On Controllability and Persistency of Excitation in Data-Driven Control: Extensions of Willems' Fundamental Lemma
Yue Yu, Shahriar Talebi, Henk J. van Waarde, Ufuk Topcu, Mehran Mesbahi, Behçet Açıkme?e
OOn Controllability and Persistency of Excitation in Data-DrivenControl: Extensions of Willems’ Fundamental Lemma
Yue Yu, Shahriar Talebi, Henk J. van Waarde, Ufuk Topcu, Mehran Mesbahi, and Behc¸et Ac¸ıkmes¸e
Abstract — Willems’ fundamental lemma asserts that all tra-jectories of a linear time-invariant (LTI) system can be obtainedfrom a finite number of measured ones, assuming controllabilityand a persistency of excitation condition hold. We show thatthese two conditions can be relaxed. First, we prove that thecontrollability condition can be replaced by a condition on thecontrollable subspace, unobservable subspace, and a certainsubspace associated with the measured trajectories. Second, weprove that the persistency of excitation requirement can bereduced if the degree of certain minimal polynomial is known ortightly bounded. Our results shows that data-driven predictivecontrol using online data is equivalent to model predictivecontrol, even for uncontrollable systems. Moreover, our resultssignificantly reduce the amount of data needed in identifyinghomogeneous multi-agent systems.
I. I
NTRODUCTION
Willems’ fundamental lemma provides a data-based pa-rameterization of trajectories generated by linear time invari-ant (LTI) systems [1]. In particular, consider the LTI system x t ` “ Ax t ` Bu t , (1a) y t “ Cx t ` Du t , (1b)where u t P R m , x t P R n , y t P R p denote the input, stateand output of the system at discrete time t , respectively.If system (1) is controllable, the lemma asserts that everylength- L input-output trajectory of system (1) is a linearcombination of a finite number of measured ones. Thesemeasured trajectories can be extracted from one singletrajectory with persistent excitation of order n ` L [1],or multiple trajectories with collective persistent excitationof order n ` L [2]. By parameterizing trajectories of thesystem (1) using measured data, the lemma has profoundimplications in system identification [3], [4], [5], and inspireda series of recent results including data-driven simulation [3],[6], output matching [7], control by interconnection [8], set-invariance control [9], linear quadratic regulation [10], andpredictive control [11], [12], [13], [14], [15], [16], [17].On the other hand, whether the conditions of controlla-bility and persistency of excitation in Willems’ fundamentallemma are necessary has not been investigated to depth. The Yue Yu, Shahriar Talebi, Mehran Mesbahi and Behc¸et Ac¸ıkmes¸e arewith the Department of Aeronautics and Astronautics, University ofWashington, Seattle, WA 98195 USA. Henk J. van Waarde is withthe Bernoulli Institute for Mathematics, Computer Science, and Arti-ficial Intelligence, University of Groningen, Nijenborgh 9, 9747 AG,Groningen, The Netherlands. Ufuk Topcu is with the Department ofAerospace Engineering, University of Texas at Austin, Austin, TX 78712USA. (emails:[email protected], [email protected], [email protected],[email protected], [email protected], [email protected],) current work aims to address this issue by answering thefollowing two questions. First, with persistency of excitation,to what extent can the linear combinations of finite number ofmeasured input-output trajectories parameterize all possibleones? Second, can the order of persistent excitation bereduced?Recent results on system identification has shown that,assuming sufficient persistency of excitation, the linear com-binations of a finite number of measured input-output tra-jectories contain