4.5. Consider the system i = Ax + Bu, where%3D0 -60 - 1-131-21-1-1-216 -210 -1A =1B =-12-121120 0(a) Show that -1 is an uncontrollable eigenvalue.-20 -2-1

For a system x.=Ax+Bu, an eigen value λ of matrix An×n is said to uncontrollable if rankA-λI B≠n.…

Answered: Consider the system i = Ax + Bu, where…

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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Related questions

Question

4.5. Consider the system i = Ax + Bu, where
%3D
0 -6
0 - 1
-1
3
1
-2
1
-1
-1
-2
1
6 -2
1
0 -1
A =
1
B =
-1
2
-1
2
1
1
2
0 0
(a) Show that -1 is an uncontrollable eigenvalue.
-2
0 -2
-1

Expert Solution

Uncontrollable Eigenvalue

For a system $\dot{x} = A x + B u$ , an eigen value $λ$ of matrix $A_{n \times n}$ is said to uncontrollable if $r a n k [A - λ I B] \neq n$ .

Given system is $\dot{x} = A x + B u$ , where:

$A = [\begin{matrix} - 1 & 0 & 0 & - 6 & 3 & - 1 \\ 1 & - 2 & 1 & 0 & - 1 & - 1 \\ 1 & 1 & 0 & 6 & - 2 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 2 & 1 \\ - 2 & 0 & 0 & - 2 & 0 & - 1 \end{matrix}], B = [\begin{matrix} 0 & 1 \\ - 1 & - 2 \\ 0 & - 1 \\ 0 & 0 \\ 1 & 2 \\ 0 & 0 \end{matrix}]$