1. Mathematical Descriptions of Systems
Chapter 1
Mathematical Descriptions of Systems

2. Linear system theory studies physical systems that can be described by linear differential equations. Any system can be expressed as the following block diagram.
Inputs
System
Outputs
To understand the system behavior, a mathematical model that is able to describe faithfully the dynamics of the system is necessary.

3. Consider the following R-C network
whose mathematical model is
It is clear that we can compute the response under any input once we obtain the mathematical model of the system.

4. §3-1 The Input-Output Description
For the input-output description, the knowledge of the internal structure of a system is assumed to be unavailable; the only access to the system is the input and output. Under this assumption, a system may be considered as a “black box”
Black Box u y
In this case, what we can do is to apply a series of typical inputs and observe the corresponding outputs

5. A system may have more than one input terminals or more than one output terminals.
Black Box u y
Definition: A system is said to be a single-variable system if and only if it has only one input and one output.

6. A system is said to be a multivariable system if it has more than one input terminals or more than one output terminals.
2.Relaxedness
Question
What conditions should be satisfied if the output can be excited solely and uniquely by the input?

7. Example:
Consider the following second-order system:
whose initial conditions are:
Assume that only input and output signals are available. If the initial condition is unknown, it is clear that the output can not be determined uniquely by the input u. Because the input u is a control signal, hence, it is impossible to judge which part of the output signal is excited by the input and which part

8. is excited by the initial condition.
In classic control theory, we always assume that all the initial conditions of a given system are zero and therefore, the output can be excited by input uniquely:
When the concept “energy” is applicable to a system, the system is said to be relaxed at time t0, if no energy is stored in the system at that instant. Therefore, represent the energy the

9. system stores from time to time t=0.
For an arbitrary physical system, we assume that it stores no energy at time In other words, it is relaxed or static at time Then, we can give the definition of the initial relaxedness of a system:

10. Definition: A system is said to be a initially relaxed system if it is relaxed or static at time .
For a relaxed system, we have
y=Hu (1-1)
where H is some operator that determines uniquely the output y in terms of the input u of the system. The equation (1-1) can be expressed in the following form:
Generally, represents a function defined over (t1, t2).

11. Operator H:
Transfer function is a linear operator.
The transfer function maps the signal into
which is expressed as
In the real number field, the operator H is a convolution:

12. where h(t) is an impulse function.

13. Linearity 1. Definition of linearity
Definition A relaxed system is said to be linear if and only if
for any inputs u1 and u2 and real numbers 1 and 2. Otherwise, the relaxed system is said to be nonlinear.
In engineering literature, the condition of (1-3) if often written as
Additivity:
Principle of superposition
Homogeneity:

14. Example. Consider the Laplace transform：
By definition, it is easy to verify that the Laplace transform is a linear system.
Example. Consider a system described by the following differential equation:
which is a linear system. In fact, we have
It is easy to verify that

15. Example. Consider the following system whose input output are related by
It can be checked that the input-output pair satisfies the property of homogeneity but not the property of additivity. Therefore, the system is not a linear system.

16. 2 Impulse response function of a relaxed system
We need the concept of  function or impulse function, which can be derived by introducing a pulse function (tt0) shown in the following figure.
Pulse function: t
1/△
t1 t1+△
As  approaches zero, the limiting “function” is called impulse function, or Dirac delta function, or simply -function:

17. -function: -function has the properties that
for any positive  and that

18. for any function f that is continues at t1.
As shown in the following figure, every piecewise continuous input can be approximated by a series of pulse functions. Since every pulse function can be described by t
1/△u(tn)△
= u(tn)
tn tn+△ tn
u(t)

19. Impulse response function
Let Then , the summation becomes an integration and becomes
. Consequently, as 0, (1-7)becomes
(1-8)

20. The physical meaning of H  (t) is that it is the output of the relaxed system due to an impulse function input applied at time .

21. Hence, the impulse function can be expressed as a function with two variables
the variable denotes the time at which the -function is applied and the variable t denotes the time at which the output is observed.

22. where,  is the time at which the output is observed
where,  is the time at which the output is observed. Using (1-9-1) we can write the equation (1-8) in the following form

23. Impulse-response matrix If a system has p input terminals and q output terminals, and if the system is initially relaxed, the input-output description (1-10) can be extended to
where

24. and gij(t, ) is the response at time t at the ith ouput terminal due to an impulse function applied at time  at the jth input terminal with the inputs at other terminals being identically zero.

