Question 3 (a) Consider the continuous-time linear system X1 = x1 + 2x2 X2 = 3x1 + 2x2 (1) Show that this system has eigenvalues 2, = 4 and 2, = -1, and hence determine the stability of the system. (ii) Find the corresponding eigenvectors, and hence find the solution of this system for the initial condition x, (0) = 0, x2 (0) = 2 (b) Now consider a corresponding state-feedback control system of the form X1 = x1 + 2x2 + u, x2 = 3x1 + 2x2 u = fix1+ f2X2 (i) Write down the matrices A, B and F when this system is written in the state- space form * = Ax + Bu, u = Fx (ii) Determine necessary and sufficient conditions on fi and f2 for the closed-loop system to be stable. (Hint: make use of the Routh-Hurwitz Criterion.) (iii) Find the values of fi and f2 such that the eigenvalues (poles) of the closed-loop system are both at -2.

Question 3 (a) Consider the continuous-time linear system X1 = x1 + 2x2 X2 = 3x1 + 2x2 (1) Show that this system has eigenvalues 2, = 4 and 2, = -1, and hence determine the stability of the system. (ii) Find the corresponding eigenvectors, and hence find the solution of this system for the initial condition x, (0) = 0, x2 (0) = 2 (b) Now consider a corresponding state-feedback control system of the form X1 = x1 + 2x2 + u, x2 = 3x1 + 2x2 u = fix1+ f2X2 (i) Write down the matrices A, B and F when this system is written in the state- space form * = Ax + Bu, u = Fx (ii) Determine necessary and sufficient conditions on fi and f2 for the closed-loop system to be stable. (Hint: make use of the Routh-Hurwitz Criterion.) (iii) Find the values of fi and f2 such that the eigenvalues (poles) of the closed-loop system are both at -2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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