信號與系統(tǒng)英文版教學課件：ch8 StateModelRepresentation

1、 There are two types of mathematical models of systems: input/output representation and state-variable representation. The input/output representation describes the input/output behavior of systems. The state-variable representation describes the internal behavior of systems. The objective of this c

2、hapter: define the state model and study the basic properties of this model for both continuous-time and discrete-time systems.INTRODUCTION11. State model For a single-input single-output causal continuous-time system, its input : v(t) output: y(t) Obviously it is not. The reason is that the applica

3、tion of the input v(t) for may put energy into the system that affects the output response for .Consider the question: At a value t1 of the time variable t , is it possible to compute the output response y(t) from only the knowledge of the input v(t) for ?2 If the system is zero at t1, y(t) can be c

4、omputed from v(t) for .If the system is not zero at t1, knowledge of the state is necessary to be able to compute the output y(t).For any time point t1, the state x(t) of the system at time is defined to be that portion of the past history of the system required to determine the output response y(t)

5、 for all given the input v(t) for . A nonzero state at time indicates the presence of energy in the system at time . 3Example 8.1 Consider the circuit in following figure. compute the currents in L1 and L2, and the voltage on C.Let x1 be the current in L1, x2 be the current in L2, x3 be the voltage

6、on C,4Rewrite the former equations, respectively, as5If we get x1, x2, x3,we can get all the information about the system. So they are necessary and enough.Matrix form representaiton:6The components x1(t),x2(t).xN(t) are called the state variables of the system. From the example, if the given system

7、 is finite dimensional, the state x(t) of the system at time t is an N-element column vector given by: 72. State EquationsFor a single-input single-output N-dimensional continuous-time system with state x(t) given by : It can be modeled by the state equation given by :derivative of the state vectoro

8、utput equation8The state equations describes the state response,while output equation gives the output response. The two parts correspond to cascade decomposition of the system as illustrated in the Figure below. Cascade structure corresponding to state model 9 In (1), A(t) is a NN matrix, B(t) is a

9、n N-element column vector. In (2), C(t) is an N-element row vector and D(t) is a real-valued function of time. The number N of state variables is called the dimension of the state model (or system). (1)(2)If f and g are linear, the state equations can be written in the form:If the system is time inv

10、ariant, then A(t), B(t), C(t) and D(t) are constant.10In this case, the state model is given by: (3) (4) With aij equal to the ij entry of A and bi equal to the ith component of B, (3) can be written in the expanded form :11 With , the expanded form of (4) is: From the expanded form of the state equ

11、ations, it is seen that the derivative of the ith state variable and the output y(t) are equal to linear combinations of all the state variables and the input.123. Construction of State Models Consider a single-input single-output continuous-time system given by the first-order input/output differen

12、tial equation: Defining the state x(t) of the system to be equal to y(t) results in the state model: If the given system is linear and time invariant so that:13 a and b are constants,then the state model is : Now suppose that the given system has the second-order input/output differential equation:D

13、efining the state variables by : yields the state model:14Example 8.2 Consider a continuous-time second-order LTI system described by the following input-output equation:Construct its state model.Let to obtain:Thus,15Example 8.3 If the input-output equation for the system in example 8.2 is:Construct

14、 its state model.The system function for the system is:y(t)v(t)1/s-a11/sb0b1-a0 x1(t)x2(t)Then16Rewrite the system function asLet , where z(t) is the output of H1(s).ThenAndThus, the state model is:17The resulting state equations are: The defining of state variables in terms of the output and deriva

15、tives of the output extends to any system given by the Nth-order input/output differential equation: with the state variables defined by18Now consider the linear time-invariant system given by the Nth-order input/output differential equation:Its block diagram representation is:y(t)v(t)1/sa1 bN-1a01/

16、sb0b11/saN-1:x1(t)x2(t)xN(t)19 This system has the N-dimensional state model where : 20Example 8.4 Consider a continuous-time LTI system with transfer function Draw the direct-form, cascade-form and parallel form signal flow graph of the system, respectively. And construct the state models of the sy

