Consider the digital feedback control system shown in Figure 4.3 where R(s) and C(s) are reference input and system output respectively, E(s) is tracking error, and the sample period is T = 0.2 secs. [Diagram showing a feedback control loop with R(s), a comparator, E(s), a sampler T, D(z), ZOH, G(s), and output C(s), with G?(z) indicated below ZOH and G(s)] Figure 4.3 The discrete transfer function of the plant G(s) when preceded by a zero order hold is given by G?(z) = (0.002(z + 0.9934)) / ((z - 1)(z - 0.9802)) A continuous time forward path compensator, D(s) = 2.1((s + 0.18) / (s + 1.83)) has been designed applying continuous controller design methods to the process transfer function G(s). (a) Use the matched pole-zero mapping method to show a corresponding candidate digital forward path compensator is given by D(z) = 1.806((z - 0.965) / (z - 0.694)) [7 marks] (b) Determine the steady-state tracking error of the feedback control system in Figure 4.3 in response to a unit step reference signal.
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Consider the digital control system above with G(s) = 0.6 e^{-T_d s} / (0.3 s + 1) where the time-delay is T_d = 20 ms and the sampling period is T = 20 ms. Then, answer the following questions. a) Find the discrete-time open loop transfer function D(z)Ḡ(z) where D(z) = K and Z {G_{ZOH}(s)G(s)} = Ḡ(z). ( Hint: Use Z {0.6 / (0.3s+1) s} = 0.6 z / (z-1) - 0.6 z / (z-0.935) for T = 20 ms. ) b) Sketch the change of the location of the closed loop poles of the system (root locus plot) in z-plane with respect to K for 0 < K < ∞. c) Find the critical gain value K_{cr} such that the closed-loop system is unstable for K > K_{cr}. Use the root locus plot you found in (b). d) Design a digital controller D(z) which satisfies the closed-loop response without overshoot and makes the steady-state error zero to step inputs.
Adi S.
Q4) Consider the system shown in Figure Q3. This is a PID control of a second-order plant G(s). Assume that disturbances d̄(s) enter the system as shown in the diagram. It is assumed that the reference input r̄(s) is normally held constant, and the response characteristics to disturbances are a very important consideration in this system. a) In the absence of the reference input i.e. r̄(s) = 0, derive the closed-loop transfer function between ȳ(s) and d̄(s). b) The performance specification requires that the unit step disturbance response be such that the settling time be approximately half a second and the system has reasonable damping. We may interpret the specification as Ζ = 0.8 and ωn = 8 for the dominant closed-loop poles. We may choose the third pole at s = -10 so that the effect of this real pole on the response is small. Derive the required characteristic polynomial that satisfies the above performance specification. c) Using the result in a) and b), calculate the controller parameters ab, a + b and K. Hence write down the controller transfer function C(s).
Consider the second-order plant with the transfer function G(s) = 1 / ((s + 1)(5s + 1)) and in a unity feedback structure. (a) Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P [Dc = kp], PD [Dc(s) = kp + kDs], and PID [Dc(s) = kp + kI/s + kDs] controllers. Let kp = 19, kI = 0.5, and kD = 4/19. (b) Determine the system type and error constant of the system with respect to disturbance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances w(t) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your response briefly. (d) Verify your results for parts (a) and (b) using Matlab by plotting unit-step and -ramp responses for both tracking and disturbance rejection.
Sri K.
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