Faculty of Engineering Digital Control

Elect. Power & Machines Dept. The State Space

Third Year Approach Tutorial

 Discretization of State Space

1) Find the state-space representation for the shown system. Choose suitable values
for R and C and obtain the corresponding discrete state-space form.
R R

C C
Vo
Vin

2) Find the state-space representation for the armature controlled DC motor

considering the motor speed as the output. Choose suitable values for the
parameters of the system and obtain the corresponding discrete state-space form.

3) Obtain the corresponding discrete state-space form for the following continuous
state-space representations.
 0 1 0
X   X   u , Y  1 1X .
a) - 2 - 3  1 

 0 - 4 1 
X   X   u , Y  0 1X .
b) 1 -4  2

4) Given X(k+1) = G(T) X(T) + H(T) U(k), What is the effect of decreasing the
sampling period T on G(T), H(T), and the required U(k)?

5) Find the discrete state space representation for the following system
1 0 0 1
̇
𝑋 = [0 1 1] 𝑋 + [ 1 ] 𝑢, 𝑦 = [1 0 1]𝑋
0 1 1 −1
 State Space Canonical Representation
6) Obtain a state-space representation of the following pulse-transfer-function system
in the controllable canonical form. Draw the corresponding block diagram.
Y ( z) z 1  2 z 2

U ( z ) 1  4 z 1  3z  2

7) Obtain a state-space representation of the following pulse-transfer-function system

in the observable canonical form. Draw the corresponding block diagram.

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Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

Y ( z) z 2  4 z 3

U ( z ) 1  6 z 1  11z  2  6 z  3

8) Obtain a state-space representation of the following pulse-transfer-function

systems in the diagonal canonical form. Draw the corresponding block diagram.
Y ( z ) 1  6 z 1  8 z 2

1 2
c) U ( z ) 1  4 z  3z
Y ( z) z 1  2 z 2

1 2
d) U ( z ) 1  0.7 z  0.12z
9) Write the state-space representation of the following systems in the Jordan form.
Draw the block diagram for systems "b" and "c".
Y ( z) az  b

U ( z ) ( z  0.1) 2 ( z  0.2)
a)
Y ( z) az 2  bz  c

U ( z ) ( z  0.5) 3 ( z  0.3) 2
b)
Y ( z) az 4  bz 3  cz 2  d

U ( z ) ( z 2  0.2 z  0.4)( z 2  0.5 z  0.9)
c)
Y ( z) az 4  bz 3  cz 2  dz  e

U ( z ) ( z 2  z  1)( z  a1 )( z  a 2 )
d)

10) Write the possible state-space representations of the system described by the
equation
y (k  2)  y (k  1)  0.16 y (k )  u (k  1)  2u (k )

11) Draw the state diagram for the following system

0 1 0 0 0
0.01 0.2 0 0 2
𝑋(𝐾 + 1) = [ ] 𝑋(𝐾) + [ ] 𝑢(𝐾),
0 0 0.3 1 0
0 0 0 0.3 2
𝑦(𝐾) = [0.3 0.4 1 1]𝑋(𝐾) + 0.5𝑢(𝑘)
By inspection, get the system transfer function.

12) The following are two state space representations for the same system
e 1 0 0
X1 (K + 1) = [0 e 1] X1 (K) + [0] u(K), y(K) = [2 4 1]X1 (K)
0 0 e 1
0 0 1 d
X2 (K + 1) = [1 0 −3] X2 (K) + [ c ] u(K), y(K) = [0 0 1]X2 (K)
0 1 3 b
Find the values of the unknowns and draw the state diagram for the controllable
canonical form for the system.

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Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

13) Obtain the state equation and the output equation for the following systems.

a)

b)

c)

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 Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

d)

 Transformation and Time response

14) Find the diagonal form for the system given in problem (3).a and the Jordan form
for the system given in problem (3).b. Then find their corresponding discrete
state-space forms.
a) Is G(T) that corresponds to the diagonal form equal to [P-1 G(T)problem (3).a P]
where P is the transformation matrix that transforms the system of problem
(3).a into diagonal form?
b) Is G(T) that corresponds to the Jordan form equal to [P-1 G(T)problem (3).b P]
where P is the transformation matrix that transforms the system of problem 3-
a into diagonal form?
c) What is the relation between G(T) this problem and G(T) problem 3?
d) Can you make a general proof for this relation?

15) Find the pulse-transfer-function matrix for the systems given in problems (13).c
and (13).d.

16) Referring to problem (3)

a) Find the pulse-transfer-function matrices for the resulting discrete systems.
b) Find the continuous pulse-transfer-function matrices, G(s).
c) Find G(z) where G(z) is the Z-transform of G(s) when it is preceded by a
ZOH.
d) Show that the results of "a" and "c" are equal.

17) Show that neither the pulse-transfer-function matrix nor the closed-loop poles of
any system are affected by the state transformation matrix P.

18) For the following systems, find X(k) and y(k) for k = 1,…,5 using two different
approaches. Assume x1(k) = x2(k) =1 and the input is a unit impulse.

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 Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

 0 1 0 
a) X (k  1)    y(k)  1 0X (k )
- 0.6  1 u (k ),
X ( k )
- 0.05  

 0.2 0 1
b) X (k  1)    X (k )   u (k ), y(k)  1 1X (k )
 0 0.4 1

k 1 1
i. Is G   {( zI  G ) } ?
k 1
ii. Is  1{( zI  G) 1 HU ( z )}   G k 1 j Hu( j) ?
j 0
iii. Draw on the state space the calculated values of x1(k) and x2(k).

