3-5 Linear System Stability AnalysisIn this chapter：qqqqqDefinition for StabilityIFF Conditions for LSSRouth-Hurwitz CriterionRouth CriterionSpecial CasesRouth CriterionApplications1CEPECH3 Time Domain Analysis 1. Definition for StabilityStability：disturbance on equilibrium deviation errorcreation disturbance off recover to equilibriumEx1. Pendulum stable equilibriumvsunstable equilibriumglobal stabilityLinear Systemvslocal stability2CEPECH3 Time Domain Analysis Definitions:一个稳定的系统定义为在有界输入作用下，其输出响应也有界的动态系统。A stable system is a dynamic system with a bounded responseto a bounded input.Stability A performance measure of a system. A system is stable if allthe poles of the transfer function have negative real parts.The concept of stability can be illustrated by considering a right circularcone placed on a plane horizontal surface. If the cone is resting on its baseand is tipped slightly, it returns to its original equilibrium position. Thisposition and response are said to be stable. If the cone rests on its side andis displaced slightly, it rolls with no tendency to leave the position on its side.This position is designated as the neutral stability. On the other hand, if thecone is placed on its tip and released, it falls onto its side. This posi-tion issaid to be unstable. These three positions are illustrated in Fig. 6.1.FIGURE 6.1 Thestability of a cone.3CEPECH3 Time Domain Analysis Example of an unstable system is shown in Fig. 6.3. The firstbridge across the Tacoma Narrows at Puget Sound, Washington, wasopened to traffic on July 1, 1940.塔科马峡谷FIGURE 6.3 TacomaNarrows Bridge (a)as oscillation beginsand (b) at(a)(b)catastrophic failure.4CEPECH3 Time Domain Analysis 2. IFF Conditions for LSSThe bridge was found to oscillate whenever the wind blew. After fourmonths on November 7, 1940, a wind produced an oscillation thatgrew in amplitude until the bridge broke apart. Figure 6.3(a) showsthe condition of beginning oscillation; Fig. 6.3(b) shows thecatastrophic failure.Equilibrium Stability 李雅普诺夫（1892，Lyaponov)Motion Stability 无外作用下齐次方程解的行为Linear System Stability (LSS)stabilityunit impulse responsecharacteristic roots /eigenvaluesIFF Conditions for Linear System Stabilityall the roots of closed loop characteristic equation has negative realorall the closed loop poles locate on left-half S plane5CEPECH3 Time Domain Analysis 3. Routh-Hurwitz Criterion（1）Hurwitz CriterionD s = a s + a sn-1 +K+ a s + a = a nCharacteristic equation： ( )0,( 0)001n-1na a a135a a a024A necessary condition:a1 a3a a .02a ,a ,K,a 01 2 n.a0.an6CEPECH3 Time Domain Analysis 3. Routh-Hurwitz CriterionPrime sequence for each order:a a a513D1= a1a a a4a1 a302a1 a3D2=a0 a2a a .02a a a513a1 .D3=a0.0a1 a3.LanDnHurwitz criterion:IFF Stability Condition: D =0, i 1,2,Lni7CEPECH3 Time Domain Analysis 3. Routh-Hurwitz Criterion4+3+2+ + =Ex3 2s s 3s 5s 10 0=1, D = 153 = -7,12005 0 03 10 01 5 02 3 10D1D322D4=15 0= 2 3 10 = -45, D = -450401 5unstable system4+2+ + =Ex4 2s 3s 5s 10 0Qa1 = 0, system is unstable8CEPECH3 Time Domain Analysis 3. Routh-Hurwitz Criterion-2 - s - =Ex 5 2 s 1 0 2 + s + =2s1 0= 101= 1, D = 11D 22stable systemai 0，then system is stableFor 2nd order system, if9CEPECH3 Time Domain Analysis 3. Routh-Hurwitz CriterionD s = a s + a sn-1 +L+ a + a = a n0 ( 0)(2)Routh Stability Criterion:()01n-1n0s ns n -1a 2a 3a a 04a 4a 5a a 06a 0a1a a 02a6a7LLs n - 2c13 =3a a1a1c23 =5a a1a1c33 =7a a1a1a a13a a15c14 = cc13c24 = cc132333c13c13Routh tableMMM0s a n10CEPECH3 Time Domain Analysis 3. Routh-Hurwitz CriterionRouth Criterion：(1) Stability First column in Routh table 0;2) If first column has element 0s04System is unstablewith 2 roots with PRP13CEPE