A Data-Driven Convex Programming Approach to Worst-Case Robust Tracking Controller Design
Liang Xu, Mustafa Sahin Turan, Baiwei Guo, Giancarlo Ferrari-Trecate
11 A Data-Driven Convex Programming Approach toWorst-Case Robust Tracking Controller Design
Liang Xu*, Mustafa Sahin Turan*, Baiwei Guo, Giancarlo Ferrari-Trecate
Abstract —This paper studies finite-horizon robust trackingcontrol for discrete-time linear systems, based on input-outputdata. We leverage behavioral theory to represent system trajec-tories through a set of noiseless historical data, instead of usingan explicit system model. By assuming that recent output dataavailable to the controller are affected by noise terms verifying aquadratic bound, we formulate an optimization problem with alinear cost and LMI constraints for solving the robust trackingproblem without any approximations. Our approach hinges ona parameterization of noise trajectories compatible with thedata-dependent system representation and on a reformulationof the tracking cost, which enables the application of the S-lemma. In addition, we propose a method for reducing thecomputational complexity and demonstrate that the size of theresulting LMIs does not scale with the number of historicaldata. Finally, we show that the proposed formulation can easilyincorporate actuator disturbances as well as constraints on inputsand outputs. The performance of the new controllers is discussedthrough simulations.
I. I
NTRODUCTION
Due to the recent advances in sensing, communication, andcomputation, data availability for control design is steadilyincreasing. This has motivated a renewed interest in systemanalysis and control design methods relying on finite-lengthdata sequences [1–4]. Several recent works propose to useraw measurements for representing discrete-time systems, andsolving system analysis and control design problems [1, 2, 5–16]. As mentioned in [1], the main feature of these approachesis to bypass explicit system identification that is usuallyrequired in standard control design. Moreover, data-basedsystem representations can be easier to update when newdata are available [17], hence facilitating the deployment ofadaptive control systems.All above works assume the availability of historical data ,i.e., finite-length trajectories produced by the open-loop systemand measured offline. The works [1, 2, 8–13] consider systemrepresentations based on input-state historical data. Data-basedparameterizations of linear state-feedback and linear quadraticregulators are developed in [1], under the assumption thathistorical input data are persistently exciting. This assumptionfurther implies that a state-space model of the system canbe perfectly reconstructed for control design. The persistenceof excitation requirement is relaxed in [2], where the authors
Authors are with the Institute of Mechanical Engineering (IGM),EPFL, Switzerland. Email: { liang.xu, mustafa.turan,baiwei.guo, giancarlo.ferraritrecate } @epfl.ch *The first two authors contributed equally to this work. This work hasbeen supported by the Swiss National Science Foundation under the COFLEXproject (grant number 200021 169906) and the National Centre of Compe-tence in Research (NCCR) in Dependable and Ubiquitous Automation. provide necessary and sufficient conditions about the infor-mativity of historical data for testing system properties andbuilding stabilizing control laws. The informativity frameworkis further extended to sub-optimal control design in [8], as wellas tracking and regulation problems in [9]. The presence ofnoise in historical data, which prevents from unambiguouslyidentifying the system dynamics, is considered in [10–13].[13] extends the optimal control method in [1] to accountfor noisy data and derives sufficient conditions to ensure thatthe proposed method returns a stabilizing controller. In [10–12], data are first used for representing all systems that arecompatible with available prior knowledge on the noise andthen for developing different kinds of state-feedback regula-tors, including robustly stabilizing, H , and H ∞ controllers.In certain applications, the system state is not accessibleand only input-output data can be collected. In this scenario,Willems’ Fundamental Lemma states that the whole set ofinput-output trajectories generated by a discrete-time linearsystem can be represented by finitely many historical datacoming from sufficiently excited dynamics [18]. In view ofthis result, [19] proposes to predict the system output from agiven time t onwards by using a set of collected historical dataand a finite amount of recent past data , i.e., an input-outputtrajectory measured right before time t . This approach is alsoused in the data-enabled predictive control (DeePC) schemedescribed in [14]. While originally developed for noiselessdata, DeePC has been recently extended to noisy trajectoriesin [15, 16]. In [15], slack variables are introduced in thedata-dependent system representation to account for noisymeasurements. The modified control scheme is shown to berecursively feasible and practically exponentially stable; how-ever, the tracking performance is not analyzed. The authorsin [16] propose a distributionally robust variant of DeePCbased on semi-infinite optimization. They then formulate afinite and convex program, whose optimal value is an upperbound to that of the original optimization problem. Thework [20] considers using noiseless historical data and noisyrecent output data to minimize the energy of the control inputwhile robustly satisfying input/output constraints. The authorspropose to separate the problems of estimation of the initialcondition and control design, and show that the solution tothe formulated problem is computed by consecutively solvingtwo optimization problems.In safety-critical applications, such as power networks andindustrial control systems, it is sometimes required to adopta bounded-error perspective by enforcing robustness