controller via the RS232 channel,
as was the value of ?(t) itself. At PC2, the plant?s outputs and the communication
sequence were first sent to the ?select x? block, which calculated ?x1 and ?x2 based
on the communication sequence and the states x1 and x2. The control signals ?u1
125
and ?u2 were then calculated by multiplying ?x1 and ?x2 with the feedback gains K1
and K2. Depending on the value of the communication sequence, one of the control
signals, ?u1 and ?u2 , was selected and sent to PC1 via the RS232 channel according
to the communication sequence. In order to account for the time delays in the
RS232 channel, we also transmitted the communication sequence ? back from the
controller (PC2) to the plant (PC1) via the RS232 channel. By doing so, the control
signal and the input communication sequence could be synchronized.
Figure 7.9: The WFD rule switching logic.
The switching logic of the scheduler under the WFD rule is shown in Fig. 7.9,
in which, the output of the scheduler is a binary valued signal with ?1? meaning
??(t) = ??(t) = 1, i.e., pendulum 1 should access the medium, and ?0? meaning
??(t) = ??(t) = 2, i.e., pendulum 2 should access the medium. Also in this subsystem,
V1 = xT(t)(AT1P +PA1)x(t)
V2 = xT(t)(AT2P +PA2)x(t)
The switching logic of the scheduler under the GD rule is implemented via a D
flip-flop with asynchronous clear(CLR). The detailed diagram of the GD scheduling
126
logic is shown in Fig. 7.10 in which
V1 = xT(t)(AT1P +PA1)x(t)
V2 = xT(t)(AT2P +PA2)x(t)
e = ??0xT(t)x(t)
Figure 7.10: The GD rule switching logic.
7.2.4 Experiment Results
Each run of the experiment was performed as follows: i) start the real time code in
PC1 and PC2, ii) hold the two pendulum rods at their up-right position, iii) release
the two pendulum rods to let the controller ?take over?. After a transient period
of 5 second, states of the two pendulums and the communication sequence began
to be recorded. The system sampling rate was set at 0.005 sec, and the experiment
was stopped when 5000 data points (corresponding to 25 sec) were collected.
WFD Rule
We first used the WFD medium access scheduling rule and experimented with three
different choices of medium access priorities (w1, w2). The experiment data plotted
127
in Fig. 7.11 and Fig. 7.12 illustrate the evolution of the states and the communication
sequence when the access priorities were chosen as w1 = 0.1 and w2 = 0.9; in
Fig. 7.13 and Fig. 7.14, the priorities were chosen as w1 = w2 = 0.5; in Fig. 7.15 and
Fig. 7.16, we chose w1 = 0.9 w2 = 0.1. Each curve was plotted based on the data
collected in a 5 sec time window, corresponding to 1000 data points. From these
data, it is clear that, i) the two inverted pendulums were effectively stabilized by the
controller and the accompanying WFD switching rule, and ii) the medium access
percentage of each pendulum could be adjusted by changing their medium access
priorities. Table 7.3 gives the average medium access percentages under different
choices of access priorities; each number was calculated based on the data collected
from three runs of the experiment (5000 data points in each run).
w1 = 0.1 w1 = 0.5 w1 = 0.9
w2 = 0.9 w2 = 0.5 w2 = 0.1
Medium access percentage
of Pendulum 1 28.28% 48.64% 64.89%
Medium access percentage
of Pendulum 2 71.72% 51.36% 35.11%
Table 7.3: Average medium access percentage of two pendulums under the WFD
rule. The average shown are taken over time and over three runs of the experiments
(5000 data points in each run).
GD Rule
We then used the GD medium access scheduling rule while choosing ?0 = 0.05. The
plots in Fig. 7.17 and Fig. 7.18 illustrate the state evolution and the communication
sequence generated (in a 5 sec time window, 1000 data points) under the GD rule.
The communication sequence in Fig. 7.18 is to be compared with those in the WFD
rule cases, where the ?(t) switches much more frequently.
Table 7.4 compares the average dwell-time of the GD rule and the WFD rule.
128
25 26 27 28 29 30?0.05
0
0.05
Pendulum 1 angles (rad)
?1
?1
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 1 angular velocities (rad/s)
d?1/dt
d?1/dt
(a)
25 26 27 28 29 30?0.05
0
0.05
Pendulum 2 angles (rad)
?2
?2
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 2 angular velocities (rad/s)
d?2/dt
d?2/dt
(b)
Figure 7.11: Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.1, w2 = 0.9.
129
25 26 27 28 29 300.5
1
1.5
2
2.5
Time (s)
??(t)=??(t)
Communication sequence
Figure 7.12: Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 = 0.1, w2 = 0.9.
