Signal Processing and Linear Systems-B.P.Lathi copy

# T 194exp 2t9exp t 0 assessment o f t he classical

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Unformatted text preview: e, were t o b e d isturbed slightly, i t w ould eventually r eturn t o i ts original equilibrium s tate if left t o itself. I n t his case, t he cone is s aid t o b e in s table e quilibrium. I n c ontrast, i f t he cone s tands o n i ts a pex, t hen t he s lightest d isturbance will cause t he cone t o move f arther a nd f arther away from i ts e quilibrium s tate. T he cone in t his case is said t o b e i n a n u nstable e quilibrium. T he cone lying o n i ts side, if 148 2 Time-Domain Analysis of Continuous-Time Systems disturbed, will neither go back to the original s tate nor continue t o move farther away from t he original state. The cone in this case is said t o be in a n eutral e quilibrium. Let us a pply these observations t o systems in general. I f, in t he absence of a n e xternal input, a system remains in a particular s tate (or condition) indefinitely, then t hat s tate is said t o be an e quilibrium s tate o f t he s ystem. For an LTI system this equilibrium s tate is t he zero state, in which all initial conditions are zero. Now s uppose an LTI system is in equilibrium (zero state) and we change this s tate by creating some nonzero initial conditions. By analogy with the cone, if the system is s table i t should eventually r eturn t o zero state. In other words, when left to itself, t he s ystem's o utput due t o t he nonzero initial conditions should approach o as t ~ 00. B ut t he system o utput generated by initial conditions (zero-input response) is m ade up of its characteristic modes. For this reason we define stability as follows: a s ystem is ( asymptotically) s table if, and only if, all its characteristic modes ~ 0 as t ~ 0 0. I f any of t he modes grows without bound as t ~ 0 0, t he system is u nstable. T here is also a borderline situation in which the zero-input response remains bounded (approaches neither zero nor infinity), approaching a constant or oscillating with a constant amplitude as t ---+ 0 0. For this borderline situation, t he s ystem is said t o b e m arginally s table or just stable. I f a n LTIC system has n d istinct characteristic roots AI, A2, . .. , An, t he zeroinput response is given by n Yo(t) = 2:~&gt;jeAJt (2.62) j =1 t Jmag R eA&lt;O stable marginally stable .... R eA=O F ig. 2 .15 Characteristic roots location and system stability. t ~ 0 0, t ke At ---+ 0, if ReA &lt; 0 (A in LHP). Therefore, repeated roots in LH~ do not cause instability. B ut when the repeated roots are on the imaginary aXIs (A = jw), t he corresponding modes t ke iwt ~ppro~~h infi?ity as t ---+ 0 0. Therefore, repeated roots on t he imaginary axis cause instabIlity. ~Igure 2.1.6 sh.ows characteristic modes corresponding t o characteristic roots a t various locatIOn I n t he com?le.x plane. Observe the central role played by t he c haracteristic roots or characteristIc modes in determining the system's stability. We have shown elsewhere [see Eq. (B.14)] lim t -+oo eM = {O 00 149 2.6 System Stability To summarize: Re A &lt; 0...
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## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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