Page 699 - Mathematical Techniques of Fractional Order Systems

Index 671

tent map with shaping parameters, Linear fractional distributed order systems,
386 387 exact solution of, 103
with signed control parameter, 380 exact response of distributed order RC
Integer order SMCPD (IOSMCPD) controller, electrical circuit, 123 124
252, 257 259, 261f, 266 270, 267t numerical examples, 121 122
Integer order systems (IOS), 75 76, 84, 163, preliminaries, 105 108
633 results, 109 121
Integer orders, intuitive results for, 4 5 stability, 125 129
Integral of absolute error (IAE), 266 269 Linear matrix inequalities (LMIs), 150, 284
Integro-differential operator, 563 564 formulation, 134, 154
Interior Point techniques, 148 149 LMI problem, solvability of, 152
Inverse Laplace transformation, 191 192, Linear time invariant (LTI) distributed order
192, 203, 203 204 differential equations, 103 105
Liouvillian chaotic systems, 284 285
K Logistic map with shaping parameters, 386
Kalman Yakubovich Popov lemma, Longest diagonal lines, 632, 632t
170 171 Lorenz attractor, 410f
Lorenz chaotic system, 286, 299
Kaplan Yorke dimension of the system, 328,
Lorenz hyperchaotic system, 288 289, 294,
480
298
Koper Gaspard model of electrochemical
fractional order, 362 363
oscillatory reactions, 482 483
fractional order complex Lorenz
Kreisselmeier-type of adaptive observer, 160
hyperchaotic system
stabilization of, 296 297
L phase portraits of, 364f
Lag synchronization, 420 terminal sliding mode control law for, 292
Lagrange optimal control problem, 202 Lorenz system, 285, 301
Lagrange problem, 206, 208, 222 Lorenz Stenflo system, 561 563, 568 569,
Lagrangian-Euler formula, 252 253, 255 571f, 573f, 574 576, 579, 582 585,
Lambert W functions, 105 106, 126 127, 587 588
130 Lu ¨ fractional order system, 319 321
Lamirarity rate, 631, 632t Lu hyperchaotic system
Lanczos’ iteration method, 93 94 fractional order, 352 354, 356 357, 362
Laplace convolution operator, 232 phase portraits of, 357f
Laplace transform, 191, 202 204, 232, 324, Lu system, 561 562
565 phase portraits of, 353f
Laplace-type integral, 118 119, 129 130 Lyapunov and Lyapunov Krasovskii
Lebesgue dominated convergence theorem, approaches, 154 155
201, 243 244 Lyapunov direct method, 163 164, 563, 567
Leibniz’s rule for fractional differentiation, Lyapunov exponent (LE), 283 284, 327, 411,
565 491, 516 517, 638
Length, uncertainty in, 275 Lyapunov Exponent Toolbox (LET), 516 517
Leray Schauder fixed point theorem, Lyapunov fractional order stability theorem,
186 187 284
Leray Schauder nonlinear alternative fixed Lyapunov function, 172, 293, 487, 567 568,
point theorem, 208 570 571, 574 580, 606, 638 639
Linear and nonlinear control techniques, Lyapunov stability, 257, 260 261, 567,
626 627 582 585
Linear feedback control technique, 418 Lyapunov stability theory, 138, 350,
Linear forced fractional order time-varying 562 563, 568 569
delay system, 139 Lyapunov’s method, 134, 139, 163 164