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 |  موضوع: كتاب Optimal Networked Control Systems with MATLAB الأربعاء 27 أكتوبر 2021, 11:20 pm | |
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Optimal Networked Control Systems with MATLAB
Jagannathan Sarangapani
Missouri University of Science and Technology
Rolla, Missouri, USA
Hao Xu
Texas A&M University - Corpus Christi
Corpus Christi, Texas, USA
و المحتوى كما يلي :
Contents
Preface xv
Authors xix
Chapter 1 Introduction to Networked Control Systems 1
1.1 Overview of Networked Control Techniques 2
1.2 Challenges in Networked Control Systems .3
1.2.1 Network Imperfections .4
1.2.1.1 Network-Induced Delay 4
1.2.1.2 Packet Dropouts 4
1.2.2 Quantization .5
1.2.3 Network Protocol Effects .6
1.3 Current Research . 13
1.3.1 Energy Efficiency . 13
1.3.2 Spectrum Management 13
1.3.3 Game Theory . 13
1.3.4 Optimal Control . 14
1.3.5 Event-Sampled Control 14
References 15
Chapter 2 Background on Lyapunov Stability and Stochastic Optimal Control .19
2.1 Deterministic Dynamical Systems 19
2.1.1 Discrete-Time Systems . 19
2.1.2 Brunovsky Canonical Form .20
2.1.3 Linear Systems . 21
2.1.3.1 Analysis 22
2.1.3.2 Simulation .22
2.2 Mathematical Background 23
2.2.1 Vector and Matrix Norms 23
2.2.1.1 Singular Value Decomposition .24
2.2.1.2 Quadratic Forms and Definiteness .25
2.2.2 Continuity and Function Norms .25
2.3 Properties of Dynamical Systems .26
2.3.1 Asymptotic Stability .27
2.3.2 Lyapunov Stability .27
2.3.3 Boundedness .28
2.3.4 A Note on Autonomous Systems and Linear
Systems .28
2.4 Nonlinear Stability Analysis and Controls Design .29
2.4.1 Lyapunov Analysis for Autonomous Systems 29
2.4.2 Controller Design Using Lyapunov Techniques . 33
x Contents
2.4.2.1 Lyapunov Analysis and Controls
Design for Linear Systems 35
2.4.2.2 Stability Analysis 35
2.4.2.3 Lyapunov Design of LTI Feedback
Controllers 36
2.4.3 Lyapunov Analysis for Nonautonomous Systems 37
2.4.4 Extensions of Lyapunov Techniques and
Bounded Stability .39
2.4.4.1 UUB Analysis and Controls Design .39
2.5 Stochastic Discrete-Time Control 42
2.5.1 Stochastic Lyapunov Stability 42
2.5.1.1 Asymptotic Stable in the Mean Square 43
2.5.1.2 Lyapunov Stable in the Mean Square . 43
2.5.1.3 Bounded in the Mean Square . 43
2.5.1.4 Bounded in the Mean 43
2.5.2 Stochastic Linear Discrete-Time Optimal Control 43
2.5.3 Stochastic Q-Learning .48
2.5.3.1 Q-Function Setup 48
2.5.3.2 Model-Free Online Tuning Based on
Adaptive Estimator and Q-Learning 50
2.5.4 Stochastic Nonlinear Discrete-Time Optimal
Control 51
2.5.5 Background on Neural Networks .54
2.5.6 Two-Layer Neural Networks 55
2.5.7 NN Function Approximation . 59
2.5.7.1 Functional Link Neural Networks 60
Problems . 61
References 62
Chapter 3 Optimal Adaptive Control of Uncertain Linear Network Control
Systems .65
3.1 Traditional Control Design and Stochastic Riccati
Equation-Based Solution . 67
3.2 Finite-Horizon Optimal Adaptive Control 69
3.2.1 Background 69
3.2.2 Stochastic Value Function 72
3.2.3 Model-Free Online Tuning of Adaptive Estimator 73
3.2.4 Closed-Loop System Stability 78
3.2.5 Simulation Results 82
3.2.5.1 LNCS State Regulation Error and
Performance 83
3.2.5.2 Bellman Equation and Terminal
Constraint Errors 83
3.2.5.3 Optimality Analysis of the Proposed
Scheme 85Contents xi
3.3 Extensions to Infinite Horizon .88
3.3.1 Adaptive Estimation for Optimal Regulator Design 88
3.3.2 Simulation Results 92
3.4 Conclusions 96
Problems .97
Appendix 3A 98
Appendix 3B 98
Appendix 3C 99
Appendix 3D 101
References 102
Chapter 4 Optimal Control of Unknown Quantized Network Control
Systems . 105
4.1 Background 108
4.1.1 Quantized Linear Networked Control Systems . 108