any trajectory whose initial state is in thecontrollable subspace [18], [19]. As we show subsequently,such results only partially answer the first question above:trajectories with initial state outside the controllable subspacecan also be contained in the said linear combinations.We answer the aforementioned questions by introducingextensions of Willems’ fundamental lemma. First, we showthat, with sufficient persistency of excitation, any length- L input-output trajectory whose initial state is in some subspaceis a linear combination of a finite number of measured ones.The said subspace is the sum of the controllable subspace,unobservable subspace, and a certain subspace associatedwith the measured trajectories. Second, we show that the or-der of persistent excitation required by Willems’ fundamentallemma can be reduced from n ` L to δ min ` L , where δ min isthe degree of the minimal polynomial of the system matrix A . Our first result completes those presented in [18], [19] byshowing exactly which trajectories are parameterizable by afinite number of measured ones for an arbitrary LTI systems.Furthermore, this result show that data-driven predictivecontrol using online data is equivalent to model predictivecontrol, not only for controllable systems, as shown in[12], [13], but also for uncontrollable systems. Our secondresult, compared with those in [2], reduces the amount ofdata samples used in identifying homogeneous multi-agentsystems by one order of magnitude.The rest of the paper is organized as follows. We firstprove an extended Willems’ fundamental lemma and discussits implications in Section II. We provide ramifications ofthis extension for a representative set of applications in Sec-tion III before providing concluding remarks in Section IV. Notation:
We let R , and N and N ` denote the set ofreal numbers, non-negative integers, and positive integers,respectively. The image and right kernal of matrix M isdenoted by im M and ker M , respectively. The Moore-Penrose inverse of M is denoted by M : . When appliedto subspaces, we let ` and ˆ denote the sum [20, p.2]and Cartesian product operation [20, p.370], respectively.We let b denote the Kronecker product. Given a signal a r X i v : . [ ee ss . S Y ] F e b : N Ñ R q and i, j P N with i ď j , we denote f r i,j s “ “ f J i f J i ` ¨ ¨ ¨ f J j ‰ J , and the Hankel matrix ofdepth d ( d ď j ´ i ` ) associated with f r i,j s as H d p f r i,j s q “ »———– f i f i ` ¨ ¨ ¨ f j ´ d ` f i ` f i ` ¨ ¨ ¨ f j ´ d ` ... ... ... f i ` d ´ f i ` d ¨ ¨ ¨ f j fiffiffiffifl . II. E
XTENSIONS OF W ILLEMS ’ FUNDAMENTAL LEMMA
In this section, we introduce extensions of Willems’ fun-damental lemma. Throughout we let p u i r ,T i ´ s , x i r ,T i ´ s , y i r ,T i ´ s q denote a length- T i ( T i P N ` ) input-state-output trajectorygenerated by system (1) for all i “ , , . . . , τ , where τ P N ` is the total number of trajectories. We let t u i r ,T i ´ s u τi “ , t x i r ,T i ´ s u τi “ , t y i r ,T i ´ s u τi “ , (2)denote the set of input, state, and output trajectories, respec-tively. We will use the following subspaces R “ im “ B AB ¨ ¨ ¨ A n ´ B ‰ , O “ ker “ C J p CA q J ¨ ¨ ¨ p CA n ´ q J ‰ J , K r x , x , . . . , x τ s “ im “ X AX ¨ ¨ ¨ A n ´ X ‰ , (3)where X “ “ x x ¨ ¨ ¨ x τ ‰ . In particular, R is knownas the controllable subspace, O is known as the unobservablesubspace, K r x , x , . . . , x τ s is the controllable subspace with B replaced by X . We say system (1) is controllable if R “ R n . One can verify that R , O and K r x , x , ¨ ¨ ¨ , x τ s are allinvariant subspace of matrix A .We will also use the following definitions that streamlinethe subsequent analysis. Definition 1.
