Mathematical Descriptions of Systems Chapter 1 Mathematical Descriptions of Systems

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.

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.

§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

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.

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?

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

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

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:

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).

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:

where h(t) is an impulse function.

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:

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

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.

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:

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

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)

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

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 .

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.

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

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

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.

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

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

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 .

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.

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

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.

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.

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

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.

t0 t01 u(t) t0 t01 y(t)

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

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

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

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.

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.

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

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.

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.

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.

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

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 .

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

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,

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

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 

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

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

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

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

and the transfer function is

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 =

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

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

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

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

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.

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

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

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

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).

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 .

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).

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.

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.

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

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

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

Summary Relaxed at  Linearly Causality

Relaxed at t0 Time-invariance t0=0

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