4.1.2 Quantizer Representation . 110
4.1.3 Quantized Nonlinear Networked Control System .112
4.2 Finite-Horizon Optimal Control of Linear QNCS 114
4.2.1 Action-Dependent Value-Function Setup . 115
4.2.2 Model-Free Online Tuning of Action-Dependent
Value Function with Quantized Signals . 117
4.2.3 Estimation of the Optimal Feedback Control 122
4.2.4 Convergence Analysis 122
4.2.5 Simulation Results 124
4.3 Finite-Horizon Optimal Control of Nonlinear QNCS 127
4.3.1 Observer Design . 129
4.3.2 Near-Optimal Regulator Design 132
4.3.2.1 Value Function Approximation 133
4.3.2.2 Control Input Approximation . 135
4.3.2.3 Dynamic Quantizer Design 137
4.3.2.4 Stability Analysis 138
4.3.2.5 Simulation Results 140
4.4 Conclusions 141
Problems . 143
Appendix 4A 144
Appendix 4B 145
Appendix 4C 148
Appendix 4D 149
References 151
Chapter 5 Optimal Control of Uncertain Linear Networked Control
Systems in Input–Output Form with Disturbance Inputs 155
5.1 Traditional Two-Player Zero-Sum Game Design and
Game-Theoretic Riccati Equation-Based Solution 156xii Contents
5.2 Infinite-Horizon Optimal Adaptive Design . 158
5.2.1 Background 159
5.2.1.1 LNCS Quadratic Zero-Sum Games . 159
5.2.1.2 LNCS Quadratic Zero-Sum Games
in Input–Output Form . 161
5.2.2 Stochastic Value Function 164
5.2.3 Model-Free Online Tuning . 167
5.2.4 Closed-Loop System Stability 171
5.2.5 Simulation Results 173
5.3 Conclusions 177
Problems . 178
Appendix 5A 178
Appendix 5B 180
Appendix 5C 183
References 184
Chapter 6 Optimal Control of Uncertain Nonlinear Networked Control
Systems via Neurodynamic Programming 187
6.1 Traditional Nonlinear Optimal Control Design and HJB
Equation-Based Solution . 188
6.2 Finite-Horizon Optimal Control for NNCS 190
6.2.1 Background 191
6.2.2 Online NN Identifier Design 192
6.2.3 Stochastic Value Function Setup and Critic NN
Design . 195
6.2.4 Actor NN Estimation of Optimal Control Policy . 198
6.2.5 Closed-Loop Stability 200
6.2.6 Simulation Results 202
6.2.6.1 State Regulation Error and Controller
Performance 204
6.2.6.2 HJB Equation and Terminal Constraint
Estimation Errors 207
6.2.6.3 Cost Function Comparison .207
6.3 Extensions to Infinite Horizon .209
6.3.1 Optimal Stochastic Value Function
Approximation and Control Policy Design 210
6.3.2 Simulation Results 214
6.4 Conclusions 218
Problems . 219
References 219
Chapter 7 Optimal Design for Nonlinear Two-Player Zero-Sum Games
under Communication Constraints 221
7.1 Traditional Stochastic Optimal Control Design for
Two-Player Zero-Sum Game .223Contents xiii
7.2 NNCS Two-Player Zero-Sum Game .225
7.3 Finite-Horizon Optimal Adaptive Design .227
7.3.1 Online NN Identifier Design 227
7.3.2 Stochastic Value Function 230
7.3.3 Approximation of Optimal Control
and Disturbance 235
7.3.4 Closed-Loop System Stability 239
7.4 Simulation Results .242
7.4.1 State Regulation and Control and Disturbance
Input Performance 242
7.4.2 Hamilton–Jacobi–Isaacs and Terminal
Constraint Errors 244
7.4.3 Optimal Performance of the Proposed Design 247
7.5 Conclusions 247
Problems .247
Appendix 7A .248
Appendix 7B 249
References 255
Chapter 8 Distributed Joint Optimal Network Scheduling and Controller
Design for Wireless Networked Control Systems 257
8.1 Background of Wireless Networked Control Systems 258
8.2 Wireless Networked Control Systems Codesign .259
8.2.1 Overview 259
8.2.2 Plant Model 260
8.2.3 Stochastic Optimal Control Design 261
8.2.4 Optimal Cross-Layer Distributed Scheduling
Scheme .264
8.2.5 Numerical Simulations .269
8.3 Conclusions 272
Problems . 272
References 272
Chapter 9 Event-Sampled Distributed Networked Control Systems 275
9.1 Distributed Networked Control Systems .277
9.2 Optimal Adaptive Event-Sampled Control 279
