Lect_10 - Nonlinear Systems and Control Lecture # 10 The...

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Nonlinear Systems and Control Lecture # 10 The Invariance Principle – p.1/17

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Example: Pendulum equation with friction ˙ x 1 = x 2 ˙ x 2 = a sin x 1 bx 2 V ( x ) = a (1 cos x 1 ) + 1 2 x 2 2 ˙ V ( x ) = a ˙ x 1 sin x 1 + x 2 ˙ x 2 = bx 2 2 The origin is stable. ˙ V ( x ) is not negative definite because ˙ V ( x ) = 0 for x 2 = 0 irrespective of the value of x 1 However, near the origin x 1 n = 0 , the solution cannot stay identically in the set { x 2 = 0 } – p.2/17
Definitions: Let x ( t ) be a solution of ˙ x = f ( x ) A point p is said to be a positive limit point of x ( t ) if there is a sequence { t n } , with lim n →∞ t n = , such that x ( t n ) p as n → ∞ The set of all positive limit points of x ( t ) is called the positive limit set of x ( t ) ; denoted by L + If x ( t ) approaches an asymptotically stable equilibrium point ¯ x , then ¯ x is the positive limit point of x ( t ) and L + = ¯ x A stable limit cycle is the positive limit set of every solution starting sufficiently near the limit cycle – p.3/17

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A set M is an invariant set with respect to ˙ x = f ( x ) if x (0) M x ( t ) M, t R Examples: Equilibrium points Limit Cycles A set M is a positively invariant set with respect to ˙ x = f ( x ) if x (0) M x ( t ) M, t 0 Example: The set Ω c = { V ( x ) c } with ˙ V ( x ) 0 in Ω c – p.4/17
The distance from a point p to a set M is defined by dist( p,M ) = inf x M b p x b x ( t ) approaches a set M as t approaches infinity, if for each ε > 0 there is T > 0 such that dist( x ( t ) ,M ) < ε, t > T Example: every solution x ( t ) starting sufficiently near a stable limit cycle approaches the limit cycle as t → ∞ Notice, however, that x

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This note was uploaded on 07/25/2008 for the course ME 859 taught by Professor Choi during the Spring '08 term at Michigan State University.

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Lect_10 - Nonlinear Systems and Control Lecture # 10 The...

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