130
25 26 27 28 29 30?0.05
0
0.05
Pendulum 1 angles (rad)
?1
?1
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 1 angular velocities (rad/s)
d?1/dt
d?1/dt
(a)
25 26 27 28 29 30?0.05
0
0.05
Pendulum 2 angles (rad)
?2
?2
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 2 angular velocities (rad/s)
d?2/dt
d?2/dt
(b)
Figure 7.13: Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.5, w2 = 0.5.
131
25 26 27 28 29 300.5
1
1.5
2
2.5
Time (s)
??(t)=??(t)
Communication sequence
Figure 7.14: Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 = 0.5, w2 = 0.5.
132
25 26 27 28 29 30?0.05
0
0.05
Pendulum 1 angles (rad)
?1
?1
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 1 angular velocities (rad/s)
d?1/dt
d?1/dt
(a)
25 26 27 28 29 30?0.05
0
0.05
Pendulum 2 angles (rad)
?2
?2
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 2 angular velocities (rad/s)
d?2/dt
d?2/dt
(b)
Figure 7.15: Experiment data: Evolution of the states under the WFD rule. The
access priorities were chosen as w1 = 0.9, w2 = 0.1.
133
25 26 27 28 29 300.5
1
1.5
2
2.5
Time (s)
??(t)=??(t)
Communication sequence
Figure 7.16: Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the WFD rule. The access priorities were chosen as w1 = 0.9, w2 = 0.1.
134
25 26 27 28 29 30?0.1
?0.05
0
0.05
0.1
Pendulum 1 angles (rad)
?1
?1
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 1 angular velocities (rad/s)
d?1/dt
d?1/dt
(a)
25 26 27 28 29 30?0.1
?0.05
0
0.05
0.1
Pendulum 2 angles (rad)
?2
?2
25 26 27 28 29 30
?0.2
0
0.2
Time (s)
Pendulum 2 angular velocities (rad/s)
d?2/dt
d?2/dt
(b)
Figure 7.17: Experiment data: Evolution of the states under the GD rule with
?0 = 0.05.
135
25 26 27 28 29 300.5
1
1.5
2
2.5
Time (s)
??(t)=??(t)
Communication sequence
Figure 7.18: Experiment data: Evolution of the communication sequence (??(t) =
??(t)) under the GD rule with ?0 = 0.05.
136
Each number in this table was calculated based on the data collected from three
runs of the experiment (5000 data points in each run). It can be seen that, compared
with the WFD rule, the GD rule effectively increased the average dwell-time of the
NCS.
Average dwell-time (s)
WFD (w1 = w2 = 0.5) 0.047
GD (?0 = 0.05) 0.115
Table 7.4: Average dwell-times under the WFD rule and GD rule. The average
shown are taken over time and over three runs of the experiments (5000 data points
in each run).
7.3 Experiment 2: LQG Control under
Static Access Scheduling
In a second experiment, we set out to stabilize an MIMO plant via a shared RS232
channel by using static medium access scheduling and the LQG controller presented
in Chapter 6. The NCS configuration of this experiment is shown in Fig. 7.19, where
the plant was a discrete time 8-th order LTI system with 4 inputs, 4 outputs, and
subject to additive Gaussian noise. The medium access constraints in this NCS were
designed such that, at any one time, only one of the inputs and one of the outputs
were allowed to access the shared communication medium. This time, the dynamics
oftheplant were simulated inPC1, andtheLQGcontroller was implemented inPC2.
The controller consisted of a periodic Kalman filter and a periodic LQ regulator.
In order to synchronize the dynamics of the plant and the controller, a global clock
was generated at PC1 and also sent to PC2 via the RS232 channel.
137
RS232 RS232
PC1
RS232 RS232
Kalman FilterPC2
Simulated Plant
LQ Gains
Global 
Clock
Figure 7.19: NCS configuration of Experiment 2: the plant is a simulated LTI system
with 4 inputs and 4 outputs; the plant communicates with an LQG controller via
the shared RS232 communication channel. A global clock is used to synchronize the
dynamics of the plant and the controller.
The LTI plant simulated in PC1 was
x(k + 1) = Ax(k) +Bu(k) +v(k),
y(k) = Cx(k) +w(k),
138
with
A =
?
??
??
??
??
??
??
??
??
??
??
?
3 ?1 0 0 0 0 ?1 1.5
0 3 ?2 0 0 ?2 1.5 0
0 0 1 ?1 ?1 0.5 0 0
?1 0 0 1 0.5 0 0 ?1
2 0 0 ?1 ?0.5 0 0 2
0 0 ?1 2 2 ?0.5 0 0
0 ?3 4 0 0 4 ?1.5 0
?3 2 0 0 0 0 2 ?1.5
?