9.2.1 ZOH-Based Event-Triggered Control System 279
9.2.2 Optimal Adaptive ZOH-Based Event-Triggered
Control 280
9.2.2.1 Value Function Setup 280
9.2.2.2 Model-Free Online Tuning of Value
Function 281
9.2.3 Cross-Layer Distributed Scheduling Design 284
9.2.3.1 Cross-Layer Design 284
9.2.3.2 Distributed Scheduling .284xiv Contents
9.3 Simulation 291
9.4 Conclusions 294
Problems .295
References 295
Chapter 10 Optimal Control of Uncertain Linear Control Systems
under a Unified Communication Protocol .297
10.1 Optimal Control Design under Unified Communication
Protocol Framework 298
10.1.1 Observer Design .299
10.1.2 Stochastic Value Function 302
10.1.3 Model-Free Online Tuning of Adaptive Estimator 304
10.2 Closed-Loop System Stability .307
10.3 Simulation Results .309
10.3.1 Traditional Pole Placement Controller
Performance with Network Imperfections . 310
10.3.2 NCS under TCP with Intermittent
Acknowledgment 310
10.3.3 NCS under TCP with Full Acknowledgment . 312
10.3.4 NCS under UDP with No Acknowledgment 313
10.4 Conclusions 315
Problems . 316
Appendix 10A 317
Appendix 10B 321
Appendix 10C 324
References 327
Index 329
329
Index
A
Action-dependent value function, 115; see also
Finite-horizon optimal control of
linear QNCS
certainty-equivalent stochastic value
function, 115
control inputs, 116
estimated value function, 120
model-free online tuning of, 117–122
standard Bellman equation, 116, 118
terminal constraint error vector, 120
update law, 120
Actor NN weights tuning law, 136; see also Nearoptimal regulator design
Actuator saturation, 105
A/D, see Analog-to-digital (A/D)
Adaptive control, see Optimal adaptive
control
Adaptive dynamic programming (ADP), 14, 65,
105, 155
Adaptive estimator (AE), 48, 66; see also
Optimal adaptive control
model-free online tuning of, 73
for optimal regulator design, 88–92
Adaptive event-sampled optimal control,
279; see also Event-sampled
distributed networked control
systems
Bellman equation, 281
cross-layer design, 284, 285
distributed scheduling, 284–289
measurement error, 283
parameter estimation error dynamics, 282
scheduler performance, 289–291
value function setup, 280–281
value function tuning, 281–283
ZOH-based event-triggered control, 279, 280,
283–284
ADP, see Adaptive dynamic programming
(ADP)
AE, see Adaptive estimator (AE)
Algebraic Riccati equation (ARE), 46
Analog-to-digital (A/D), 5
ARE, see Algebraic Riccati equation (ARE)
B
Back-off interval (BI), 265
Bellman recursion, 45
BI, see Back-off interval (BI)
C
CAN, see Controller area network (CAN)
Carrier sense multiple access (CSMA), 258, 275
Certainty-equivalent Riccati equation, 45
Certainty-equivalent SRE, 68
Cognitive radio (CR), 13
Controller area network (CAN), 257
Controller design, 33; see also Nonlinear stability
analysis
example, 34–35
for linear systems, 35
Lyapunov design of LTI feedback controllers,
36–37
Lyapunov theorem for linear systems, 36
problems, 61–62
signum function, 33, 34
stability analysis, 35–36
Control systems, 1
CR, see Cognitive radio (CR)
Cross-layer design, 276, 284, 285
Cross-layer network protocol designs, 275
CSMA, see Carrier sense multiple access
(CSMA)
D
D/A, see Digital-to-analog (D/A)
Digital-to-analog (D/A), 5
Discrete-time single-input Brunovsky form, 20
Distributed networked control systems
(DNCS), 14, 15, 277; see also
Event-sampled distributed
networked control systems
advantages of event-triggered control for,
278–279
basic structure of, 277
communication protocol design, 275
cross-layer scheme for system in, 276
revolutionary scheme for, 275
system dynamics, 278
DP, see Dynamic programming (DP)
Dynamical systems, 19; see also Nonlinear
stability analysis; Stochastic discretetime control
