With an ideal op-amp, we have vo = A(vin − v− ), The circuit diagram is shown in Figure E2.5. The final value is computed using the final value theorem: 6(s + 50) =1. Using the Laplace transform table, we find that y(t) = 1 + 0.2e−30t − 1.2e−10t. The partial fraction expansion of Y (s) is given by A1 A2 A3 Y (s) = + + s s + 30 s + 10 where A1 = 1, A2 = 0.2 and A3 = −1.2. 2 1.5 Spring b reaksįIGURE E2.3 Spring force as a function of displacement. The spring constant for the equilibrium point is found graphically by estimating the slope of a line tangent to the force versus displacement curve at the point y = 0.5cm, see Figure E2.3. Thus, the linear approximation is computed by considering only the first-order terms in the Taylor series expansion, and is given by ∆R = −135∆T. A plot y versus r is shown in Figure E2.1.ĭefine f (T ) = R = R0 e−0.1T and ∆R = f (T ) − f (T0 ), ∆T = T − T0. įIGURE E2.1 Plot of open-loop versus closed-loop.įor example, if r = 1, then e2 + e − 1 = 0 implies that e = 0.618. Modern Control Systems 13th Edition Dorf SOLUTIONS MANUAL Full clear download (no formatting errors) at: Įxercises We have for the open-loop y = r2 and for the closed-loop e = r − y and y = e2.
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