现代控制理论chapter3-ControllabilityandObservabilityofControlSystem

1、Modern Control TheoryControllability and Observability of Linear Control SystemChapter 3Controllability andObservability ofLinear Control SystemsControllability and Observability of Linear Control SystemreviewSystem analysisquantitativequalitativequantitativequalitativeThe exact response of a system

2、The general properties of a systemModern Control TheoryModern Control TheoryControllability and Observability of Linear Control SystemControllability and Observability are two very importantconcepts in modern control theory. They belong toqualitative analysis.Outline of this Chapter:1.2.3.Introducti

3、onControllabilityObservability4. Controllability and Observability of discrete system5.Duality system（能控性）（能观测性）Controllability and Observability of Linear Control System6. Controllable canonical form andObservable canonical form7. Controllability , Observability and transfer function8.9.Decompositi

4、on of a systemRealization problem10.How to use MATLAB to determine the Controllabilityand ObservabilityModern Control TheoryState variable?Controllability and Observability of Linear Control System3.1IntroductionFirst, introduce the concepts by using the network asfollowing.The state variables areno

5、t controlled by inputThis system is notControllableModern Control Theoryx Ax bu x ux(0) 2 et e3tt 3t x(t ) e1Modern Control TheoryControllability and Observability Linear Control Systemas state variables，x1 uC1 ， x2 uC 2 ，the output is thevoltage of C2 ， y x2 ，then the state-space representationis a

6、s:111 2 2 1y Cx 0 1xThe state transition matrix0If the initial state:0e t e3t e e 1 e t e3te AtThe solution is:u( ) d 1 t (t )0So whatever input signal，wecan getx1 x2x Ax Buy Cxx Ax Bu x u2 e e3tt 3t Controllability and Observability of Linear Control（1）101 -2-2 1y Cx 1 1xThe state transition matrix

7、e t e 3t e e 1 e t e 3t te AtControllability of a system depends on matrix B and AObservability of a sysyem is to research whether the outputaffects the states.Example the input is u(t ), the ouput is y(t )The solution ist0Modern Control TheoryControllability and Observability of Linear Control Syst

8、emFor convinience，u(t ) 0hencex (t ) e At x(0)y(t ) C e At x(0) x1 (0) x2 (0) e3tFrom the example, we can see that regardless of the initialstate value, the output only depends onx1(0) x2 (0).When x1(0) x2 (0) , then the output identically equals zero.Clearly, we cannot determine the initial state b

9、y observing theoutput, saying such a system can not be observed .thesystem is not observable.Observablity of a system depends on matrix Cand AModern Control TheoryBuAxx （2）Modern Control TheoryControllability and Observability of Linear Control System3.2 Controllability3.2.1 Controllability And Crit

10、erions of LTIS1. DefinitionThe state-space representationof a LTIS isNOTE:1） Initial state is arbitrary finite non-zero in the state space.The control objective is zero state.system is completely controllable。Given the initial state x (t0 ) ，if at some finite time t1 t0 ，thatis to say in the time of

11、 t 0 , t 1 exists a input u(t ) ，tomakex(t1 ) 0 ，then the initial state x (t0 ) is controllable；Ifthis is true for all initial times t0 , and all initial states x (t0 ),thex (0) et10Bu( ) d AControllability and Observability of Linear Control System2）Only all the initial states are controllable ,the

12、n thesystem is controllable3）The state that meet equation（3），must becontrollable（3）x Ax Bu f (t )4）When exist the interference f (t ) that do not rely on u(t ) ，which will not change the controllability（4）Modern Control Theory0 ai ( )u( ) d i 2 i ir Controllability and Observability of Linear Contro

13、l SystemTheorem1 The necessary and sufficient condition ofcontrollability is the following n nr-dimensional（6）（7）controllability matrix full rank.QC B AB A2 B An1 Brank QC n i1 t1( i 0 ,1, , n 1)Modern Control Theoryt1n 01i 0n 1i0AB An-1 B 1 n 1 Controllability and Observability of Linear Control Sy

