System TheoryContentsOverviewLinear SystemsNyquistNyquistFunctional Differential EquationsFDEFDESystem TheoryEECS 228aJean WalrandUniversity of CaliforniaBerkeley2ContentsnOverviewnLinear SystemsnNyquistnFunctional Differential Equations3Overviewu(t)y(t)x(t)Examplex(t) = statesisoGeneralu(t) = inputy(t) = outputMIMOu(t)y(t)mk4Overview …Feedback Control SystemGoals of Control•Path following•Stabilization •Disturbance rejection•Performance improvement•Robustness w.r.t. System uncertainties or imperfections5Linear SystemsLaplaceTransform6Linear Systems7Linear SystemsExample8Linear SystemsExample9Linear SystemsObservationsuvDArg(s - u)Arg(s - v)+10NyquistSlide from a takby Glenn Vinnicombe11NyquistsDR >> 112NyquistSlide from a takby Glenn Vinnicombe13NyquistMIMO Case:14NyquistExample 1èClosed-Loop is stable15NyquistExample 2- 1016NyquistExample 2 …è Stable if T < 1.35s17Functional Differential Equations18Functional Differential EquationsExample:-2024681012-3 -2 -10 1 2 31.210.819FDE-60-40-2002040600 500 1000 1500 2000 2500 3000 3500 4000 45001.021.0420FDE-4-3-2-101234-20 2 4 6- 10-4-3-2-101234-20 2 4
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