chap13

Modern Control Engineering Chapter 13: LyapunovLyapunovStability Analysis Dr.Y. X. Liu Department of Automation Inner Mongolia University 2004-4-26Y. X. Liu, Inner Mongolia University2 Lecture Content LyapunovLyapunov Stability analysis Linear, Time-Invariant system Model-reference control system Quadratic optimal control 2004-4-26Y. X. Liu, Inner Mongolia University3 LyapunovStability Analysis For a given control system, stability is usually the most important thing to be determined. For linear and time invariant ?Rouths stability criterion ?Nyquiststability criterion For nonlinear and/or time-varying ?The stability criteria above do not apply ?The second method of Lyapunov(Lyapunovdirect method) The second method of Lyapunov ?Determination stability of linear, nonlinear, time invariant, and time-varying ?Solving quadratic optimal control problem 2004-4-26Y. X. Liu, Inner Mongolia University4 Lyapunovmethods In 1892 A. M. A. M. LyapunovLyapunovpresented. Determining the stability of dynamic system described by ordinary differential equations The first method of Lyapunov (Lyapunov Indirect Method) ?Determining the stability of dynamic system depend on the form of solutions of the ordinary differential equations. The Second method of Lyapunov (Lyapunov Direct Method) ?Determine the stability of the system without solving the state equation. A.M.Lyapunov 2004-4-26Y. X. Liu, Inner Mongolia University5 The system defined by Equilibrium state ?A state xewhere f(xe,t)=0 for all t. is called an equilibrium state. ?If the system is linear time invariant (f(x,t)=Ax ), then there exists only one equilibrium state if A is nonsingular, and there exist infinitely many equilibrium states if A is singular. ?For nonlinear system, there may be one or more equilibrium states, and correspond to the constant solution of the system (x=xefor all t). ?Any isolated equilibrium state can be shift to the origin of the coordinates, or f(0,t)=0 for all t, by translation of coordinates. () ()() () ()() 00000 n ,t;t,tt; xxxvectornt vectorn t xxx xf x xfx = = = = 0 21 thussolution denote 1 .13 , , L & 2004-4-26Y. X. Liu, Inner Mongolia University6 Stability in the sense of Lyapunov LyapunovStability: 2004-4-26Y. X. Liu, Inner Mongolia University7 Definiteness Positive Definite Positive Semi-definite Negative Definite and Negative Semi-definite Indefiniteness ?V(x) Both positive and negative values, no matter how small the region D is. 2004-4-26Y. X. Liu, Inner Mongolia University8 Hermitian(Quadratic) form Hermitianform ?Here x is complex n-vector and P is a Hermitian matrix. (For a real vector x and a real symmetric matrix P, Hermitian form x*Px becomes equal to Quadratic form xTPX.) ( ) = nnnn n n x x pp pp xxPxVM L MM L L 1 1 111 1 * xx 2004-4-26Y. X. Liu, Inner Mongolia University9 Sylvesters criterion 000 1 111 2212 1211 11 nnn n pp pp pp pp p L MM L L Positive definite ?V(x) =x*Px be positive definite is that all the successive principal minors of P be positive, that is Positive Semi-definite ?P is singular and all the principal minors are nonnegative. Negative definite ?the successive principal minors of P be negative and positive in turn. negative Semi-definite 2004-4-26Y. X. Liu, Inner Mongolia University10 LyapunovDirect If x = 0 is an equilibrium for the system and there exists a continuously differentiable function, V(x), which contains the equilibrium: ?x=0 is stable ?x=0 is asymptotically stable If x= 0 is asymptotically stable and, in addition, V is radially unbounded then the x is globally asymptotically stable 2004-4-26Y. X. Liu, Inner Mongolia University11 Krasovskiismathod ( ) () () ( )( )( )xFxFxFdefine x f x f x f x f x f x f x f x f x f xx ff xF n nn n n n n += = = * 2 2 1 2 2 2 1 2 1 2 1 1 1 1 1 , , L L The system and assume that f(0)=0 and f(x) is differentiable with respect to xi,i=1,2,n. Define the Jacobian matrix: ?is Hermitian (symmetric if F(x) is real). If is negative definite, then the equilibrium state x=0 is asymptotically stable. ( )xF ( )xF ( )( ) ( ) ( ) ( )largein stateally asymptotic is0 function Lyapunov * = = xxasxfxf xfxfxV * 2004-4-26Y. X. Liu, Inner Mongolia University12 Lyapunovstability analysis of linear time-invariant system For system , a necessary and sufficient condition that the equilibrium state x=0 be asymptotically stable in the large is that, given any positive-definite Hermitian (or real symmetric) matrix Q, there exits a positive- definite Hermitian (or real symmetric) matrix P such that ?Lyapunovfunction for this system is x*Px. ?If does not vanish identically along any trajectory, then Q may be choose semidefinite, and Q satisfiers: xxA=& QPAPA=+ * ( )xx QxV * = & n AQ AQ Q rank n = 1 2 1 2 1 2 1 2004-4-26Y. X. Liu, Inner Mongolia University13 Model-reference control system 1.Is a negative-definite matrix. 2.The control vector u can be chosen to make the scalar quantity M nonpositive () () ( )( )() ()()BvtuAPM MPAPAxVPVLyapunov BvtuAA BvA tu d dd += += += += = , 2 ,define referenceModel ,equationPlant * * xfxe eeeee xfxeexxe xx xfx & & & & QPAPA=+ * x uxdCommand input v Plant Model-reference system controller 2004-4-26Y. X. Liu, Inner Mongolia University14 Parameter-optimization System, where all eigenvalues of A have negative real parts, or origin x=0 is asymptotically stable. It is desired to minimize the following performance index: ?Since X(0) is the given initial condition and Q is also given. P is a function of the elements of A. Hence, to minimize the performance index Jwill result in the optimal value of the adjustable parameter. xxA=& ( )( )( )000 * * 0 * xx xxxx xx PJthereforexfor QPAPAobtainP dt d Qassume dtQJ = =+= = 2004-4-26Y. X. Liu, Inner Mongolia University15 Quadratic optimal control BuA += x x & -K xu Given the system equations, determine the matrix K of the optimal control vector u, so as to minimize the performance index J. () ()() ()()()()() ( )( )( ) PBRK QPBPBRPAPA dtPJ RKKQBKAPPBKAP dt d RKKQ dtRKKQJBKA dtRuuQJKuB