14、stemSo 0 x (0) B（9）If it is completely controllable，we must get theABrank QC rank BA2 B An 1 B nModern Control Theorymeetx x B un （11）0 2 1 0Theorem3 A all eigenvalues i are distinct,The system is transformed into diagonal matrices bynon-singular linear transformationThe necessary and sufficient con

15、dition of controllability isthat the matrix B does not contain a full zero line.Modern Control TheoryControllability and Observability of Linear Control SystemTheorem2 （PBH） The necessary and sufficienti（10）condition of controllability is that A all eigenvalues(特征根)rank i I A B n ( i 1, 2 , , n )x x

16、 B un Modern Control Theory0 2 1 0Controllability and Observability of Linear Control SystemreviewQC B AB A2 B An1 Brank QC nrank i I A B n x 0u x 4 0uControllability and Observability of Linear Control SystemExample Given the two LTIS，please determine thecontrollability 50 2 1 9 50 0 1 1 7 5 7x 0 7

17、（1） x 0（2）Answer :According to Theorem ，（1）not controllable ； （2）controllable。Modern Control Theory lx x B uJ k J i i 1 i Modern Control Theory0 J 2J1 0 i 01 0 （12）Controllability and Observability of Linear Control SystemTheorem4 The matrix A has the eigenvalues aswhichare re-rootl 1 l 2 l 3 l k且ik

18、i 1 n ，i j ，( i j ) The system is transformed into Jordancanonical form by non-singular lineartransformation 1 2 3 kelementsThe necessary and sufficient condition of controllability isthat matrix A s each sub-block does not contain a full zeroline at the bottom line corresponding to theBControllabil

19、ity and Observability of Linear Control SystemExample Given the two LITS，please determine the controllability 1 400 00 x 4u 2 3 1 400 4 20 x 0 0u 2 3 0 4（1） x 0 0 4（2） x 0 0AnswerAccording to Theorem ，（1） controllable ； （2） not controllable。Modern Control TheoryControllability and Observability of L

20、inear Control System3.3.1 Observability And Criterions of LTIS（18）1. DefinitionThe equation isA linear system is said to be observable at t0 ,if x (t0 ) can bedetermined from the output y(t ) , for t 0 , t 1 t1 t0 . If this istrue for all t0 and x (t0 ),the system is said to be completelyobservableN

21、OTE：y(t )测的目标是为了确定 x(t0 )。y(t )状态 x(t1 ) ，则称系统是可检测的。连续系统的能观测性和能检测性等价。Modern Control Theoryx Ax Buy CxModern Control TheoryControllability and Observability of Linear Control System3）Only all the states are observable ,then the system isobservable 。which will not change the observabilityx Ax Bu f (t

22、)4）When exist the interference f (t ) that do not rely on u(t ) ， CA QO rank QO n CAn 1 nmny(t ) C ai ( ) Ai x(0)Modern Control TheoryControllability and Observability of Linear Control SystemTheorem1 The necessary and sufficient condition ofobservability is the following n nr-dimensionalobservabili

23、ty matrix full rank. C Prove: u(t ) 0， the solution of system isx(t ) e At x(0)y(t ) Cx (t ) C e At x(0)（23）n 1i 0ia i ( ) AA Using Cayley Hamilton theorem, there isen 1Soi 0 x x 2 u C CA rank 1 C )Controllability and Observability of Linear Control System 由于ai (t ) 是已知函数，因此，根据有限时间 0 , t 1 内的 y(t )能

24、够唯一地确定初始状态 x(0 的充分必要条件为 QO 满秩。Theorem2 （PBH） The necessary and sufficient conditionof observability is that A all eigenvalue i meet C rank ExampleGiven the system, please determin the observability0 1 5 2 01 xy 00 1 0 5Solution:rank Not observableModern Control Theoryx x B un The necessary and suffi

25、cient condition of observability is thatthe matrix C does not contain a full zero column.Modern Control TheoryControllability and Observability of Linear Control Systemare distinct,The system is transformed into diagonal matrices bynon-singular linear transformation0 2 1 0（1） x x xy xControllability

26、 and Observability of Linear Control SystemExample Given the two systems,please determine the observability 50 1 7 0y 0 4 5x（2） 50 1 7x 03 2 00 3 1Solution: According to Theorem ，（1） not observable ； （2） observableModern Control Theory lx x B uJ k J i i 1 i 0 J 2J1 0 i 01 0 （12）Controllability and O

