1. Discription of Computer-
Chapter 4
Discription of Computer-
Controlled System

2. Contents 4.1 Description of Linear Discrete Systems
4.2 Pulse Response Function
4.3 Pulse Transfer Function
4.4 Open/closed-loop Pulse Transfer Function
4.5 Response of the CCS
4.6 Performance Specifications of the CCS
4.7 State Space Description of the CCS

3. 4.1 Description of Linear Discrete Systems

4. 4.1 Description of Linear Discrete Systems
The system considered in the class is the linear time-invariant system, i.e. the relation between the output and input is unchangeable over time.
r(kT)→y(kT); r(kT-iT)→y(kT-iT), k=0,1,2,…; i=…,-2,-1,0,1,2,…

5. 4.2 Pulse Response Function
Pulse response function is the basis for studying pulse transfer function.
G(s)
x(t)
x*(t)
y(t)
y*(t)
Fig. 4.1 Continuous system with impulse sampling signal input

6. 4.2 Pulse Response Function

7. 4.2 Pulse Response Function

8. 4.2 Pulse Response Function

9. 4.2 Pulse Response Function

10. 4.3 Pulse Transfer Function

11. 4.3 Pulse Transfer Function
G(z)
X(z)
Y(z)
Fig Diagram of Pulse Transfer System

12. 4.3 Pulse Transfer Function

13. 4.3 Pulse Transfer Function

14. 4.4 Open/closed-loop Pulse Transfer Function
4.4.1 Laplace transform of the sampled signal
G(s)
x(t)
x*(t)
y(t)
Y(s)
X*(s)
X(s)

15. 4.4 Open/closed-loop Pulse Transfer Function

16. 4.4 Open/closed-loop Pulse Transfer Function

17. 4.4 Open/closed-loop Pulse Transfer Function
4.4.2 Properties of X*(s)

18. 4.4 Open/closed-loop Pulse Transfer Function

19. 4.4 Open/closed-loop Pulse Transfer Function
4.4.3 How to get pulse transfer function
(1) System with sampler
G(s)
x(t)
x*(t)
y(t)
y*(t)
Fig. 4.3 system with sampler

20. 4.4 Open/closed-loop Pulse Transfer Function
(2) System without sampler
G(s)
x(t)
y(t)
X(s)
Y(s)
Fig. 4.4 system without sampler

21. 4.4 Open/closed-loop Pulse Transfer Function
(3) The methods to get pulse transfer function
G(s)
x(t)
x*(t)
y(t)
y*(t)

22. 4.4 Open/closed-loop Pulse Transfer Function
4.4.4 Pulse transfer function and difference equation
Pulse transfer function can be converted to the difference equation, and vice versa.

23. 4.4 Open/closed-loop Pulse Transfer Function

24. 4.4 Open/closed-loop Pulse Transfer Function

25. 4.4 Open/closed-loop Pulse Transfer Function

26. 4.4 Open/closed-loop Pulse Transfer Function
4.4.5 Pulse transfer function of the system with ZOH
The transfer function of the zero order holder is
The transfer function of the system with zero order holder is

27. 4.4 Open/closed-loop Pulse Transfer Function

28. 4.4 Open/closed-loop Pulse Transfer Function
4.4.6 Open loop pulse transfer function of the system
(1) Pulse transfer function of cascaded elements without sampler between them
G1(s)
G2(s)
R(s)
C1(s)
C*(s)
C(s)
C(z)
R(z)
R*(s)
G(z) T
(s)
Fig. 4.5 cascade connection without sampler

29. 4.4 Open/closed-loop Pulse Transfer Function

30. 4.4 Open/closed-loop Pulse Transfer Function
(2) Pulse transfer function of cascaded elements with sampler between them
G1(s)
G2(s)
R(s)
C1(s)
C*(s)
C(s)
C(z)
R(z)
R*(s)
G(z) T
C1*(s)
C1(z)
Fig. 4.6 cascade connection with sampler

31. 4.4 Open/closed-loop Pulse Transfer Function

32. 4.4 Open/closed-loop Pulse Transfer Function

33. 4.4 Open/closed-loop Pulse Transfer Function
(3) Pulse transfer function of parallel elements
G1(s)
U*(s)
G2(s)
Y(s)
Y*(s)
U(s) 
Y1(s)
Y2(s)

34. 4.4 Open/closed-loop Pulse Transfer Function
G1(s)
U*(s)
G2(s)
Y(s)
Y*(s)
U(s) 
Y1(s)
Y2(s)

35. 4.4 Open/closed-loop Pulse Transfer Function
4.4.7 Closed-loop pulse transfer function of the system
(1) Sampler is located after the comparator
G (s)
E(s)
C(s)
E(z)
E*(s)
Ф(z)
H (s)
R(s)
R*(s)
R(z)
C*(s)
C(z)
B(s) -
Fig. 4.7 sampler is located after the comparator

36. 4.4 Open/closed-loop Pulse Transfer Function

37. 4.4 Open/closed-loop Pulse Transfer Function

38. 4.4 Open/closed-loop Pulse Transfer Function
(2) Sampler is located at the feedback channel
G (s)
E(s)
C(s)
E*(s)
H (s)
R(s)
R*(s)
R(z)
C*(s)
C(z)
B(s) -
Fig Sampler is located at the feedback channel

