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Q5. (a) Consider the following difference equation i) ii) iii) y(k) = 0.9y(k − 1) + y(k − 2) + u(k − 1) Use the z-transform to obtain a discrete-time transfer function. Determine the poles and zeros of the discrete-time system and plot them in a pole-zero map. Discuss the stability of the discrete-time system.

Q5. (a) Consider the following difference equation i) ii) iii) y(k) = 0.9y(k − 1) + y(k − 2) + u(k − 1) Use the z-transform to obtain a discrete-time transfer function. Determine the poles and zeros of the discrete-time system and plot them in a pole-zero map. Discuss the stability of the discrete-time system.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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Q5. (a) (b) Consider the following difference equation i) ii) iii) i) Consider now the above discrete-time system in state space form given by ii) y(k) = 0.9y(k − 1) + y(k − 2) + u(k − 1) Use the z-transform to obtain a discrete-time transfer function. iii) Determine the poles and zeros of the discrete-time system and plot them in a pole-zero map. Discuss the stability of the discrete-time system. [x₁(k + 1)] [x₂ (k+1)] = [ y(k) = [0_1][x₂(k)] Verify that this discrete-time state space system can be stabilised by a feedback controller. 1 0.91 [x₂] + []u(k) Design a state feedback controller u(k) = −Kx(k), such that the closed-loop system poles are placed at A₁ = 0.05, λ₂ = 0.12 Transform the closed-loop state-space system into transfer function form (from the reference signal r to the output y), and determine the steady state value of the output using a unit step input r(k).
Transcribed Image Text:Q5. (a) (b) Consider the following difference equation i) ii) iii) i) Consider now the above discrete-time system in state space form given by ii) y(k) = 0.9y(k − 1) + y(k − 2) + u(k − 1) Use the z-transform to obtain a discrete-time transfer function. iii) Determine the poles and zeros of the discrete-time system and plot them in a pole-zero map. Discuss the stability of the discrete-time system. [x₁(k + 1)] [x₂ (k+1)] = [ y(k) = [0_1][x₂(k)] Verify that this discrete-time state space system can be stabilised by a feedback controller. 1 0.91 [x₂] + []u(k) Design a state feedback controller u(k) = −Kx(k), such that the closed-loop system poles are placed at A₁ = 0.05, λ₂ = 0.12 Transform the closed-loop state-space system into transfer function form (from the reference signal r to the output y), and determine the steady state value of the output using a unit step input r(k).
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