We show how to compute a controller directly from data for a class of linear-time invariant systems. To do this, we use the interconnection paradigm where the control variables and to-be-controlled variables coincide, i.e. full interconnection. We also illustrate this process with an example

Maupong, Thabiso

Rapisarda, Paolo

Southampton (e-Prints Soton)

Data-Driven Control: The Full Interconnection CaseThabiso Maupong and Paolo RapisardaAbstract—We show how to compute a controller directlyfrom data for a class of linear-time invariant systems. To dothis, we use the interconnection paradigm where the controlvariables and to-be-controlled variables coincide, i.e. full inter-connection. We also illustrate this process with an example.Index Terms—Data-driven control, behaviours, interconnec-tion, annihilators.AMS Subject—93C05, 93C55.I. INTRODUCTIONConsider given: observed noise free trajectories from a li-near time-invariant unknown system; an “example trajectory”of the desired controlled system. We show how to find arepresentation of a controller that implements the desiredcontrolled system. We call this approach data-driven control,as in [3] .Data-driven control has been studied from different pointsof view and different names, for example data-based control[6]; model-free control [1] and unfalsified control [5]. In [1],[3], [6] system data is used to find control inputs, whereasin [5] input/output data is used to falsify a control lawfrom a set of available admissible control laws. We haveshown in [4], that under suitable conditions, we can finda controller that implements a desired controlled systemdirectly from data. We use the interconnection paradigm, see[9], to find a controller directly from data for both the generalinterconnection, i.e., when the system variables are split intocontrol variables and to-be-controlled variable and the fullinterconnection case. In this paper, we further study thefull interconnection case. We prove necessary and sufficientconditions suitable for finding a controller from data usingfull interconnection. Then, under such conditions, we presentan algorithm. We also present an example to illustrate thisprocedure.The structure of this paper is as follows. In SectionII we introduce some background material including thenotation used and some relevant concepts of the behavioralapproach. In Section III we cover some aspects of controlas interconnection, focusing mainly on full interconnection.Then, in Section IV we formally state the full interconnectiondata-driven control problem and present our solution to theproblem. In Section we V provide an example. Finally, inSection VI we give some conclusions.Both Thabiso Maupong and Paolo Rapisarda are with the VLC group,School of Electronics and Computer Science, University of Southampton,Great Britain {tmm204,pr3}@ecs.soton.ac.ukII. BACKGROUNDA. NotationWe denote the space of w dimensional real vectors byRw and that of g × w real matrices by Rg×w. colspan(A)denotes the subspace consisting of all linear combination ofthe columns of A and leftkernel(A) denotes the subspacespanned by all vectors v such that vA = 0. col(A,B) isthe matrix obtained by stacking the matrix A ∈ Rg1×w overB ∈ Rg2×w. The ring of polynomials with real coefficientsin the indeterminate ξ is denoted by R[ξ] and the set of g×wmatrices in the indeterminate ξ is denoted by Rg×w[ξ]. LetR := R0+ · · ·+RLξL ∈ Rg×w[ξ] with RL 6= 0 then L is thedegree of R, denoted by deg(R). The set of all maps fromZ to Rw is denoted by (Rw)Z. The collection of all linear,closed, shift-invariant subspaces of (Rw)Z equipped with thetopology of pointwise convergence is denoted by L w. Thebackward shift σ is defined by (σf)(t) := f(t+ 1).B. Linear difference behaviorsWe define a