Nonlinear Control Sy.. - Free

346 LIST OF FIGURES
3.1 Stable equilibrium point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Asymptotically stable equilibrium point . . . . . . . . . . . . . . . . . . . . . 67
3.3 Mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Pendulum without friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 The curves V(x) =,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 System trajectories in Example 3.21 . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 (a) Continuous-time system E; (b) discrete-time system Ed. . . . . . . . . . 126
4.2 Discrete-time system Ed .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.3 Action of the hold device H . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1 (a) The system (5.7)-(5.8); (b) modified system after introducing -O(x); (c)
"backstepping" of -O(x); (d) the final system after the change of variables. 142
5.2
6.1
Current-driven magnetic suspension system .. . . . . . . . . . . . . . . . . . 151
The system H . .. . . . . .. ..... . .. . ... . .. ........... 155
6.2 Experiment 1: input u(t) applied to system H. Experiment 2: input u(t) =
uT(t) applied to system H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Causal systems: (a) input u(t); (b) the response y(t) = Hu(t); (c) truncation
of the response y(t). Notice that this figure corresponds to the left-hand side
of equation (6.8); (d) truncation of the function u(t); (e) response of the
system when the input is the truncated input uT(t); (f) truncation of the
system response in part (e). Notice that this figure corresponds to the righthand
side of equation (6.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.4 Static nonlinearity N(.) . . ..... .......... ............. 163
6.5 The systems Hlu = u2, and H2u = e1" 1 . . . . . . . . . . . . . . . . . . . . . 163
6.6 Bode plot of JH(yw)j, indicating the IIHIIoo norm of H...... ...... . 169
6.7 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.8 The nonlinearity N(.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.9 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.10 The Feedback System SK ......................... .... 176
6.11 The Feedback System 5M .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177