118 brPage 2br Exponential Stability The origin of is exponentially stable if and only if the linearization of at the origin is Hurwitz Theorem Let be a locally Lipschitz function de64257ned over a domain Let be a continuously differentiable functi ID: 24215 Download Pdf

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118 brPage 2br Exponential Stability The origin of is exponentially stable if and only if the linearization of at the origin is Hurwitz Theorem Let be a locally Lipschitz function de64257ned over a domain Let be a continuously differentiable functi

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Nonlinear Systems and Control Lecture # 11 Exponential Stability Region of Attraction p. 1/18

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Exponential Stability: The origin of is exponentially stable if and only if the linearization of at the origin is Hurwitz Theorem: Let be a locally Lipschitz function deﬁned over a domain . Let be a continuously differentiable function such that for all , where , and are positive constants. Then, the origin is an exponentially stable equilibrium point of . If the assumptions hold globally, the origin will be globally exponentially stable p. 2/18

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Proof: Choose c > small enough that } } ⊂ { } is compact and positively invariant; (0) dV dt )) (0)) /k p. 3/18

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k )) /a (0)) /k /a (0) /k /a /a γt (0) , p. 4/18

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Example yh y, c , c ) = 1 1 1 2 + 2 dy dy = [ + 2 )] + [ + 2 ][ The origin is globally exponentially stable p. 5/18

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Region of Attraction Lemma: If = 0 is an asymptotically stable equilibrium point for , then its region of attraction is an open, connected, invariant set. Moreover, the boundary of is formed by trajectories p. 6/18

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Example + ( 1)

−4 −2 −4 −2 p. 7/18

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Example −4 −2 −4 −2 p. 8/18

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Estimates of the Region of Attraction : Find a subset of the region of attraction Warning: Let be a domain with such that for all is positive deﬁnite and is negative deﬁnite Is a subset of the region of attraction? NO Why? p. 9/18

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Example: Reconsider ) = 1 1 1 2 + 2 dy ) = (1 { < x Is a subset of the region of attraction? p. 10/18

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The simplest estimate is the bounded component of < c , where = min ∂D

For ) = Px , where , the minimum of over ∂D is given by For {k < r min Px min For {| < r min Px For {| < r , i = 1 , . . . , p Take = min min ∂D Px p. 11/18

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Example (Revisited) + ( 1) ) = 1 ) = for def Take min = 0 691 = 0 618 < c is an estimate of the region of attraction p. 12/18

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cos θ, x sin cos sin (2 sin cos cos sin | | 2 sin cos 3849 2361 + 0 861 for 861 Take min 691 861 = 0 803 Alternatively, choose as the largest constant such that Px < c is a subset of p. 13/18

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−2 −1 −2 −1 (a)

−2 −3 −2 −1 (b) (a) Contours of ) = 0 (dashed) ) = 0 (dash-dot), ) = 2 25 (solid) (b) comparison of the region of attraction with its estimate p. 14/18

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If is a domain where is positive deﬁnite and is negative deﬁnite (or is negative semideﬁnite and no solution can stay identically in the set ) = 0 other than = 0 ), then according to LaSalles theorem any compact positively invariant subset of is a subset of the region of attraction Example 4( (0) = 0; uh ∀ | | p. 15/18

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) = Px 2 1 1 1 = 2 + 2 ) = (4 + 2 )

+ 2( ) 6( 2( 6( ∀ | | 8 6 6 6 is negative deﬁnite in {| | = [1 1] , c = min =1 Px = 1 p. 16/18

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dt = 2 σx σh σx ∀ | | On = 1 dt On dt ,x =4 = 10 , c =3 ,x = 10 Γ = 10 and | is a subset of the region of attraction p. 17/18

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−5 −5 (−3,4) (3,−4) V(x) = 10 V(x) = 1 p. 18/18

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