European Embedded Control Institute. Graduate School on Control Spring The Behavioral Approach to Modeling and Control.

Transcription

1 p. 1/50 European Embedded Control Institute Graduate School on Control Spring 2010 The Behavioral Approach to Modeling and Control Lecture VI CONTROLLABILITY

2 p. 2/50 Theme Behavioral controllability, the property of being able to choose the future of a trajectory regardless of the past, is an important and regularizing system property. In this lecture we obtain (i) various tests for controllability of LTIDSs, (ii) special system representations related to controllability, (iii) and generalizations, e.g. to stabilizability.

3 p. 3/50 Outline Controllability of LTIDSs Image representations Autonomous LTIDSs Stability and stabilizability The controllable part of a behavior The three main theorems for LTIDSs and their discrete-time and PDE counterparts Observability

4 Controllability p. 4/50

5 p. 5/50 Reminder R w w 1,w 2 B w 1 0 w 2 time R w transition R w w B w 1 w 0 w t time σ t w 2 w controllability : concatenability of trajectories after a delay

6 p. 6/50 Controllability of LTIDSs Theorem The following are equivalent for B L w. 1. B is controllable. 2. Let R ( d dt) w = 0 be a kernel representation B. Then R(λ) has the same rank for all λ C. Hence, R ( d dt) w = 0 is a minimal kernel representation of a controllable LTIDS if and only if R is left prime. B L is controllable if and only if it has a kernel representation R ( d dt) w = 0 with R left prime over R[ξ].

7 p. 7/50 Proof in telegram-style First prove that if U R[ξ] w w is unimodular, then B is controllable if and only if U ( d dt ) B is controllable. Consequently, we may assume that B has a minimal kernel representation with R in Smith form, ] R = [diag(d 1,d 2,...,d r ) 0 r (w r) Observe, using the theory of autonomous systems, that B is controllable if and only if all the invariant polynomials d 1,d 2,...,d r of R are equal to one. Equivalently, if and only if 2. holds.

8 Examples Consider the single-input/single output system ( ) ( ) [ ] d d u p y = q u, w =, dt dt y with p,q R[ξ]. This system is controllable if and only if p and q are coprime. The interpretation of common factors in p and q has been a long-standing question in the field. Behavioral controllability demystifies common factors. We now understand that common factors correspond exactly to lack of controllability, nothing more, nothing less. In the exercises several algorithms for coprimeness of polynomials are discussed. p. 8/50

9 Examples Applying the controllability theorem and the relation between behavioral and state controllability discussed in Lecture I, shows that the system u d x = Ax+Bu, y = Cx+Du, w = y dt x is controllable if and only if ([ ]) rank A I n n λ B =n for all λ C. This state version of controllability is the PBH (Popov-Belevitch-Hautus) test. The controllability theorem is a far-reaching generalization of this classical result. V.M. Vitold Malo Popov Belevitch Hautus (1928 ) ( ) (1940 ) p. 9/50

10 p. 10/50 Examples The port behavior of the RLC circuit (see Lecture IV) I V + R C L C R L is controllable unless CR C = L R L and R L = R C. This shows that lack of controllability can occur in non-degenerate physical systems.

11 p. 11/50 More tests for controllability Theorem The following are equivalent for B L w. 1. B is controllable. 3. N B, the R[ξ]-module of annihilators of B, is closed. 4. B has a direct summand, i.e., there exists B L w such that B B = C (R,R w ). The closure of the R[ξ]-submodule M of R[ξ] w is defined as M closure := { m R[ξ] w π R[ξ],π 0, and m M such that m = π m}. M is said to be closed if M = M closure.

12 p. 12/50 Proof in telegram-style First prove that if U R[ξ] w w is unimodular, then B is controllable if and only if U ( d dt ) B is controllable. Consequently, we may assume that B has a minimal kernel representation with R in Smith form, ] R = [diag(d 1,d 2,...,d r ) 0 r (w r) Observe that N B = [ ] R[ξ]d 1 R[ξ]d r 0 0. Hence N B is closed if and only if all the invariant polynomials d 1,d 2,...,d r of R are equal to one. Equivalently, if and only if 2. holds.