We say a length- L input-output trajectory p u r ,L ´ s , y r ,L ´ s q with L P N ` is parameterizable by t u r ,T i ´ s , y r ,T i ´ s u τi “ if there exists g P R ř τi “ p T i ´ L ` q such that „s u r ,L ´ s s y r ,L ´ s “ « H L p u r ,T ´ s q ¨ ¨ ¨ H L p u τ r ,T τ ´ s q H L p y r ,T ´ s q ¨ ¨ ¨ H L p y τ r ,T τ ´ s q ff g. (4)As an example, if τ “ , L “ , T “ , T “ , thenequation (4) becomes the following »——– s u s u s y s y fiffiffifl “ »——– u u u u u u u u u u y y y y y y y y y y fiffiffifl g. Definition 2 (Collective persistent excitation [2]) . We say t u r ,T i ´ s u τi “ is collectively persistently exciting of order d P N ` if d ď T i for all i “ , , . . . , τ and the mosaic-Hankel matrix, defined as ” H d p u r ,T ´ s q ¨ ¨ ¨ H d p u τ r ,T τ ´ s q ı , (5) has full row rank. If τ “ , then Definition 2 reduces to the traditional notionof persistency of excitation.The Willems’ fundamental lemma asserts that: if system(1) is controllable and t u i r ,T i ´ s u τi “ is collectively persis-tently exciting of order n ` L , then p s u r ,L ´ s , s y r ,L ´ s q isparameterizable by t u r ,T i ´ s , y r ,T i ´ s u τi “ if and only if itis an input-output trajectory of system (1) [1], [2]. As ourmain contribution, the following theorem shows that not onlythis lemma can be extended to an arbitrary LTI system, butalso the required order of collective persistent excitation canbe reduced from n ` L to δ min ` L , where δ min is the degreeof the minimal polynomial of matrix A . Theorem 1.
Let δ min be the degree of the minimal poly-nomial of matrix A , and δ P N ` with δ ě δ min . Let t u i r ,T i ´ s , x i r ,T i ´ s , y i r ,T i ´ s u τi “ be the set of input-state-output trajectories generated by system (1) . If t u r ,T i ´ s u τi “ is collectively persistently exciting of order δ ` L , then im « H p x r ,T ´ L s q ¨ ¨ ¨ H p x τ r ,T τ ´ L s q H L p u r ,T ´ s q ¨ ¨ ¨ H L p u τ r ,T τ ´ s q ff “ p R ` K r x , x , . . . , x τ sq ˆ R mL . (6) Further, p s u r ,L ´ s , s y r ,L ´ s q is parameterizable by t u r ,T i ´ s , y r ,T i ´ s u τi “ if and only if there exists astate trajectory s x r ,L ´ s with s x P R ` O ` K r x , x , . . . , x τ s , (7) such that p s u r ,L ´ s , s x r ,L ´ s , s y r ,L ´ s q is an input-state-output trajectory of system (1) .Proof. See the Appendix.
Remark 1.
The equality in (6) generalizes [18, Lem. 2] byproving stronger results using weaker assumptions. Particu-larly, the assumption of n ` L order of persistently excitationin [18, Lem. 2] is reduced to δ ` L with δ ě δ min , and thecontrollable subspace R used in [18, Lem. 2] is extended toits superset R ` K r x , x , . . . , x τ s . Remark 2.
Theorem 1 generalizes [2, Thm. 2] by provingthe same results using weaker assumptions. Particularly, toensure all input-output trajectories generated by system (1) are parameterizable by a finite number of them, [2, Thm. 2]assumes R “ R n . In comparison, using Theorem 1 one onlyneed to assume that R ` O ` K r x , x , . . . , x τ s “ R n toensure the same results. The first statement in Theorem 1 leads to a new result insystem identification, summarized by the following corollary.
Corollary 1.
Let t u i r ,T i ´ s , x i r ,T i s u τi “ be input-state tra-jectories generated by system (1a) , and input sequences t u i r ,T i ´ s u τi “ are collectively persistently exciting of order δ ` with δ ě δ min , where δ min is the degree of the minimalpolynomial of A . Define “ ˆ A ˆ B ‰ : “ XS : , where, X “ ” H p x r ,T s q ¨ ¨ ¨ H p x τ r ,T τ s q ı , (8a) The minimal polynomial of a matrix A is the unique monic polynomialof minimum degree that annihilates matrix A [20, Def. 3.3.2]. “ « H p x r ,T ´ s q ¨ ¨ ¨ H p x τ r ,T τ ´ s q H p u r ,T ´ s q ¨ ¨ ¨ H p u τ r ,T τ ´ s q ff . (8b) Then B “ ˆ B and Az “ ˆ Az for any z P R ` K r x , x , . . . , x τ s .Proof. See the Appendix.
