4. Sheet 3
1. You are given the following transfer functions for a single-input single-output system:
a) 𝐺( 𝑠) =
𝑌(𝑠)
𝑈(𝑠)
=
3𝑠+2
𝑠2+7𝑠+12
b) 𝐺( 𝑠) =
𝑌(𝑠)
𝑈(𝑠)
=
5𝑠2+8𝑠+1
𝑠3+8𝑠2+20𝑠+16
In each case:
i. Obtain the differential equation
ii. Derive the standard companion state model from the differential equation
2. A single-input, single-output system is represented by the following transfer function:
𝐺( 𝑠) =
𝑠 + 5
𝑠2 + 4𝑠 + 3
Derive the controllable, the observable, and the diagonal/ Jordan canonical state space representation.
3. For the transfer functions of Problem (1) derive the following state models:
i. Controllable canonical state model
ii. Observable canonical state model
iii. Diagonal or Jordan canonical state model, whichever seems applicable.