analysis, 22
asymptotic stability, 27
autonomous systems and linear systems,
28–29
boundedness, 28
Brunovsky canonical form, 20–21330 Index
Dynamical systems (Continued)
continuity and function norms, 25–26
discrete-time single-input Brunovsky
form, 20
discrete-time systems, 19–20
example, 22
linear systems, 21
Lyapunov stability, 27–28
mathematical background, 23
problems, 61–62
properties of, 26–27
quadratic forms and definiteness, 25
simulation, 22–23
singular value decomposition, 24
uniform ultimate boundedness, 28
vector and matrix norms, 23–24
Dynamic programming (DP), 44, 67
Dynamic quantizer, 105
scheme, 111–112
E
Embedded intelligent control, 275
Embedded round robin (ERR), 269
ERR, see Embedded round robin (ERR)
Event-sampled distributed networked control
systems, 275, 294; see also Adaptive
event-sampled optimal control;
Distributed networked control systems
(DNCS)
communication protocol design, 275
cross-layer design, 276
example, 291
NCS control at cyber layer, 276
problems, 295
simulation, 291–294
Event-triggered control techniques, 14
EXCLUSIVE OR (X-OR), 56
F
Fairness index (FI), 269
FI, see Fairness index (FI)
Finite-horizon optimal adaptive control, 69; see
also Optimal adaptive control
auxiliary error vector dynamics, 75
auxiliary residual error vector, 74–75
background, 69–71
bellman equation, 83–85
Bellman equation residual error, 74
Bellman error, 85
boundedness in mean of AE errors, 76–78,
98–99
closed-loop system stability, 78–82
convergence of optimal control signals,
80–82, 99–100
cost-to-go term, 76
estimated LNCS control signals, 84
estimation error, 74
example, 83
finite-horizon stochastic optimal design, 82
finite-horizon stochastic optimal regulator, 79
lemma, 71, 79–80, 98
LNCS state regulation error and
performance, 83
model-free online tuning of AE, 73–76
optimality analysis of proposed scheme,
85–88
simulation results, 82
state regulation error for NCS, 84
stochastic linear time-varying system, 70
stochastic value function, 72–73
terminal constraint errors, 83–85
terminal constraint estimation error vector,
74–75
update law for AE, 75
Finite-horizon optimal adaptive design for
NNCS, 227; see also Nonlinear
networked optimal control system
action NNs estimation errors, 237–238
auxiliary error dynamics, 230
boundedness in mean, 230, 234–235,
249–255
closed-loop system stability, 239
control and disturbance approximation,
235–239
critic NN approximation, 231
estimated action NNs, 236
estimation error, 233, 236
flowchart of, 240
HJI equation TD error dynamics, 232
identification error, 229, 230
identifier weight estimation error
dynamics, 230
Lyapunov function, 234
NNCS two-player zero-sum game system
state, 229
online NN identifier design, 227–230
stochastic optimal control input and
disturbance, 240, 248
stochastic optimal design, 241
stochastic value function, 230–234
terminal constraint estimation error, 233
update law for critic NN weight, 233
update law for estimated action NNs weight
matrices, 236
update law for NN identifier, 229
value function estimation error, 232
Finite-horizon optimal control, 105
Finite-horizon optimal control for NNCS,
190–192; see also Nonlinear
networked optimal control system
actor NN estimation, 198–200
auxiliary identification error vector, 193–194Index 331
boundedness in mean, 195, 198
closed-loop stability, 200–202
controller performance, 204–207
cost function comparison, 207–209, 210
critic NN design, 195
estimation error, 197
example, 202–204
flowchart of, 201
HJB equation and terminal constraint errors,
207, 209
ideal input, 198
identification error, 193, 209
NNCS block diagram, 191
NNCS internal dynamics, 192–193
NNCS system state, 193