13 p. 12/50 Proof in telegram-style Consequently, we may assume that B has a minimal kernel representation with R in Smith form, ] R = [diag(d 1,d 2,...,d r ) 0 r (w r) Take for B the system with kernel representation R ( ) ] d dt w = 0, with R = [0 w r r I (w r) (w r).

14 p. 12/50 Proof in telegram-style Consequently, we may assume that B has a minimal kernel representation with R in Smith form, ] R = [diag(d 1,d 2,...,d r ) 0 r (w r) Note that [[B B = C (R,R w )]] [[N B N B = R[ξ] 1 w ]]. Let R ( ) d dt w = 0 be a minimal kernel representation of B. [ ] Then N B N B = R[ξ] w R implies that the rows of R form a [ ] basis for R[ξ] 1 w R. Equivalently, that R is unimodular. Hence that the invariant polynomials of R are all equal to one.

15 p. 13/50 Geometric interpretation of controllability Attach to each point λ C of the Riemann sphere (think of the Riemann sphere as C), kernel(r(λ)). This associates with each λ C a linear subspace of C w. In general, this yields a picture shown below. Since the dimension of the subspace attached may change, we obtain a sheaf. λ kernel(r(λ)) Riemann sphere Bernhard Riemann

16 p. 13/50 Geometric interpretation of controllability Attach to each point λ C of the Riemann sphere (think of the Riemann sphere as C), kernel(r(λ)). This associates with each λ C a linear subspace of C w. The dimension of the subspace is constant, that is, we obtain a vector bundle over the Riemann sphere, if and only if the system is controllable. λ Riemann sphere kernel(r(λ)) Bernhard Riemann

17 Representations of behaviors p. 14/50

18 p. 15/50 Kernels, images, and projections A model B is a subset of U. There are many ways to specify a subset. For example, as the set of solutions of equations, as the image of a map, as a projection.

19 p. 15/50 Kernels, images, and projections A model B is a subset of U. There are many ways to specify a subset. For example, as the set of solutions of equations: f : U, B = {w U f(w) = 0}, as the image of a map: f : U, B = {w U l such that w = f(l) }, as a projection: B extended U L, B = {w U l L such that (w,l) B extended }.

20 Kernels, images, and projections A model B is a subset of U. There are many ways to specify a subset. For example, as solutions of equations: kernel representation f : U, B = {w U f(w) = 0}, as the image of a map: image representation f : U, B = {w U l such that w = f(l) }, as a projection: latent variable representation B = {w U l L such that (w,l) B extended }, w s manifest variables: the variables the model aims at, l s latent variables: auxiliary variables. p. 15/50

21 p. 16/50 Image as the image of a map: image representation f : U, B = {w U l such that w = f(l) }. f B U

22 p. 17/50 Controllability and image representations

23 p. 18/50 Images We have seen kernels and projections of linear constant-coefficient differential operators. It is time for images! w = M ( d dt) l. M ( ) d dt C (R,R w ) C (R,R ) image ( M ( )) d dt

24 p. 18/50 Images We have seen kernels and projections of linear constant-coefficient differential operators. It is time for images! w = M ( d dt) l. M ( ) d dt C (R,R w ) C (R,R ) image ( M ( )) d dt Elimination theorem (see Lecture IV) an image is a kernel. What is special about images?

25 p. 19/50 The image representation theorem Theorem The following are equivalent for B L w. 1. B is controllable. 5. B has a image representation w = M ( d dt) l. Images have a nice system theoretic interpretation!

26 p. 20/50 Proof in telegram-style By controllability, the invariant polynomials of R, with R ( d dt) w = 0 a minimal kernel representation of B, are equal ] to one. Therefore, R = V [I r r 0 r (w r) U, with U,V ] unimodular. It follows that w = U 1( d dt ) [ 0 r (w r) is an image representation of B I (w r) (w r) The extended behavior {(w,l) w = M ( ) d dt l} is controllable, [ ] since I w w M(λ) has rankwfor all λ C. This implies that the projection is controllable. l

27 Algorithm for controllability p. 21/50

28 p. 22/50 Algorithm Start with R R[ξ] w parametrizing the LTIDS R ( d dt ) w = 0. Problem: Verify if this system is controllable.