??
??
??
??
??
??
??
??
??
??
?
,
B =
?
??
??
??
??
??
??
??
??
??
??
?
1 0 0 ?1
0 1 ?1 0
0 ?1 1 0
?1 0 0 1
2 0 0 ?1
0 2 ?1 0
0 ?1 2 0
?1 0 0 2
?
??
??
??
??
??
??
??
??
??
??
?
, C =
?
??
??
??
??
1 0 0 ?1 2 0 0 ?1
0 1 ?1 0 0 2 ?1 0
0 ?1 1 0 0 ?1 2 0
?1 0 0 1 ?1 0 0 2
?
??
??
??
??
.
The random disturbances, v(?) and w(?), are designed to be Gaussian i.i.d., with
v(?) ? N(0,I8?8) and w(?) ? N(0,I4?4). The updates of the plant and controller
were synchronized by the global clock signal whose frequency was set at 10 Hz.
7.3.1 NCS Design
We designed an LQG controller to minimize the quadratic cost function:
J = E[xT(N)Qx(N) +
N?1summationdisplay
k=0
xT(k)Qx(k) + ?uT(k)?u(k)] (7.8)
139
with Q = 4I8?8.
The plant dynamics (A,B,C) were chosen such that the obvious ?round-robin?
communication sequences ??(?) = 1,2,3,4,1,2,3,4??? and ??(?) = 1,2,3,4,1,2,3,4???
would not preserve the stabilizability and detectability of the plant. Based on the
algorithms introduced in Section 4.2.1, the input and output communication se-
quences were chosen as the 5-periodic sequences
??(?) = ??(?) = 2,4,3,1,1,2,4,3,1,1,??? (7.9)
At each time k, the controller performs the following LQG control algorithm:
1. Kalman filter time update:
?x(k?) = A?x(k?1)+ ?B(k?1)?u(k?1) (7.10)
2. Kalman filter measurement update:
?x(k) = ?x(k?) +H(k)(?y(k)? ?C(k)?x(k?)) (7.11)
3. Feedback computation:
?u(k) = ?L(k)x(k) (7.12)
The periodic time-varying gains H(k)?s and L(k)?s were computed off-line by
solving their associated Riccati equations (see Section 6.4). Because the extended
plant was both stabilizable and detectable under the periodic communication se-
quences (7.9), H(k) and L(k) each converged to a steady-state solution with a
140
period of 5 steps. The periodic gains we calculated were, for i ? Z:
H(5i) = [0.2578,0.2604,0.0091,?0.0940,0.1872,?0.0187,?0.1794,?0.2735]T ,
H(5i + 1) = [?0.0879,0.2756,?0.1118,?0.1894,0.3805,0.2214,?0.1695,0.2031]T ,
H(5i + 2) = [?0.2439,0.0842,0.0380,0.0512,?0.1018,?0.0750,?0.0671,0.3014]T ,
H(5i + 3) = [?0.0198,?0.2497,?0.0027,?0.0168,0.0343,0.0078,0.3803,0.0770]T ,
H(5i + 4) = [0.2418,0.0791,?0.0754,?0.1145,0.2266,0.1509,?0.0497,?0.1898]T .
L(5i) = [3.6090,1.4186,?0.1438,?0.0271,?0.0184,?0.1428,0.6527,1.6510],
L(5i + 1) = [0.5889,3.1470,?0.0080,?0.0689,?0.0682,?0.0070,1.5464,0.2793],
L(5i + 2) = [?0.0675,0.7660,0.0221,0.0109,0.0099,0.0232,0.8818,?0.0183],
L(5i + 3) = [?3.8455,?4.7714,1.8934,?0.1303,?0.1064,1.8501,?1.9300,?1.7821],
L(5i + 4) = [2.6672,2.8737,?0.0655,0.0110,0.0124,?0.0659,1.1564,1.4160].
