1、2022/9/21Chapter 3-61CHAPTER 3Time Domain Analysis of Control System and Its Characteristics By Hui Wang2022/9/21Chapter 3-62Outline of this chapterIntroductionSolution of linear differential equationsDynamic of first and second order systems Time response specifications(indices)Rouths stability cri

2、terionSteady-state error analyzingSolution of the state Equation2022/9/21Chapter 3-63Solution of the state Equation (see P82-89,Ch.P103-115) state-variable equations state transition matrix(STM) evaluating the state transition matrix complete solution of the state equation stability of state space m

3、odel 2022/9/21Chapter 3-64State-variable equationsSystems differential equation is an input/output model, it only expresses the relationship of input and output. -classical control theory modelThe state space model can depict the interior variable of the system, can describe multi-variable systems a

4、nd nonlinear systems. And state space model can easily be used by computer. -modern control theory modelA feature of the state-variable method is that it poses a complex system into a set of smaller systems that can be normalized to have a minimum interaction and that can be solved individually. 202

5、2/9/21Chapter 3-65State-variable equationsState equationOutput equationState space modelThe diagram of state space model Sx1,x2, xn u1u2umy1y2ylThe variables x(t), u(t), and y(t) are column vectors, and A, B, C and D are matrices having constant elements for a LTI system. The system has m inputs, l

6、outputs, and n state variables.To determine the output of the system y(t), the state equation is solved first.2022/9/21Chapter 3-66State transition matrix(STM) If n=1 and initial condition x(t=0)=x(0), the state equation is a scalar equation, which represents a first-order system. It is easy to get

7、the solution of the scalar equation.Similary the general solution of a linear differential equation, the solution of a state equation is the sum of two components. One of them is the solution of its corresponding homogeneous equation, which is the transient component of the solution.Consider the hom

8、ogeneous state equation firstly, which with the input u(t)=0, i.eFor any other initial condition, at time t0, x(t0) is known, then 2022/9/21Chapter 3-67State transition matrix(STM) Comparing the solutions obtained from different ways, they should be equal. Therefore, we have When we find the solutio

9、n in the S domain, then Consider the solution of the scalar equationwhere(請復(fù)習(xí)冪級數(shù)ex的展開(- x) LTIt is useful and interesting!2022/9/21Chapter 3-68State transition matrix(STM) Comparing the scalar and the state equation, the solution of the state is analogous to that of scalar, or Laplace method is used

10、LTSoWhere A are a square matrix, expAt is a square matrix of the order as A, which is an infinite series. But it has closed form, which is called state transition matrix (STM) or fundamental matrix of the system and is denoted by n=nn=12022/9/21Chapter 3-69State transition matrix(STM) For LTI system

11、, but they have difference in concept. Matrix exponent（矩陣指數(shù)）, mathematics function STM, has physical meaning and is applicable to time-varying and/or discrete systemAs a mathematics function, has the following properties:1) If A is a diagonalizing matrix, then expAt is too. I.e 2) 3) If t and s are

12、independent variables, then2022/9/21Chapter 3-610State transition matrix(STM) 4)5) always nonsingular matrix, and its inverse matrix is 6) For nn square matrixes A and B, if AB=BA, then 7) If there is any nonsingular matrix T, then These properties can be proved by the definition of .See Ch. P104-10

13、52022/9/21Chapter 3-611State transition matrix(STM) The term state transition matrix (STM狀態(tài)轉(zhuǎn)移矩陣) is descriptive of the unforced or natural response and is the expression preferred by engineers. The STM has the following properties: 2022/9/21Chapter 3-612There are several methods for evaluating the S

14、TM in closed form for a given matrix A.Evaluating the state transition matrix1) Directly calculation 2) Using Laplace transform (See P126-128, 4.17)3) Canonical(diagonalizing) matrix A(See P168-169, 5.11)See Ch. P106-1084) Cayley-Hamilton theorem(See P85-88, 3.14)2022/9/21Chapter 3-613Evaluating the

15、 STM: 1) Directly calculation Solution:Example 1. Assuming matrix , find exp(At) by method 1. andandSee Ch. P106-1072022/9/21Chapter 3-614Evaluating the STM: 2) Laplace tranform Example 2. Assuming matrix , find exp(At) by method 2. See Ch. P127Solution:2022/9/21Chapter 3-615Evaluating the STM: 2) L

16、aplace tranform Solution:Example 3. Assuming matrix , find exp(At) by method 2. See Ch. P107The result is the same as that given in Ex.1. 2022/9/21Chapter 3-616Evaluating the STM: 3) diagonalizing matrix A For a non-diagonal matrix A, we can diagonalize it first, that means to evualute the modal mat

17、rix T for diagonalizing A. Then Four methods are given in 5.10.P162-168. Method 1 For a companion form matrix A(A=AC), when eigenvalues i are distinct, T is easily obtained, which is called Vandermonde matrix. 2022/9/21Chapter 3-617Evaluating the STM: 3) diagonalizing matrix A Method 2 It is not req

18、uired A to be a companion form matrix. The modal matrix T is defined by , yields Eigenvector vi is proportional to any nonzero column of adji I-A2022/9/21Chapter 3-618Evaluating the STM: 3) diagonalizing matrix A Method 3 It is an alternate method for evaluating the n elements of each vi is to form

