Adaptive Control System Part 15 pdf

partial adaptive pole placement, 10, 13,
14, 18
See also Model reference adaptive
control (MRAC)
Chattering behaviour, 56
Cholesky decomposition, 69, 71
Concave covers, 223±8
de®nitions, 223
examples of, 224±8
properties of, 223±4
Concave functions, 217, 221±3
de®nitions, 221
properties of, 221±3
Continuous-time Lyapunov design,
268±9
Control Lyapunov function (clf), 188
Controlled Dung equations, 135±6
Converging-output converging-state
(COCS) function, 128, 132
Converse Lyapunov Theorem, 130
Convex functions, 217, 221±3
de®nition, 221
properties of, 221±3
Dead zone based methods, 23±5, 260±2,
267, 277
inverse of dead zone characteristic, 264
ordinary direct adaptive control with
dead zone, 27±8
robust direct adaptive control, 28±31
See also Actuator compensation
Deadbeat control strategy, 67, 160, 163,

166
Dense search method, 83±4
Didinsky, G., 186
Direct adaptive control, 28±31
multi-input multi-output systems,
289±96
with dead zone, 27±8
Direct localization principle, 89±101
localization in the presence of
unknown disturbance, 99±101
optimal localization, 94±9
Discrete-time nonlinear systems:
active identi®cation, 159±82
avoidance of nonlinear
parametrization, 164±6
®nite duration, 178
input selection for identi®cation,
173±8, 179, 180
pre-computation of projections,
168±73, 180
problem formulation, 161±6
second-order example, 162±4
adaptive stabilization, 80±115, 245
direct localization principle, 89±101
indirect localization principle, 101±8
problem statement, 86±9
simulation examples, 108±12
See also Nonlinear systems
Dung equations, 135±6
Dynamic feedback, 187, 204, 209

Dynamic normalization technique, 151±5
Feedback passivation, see Adaptive
passivation
Finite escape time, 162, 166
Freeman, R.A., 185
Friction models, 236±40
Fuzzy systems, 287
applications, 299±305
two-link robot arm, 301±5
direct adaptive control, 290±2
indirect adaptive control, 297±9
Globally adaptively stabilizable systems,
128, 129
Globally uniformly asymptomatically
stable (GUAS) systems, 124
Gradient projection adaptive law, 266±8
H
2
optimal control:
adaptive internal model control
schemes, 11, 13±14
simulation example, 18±19
nonadaptive control schemes, 4±5
H
I
optimal control:
adaptive internal model control
schemes, 11±12, 13, 14
simulation example, 18±19
nonadaptive control schemes, 5±6

Index 331
Hamilton±Jacobi±Bellman (HJB) pde,
184, 185
Hurwitz polynomial of degree, 7±8, 9
Hysteresis, 260±2, 277
inverse, 264
See also Actuator compensation
Hysteresis switching, 11, 84
Identi®cation scheme, see Active
identi®cation scheme
Indirect adaptive control, 63±79
adaptive control law, 69±74
properties of identi®cation scheme,
69±70
strategy, 70±4
adaptive control scheme, 68±9
multi-input multi-output systems,
296±9
problem formulation, 65±7
simulation examples, 74±8
system with noise, 75±8
system without noise, 74±5
Indirect localization principle, 101±8
localization in the presence of
unknown disturbance, 108
Input-to-output practically stable (IOpS)
systems, 125, 126
Input-to-output stable (IOS) systems, 125
Input-to-state practically stable (ISpS)
systems, 125, 126

Input-to-state stable (ISS) systems, 120,
124±5
Internal model control (IMC) schemes,
1±2
IMC ®lter, 5, 11
known parameters, 2±7
H
2
optimal control, 4±5
H
I
optimal control, 5±6
model reference control, 3±4
partial pole placement control, 3
robustness to uncertainties, 6±7
See also Adaptive internal model
control schemes
Inverse optimal control, 185±6
inverse optimal adaptive tracking, 197±
202
inverse optimality via backstepping,
202±5
See also Actuator compensation,
adaptive inverse approach
Ioannou, P.A., 201
Isidori, A., 123, 136
Kharitonov Theory, 10
Kokotovic, P.V., 185, 186
Krstic, M., 185, 186, 196±7, 205±8, 210
Kuhn Tucker theorem, 249

