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The application of deadbeat control in discrete-time systems, focusing on the calculation of c(z) to control p(z) and the ideal closed-loop response. The text also covers potential problems and sample rate selection. The methods discussed are analogous to continuous-time methods but have different interpretations in the z-plane.

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Download Direct Digital Design: Deadbeat Control in Discrete-Time Systems and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Direct digital design Design methods for P (z) • Pole placement for 1 1 + P (z)C(z) • Frequency response based designs (loopshaping) • Root locus • Tuning PID controllers The above methods are analogous to the continuous time methods (except for the di!erent interpretation between s-plane and z-plane). The following does not have a continous time analogue. • Finite settling time (deadbeat) control. Roy Smith: ECE 147b 7: 2 Direct digital design Calculating C(z) to control P (z) P(s) C(s) C(z)P(z) Approximation of C(s) with C(z) Model P(s), and sample/hold as P(z) Continuous-time design Discrete-time design • ZOH equivalence gives P (z). • Design C(z) for good closed-loop control of P (z) Roy Smith: ECE 147b 7: 1 Deadbeat control Approach: P (z) C(z) !" #$ + ! % !!!! " r(z)y(z) The closed-loop response is: M(z) = P (z)C(z) 1 + P (z)C(z) . So: • Choose M(z), the desired closed-loop response. • Solve the above to get C(z). Roy Smith: ECE 147b 7: 4 Deadbeat control Ideal closed-loop response: M(z)& & r(k)y(k) Consider the best possible closed-loop response: M(z) = z!1 A step input, r(z) = z z ! 1 Gives y(z) = 1 z ! 1 (a delayed step output). The error (e(z) = r(z) ! y(z)) is simply a unit pulse: e(z) = 1. Roy Smith: ECE 147b 7: 3 Deadbeat control Sample rate selection P (s) = 5 (s2 + 2s + 5) , desired response: M(z) = z!1 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time: seconds Controller output, u(k) Plant input, u(t) Sampled plant output, y(k) Plant output, y(t) Sample period: T = 1.0 seconds Roy Smith: ECE 147b 7: 10 Deadbeat control Intersample behavior P (s) = 1 s2 desired response: M(z) = z!1 5 10 15 20 25 -1.5 -1 -0.5 0 0.5 1 1.5 2 Time: seconds Controller output, u(k) Sampled plant output, y(k)Plant output, y(t) Plant input, u(t) Roy Smith: ECE 147b 7: 9 Deadbeat control Sample rate selection P (s) = 5 (s2 + 2s + 5) , desired response: M(z) = z!1 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time: seconds Controller output, u(k) Sampled plant output, y(k) Plant output, y(t) Plant input, u(t) Sample period: T = 2.0 seconds Roy Smith: ECE 147b 7: 12 Deadbeat control Sample rate selection P (s) = 5 (s2 + 2s + 5) , desired response: M(z) = z!1 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 1.5 2 2.5 Time: seconds Controller output, u(k) Sampled plant output, y(k) Plant output, y(t) Plant input, u(t) Sample period: T = 0.5 seconds Roy Smith: ECE 147b 7: 11 Deadbeat control Sample rate selection comparison P (s) = 5 (s2 + 2s + 5) , desired response: M(z) = z!1 Plant input: 0 1 2 3 4 5 6 7 8 9 10 -1 -0.5 0 0.5 1 1.5 2 2.5 Time: seconds Plant input, u(t), T = 0.5 seconds Plant input, u(t), T = 1.0 seconds Plant input, u(t), T = 2.0 seconds Roy Smith: ECE 147b 7: 14 Deadbeat control Sample rate selection comparison P (s) = 5 (s2 + 2s + 5) , desired response: M(z) = z!1 Plant output: 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Time: seconds Plant output, y(t), T = 0.5 seconds Plant output, y(t), T = 1.0 seconds Plant output, y(t), T = 2.0 seconds Roy Smith: ECE 147b 7: 13