dynamical system as Σ = (T,W,B), with Tthe time axis, W the signal space and B ⊆WT the behavior.Let T = Z and W = Rw. We consider a class of systemswhose behavior is a subspace of L w, i.e., B is linear, shiftinvariant and closed. It has been proven in Prop. 4.1A pp.232-233 of [8] that if B ∈ L w then there exists R ∈ Rg×w[ξ]such thatB := {w : Z→ Rw|R(σ)w = 0},where the operator R(σ) is called polynomial operator inthe shift and R(σ)w = 0 is called kernel representation.Henceforth we write B = ker(R(σ)). R induces a minimalrepresentation if no other kernel representation of B has lessthan g rows. It has been proven in [9] Prop. 1, p. 331 that ifR,R′ ∈ R•×w[ξ] are both minimal, then B = ker(R(σ)) =ker(R′(σ)) iff there exists a unimodular matrix (see Lemma6.3-1, p. 375 of [2]) U ∈ R•×•[ξ] such that R = UR′.B ∈ L w is controllable if for any two trajectoriesw1, w2 ∈ B there exists t1 ≥ 0 and w ∈ B such thatw(t) = w1(t) for t ≤ 0 and w(t) = w2(t−t1) for t ≥ t1. Wedenote by L wcontr the collection of all controllable elementsof L w.Let L ∈ N. The restriction of trajectories of B on theinterval [1, L] is defined byB|[1,L] :={w : [1, L]→ Rw|∃w′ ∈ B s.t.w(t) = w′(t) for all 1 6 t 6 L}.The integer L in the above equation is called the lag.We denote by L(B) the smallest L such that [w|[t,t+L] ∈B|[t,t+L] for all t ∈ T] ⇒ [w ∈ B]. Equivalently, L(B)is the smallest degree over all R such that B = ker(R(σ)).We also use the following integer invariants: n(B), McMillandegree, the smallest state-space dimension among all possi-ble state representations of B; and l(B), the shortest lagdescribed as follows. Let B = ker(R(σ)) and define thedegree of each row of R to be the largest degree of theentries. Then the minimum of degrees of the rows of R isthe minimal lag associated with R and l(B) is the smallestpossible minimal lag over all R such that B = ker(R(σ)).Hence, all kernel representations of B has rows of lag atleast l(B).Let w ∈ B. A partition of w := (w1, w2) is aninput/output partition if w1 is maximally free i.e., piw1(B) =(Rm)Z, where piw1(B) := {w1| ∃ w2 s.t (w1, w2) ∈ B}, andw2 contains no free components (see pp. 243-244 of [8]).Then w1 is an input and w2 an output. We denote by p(B)the output cardinality, i.e. number of outputs and m(B) inputcardinality, the number of inputs.C. Annihilators and fundamental lemmaThe module of annihilators of B is defined by NB :={n ∈ R1×w[ξ]|n(σ)B = 0}. Let B = ker(R(σ)), then NBis equal to the R[ξ]-submodule of R1×w[ξ] generated by therows of R, see [10] pp. 83-84. The set of annihilators ofB of degree less than j ∈ Z+ is defined by NjB := {r ∈R1×w[ξ]|r ∈ NB and r has degree 6 j}. Let r1, . . . ri ∈NjB and r˜1 . . . r˜i be the coefficients of r1, . . . ri. Then N˜jBis the set containing r˜1 . . . r˜i.Definition 1: Let L ∈ N. The Hankel matrix associatedwith a vector w(1), . . . w(T ) for T > L is defined byHL(w) :=w(1) w(2) . . . w(T − L+ 1)w(2) w(3) . . . w(T − L+ 2)...... . . ....w(L) w(L+ 1) . . . w(T ) .Definition 2: A vector u˜ = u˜(1), u˜(2), . . . , u˜(T ) is per-sistently exciting of order L if HL(u˜) is full row rank.Now we state the fundamental lemma cf. [11].Lemma 1: Assume B ∈ L wcontr. Let w˜ =w˜(1), w˜(2), . . . , w˜(T ) := col(u˜, y˜), w˜ ∈ B|[1,T ] suchthat u˜(k) ∈ Rm(B) and y˜(k) ∈ Rp(B) for 1 6 k 6 T .Finally, let L ∈ N such that L > L(B). If u˜ is persistentlyexciting of order L+ n(B), then colspan(HL(w˜)) = B|[1,L]and leftkernel(HL(w˜)) = N˜LB.Proof: See Th. 1 pp. 327-328 of [11].Under the conditions of Lemma 1, for all w˜′ ∈ B|[1,L]there exists υ˜ ∈ RT−L+1 such that w˜′ = HL(w˜)υ˜. Moreover,we can recover from w˜ the laws of the system B thatgenerated w˜. This leads us to following definition.Definition 3: Let L ∈ N such that L > L(B) andT ∈ N such that T  