Remark 3. If δ “ n and system (1) is controllable, thenCorollary 1 reduces to Theorem 1 in [10]. Another implication of Theorem 1 is that the order ofpersistent excitation required by trajectory parameterizationonly depends on the degree of the minimal polynomial ofmatrix A in (1), instead of its dimension. In general, it isdifficult to establish a bound of the degree of the minimalpolynomial of a matrix tighter than its dimension. However,the following corollary shows an exception example; itsusefulness will be illustrated later in Section III-B. Corollary 2.
If there exists s A P R nN ˆ nN such that A “ I N b s A , then Theorem 1 holds with δ “ nN .Proof. The results directly follow from Theorem 1 and thefact the minimal polynomial of A is the same as the one of s A , which has degree at most nN .III. A PPLICATIONS
In this section, we provide two examples that illustratedistinct implications of Theorem 1.
A. Online data-driven predictive control
Model predictive control (MPC) provides an effectivestrategy for systems with physical and operational constraints[21], [22]. In particular, consider system (1a). At eachsampling time t , MPC solves the following optimization toobtain the input u t minimize s u r t,t ` L ´ s s x r t,t ` L ´ s ř t ` L ´ k “ t ` (cid:107) s x k ´ r k (cid:107) Q ` (cid:107) s u k (cid:107) R ˘ subject to s x k ` “ A s x k ` B s u k , s x t “ ˆ x t , s x k P X , s u k P U , k “ t, . . . , t ` L ´ , (9)where ˆ x t P R n is the current state and L P N ` is the planninghorizon. Closed convex sets X Ă R n and U Ă R m describefeasible states and inputs, respectively. Symmetric positivesemi-definite weighting matrices Q P R n ˆ n and R P R m ˆ m ,together with reference state trajectory r r t,t ` L ´ s , define thequadratic tracking cost function.Recently, [12], [13] proposed data-driven predictive con-trol (DPC) that replaces optimization (9) withminimize g, s u r t,t ` L ´ s s x r t,t ` L ´ s ř t ` L ´ k “ t ` (cid:107) s x k ´ r k (cid:107) Q ` (cid:107) s u k (cid:107) R ˘ subject to „s u r t,t ` L ´ s s x r t,t ` L ´ s “ „ H L p u r ,T ´ s q H L p x r ,T ´ s q g s x t “ ˆ x t , s x k P X , s u k P U , k “ t, . . . , t ` L ´ , (10)where p u r ,T ´ s , x r ,T ´ s q is an input-state trajectory ofsystem (1a) and generated offline . If u r ,T ´ s is persistently exciting of order n ` L and system (1a) is controllable,Willems’ fundamental lemma guarantees that optimization(10) is equivalent to the one in (9).However, testing the controllability of system (1a) usingits input-state data p u r ,T ´ s , x r ,T ´ s q is expensive: its com-putation time scales cubically with T [23], [18]. Further,if system (1a) is uncontrollable, then Willems’ fundamentallemma provides no guarantee on the equivalence betweenoptimization (10) and (9).On the other hand, Theorem 1 shows that the assumptionof p A, B q being controllable can be replaced by ˆ x t P R ` K r x s . In particular, if there exists an input sequence ˜ u , ˜ u , . . . , ˜ u k ´ P R m such that ˆ x t “ A k x ` ř k ´ j “ A k ´ j ´ B ˜ u j . (11)Using the Cayley-Hamilton theorem, one can verify that ˆ x t P R ` K r x s for any k P N ` . With this observation, we proposethe following online DPC algorithm:1) At time