29 p. 22/50 Algorithm Start with R R[ξ] w Problem: Verify if this system is controllable. The set { f R[ξ] w R f = 0} is called the right syzygy of R. It is obviously an R[ξ]-module. Compute a basis for the right syzygy of R. Let F be a matrix whose rows form a basis for this syzygy.

30 p. 22/50 Algorithm Start with R R[ξ] w Problem: Verify if this system is controllable. Compute a basis for the right syzygy of R. Let F be a matrix whose rows form a basis for this syzygy. Compute a basis for the left syzygy of F. Let R be a matrix whose rows form a basis for this syzygy. Computing such a basis is a standard problem in computer algebra

31 p. 22/50 Algorithm Start with R R[ξ] w Problem: Verify if this system is controllable. Compute a basis for the right syzygy of R. Let F be a matrix whose rows form a basis for this syzygy. Compute a basis for the left syzygy of F. Let R be a matrix whose rows form a basis for this syzygy. The system is controllable R = R. Verifying this equality is a standard problem in computer algebra.

32 Autonomous LTIDSs p. 23/50

33 p. 24/50 Reminder W time PAST W time FUTURE autonomous : the past implies the future.

34 p. 25/50 Characterization of autonomous LTIDSs Theorem The following are equivalent for B L w. B is autonomous. B is a finite-dimensional subspace of C (R,R w ). B has a minimal kernel representation R ( d dt) w = 0 with R square and determinant(r) 0. m(b) = 0, equivalently, p(b) =w(b).

35 p. 25/50 Characterization of autonomous LTIDSs Theorem The following are equivalent for B L w. B is autonomous. B is a finite-dimensional subspace of C (R,R w ). B has a minimal kernel representation R ( d dt) w = 0 with R square and determinant(r) 0. m(b) = 0, equivalently, p(b) =w(b). The proof follows readily from the Smith form and Propositions 1 and 3 of the section on differential operators (see Lecture II).

36 p. 25/50 Characterization of autonomous LTIDSs Theorem The following are equivalent for B L w. B is autonomous. B is a finite-dimensional subspace of C (R,R w ). B has a minimal kernel representation R ( d dt) w = 0 with R square and determinant(r) 0. m(b) = 0, equivalently, p(b) =w(b). With R minimal, there holds, for w = 1, dimension(b) = degree(r), for w > 1, dimension(b) = degree(determinant(r)).

37 p. 26/50 Autonomous LTIDSs Each trajectory w of an autonomous B L w is a sum of products of a polynomial and an exponential in the complex case, w(t) = π 1 (t)e λ 1t + π 2 (t)e λ 2t + +π r (t)e λ rt, with π k C[ξ] w and λ k C. In the real case, it is a sum of products of a polynomial, an exponential, and a trigonometric function, w(t) = π 1(t)e λ 1t cos(ω 1 t)+π 1(t)e λ 1t sin(ω 1 t) + π 2(t)e λ 2t cos(ω 2 t)+π 2(t)e λ 2t sin(ω 2 t) + +π r(t)e λ rt cos(ω r t)+π r(t)e λ rt sin(ω r t), with π k,π k R[ξ] w,λ k R, and ω k R.

38 Stability p. 27/50

39 p. 28/50 Reminder W time stability : all trajectories go to 0.

40 p. 28/50 Reminder W time stability : all trajectories go to 0. For B L w, there holds [[B stable]] [[B autonomous]].

41 Stability of LTIDSs Theorem The following are equivalent for B L w. B is stable. Every exponential trajectory t e λt a, a C w, in B (complexified) has Real(λ) < 0. B has a minimal kernel representation R ( ) d dt w = 0 with R Hurwitz. A polynomial C[ξ] is said to be Hurwitz if all its roots are in {λ C Real(λ) < 0}. P C[ξ] n n is said to be Hurwitz if it is square and determinant(p) is Hurwitz. p. 29/50

42 p. 29/50 Stability of LTIDSs Theorem The following are equivalent for B L w. B is stable. Every exponential trajectory t e λt a, a C w, in B (complexified) has Real(λ) < 0. B has a minimal kernel representation R ( ) d dt w = 0 with R Hurwitz. The proof follows readily from the Smith form and Propositions 1 and 3 of the section on differential operators (see Lecture II).