19、a set of n equations from the matrix equation Each value of i is inserted in this matrix, and the corresponding elements are equated to form n equations. Method 4 See P166.2022/9/21Chapter 3-619Evaluating the STM: 3) diagonalizing matrix A Example 4. Assuming matrix , find exp(At) by method 3.Soluti

20、on:We can find the eigenvalues of A:Since A is not diagonal matrix, it is required to evualute the modal matrix for diagonalizing A. For this example, A is not in companion form, the modal matrix T should be evualuted. Here using method 2 given in 5.10 The result is the same as that given in Ex.2. S

21、ee P1692022/9/21Chapter 3-620Review: Characteristic valuesThe roots i is of the characteristic equation are called the characteristic values or eigenvalues of A.Consider a system of equations represented byCharacteristic equationThe polynomial Q() may be written in factored form asThe procduct of th

22、e eigenvalues of a matrix A is equal to its determinant, that is, 2022/9/21Chapter 3-621Cayley-Hamilton theorem(see P85-86)凱萊-哈密爾頓(Cayley-Hamilton)定理考慮nn維矩陣A及其特征方程凱萊-哈密爾頓定理指出，矩陣A滿足其自身的特征方程，即推論1：矩陣A的 k(kn) 次冪可表示為A的 (n-1) 階多項式推論2：矩陣指數(shù)eAt可表示為A的 (n-1) 階多項式詳細(xì)的中文證明可參見胡壽松書P437440。2022/9/21Chapter 3-622Cayl

23、ey-Hamilton theorem(see P85-86)為了證明Cayley-Hamilton定理，注意到（IA）的伴隨矩陣adj(IA)是的n1次多項式，即可得式中，B1=I 。由于從上式可看出，A 和 Bi (i=1,2,，n)相乘的次序是可交換的。因此，如果(I-A)及其伴隨矩陣adj(I-A)中有一個為零，則其乘積為零。如果在上式中用A代替，顯然I-A為零。這樣即證明了凱萊-哈密爾頓定理。2022/9/21Chapter 3-623Cayley-Hamilton theorem(see P85-86)Consider a general polynomial of the for

24、mThe matrix polynomial N(A) (where A is variable) When the polynomial N() is divided by the characteristic polynomial Q(), the result: The remainder, maximum order is n-1, or 1 less than the order of Q() For =i the value Q(i)=0; thusor2022/9/21Chapter 3-624Cayley-Hamilton theorem(see P85-86)Cayley-H

25、amilton theorem: “every square matrix A satisfies its own characteric equation.” There are n equations The analogous matrix equation is is a null matrix with the same order as Q(A) Characteristic polynomial Q()2022/9/21Chapter 3-625Cayley-Hamilton theorem(see P85-86)For there are n roots i , which y

26、ield n equations that can be solved simultaneously for the coefficients k , then they can be used in The matrix polynomial N(A) is therefore The exponential function N()=e t is convergent in the region of analyticity, it can be expressed in closed form by a polynomial in of drgree n-1. For each eige

27、nvalueSo, for the STM 2022/9/21Chapter 3-626Evaluating the STM: 4) Cayley-Hamilton theoremExample 5. Assuming matrix , find exp(At) by method 4.The roots of this equation areSolution: The characteristic equation isFor A is second order matrix, thusThe result is the same as that given in Ex.2.2022/9/

28、21Chapter 3-627Complete solution of the state equationFor e-At is a square matrix, x(t) is a n1 state vector, soUsually there are two methods for evaluating the complete solution of the state equation: direct solving equation (time domain) and using Laplace transformation (S domain). Method 1 Review

29、: derivative of a product of matrices rule:For state equation:Integrating this equation from time 0 and tHow to get x(t)?(See P88-89, Ch.111-112)2022/9/21Chapter 3-628Complete solution of the state equationUsing the STM and generalizing for initial conditions at time t=t0, gives the solution to the

30、state-variable equation with an input u(t) as (See P88-89, Ch.111-112) From equation , we want to get x(t). state transition equation Let: =t- 2022/9/21Chapter 3-629Complete solution of the state equation(See P88-89, Ch.111-112)zero-input response: u(t)=0zero-state response: x(t0)=02022/9/21Chapter

31、3-630Complete solution of the state equation(See P88-89, Ch.111-112)Example 6 Solve the following state equation for a unit step-function scalar input u(t)=1. Solution:The STM for the matrix A is given by Ex.2,4,5. If the inintial conditions and output equation is given, then not only the state traj

32、ectory but also output response could be easily obtained.See the example in Ch. P112.2022/9/21Chapter 3-631Complete solution of the state equation Method 2 By Laplace transformation (See P126-128, Ch.112-113)state equation:LTLT-12022/9/21Chapter 3-632Complete solution of the state equation(See P88-8

33、9, Ch.111-112)Example 7 Solve the following state equation for a unit step-function scalar input u(t)=1 by Laplace mrthod. Solution:The (s) for the matrix A is given by Ex.2The result is the same as that given in Ex.6.LT-12022/9/21Chapter 3-633Stability of state equation model (See Ch.113-114)We have learnt the stability criterion for a control system, however, is it useful to a system which represented by a state space model? Yes. It is because of the characteristic equation for a system b