t “ , , . . . , T ´ , generate input-state data p u r ,T ´ s , x r ,T ´ s q such that u r ,T ´ s is persistentlyexciting of order n ` L ,2) At time t “ T, T ` , . . . , compute the input u t bysolving optimization (10) given the current state ˆ x t andinput-state data p u r ,T ´ s , x r ,T ´ s q .Notice that this online DPC algorithm ensures that ˆ x t “ A t x ` ř T ´ i “ A T ´ i ´ Bu i ` ř tj “ T A t ´ j ´ u j , (12)for all t ě T , i.e. , condition (11) is satisfied with inputsequence u , u , . . . , u T ´ , u T , u T ` , . . . , u t . From Theo-rem 1, we know optimization (10) solved in the aboveonline DPC algorithm is equivalent to (9), regardless of thecontrollability of system (1a).Consider the system in (1a), for example, with A “ »——– . . . fiffiffifl , B “ »——– . . fiffiffifl . (13)Let r x s i denote the i -th coordinate of the state. One canverify that r x s and r x s are uncontrollable modes, i.e. , theyevolve independently from the inputs. We choose L “ andgenerate online input-state trajectories p u r ,T ´ s , x r ,T ´ s q ,where u t is sampled uniformly from r´ . , . s for all t “ , . . . , T ´ , until u r ,T ´ s is persistently exciting of order .At time t “ T, T ` , . . . , given the current state ˆ x t , we obtainthe input u t by solving (10) where Q “ I , R “ . , X “ R and U “ t u P R | ´ . ď u ď . u . Fig. 1 shows that,although the system contains uncontrollable modes, onlineDPC still ensures that the controllable modes, e.g. , r x s , trackthe reference trajectory as desired. B. Identification of homogeneous multi-agent systems
Consider a network of N agents with the same LTIdynamics [24]. Further, agent i can measure the state of agent j in a local coordinate system if p i, j q is an edge of a directed ig. 1. Online DPC for p A, B q in (13). The blue and red curves denotedata and online DPC trajectories, respectively; the solid and dashed linesdenote input bounds and state reference values, respectively. graph G , which is composed of N nodes and M edges. Thedynamics of this multi-agent system is given by (1) with A “ I N b s A, B “ I N b s B, C “ E b I s n , D “ M s n ˆ N Ď m , (14)where s A P R s n ˆ s n and s B P R s n ˆ Ď m describe the dynamicsof an individual agent. Each row of matrix E P R M ˆ N isindexed by an directed edge, i.e. , an edge with an head anda tail, in graph G : the i -th entry in each row is “ ” if node i is the head of the corresponding edge, “ ´ ” if it is the tail,and “ ” otherwise. We assume that s B is a non-zero matrixand p s A, s B q is controllable.If at least one non-zero entry in matrix E is known, thensystem matrices in (14) can be computed using the following Markov parameters [25, Sec. 3.4.4] M k “ CA k ´ B ` D “p E b I s n qp I N b s A q k ´ p I N b s B q“ E b p s A k ´ s B q , @ k “ , , . . . , s n ` . (15)In particular, let p M k q ij denote the ij -th s n ˆ s m block of M k .If we know E ij “ (the case of “ ´ is similar), then (15)implies p M k q ij “ s A k ´ s B . For example, if E “ “ ´ ‰ ,then (15) says M k “ “ s A k ´ s B ´ s A k ´ s B ‰ . Hence given theMarkov parameters (15) and that E ij “ , we know E kl is“ ” if p M q kl “ p M q ij , “ ´ ” if p M q kl “ ´p M q ij , and“0” otherwise. Further, s B “ p M q ij and s