NN weight estimation error dynamics 194
online NN identifier design, 192–195
packet losses distribution, 205
residual error dynamics, 196
simulation results, 202–204
state regulation errors, 206, 207
stochastic optimal control inputs, 202, 206
stochastic update law, 194, 197
stochastic value function, 195, 196
Finite-horizon optimal control of linear QNCS,
114; see also Action-dependent value
function; Quantized networked
optimal control system
adaptive estimation error convergence,
122–124, 144–149
adaptive linear quadratic regulator, 123
Bellman equation error, 115
boundedness of closed-loop system, 124,
149–151
control inputs, 126
convergence analysis, 122
cost, 129
error history, 129
example, 125
feedback control estimation, 122
quantization error, 127, 128
simulation results, 124–127
system response, 126
Finite-horizon optimal control of nonlinear
QNCS, 127; see also Quantized
networked optimal control system
boundedness of observer error, 132
extended Luenberger observer, 130
observer design, 129
persistence of excitation, 131
state estimation error, 130–131
system dynamics, 129
system state vector, 130
FLNN, see Functional link neural net (FLNN)
Frobenius norm, 24
Functional link neural net (FLNN), 59, 60–61;
see also Neural networks (NN)
G
Game-theoretic Riccati equation (GRE), 156; see
also Infinite-horizon optimal adaptive
design
-based solution, 157
Game theory, 13
GAS, see Globally asymptotically stable
(GAS)
Generalized energy approach, 29
Globally asymptotically stable (GAS), 27
Globally UUB (GUUB), 28
GRE, see Game-theoretic Riccati equation
(GRE)
GUUB, see Globally UUB (GUUB)
H
Hamilton–Jacobi–Bellman equations (HJB
equations), 14, 105
Hamilton–Jacobi–Isaacs equation (HJI equation),
222
HJB equations, see Hamilton–Jacobi–Bellman
equations (HJB equations)
HJI equation, see Hamilton–Jacobi–Isaacs
equation (HJI equation)
I
Infinite-horizon optimal adaptive design, 158,
159; see also Optimal uncertain linear
networked control systems; Twoplayer zero-sum game
action-dependent value function, 165
auxiliary residual error vector, 168
closed-loop system stability, 171–172
example, 173–174
excitation persistence, 169
gain matrix, 166
LNCS quadratic zero-sum games, 159–164,
178–183
model-free online tuning, 167–171
optimal control policies and worst-case
disturbance signals, 172–173,
183–184
performance of adaptive estimation-based
optimal strategy, 175–176
simulation results, 173–177
stochastic cost function, 163
stochastic optimal control scheme, 171
stochastic optimal design for LNCS with
disturbance, 173
stochastic value function, 164–167
Input-to-state stable (ISS), 14
Internet protocol (IP), 5
IP, see Internet protocol (IP)
ISS, see Input-to-state stable (ISS)332 Index
K
Kalman gain, 45
L
LAS, see Locally asymptotically stable (LAS)
Linear in the tunable parameters (LIP), 60
Linear in the unknown parameters (LIP), 50
Linear NCS (LNCS), 66, 69; see also Optimal
adaptive control
control with network imperfections, 66
finite-horizon stochastic optimal regulator
for, 79
state regulation error and performance, 83
Linear quadratic regulator (LQR), 45
design technique, 37
regulation, 105
Linear time invariant (LTI), 21
LIP, see Linear in the tunable parameters (LIP);
Linear in the unknown parameters
(LIP)
LNCS, see Linear NCS (LNCS)
Locally asymptotically stable (LAS), 27
LQR, see Linear quadratic regulator (LQR)
LTI, see Linear time invariant (LTI)
Lyapunov approach, 29
Lyapunov theorem for linear systems, 36
M
MAC, see Medium-access control (MAC)
MAD, see Maximum allowable delay (MAD)
MATI, see Maximum allowable transfer interval
(MATI)
Maximum allowable delay (MAD), 3
Maximum allowable transfer interval (MATI), 3