7.3.2 Simulink Implementation
The Simulink implementation of our second experiment is illustrated in Fig. 7.20. In
order to synchronize the controller and the plant, a global clock signal ,?CLK?, was
generated at the plant side and was sent to the controller over the communication
medium. At the plant side, ?CLK? triggered the updates of the output communica-
tion sequence ?(?) and the LTI plant. At the controller side, ?CLK? triggered the
updates of the Kalman Filter, LQ regulator, and the input communication sequence
?(?). Based on the output communication sequence ? generated at PC1, at each
141
sigma
y
~y
select
System Clock
~u
rho
u
Select
In1Out1
RS232 in 1
In1 Out1
RS232 in 
In1Out1
RS232 Out1
In1 Out1
RS232 Outu y
Plant
^x(k)
rho
~u
LQ gain
sigma
~y
~u
rho
^x(k)
Kalman Filter
Demux
Demux
CLKrho
Comm. Sequence1
CLK Sigma
Comm. Sequence
~y~u
rho_in
CLK
CLK
Sigma
Sigma
RS232 115,200bps
~y
sigma
^x(k)
CLK
RS232 115,200bps
~u
~u
Figure 7.20: Simulink implementation of Experiment 2.
time step, one of the 4 outputs of the plant was selected (by the ?select? block) and
sent to the controller. The output communication sequence was also sent to the
controller from the PC1 via the RS232 channel. From the output ?y and the output
communication sequence, the LQG controller calculated the optimal control ?u and
transmitted it to the plant. The input communication sequence ? was generated at
the controller side and was used to calculate the LQ gains. As Experiment 1, we ac-
counted for the transmission delays in the RS232 channel by sending ? back to PC1
in order to synchronize the control signal and the input communication sequence.
The sampling periods of the real-time code running in PC1 and PC2 were both set
to 0.005 second; the period of the global update clock, ?CLK?, was set to 0.1 second.
The reason for doing so was to account for the delays caused by the RS232 channel
and the computation times needed by the controller and the simulated plant (we
measured that, the total delays was up to 8 times of the sampling period of the
real-time code).
142
7.3.3 Experiment Results
We set the initial condition of the plant at x(0) = [15,50,7,6,100,20,4,3]T, and
the initial condition of the Kalman filter at ?x(0) = [1,1,3,4,1,1,0,0]T. Fig. 7.21(a)
shows the state evolution of the NCS under the selected communication sequences
and the LQG controller. It can be seen that, after an initial transient process
of about 15 steps (1.5 sec), the state of the plant was effectively attenuated, and
remained stable afterwards. The evolution of the Kalman filter estimation error is
shown in Fig. 7.21(b).
143
0 5 10 15 20 25 30 35 40 45 50?10000
?5000
0
5000
k
X(k)
(a) States evolution.
0 5 10 15 20 25 30 35 40 45 50?400
?300
?200
?100
0
100
200
300
400
500
k
e(k)
(b) Evolution of the Kalman filter?s one-step prediction error e(k) =
x(k)? ?x(k?).
Figure 7.21: Experiment data: Stabilization of an NCS using LQG controller. The
plant was an LTI system with 2 inputs and 2 outputs. The communication medium
provided 1 input channel and 1 output channel.
144
Chapter 8
Conclusions and Future Work
This thesis studied networked control systems (NCSs) in which the sensors and
actuators of an MIMO plant communicate with a feedback controller via a network
or other shared communication medium. The medium can only provide a limited
number of simultaneous access channels, so that the plant?s sensors and actuators
must be scheduled to gain access to the shared medium. The fundamental point
made in this thesis is that, in an NCS, the sensors and actuators that are not
actively accessing the communication medium should be ignored by the controller
and the plant. Doing so has the effect of greatly simplifying matters by decoupling
the selection of communication policies from that of controllers.
In this spirit, an MIMO plant with medium access constraints at its inputs and
outputs can be effectively modeled by an ?extended plant? which is constructed
by cascading the plant dynamics with a pair of time-varying matrix operators that
encode the medium access policy governing controller-plant communication. The
extended plant has fewer effective input and output ports than the original plant,
and describes the ?image? of the original plant viewed over the communication
medium. Moreover, the state of the extended plant coincides with that of the plant,
145
hence the problem of stabilizing an MIMO plant under medium access constraints
can be solved by designing a feedback controller that stabilizes the extended plant.
Our results show that:
? If the plant is a continuous-time LTI system, the controllability and observ-
ability of the plant can be preserved if the communication sequences grant
each sensor and actuator some finite amount of medium access periodically;
? If the plant is a reversible (A invertible) discrete-time LTI system, there always
exist periodic communication sequences that preserve the reachability and
observability of the plant;
? If the plant is an irreversible (A not invertible) discrete-time LTI system, there
always exist periodic communication sequences that preserve the controllabil-
ity (or stabilizability) and reconstructibility (or detectability) of the plant.
Such communication sequences can be obtained via the algorithm presented in Sec-
tion 4.2.1, and enable the solution of state estimation and feedback control of NCSs
when combined with the controller design tools introduced in Chapters 4 and 6.