17、stem on the signal flow graph, respectively.Direct-formstate model21cascade-formstate modelThus,22parallel-formstate modelThus,23Example 8.5 Integrator Realization Consider a two-dimensional state model with arbitrary coefficients; that is,Draw the signal flow graph of the system.Step 1: Define the

18、output of each integrator in the interconnection to be a state variable. Then if the output of the ith integrator is , the input to this integrator is . x2(t)v(t)y(t)x1(t)24Step 2: Realize the state equationx2(t)v(t)y(t)x1(t)a11a12a21a22b1b225Step 3: Realize the output equationb1x2(t)v(t)y(t)x1(t)b2

19、a11a12a22a21c1c2264. Multi-Input Multi-Output Systems The state model of a p-input r-output LTI Nth-order continuous-time system is given by: where now B is a Np matrix of real numbers, C is a rN matrix of real numbers, and D is a rp matrix of real numbers. The matrix A is still NN.27From the figure

20、,Example 8.6 Two-Input Two-Output SystemConsider the two-input two-output system that is shown in the Figure-3Inserting the expression for into the expression for gives 28Putting these equations in matrix from results in the state model295. Solution of State Equations matrix exponential :For each re

21、al value of t, is defined by the matrix power series: where I is the NN identity matrix. Properties of :for any real numbers t and , always has an inverse, which is equal to the matrix 30As a result of this property, The matrix is called the state-transition matrix of the system.it is seen that the

22、state at time t resulting from state at time with no input applied for can be computed by multiplying by the matrix .From the derivative property of matrix exponential, we have that the solution of is: The derivative of the matrix exponential31 For the eqution , Multiplying both sides on the left by

23、 and rearranging terms yields: From the derivative property of the matrix exponential,we can getThis is the complete solution of the state equation resulting from initial state and input applied for . 32 6. Output ResponseFrom and the solution for the state equations, we can get: From the definition

24、 of the unit impulse,we can rewrite the former equation as: And the impulse response matrix isWhere the zero-input response and the zero-state response are:337. Solution via the Laplace TransformWhere is the Laplace transform of the state-transition matrix . from this we can get:Taking the Laplace t

25、ransform of the state equation gives:34 where H(s) is the transfer function matrix of the system given by Taking the Laplace transform of the output equation yields: From the Laplace transform solution for state variable x(t), we can get:If x(0)=0, then35Example 8.7 Consider the two-input three-outp

26、ut two-dimensional system with state model , whereif the initial state and input , compute the output y(t).First compute the state-transition matrix: Since36Thus, state-transition matrixThe state response x(t) resulting from the initial state x(0) with zero input is given by , so 37SinceThe state re

27、sponse x(t) resulting from the input is to be computed.From , we have38Taking the inverse Laplace transform of Xzs(s) yields Then the state variableThe output response398. Discrete-Time Systems A p-input r-output finite-dimensional linear time-invariant discrete-time system can be modeled by the sta

28、te equations: The state vector xn is the N-element column vector:40The matrix A, B, C, and D are NN, Np, rN, and rp respectively.The input vn and output yn are the column vectors:419. Construction of State Models For a single-input single-output linear time-invariant discrete-time system with the in

29、put/output difference equation ：The system function for the system is:Rewrite the system function as42Defining the state variables：Where zn is the output of the first sub-system H1(z).Then43Thus, the state model is:where4410. Solution of State EquationsSetting n=0 in (5) givesConsider the p-input r-

30、output discrete-time system with the state model:(5)(6)Setting n=1 in (5) givesIf this process is continued, for any integer value of , 45The right-hand side of the former equation is the state response resulting from initial state and input applied for . Note that if for , then it is seen that the state transition from initial state to state at time n (with no input applied) is equal to times the matrix . Therefore, in the discrete-time case the state-transition matrix is th