Systems and Control Theory Lecture Notes. Laura Giarré

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1 Systems and Control Theory Lecture Notes Laura Giarré L. Giarré

2 Lesson 14: Rechability I Reachability (DT) I Reachability theorem (DT) I Reachability properties (DT) I Reachability gramian (DT) I Reachability from an Arbitrary Initialm state (DT) I Controllability vs. Reachability I Reachability (CT) I Reachability properties (CT) L. Giarré Systems and Control Theory

3 Reachability (DT) I We now turn to a more detailed examination of how inputs a ect states; consider a n dimensional DT system: I Recall that x(i + 1) =Ax(i)+Bu(i) 2i=0 Xk 1 x(k) =A k x(0)+c A k i 1 Bu(i) 0 =A k x(0)+[a k 1 B A k 2 B... B] =A k x(0)+r k U k ~ u(0) u(1). u(k 1) Ua AEI L. Giarré Systems and Control Theory C A

4 Reachability (DT) I Consider whether and how we may choose the input sequence u(i), i 2 [0; k 1], so as to move the system from x(0) =0to a desired target state x(k) =d at a given time k. Ifthereis such an input, we say that the state d is reachable in k steps. I The kreachable set R k is the set of the states reachable from the origin in k steps: R k. = Ra(Rk ) I This set is a subspace. I The matrix R k is said the kstep reachability matrix. L. Giarré Systems and Control Theory

5 Reachability Theorem (DT) Theorem: For k apple n apple l Ra(R k ) Ra(R n)=ra(r l ) Proof: The fact that Ra(R k ) Ra(R n ) for k apple n follows trivially from the fact that the columns of R k are included among those of R n.to show that Ra(R n )=Ra(R l ) for l n, note from the CayleyHamilton theorem that A i for i n can be written as a linear combination of A n 1,...A, I, so all the columns of R l for l n are linear combinations of the columns of R n. L. Giarré Systems and Control Theory

6 Reachability properties (DT) I The subspace of states reachable in n steps, i.e. Ra(R n ),is referred to as the reachable subspace, and will be denoted simply by R. I Any reachable target state, i.e. any state in R, isreachablein n steps (or less). I The system is termed areachablesystemif all of R n is reachable, i.e. if rank(r n )=n. I The matrix R n =[A n 1 B A n 2 B... B] is termed the reachability matrix (often written with its block entries ordered oppositely to the order that we have used here). L. Giarré Systems and Control Theory

7 Reachability Gramian (DT) I Let us first define the kstep reachability Gramian P k by kx P k = R k Rk T = A i BB T (A T ) i I This matrix is therefore symmetric and positive semidefinite. i=0 Lemma Ra(P k )=Ra(R k )=R k L. Giarré Systems and Control Theory

8 Reachability from an Arbitrary Initial State I Note that getting from a nonzero starting state x(0) =s to a target state x(k) =d requires us to find a U k for which air d A k s = R k U k a I For arbitrary d, s, the requisite condition is the same as that for reachability from the origin. = I Thus we can get from an arbitrary initial state to an arbitrary final state if and only if the system is reachable (from the origin); and we can make the transition in nsteps or less, when the transition is possible. L. Giarré Systems and Control Theory

9 Controllability versus Reachability I Now consider what is called the controllability problem: bringing an arbitrary initial state x(0) to the origin in a finite number of steps: A k x(0) =R k U k Ah EYE I If A is invertible and x(0) is arbitrary, then the left side is arbitrary, so the condition for controllability of x(0) to the origin is rank(r k )=n for some k, i.e.justthereachability condition that rank(r n )=n. I If, on the other hand, A is singular (i.e. has eigenvalues at 0), then the left side will be be confined to a subspace of the state space and we can prove that the system is controllable, i Ra(A n ) Ra(R n ) L. Giarré Systems and Control Theory

10 " A = 1oz If B = [9) A I 1 3 I At B B A = h 1 U 2 deter ) = I to Baek ( R ) = 2 R an. ihd present. 2 columns

11 Assure target is your 131 =D then :D ' Eni.. it :X :L Ed

12 The signee is reachable ( cryakg) Es 2 A 5%713=17) for * I ;) " til 41) rank Ck )

13 . He " This is not cuyeelelg Reachable din I Ra CR ) ) = ' I Reachable subspace hias di wee Wen z K I Raf Rnz

