现代控制理论精品ppt课件(英文版)Chapter.ppt

Chapter 4 State Space Analysis ofLinear Control SystemState space analysis of linear systemControllability and CriterionObservability and Criterion Duality PrincipleControllable&Observable Canonical FromSystem structure decompositionSystem realization1.Controllability and Observability Definition of controllability:For linear system given the initial state at if there exists finite time interval and admissible input u(t)thatcould transit to any state within time ,then the system is controllable atExplanation1)Input affected state is controllable2)u(t)satisfies unique solution condition3)Definition domain is finite interval Controllable Criterion:1)For any LTI continuous system with n dimension stateThe necessary and sufficient condition of system being completely controllable is 2)If the system has distinct eigenvalue ,the necessary and sufficient condition of system being completely controllable is matrix B does not contain row with all 0 element in diagonal canonical form obtained through equivalent transformOutput controllableDefinition:For linear system there exists admissible input u(t)that could transit any given to within finite timeinterval then the system is output controllable.Criterion:For any LTI continuous system with m dimension outputThe necessary and sufficient condition of system being completely output controllable is Example.Known system with block diagram as following,please study the state and output controllability.Solution:System state descriptionSo the system(state)is not complete controllableThe output is completely controllable2.Observability and Criterion Definition of Observability:For linear systemGiven ,if the initial state could be uniquely determined according to the measurable output over interval then the system is observable.Explanation:1)Output reflected state is observable2)Considering only the system free motion when studying observability Observabiltiy Criterion:1)For any LTI continuous system with m dimension outputThe necessary and sufficient condition of system being completely observable is Observabiltiy Criterion:2)If the system has distinct eigenvalues the necessary and sufficient condition of system being completely observable is does not contain column with all 0 element in diagonal canonical form obtained through equivalent transformExample.Please examine the system observability.Solution:So the system is observableSolution:A is diagonal form with distinct eigenvalue.has no column with all elements are zero.So the system is observableObservabiliy and controllability of discrete system (not required)3.Duality principle For linear system S1and system S2System S1 and S2 are called dual systemsBlock diagram of dual systemsNote the relationship between dual systemsDuality principle:the system S1 is completely controllable(or observable)if its dual system S2 is completely observable(or controllable).4.Controllable and Observable Canonical From 1)Controllable canonical form SISO systemThen the state space model is called controllable canonical formTheorem:if system(A,B,C)is completely controllable,thenthere exists a nonsingular linear transformation makingsystem(A,B,C)to be controllable canonical form.Matrix P is determined asStep of transform state space model to controllable canonical form:1)Calculate matrix2)Calculate invert of 3)Set 4)Calculate 5)Controllable canonical form2)Observable canonical form SISO systemThen the system is called observable canonical formTheorem:if system(A,B,C)is completely observable,thenthere exists a nonsingular linear transform makingsystem(A,B,C)to be observable canonical form.Matrix T is determined asStep of transform system to observable canonical form:1.Calculate matrix2.Calculate invert of 3.Set 4.Calculate 5.Observable canonical form5.System structure decomposition If the LTI system is not completely controllable or observable.For LTI system we could resort the state variable ascalled system structure decompositionSystem structure decomposition could be started from controllability decomposition to observability decomposition 1)Controllability structure decompositionTheorem:if the n-dimension LTI system(A,B,C)is not completely controllablethen there exists a nonsingular linear transform making thesystem to beThe k-dimension subsystemis completely controllable.The(n-k)dimension subsystemis uncontrollable.Block diagram of new systemThe nonsingular matrix Where are k irrespective column vectors of matrixAnd are another n-k column vectors making the matrix nonsingularWe can get the new system through equivalent transformationCharacteristics of decomposed system(1)Decomposition does not change the system controllability or observabilityAs to the equivalent transformed system(2)transfer function matrix of systemWe could getSo the TF matrix of controllable subsystem is the same as the whole system TF matrix while the dimension is reduced.(3)Input go through only the controllable subsystem to affect output(4)Uncontrollable subsystem is related to the system stability and response(5)The structure decomposition form is not unique2)Observability structure decompositionTheorem:if n-dimension system(A,B,C)is not completely observableThen there exists a nonsingular linear transform making the system to beBlock diagram of decomposed systemHere,the -dimension subsystemis completely observable.And the dimension subsystemis unobservable.The nonsingular matrix Where are irrespective row vectors of matrix And are row vectors making the matrix nonsingularThe decomposed system(3)Controllability&observability structure decompositionTheorem:if n-dimension system(A,B,C)is not completely controllable and observablethen there exists a nonsingular linear transform making the system to bewhereThe system is decomposed into 4 subsystemsThe 4 subsystems(1)Controllable and observable system(2)Controllable but unobservable system (3)uncontrollable but observable system(4)uncontrollable and unobservable system Actually,all linear systems are consist of all or part of the four above 4 SubsystemsStep of system controllable and observable structuredecomposition1.Controllable structure decomposition 2.Decompose the controllable subsystem into observable and unobservable systems with transform matrix 3.Decompose the uncontrollable subsystem into observable and unobservable systems with matrix 4.Get the transform matrix whereThe decomposed systemTransfer function matrixConclusion:Transfer function matrix reflects only the controllable and observable part of the whole system6.System realization For complex systems,it is difficult to get the state space description directly.It is much easier to get the system transfer function(matrix)firstand then find the proper state description of the complex system.Definition:For any system with given transfer function ,find the proper state description as following which satisfiesthen the description(A,B,C)is called the realization of system Basic characteristics of realization(1)Existence of physically realizable system(2)The realization is not unique1)Canonical form realizationDefinition:To realize the system transfer function with statespace description of controllable or observable canonical form(1)SISO system Where The controllable canonical form is And the observable canonical formNote:dual system(2)MIMO system Where The controllable canonical form of MIMO system is And the observable canonical formNote:not dual system if Usually we prefer the realization with less dimensions So controllable and observable canonical form realization are dual system2)minimum realizationNote:(1)The realization of is not unique and the dimension of different realization varies.(2)Usually realization with less dimension is expected.Definition:Minimum realization is the realization of system with the least dimension and the simplest structureTheorem:The realization(A,B,C)of system is the minimum realization when(A,B,C)is both controllable and observable.Steps of system minimum realization(1)Find one realization of ,usually,we will choose the controllable or observable canonical form.(2)Perform the controllable(observable)structure decomposition on the observable(controllable)canonical form realization.