《现代控制理论》课程教学资源课程教学资源——教学课件_Chap3_3.3 Lyapunov Stability Theory

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

3.3 Lyapunov Stability TheoryLyapunov's work about the stability includes two methodsTesting for stability by considering the linear approximationto a differential equation is referred to as Lyapunov firstmethod (i.e. the linearization method or the indirect method)Using the idea of the Lyapunov function for a direct attack onthe stability question is Lyapunovmethod.secondCorrespond with the linearization method, the method iscalled the direct method

3.3 Lyapunov Stability Theory

3.3.1 Lyapunov First MethodAs the discussion above, a nonlinear system may have morethan one equilibrium point. The nonlinear system can beexpanded in a Taylor series about the equilibrium point (thein a smallorigin is always selected in this chapter)neighborhood of it.Assume that the nonlinear system described byX(t) = f[X(t),t)can be expanded about the equilibrium point X。 in thefollowing Taylor series

3.3.1 Lyapunov First Method

3.3.1 Lyapunov First MethodX(t) = f[X(t),t)afX = f(X.)+.(X - X.)+ g(X)axtIX=Xwhere g(X) is the summation of the higher-order terms intheTaylorseries.aX= X-XLettingand neglecting the summation of the higher-order terms in theTaylor series yield the linearized differential equation X=_af: A = J.AXaxtX=Xe

3.3.1 Lyapunov First Method

3.3.1 Lyapunov First MethodX(t) = f[X(t),t)afX = f(X.)+.(X -X.) + g(X)axtIX=XeA=_f:AX =J.AXLetting ^X = X- XaxtX=Xeaf1aSwhereax=OxnJx=xis called a Jacobimatrix, and f is the ith row of f(X)

3.3.1 Lyapunov First Method

3.3.1 Lyapunov First MethodTheorem 3.6 For a nonlinear system described byX(t) = f[X(t),t](1)If all eigenvalues of the linearized differential equationafir=_of.^X = J. AXJaxTX=XeaxAX = X-XoxnX=Xhave negative real part, then the equilibrium point X。 isasymptotically stable in a small neighborhood of it

3.3.1 Lyapunov First Method

3.3.1 Lyapunov First MethodTheorem 3.6 For a nonlinear system described byX(t) = f[X(t),t)(2) If there is at least one eigenvalue of the linearizeddifferential equation4=.0f. ax =J.AXIaxtIX-X.Ox,X=Xhas positive real part, then the equilibrium point X。 isunstable in a small neighborhood of it

3.3.1 Lyapunov First Method

3.3.1 Lyapunov First MethodTheorem 3.6 For a nonlinear system described byX(t) = f[X(t),t)(3) If some eigenvalues of the linearized differentialX=fequation .ax =J.axaxtIX=Xehave zero real part and others have negative real part, thestability of the equilibrium point X。 is related to thesummation of the higher-order terms in the Taylor seriesg(X). In the case, the equilibrium point X。 may be stable.asymptotically stable or unstable

3.3.1 Lyapunov First Method

Example 3.7Consider the system described byX=X-XX2X2 =-X2 + XiX2Determine the equilibrium points and the stability of thesystem on them..Obviously, the system is a non-linear system.SolutionBy letting X = [xi x2」, the system can be described by[fi(X)]_[xi -xix2X = f(X) :fi(X)]L-x2 + xix2Letting X = O, two equilibrium points of the system can be[1][o]obtained as.X.Xel =2[1][0

Example 3.7Consider the system described byX=X-XX2X2=-X2+XiX2SolutionThe Jacobi matrix can be calculated asafiafiarOx,1-x,-Xi1af2af.-1+xiX2axOx.The linearized differential equation of the system on Xcan be obtained as[o]0aXX.:aX00