Furthermore, for continuous-time NCSs in which all states can be measured
(although not simultaneously), we have proposed a class of ?dynamic? scheduling
policies that decide on-line (based on state feedback) which sensors and actuators
should be granted access. These policies are accompanied by a design method
for computing stabilizing feedback control gains, based on solving a standard pole
placement problem. The proposed feedback-based scheduling policies also allow one
to assign different medium access priorities to different inputs and outputs and to
ensure a minimum dwell-time between each change in medium access.
146
8.1 Opportunities of Future Work
In designing periodic communication sequences, in this thesis, we have been guided
by the requirement that they must preserve the controllability (or reachability, sta-
bilizability) and observability (or reconstructibility, detectability) of the plant. Al-
though we have proved the existence of such sequences, their design requires further
investigation. In fact, it is easy to show that the number of communication se-
quences satisfying our requirements guideline is infinite; it is thus important to
study optimal communication sequence design, which involves both the choice of
communication period (i.e., the length of ?T?) and that of the communication pat-
tern (e.g., choice between ?122,122,...? and ?112,112,...?). The communication
sequence design problem becomes particularly interesting when the plant is subject
to random disturbances. In that case, choices of communication sequence will affect
the optimal control cost and the variance of the state estimation. The methods
introduced in [54] and [55] show that an optimal communication sequence that min-
imize an LQ cost in a finite horizon can be found via exhaustive search or dynamic
programming. However, the problem of designing optimal periodic communication
sequences over an infinite time horizon is still open.
Furthermore, if an NCS?s feedback controller has been designed in advance, one
can treat the choices of communication sequences as inputs to the closed-loop sys-
tem; these inputs take values on a finite set. Thus, an optimal communication
sequence may be obtained by solving an optimal control problem (finite or infi-
nite horizon) via the Maximum Principle (continuous-time) or Dynamic Program-
ming (discrete-time). In both cases, the resulting optimal communication sequences
should yield a feedback-based switching law. This sequence design method may also
reveal some fundamental rules that have yet been discovered for dynamic medium
147
access scheduling.
Throughout this thesis, we have assumed that the number of input and output
channels (i.e., ?w?? and ?w??) are fixed. However, in a real-world NCS, the commu-
nication medium may be shared by the inputs and outputs jointly. Thus, one may
need to determine (either off-line or on-line) how many channels should be assigned
to the inputs versus the outputs. Furthermore, many communication protocols, such
as CAN and TCP/IP, are essentially serial, and provide only one physical channels
to all users. It is thus important to study the case where a single communication
channel is shared by both inputs and outputs of the plant. Using the theory de-
veloped in Chapter 4, one can easily find a communication sequence that preserves
the controllability (reachability, stabilizability) and observability (reconstructibility,
detectability) of the plant. However, a scenario worth studying involves a plant
which is subject to random disturbances and modeling uncertainties. In that case,
more medium access granted to the inputs yields a lower control cost, while more
medium access granted to the outputs yields smaller state estimation covariance.
The dynamic access scheduling and controller design schemes introduced in
Chapter 5 can not be used to stabilize an NCS if the inputs and outputs share
a single communication channel. This is because the controller uses a constant
feedback gain; when the shared channel is used to transmit an input, no output
information can be used to calculate the control, therefore the input to the plant
is constantly zero. In that case, it might be desirable to let the controller have a
?state? (i.e., be a dynamical system) that retains certain information from the plant.
In the dynamic scheduling policies introduced in Chapter 5, one has the freedom
to choose the parameters ?ij?s, and wij?s. The choice of ?ij affects the value of
the (i,j) entry of the feedback gain matrix K, while the wij affects the medium
access priorities of the i-th inputs and the j-th outputs. It would be interesting to
148
investigate how these parameters could be chosen to meet various design criteria,
e.g., system trajectory bound, avoidance of actuator saturation, or minimizing the
feedback gain.
One limitation of our proposed dynamic scheduling policies is that they all re-
quire real-time state information of the plant which is not always available in real-
world NCSs. A straightforward extension is to address the output feedback case,
where the plant output is y(t) = Cx(t), with C a p-by-n matrix (p < n). In
this case, an observer may be used to reconstruct the state at either the plant or
the controller side of the communication medium. For that purpose, it will be
necessary to develop theoretical results on the stability of NCSs having this config-
uration. Moreover, in this thesis, we have only discussed dynamic access scheduling
in continuous-time. Most NCSs operate in discrete-time (under the sampled-data
configuration); it would thus be useful to find the discrete-time counterpart of the
NCS design method developed in Chapter 5. The results presented in [77] and the
references therein may be a useful starting point in studying quadratic stability of
a discrete-time switched system.
