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