Medium-access control (MAC), 13
Multilayer perceptron, 55
N
NCS, see Networked control systems (NCS)
NDP, see Neurodynamic programming (NDP)
Near-optimal regulator design, 132; see also
Quantized networked optimal control
system
actor NN weights tuning law, 136
Bellman equation, 133
boundedness of closed-loop system, 139
bounds on optimal closed-loop dynamics, 138
certainty-equivalent time-varying stochastic
value function, 133
control input approximation, 135–137
control input error, 135
dynamic quantizer design, 137–138
dynamic quantizer for control input, 137
error dynamics for actor NN weights, 137
example, 140–141, 142, 143
finite-horizon near-optimal regulator, 139
scaling parameter, 138
simulation results, 140
stability analysis, 138–140
terminal constraint error, 135
terminal constraint of value function, 132
update law for critic NN, 134
value function approximation, 133–135
value function at terminal stage, 133
Networked control systems (NCS), 1, 2–3,
65, 155
challenges in, 3
communication packet in, 1
control at cyber layer, 276
current research, 13
distributed, 15
energy efficiency, 13
event-sampled control, 14–15
game theory, 13
network imperfections, 4
network-induced delay, 4
network protocol effects, 6–12
optimal control, 14
packet dropouts, 4–5
QNCS with lossless network, 5
quantization, 5–6
spectrum management, 13
stability region for packet losses, 3
state regulation error for, 84
under TCP, 8, 9
timing diagram of signals in, 4
Network imperfections, 3, 155
Neural networks (NN) , 19, 54–55; see also
Stochastic discrete-time control
example, 58–59
functional link, 60–61
function approximation, 59–60
identifier, 188
multilayer perceptron, 55
online NN identifier design, 192
two-layer, 55–58
Neurodynamic programming (NDP), 187
NNCS, see Nonlinear networked control systems
(NNCS)
Nonlinear networked control systems (NNCS),
187, 191
infinite-horizon optimal control for, 221
network imperfections, 191
Nonlinear networked optimal control system,
187, 218; see also Finite-horizon
optimal control for NNCS
actor NN estimation error dynamics, 213
affine nonlinear discrete-time system, 188
boundedness in mean of critic NN estimation
errors, 212–214Index 333
control signal convergence, 214
discrete-time HJB Equation, 189
example, 214–215
extensions to infinite horizon, 209
HJB equation-based solution, 188–190
optimal control input derivation, 189
optimal stochastic value function and control
policy, 210–212
problems, 219
residual error or cost-to-go error, 211
simulation results, 214–218
traditional nonlinear optimal control
design, 188
update law for actor NN weights, 212
Nonlinear stability analysis, 29; see also
Controller design; Dynamical
systems; Stochastic discrete-time
control
asymptotic stability, 30–31
autonomous dynamical system, 29
decrescent function, 38
example, 31–33, 38, 40–42
extensions of Lyapunov techniques and
bounded stability, 39
global stability, 31
Lyapunov analysis for autonomous
systems, 29
Lyapunov analysis for linear systems, 35
Lyapunov analysis for nonautonomous
systems, 37–39
Lyapunov stability, 30
problems, 61–62
UUB analysis and controls design, 39
UUB by Lyapunov analysis, 39–40
UUB of closed-loop system, 41–42
UUB of linear systems with disturbance,
40–41
Nonlinear two-player zero-sum game design,
221, 247; see also Finite-horizon
optimal adaptive design
augment state variable, 227
certainty-equivalent discrete-time HJI,
223, 225
delay distribution in NNCS, 243
designing control and disturbance
policies, 222
estimated critic and two action NNs
weights, 245
example, 242
HJI and terminal constraint errors, 244–246
effect of increasing final time, 246
iteration-based NDP methods, 221–222
NNCS two-player zero-sum game,