39. 4.4 Open/closed-loop Pulse Transfer Function

40. 4.4 Open/closed-loop Pulse Transfer Function
(3) Sampler is located at the forward channel T
R*(s)
R(z)
C*(s)
R(s)
E*(s)
E(s)
G (s)
E(z) -
C(z)
B(s)
H (s)
Fig. 4.9 sampler is located at the forward channel

41. 4.4 Open/closed-loop Pulse Transfer Function

42. 4.4 Open/closed-loop Pulse Transfer Function

43. 4.4 Open/closed-loop Pulse Transfer Function
G2(s)
E(s)
C(s)
R(s) -
G1(s)
E*(s)
U(s)

44. 4.4 Open/closed-loop Pulse Transfer Function

46. Fig. 4.10 Open loop sampling system
4.5 Response of the CCS
(1) System response at sampling instant
Gh(s)
G1(s)
Gp(s)
H(s)
x(t)
x*(t)
y*(t)
Fig Open loop sampling system

47. 4.5 Response of the CCS

48. 4.5 Response of the CCS

49. 4.5 Response of the CCS

50. The response of the open-loop control system
4.5 Response of the CCS
The response of the open-loop control system

51. 4.5 Response of the CCS
G2(s)
E(s)
C(s)
R(s) -
G1(s)
E*(s)
U(s)
H(s)

52. 4.5 Response of the CCS

53. 4.5 Response of the CCS
0.5
0.5

54. The response of the open-loop control system
4.5 Response of the CCS
The response of the open-loop control system

55. 4.5 Response of the CCS
(2) System response between consecutive sampling
instants-Laplace transform method
G (s)
E(s)
C(s)
E(z)
E*(s)
Ф(z)
H (s)
R(s)
R*(s)
R(z)
C*(s)
C(z)
B(s) -

56. 4.5 Response of the CCS

57. 4.5 Response of the CCS
Example 4.9: Assume r(t)=1(t), T=1s, find the analyzing
solution of system output c(t).
G2(s)
E(s)
C(s)
R(s) -
G1(s)
E*(s)
U(s)

58. 4.5 Response of the CCS

59. 4.5 Response of the CCS

60. 4.5 Response of the CCS

61. 4.5 Response of the CCS

62. 4.5 Response of the CCS

63. 4.5 Response of the CCS

64. 4.5 Response of the CCS
Continuous output

65. 4.6 Performance Specifications of CCS
Dynamic specifications:
Delay time (td) : the time for the response to get to the half of the final value
Rising time (tr) : the time for the response to get to 90% of the final value from 10% of the final value for the over-damped system and transmission lag system; the time for the response to get to 100% of the final value from 0 for under-damped systems

66. 4.6 Performance Specifications of CCS
Peak time (tp) : the time for the response to get to the first peak
Overshoot (σp) : c(tp)-c(∞)/ c(∞)x100%
Settling time (ts) : if t≥ts, |c(t)- c(∞)|<Δ(Δ=0.02 c(∞) or 0.05 c(∞)), ts is set as the settling time, which is related to the maximum time constant of the system

67. 4.6 Performance Specifications of CCS
Dynamic specifications

68. 4.6 Performance Specifications of CCS
Stable state specifications:
stable state error
Integrated specifications: different performance specifications can be defined in the optimal control.

69. 4.7 State Space Description
4.7.1 Sampling continuous-time signals
The sampled version of the signal f is the sequence {f(tk):kZ}, where tk=kh, it is called periodic sampling and h is sampling period. The corresponding frequency fs=1/h(Hz) or s=2/h(rad/s) is called sampling frequency. Nyquist frequency is a very useful frequency fN=1/(2h)(Hz) or N=/h(rad/s).

70. 4.7 State Space Description
4.7.2 Sampling a continuous-time state-space system
To find the discrete-time equivalent of a continuous-time system is called sampling a continuous-time. The model obtained is called a stroboscopic model.
Consider the system (4.7.1)
(4.7.1)

71. 4.7 State Space Description
(1) Zero-order-Hold sampling of a system
A common situation in computer control is that the D-A converter is a zero-order-hold circuit, i.e. it holds the analog signal constant until a new conversion is commanded.

72. 4.7 State Space Description
The state at some future time t is obtained by solving (4.7.1). The state at time t, where tk t  tk+1 ,is

73. 4.7 State Space Description
The system equation of the sampled system at the sampling instants is then
(4.7.2)
The model in (4.7.2) is called a zero-order-hold sampling of the system in (4.7.1), can also be called the zero-order-hold equivalent of (4.7.1).

74. 4.7 State Space Description
For periodic sampling with period h, the model of (4.7.2) simplifies to the time-invariant system:

75. 4.7 State Space Description
(2) How to compute  and 
i. Using the equation directly for 1-order system
Example 4.10 First-order system
where a  0. we get
The sampled system thus becomes

76. 4.7 State Space Description
ii. Numerical calculation in MATLAB or MATRIX
when a = 1, b = 3, c = 1, d = 1, h = 0.1 in Example 4.10, then
a. Using Matlab command line, the result is as follows:
>> sys=ss(1,3,1,1);
>> sysd=c2d(sys,0.1)
The result is  = 1.105,  = , c = 1, d = 1.