19) Given the system

0.1 0 0.1 0.9
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘)
0 0.2 0 1
1 𝛿(𝑘)
With 𝑋(0) = [ ] and 𝑢(𝑘) = [ ], find the closed form of 𝑋(𝑘) in terms of k.
1 𝑢(𝑘)

0.1 0.2 −1
20) Given the system 𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘)
0 0.2 1
a) Find an expression for 𝑋(𝑘) in terms of 𝑘 knowing that the initial state of the
1 1 ∀ 𝑘≥0
system is [ ] and the input is 𝑈(𝑘) = {
1 0 ∀ 𝑘<0
b) Write an expression for 𝑋(𝑘) if the input becomes 3𝑈(𝑘) − 2𝑈(𝑘 − 1)

 Design via state space approach

21) Check the controllability and the observability of the following systems

 0.1 - 0.2 - 0.2 2

X (k  1)   0 - 0.1 
0.1  X (k )  0 u (k ),
a)
 0.1 0 - 0.1  1 
y(k)   1 1 0 X (k )

 0.2 0 0  0 1
X (k  1)   0 0.2 0  X (k )  1
 0u (k ),
b)  0 0. 3 0.1  0 1
1 0 0 
Y(k)  
0 
X (k )
0 1

0 1 0 0 
X (k  1)   0 0 1  X (k )  0u (k ),

c)
0.006 0.11 0.6  1 
y(k)  2 0. 9 0.1X (k )

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Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

0 0 0.006 2 
X (k  1)   1 0 
0.11  X (k )  0.9u (k ),
d)
 0 1 0.6  0.1
y(k)   0 0 1 X (k )

 0 .2 - 0.2 4 
X (k  1)    X (k )   u (k ),
e)  0.1 0.5    2
y(k)  3 3X (k )

i. Comment on the results of both "c" and "d"

ii. For part "e", find the pulse transfer function and diagonalize the system
and comment on the results.

22) Show that the following system is complete state observable and it can be
stabilized.
2 0 1 0
𝑋(𝑘 + 1) = [0 0.5 0] 𝑋(𝑘) + [0] 𝑈(𝑘), 𝑦(𝑘) = [1 1 0]𝑋(𝑘)
0 0 2 1
23) Given the system,

0.2 0.1 1
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘), 𝑌(𝑘) = [2 −1]𝑋(𝑘)
0.3 0.4 1

a) Find 𝑋(2) given that 𝑢(0) = 𝑢(1) = 𝑢(2) = 1, and 𝑦(2) = 0.3 and 𝑦(3) =
1.
0.5
b) Find the required control vector that results in 𝑋(5) = [ ].
0.5
24) Find the state feed back gain for each of the following systems using two different
methods such that the closed loop poles are as given below:

0.1 0.2  1 
a) X (k  1)    X (k )   u (k ), Required poles are at -0.1, -0.1.
 0.4 - 0.1  2

0.2 - 0.3   2
b) X (k  1)   
0.1 0 u (k ), Required poles are at -0.2, -0.3.
X ( k )
 - 0.4  

0.1 0.1  1 
c) X (k  1)    X (k )   u (k ), Required poles are at -0.2±j0.2.
0.1 - 0.1  2

25) Given the system,

0.1 0.5 1
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘)
0.1 0.3 0

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Faculty of Engineering Digital Control
Elect. Power & Machines Dept. The State Space
Third Year Approach Tutorial

Which of the following state feedback gain vectors is suitable for the system? Justify
your choice.
i. [−1.4 12.2]
ii. [0.2 1]
iii. [−0.6 3.4]

26) Using two different approaches show that the following system can not be
stabilized via state feed back.

0 1 1 
X (k  1)    X (k )   u (k ).
0 2 0 

27) Find the state feedback gain that is required to assign the poles of the following
system to Z = 0.5.

2 0 0
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘), 𝑌(𝑘) = [0 1]𝑋(𝑘).
1 0.5 1

Design a full order observer for the system.

28) Given the system,

a 0.2 1
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘), 𝑌(𝑘) = [2 1]𝑋(𝑘)
0.3 b −1

a) Find the range of values of 𝑎 and 𝑏 for which the system is completely state
controllable and completely state observable.
b) For 𝑎 = 0.3 , 𝑎𝑛𝑑 𝑏 = 0.4 :
i. Find the system transfer function and identify the uncontrollable pole.
ii. Find the state feedback vector that places the controllable pole at the same
place of the uncontrollable.
1
iii. Is the observer of 𝐿 = [ ] suitable for the system? Why?
0.4
29) Given the system,

0.1 0.4 a
𝑋(𝑘 + 1) = [ ] 𝑋(𝑘) + [ ] 𝑢(𝑘), 𝑌(𝑘) = [−14 8]𝑋(𝑘)
−0.8756 1.3 b

a) Find the range of values of 𝑎 and 𝑏 for which the system is completely state
controllable.
b) For 𝑎 = 𝑏 = 1, design a state feedback controller to speed up the system by a
factor of 2.
c) Design a full order observer for the system with appropriate poles location.