Lagrangian function, 249
LaSalle's theorem, 191
Levy±Desplanques theorem, 298
Linear-quadratic-regulator (LQR), 201
Linearly parametrized systems, 215±16
Localization method, 80, 85
direct localization principle, 89±101
localization in the presence of
unknown disturbance, 99±101
optimal localization, 94±9
indirect localization principle, 101±8,
111±12
localization in the presence of
unknown disturbance, 108
simulation examples, 108±12
constant parameters, 109±10
indirect localization, 111±12
parameter jumps, 110±11
LU decomposition, 69
Magnetic bearing system, 242±5
adaptive control based on nonlinear
parametrization, 243±5
Martensson, B., 81±3
Min-max optimization, 215, 217, 245
Model reference adaptive control
(MRAC), 10±11, 41
output feedback design, 272
simulation example, 18
stability and robustness analysis, 13, 14
variable structure MRAC system, 41±3

MRAC based error model, 44±6
See also Adaptive variable structure
control
332 Index
Model reference control, nonadaptive
control schemes, 3±4
Morse, A.S., 84
Multi-input multi-output (MIMO)
systems, 287±306
direct adaptive control, 289±96
indirect adaptive control, 296±9
See also Fuzzy systems
Multivariable systems, 275±80
nonlinearity model and its inverse,
277
output feedback designs, 280
state feedback designs, 278±80
Neural networks, 216, 287
See also Fuzzy systems
Newton±Raphson method, 227
Nonadaptive control schemes, 2±7
H
2
optimal control, 4±5
H
I
optimal control, 5±6
model reference control, 3±4
partial pole placement control, 3
robustness to uncertainties, 6±7

See also Internal model control (IMC)
schemes
Nonlinear gain, 124±7
input-to-state stable (ISS) systems,
124±5
nonlinear small gain theorems, 125±7
Nonlinear parametrization (NP), 216±18,
258, 277
avoidance of, 164±6
active identi®cation, 165
orthogonalized projection
estimation, 166
projection measurements, 165
regressor subspace, 164±5
concave covers, 223±8
de®nitions, 223
examples of, 224±8
properties of, 223±4
concave/convex functions, 220±3
de®nitions, 221
properties of, 221±3
problem statement, 218±20
stable adaptive NP systems, 228±35
adaptive controller, 230±4
adaptive observer, 234±5
extensions to the vector case, 229
low-velocity friction model, 236±40
magnetic bearing system, 242±5
new error model, 228±9
scalar error model with linear

parameters, 229±30
stirred tank reactors (STRs), 240±2
structure of adaptive controller,
218±20
See also Nonlinear systems
Nonlinear systems:
adaptive robust control for uncertain
dynamical systems, 308±27
adaptive robust control with -
modi®cation, 312±17
application to PM synchronous
motors, 317±20
problem formulation, 310±12
nonsmooth nonlinearities, 260±1
designs for, 281±4
See also Actuator compensation,
adaptive inverse approach optimal
adaptive tracking, 184±213
adaptive backstepping, 193±7
adaptive tracking control Lyapunov
function (atclf), 188±93
design for strict-feedback systems,
205±9
inverse optimal adaptive tracking,
197±202
inverse optimality via backstepping,
202±5
problem statement, 186±8
transient performance, 209±10
See also Adaptive passivation;

Discrete-time nonlinear systems;
Nonlinear parametrization; Small
gain techniques
Nonsmooth nonlinearities, 260±1
designs for, 281±4
See also Actuator compensation,
adaptive inverse approach;
Nonlinear systems
Nussbaum, R.D., 81±2
Index 333
Orthogonalized projection scheme, 160,
166
algorithm, 169±70
for output-feedback systems, 170±1
Output feedback systems, 271±5
adaptive output feedback passivation,
136±8
for multivariable systems, 280
inverse control, 269±71
adaptive law, 271
error model, 270±1
reference system, 269±70
model reference design, 272
orthogonalized projection for,
170±1
parametric-output-feedback form,
161±2
PID design, 273±5
pole placement design, 272±3
Overparametrization, 216

