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Lect_1 - Nonlinear Systems and Control Lecture 1...

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Nonlinear Systems and Control Lecture # 1 Introduction – p.1/19
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Nonlinear State Model ˙ x 1 = f 1 ( t, x 1 , . . . , x n , u 1 , . . . , u p ) ˙ x 2 = f 2 ( t, x 1 , . . . , x n , u 1 , . . . , u p ) . . . . . . ˙ x n = f n ( t, x 1 , . . . , x n , u 1 , . . . , u p ) ˙ x i denotes the derivative of x i with respect to the time variable t u 1 , u 2 , . . . , u p are input variables x 1 , x 2 , . . . , x n the state variables – p.2/19
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x = x 1 x 2 . . . . . . x n , u = u 1 u 2 . . . u p , f ( t, x, u ) = f 1 ( t, x, u ) f 2 ( t, x, u ) . . . . . . f n ( t, x, u ) ˙ x = f ( t, x, u ) – p.3/19
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˙ x = f ( t, x, u ) y = h ( t, x, u ) x is the state, u is the input y is the output ( q -dimensional vector) Special Cases: Linear systems: ˙ x = A ( t ) x + B ( t ) u y = C ( t ) x + D ( t ) u Unforced state equation: ˙ x = f ( t, x ) Results from ˙ x = f ( t, x, u ) with u = γ ( t, x ) – p.4/19
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Autonomous System: ˙ x = f ( x ) Time-Invariant System: ˙ x = f ( x, u ) y = h ( x, u ) A time-invariant state model has a time-invariance property with respect to shifting the initial time from t 0 to t 0 + a , provided the input waveform is applied from t 0 + a rather than t 0 – p.5/19
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Existence and Uniqueness of Solutions ˙ x = f ( t, x ) f ( t, x ) is piecewise continuous in t and locally Lipschitz in x over the domain of interest f ( t, x ) is piecewise continuous in t on an interval J R if for every bounded subinterval J 0 J , f is continuous in t for all t J 0 , except, possibly, at a finite number of points where f may have finite-jump discontinuities f ( t, x ) is locally Lipschitz in x at a point x 0 if there is a neighborhood N ( x 0 , r ) = { x R n | bardbl x x 0 bardbl < r } where f ( t, x ) satisfies the Lipschitz condition bardbl f ( t, x ) f ( t, y ) bardbl ≤ L bardbl x y bardbl , L > 0 – p.6/19
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A function f ( t, x ) is locally Lipschitz in x on a domain (open and connected set) D R n if it is locally Lipschitz at every point x 0 D When n = 1 and f depends only on x | f ( y ) f ( x ) | |
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