27、bservability of Linear Control SystemTheorem4 The matrix A has the eigenvalues aswhichl 1 l 2 l 3 l k are re-root且iki 1 n ，i j ，( i j ) The system is transformed into Jordancanonical form by non-singular lineartransformation 1 2 3 kThe necessary and sufficient condition of observability is formatrix

28、 A s each sub-block does not contain a full zero line atthe first column corresponding to the elements of CModern Control Theory0 x 0y xControllability and Observability of Linear Control SystemExample Given the system3000 0 0 x1 -21 0 03 1 00 3 00 0 -20 0 01 1 1 1 00 1 1 0 0Please judge the observa

29、bilitySolution : observableModern Control Theoryy(k ) Cx(k )Controllability and Observability of Linear Control SystemObservability of discrete timesystemThe equation of linear time-invariant system（29）x(k 1) Gx(k ) Hu(k ) x(DefinitionGiven the initial state x(0) ，if exists k 0 ，that is to say inthe

30、 time of 0 , k exists a input u(k ) ，to make x (k ) 0 ，the system is completely controllable。0)Modern Control Theory 16 3Modern Control TheoryGH G n1 H nrank QC rankH Example The state equation of a discrete system 1 0 0 1 2 1 1 0 1 1 1 2 1 1 Please judge the controllabilityrank Q C rank H GH G 2 H

31、rank 0The system is controllable。 1Controllability and Observability of Linear Control System3.4.2 Controllability CriterionTheorem1 The necessary and sufficient condition of QCcontrollability is the following n nr-dimensionalcontrollability matrix full rank.Theorem1 The necessary and sufficient con

32、dition ofcontrollability is the following n nr-dimensionalQOcontrollability matrix full rank. C Modern Control TheoryControllability and Observability of Linear Control System3.4.3 DefinitionGiven the initial state x (0) ，If according to finite theinitial state x(0) is can be determined ,then the sy

33、stem iscompletely observable。3.4.4 Observability CriterionControllability and Observability of Linear Control SystemExample the state equation of a discrete system 1 0 0 1 x (k 1) 0 2 2 x(k ) 0 u(k ) 1 1 0 1y(k ) 1 1 1x(k ) Please judge the observability C 1 1 1 rank QO rank CG rank 0 3 2 3CG 2 2 4

34、6The system is observableModern Control Theoryx Ax Buy Cxy ( k ) Cx ( k )G eH T e AT d t BControllability and Observability of Linear Control System3.4.5Controllability And Observability of Linear Discrete-time SystemSystem modeldiscretizationx ( k 1) Gx ( k ) Hu ( k ) （31）（32）其中AT 0 T 是采样周期定理3-19 如

35、果线性定常系统（31）不能控（不能观测），则离散化后的系统（32）必是不能控（不能观测）。其逆定理一般不成立。定理3-20 如果线性离散化后系统（32）能控（能观测），则离散化前的连续系统（31）必是能控（能观测）。其逆定理一般不成立。Modern Control TheoryControllability and Observability of Linear Control System定理3-21 如果连续系统（31）能控（能观测），A 的全部特征值互异，i j，并且对Re i j 0的特征值，如果 Imi j 与采样周期的关系满足条件2kImi j T k 1, 2 , （33）则离散

36、化后的系统仍是能控（能观测）的。Modern Control Theoryx Ax Buy CxControllability and Observability of Linear Control System3.5 DualityConstruct a systemT（34）and（35）is adual systemModern Control TheoryThe controllability and observability described above are very similar on Conceptand form. Maybe they are linked to e

37、ach other. The link is the dual principle .The system isidea（34）Characteristic?QC 2 QModern Control TheoryControllability and Observability of Linear Control SystemTwo basic characteristics of dual system :1. The transfer functions is transpose matrix.G1 (s) CsI A1 BG2 (s) BT sI AT 1 C T C (sI A)1 B

