Time Response*, ME451Instructor: Jongeun Choi* This presentation is created by Jongeun Choi and Gabrial GomesZeros and poles of a transfer function• Let G(s)=N(s)/D(s), then– Zeros of G(s) are the roots of N(s)=0– Poles of G(s) are the roots of D(s)=0Re(s)Im(s)Theorems• Initial Value Theorem• Final Value Theorem– If all poles of sX(s) are in the left half plane (LHP), thenDC gain or static gain of a stable system0 0.5 1 1.5 2 2.5 300.20.40.60.811.21.4DC Gain of a stable transfer function • DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response• Use final value theorem to compute the steady state of the unit step responsePure integrator • ODE :• Impulse response :• Step response :• If the initial condition is not zero, then :Physical meaning of the impulse responseFirst order system• ODE :• Impulse response :• Step response :• DC gain: (Use the final value theorem)RCFirst order system• If the initial condition was not zero, thenPhysical meaning of the impulse responseMatlab Simulation• G=tf([0 5],[1 2]); • impulse(G)• step(G)• Time constant0 0.5 1 1.5 2 2.5 300.511.522.533.544.55Impulse ResponseTime (sec)Amplitude0 0.5 1 1.5 2 2.5 300.511.522.5Step ResponseTime (sec)AmplitudeFirst order system responseSystem transfer function :First order system responseSystem transfer function : Impulse response :First order system responseSystem transfer function : Impulse response :First order system responseSystem transfer function : Impulse response : Step response : 0 100 200 300 400 500 6000102030405060708090100Step ResponseTime
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