Systems and Control Theory Lecture Notes. Laura Giarré

Transcription

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

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

3 Reachability (DT) We now turn to a more detailed examination of how inputs affect states; consider a n dimensional DT system: Recall that x(i + 1) = Ax(i) + Bu(i) k 1 x(k) =A k x(0) + C A k i 1 Bu(i) 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) L. Giarré- Systems and Control Theory

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

5 Reachability Theorem (DT) Theorem: For k n l Ra(R k ) Ra(R n) = Ra(R l ) Proof: The fact that Ra(R k ) Ra(R n ) for k 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 Cayley-Hamilton 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) 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. Any reachable target state, i.e. any state in R, is reachable in n steps (or less). The system is termed a reachable system if all of R n is reachable, i.e. if rank(r n ) = n. 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) Let us first define the k-step reachability Gramian P k by k P k = R k Rk T = A i BB T (A T ) i This matrix is therefore symmetric and positive semi-definite. i=0 Lemma Ra(P k ) = Ra(R k ) = R k L. Giarré- Systems and Control Theory

8 Reachability from an Arbitrary Initial State 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 d A k s = R k U k For arbitrary d, s, the requisite condition is the same as that for reachability from the origin. 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 n-steps or less, when the transition is possible. L. Giarré- Systems and Control Theory

9 Controllability versus Reachability 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 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. just the reachability condition that rank(r n ) = n. 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, iff Ra(A n ) Ra(R n ) L. Giarré- Systems and Control Theory

10 Reachability (CT) Given a system described by the (n-dimensional) state-space 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 [0, L] u(t) such that x(l) = x d. Given an input signal over [0, L], one can compute x(l) = L 0 e A(L t) Bu(t)dt = L 0 F T (t)u(t)dt. = F, u L where F T (t). = e A(L t) B. The set R of all reachable points is a linear (sub)space: if x a and x b are reachable, so is αx a + βx b. 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

11 Reachability (CT): Properties The Reachable subspace R is related to the Reachability Gramian (at time L): Theorem. 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 iff rank(r n ) = n Notice that this condition does not depend on L! Controllability and reachability coincides for CT systems (e At is always invertible). L. Giarré- Systems and Control Theory

12 Lesson 15: Modal Aspects A-invariance Standard Kalman form Modal Reachability tests L. Giarré- Systems and Control Theory

13 A-invariance Corollary The reachable subspace Ris A-invariant, i.r. x R Ax R: AR = R L. Giarré- Systems and Control Theory

14 Standard (Kalman) Form for an unreachable system Let r = rank(r), then the subspace of reachable states has dimensions dim(r = r, ra(r) = r, and the system presents n r unreachable states [ ] Let z = T 1 zr x =. z r In these new coordinate the system will take the form [ ] ] [ ] [ zr (k + 1) zr (k) Br + z r (k + 1) z r (k) 0 [ Ar A = r r 0 A r ] u(k) L. Giarré- Systems and Control Theory

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

16 Reachable/unreachable eigenivalues The motion of z r (k) is described by the rth-order reachable state-space model z r (k + 1) = A r z r (k) + B r u(k) that is called the reachable subsystem. The eigenvalues of A r are the reachable eigenvalues the eigenvalues of A r are called the unreachable eigenvalues. L. Giarré- Systems and Control Theory

17 Modal Reachability Tests Theorem The system is unreachable if and only if w T B = 0 for some left eigenvector w T of A. We say that the corresponding eigenvalue λ is an unreachable eigenvalue. Proof If w T B = 0 and w T A = λw T with w T 0, then w T AB = λw T B = 0 and similarly w 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 λ. Then w 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

18 Modal Reachability Tests Corollary The system is unreachable if and only if [zi A B] loses rank for some z = λ. This λ is then an unreachable eigenvalue. Proof The matrix [zi A B] has less than full rank at z = λ iff w T [si A B] = 0 for some w T 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

19 Thanks DIEF- Tel: giarre.wordpress.com