A is the uniquesolution to the following linear equations s A “ p M q ij ¨ ¨ ¨ p M s n q ij ‰ “ “ p M q ij ¨ ¨ ¨ p M s n ` q ij ‰ . (16)Therefore, given at least one non-zero entry in matrix E ,in order to compute the system matrices in (14), it sufficesto know the Markov parameters (15). These parameters canbe computed using Corollary 2 via a data-driven simulationprocedure [3], [7]. We include the detailed steps of thisprocedure in the Appendix for completeness.In numerical simulations, we consider the homogeneousmulti-agent system used in [24, Example 3], discretized witha sampling time of 0.1s such that the system dynamics isgiven by (14) where s A “ „ . . ´ . ´ . . . ´ . ´ . . . . . . . . . , s B “ „ . . . ´ . ´ . . ´ . . . Since p s A, s B q is controllable, matrix “ p M q ij ¨ ¨ ¨ p M s n ´ q ij ‰ “ “ s B ¨ ¨ ¨ s A s n s B ‰ has full column rank and (16) has a unique solution. In addition, we let E “ “ N ´ ´ I N ´ ‰ , where N ´ P R N ´ is the vector of all ’s.We compare Corollary 2 against the results in [2, Thm.2] in terms of the least amount of input-output data neededto compute matrices in (14). In particular, since matrix s A has spectrum radius . , we use use input-output trajec-tories t u i r ,T ´ s , y i r ,T ´ s u τi “ with relatively short length T “ to avoid numerical instability [2]. The entriesin t u i r ,T ´ s u τi “ are sampled uniformly from r´ . , . s .Using Corollary 2, the data-driven simulation procedurerequires t u i r ,T ´ s u τi “ to be collectively persistently excitingof order p N ` q s n ` . In other words, matrix (5) with d “ p N ` q s n ` has full row rank, hence it must haveat least as many columns as rows, i.e. , τ ě pp N ` q s n ` q N Ď mT ´p N ` q s n “ N ` N ´ N . In comparison, if we use [2, Thm. 2] instead of Corollary 2,we need t u i r ,T ´ s u τi “ to be collectively persistently excitingof order N n ` (see [2, Sec. IV-A]). In other words, matrix(5) with d “ N n ` has full row rank, which implies τ ě p N s n ` q N Ď mT ´ N s n “ N ` N ´ N . In Fig. 2, we show the minimum number of input-outputtrajectories required to compute matrices in (14) in numericalsimulations. The results tightly match the aforementionedtwo lower bounds, and the number of trajectories requiredby Corollary 2 is one order of magnitude less than that of[2, Thm. 1] when N “ . Fig. 2. Minimum number of length T “ input-output trajectoriesrequired to identify system (14). IV. C
ONCLUSIONS
We introduced an extension of Willems’ fundamentallemma by relaxing the controllability and persistency ofexcitation assumptions. We demonstrate the usefulness ofour results in the context of DPC and identification ofhomogeneous multi-agent system. Future directions includegeneralizations to noisy data and nonlinear systems.A
PPENDIX
Proof of Theorem 1:
We start by proving the firststatement using a double inclusion argument. It is trivial toshow that the left hand side of (6) is included in its rightand side. To show the other direction, we show that the leftkernel of matrix « H p x r ,T ´ L s q ¨ ¨ ¨ H p x τ r ,T τ ´ L s q H L p u r ,T ´ s q ¨ ¨ ¨ H L p u τ r ,T τ ´ L s q ff (17)is orthogonal to p R ` K r x , x , . . . , x τ sqˆ R mL , To this end,let v J “ “ ξ J η J η J ¨ ¨ ¨ η J L ‰ (18)be an arbitrary row vector in the left kernel of matrix (17),where ξ P R n , η , η , . . . , η L P R m . Since δ ě δ min , using[20, Def. 3.3.2] we know there exists α k , α k , . . . , α δ ´ ,k P R such that A k ` ř δ ´ j “ α jk A j “ n ˆ n , @ k “ δ, δ ` , . . . (19)The above equation implies that A k B “ ´ ř δ ´ j “ α jk A j B and A k x i “ ´ ř δ ´ j “ α jk A j x i for all k “ δ, . . . , n ´ and i “ , . . . , τ. Therefore in order to show the left kernel ofmatrix (17) is orthogonal to p R ` K r x , x , . . . , x τ sqˆ R mL ,it suffices to show the following η J “ η J “ ¨ ¨ ¨ “ η J L “ J m , (20a) ξ J B “ ξ J AB “ ¨ ¨ ¨ “ ξ J A δ ´ B “ J m , (20b) ξ J x i “ ξ J Ax i “ ¨ ¨ ¨ “ ξ J A δ ´ x i “ . @ i “ , . . . , τ. (20c)In order to prove (20a), we let w , w , . . . , w δ P R n ` m p δ ` L q be such that w “ “ v J J mδ ‰ J and w j equals ” ξ J A j ξ J A j ´ B ¨ ¨ ¨ ξ J B η J ¨ ¨ ¨ η J L J m p δ ´ j q ı J , for j “ , . . . , δ . Since v J is in the left kernel of matrix(17), using (1) one can verify that w J , w J , . . . , w J δ are inthe left kernel of « H p x r ,T ´ δ ´ L s q ¨ ¨ ¨ H p x τ r ,T τ ´ δ ´ L s q H δ ` L p u r ,T ´ s q ¨ ¨ ¨ H δ ` L p u τ r ,T τ ´ s q ff . (21)Let α δδ “ and k “ δ in (19), we have n ˆ n “ ř δj “ α jδ A j . Hence ř δj “ α jδ w J j “ ”ř δj “ α jδ ξ J A j r J ı “ “ n r J ‰ , (22)for some vector r P R m p δ ` L q . Since row vectors w J , w J , . . . , w J δ are in the left kernel of matrix (21), equa-tion (22) implies that r J is in the left kernel of matrix ” H δ ` L p u r ,T i ´ s q ¨ ¨ ¨ H δ ` L p u τ r ,T i ´ s q ı . (23)Since t u i r ,T i ´ s u τi “ is collectively persistently exciting oforder δ ` L , matrix (23) has full row rank. Therefore r “ m p δ ` L q . (24)Observe that the last m entries of r are given by α δδ η L “ η L ,hence equation (24) implies that η L “ m . Then the last m entries of r are given by “ α δδ η J L ´ ` α p δ ´ q δ η J L α δδ η J L ‰ J .Since η L “ m and α δδ “ , equation (24) also implies that η L ´ “ m . By repeating similar induction we can provethat (20a) holds. Next, since (20a) holds, the first mδ entries in r are „ δ ř j “ α jδ ξ J A j ´ B δ ř j “ α jδ ξ J A j ´ B ¨ ¨ ¨ α δδ ξ J B J . By combining this with (24) we get that J m “ ř δj “ k α jδ ξ J A j ´ k B, @ k “ , . . . , δ. (25)Since α δδ “ , considering k “ δ in (25) implies that ξ J B “ m . Substitute this back into (25) and considering k “ δ ´ implies that ξ J AB “ m . By repeating a similar inductionwe can prove that (20b) holds.Further, by using (19) and (20b) we can show that ξ J A k B “ m for all k ě . Combining this together withthe fact that row vector (18) is in the left kernel of matrix(17) and that (20a) also holds, it follows that, “ ξ J x ik “ ξ J ` A k x i ` ř k ´ j “ A k ´ j ´ Bu ij ˘ “ ξ J A k x i , @ k “ , . . . , T i ´ L, i “ , . . . , τ. Since T i ě δ ` L for i “ , . . . , τ by assumption, weconclude that (20c) holds.We now prove the second statement. Given the input, state,and output trajectories in (2), suppose (4) holds. Let s x “ ” H p x r ,T ´ L s q ¨ ¨ ¨ H p x τ r ,T τ ´ L s q ı g, s x t ` “ A s x t ` B s u t , ď t ď L ´ . (26)Then from (6) we know s x satisfy (7), and one can verify that p s u r ,L ´ s , s x r ,L ´ s , s y r ,L ´ s q is indeed an input-state-outputtrajectory of system (1).Conversely, let p s u r ,L ´ s , s x r ,L ´ s , s y r ,L ´ s q be an input-state-output trajectory of system (1) with s x P R ` O ` K r x , x , . . . , x τ s . Then there exists s x a P R ` K r x , x , . . . , x τ s , s x b P O , (27)such that s x “ s x a ` s x b and „s u r ,L ´ s s y r ,L ´ s “ „ IO L T L „s x a ` s x b s u r ,L ´ s (28)where T L “ »—————– D ¨ ¨ ¨ CB D ¨ ¨ ¨ CAB CB D ¨ ¨ ¨ ... ... ... . . . ... CA L ´ B CA L ´ B CA L ´ B ¨ ¨ ¨ D fiffiffiffiffiffifl ,O L “ “ C J p CA q J p CA q J ¨ ¨ ¨ p CA L ´ q J ‰ J . (29)Further, using the Cayley-Hamilton theorem one can showthat O Ă ker O L for any L P N ` . Hence (27) implies O L x b “ Lp . (30)Since s x a P R ` K r x , x , . . . , x τ s , the first statement impliesthat there exists g P R ř τi “ p T i ´ L ` q such that „ s x a s u r ,L ´ s “ « H p x r ,T ´ L s q ¨ ¨ ¨ H p x τ r ,T τ ´ L s q H L p u r ,T ´ s q ¨ ¨ ¨ H L p u τ r ,T τ ´ s q ff g. (31)otice that „ IO L T L « H p x i r ,T i ´ L s q H L p u i r ,T i ´ s q ff “ « H L p u i r ,T i ´ s q H L p y i r ,T i ´ s q ff . (32)for i “ , . . . , τ . Substituting (30), (31) and (32) into (28)gives (4), thus completing the proof. Proof of Corollary 1 :
The first statement of Theorem 1implies that, for any z P R ` K r x , x , . . . , x τ s and v P R m ,there exists g P R ř τi “ T i such that „ zv “ Sg . Using thedefinition of matrix S and X we can show “ A B ‰ „ zv “ “ A B ‰ Sg “ Xg.
Further, notice that “ ˆ A ˆ B ‰ „ zv “ XS : Sg “ Xg “ “ A B ‰ „ zv . This completes the proof by considering the following twocases where: 1) z “ n and v ranges over R m , and 2) v “ m and z ranges over R ` K r x , x , . . . , x τ s . Markov parameters computation in Section III-B:
Let t u i r ,T i ´ s , y i r ,T i ´ s u τi “ be input-output trajectories of thesystem described by (14), such that inputs t u i r ,T i ´ s u τi “ arecollectively persistently exciting of order p N ` q s n ` . Let n “ N s n, m “ N s m, p “ M s n . Since p s A, s B q in (14) is con-trollable, one can verify that R ` K r x , x , . . . , x τ s “ R n ,regardless of the values of x , x , . . . , x τ . Using Corollary 2we know that there exists a matrix G k P R p ř τi “ p T i ´ n qqˆ m such that, « H n ` p u r ,T ´ s q ¨ ¨ ¨ H n ` p u τ r ,T τ ´ s q H n ` p y r ,T ´ s q ¨ ¨ ¨ H n ` p y τ r ,T τ ´ s q ff G k “ ” J m p n ´ k qˆ m I m Jp pn ` km ´ kp qˆ m M J ¨ ¨ ¨ M J k ı J (33)for all k “ , , . . . , s n ` , where M “ p ˆ m and M k is given by (15); also see [7, Sec. 4.5] for details. Next,given M , . . . , M k ´ , we can compute M k by first solvingthe first m p n ` q ` pn equations in (33) for matrix G k , thensubstituting the solution into the last p equations in (33). Byrepeating this process for k “ , . . . , s n ` we obtain Markovparameters (15). Using Kalman decomposition one can verifythat Markov parameters of system (14) are the same as thoseof a reduced order controllable and observable system withstate dimension less than n . Hence matrix M k obtained thisway is unique; see [7, Prop. 1].R EFERENCES[1] J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. De Moor, “Anote on persistency of excitation,”
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