25. 4.Causality
Definition: A system is said to be causal if the output of the system at time t does not depend on the input applied after time t; it depends only on the input applied before and at time t. In short, the past affects the future, but not conversely.
If a relaxed system is causal, the output is identically zero before any input is applied. Hence, a linear system is causal if and only if

26. Consequently, the input-output description of a linear, causal, relaxed system becomes
Example: We often use the truncation operator to express the causality in mathematics. The definition of truncation operator is as follows

27. which is shown in the following figure
past
future
present
A system is causality if the following equation holds:
Equation (*) means that the input in can not affect output in

28. The future input can not affect the passed and the present output.y t t u y
The future input can not affect the passed and the present output.

29. 5. Relaxedness at time t0 definition of relaxedness at time t0
Definition 1-3 A system is said to be relaxed at time t0 if and only if the can be determined by solely and uniquely.
If a system is known to be relaxed at t0, then its input-output relation can be written as
Definition: A linear system is said to be relaxed at t0 if and only if

30. In particular, if the system is causal, we have
Example: Consider the system
we have
It is clear that the system is a relaxed and causal system at t0, because can be determined by solely and uniquely.

31. Example: Consider the system
we have
Although the output can be determined uniquely, it is not a relaxed system at t0, that is because can not be determined by solely.

32. Example: If a linear system satisfies , then it is relaxed at t0
Example: If a linear system satisfies , then it is relaxed at t0. In fact

33. is only a sufficient condition of relaxedness at t0,
Example: Consider a unit-time-delay system
It is easy to verify that H is a linear operator. Although we have if
the system is still relaxed at t0. In order to determine by uniquely, it suffices to know that is zero.

34. t0
t01
u(t) t0
t01
y(t)

35. Criterion
Theorem: The system that is described by
is relaxed at t0 if and only if
implies
Proof.
Necessity. If a system is relaxed at t0, the output y(t) for t t0 is given by

36. Sufficiency: We show that if
then the system is relaxed at t0. Since
The assumption u[t0, +)0, y[t0, +)0 implies that

37. In words, the net effect of u(, t0) on the output y(t) for t t0 is zero. Hence,
Q.E.D.

38. The relaxedness of the system can be determined from the behavior of the system after t0 without knowing the previous history of the system. Certainly, it is impractical or impossible to observe the output from time t0 to infinity; fortunately, for a large class of systems, it is not necessary to do so.

39. The following corollary gives a more applicable criterion for a system that is relaxed at t0.
Corollary: If the impulse response matrix
can be decomposed into
and if every element of M(t) is analytic on (, ), then the system is relaxed at t0 if for a fixed positive , implies
Remark: Because  is a fixed positive number, the corollary is applicable.

40. Example. Consider the following systemIf
using the property of matrix exponential , G(t,) can be decomposed into
where M(t)=eAt is an analytic function.

41. Appendix: Analytic Function
Let D be an open interval in the real line R and let f(·) be a function defined on D; that is, to each point in D, a unique number is assigned to f.
A function of a real variable, f(·), is said to be an element of class Cn on D if its nth derivatives f(n)(t), exists and is continuous for all t in D. C is a class of functions having derivatives of all orders.

42. Definition: a function of real variable, f (t), is said to be analytic on D if f(t) is an element of C and if for each t0 in D there exists a positive real number 0, such that for all t in (t00, t0 +0), f (t) is representable by a Taylor series about the point t0
For instance, polynomials, exponential functions, and sinusoidal functions are analytic in the entire real line.

43. Theorem: If a function f is a analytic on D and if f is known to be identically zero on an arbitrarily small nonzero interval in D, then the function f is identically zero on D.
Proof. If the function is identically zero on an arbitrarily small nonzero interval, say, (t0, t1), then the function and its derivatives are all equal to zero on (t0, t1). By analytic continuation, the function can be shown to be identically zero.
Q.E.D.

