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Question 3 Consider a control system described by the following state variable model x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with A = - [ - K₁ K₁²₂]. B = [K₂₁₂ -k₂ k₂ 2₁], C = [] and D = | ► = [8]. (a) Determine the dimension of the input u(t) and output y(t) respectively and obtain the characteristic equation of the system. (b) Find the state-transition matrix for k₁ = 1 and k₂ = 1. (c) For system in (b), assume the initial condition is x(0) = [¹], determine the unforced time response (for u(t) = 0) of the state variables x(t) and the output y(t). (d) For another system with the following characteristic question A(s) = s³ + ks² + ks + 8 = 0 Determine the range of k to make the system stable using Routh-Hurwitz criterion.

Question 3 Consider a control system described by the following state variable model x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with A = - [ - K₁ K₁²₂]. B = [K₂₁₂ -k₂ k₂ 2₁], C = [] and D = | ► = [8]. (a) Determine the dimension of the input u(t) and output y(t) respectively and obtain the characteristic equation of the system. (b) Find the state-transition matrix for k₁ = 1 and k₂ = 1. (c) For system in (b), assume the initial condition is x(0) = [¹], determine the unforced time response (for u(t) = 0) of the state variables x(t) and the output y(t). (d) For another system with the following characteristic question A(s) = s³ + ks² + ks + 8 = 0 Determine the range of k to make the system stable using Routh-Hurwitz criterion.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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Question 3 Consider a control system described by the following state variable model x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) [-k₁ k₂ [k₁ -k₂ with A =[^].B= 22.C=[] and D = | [8]. - −k₂] k₁ k₂ (a) Determine the dimension of the input u(t) and output y(t) respectively and obtain the characteristic equation of the system. (b) Find the state-transition matrix for k₁ = 1 and k₂ = 1. (c) For system in (b), assume the initial condition is x(0) = [¹], determine the unforced time response (for u(t) = 0) of the state variables x(t) and the output y(t). (d) For another system with the following characteristic question A(s) = s³ + ks² + ks + 8 = 0 Determine the range of k to make the system stable using Routh-Hurwitz criterion.
Transcribed Image Text:Question 3 Consider a control system described by the following state variable model x(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) [-k₁ k₂ [k₁ -k₂ with A =[^].B= 22.C=[] and D = | [8]. - −k₂] k₁ k₂ (a) Determine the dimension of the input u(t) and output y(t) respectively and obtain the characteristic equation of the system. (b) Find the state-transition matrix for k₁ = 1 and k₂ = 1. (c) For system in (b), assume the initial condition is x(0) = [¹], determine the unforced time response (for u(t) = 0) of the state variables x(t) and the output y(t). (d) For another system with the following characteristic question A(s) = s³ + ks² + ks + 8 = 0 Determine the range of k to make the system stable using Routh-Hurwitz criterion.
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