## 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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