77. 4.7 State Space Description
b. The second method using Matlab, the command is expm.
From the following equation,
We know that so
>> expm([1 3;0 0]*0.1)
ans =

78. 4.7 State Space Description
iii. Series expansion of the matrix exponential
Example 4.11 Double integrator
The double integrator is described by
Hence
The discrete-time model of the double integrator is

79. 4.7 State Space Description
iv. The Laplace transform of exp(At) is (sI-A)-1
Example 4.12 Motor
Simple normalized model of an electrical DC motor is given by
The Laplace transform method gives
Hence
where L-1 is the inverse of the Laplace transform.

80. 4.7 State Space Description
(3) The inverse of sampling
Get continuous-time system from a discrete-time description.
where ln(.) is the matrix logarithmic function. The continuous-time system is thus obtained by taking the matrix logarithm function of a block matrix.
*只有当矩阵Ф具有非负实数特征值时，该矩阵才存在对数。

81. 4.7 State Space Description
Example 4.13 Inverse sampling

82. 4.7 State Space Description

83. 4.7 State Space Description
(4) State-space block diagram
We can draw the state-space block diagram for a discrete-time control system, for example:

84. 4.7 State Space Description
(5) Solution of the system equation
Time-invariant discrete-time systems can be described by the difference equation
(4.7.3)
For simplicity the sampling time is used as the time unit, h = 1. Assume that the initial condition x(k0) and the input signals u(k0), u(k0+1),…are given. How is the state then evolving? It is possible to solve (4.7.3) by simple iterations.

85. 4.7 State Space Description

86. 4.7 State Space Description
Example 4.14 Solution of the difference equation
If |λi|<1,j=1,2,then x(k) will converge to the origin. If one of the eigenvalues of Ф has an absolute value larger than 1, then one or both of the states will diverge.

87. 4.7 State Space Description
4.7.3 Changing coordinates in state-space models
Assume that T is a nonsingular matrix and define a new state vector z(k) = Tx(k). Then

88. 4.7 State Space Description
The invariants under the transformation are of interest.
Theorem 3.1 Invariance of the characteristic equation. The characteristic equation
is invariant when new states are introduced through the nonsingular transformation matrix T.
Coordinates can be chosen to give simple forms of the system equations.

89. 4.7 State Space Description
(1) Diagonal Form
Assume that  has distinct eigenvalues. Then there exists a T such that
where i are the eigenvalues of .

90. 4.7 State Space Description
Consider the motor in example 4.12 with h=1. Using the transformation

91. 4.7 State Space Description
(2) Jordan Form
If  has multiple eigenvalues, then it is generally not possible to diagonalize . Let  be a n  n matrix and introduce the notation
where Lk is a k  k matrix.
Then there exists a matrix T such that
where k1 + k kr = n.

92. 4.7 State Space Description
4.7.4 Transforms between state space model and pulse transfer function

93. 4.7 State Space Description
(1) Pulse-transfer function to state-space models
If the pulse-transfer function is as following
its state-space equation is

94. 4.7 State Space Description

95. 4.7 State Space Description
Define state variable:
With inverse z-transform
Together with

96. 4.7 State Space Description
(2) Difference equation to state-space models
If the forward formation is
or the backward formation is

97. 4.7 State Space Description
Then the state-space equation is

98. 4.7 State Space Description

99. 4.7 State Space Description
Define state variable:
With inverse z-transform
Together with

100. 4.7 State Space Description
(3) Discrete state-space to Z transfer function

101. 4.7 State Space Description
Example:
Sampling this continuous-time system to obtain the corresponding discrete system, T = /2. When the input signal u(t) is a unit step, determine the analyzing solution of the y(t).
Solution:
1. Compute discrete-time system

102. 4.7 State Space Description
When T = /2,
The discrete-time system is

103. 4.7 State Space Description
2. Compute the analyzing solution of y(t)
The pulse-transfer function can now be introduced

104. 4.7 State Space Description
(z-transform of sin kT is )
Step response of the continuous-time system is

105. Summarization

106. Summarization
ComputeΦ, Γ
Inverse sampling
Draw block diagram

107. Summarization (2) Changing coordinates in state-space models
(3) Transforms between state space model and pulse transfer function

108. Homework
1. Find the pulse transfer function of the following systems:

109. Homework
2. Obtain the state-space equation for the following pulse-transfer function, and then draw the block diagram for the state-space form.
3. Obtain the state-space equation for the following difference equation, and then draw the block diagram for the state-space form.(y(0)=y(1)=u(0)=0)

110. Homework 4. Compute the z-pulse transfer function.
5. The system’s difference equation is
It is assumed that y(k)=0 k<0. when the input is 1(t), determine the
analyze solution of y(k).

111. Homework
6. Try to compute the transform matrix T when the original system and the changed system are described as follows: