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Lectures on Dynamic Systems and  Con trol  Mohammed Dahleh Mun ther A Lectures on Dynamic Systems and  Con trol  Mohammed Dahleh Mun ther A

Lectures on Dynamic Systems and Con trol Mohammed Dahleh Mun ther A - PDF document

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Lectures on Dynamic Systems and Con trol Mohammed Dahleh Mun ther A - PPT Presentation

Dahleh George erghese Departmen of Electrical Engineering and Computer Science Massac uasetts Institute of ec hnology brPage 2br Chapter 24 Observ abilit 241 In tro duction Observ abilit is notion that pla ys ma jor role in ltering and reconstructi ID: 25950

Dahleh George erghese Departmen

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Lectures on Dynamic Systems and  Con trol  Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massac huasetts Institute of T ec hnology 1 1 \r c 24.3.2 The Observ ability Gramian We begin by de\fning the k-step observ ability Gramian as Qk = O k T Ok k �1 = ( A i ) T C T CA : i The unobservable space o ver k steps is evidently the nullspace of Q k. The system is observable if and only if rank ( Q n )= n. If the system is stable, then we can de\fne the observability Gramian as X i =0 1 X Q = lim Qk = ( A i ) T C T CA : i k!1 i=1 Q satis\fes a Ly apunov equation that is quite similar to the reac hability gramian, i.e., A T QA �Q = � C T C: 24.4 Con tinuous-Time Analysis As with reac habilit y, the discussion of observ ability for con tinuous-time systems is algebraically very similar to the discussion for discrete-time systems. We begin with a theorem. Theorem 24.3 For con tinuous-time systems, the following conditions are equiv alent: 1. x(0) is unobservable in time T . 2. x(0) is unobservable in any time. 2  3  C 3. O n x(0) = 6 6 64 CA . . 7 7 7 x(0) = 0:  .  C A n �1 Proof: 1) =) 2): If  is unobservable in time T , then for x(0) = , if u ( t) = 0 then y ( t ) = 0; 0  t T . It follows that 0= C e At ; 0  t T From this, it follows that C e A 0  = C =0 d C e At dt \f \f \f\f \f \f \f t=0 . . . = CA =0 dk C e At = C A k  = 0: dt k  t=0 Now, since the p o wer series representation is valid 8 t  0, this implies that C e At  = 0; 8 t  0: Hence the result. 2) =) 1): This is immediate. 2) () 3): This follows from the T aylor series argument, since by C ayley-Hamilton, 2 C C 3 2 3 7 7 75 N 6 6 6 7 7 7 = N 6 6 6 C A  .  .  C A  .  .  8 k  0: .  .     C A n �1 C A n +k 24.4.1 The Observ ability Gramian De\fne Z t Qt = (e A ) T C T C e A d: 0 The system is then observable if and only if rank ( Q t )= n; 8 t� 0. If A is stable, then we may de\fne the observ ability Gramian as Z 1 Q = (e A ) T C T C e A d: 0 Once again, Q satis\fes a Ly apunov equation: A T Q + QA = � C T C: 24.5 Further Results In view of dualit y, we can use our reac hability results to immediately derive various conclusions, tests, standard and canonical forms, etc., for observable and unobservable systems. We shall simply list the main results: Fact 1: The unobservable subspace is A -in v ariant (and is in fact the largest A -in v ariant subspace contained in the nullspace of C). Fact 2: An unobservable pair (C; A ) can be taken by a similarity transformation to the form     A ! A1 0 A12 A2 (24.8)     C ! 0 C2 (24.9) where ( C 2 ; A 2) is observable. The unobservable subspace of the system associated with (24.8), (24.9)      is given by states of the form . The eigen values of A1 are the unobservable eigen values of the 0 system, while those of A2 are the observable eigen values. Fact 3: The system is unobservable i Cv = 0 for some right eigen vector v of A, associated with an eigen value ; or equiv alently i     sI � A (24.10) C drops rank for some s = . This  is an unobservable eigen value of the system. Fact 4: The dual of the problem of con trollability to the origin is referred to as constructability of the \fnal state, i.e. instead of trying to uniquely determine the initial state x(0) from input/output measurements o ver an in terval, we wish to determine the \fnal state. In CT, the condition for this reduces to the observ ability condition, but in DT it turns out that x ( k) (for k  n) can be determined from u ( i ) ; y ( i) o ver [0 ;k � 1] i N ( O n) N ( A n ): (24.11) We l e a ve you to pro ve this. Note that for in vertible A we reco ver the observ ability condition. Fact 5: Any single-output (SO) observable n th -order system is similarity transformable to an observ - ability canonical form (dual to the reac habilit y/con trollability canonical form presented last lecture) or to an observer canonical form (dual to the controller canonical form of last lecture). 24.5.1 Standard Form for Unobservable Systems Given an arbitrary system, we can construct a rectangular matrix T o n r , whose columns span the nullspace of the observ ability matrix O n. We may then construct T 0 by selecting (n � r) linearly indep endent vectors, such that     rank (T n  n ) = rank To T 0 = n: Since T is in vertible, we can perform a similarity transform to generate an equiv alent system, where we ha ve:         AT = A To T 0 = TA = To T 0 A1 A2 0 A3 (24.12)    CT = C To T 0 = C =0 C1 (24.13) The presence of the zero blo cks in the transformed system and output matrices follows from an argu - ment similar to that used for the reachable canonical form coupled with the fact the the unobservable space is also A -in v ariant. It follows from (24.12) and (24.13) that 3 2 3 2 CT 0 C1 On = 6 6 64 7 7 7 =  6 6 6 7 7 75 C AT  .  .  0 C 1 A3 . .  . .  :   . ..    C A n � 1 T 0 C 1 A n 3 �1 Now since the transformed system is equiv alent to the original one, rank ( O n) = rank ( O n ): Theorem 24.4 ( Modal T est) A c o n tinuous-time system is observable if and only if  I � A rank = n; 8  2 C C Proof: The proof follows from the observable canonical form. Exercises Exercise 24.1 (a) Given the observable nth-order DT system x (k +1) = Ax ( k )+ B u ( k), y ( k )= C x ( k )+ D u ( k), show that we can uniquely determine the initial condition x(0) from output me asurements alone, i.e. without knowledge of the inputs u ( i), if D = 0, CB = 0, CAB = 0 ; : : : ; C A n � 2 B = 0. (b)  (Optional) Pro ve that the sucient condition in (a) is also ne cessary in the case where the output y is scalar. (c)  Verify in the case of a single-input, single-output (SISO) system that the condition i (a) corre - sponds to the transfer function of the system having no (\fnite) zeros, only poles. Exercise 24.2 Consider the system x (k +1) = Ax ( k), y ( k )= C x ( k), and suppose that A and C ha ve the following form: 0  A11 A12 0 : :: 0 1 A = B B B B@ A21 A22 A23 : :: 0 . . . .. . . . .. . . . .. A k � 1 ;1 A k � 1 ;2 A k � 1 ;3 : :: A k � 1;k C C C C  A k1 A k2 A k3 : :: Akk C =( C1 00 : :: 0) with the A i � 1;i and C1 all having full column rank, and with the Aii all square. (a)  Show that the system is observable. (Hint: Show \frst that if P and Q ha ve f u ll column rank, then PQ has full column rank.) (b) What can you say about the system if A k � 1 ;k, instead of having full column rank, is actually 0? [The results of this problem form the basis for one of the best numerical methods for c hecking observ ability (or, using a dual set of results and computations, for c hecking reac hability). The p o i nt is that orthogonal (and therefore numerically well b eha ved) similarity transformations can be used to bring a ( C;A ) pair to the structured form ab o ve.] Exercise 24.3 (a) Consider the CT single-output nth-order observable system x _ ( t )= Ax ( t) ; y ( t )= cx ( t), where c is an n -comp onent vector. Suppose we observe o n ly samples of the output, taken at times t = kT for integer k and some \fxed T . Write down a DT state-space model that describes the evolution of these output samples. (b)  Find a necessary and sucient condition, stated in terms of the eigen values of A, for your DT model in (a) to be observable. (Hint: First show that there is no loss of generality in taking A to be in Jordan form.) Exercise 24.4 Consider an observable single output LTI system. Find the smallest perturbation of the C row vector, k  k 2, so that the system becomes unobservable. Exercise 24.5 Consider a SISO, stable, discrete-time system with a state-space description given by (A; B; C; 0), both reachable and observable. The dimension of the state vector is equal to n. We are interested in studying the e ects of past inputs on future outputs in a precise fashion. It turns out that this information is quite valuable in model reduction problems, but we will not look into that here. Recall that the energy of a signal u is given by 1 X 2 2 k u k2 = (u T u) 11 =( u ( t ) 2 ) t =�1 (Don't be afraid to transpose in\fnite vectors or matrices). The precise quan tity that we would like to compute is: 1 � 1 XX = maxf y ( t ) 2 j u ( t ) 2  1 ; u ( k ) =0 8k  0 g u t=0 t =�1 In words, we w ant to \fnd the maximal ac hievable energy of the output after t = 0 w h ich is the response to an input starting at �1 and ending at �1, i.e., the has the following form T u =( : :: u ( � N) u ( �N + 1) : :: u ( �1) 0 : :: ) with energy less than or equal to one. (a)  Find an expression for x(0) in terms of the ab o ve input. Does the value x ( �1) enter your expression. Explain. (b)  Can any  2 R n be a c hieved by s o me choice of an input of the ab o ve fo r m? If so, \fnd an expression of the minimum energy input, umin that ac hieves the value x(0) = . Compute the square of the energy of u min. Write this expression in terms of the Reac hability Gramian, and denote it by 1 ( ). (c)  If some input umin results in x(0) = , write an expression of the output for t  0. Compute t square of the energy of the output (for t  0) as a function of . Write this in an expression in volving the Observ ability Gramian and denote it by 2 ( ). (d  ) Argue that is given by = max f 2 ( ) j 1 ( )  1 g  . (e) Pro ve that =  max fQPg where P and Q are the reac hability and observ ability gramians of the system, and max denotes the maximum eigen value of a matrix.  (Hint: Use the fact that any symmetric p ositive semi-de\fnite matrix can be written as M T M .  Also you may need the fact that  max ( M T GM )=  max ( GM M T )).