L. Then w˜ ∈ B|[1,T ] is sufficientlyinformative about B if colspan(HL(w˜)) = B|[1,L] andleftkernel(HL(w˜)) = N˜LB.III. FULL INTERCONNECTIONIn the following discussion we assume that the plantbehavior, the controller behavior and the controlled behaviorare all elements of L wcontr. Let the to-be-controlled plantbehavior be described byP := {w : Z→ Rw|w satisfies the plant equations}and a to-be-designed controller defined by the control beha-viorC := {w : Z→ Rw|w satisfies the controller equations}.The interconnection of the plant and the controller throughthe w denoted by P ∧w C results in the w’s obeying both thelaws of the plant and the controller. Therefore the controlledbehavior is defined byD := {w : Z→ Rw |w ∈ P and w ∈ C} = P ∩ C.It has been shown in [12], Theo.1 p. 62 that a controllerC implementing D exists iff D ⊆ P .Let NP ,NC and ND be the module of annihilators asso-ciated with P, C and D, respectively. To prove necessaryand sufficient conditions for C to implement D via fullinterconnection consider the follow lemmas.Lemma 2: Let RC ∈ Rc×w[ξ], RP ∈ Rp×w[ξ] and RD ∈Rg×w[ξ] where c, p, g ∈ N, be such that C = ker(RC(σ)),P = ker(RP (σ)) and D = ker(RD(σ)). Then the followingstatements are equivalent,1) NP +NC = ND2) there exist F1 ∈ Rg×c[ξ] and F2 ∈ Rg×p[ξ] such thatF1RC + F2RP = RD.Lemma 3: Let r1, . . . , rp and c1, . . . , cc be bases gene-rators of NP and NC , respectively. Assume that NP +NC = ND, then {r1, . . . , rg, c1, . . . , cc} is a basis of NDiff NP ∩NC = {0}.In the following theorem we prove necessary and sufficientconditions for C to implement D via full interconnection.Theorem 1: Let C = ker(RC(σ)), P = ker(RP (σ)) andD = ker(RD(σ)). Assume that RD, RP and RC induceminimal representations, and that NP ∩ NC = {0}. Thena controller C implements D iff NP +NC = ND.Proof: (Only if) Assume that C implements D, thenC ∩ P = ker([RC(σ)RP (σ)]). Since D = ker(RD(σ)) thenker(RD(σ)) = ker([RC(σ)RP (σ)]). Therefore there exists F ∈R•×•[ξ] such that RD = F[RCRP]. Partition the columnsof F := [F1 F2] accordingly with respect to the rows ofRC and RP . Then RD = F1RC + F2RP . Now it followsfrom Lemma 2 that RD = F1RC + F2RP which impliesthat NP +NC = ND. (If) From Lemma 2 NP +NC = NDimplies that there exist F1 ∈ Rg×c[ξ] and F2 ∈ Rg×p[ξ] suchthat F1RC + F2RP = RD. Now, RD =[F1 F2] [RCRP]with[F1 F2] ∈ Rg×(c+p)[ξ]. By the assumption thatRD, RP and RC are minimal, and that NP ∩NC = {0} thenc + p = g moreover, since ND = NP + NC then [F1 F2]is unimodular. Consequently, ker(RD(σ)) = ker([RC(σ)RP (σ)])which implies that D = P ∩ C.IV. DATA-DRIVEN FULL INTERCONNECTIONWe state the data-driven full interconnection problem andpresent our solution.Problem 1: Let T ∈ N be “sufficiently large”. Givensufficiently informative trajectories w˜ ∈ P and w˜d ∈ D,both of length T . Find a minimal representation of C suchthat P ∧w C = D.Let NP ,NC and ND be the module of annihilators ofP , C and D, respectively. Under assumption of Theo. 1, tofind C we first find bases generators of NP and ND. Then,under the assumptions of Lemma 3, we compute a basisgenerator for NC . To find bases of NP and ND we usethe fact that w˜ and w˜d are sufficiently informative, thereforeleftkernel(HL(w˜)) = N˜LP and leftkernel(HL(w˜d)) = N˜LDwhere L ∈ Z+ is greater than both L(P) and L(D). Aprocedure for finding minimum lag bases generators has beenillustrated in Algorithm 2, p. 679 of [7].Our solution to Problem 1 is summarised in Algorithm1. Note that in Algorithm 1 we denote by NnC a set ofannihilators of C of degree n.Remark 1: In Problem 1, we assumed that observed tra-jectories are of sufficiently large length T , whereas Algo-rithm 2, p. 679 of [7] is under the assumptions that observedtrajectories are of infinite length. This brings