1、Chapter 4 State Space Analysis ofLinear Control SystemState space analysis of linear system Controllability and Criterion Observability and Criterion Duality Principle Controllable & Observable Canonical From System structure decomposition System realization1.Controllability and Observability Defini
2、tion 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 condit
3、ion3)Definition domain is finite intervalxAxBu 0 x t0t0ftt 0 x t fx t0ftt0t0ftt 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 an
4、d 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 transform21 ,nrank B AB A BABn rank criterion x tAx tBu t123,n Output controllableDefinition: For linear system there exists admis
5、sible input u(t) that could transit any given to within finite timeinterval then the system is output controllable.1122001,20innbbxxubinbxAxBu 0y t fy t0fttCriterion: For any LTI continuous system with m dimension outputThe necessary and sufficient condition of system being completely output control
6、lable is x tAx tBu ty tCx tDu t1,nrank CB CABCAB DmExample. Known system with block diagram as following, please study the state and output controllability.Solution: System state description 0010011 1x tx tu ty tx t u t y t 1x t 2xt 1x t 2xtSo the system (state) is not complete controllableThe outpu
7、t is completely controllable10 ,110rank B ABrankn ,2001rank CB CAB Drankm 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
8、:1)Output reflected state is observable2)Considering only the system free motion when studying observability x tAx tBu ty tCx tDu t 0 x t0ftt0,ftt y t Observabiltiy Criterion: 1) For any LTI continuous system with m dimension outputThe necessary and sufficient condition of system being completely ob
9、servable is 21nCCAranknrank crierionCACA x tAx tBu ty tCx tDu t 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 equ
10、ivalent transform123,n 12112000nmiAP APCCPccccxPxCExample. Please examine the system observability.Solution: So the system is observable 2111)1311010 x tx tu ty tx t101022121CrankrankCASolution: A is diagonal form with distinct eigenvalue. has no column with all elements are zero. So the system is o
11、bservableObservabiliy and controllability of discrete system (not required) 72)51320031x tx ty tx tC3. Duality principle For linear system S1and system S2System S1 and S2 are called dual systems 11111x tAx tBu ty tCx t111111urxnym 22222TTTxtA xtC utytB xt222111umxnyr Block diagram of dual systemsNot
12、e the relationship between dual systemsDuality principle: the system S1 is completely controllable ( or observable) if its dual system S2 is completely observable ( or controllable). 1u t 1y tBCA 1x t 1x t1:S 2ut 2ytTCTBTA 2xt 2xt2:S4. Controllable and Observable Canonical From 1) Controllable canon
13、ical form SISO systemThen the state space model is called controllable canonical form x tAx tBu ty tCx tDu t1210100000100000011nnnif ABaaaa Theorem: if system (A, B, C) is completely controllable, thenthere exists a nonsingular linear transformation makingsystem (A, B, C) to be controllable canonica
14、l form.Matrix P is determined as1112111npp APp Ap AxPx121121001,nnwherepB AB A BABthat is thelast rowofB AB A BABStep of transform state space model to controllable canonical form:1) Calculate matrix2) Calculate invert of 3) Set 4) Calculate 5) Controllable canonical form11111npp APp A21 , nSB AB A
15、BAB and whether rank SnS11001pS11,AP AP BP B CCP2) Observable canonical form SISO systemThen the system is called observable canonical form x tAx tBu ty tCx tDu t1021211000100010,00010001nnnaaifABaCTheorem: if system (A, B, C) is completely observable, thenthere exists a nonsingular linear transform
16、 makingsystem (A, B, C) to be observable canonical form.Matrix T is determined as211111 ,nTT AT A TATxTx11111001nnCCCACAWhereTthelast column ofCACA Step of transform system to observable canonical form:1.Calculate matrix2.Calculate invert of 3.Set 4.Calculate 5.Observable canonical form1 nCCAVand wh
17、ether rank VnCAV11001TTV11,ATATBT B CCT211111 ,nTT AT A TAT5. System structure decomposition If the LTI system is not completely controllable or observable. x tAx tBu ty tCx tDu t2121 ,nnif rank B AB A BABnnot controllableCCAranknnot observableCACA For LTI system we could resort the state variable a
18、scalled system structure decompositionSystem structure decomposition could be started from controllability decomposition to observability decompositioncoccococcoxxxxxxxcocococoxxxxx:cocococoxcontrollable and observablexcontrollablebut unobservablexuncontrollablebut observablexuncontrollable and unob
19、servable 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 be 21 ,nrank Srank B AB A BABknxPx 11111212222112200 x tx tAABu txtxtAx ty tCCxtThe k-dimension subsy