38、T G1T (s)2. The two systems have same eigenvalues .detsI A detsI AT 3.Dual theory： Controllability of system (34) is equivalent toobservability of system (35); observability of system (34) isequivalent to controllabilityof system(35).TO1QO 2 QCT 1Controllability and Observability of Linear Control S

39、ystemExampleGiven the system, please judge thecontrollability and observability 0 0 1 1x Ax Bu 1 0 0 x 0u0 1 0 0The dual system is0 1 0 0 AT C T 0 0 1 0 1 0 0 1y Cx 0 0 1x BT 1 0 0Controllability0 1 0QC 0 0 11 0 0rank QC 3The dual system is controllable，so the givensystem is observable。Modern Contro

40、l Theoryx Ax bu y Cx du 0 x u1 0Modern Control Theory（36）Controllability and Observability of Linear Control SystemCanonical Form3.6.1 Controllable Canonical FormThe systemSuppose A polynomialdet I A n an 1 n 1 a1 a0QC b Ab An1bTheorem If the system（36）is controllable，then it can transformed intothe

41、 following : 1 0 00 1 0 a1 an 1 11 n 1 x du 0 x 0 a0y 0（37）How manycanonicalforms?Controllability and Observability of Linear Control SystemExampleGiven the controllable system 0101 0 0 x 1 u0 1 1x 0 1y 1 1 0 x0 1 1A2 b 1 1 11 0 1Ab（1）QC brank QC 3controllable（2）A polynomialdet I A 3 2 2 1Modern Con

42、trol Theory 11 1 10 1x 0 0 1x 0uControllability and Observability of Linear Control System 1 1 1 00 1 1 10 1 2 1a210 a1A2 ba2 1Ab（3）compute P p1 p2 p3 b 11p3 1 1 1 1 2 1 1 1 2 1 3 1 2p2P p10 10 1 21 1 1 2（4）computeC 1C CP 1 1 1 0 1（5）the controllable canonical form 0 1 0 0 1 0 2 1y 2 0 1xModern Cont

43、rol Theory a1 1 0 x 0 1 x u（38）Controllability and Observability of Linear Control System3.6.2 Observable Canonical Form C Theorem If the system（36）is observable，then it cantransformed into the following : 1 0 0 1 an1 n1 0 a0 0 0 y 0 0 1xModern Control Theory aP an 1a2 an 1 1 C CA CAn 1 0Controllabi

44、lity and Observability of Linear Control SystemThe transformation matrix is 11 a1 2 1（39）Modern Control Theoryx Ax Bu y Cxg (s) CsI A b Controllability and Observability of Linear Control System3.7 The relation of Controllability、Observability and Transfer FunctionSISO:（40）Transfer function（41）N (s)

45、D(s)1C adjsI A bdetsI AThe numerator and the denominator areN (s) C adjsI A bD(s) detsI AIn the absence of zero-pole cancellation, the roots oftransfer function are same to the eigenvalue of A.Modern Control Theoryx x 1ug (s) CsI A1 b C adjsI A b s 2QC bAb 2 rank QC 2 nQO 1 1Controllability and Obse

46、rvability of Linear Control SystemTheorem If there is no zero-pole cancellation ,SISOsystem (40) can be controllable and observableExample compute the transfer function and judge thecontrollability and observability-1 -3 0 0 2 y 1 1 x1s 10 31Transfer function C 1 1 CA detsI A (s 1)(s 2)rank QO 1 nCo

47、ntrollable ,not observableThe theorem does not be applied to the MIMO system 。Modern Control TheoryG (s) CsI A B y x1 0 001 Modern Control Theory 1 0 0 1 3 2 rank QO rank 3 n01 15 10 0 0 1 Controllability and Observability of Linear Control SystemExample 1 3 2 0 1x 0 4 2 x 0 0u0 0 1 1 011 0 00 0 1s

48、1 2 s 4(s 1) 2 (s 4) s 4 0 0 1 2 1 10 1rank QC rank 0 0 2 0 10 0 3 n1 0 1 0 1 03.8 Decomposition of a LTISControllability and Observability of Linear Control SystemideaFrom a structural point a system that is not controllable andobservable must include the subsystems that is notcontrollable and obse