225–227, 244
nonlinear two-player zero-sum game, 223
optimal adaptive control scheme using timebased NDP, 222
optimal control and disturbance, 224
optimal performance of proposed design,
246, 247
packet loss distribution in NNCS, 243
performance analysis, 242–244
problems, 247–248
proposed finite-horizon stochastic optimal
strategies, 244
simulation results, 242
stochastic optimal control design, 223–225
O
Open system interconnection (OSI), 275
Optimal adaptive control, 65, 96; see also
Finite-horizon optimal adaptive
control; Linear NCS (LNCS); Optimal
regulator design
ADP approaches, 65–66
Bellman equation in discrete time, 67
certainty-equivalent SRE, 68
certainty-equivalent stochastic, 66
discrete-time HJB equation, 68
example, 92–93
extensions to infinite horizon, 88
LNCS control with network imperfections, 66
problems, 97
simulation results, 92–96
stochastic linear discrete-time system, 67
stochastic optimal control derivation, 66
stochastic Riccati equation-based solution, 67
traditional control design, 67
Optimal regulator design, 88; see also Optimal
adaptive control
auxiliary residual error vector, 89
auxiliary vector dynamics, 89
control signal convergence, 91, 101–102
cost AE error asymptotic stability, 90–91
infinite-horizon stochastic optimal design, 92
parameter estimation error, 90
residual dynamics, 89
value function, 88
Optimal uncertain linear control systems, 297,
315–316
action-dependent value function, 303
adaptive estimator tuning, 304–306, 321–324
closed-loop system stability, 307–309,
324–327
estimation error dynamics, 301
example, 309
NCS under TCP, 310–313
NCS under UDP, 313–315
networked control system, 299
observability criterion, 302
observed state, 301
observer design, 299–302, 317–321
optimal control design, 298334 Index
Optimal uncertain linear control systems
(Continued)
pole placement controller performance, 310
prediction error, 300
problems, 316
simulation results, 309
state regulation errors of control, 310
stochastic optimal regulator for LNCS, 307
stochastic value function, 302–303
system state vector, 299
unified framework, 297–298
update law for parameter vector, 301, 305
value function with system states, 305
Optimal uncertain linear networked control
systems, 155, 177–178; see also
Infinite-horizon optimal adaptive
design
game-theoretic Riccati equation-based
solution, 157–158
linear timevarying discrete-time system ,
160, 180
linear time-varying stochastic discrete-time
system, 160, 178–180
positive definite Lyapunov function, 183–184
problems, 178
stochastic linear discrete-time system, 156
two-player zero-sum game design, 156–158
OSI, see Open system interconnection (OSI)
P
PE, see Persistency of excitation (PE)
Persistency of excitation (PE), 121, 131, 230
Q
QCS, see Quantized control system (QCS)
Q-function setup, 48; see also Stochastic
Q-learning
Q-learning, 155
QNCS, see Quantized networked control system
(QNCS)
Quantized control system (QCS), 108
with lossless network, 109
Quantized networked control system (QNCS), 5
Quantized networked optimal control system,
105, 108, 141–143; see also Finitehorizon optimal control of linear
QNCS; Finite-horizon optimal control
of nonlinear QNCS; Near-optimal
regulator design
adaptive estimation error convergence, 144
boundedness of closed-loop system,
124, 149
dynamic quantizer scheme, 111–112
ideal and realistic quantizer, 111
linear networked control systems, 108–110
nonlinear networked control system, 112–114
problems, 143–144
QCS with lossless network, 109
quantization error converges, 124, 145, 148
quantized system with input saturation, 112
quantizer representation, 110
time-invariant linear discrete-time
system, 108
