MECH 466 - Lecture6-Stability-2009W

Zero roots of ns zero ns system is bibo stable pole

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Unformatted text preview: : roots of n(s) Zero n(s) system is BIBO stable Pole : roots of d(s) Pole d(s) All the poles of G(s) are in the open left G(s) half of the complex plane. Characteristic polynomial : d(s) Characteristic d(s) system is asymptotically stable Characteristic equation : d(s)=0 Characteristic d(s)=0 2008/09 MECH466 : Automatic Control 7 2008/09 MECH466 : Automatic Control 8 2 Idea of stability condition Remarks on stability Example For a general system (nonlinear etc.), BIBO For stability condition and asymptotic stability condition are different. For linear time-invariant (LTI) systems (to which For timewe can use Laplace transform and we can obtain a transfer function), the conditions happen to be the same. In this course, we are interested in only LTI In systems, we use simply “stable” to mean both stable” BIBO and asymptotic stability. Asym. Stability: Asym. (U(s)=0) U(s)=0) BIBO Stability: (y(0)=0) Bounded if Re(α)>0 (α)> 2008/09 MECH466 : Automatic Control 9 2008/09 Remarks on stability (cont’d) G(s) has no pole in the open RHP (Right Half Plane), & G(s) G(s) has at least one simple pole on jw-axis, & G(s) jwG(s) has no multiple pole on jw-axis. G(s) jw- NOT marginally stable Unstable if a system is neither stable nor Unstable marginally stable. 2008/09 MECH466 : Automatic Control 10 Stability summary Marginally stable if Marginally Marginally stable MECH466 : Automatic Control Let si be poles of G. Then, G is … (BIBO, asymptotically) stable if (BIBO, Re(si)<0 for all i. marginally stable if marginally Re(si)<=0 for all i, and Re(s simple root for Re(si)=0 simple unstable if unstable it is neither stable nor marginally stable. 11 2008/09 MECH466 : Automatic Control 12 3 Mechanical examples: revisited K f(t) f(t) M Examples Stable/marginally stable /unstable f(t) f(t) M x(t) x(t) ? x(t) x(t) Poles= stable? Poles= stable? ? K B M f(t) f(t) x(t) x(t) 2008/09 M B Poles= stable? ? f(t) f(t) ? x(t) x(t) Poles= stable? MECH466 : Automatic Contr...
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