49、rvable. How to decompose them?.We know linear transformation does not changethe controllability and observability. Therefore, thelinear transformation method can be used todecompose the subsystems. Here three issuesmust be addressed1、how to decompose？2、what form is the system decomposed？3、how to det

50、ermine the transformation matrix？Modern Control Theory A12 xC BC AC xC 0 xC Modern Control Theoryx Ax Buy CxControllability and Observability of Linear Control SystemGiven the system（43）3.8.1 Controllable DecompsitionTheorem If system（43）is not controllable，and thestate x has n1 components that are

51、controllable，henceexist x PC x to make the equation transformed intofollowing form， u xC AC xC 0 xC CC y1 CC（44）G (s) C sI A B C sI A BsI AC A12 BC sI AC 0 BCModern Control TheoryControllability and Observability of Linear Control SystemThe transfer function matrix1 1 CC CC CC sI AC0（46）1Controllabl

52、esubsystemx C AC xC A12 xC BC uy1 CC xC（45）Modern Control TheoryControllability and Observability of Linear Control Systemhow to determine the transformation PCvectors, which constitutes a non-singularrankQC rankB AB A2 B n1 nThat the matrix QC has n1 independentcolumn vectors , and then add(n n1 )

53、columnmatrix PCAb rank 1 n 2 x 0uy 1 1Controllability and Observability of Linear Control SystemExample please decompose the given system 2 1 1 1 2 A PC APC1B PC BC CPC1 xC 0 3 C xC 1 1 xC 1 x C x C Modern Control Theorysolution1 11 1rank QC rankbuncontrollableQC has a independent column0 111 1 Mode

54、rn Control TheoryObservablesubsystem（48）x O AO xO BO uy CO xOTransfer function G (s) C (sI A) 1 B CO (sI AO ) 1 BO（49）Controllability and Observability of Linear Control System3.8.2 Observable DecompsitionTheorem If system（43）is not observable， and thestate x has n1 components that are observable，he

55、nceexist x PO x to make the equation transformed intofollowing form， u（47） xO 0 CAnn 2rankrank QO n 1CAModern Control TheoryControllability and Observability of Linear Control System C vectors, which constitutes a non-singular)That the matrix QO has n2 independent rowvectors , and then add (n n2 row

56、matrix POhow to determine the transformation POPO-1 0 x O 2 2 0 xO 1u xO x 2 2 1 O 1y 1 0 0 1 1 0PO 0 1 10 0 10 11 1 110 0A PO APO1B PO BC CPO1 0 1 0 0CO xO xO Modern Control TheoryControllability and Observability of Linear Control SystemExample please decompose the given system 0 1 0 00 y 1 1 0 x

57、2 4 3 1 C 1 1 0 1CA2 2 4 21 1 0add a independent row x xC O xC O Modern Control TheoryControllability and Observability of Linear Control System3.8.3Decomposition According to Controllability And Observability xCO 0 CO y CCO 0 CC OTransfer function matrixG (s) C (sI A) 1 B CCO (sI ACO ) 1 BCOTheorem

58、 If the system（43）is not controllable and notobservable，exist x Px to make the equation transformedinto following form， xCO ACO 0 A13 0 xCO BCO u0 A43 （50）（51）Modern Control TheoryControllability and Observability of Linear Control SystemModern Control Theoryx Ax Buy Cx（52）Controllability and Observ

59、ability of Linear Control System3.9 Realization ProblemIf given the transfer functiong (s) ，we can obtain asystem equation（53）orx Ax buy Cx dus an 1s a1s a0 nControllability and Observability of Linear Control System3.9.1 Controllable Realizationy(s)u(s)Transfer function g (s) n 1s n 1 n 2 s n 2 1s

60、0n n 1y(s)u(s)1.There is no zerosg (s) 0s an 1s n 1 a1s a0s n y(s) an 1s n 1 y(s) a1sy(s) a0 y(s) 0u(s)Laplace transformSelect the statesy ( n ) an 1 y ( n 1) a1 y a0 y 0ux1 y / 0 x2 y / 0 xn y ( n 1) / 0Modern Control Theoryx1 ，2 x2 ， xn 1 ，n 00b Modern Control TheoryControllability and Observabili