Quantizer, 111
R
Riccati equation, 36, 37
S
SGRE, see Stochastic game-theoretic Riccati
equation (SGRE)
Short network-induced delay, 3
Signal-to-interference-plus-noise ratio
(SINR), 4
Signum function, 33, 34
Singular value decomposition (SVD), 24
SINR, see Signal-to-interference-plus-noise ratio
(SINR)
SISL, see Stable in the sense of Lyapunov
(SISL)
SRE, see Stochastic Riccati equation (SRE)
Stable in the sense of Lyapunov (SISL), 27
Static quantizer, 105
Stochastic discrete-time control, 42; see also
Dynamical systems; Neural networks
(NN); Nonlinear stability analysis;
Stochastic Lyapunov stability;
Stochastic Q-learning
affine nonlinear discrete-time system, 52
Bellman recursion, 45, 47
certainty-equivalent Riccati equation, 45
certainty-equivalent stochastic optimal value
function, 46
discrete-time HJB Equation, 52, 53
example, 54, 55
Kalman gain, 45
problems, 61–62
steady-state Kalman gain sequence, 46
stochastic linear discrete-time optimal
control, 43–48
stochastic Lyapunov stability, 42–43
stochastic nonlinear discrete-time optimal
control, 51
system response, 52
value function increment, 45
Stochastic game-theoretic Riccati equation
(SGRE), 164
Stochastic Lyapunov stability, 42; see also
Stochastic discrete-time control
in mean square, 43Index 335
Stochastic Q-learning, 48; see also Stochastic
discrete-time control
assumption, 50
example, 51
optimal action-dependent value function, 49
Q-function setup, 48–50
tuning based on adaptive estimator and
Q-learning, 50
Stochastic Riccati equation (SRE), 47
SVD, see Singular value decomposition (SVD)
T
TCP, see Transmission control protocol (TCP)
TD, see Temporal difference (TD)
TDE, see Temporal difference error (TDE)
Temporal difference (TD), 232
Temporal difference error (TDE), 115
Terminal constraint error, 105
Time-based ADP approach, 65, 156
Traditional feedback control systems, 105
Traditional optimal control theory, 105
Transmission control protocol (TCP), 3
Transmission delay, 4
Two-player zero-sum game, 156; see also
Infinite-horizon optimal adaptive
design
certainty-equivalent value function, 157
gain matrix, 157–158
optimal action-dependent value function,
157
state-dependent dynamics, 156
U
UAV, see Unmanned aerial vehicles (UAV)
UDP, see User datagram protocol (UDP)
Uniformly ultimately bounded (UUB), 28
Unmanned aerial vehicles (UAV), 1
User datagram protocol (UDP), 3
UUB, see Uniformly ultimately bounded (UUB)
V
Value function increment, 45
Value iterations (VIs), 65
VIs, see Value iterations (VIs)
W
Wireless networked control systems (WNCS),
257, 258, 272
algorithms, 263–264
codesign, 259
components, 258
cross-layer distributed scheduling, 265, 266
CSMA-based distributed scheduling, 258
example, 269
fairness comparison, 270
issues for control design, 257
multiple WNCS pairs, 259
network structure for cross-layer design, 260
numerical simulations, 269–272
optimal cross-layer distributed scheduling
scheme, 264–269
optimal cross-layer in performance
scheduling algorithm, 271
optimal distributed scheduling problem, 265
plant model, 260–261
problems, 272
state regulation errors with stochastic control,
270
stochastic optimal controlled design, 261–264
structure of, 259
theorem, 264, 267–269
utility comparison, 271
utility function for WNCS, 264
WNCS pair, 259
Wireless sensor networks (WSN), 13
WNCS, see Wireless networked control systems
(WNCS)
WSN, see Wireless sensor networks (WSN)
X
X-OR, see EXCLUSIVE OR (X-OR)
Z
Zero-order-hold (ZOH), 279; see also Adaptive
event-sampled optimal control
ZOH, see Zero-order-hold (ZOH)
ZOH-based event-triggered control system, 279;
see also Adaptive event-sampled
optimal control
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