自控课件自动控制理论

1、1Ch9 State space and state equation9.1 State vector and state space9.2 State equation and output equation9.3 State Differential equation9.4 Signal-flow Graph State Model9.5 Realization and Minimum Realization9.6 The transfer function of the state Equation.9.7 The State Transition Matrix29.1 State ve

2、ctor and state spaceStatethe minimum set of variables (called the state variables) which at some initial time t0 , together with the inputs signal u(t) for time , suffices to determine the future behavior of the system for time . x1(t), x2(t), , xn(t).3The State variables describe the present config

3、uration of a system and can be used to determine the future response, given the excitation inputs and the equations describing the dynamics.4Example 9.1Define a set of state variables:The equation can be rearranged as:5State vector Vector which consists of n state variables that described entirely t

4、he dynamics action of a known system. that is described as 6State space is n dimensions space which takes x1,x2xn as coordinate.The manner in which the state variables change as a function of time may be thought of a trajectory in the n dimensional state space.7The properties of state variablesLinea

5、r independent each otherNot unique and may be selected to suit the problem being studied.The number of state variables is uniqueThe method, how to select the set of state variables, is not unique. 8Example 9.29A choice of state variables:Another choice :10Conclusion: For a passive network, the numbe

6、r of state variables required is equal to the number of independent energy storage elements.The state variables that describe a system are not a unique set. (linear transformation)A widely used choice is a set of state variables that can be readily measured.119.2 State equation and output equationSt

7、ate equation A set of first-order differential equations. Output equation A set of algebraic equation which describe the relationship of output vector and input vector, state vector.12Standard form of linear time-invariant system state equation and output equation:State vectorInput vectorOutput vect

8、or13System (state) matrixInput (control) matrix14Input-output matrixState-output matrix15Fig. Diagram of state variablesy(t) B C Au(t)x(t)x(t)+ D16A time-varying control system is a system for which one or more of the parameters of the system may vary as a function of time.17Nonlinear time-unvarying

9、 system Nonlinear time-varying system18Classical theoryModern theorySISO, linear, time-unvarying systemLaplace transformTime, frequency domainOutputMIMO, nonlinear, time-Varying systemMatrix, vector, linear algebraTime domainState19In example 9.220219.3 State Differential equationDifferential equati

10、on Transfer function Block diagram State-space model 22State-space model from differential equation(1). No differential of input2324matrix：D0可观测规范II型。b0=1时为能控规范型。25Example 9.3 DC motorset26(2). differential of input27Method 1:Take28We can have29From which we get30So the state equation31The matrix:可观

11、测规范II型。32Example 9.433可观测规范II型。34Method 2:35Take :State equation：能控规范型。36Example 9.5能控规范型。379.4 Signal-flow Graph State Model(1) phase variable format 相变量模型 3839Let4041State equation：能控规范型。42If m=n 43The state equation：能控规范型。44Example 9.6 n=4454647The state equation: 可控规范型48 (2) Input feedforward fo

12、rmat 4950The matrix:So 51(3). Observable canonical form 可观测规范型 52So The matrix:可观测规范型53Example 9.7 open-loop transfer function find out the state equation. 54Observable form:Controllable form:55(4) Alternative state Models(a). Diagonal canonical form. 56575859Example 9.860(b) Jordan canonical formWh

13、en there are multiple roots in characteristic equation, such as :61626364Jordan canonical form659.5 Realization and Minimum Realization1.Modeling methodsmechanism modeling(机理建模): construct systems differential equations (state equations and transfer function).identity modeling（辨识建模）: time-domain ide

14、ntity modeling and frequency-domain identity modeling.662. Realization of systemDefinition: for the transfer function G(s) or impulsive responding function g(t) of a specific system, if we can find a state space equation A, B, C to satisfy that: or then we call A, B, C as a realization of the system

15、 with the transfer function G(s) or impulsive responding function g(t).6768 3. Properties of realization Suppose A, B, C is a realization of the specific transfer function G(s), then it is the minimum realization if and only if it is completely controllable and observable.If G(s) has no zero-poles c

16、ancellation, then there exists the minimum realization A, B, C of the system, which is completely controllable and observable. 69If G(s) have the same zero and pole, then system can not be realized to be completely controllable and observable.If A1, B1, C1 and A2, B2, C2 are two minimum realization

17、of G(s), then A1, B1, C1 is algebraic equivalent to A2, B2, C2 .704. The equivalence of minimum realization and real structure of systemIf the system is controllable and observable, the minimum realization is algebraic equivalent to real structure of the system.If the system is noncontrollable or no

18、nobservable, the minimum realization is partly algebraic equivalent to real structure of the system.719.6 The transfer function from the state Equation(1) Solution of the state equation72somethod 173Then74method 275Thenor76非齐次状态方程的解由两部分组成：第一部分是初始状态所引起的状态响应，称为零输入响应；第二部分是由输入所引起的状态响应，称为零状态响应。 也可理解为：第一部分初态的转移。第二部分外加输入引起的转移。77State equation:(2) Transfer Function Matrix from State Space Model 7879809.7 The State Transition MatrixThe Solution of first-order differential equation81Let82when u=0 , unforced systemState transition matrix 83Characters： State Transition Matrix84Method1Method2Evaluation of