about the issuesof how large is sufficiently large and of consistency. We donot address these issues in this paper, but reserve them forfuture research, along with the effect of noise on observeddata.We now prove the correctness of Algorithm 1.Proposition 1: Let w˜ ∈ P and w˜d ∈ D be sufficientlyinformative about their respective behaviors. Assume thatr1, . . . , rg and a1, . . . , at in Algorithm 1 are minimum lagbases of NP and ND, respectively, and that C implementsD then NC in Algorithm 1 is a module of annihilators of Cthat implements D.Proof: The fact that r1, . . . , rg and a1, . . . , at inAlgorithm 1 are minimum lag bases of NP and ND followsfrom Theo. 14, p. 679 of [7]. Furthermore, the fact that Cimplements D follows from Theo. 1. Now, let al′1 , . . . , al′qand rl1 , . . . , rlk as in step 1 of Algorithm 1. Denote byNnD and NnP a set containing al′1 , . . . , al′q and rl1 , . . . , rlk ,respectively and NnC set of annihilators of C of degree n.Since a1, . . . , at is a basis of ND then NC ∩ NP = {0}which implies that NnC ∩NnP = {0} therefore in Algorithm1 if k = q then al′1 , . . . , al′q ∈ NnP . Hence, NnC = {0}.Now, if k = 0 and q 6= 0, then al′1 , . . . , al′q ∈ NnD suchthat al′1 , . . . , al′q /∈ NnP implies that al′1 , . . . , al′q ∈ NnC ,therefore NnC = {al′1 , . . . , al′q}. Finally, k < q means NnD hasmore annihilators of degree n than NnP , therefore some ofthem belong to NnC . Denote by N˜nP and N˜nD sets containinga˜l′1 , . . . , a˜l′q and r˜l1 , . . . , r˜lk , respectively. Since NnC ∩NnP ={0} then N˜nC ∩ N˜nP = {0}. Moreover, NP + NC = NDimplies that NnP +NnC = NnD hence N˜nP + N˜nC = N˜nD. SinceN˜nC ∩N˜nP = {0}, N˜nP+N˜nC = N˜nD, and r˜l1 , . . . , r˜lk is a basisof N˜nP then the projection matrix P exists and u˜>1 , . . . u˜>x arethe coefficients vectors of annihilators of C of lag n. HenceInput : w˜ ∈ P and w˜d ∈ DOutput : NCAssumptions: Theorem 1 and Lemma 31 Determinations of bases of NP and NDi. Using Algorithm 2, p. 679 of [7] determine minimumlag bases r1, . . . , rg and a1, . . . , at of NP and ND,respectively.ii. Define r˜1, . . . , r˜g and a˜1, . . . , a˜t as the coefficients ofr1, . . . , rg and a1, . . . , at respectively.iii. Define dm := deg(am) for m = 1, . . . , t,t := {1, 2, . . . , t} and g := {1, 2, . . . , g}. Letd = max(d1, . . . dm).2 Compute steps 3-4 recursively starting from n = 0 to d.3 Classifying r˜1, . . . , r˜g and a˜1, . . . , a˜t by their lagsi. choose l1, . . . lk ∈ g such that r˜l1 , . . . , r˜lk are all of lagn. If there is no r˜l1 , . . . , r˜lk of lag n set k = 0 .ii. choose l′1, . . . l′q ∈ t such that a˜l′1 , . . . , a˜l′q are all oflag n. If there is no a˜l′1 , . . . , a˜l′q of lag n set q = 0.4 Compute coefficients of NnC as followsif k = q thenNnC := {0}else if k = 0 and q 6= 0 thena˜l′1 , . . . , a˜l′q defines the coefficients ofannihilators of C of degree n, therefore defineNnC := {al′1 , . . . , al′q}.else if k < q then Define the matrix A whosecolumns are r˜l1 , . . . , r˜lk asA :=r˜0l1 . . . r˜0lk... . . . ...r˜nl1 . . . r˜nlk ;Define a projection matrix P := A[A>A]−1A>;Define H := [a˜l′1 − P a˜l′1 , . . . , a˜l′q − P a˜l′q ];Compute x rank of H and compute the SVD ofH = UΣV >;Partition U = [U1 U2] where U1 has x columns;The columns of U1, u˜>1 , . . . u˜>x defines thecoefficients of annihilators of C of degree n,therefore define NnC := {u1, . . . , ux};end ;5 Specification of NCi. Define NC :=d⋃k=0NkCAlgorithm 1: Solution of Problem 1NnC = {u1, . . . , ux}.V. EXAMPLEWe consider a simple example of power factor rectificati-on. Let the circuit in Fig. 1 be the to-be-controlled systemwith w = col(iz, iv, is, v), m(P) = 2 and p(P) = 2. Theinput/output variables are is, v and iz, is, respectively. Togenerate w˜ ∈ P with T = 200000 samples, the circuit inFig. 