44. D1 D , D1 is the interval on which f(t)0
f (t) is analytic on D
D1 D , D1 is the interval on which f(t)0 D

45. Corollary 1-1. If the imponse response matrix
can be written as
and each element of is analytic over , then the system is relaxed if for a constant , implies that

46. Proof We only need to prove that implies i.e.
let , then

47. Because
is a constant, then the assumption that M(t) is a analytic function implies that y(t) is analytic over The equation implies that Hence,
Using the above theorem,

49. Example: consider the system
where A and B are constant matrices with proper dimension. We have
If x(t0)=0, the system is relaxed at t0. Hence

50. 6. Time invariance
If the characteristics of a system do not change with time, then the system is said to be time invariant. t t u y  t u y t 

51. which is relaxed at time t=0.
Example: Consider the linear system
which is relaxed at time t=0.
Consider the responses of the system with two inputs u=1(t) and u=1(t1), respectively.
Response of u=1(t): t
Response of u=1(t-1): From the Laplace transform theorem t
we have

52. 1. Definition of shifting operator and time-invariant system
Conception of Q.
The effect of the shifting operator Q is illustrated in Figure 1-5. the output of Q is equal to the input delayed by  seconds. u t α
Figure 1-5

53. Definition 1-4 A relaxed system is said to be time invariant if and only if
for any u and any . Otherwise the relaxed system is said to be time varying.  t u y t  t  t

54. Example: Show that for a constant , the shifting operator Q is a linear time-invariant system. Then, compute its impulse function and transfer function.
Proof: It is clear that Q is linear. From the definition of linearity, we only need to prove that
for any real number . In fact
Hence, the system is LTI. Its impulse function is

55. and the transfer function is

56. 2. Impulse response function of time invariant system
For relaxed linear system, if it is time invariant, the impulse response function of the system becomes
In fact, from the property of time invariance, we have
Right hand side =
Left hand side =

57. which implies that
for any t, ,  .
That is, if the input is shifted by  seconds, the output is shifted by  seconds also.

58. In particular, if , we have
For simplicity, we write as
It shows that for a time-invariant system the impulse response is only determined by the difference between t and  .
Example: Consider the linear relaxed causal time-invariant system

59. 3. multi-variable system
For all t and , we have
Hence,the input-output pair of a linear causal time-invariant system which is relaxed at t0 satisfies
For time invariant systems, without loss of generality, we can choose t0 =0. Now the equation (1-19) can be written as

60. Example: The system
is a linear time-invariant system which is relaxed at zero. In fact, the solution of the differential equation is

61. 7. transfer-function matrix, poles and zeros
1. transfer-function matrix Taking the Laplace transform of y(t), we have
From the convolution integral theorem of the Laplace transform, we have
where
is the Laplace transform of impulse response which is said to be the transfer-function matrix of the system.

62. In this course, all the elements of transfer-function matrix are rational of s.
Example: The impulse response matrix of a system is
Find its transfer-function matrix.
Taking the Laplace transform for each element of G(t) yields

63. Proper and strict proper We assume that the numerator polynomial and denominator polynomial of every elements of G(s) have no Common divisors.
Definition A rational transfer function G(s) is said to be proper if is nonzero, and G(s) is said to be strict proper if
Zeros and poles of the transfer functions

64. Assumption G(s) is a q×p rational function matrix with rank r, where the rank of a transfer matrix is the highest order of G with non-zero minor.
Example Consider the following transfer function matrices

65. Definition 1-5 The pole polynomial of G(s) is defined as the least common denominator of all minors of G(s). All zeros of pole polynomial are called poles of G(s).

66. Example Consider the following transfer function matrix
From the definition 1-5, the common denominator of minors of order 1 is (s+1)(s1)(s+2). The minors of order 2 are
The common denominator of minor of order 2 is

67. Hence, the characteristic polynomial of G(s) is
That is, G(s) has four poles, 1, 2, 2 and +1.
Definition 1-6 When the denominator of all the r order minors of G(s) are substituted by the characteristic polynomial, the common factor of the numerator is saied to be the zero polynomial of G(s). The roofs such that the zero polynomial equals to zero are called zeros of G(s).

68. Example: Consider the following transfer function matrix
The characteristic polynomial of G(s) is
The three minors of order 2 are
whose common factor is(s1). Hence the zero polynomial of G(s) is (s1), and G(s) has one zero s=1.

69. Example Consider the following transfer function matrices
Find their characteristic polynomials.
The four minors of order 1 of G1 are
The minor of order 2 is zero. Therefore, the characteristic polynomial is s+1.

70. The four minors of order 1 of G2 are
The minor of order 2 is
Therefore, its characteristic polynomial is (s+1)2.

71. Example Consider the following transfer function matrix
The minors of order 1 are the elements of G(s). The minors of order 2 are

72. The least common denominator of all minors of G(s) is s(s+1)2(s+2)(s+3).

73. Summary
Relaxed at 
Linearly
Causality

74. Relaxed at t0
Time-invariance
t0=0

75. Laplace transform
SISO
g(s) is a transfer function that we studied in classical control theory.