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
3.2 Lyapunov StabilityThe concept of stability is extremely important becausealmost every workable system is designed to be stable. If asystem is not stable, it is usually of no use in practice. +The most important approach for studying thestability of control system is theLyapunovstability theory, which is introduced by theRussian mathematician Alexandr MikhailovichLyaponov in the late 19th century
3.2 Lyapunov Stability
3.2 Lyapunov Stability3.2.1 Equilibrium PointThe concept of stability plays an important role for thesystem analysis and synthesis. In the general theory, where,thetime varying and nonlinear systems are considered,definitions of stability are rather involved and the distinctionsare subtle.In the studying for the stability theory, the equilibrium pointof a system is an important concept. In fact, Lyapunovintroduced the concepts of the stability in the vicinity of anequilibrium point
3.2 Lyapunov Stability 3.2.1 Equilibrium Point
3.2.1 Equilibrium PointDefinition 3.6 Suppose an autonomous (or unforced) systemis described by.X(t) = f[X(t),0,t)X(t) = f[X(t),t)orThe state X。 is called the eequilibriumpointtof the systemif it satisfies -for t ≥tof(X.,t) = 0It should be noted that the definition is applicable for bothlinear system and non-linear system
3.2.1 Equilibrium Point
3.2.1 Equilibrium PointConsider the LTI systemX(t) = AX(t)onlyif matrix A is nonsingular, then the system has theequilibrium point X。=O. Otherwise, the system may havemany equilibrium points. +Generally, the non-linear system may have many equilibriumpoints also
3.2.1 Equilibrium Point
Example 3.5 Determine the equilibrium points of thefollowing systemxi = -X2X2 = -Xi - X2Solution The system is a LTI system. By lettingX = [xix, J , the system can be described by.0X = AX =X-1 Obviously, the system matrix A is nonsingular. By lettingX = 0, it can be deduced that the origin X。 = 0 of the statespace is the only equilibrium point of the system
Example 3.6 Determine the equilibrium points of thefollowing systemX = -XiX2 = Xi + X2 -x3SolutionObviously, the system is a non-linear systemg X=[x x,] and letting X=o ， threeBy lettingequilibrium points of the system can be obtained as[o][o]0X.XX001
3.2.1 Equilibrium PointIn fact, the origin of the state space is always an equilibriumpoint for systems, although it needs not to be the only one.Only isolated equilibrium points will be considered in thischapter and any nonzero isolated equilibrium point can betransferred to the origin by a change of variableX=X-XFor this reason it is assumed in the sequel that X。 =0 andit will only be considered in this chapter
3.2.1 Equilibrium Point
3.2.2 Concepts of Lyapunov StabilityFor the continuous-time system with zero input, the studyingfor the stability deals with the question that whether the statewhich is perturbed from its equilibrium point Xe at time to,return to X., or remain close to X., or diverge from it.Definition 3.7 The equilibrium point X is said to be stablei.s.L,if for any given real ε>O, there exists a realS(e,t.) > 0 so that if the initial state satisfy X。- Xe≤8(8,to)forthen.Vt ≥toX(t;Xo,to) - Xel≤8
3.2.2 Concepts of Lyapunov Stability
3.2.2 Concepts of Lyapunov StabilityLyapunov stability means that we are able to select a boundon initial condition that will result in the state trajectoryremains within a chosen finite limit. The geometrical implication of Lyapunov stability is shownX。-Xel ≤ S(c,to)statetrajectoryX(t; Xo,to)- Xell≤ ?
3.2.2 Concepts of Lyapunov Stability