In Chapters 4 and 5, we have addressed the NCS design problem in the classical
system theoretical framework where stability is the only design objective. The
modern view of control sees feedback as a tool for uncertainty management [78], it
is necessary to re-examine the NCS design problem from the modern robust control
view point. The LQG design method developed in Chapter 6 gives a good starting
point for H2 synthesis of NCSs; it is also desirable to investigate NCS design (both
the communication sequence and the controller ) in the H? framework.
This thesis did not address transmission delays in the communication medium.
If the transmission delays are always known to the controller, it may be possible to
design a model-based predictor (e.g., the one introduced in [25]) that compensates
149
for the delay. The case where the delays are random and not known by the controller
is more complex and requires further study.
Finally, it is worth studying the controllability and observability of a non-linear
NCS using the extended plant formulation introduced in Section 3.6. In that case,
the analytical method introduced in the proof of Theorem 4.1 may be a good starting
point.
150
Appendix A
Calculating the Time-varying
Feedback and Observer Gains
In Chapter 4, we show that designing the feedback controller that exponentially
stabilize the continuous time extended plant (4.2) involves calculating periodic time
varying gains having the forms of
K(t) = ??BT(t)W?1a (t,t+T), (A.1)
H(t) = [e?ATTMa(t?T,t)e?AT]?1 ?CT(t), (A.2)
where the two Gramians Wa and Ma are defined as follows [64]
Wa(t0,tf) defines
integraldisplay tf
t0
2e4a(t0??)eA(t0??) ?B(?) ?BT(?)eAT(t0??)d?, (A.3)
Ma(t0,tf) defines
integraldisplay tf
t0
2e4a(??tf)eAT(??t0) ?CT(?) ?C(?)eA(??t0)d?. (A.4)
In this appendix, we derive the ordinary differential equations (ODEs) that are
are needed for calculating these gains. With these ODEs, the gains can be calculated
151
numerically off-line using standard ODE solvers (e.g., Matlab ?ode45?), and then
stored in a time-indexed look-up table (since both gains are T-peridic) when the
system is running.
A.1 Solving for the Gramian Wa(t,t + T)
A.1.1 Preliminaries
To solve for the Gramian Wa(t,t + T), it is helpful to first study the derivative
of the controllability Gramian of the extended plant (4.2) during the time period
[t,t + T]. The controllability Gramian of the extended plant (4.2) during [t0,tf],
denoted W(t0,tf), is defined as follows:
W(t0,tf) =
integraldisplay tf
t0
?(t0,?) ?B(?) ?BT(?)?T(t0,?)d?, (A.5)
where ?(t0,?) is the transition matrix of the extended plant (4.2), defined as
?(tf,t0) = eA(tf?t0). (A.6)
Note that
W(t,t +T) =
integraldisplay t+T
t
?(t,?) ?B(?) ?BT(?)?T(t,?)d?
= ?(t,t0)?
integraldisplay t+T
t
?(t0,?) ?B(?) ?BT(?)?T(t0,?)d? ??T(t,t0), (A.7)
where t0 could be any real number. It is well known that the transition matrix
?(t,t0) satisfies
d
dt?(t,t0) = A??(t,t0). (A.8)
152
Hence we have
d
dtW(t,t +T) = AW(t,t +T) +W(t,t +T)A
T
+ ?(t,t0)[?(t0,t+T) ?B(t +T) ?BT(t +T)?T(t0,t +T) (A.9)
??(t0,t) ?B(t) ?BT(t)?T(t0,t +T)]?T(t,t0).
Remark A.1. To obtain the above equation, we need to use the following fact: ?For
an integrable function f(?), ddt integraltextt+Tt f(?)d? = f(t+T)?f(t)?.
Based on the spirit of Lemma 4.1, we will always let communication sequence
?(?) be T-periodic. Therefore ?B(t) is also T-periodical, then the above reduce to
d
dtW(t,t +T) = AW(t,t +T) +W(t,t +T)A
T
+ ?(t,t +T) ?B(t) ?BT(t)?T(t,t+T)? ?B(t) ?BT(t). (A.10)
Using the fact that for the extended plant, ?(t,t + T) = e?AT, we can further
reduce the above ODE to
d
dtW(t,t +T) = AW(t,t+T) +W(t,t +T)A
T
+e?AT ?B(t) ?B(t)Te?ATT ? ?B(t) ?B(t)T. (A.11)
Equation (A.11) is the ODE satisfied by the controllability Gramian W(t,t+T).
In order to solve it numerically, we also need to know the initial conditions of the
matrix valued function W(t,t +T) at t = 0, which is W(0,T).
