This document provides instructions for linear transformations to diagonalize 4 different linear time-invariant systems represented in state space. For the first two systems, the user is asked to diagonalize the given A and B matrices. For the third problem, the user is asked to derive a general transformation to diagonalize any observable linear system and show the eigenvalues are preserved. The fourth problem applies this general transformation to a given system.

1. Modern Control Systems
Supplementary Sheet (2)
1. A linear time-invariant system has the following companion state space
representation:
𝐀 = [
0 1
−10 −7
] 𝐁 = [
0
1
] 𝐂 = [1 0] 𝐃 = 𝟎
Use an appropriate linear transformation to diagonalize the system.
2. A linear time-invariant system has the following controllable state space
representation:
𝐀 = [
0 1
−20 −9
] 𝐁 = [
0
1
] 𝐂 = [3 1] 𝐃 = 𝟎
Use an appropriate linear transformation to diagonalize the system.
3.a) Derive a linear transformation that diagonalizes any observable linear time-
invariant state model.
b) Show that the eigenvalues of the observable and diagonalized state models
of part (a) are the same.
4. Use the results of Problem (3) to diagonalize the following observable linear
time-invariant system:
𝐀 = [
0 −12
1 −8
] 𝐁 = [
4
1
] 𝐂 = [0 1] 𝐃 = 𝟎