Pan, Z., 186
Parametric strict-feedback system, 139,
185±6, 205±9, 281
Parametric-output-feedback form, 161±2
Parseval's Theorem, 4
Partial pole placement control:
adaptive internal model control
schemes, 10, 13, 14
simulation example, 18
nonadaptive control schemes, 3
Passivation, 120, 123
See also Adaptive passivation
Passivity, 121±3
strict passivity, 122
Pendulum example, 147±51
Permanent magnet synchronous (PMS)
motor example, 309±10, 317±20
control design, 318±20
model of PMS motor, 317±18
simulation study, 320
PID design, 273±5
Piecewise-linear characteristic, 260±2, 277
inverse, 264
See also Actuator compensation
Polar decomposition, 65, 69
Pole placement design, 272±3
Polynomial dierential operators, 81
Recursive adaptive passivation, 1315
Riccati equations, 184, 185, 201
Robots, 289

two-link robot arm, 301±5
Robust adaptive control:
for nonlinear uncertain dynamical
systems, 308±27
adaptive robust control with -
modi®cation, 312±17
application to PM synchronous
motors, 317±20
problem formulation, 310±12
with less prior knowledge, 23±38
direct adaptive control, 28±31
problem formulation, 25±7
simulation example, 36±8
with least prior knowledge, 32±6
Robustness, 261
adaptive internal control schemes, 9±
10
adaptive algorithms, 81
robust adaptive law design, 7±9,
313±14
robustness analysis, 12±18
adaptive variable structure control, 47±
8, 52±5
nonadaptive control schemes, 6±7
Saturation function, 221
Self-excitation technique, 64
Separable sets, 95±6
Separation Principle, 148
Sepulchre, R., 185
Singular value decomposition (SVD), 65,

69
Small control property, 190
Small gain techniques, 6±7, 119±20, 138±
55
adaptive controller design, 139±45
initialization, 139±40
recursive steps, 140±4
small gain design step, 144±5
class of uncertain systems, 138±9
334 Index
examples and discussions, 14755
pendulum example, 14751
robusti®cation via dynamic
normalization, 151±5
nonlinear small gain theorems, 125±7
stability analysis, 146±7
Smooth control law, 203
Sontag, E.D., 188, 189±90, 196, 200, 202
Stability analysis, 12±18
adaptive variable structure control,
47±8, 52±5
small gain approach, 146±7
Stabilization, 185
See also Adaptive stabilization
Stabilizing sets, 101±2
State feedback designs, 265±9, 278±80
continuous-time Lyapunov design,
268±9
error model, 265±6
gradient projection adaptive law,

266±8
multivariable systems, 278±80
Stirred tank reactors (STRs), 240±2
adaptive control based on nonlinear
parametrization, 241±2
Strict passivity, 122
Strict-feedback systems, 139, 185±6, 205±
9
Sun, J., 201
Supervisory control, 84, 89
Switching control, 80±6
algorithms, 81
conventional switching control, 88±9
direct localization principle, 89±101
localization in the presence of
unknown disturbance, 99±101
optimal localization, 94±9
indirect localization principle, 101±8
localization in the presence of
unknown disturbance, 108
problem statement, 86±9
simulation examples, 108±12
constant parameters, 109±10
indirect localization, 111±12
parameter jumps, 110±11
supervisory switching control, 84, 89
Sylvester resultant matrix, 64±5
Takagi±Sugeno fuzzy systems, see Fuzzy
systems
Tracking performance, adaptive variable

structure control, 47±9, 52±6
See also Nonlinear systems
Trajectory initialization, 210
Tuning functions, 120, 186, 191, 233
adaptive backstepping with, 120, 127,
193±7
Unboundedness observable (UO)
function, 128
Uncertain discrete-time systems, see
Discrete-time nonlinear systems
Variable structure design (VSD), 41±2
See also Adaptive variable structure
control
Willems, J.C., 123
Youla parameter, 3
Young's inequality, 137
Zero-state detectability, 122
Index 335