## Description

1. (10 Points) Modal reachability test states that A DT system represented in state space form with G is

the system matrix and H is the input matrix is un-reachable if w

T H = 0 for a left eigenvector of G.

A left eigenvalue, eigenvector pair of G is defined as

w

T G = λwT

, w ∈ R

n

In this context, show that if w

T H = 0 for a left eigenvector of G, then the system is un-reachable.

2. (10 Points) Modal observability test states that A DT system represented in state space form with G

is the system matrix and C is the output matrix is un-observable if Cv = 0 for a (right) eigenvector of

G. A right eigenvalue, eigenvector pair of G is defined as

Gv = λv , v ∈ R

n

In this context, show that if Cv = 0 for a left eigenvector of G, then the system is un-observable.

3. (10 Points) Given a SISO DT dynamical system

x[k + 1] =

0.5 0 0

0 0.5 1

0 0 0.5

x[k] + Hu[k]

y[k] = Cx[k]

(a) If possible find an input vector H ∈ R

n, such that the system is fully reachable. If it is not

possible to find such an H, then show it why?

(b) If possible find an output row-vector C ∈ R

1×n, such that the system is fully observable. If it is

not possible to find such an C, then show it why?

∗This document c M. Mert Ankarali

1

4. (25 Points) Consider the discrete-time plant given by

x[k + 1] =

0 1

1 −1

x[k] +

0

1

u[k]

y[k] =

1 1

x[k]

(a) Assume that we “close the loop” around the plant using constant output feedback control law

u[k] = αy[k]

Show that the closed-loop system can not be stabilized regardless of the choice of α.

The moral here is that the system cannot be stabilized using static output feedback gain.

(b) Now we will consider a time-dependent output feedback control policy. The control policy is given

by

u[k] = α[k]y[k] where

α[k] =

−1 k is even

3 k is odd

Show that the state trajectories corresponding to any initial condition return to the origin in at

most 4 time steps.

5. (25 Points) Consider the following DT state evolution equation

x[k + 1] =

−3/4 7/4

−1/4 −3/4

x[k] +

−1

1

u[k]

(a) Design a state-feedback controller, u[k] = −K∗x[k] such that the closed-loop behavior shows a

dead-beat response. Then, use Matlab to compute and plot the response of each of the state

variables from k = 0 to k = 10 assuming x[0] =

4

0

.

(b) Now suppose that there is an inevitable delay such that the control action effects the system after

one sample delay

u[k] = −K∗x[k − 1]

i. Find a state-space model for the closed-loop system in this case (Hint: controller now has a

memory). Use the same gain you computed in previous part.

ii. Compute the eigenvalues of this new closed-loop system.

iii. Agin, use Matlab to compute and plot the response of each of the state variables (for this

case) from k = 0 to k = 10 assuming x[0] =

4

0

(and x[−1] =

0

0

).

iv. Comment on the results.

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6. (35 Points) In this problem you will going to analyze many different aspects of state-space design and

analysis of DT control systems using the CT plant, which is illustrated below.

u(t)

x2

(t)

x1

(t)

(a) Derive a state-space representation for the given CT LTI system. Use x(t) =

x1(t) x2(t)

T

as the state vector, and output equation is simply given as y(t) = x(t) (i.e. C = I).

(b) In this part you will analyze the following digital (discretized) control system. Given that T =

0.5 s., derive a discrete-time state-space representation, where the input, state-vector, and output

vector are defined as u[k], x[k] =

x1[k] x2[k]

T

, and y[k] = x[k], respectively.

u(kT) x1

(kT) ZOH u[k]

T

x1

[k]

x2

(kT)

x2

[k]

T

(c) In this part you will analyze the digital system that is controlled via a state-feedback control law,

as illustrated in the Figure below. Choose a (k1, k2) pair such that closed-loop system rejects all

initial conditions in finite-time (i.e. dead-beat behavior).

Then, simulate the whole closed-loop system in Simulink, starting from different initial conditions.

(Hint: It is possible to assign initial conditions in Simulink’s transfer function blocks ).

u(kT) x1

(kT) ZOH u[k]

T

x1

[k]

x2

(kT)

x2

[k]

T

k1

k2

−

−

3