20、stemis completely controllable.The (n-k) dimension subsystemis uncontrollable. 2222222xtA xtytC xt 111 1122111 1x tA x tA xtBu ty tC x tBlock diagram of new system u t 1y t1B1C11A 1x t 1x t 2yt12A 2xt 2xt2C22A y tThe nonsingular matrix Where are k irrespective column vectors of matrixAnd are another
21、 n-k column vectors making the matrix nonsingularWe can get the new system through equivalent transformation121,kknPs ss ssP12,ks ss1knssS11AP APBP BCCPPCharacteristics of decomposed system(1) Decomposition does not change the system controllability or observability2111111112121 , , ,nnnnrank B AB A
22、 BABrank P B P AP P BP APP Brank PB AB A BABrank B AB A BAB11111nnnCPCCCPP APCACArankrankPrankCACACP P APAs to the equivalent transformed system(2) transfer function matrix of system211111112111111111211111111111 ,0000,nnnnrank B AB A BABrank P B P AP P BP APP BBA BA BABrankrank B A B A BABk 1111111
23、121122200G sC sIABCP sIP APP BsIAABCCsIAA B CWe 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 stab
24、ility and response(5) The structure decomposition form is not unique 111111G sC sIABCsIAB2) Observability structure decompositionTheorem: if n-dimension system (A, B, C) is not completely observableThen there exists a nonsingular linear transform making the system to be 21nCCArank VranklnCACA x tTx
25、t 11111222212211200 x tx tBAu txtxtBAAx ty tCxtBlock diagram of decomposed system u t y t1B1C 1x t 1x t21A 2xt 2xt22A2B11AHere, the -dimension subsystemis completely observable.And the dimension subsystemis unobservable. 221 122222200 xtA x tA xtB u tytxt 111 1111 1x tA x tBu ty tC x tlnlThe nonsing
26、ular matrix Where are irrespective row vectors of matrix 1211llnTTTTTTT12lTTTlVAnd are row vectors making the matrix nonsingularThe decomposed system1llnTTT11ATATBT BCCTnlT (3)Controllability & observability structure decompositionTheorem: if n-dimension system (A, B, C) is not completely controllab
27、le and observablethen there exists a nonsingular linear transform making the system to be 21nCCArank VranklnCACA 21 ,nrank Srank B AB A BABknxTxwhere x tAx tBu ty tCx t111312122232423343441300000000000AABAAAABABAAACCCThe system is decomposed into 4 subsystems 1234xtxtx txtxt 1234ytyty tytytThe 4 sub
28、systems(1) Controllable and observable system(2) Controllable but unobservable system 221 122223324422200 xtA x tA xtA xtA xtB u tytxt 111 1133111 1x tA x tA xtBu ty tC x t(3) uncontrollable but observable system(4) uncontrollable and unobservable system Actually, all linear systems are consist of a
29、ll or part of the four above 4 Subsystems 44334444400 xtA xtA xtytxt 3333333xtA xtytC xtStep of system controllable and observable structuredecomposition1.Controllable structure decomposition 2.Decompose the controllable subsystem into observable and unobservable systems with transform matrix 3.Deco
30、mpose the uncontrollable subsystem into observable and unobservable systems with matrix 4.Get the transform matrix where12oooTdiag TT cccxtcontrollablex tTxtuncontrollable 1oT2oTcoTT TThe decomposed system 111312122232423343441300000000000AABAAAABx tx tu tAAAy tCCx tcA12AcAcBcCcCTransfer function ma
31、trixConclusion: Transfer function matrix reflects only the controllable and observable part of the whole system 111211111122122111110000cccccccccoG sC sIABsIAABCCsIACsIABBsIACBAsIAC sIABGs 6. System realization For complex systems, it is difficult to get the state space description directly.It is mu
32、ch 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 x tAx tBu ty tCx t G s 1G sC sIABthen the description
33、(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 ca
34、nonical form G s G s(1) SISO system Where The controllable canonical form is 111111( )( )( )nnnnnnnY sbsbsbG sU ssa sasa,iiaR bR121110100000100,000101ccnnncnnABaaaaCbbb And the observable canonical formNote: dual system11211000100,01000010001nnnnoooababABbabC,TTTcococoAABCCB(2) MIMO system Where The
35、 controllable canonical form of MIMO system is 111111( )( )( )nnnnnnnY sBsBsBG sU ssa sasa ,rmu tRy tRG sm r1211100000000,0000rrrrrrrrrrccrrrrrnrnrnrrrcnnIIABIa IaIaIa IICBBBm riBR0 ,rrIrrn rAnd the observable canonical formNote: not dual system if 112110000000,00000mmmnmnmmmnmnommmommmmommmma IBIaI
36、BAIBBIa IBCI,TTTcococoAABCCBn m0 ,mmImmmrUsually we prefer the realization with less dimensions So controllable and observable canonical form realization are dual systemminrealization demensionn m n r,TTTcococoAABCCBifmrcontrollable canonical formelse mrobservable canonical formif mr2) minimum reali
37、zationNote: (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 structure G s G sTheorem: The realization (A,
38、 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. G s G s
