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1、Chapter 4 State Space Analysis ofLinear Control System,State space analysis of linear system,Controllability and CriterionObservability and Criterion Duality PrincipleControllable&Observable Canonical FromSystem structure decompositionSystem realization,1.Controllability and Observability Definition
2、 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 atExplanationInput affected state is controllableu(t)satisfies unique solution conditionDefinition
3、 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 controll
4、able is matrix B does not contain row with all 0 element in diagonal canonical form obtained through equivalent transform,Output controllableDefinition:For linear system there exists admissible input u(t)that could transit any given to within finite timeinterval then the system is output controllabl
5、e.,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 description,So the s
6、ystem(state)is not complete controllableThe output is completely controllable,2.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:
7、Output reflected state is observableConsidering 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 syst
8、em 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 transform,Example.Please examine the system observability.Solution:So the system is observable
9、,Solution: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 systems,Block diagram of d
10、ual 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 mo
11、del is called controllable canonical form,Theorem: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 as,Step of transform state space model to controllable canonical form:1)Calcul
12、ate matrix2)Calculate invert of 3)Set 4)Calculate 5)Controllable canonical form,2)Observable canonical form SISO systemThen the system is called observable canonical form,Theorem:if system(A,B,C)is completely observable,thenthere exists a nonsingular linear transform makingsystem(A,B,C)to be observa
13、ble canonical form.Matrix T is determined as,Step of transform system to observable canonical form:Calculate matrixCalculate invert of Set Calculate Observable canonical form,5.System structure decomposition If the LTI system is not completely controllable or observable.,For LTI system,we could reso
14、rt 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
15、 exists a nonsingular linear transform making thesystem to be,The k-dimension subsystemis completely controllable.The(n-k)dimension subsystemis uncontrollable.,Block diagram of new system,The nonsingular matrix Where are k irrespective column vectors of matrixAnd are another n-k column vectors makin
16、g the matrix nonsingularWe can get the new system through equivalent transformation,Characteristics of decomposed system(1)Decomposition does not change the system controllability or observability,As to the equivalent transformed system(2)transfer function matrix of system,We could getSo the TF matr
17、ix 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 unique,2)Obs
18、ervability structure decompositionTheorem:if n-dimension system(A,B,C)is not completely observableThen there exists a nonsingular linear transform making the system to be,Block diagram of decomposed system,Here,the-dimension subsystemis completely observable.And the dimension subsystemis unobservabl
19、e.,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 n
20、onsingular linear transform making the system to be,where,The system is decomposed into 4 subsystems,The 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 syste
21、ms are consist of all or part of the four above 4 Subsystems,Step of system controllable and observable structuredecompositionControllable structure decomposition Decompose the controllable subsystem into observable and unobservable systems with transform matrix Decompose the uncontrollable subsyste
22、m into observable and unobservable systems with matrix Get the transform matrix where,The decomposed system,Transfer function matrixConclusion:Transfer function matrix reflects only the controllable and observable part of the whole system,6.System realization For complex systems,it is difficult to g
23、et 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 satisfies,then the de
24、scription(A,B,C)is called the realization of system Basic characteristics of realizationExistence of physically realizable system The realization is not unique1)Canonical form realizationDefinition:To realize the system transfer function with statespace description of controllable or observable cano
25、nical 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
26、So controllable and observable canonical form realization are dual system,2)minimum realizationNote:The realization of is not unique and the dimension of different realization varies.Usually realization with less dimension is expected.Definition:Minimum realization is the realization of system with
27、the least dimension and the simplest structure,Theorem:The realization(A,B,C)of system is the minimum realization when(A,B,C)is both controllable and observable.Steps of system minimum realizationFind one realization of,usually,we will choose the controllable or observable canonical form.Perform the controllable(observable)structure decomposition on the observable(controllable)canonical form realization.,
