Stability BoundedInput BoundedOutput BIBO Stability Asymptotic Stability Lyapunov
Stability Bounded-Input Bounded-Output (BIBO) Stability Asymptotic Stability Lyapunov Stability Linear Approximation of a Nonlinear System 1 Bounded-Input Bounded-Output (BIBO) stablility Definition: For any constant N, M >0 Any bounded input yields bounded output, i. e. u(t) N y(t) M For linear systems: T (s) p(s) C(s. I A) 1 B q(s) BIBO Stability я All the poles of the transfer function lie in the LHP. q(s) 0 Solve for poles of the transfer function T(s) Characteristic Equation 2
Asymptotic stablility When u(t) 0, i. e. the systemx& Ax x(t) 0 as t For linear systems: x& Ax Bu y Cx Asymptotically stable я All the eigenvalues of the A matrix have negative real parts (i. e. in the LHP) T (s) p(s) C adj[s. I A]B C (s. I A) 1 B q(s) s. I A 0 Solve for the eigenvalues for A matrix Note: Asy. Stability is indepedent of B and C Matrix 3 Asy. Stability from Model Decomposition Suppose that all the eigenvalues of A are distinct. A R n n Let vi the eigenvector of matrix A with respect to eigenvalue i i. e. i , satisfying Avi i vi , i 1, L , n Coordinate Matrix T [v 1 , v 2 , L , vn ] A T 1 AT & T 1 AT & 1 1 0 &2 0 2 M 0 M & 0 n 0 0 L 0 O L 0 1 0 2 M M n n B T 1 B C CT z& T 1 ATz T 1 Bu y CTz Du 1 x(t) T (t) v 1 e 1 t ξ 1 (0) v 2 e 2 t ξ 2 (0) L vn e n t ξ n (0), ξ(0) T x(0) Hence, system Asy. Stable я all the eigenvales of A at lie in the LHP 4
Asymptotic Stablility versus BIBO Stability In the absence of pole-zero cancellations, transfer function poles are identical to the system eigenvalues. Hence BIBO stability is equivalent to asymptotical stability. Conclusion: If the system is both controllable and observable, then BIBO Stability я Asymptotical Stability Methods for Testing Stability • Asymptotically stable • All the eigenvalues of A lie in the LHP • BIBO stable • Routh-Hurwitz criterion • Root locus method • Nyquist criterion • . . etc. 5 Lyapunov Stablility A state xe of an autonomous systemis called an equilibriu m state, if starting at that state the systemwill not move from it in the absence of the forcing input. In other words, consider the system x& f ( x(t), u(t)) equilibriu m state xe must satisfy f ( xe , 0) 0, Example: 1 1 x& 0 x 1 u(t) 2 3 t t 0 x 2 Equilibrium point x 1 u(t) 0, 0 1 x 1 e 0 0 we get 3 x x 2 e 2 e 0 2 Set 6
Definition: An equilibrium state xe of an autonomous system is stable in the sense of Lyapunov if for every 0 , exist a ( ) 0 such that x 0 xe x(t, x 0 ) xe for t t 0 x 2 x(t) xe x 1 x 0 7 Definition: An equilibrium state xe of an autonomous system is asymptotically stable if (i) it is stable (ii) there exist a e 0 such that x 0 xe e x(t) xe 0, as t x 2 e xe x 1 x 0 x(t) 8
Instability in 1 D Stability Concept in 1 D Asymptotic Stability in 1 D 9 Lyapunov Theorem x& f (x) Consider the system xe 0 Eq. State : (6. 1) Q f (0) 0 A function V (x ) is called a Lapunov fuction V (x ) if (1) V ( x) 0, x 0 (2) V (0) 0 for x 0 d. V ( x) f ( x) 0 (3) dt dx Then eq. state of the system (6. 1) is stable. Moreover, if the Lyapunov function satisfies d. V ( x) dt 0, x 0 and d. V ( x) 0 x 0 dt Then eq. state of the system (6. 1) is asy. stable. 10
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