W(0,T) can be calculated in two ways. First, it is easy to prove that (Ex.9.4 in
153
[64]) W(t,T) satisfies the following ODE
d
dtW(t,T) = A(t)W(t,T)?
?B(t) ?BT(t) +W(t,T)AT(t), (A.12)
withW(T,T) = 0. We can then integrate the above ODE backwards to findW(0,T)
(the Matlab function ?ode45? can do this job).
An alternative way is to observe that
d
dtW(0,t) = ?(0,t)
?B(t) ?BT(t)?T(0,t) (A.13)
= e?At ?B(t) ?BT(t)e?At. (A.14)
Using W(0,0) = 0, we can then can obtain W(0,T) by integrating the above ODE
for t ? [0,T].
A.1.2 The Gramian ODEs
Now, let?s return to the problem of calculating the Gramian Wa(t,t + T). By
definition,
Wa(t0,tf) =
integraldisplay tf
t0
2e4a(t0??)eA(t0??) ?B(?) ?BT(?)eAT(t0??)d?
= 2
integraldisplay tf
t0
e(A+2aI)(t0??) ?B(?) ?BT(?)e(A+2aI)T(t0??)d?.
Define Aa = A + 2aI, we see that Wa(t0,tf) is nothing but two times the con-
trollability Gramian (A.11) defined based on the new dynamics Aa instead of A.
To clarify notation, we will now add a subscript to the controllability Gramian
W(t0,tf), denoting which dynamics it is referred to, e.g., W(t0,tf)Aa meaning that
the controllability Gramian (A.11) is calculated based on Aa. Hence the above
154
analysis gives
Wa(t0,tf) = 2W(t0,tf)Aa. (A.15)
Differentiating the above equation, and applying the ODEs given in (A.11), we
obtain the ODE satisfied by the Gramian Wa(t,t+T).
d
dtWa(t,t+T) = 2
d
dtW(t,t +T)Aa
= 2(AaW(t,t +T)Aa +W(t,t +T)AaATa
? ?B(t) ?BT(t) +e?AaT ?B(t) ?BT(t)e?ATaT).
The above then reduces to
d
dtWa(t,t +T) = AaWa(t,t +T)+Wa(t,t +T)A
T
a
?2 ?B(t) ?BT(t) + 2e?AaT ?B(t) ?BT(t)e?ATa T, (A.16)
which is the ODE we need for calculating the Gramian Wa(t,t+T).
To solve for the initial condition of Wa(t,t + T) at t = 0, i.e., Wa(0,T), which
equals to 2W(0,T)Aa , we can directly use the ODEs we developed for W(t,T) of
W(0,t) and obtain the following:
d
dtWa(t,T) = AaWa(t,T)?2
?B(t) ?BT(t) +Wa(t,T)ATa, (A.17)
Wa(T,T) = 0. (A.18)
155
or alternatively,
d
dtWa(0,t) = 2e
?Aat ?B(t) ?BT(t)e?ATa t, (A.19)
Wa(0,0) = 0. (A.20)
A.2 Solvingfor the Gramian [e?ATTMa(t?T,t)e?AT]
A.2.1 Preliminaries
Again, to solve for the Gramian [e?ATTMa(t?T,t)e?AT], it is helpful to first study
the observability Gramian of the extended plant (4.2). The observability Gramian
M(t0,tf) of the extended plant is defined as
M(t0,tf) =
integraldisplay tf
t0
?T(?,t0) ?CT(?) ?C(?)?(?,t0)d?, (A.21)
and for the extended plant (4.2), ?(tf,t0) = eA(tf?t0). Then
Ma(t0,tf) =
integraldisplay tf
t0
2e4a(??tf)?T(?,t0) ?CT(?) ?C(?)?(?,t0)d?. (A.22)
To simplify notation, we define
N(t0,tf) defines ?T(t0,tf)M(t0,tf)?(t0,tf) (A.23)
=
integraldisplay tf
t0
?T(?,tf) ?CT(?) ?C(?)?(?,tf)d?, (A.24)
and, imitating the definitions of Wa, we also define
156
Na(t0,tf) defines ?T(t0,tf)Ma(t0,tf)?(t0,tf) (A.25)
=
integraldisplay tf
t0
2e4a(??tf)?T(?,tf) ?CT(?) ?C(?)?(?,tf)d?. (A.26)
Based on the above definition, the Gramian [e?ATTMa(t ? T,t)e?AT] can be ex-
pressed as follows
e?ATTMa(t?T,t)e?AT = Na(t?T,t). (A.27)
Using the fact that ddt integraltexttt?T f(?)d? = f(t)?f(t?T) , and similar techniques used
in Section A.1.1, we obtain that
d
dtN(t?T,t) = ?A
TN(t?T,t)?N(t?T,t)A
+ ?CT(t) ?C(t)?e?ATT ?CT(t) ?C(t)e?AT. (A.28)
A.2.2 The Gramian ODEs
Again, define Aa = A+2aI. It is easy to prove that the Gramian matrix Na(t0,tf) is
nothing but 2times thematrixN(t0,tf), calculatedbased onAa (denoted N(t0,tf)Aa),
i.e.,
Na(t0,tf) = 2N(t0,tf)Aa. (A.29)
Then using the result in (A.28) we obtain
d
dtNa(t?T,t) = ?Aa
TNa(t?T,t)?Na(t?T,t)Aa
+ 2 ?CT(t) ?C(t)?2e?ATa T ?CT(t) ?C(t)e?AaT. (A.30)
157
which is the ODE we need to calculate Na(t?T,t).
