Description
1. (30 Points)
(a) Find a minimal state-space realization for the following discrete time transfer function
G1(z) = z
2 + 0.5z
z
3 − 2.2z
2 + 1.52z − 0.32
(b) Find a minimal state-space realization for the following discrete time transfer function
G2(z) = z
2 + 0.4z − 0.12
z
2 + 0.6z − 0.4
(c) Using the answers of parts 1(a) and 1(b), find a minimal state-space realization for the following
discrete-time closed-loop system.
u[k] G y[k] 1
(z)
G2
(z)
2. (30 Points) In this problem, we will investigate the matrix exponential. Consider A, P ∈ R
nn
(a) Show that when det(P) 6= 0
e
(P
−1AP )t = P
−1
e
AtP
(b) Show that when det(A) 6= 0
Z
T
0
e
Aλdλ
= A
−1
e
AT − I
=
e
AT − I
A
−1
(c) Given that (λ , ν) is an eigenvalue and eigenvector pair of A. Based on this information, derive
the associated eigenvalue and eigenvector pair of e
At
.
You are supposed to derive the result, thus don’t just type the answer.
∗This document c M. Mert Ankarali
(d) Compute e
At for the following matrix
A =
σ ω
−ω σ
Hint: Your solution should be in terms of sinusoidal and exponential functions of ωt and σt.
(e) Compute e
At for the following matrix
A =
0 1
1 0
without using the Laplace transform domain solution method.
3. (30 Points) Consider the following CT state-space representation
x˙(t) =
0 1
0 −1
x(t) +
0
1
u(t)
(a) Based on the procedures detailed in the lecture notes, discretize this state-space formulation
under ZOH operation at the input and uniform ideal sampling at the states and compute the DT
state-space representation and associated matrixes.
x[k + 1] = Gx[k] + Hu[k]
T should exist symbolically in your matrices.
(b) Now approximate e
AT using the first order approximation given below
e
AT ≈ I + AT
and using this approximation compute the approximated discretie time state-space equation
x[k + 1] ≈ Gx˜ [k] + Hu˜ [k]
(c) Compute (G, H) and (G, ˜ H˜ ) for different values of T, compre the results, and comment on them.
4. (30 Points) Stability of CT and DT dynamical systems
(a) Consider the DT system
x[k + 1] =
0 1
α 2α − 1/2
x[k] +
0
1
u[k]
y[k] =
−2 1
u[k]
i. For what values of parameter α is the system asymptotically stable?
ii. For what values of parameter α is the system BIBO stable?
(b) Consider the CT system
x˙(t) =
0 1
α 2α − 1/2
x(t) +
0
1
u(t)
y(t) =
−2 1
u(t)
i. For what values of parameter α is the system asymptotically stable?
ii. For what values of parameter α is the system BIBO stable?
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