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 . 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,. 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. 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