In order to obtain the initial condition of Na(t?T,t) at t = 0, i.e., Na(?T,0),
we can integrate the ODE satisfied by Na(?T,t). Using the definitions of Na(t0,tf)
the following ODE can be easily obtained
d
dtNa(?T,t) = ?A
T
aNa(?T,t)?Na(?T,t)Aa + 2 ?C
T(t) ?C(t),
Na(?T,?T) = 0. (A.31)
Alternatively, Na(?T,0) can also be obtained by integrating backwards the ODE
satisfied Na(t,0):
d
dtNa(t,0) = ?2e
ATa T ?CT(t) ?C(t)eAaT,
Na(0,0) = 0. (A.32)
158
Appendix B
The Dynamics of a Rotary
Inverted Pendulum
Consider the simplified model of the rotary inverted pendulum shown in Fig. 7.7
(b). The kinetic energy of the pendulum and the rotary arm are given by:
KEpendulum = 12m[(r??cos? +l ??)2 + (l??sin?)2 + (r??sin?)2], (B.1)
KEarm = 12J ??2. (B.2)
The potential energy of the system is given by
PE = PEpendulum = mglcos?. (B.3)
Since we are considering the case when ? is very small, we omit the the terms
159
associated with sin?2, then the Lagrangian of the system can be expressed as:
L = KEpendulum +KEarm ?PE, (B.4)
= 12(mr2 +J)??2 + 12ml2 ??2 +mrlcos??? ???mglcos?. (B.5)
Using the Lagrangian formulation we obtain the following differential equations
ml2?? +mrlcos????mglsin? = 0, (B.6)
mrlcos ?? + (mr2 +J)???mrlsin?sin???2 = T. (B.7)
Linearize theabove differential equationabouttheequilibrium point [?,?, ??, ??]T =
[0,0,0,0]T, we obtain the linear model of the rotary inverted pendulum.
?
??
??
??
??
??
??
??
??
?
??
??
??
??
=
?
??
??
??
??
0 0 1 0
0 0 0 1
0 ?mrgJ 0 0
0 gJ+mr2lJ 0 0
?
??
??
??
??
?
??
??
??
??
?
?
??
??
?
??
??
??
??
+
?
??
??
??
??
0
0
1
J
? rlJ
?
??
??
??
??
T. (B.8)
In the servo plant, the arm is drive by an electrical DC motor via a gear box.
The equations that govern a DC-motor are
V = IRm +Kb?m, (B.9)
Tm = KmI, (B.10)
in which Rm is the motor?s armature resistance, Kb is the motor?s back EMF con-
stant, Km is the torque constant, Tm is the torque produced by the motor at its
shaft, ?m is the angular velocity of the motor shaft, u is voltage applied to the
160
motor, I is the motor?s current. Denote the gear ratio by Kg, we also have
?m = Kg ??, (B.11)
T = KgTm. (B.12)
Hence the relation between T and u can be expressed as
T = KmKgR
m
u? KmKbK
2
g
Rm
??. (B.13)
We thus obtain the dynamics of a rotary inverted pendulum
?
??
??
??
??
??
??
??
??
?
??
??
??
??
=
?
??
??
??
??
0 0 1 0
0 0 0 1
0 ?mrgJ ?K2gKmKbRmJ 0
0 gJ+mr2lJ K2gKmKbrJRml 0
?
??
??
??
??
?
??
??
??
??
?
?
??
??
?
??
??
??
??
+
?
??
??
??
??
0
0
KmKg
JRm
?KmKgrJRml
?
??
??
??
??
u. (B.14)
161
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