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15 Reachability (CT) I Given a system described by the (ndimensional) statespace model ẋ(t) =Ax(t)+Bu(t), x(0) =0, a point x d is said to be reachable in time L if there exists an input u : t 2 [0, L] 7! u(t) such that x(l) =x d. I Given an input signal over [0, L], one can compute 9 x(l) = 9 Z L 0 e A(L t) Bu(t)dt = Z L 0 F T (t)u(t)dt. = F, u L where F T (t). = e A(L t) B. I The set R of all reachable points is a linear (sub)space: if x a and x b are reachable, so is x a + x b. I If the reachable set is the entire state space, i.e., if R = R n, then the system is called (completely) reachable. L. Giarré Systems and Control Theory

16 Reachability (CT): Properties I The Reachable subspace R is related to the Reachability Gramian (at time L): Theorem. R L Let P L = F, F = 0 F T (t)f (t)dt. Then, R = Ra(calP. L ). Theorem 2 (reachability matrix) Ra(P L )=Ra(R N )=Ra([A n 1 B A n 2 B... B]) Corollary The system is reachable i rank(r n )=n I Notice that this condition does not depend on L! I Controllability and reachability coincides for CT systems (e At is always invertible). L. Giarré Systems and Control Theory

17 Lesson 15: Modal Aspects I Ainvariance I Standard Kalman form I Modal Reachability tests L. Giarré Systems and Control Theory

18 Ainvariance Corollary The reachable subspace Ris Ainvariant, i.r. x 2R!Ax 2R: AR = R I a Uf R MAKER L. Giarré Systems and Control Theory

19 I Standard (Kalman) Form for an unreachable system I Let r = rank(r), then the subspace of reachable states has dimensions dim(r = r, ra(r) =r, andthesystempresents n r unreachable states apple I Let z = T 1 zr x =. F? z r I In these new coordinate the system will take the form apple zr (k + 1) z r (k + 1) apple Ar = It A r r 0 A r apple zr (k) z r (k) apple Br + 0 u(k) So = SPEE ) = SHAD 0k L. Giarré Systems and Control Theory

20 ' E I F * " Extras Are

21 Standard Form: Constructing T I Let T1 n r be a matrix whose columns form a basis for the reachable subspace, i.e. Ra(T 1 )=Ra(R n ) I Let T 2 n n (n r) be a matrix whose columns are independent of each other and of those in T 1. I Then choose T =[T 1 T 2 ]. I This matrix is invertible, since its columns are independent by construction I We claim that apple Ar A[T 1 T 2 ]=T O Ā =[T 1 T 2 ] apple B = T B Br =[T 1 T 2 ] 0 A r r 0 A r I The proof os based on the Ainvariance (columns AT 1 remains ad in Ra(T 1 )) a L. Giarré Systems and Control Theory

22 rank ( R ) = t a T f. Et. huns Efe.. Y I r, in V defender of R

23 Reachable/unreachable eigenivalues I The motion of z r (k) is described by the rthorder reachable statespace model z r (k + 1) =A r z r (k)+b r u(k) that is called the reachable subsystem. I The eigenvalues of A r are the reachable eigenvalues I the eigenvalues of A r are called the unreachable eigenvalues. L. Giarré Systems and Control Theory

24 A f ] Bit R [ 3 f ] rank (K ) = I =r F [ I i% ' ' T I 4 we

25 Are 2 AF = O X r = 2 reached A F = O Wide

26 Modal Reachability Tests Theorem The system is unreachable if 1 and only if w T B = 0 for some left eigenvector w T of A. We say that the corresponding eigenvalue is an unreachable eigenvalue. Proof If w T B = 0andw T A = w T with w T 6= 0, then w T AB = w T B = 0andsimilarlyw T A k B = 0, so w T R n = 0, i.e. the system is unreachable. Conversely, if the system is unreachable, transform it to the standard form. Now let w2 T denote a left eigenvector of A r, with eigenvalue.thenw T =[0 w2 T ] is a left eigenvector of the transformed A matrix, namely Ā and is orthogonal to the (columns of the) transformed B, namely B. L. Giarré Systems and Control Theory

27 Modal Reachability Tests Corollary The system is unreachable if and only if [zi A B] loses rank for some z =.This is then an unreachable eigenvalue. Proof The matrix [zi A B] has less than full rank at z = i w T [si A B] =0 for some w T 6= 0. But this is equivalent to having a left eigenvector of A being orthogonal to (the columns of) B. L. Giarré Systems and Control Theory

28 ".. det ( A A EA ) e D. = ( A ( A 3) = 2) ( 1123 d 2) CA o.d 1? 4 agenda f

29 I A I B) =. :B 2 As, Xz Az * I :

30 left A eigenvectors wt 4kt A ) = O \ wt ( At Right = A ) v eigenvector o B o

31 Ew. wow DfE? fee 0 o 0 A I A for A = o. s T w s tf WELL ) To e D (1) = 0 O. 5 is unreachable

32 Thanks DIEF Tel: giarre.wordpress.com