CHAPTER3 STABILITY OF THE CONTROLSYSTEMCONTENT>3.1 TheBasics of StabilityTheory inMathematics> 3.2 Lyapunov Stability> 3.3 Lyapunov Stability Theory> 3.4 Application of Lyapunov 2nd Method tothe LTI System
CHAPTER3 STABILITY OF THE CONTROL SYSTEM • CONTENT 3.1 The Basics of Stability Theory in Mathematics 3.2 Lyapunov Stability 3.3 Lyapunov Stability Theory 3.4 Application of Lyapunov 2nd Method to the LTI System
3.4 Application of Lyapunov 2nd Method to theLTI SystemAs mentioned above. the Lyapunov stability theoremsprovidewhich are notonly the sufficient conditions,necessary, for the stability of the non-linear systems. Yet, for the LTI systems, the Lyapounov stability theoremscan be relaxed to the form which is sufficient and necessary
3.4 Application of Lyapunov 2nd Method to the LTI System
3.4 Application of Lyapunov 2nd Method to theLTI SystemTheorem 3.11 Consider the LTI homogeneous systemdescribed by X(t) = AX(t)with the initial condition X(O) = X。: The equilibrium pointX, = 0 is asymptotically stable i.s.L, iff for any symmetricpositive definite matrixQ , the following Lyapunovequation ATP+PA=-Qhas theuniquesymmetric positive definite solution matrix P
3.4 Application of Lyapunov 2nd Method to the LTI System
3.4 Application of Lyapunov 2nd Method to theLTI SystemProof.(1) Necessitybe omitted(2) SufficiencyBased on the symmetric positive definite matrix P , aquadratic form function can be construct as.V(X)= XIPXIt is clear that V(X) is positive definite for X+ O, andV(O) = 0. The derivativeV(X) = XT PX + X'PX = XTA'PX + X'PAX= XT(AIP+ PA)X =-XIOX
3.4 Application of Lyapunov 2nd Method to the LTI System be omitted
3.4 Application of Lyapunov 2nd Method to theLTI SystemProof.(l) Necessity(2)SufficiencyX =-XTOXBecause the matrix Q is any symmetric positive definite, soV(X) is negative definite for X+ 0 and V(0)= 0. So, the equilibrium point X,=O is asymptotically stablei.s.L and the quadratic form function V(X)= XTPX is aLyapunov function
3.4 Application of Lyapunov 2nd Method to the LTI System
Example 3.12 Consider the LTI system-11X:X2-3Determine the stability of the system at its equilibrium point.SolutionObviously, the system matrix11A=2 -3is nonsingular. So, the origin X. = 0 is the only equilibriumpoint of the system..P11P12Takeand denote P by. PQ=1_P12P22
Example 3.12 Consider the LTI system-1 1X =X2-3Determine the stability of the system at its equilibrium point.P11P12Q=I and denote P by. P=Solution TakeLP12P22 From the Lyapunov equation A' P+ PA = -Q, we have0-1 2'1./P11P12P11 Pi2+2-30-1￥1-3P12P22]P12p22The solution matrix is[7/4 5/8PL5/83/8
Example 3.12 Consider the LTI system-11X:X2 -3Determine the stability of the system at its equilibrium point.Solution[7/45/8]P=[5/8 3/8It's leading principal minor determinants are.[7/4 5/8]17△, =|7/4 >0>0△5/8 3/864According to the Sylvester criterion, P is positive definiteThen the equilibrium point X, = 0 is asymptotically stable
Example 3.12 Consider the LTI system-1 1X :X2 -3Determine the stability of the system at its equilibrium point.SolutionFurthermore, X, = 0 is the unique equilibrium point of theLTI system, so X,=0 is globally asymptotically stablei.s.L..The Lyapunov function can be constructed as357[7/4 5/822V(X) = XT PXPX,x23844[5/83/8