1 is simulated in Matlab Simulink with samplingrate of 5µs. To guarantee that w˜ is sufficiently informativeis and v are generated by current and voltage sources whichare driven by a random number generator so that both arepersistently exciting of sufficiently high order. The values ofR,L and C are 100Ω,0.01H and 0.001C, respectively.vivisizCLLCRFig. 1. To-be controlled systemThe controlled system, i.e. circuit with the correct powerfactor, is chosen as in Fig. 2. wd = col(iz, iv, is, v), withm(D) = 1 and p(D) = 3. To generate w˜d ∈ D the circuit inFig. 2 is simulated like the one above but this time with onlyv generated by voltage sources which is driven by a randomnumber generator. The values of R,L and C are the samewhile Lf = 0.001H and Rf = 0.4252ΩvivLfisRfizCLLCRFig. 2. Example of controlled systemUsing Algorithm 1, NP has 2 basis generators, one ofdegree 0 and the other 2. ND has 3 basis generators, one ofdegree 0, and others of degree 1 and 2. Hence, the generatorof ND of degree 1 belongs to NC . Consequently, a controllerrepresentation is[− 322 σ1611 + 2661155 322 σ1611 − 2261155 − 965 σ2414 + 4521155 − 151 σ10180− 15110180] izivisv=0We verify the controller above by interconnecting it withthe to-be-controlled systems then comparing the impulseresponse with that of the controlled system. The impulseresponses coincide as shown in Fig. 3.To: Out(1)-2000200To: Out(2)-2000020000 0.05 0.1 0.15 0.2 0.25 0.3 0.35To: Out(3)05001000Impulse ResponseTime (seconds)AmplitudeFig. 3. Impulse responses of the to-be-controlled system interconnectedwith computed controller (blue) and that of the controlled system(red)VI. CONCLUSIONSWe have proved necessary and sufficient data-driven con-ditions for a controller to implement the desired controlledbehavior via full interconnection. Then, under those con-ditions, we illustrated how the controller can be computeddirectly from data. As a matter of future research we intendto investigate whether such data-driven approach can beextended to two-dimensional systems and be applied toboundary condition control problems.REFERENCES[1] M. Ikeda, Y. Fujisaki, and N. Hayashi. A model-less algorithm fortracking control based on input-output data. Nonlinear Analysis:Theory, Methods & Applications, 47(3):1953–1960, 2001.[2] T. Kailath. Linear systems. Prentice-Hall, 1980.[3] I. Markovsky and P. Rapisarda. Data-driven simulation and control.International Journal of Control, 81(12):1946–1959, 2008.[4] T. M. Maupong and P. Rapisarda. Data-driven control: behavioralapproach (special issue JCW). Systems & Control Letters, Submitted.[5] M. G. Safonov and T.-C. Tsao. The unfalsified control concept andlearning. IEEE Transactions on Automatic Control, 42(6):843–847,1997.[6] G. Shi and R. E. Skelton. Markov Data-Based LQG Control. Journalof Dynamic Systems, Measurement, and Control, 122(3):551–559,2000.[7] J. C. Willems. From time series to linear system-Part II. Exactmodelling. Automatica, 22(6):675–694, 1986.[8] J. C. Willems. Models for dynamics. Dynamics reported, 2:171–269,1989.[9] J. C. Willems. On interconnections, control, and feedback. IEEETransactions on Automatic Control, 42(3):326–339, 1997.[10] J. C. Willems. The behavioral approach to open and interconnectedsystems. In IEEE Control Systems Magazine, volume 27, pages 46–99.2007.[11] J. C. Willems, P. Rapisarda, I. Markovsky, and B. L. M. De Moor.A note on persistency of excitation. Systems & Control Letters,54(4):325–329, 2005.[12] J. C. Willems and H. L. Trentelman. Synthesis of dissipative systemsusing quadratic differential forms: Part I. IEEE Transactions onAutomatic Control, 47(1):53–69, 2002.

Data-driven control: the full interconnection case

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