43 Stabilizability p. 30/50

44 p. 31/50 Reminder w W 0 w time stabilizability : all trajectories can be steered to 0.

45 p. 32/50 Stabilizability of LTIDSs Theorem The following are equivalent for B L w. 1. B is stabilizable. 2. If R ( ) d dt w = 0 is a kernel representation of B, then [[λ C,rank(R(λ)) < rank(r)]] [[Real(λ) 0]].

46 p. 33/50 Proof in telegram-style First assume that R is in Smith form. Prove that if 2. is satisfied, then the firstrcomponents of w are polynomial exponentials, with exponentials having negative real part, while the remaining components of w are free. Conclude stabilizability Conversely, if 2. is not satisfied, then there is a solution whose first component is an exponential with real part 0. This exponential cannot be steered to zero, and the system is not stabilizable.

47 p. 33/50 Proof in telegram-style Next, consider general R s. The solutions are now of the form w = U ( d dt) w with U R[ξ] w w unimodular, and w a solution corresponding to the Smith form of R. The arguments extend, since polynomial exponentials are converted by U ( ) d dt to polynomial exponentials with the same exponential coefficients.

48 More on controllability p. 34/50

49 p. 35/50 The controllable part of a LTIDS The controllable part of B L w, denoted by B controllable, is defined as the largest controllable LTIDS contained in B. That is, 1. B controllable L w, 2. B controllable B, 3. [[B L w,b B, and B controllable ]] [[B B controllable ]].

50 p. 35/50 The controllable part of a LTIDS The controllable part of B L w, denoted by B controllable, is defined as the largest controllable LTIDS contained in B. Let R ( d dt ) w = 0 be a minimal kernel representation of B. The polynomial matrix R can be factored as R = FR, with F R[ξ] p(b) p(b) and with R R[ξ] p(b) w(b) left prime: R (λ) has the same rank for all λ C. Then R ( d dt) w = 0 is a kernel representation of B controllable.

51 p. 35/50 The controllable part of a LTIDS The controllable part of B L w, denoted by B controllable, is defined as the largest controllable LTIDS contained in B. C There holds B controllable = B (R,R w(b) ) compact, where B compact denotes the set of compact support trajectories C in B, and B (R,R w(b) ) compact the closure of Bcompact in the C (R,R w(b) )-topology.

52 p. 35/50 The controllable part of a LTIDS The controllable part of B L w, denoted by B controllable, is defined as the largest controllable LTIDS contained in B. Every B L w admits a decomposition B = B 1 B 2, with B 1 L w controllable and B 2 L w autonomous. In every such a decomposition, B 1 = B controllable.

53 p. 36/50 Controllability and rational symbols B is controllable if and only if it admits a representation ( ) d w = M l dt with M R(ξ) w. In particular, therefore, the system ( ) d y = G u, dt with G R(ξ) p m, is always controllable.

54 p. 37/50 Controllability and transfer functions Consider the input/output systems ( ) ( ) ( d d d P 1 y 1 = Q 1 u 1, P 2 dt dt dt ) ( d y 2 = Q 2 dt ) u 2, with determinant(p 1 ) 0, and determinant(p 2 ) 0. These two systems have the same transfer function, P 1 1 Q 1 = P 1 2 Q 2, if and only if they have the same controllable part. Therefore, the transfer function determines the controllable part of a system, but only the controllable part.

55 The three theorems p. 38/50

56 p. 39/50 LTIDSs The following are the three main theorems for LTIDSs. Theorem 1: There exists a 1 1 relation between L w and the R[ξ]-submodules of R[ξ] w. Theorem 2: Elimination. L is closed under projection: [[B L w 1+w 2 ]] [[Π 1 B L w 1]]. Π w1 defines the projection onto the firstw 1 components. Theorem 3: A LTIDS is controllable if and only if its behavior is the image of a linear constant-coefficient differential operator.

