Control assignment#3

1. 4th
Year Biomedical Eng. Systems Engineering 2020 Assignment#3
Solve the following problems:
1- a) Draw a block diagram realization of the discrete-time system described by
𝑥(𝑘 + 1) = [
0.5 −0.5 1
−1.5 −0.5 3
0.5 0.5 3
] 𝑥(𝑘) + [
1
1
2
] 𝑢(𝑘),
𝑦(𝑘) = [ 1 1 0]𝑥(𝑘)
b) Find the pulse transfer function of the system given in part a), given that the transformation
matrix P and its inverse P-1
that transform the system in part a) to the standard controllable
form are given by
𝑃 = [
−3 −1 1
1 1 1
−3 1 2
] , 𝑃−1
= [
1/6 1/2 −1/3
−5/6 −1/2 2/3
2/3 1 −1/3
] , 𝑎𝑛𝑑 𝑃−1
GP = [
0 1 0
0 0 1
−5 3 3
]
c) Given that 𝑥1
^
(𝑘)= (𝑥1 (k) + 0.5 𝑥2 (k)) , 𝑥2
^
(𝑘)= (𝑥1 (k) - − 0.5 𝑥2 (k)) and that
𝑥3
^
(𝑘)= 𝑥3 (k), find the transformation matrix P such that x(k) = P x^
(k). Further, find the
state space equations in terms of the new state variables x^
(k).
2- The pulse transfer function of a single-input single-output linear discrete system is given by
𝑌(𝑧)
𝑈(𝑧)
=
3 𝑧2+0.04 𝑧+1.6
𝑧3−0.02 𝑧2−0.28 𝑧−0.52
a) Find the partial fraction expansion of Y(z)/U(z).
b) Using the result of part a), write the state equations in the diagonal form. Note that two
poles are complex conjugate.
c) Use a suitable transformation of the complex states so as to get a real state equations.
3- a) Find which of the following three systems constitute different representations of
the same system:
  ),(012)(),(
2
1
0
)(
572
231
120
)1( kxkykukxkx 
























  ),(332)(),(
1
2
5
)(
331
562
11145
)1( kxkykukxkx




























2.   ),(214)(),(
2
0
1
)(
210
101
100
)1(
^^^
^
kxkykukxkx 




















 

Given that the following transformations transform these systems to the standard
controllable forms
,
,
201
010
101
,
100
210
521
,
210
110
001
^^1^1
1
^
TGTTGTGTT
whereTTT












































and
^
,, GandGG

are the system transition matrices of the first, second, and third systems
respectively.
Further, find the transformation matrix P which relates the two different state
representations of the same system.
4- A third order discrete system is described by
𝑥(𝑘 + 1) = [
−𝑎1 −𝑎2 −𝑎3
(1 − 𝑎1
2
) −𝑎1 ∗ 𝑎2 −𝑎1 ∗ 𝑎3
0 (1 − 𝑎2
2
) −𝑎2 ∗ 𝑎3
] x(k)
The solution of the discrete Lyapunov equation, given by
𝑃 − 𝐺 𝑇
𝑃 𝐺 − 𝑄 = 𝑂,
For Q = 𝑉 𝑇
𝑉, 𝑤ℎ𝑒𝑟𝑒 𝑉 = [
1
𝑎1
𝑎2
] results in
P=
[
1
((1 − 𝑎1
2
)(1 − 𝑎2
2)(1 − 𝑎3
2))⁄ 0 0
0 1
((1 − 𝑎2
2)(1 − 𝑎3
2))⁄ 0
0 0 1
(1 − 𝑎3
2
)⁄ ]
Note that Q is positive semidefinite and the system will be stable according to Lyapunov if and
only if P is positive definite. Under what conditions on a1 , a2 and a3 will the system be stable?
5- a) Draw a block diagram realization of the discrete-time system described by
𝑥(𝑘 + 1) = [
0.5 0 0
0 −1.2 −0.7
0 1.4 0.9
] 𝑥(𝑘) + [
1
3
−4
] 𝑢(𝑘),

3. 𝑦(𝑘) = [ 1 1 1]𝑥(𝑘)
b) Check the controllability and observability of this system
c) Find the pulse transfer function of this system and check for pole – zero cancellation.
d) Check the stability of this system using the second method of Lyapunov knowing that the
solution of the discrete Lyapunov equation GT
P G - P + Q = 0, for Q = I yields
P = [
1.3333 0 0
0 4.9659 2.3598
0 2.3598 2.4205
]
6- For the two cascaded systems shown in Fig. 1
3
------------
(z – 0.6)
5(z -1)
------------
(z + 0.8)
2
------------
(z – 1)
U(k) X3(k) X2(k) X1(k)
3
------------
(z – 0.6)
2
------------
(z - 1)
5(z - 1)
------------
(z + 0.8)
U(k) X3(k) X2(k) X1(k)
Y(k)
Y(k)
(a) Write the state equations of the two systems shown in Fig 1.
(b) Check the controllability and observability of those systems and comment on your
results.
7- The state equation of a digital control system is given by
a) Find a control sequence that transfers the system from zero initial condition to a final state of
b) Determine the control sequence with minimum energy that transfers this system from zero
initial state to the same final state in two samples.
c) Determine the initial state x(0) given that the outputs were y(0) = 0.5, y(1) = 1, y(2) = 2.5
corresponding to u(0) = [ 1 1]T
and u(1) = [ 1.5 -1]T
.
u(k),+x(k)=1)+x(k























20
10
01
210
021
132
y(k) = [ 0 1 1 ] x(k)
 123=)tx( T
f
Fig. 3

4. 8- Given the system described by
𝑥(𝑘 = 1) = [
1 1.5 1.5
0 0.5 0
0 −1 −.5
] x(k) + [
0
1 + 𝛼
1 − 𝛼
] u(k) , y(k) = [ 1 β+2 β] x(k)
a) Under what condition on α would this system be controllable?
b) Under what condition on β would this system be observable ?
9- Consider the system described by
x(k+1) = [
−1 0 0
0 0.5 1.5
0 1 1
] x(k) + [
1
1
0
] u(k) , y(k) = [ 1 α α+1 ] x(k)
a) Under what conditions on α is this system state controllable?
b) Under what conditions on α is this system observable?
10- a) Consider the single-input third order discrete system described by
𝑥(𝑘 + 1) = 𝐺 𝑥(𝑘) + ℎ 𝑢(𝑘),
Where h = h1 + h2 and h1 and h2 are two independent eigenvectors of G corresponding
to eigenvalues α1 and α2, respectively, and α1 ≠ α2 . Check the controllability of this
system.
b) Given the unobservable system described by
𝑥(𝑘 + 1) = ⌈
0 1 −1
−5 −1 0
3 −3 4
⌉ 𝑥(𝑘) + ⌈
0
0
1
⌉ 𝑢(𝑘),
𝑦(𝑘) = [ 1 −1 1] 𝑥(𝑘),
The following outputs were observed for the given inputs
u(0) = -4, u(1) = 3, y(0) = 1, y(1) = -1, and y(2) = -5.
Determine x2(0) and x3(0) in terms of x1(0) and hence find the initial state of the system
assuming that x1(0) = 0.