(a) Let A be the matrix given by A =(i) Obtain the eigenvalues and corresponding eigenvectors of A.(ii) Hence find the solution of the linear system* = Axx(0) =[4marks(b) For each of the following input-output systems write down the corresponding matrices A,B, C and D when written in the state-space form* = Ax + Buу %3D Сх + Dи(i)(ii)x1 = 2x1 – x2 + 6ux1 = -x2 + 2u1 + uzx2 = -2x1 + 3x2x2 = 8x1- 2x2y = 3x1 + x2Yı = x1+ U2Y2 = -6x1 + x2 – u1 + 5u2(c) The state-space matrices of a given linear input-output system are5-1B =С-1 -2],D=[2]-41Show that the transfer function G(s) of the input-output system is given by2s2-20s+15G(s) =s2-6s+1Hence determine the stability of the input-output system.

Formula: The solution of system x˙=Ax is given by, x(t)=c1ueλ1t+c2veλ2t, where u, v eigenvectors of…

Answered: (a) Let A be the matrix given by A = |E…

(a) Let A be the matrix given by A = |E I (i) Obtain the eigenvalues and corresponding eigenvectors of A. (ii) Hence find the solution of the linear system * = Ax x(0) = ]

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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Question

(a) Let A be the matrix given by A =
(i) Obtain the eigenvalues and corresponding eigenvectors of A.
(ii) Hence find the solution of the linear system
* = Ax
x(0) =
[4marks
(b) For each of the following input-output systems write down the corresponding matrices A,
B, C and D when written in the state-space form
* = Ax + Bu
у %3D Сх + Dи
(i)
(ii)
x1 = 2x1 – x2 + 6u
x1 = -x2 + 2u1 + uz
x2 = -2x1 + 3x2
x2 = 8x1
- 2x2
y = 3x1 + x2
Yı = x1+ U2
Y2 = -6x1 + x2 – u1 + 5u2
(c) The state-space matrices of a given linear input-output system are
5
-1
B =
С-1 -2],
D=[2]
-4
1
Show that the transfer function G(s) of the input-output system is given by
2s2-20s+15
G(s) =
s2-6s+1
Hence determine the stability of the input-output system.

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