57 p. 40/50 Discrete-time LTIDSs These theorems remain valid for discrete-systems. The relevant ring for the case T = Z is R[ξ,ξ 1 ]. For T = Z +, it is R[ξ].

58 p. 41/50 PDEs These theorems also remain valid for systems described by linear constant-coefficient PDEs with the relevant ring R[ξ 1,ξ 2,...,ξ n ], the appropriate notion of controllability.

59 p. 42/50 n-d controllabilty [[(R n,r w,b) is controllable ]] : [[ w 1,w 2 B, and open subsets O 1,O 2 R n with non-intersecting closures, w B such that w O1 = w 1 O1 and w O2 = w 1 O2 ]].

60 p. 42/50 n-d controllabilty [[(R n,r w,b) is controllable ]] : [[ w 1,w 2 B, and open subsets O 1,O 2 R n with non-intersecting closures, w B such that w O1 = w 1 O1 and w O2 = w 1 O2 ]]. This definition is illustrated in the picture below. R w R w w O 1 O 2 R R w w w w R O 1 O 2 R

61 Observability and detectability p. 43/50

62 p. 44/50 Reminder observed w 1 System w 2 to-be-deduced observability : w 2 may be deduced from w 1.!!! Knowing the laws of the system!!!

63 p. 45/50 The observability theorem Theorem The following are equivalent for B L w 1+w 2, B C (R,R w 1 R w 2). 1. w 2 is observable from w 1 in B. 2. B has a kernel representation R 1 ( d dt) w1 = R 2 ( d dt) w2, with rank(r 2 (λ)) =w 2 for all λ C. 3. B has a minimal kernel representation w 2 = F ( d dt ) w1, R ( d dt ) w1 = 0.

64 p. 46/50 Proof in telegram-style [[Observability]] [[R 2 ( d dt) is injective]] [[2.]]

65 p. 46/50 Proof in telegram-style [[Observability]] [[R 2 ( d dt) is injective]] [[2.]] ( ) is obvious. To prove [ ( ), ] observe that 2. implies that R 2 I w2 w is of the form R 2 = V 2 U, with V,U unimodular. 0 w2 Therefore B admits the kernel representation ( ) ( ) ( ) d d d R 1 w 1 = U w 2, R 1 w 1 = 0, dt dt dt leading to 3.

66 p. 47/50 The detectability theorem Theorem The following are equivalent for B L w 1+w 2, B C (R,R w 1 R w 2). 1. w 2 is detectable from w 1 in B. 2. B has a kernel representation R 1 ( d dt) w1 = R 2 ( ddt ) w2, with rank(r 2 (λ)) =w 2 for λ {λ C Real(λ ) 0}. 3. B has a minimal kernel representation ( ) ( ) ( d d d H w 2 = F w 1, R dt dt dt with H Hurwitz. ) w 1 = 0,

67 Recapitulation p. 48/50

68 p. 49/50 Summary There exists tests for verifying the controllability of a LTIDS. A LTIDS is controllable if and only if it allows an image representation.

69 p. 49/50 Summary There exists tests for verifying the controllability of a LTIDS. A LTIDS is controllable if and only if it allows an image representation. There exists tests for verifying the stabilizability of a LTIDS.

70 p. 49/50 Summary There exists tests for verifying the controllability of a LTIDS. A LTIDS is controllable if and only if it allows an image representation. There exists tests for verifying the stabilizability of a LTIDS. Every LTIDS admits a decomposition as controllable autonomous.

71 p. 49/50 Summary There exists tests for verifying the controllability of a LTIDS. A LTIDS is controllable if and only if it allows an image representation. There exists tests for verifying the stabilizability of a LTIDS. Every LTIDS admits a decomposition as controllable autonomous. The transfer function determines the controllable part only.

72 p. 49/50 Summary There exists tests for verifying the controllability of a LTIDS. A LTIDS is controllable if and only if it allows an image representation. There exists tests for verifying the stabilizability of a LTIDS. Every LTIDS admits a decomposition as controllable autonomous. The transfer function determines the controllable part only. There exists tests for verifying the observability and the detectability of a LTIDS.

73 End of Lecture VI p. 50/50