Introduction. - The Second Lyapunov Method. - The First Lyapunov Method

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1 Stablty Analyss A. Khak Sedgh Control Systems Group Faculty of Electrcal and Computer Engneerng K. N. Toos Unversty of Technology February

2 Introducton Stablty s the most promnent characterstc of dynamcal systems Dynamcal Systems Stablty from: NLTV to LTV and LTI Stablty of LTI dynamcal systems: The Egenvalue concept Methods of examnng LTI dynamcal System Stablty Lyapunov : Stablty analyss from NLTV to LTV and LTI Two man Lyapunov Methods: - The Frst Lyapunov Method - The Second Lyapunov Method 2

3 Stablty Defntons x() t = f [ x(), t u(),] t t Equlbrum Pont Concept 3

4 Defnton Stable Equlbrum Pont n the sense of Lyapunov Defnton Asymptotcally Stable Equlbrum Pont 4

5 Defnton Doman of Attracton Defnton Globally Asymptotcal Stable Equlbrum Pont Defnton Unstable Equlbrum Pont Defnton Internal Stablty Defnton BIBO Stable Defnton T-Stable or Totally Stable 5

6 LTI System Stablty ( ) x () t = Ax t + Bu() t y() t = Cx() t Autonomous Systems wth Dstnct Egenvalues Yelds: x t ( ) n = µ = 1 Theorem A System s Stable n Lyapunov Sense f and only f: - All Egenvalues have Non-Postve Real Parts - Complex Egenvalues are Smple e λ t e 6

7 Theorem An LTI system s Asymptotcally Stable f and only f t s Globally Asymptotcally Stable. Theorem An LTI system s Asymptotcally Stable f and only f t s Egenvalues have Strctly Negatve Real Parts. Theorem An LTI system s BIBO (Bounden Input-Bounded Output) Stable f and only f All transfer Functon Poles have Strctly Negatve Real Parts. 7

8 Theorem If an LTI system s Controllable and Observable, then the followng terms are Equvalent - System s Totally Stable. - Systems Zero State Response s BIBO Stable. - Transfer Functon Poles have Negatve Real Parts. - Egenvalues of the State Matrx have Negatve Real Parts. 8

9 Lyapunov s Frst Method Stablty Analyss of Nonlnear Systems n Operatng Ponts After Lnearzaton Theorem System x( t) = f [ x( t ),0] and x e x () t = Ax () t Where, A = Jacoban of f at x e That s, State Matrx Asymptotc Stablty yelds Nonlnear System Asymptotc Stablty n a specfc operatng pont. 9

10 Lyapunov s Second Method A General Stablty Analyss of Dynamcal Systems from an Internal System Pont of vew. Lyapunov Theory s Common Practce n Analyss and Desgn of Dynamcal Systems. Lyapunov Canddate Functon. Exstence of Lyapunov Functon. Unqueness of Lyapunov Functon? 10

11 Mathematcal Prelmnares Quadratc Forms V ( x) n n = = 1 j= 1 a x x j j x = x1 x n (,, ) [ ] V ( x) = x x x 1 2 T x Ax = = x, Ax n aj R a11 a12 a1 n x1 a21 a22 a 2n x 2 a a a x n1 n2 nn n Symmetrc Matrx 11

12 Defnton Postve Defnte Scalar Functon V ( x) n vcnty of S Defnton Postve Sem Defnte Scalar functon V ( x) n vcnty of S Defnton Negatve Defnte Scalar Functon V ( x) n vcnty of S Defnton Negatve Sem Defnte Scalar functon V ( x) n vcnty of S Defnton Indefnte Scalar functon V ( x) n vcnty of S Scalar Functon Sgn Determnaton Methods. 12

13 Lyapunov s Second Method Classcal Mechancal Theory Prncpal: Oscllatng Systems wth No Stmulatng Input are Stable f ther Total Energy s Contnuously Decreasng. Lyapunov Theory Based on Energy Functon Generalzed Energy Functon or Lyapunov Functon Lyapunov Canddate Functon Propertes of Lyapunov Canddate Functon x = ( x1,, x n ) V ( x) ( ) dv x dt 13

14 Lyapunov Functon Illustraton 14

15 Theorem System s Asymptotcally Stable n Localty of Equlbrum Pont n orgn f there exsts a Scalar Functon such that: V ( x) > 0 for x 0 V (0) = 0 dv ( x ) dt < 0 for x 0 15

16 What f Lyapunov Functon s Dervatve s Negatve Sem Defnte? Do System Trajectores Zero Lyapunov Functon s Dervatve? Revsed Asymptotc Stablty Condton V ( x) 0, V ( x ) = 0 x f ( x( t)) 16

17 Theorem A system s Asymptotcally Stable n vcnty of Orgn f t has only One Equlbrum Pont, Furthermore there exsts a Scalar Functon so that: Functon s Contnuous on entre State Space and ts Partal Dervatves are also Contnuous V ( x) > 0 for x 0 V (0) = 0 V ( x) for V x ( x) 0, V ( x ) = 0 x f ( x( t)) 17

18 Theorem (System Instablty) A system s unstable n the vcnty of orgn f there exsts a scalar functon that: V ( x) 0, V (0) = 0, V ( x) Contnuous n S wth contnuous partal dervatves V ( x) > 0, V (0) = 0 18

19 Lyapunov Stablty Analyss for Lnear Tme Invarant Systems Lnear Tme Invarant System: x() t = Ax() t Necessary and Suffcent condtons for Lnear Tme Invarant Stablty based on Egenvalues and Characterstc Equaton Lyapunov Algebrac Method for Stablty Analyss of Lnear Tme Invarant Systems 19

20 General Structure T x() t = Ax() t and V ( x) = x Px T T V ( x) = x Px + x P x T T = ( Ax) Px + x P( Ax) T T = x ( A P + PA) x = T x Qx Symmetrc Matrx Symmetrc Matrx T A P + PA = Q Lyapunov Equaton 20

21 Stablty Evaluaton Usng Lyapunov s Second Method Step 1: PD or PSD Matrx Selecton Step 2: Solvng the Lyapunov Equaton Step 3: Sgn Determnaton of the Matrx Step 4: Stablty Evaluaton from the Sgn of the Matrx Q A T P + PA = Q P Matrx n Step 1 PD or PSD? 21

22 Theorem The followng system x () t = Ax() t s asymptotcally stable f and only f Q PD Matrx P PD Matrx Proof See Ref [1] PSD Case? 22