控制理论基础-自控a_ch9

1,Ch9 State space and state equation,9.1 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 Matrix,2,9.1 State vector and state space,Statethe 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).,3,The State variables describe the present configuration of a system and can be used to determine the future response, given the excitation inputs and the equations describing the dynamics.,4,Example 9.1,Define a set of state variables:,The equation can be rearranged as:,5,State vector Vector which consists of n state variables that described entirely the dynamics action of a known system. that is described as,6,State 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.,7,The properties of state variables,Linear 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.,8,Example 9.2,9,A choice of state variables:,Another choice :,10,Conclusion: For a passive network, the number 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.,11,9.2 State equation and output equation,State 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.,12,Standard form of linear time-unvarying system state equation and output equation:,State vector,Input vector,Output vector,13,System (state) matrix,Input (control) matrix,14,Input-output matrix,State-output matrix,15,Fig. Diagram of state variables,16,A 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.,17,Nonlinear time-unvarying system Nonlinear time-varying system,18,Classical theory,Modern theory,SISO, linear, time-unvarying systemLaplace transformTime, frequency domainOutput,MIMO, nonlinear, time-Varying systemMatrix, vector, linear algebraTime domainState,19,In example 9.2,20,21,9.3 State Differential equation,Differential equation,Transfer function,Block diagram,State-space model,22,State-space model from differential equation(1). No differential of input,23,24,matrix：D0,可观测规范II型。b0=1时为能控规范型。,25,Example 9.3 DC motor,set,26,(2). differential of input,27,Method 1:,Take,28,We can have,29,From which we get,30,So the state equation,31,The matrix:,可观测规范II型。,32,Example 9.4,33,可观测规范II型。,34,Method 2:,35,Take :,State equation：,能控规范型。,36,Example 9.5,能控规范型。,37,9.4 Signal-flow Graph State Model,(1) phase variable format 相变量模型,38,39,Let,40,41,State equation：,能控规范型。,42,If m=n,43,The state equation：,能控规范型。,44,Example 9.6 n=4,45,46,47,The state equation:,可控规范型,48,(2) Input feedforward format,49,50,The matrix:,So,51,(3). Observable canonical form 可观测规范型,52,So,The matrix:,可观测规范型,53,Example 9.7 open-loop transfer function find out the state equation.,54,Observable form:,Controllable form:,55,(4) Alternative state Models,(a). Diagonal canonical form.,56,57,58,Example 9.8,59,(b) Jordan canonical form,When there are multiple roots in characteristic equation, such as :,60,61,62,63,Jordan canonical form,64,9.5 Realization and Minimum Realization,1.Modeling methodsmechanism modeling(机理建模): construct systems differential equations (state equations and transfer function).identity modeling（辨识建模）: time-domain identity modeling and frequency-domain identity modeling.,65,2. Realization of system,Definition: 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 with the transfer function G(s) or impulsive responding function g(t).,66,67,3. 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 cancellation, then there exists the minimum realization A, B, C of the system, which is completely controllable and observable.,68,If G(s) have the same zero and pole, then system can not be realized to be completely controllable or observable.If A1, B1, C1 and A2, B2, C2 are two minimum realization of G(s), then A1, B1, C1 is algebraic equivalent to A2, B2, C2 .,69,4. The equivalence of minimum realization and real structure of system,If the system is controllable and observable, the minimum realization is algebraic equivalent to real structure of the system.If the system is noncontrollable or nonobservable, the minimum realization is partly algebraic equivalent to real structure of the system.,70,9.6 The transfer function from the state Equation,(1) Solution of the state equation,71,so,method 1,72,Then,73,method 2,74,Then,or,75,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。,76,State equation:,(2) Transfer Function Matrix from State Space Model,77,78,79,9.7 The State Transition Matrix,The Solution of first-order differential equation,80,Let,81,when u=0 , unforced system,State transition matrix,82,Characters：,State Transition Matrix,83,Method1,Method2,Evaluation of t