againstall possible noise realizations and providing worst-case per-formance guarantees. This is the setting considered in the a r X i v : . [ m a t h . O C ] F e b present paper and, for this purpose, we utilize the data-drivenprediction method in [19]. We assume the historical dataare noiseless while recent data are corrupted by noise termssatisfying a quadratic constraint similar to the one in [12]. Thisassumption corresponds to scenarios where one can utilizevery accurate (and, thus, expensive) sensors to collect offlinehistorical data, but only has relatively inaccurate and noisysensors to be used during online operations. Our goal isto provide a control design method for worst-case optimalreference tracking with explicit performance guarantees.We first characterize noises that are consistent with theinput-output data, and then reformulate the tracking cost. Thisenables us to apply the S-lemma [21] to transform the worst-case robust control problem to an equivalent minimizationproblem with a linear cost and LMI constraints. Moreover, wepropose a method for reducing the size of LMI constraints, andalso show that our formulation can easily incorporate input-output constraints as well as actuator disturbances. In contrastto [20], we consider to minimize quadratic cost on both inputsand outputs, while the method in [20] only deals with theminimization of the input energy. The main features of ourmethod are the following: (1) we consider the minimization ofthe worst-case tracking performance; (2) the proposed designprocedure is non-conservative, meaning that we obtain theoptimal tracking controllers without any approximations; (3)the complexity of the controller design procedure does notincrease with the number of historical data. To the authors’knowledge, this is the first time that the worst-case robustoptimal tracking control is considered and exactly solved in adata-driven fashion.A preliminary version of this work has been submitted to aconference [22]. With respect to it, this paper provides com-plete proofs for all intermediate results, contains new resultson reducing the LMI constraint size, illustrates how to considerinput-output constraints as well as actuator disturbances, andadds new numerical experiments. This paper is organized asfollows. In Section II, we provide preliminaries on data-drivenprediction. The problem formulation is given in Section III.The data-based robust optimal tracking control problem issolved in Section IV. Extensions for considering input-outputconstraints as well as actuator disturbances are discussed inSection V. Simulations are provided in Section VI. Concludingremarks are given in Section VII. Notation : col( { x k } jk = i ) denotes the column concatenationof the vectors x k in the sequence { x k } jk = i . For a square matrix Φ , Φ > ≥ represents that it is positive definite(semidefinite). For Q ≥ , (cid:107) x (cid:107) Q denotes (cid:112) x (cid:62) Qx . dim( V ) denotes the dimension of the vector space V . For a matrix A ∈ R n × m , ker( A ) and range( A ) denote its null space and columnspace, respectively. Moreover, N ( A ) ∈ R m × dim(ker( A )) and R ( A ) ∈ R n × rank( A ) denote matrices whose columns forma basis for ker( A ) , range( A ) , respectively. I and denoteidentity and zero matrices of suitable size. When used withsubspaces, the operator + denotes the subspace sum. Theoperator ⊗ denotes the Kronecker product. II. P RELIMINARIES ON D ATA -D RIVEN P REDICTION
We consider a controllable and observable discrete-time LTIsystem G with state-space representation x k +1 = Ax k + Bu k ,y k = Cx k + Du k , (1)where x k ∈ R n , u k ∈ R m , y k ∈ R p are the system state,input and output, respectively. In this paper, we assume thatsystem matrices ( A, B, C, D ) are unknown, the states x k arenot measurable, and only a finite set of input-output samplesof G is available. In this section, we recall how to form adata-based representation of G that allows for predicting theoutput given any input [18, 19].We start by introducing the following definitions. The lag l ( G ) is the smallest integer l such that the l -step observabilitymatrix [ C (cid:62) , A (cid:62) C (cid:62) , . . . , ( A l − ) (cid:62) C (cid:62) ] (cid:62) has full column rank.Moreover, l ( G ) ≤ n since G is observable. A sequence { u k , y k } l + T − k = l is a trajectory of G if and only if thereexists a state sequence { x k } l + Tk = l such that (1) holds for k = l, . . . , l + T − . For a sequence { v k } jk = i of vectors,we use v to denote col( { v k } jk = i ) when the start and end times i, j are clear from the context. The Hankel matrix of depth L corresponding to a sequence { v k } l + T − k = l is defined as H L ( v ) := v l v l +1 · · · v l + T − L v l +1 v l +2 · · · v l + T − L +1 ... ... . . . ... v l + L − v l + L · · · v l + T − . The sequence { v k } l + T − k = l is persistently exciting of order L ifthe Hankel matrix H L ( v ) is of full row rank.In the following, we introduce the input-output data-basedrepresentation of linear systems in [18] and the predictionmethod in [19]. Suppose a trajectory { ¯ u k , ¯ y k } t h + T d − k = t h of G is collected, where t h (cid:28) . The trajectory is called historical ,since it can be regarded as collected long before the start(indicated by time ) of any control or prediction tasks. TheFundamental Lemma proposed by Willems et al. [18] showshow to use the historical trajectory to characterize all possiblesystem trajectories of length T f . Lemma 1 (Fundamental Lemma [18]) . Suppose that { ¯ u k , ¯ y k } t h + T d − k = t h is a trajectory of G and that the input ¯ u is persistently exciting of order T f + n . Then, { u k , y k } T f − k =0 is a trajectory of G if and only if there exists g ∈ R T d − T f +1 such that (cid:20) H T f (¯ u ) H T f (¯ y ) (cid:21) g = (cid:20) uy (cid:21) . (2)For a given time t ≥ , consider the problem of us-ing (2) for computing predictions y = col( { y k } T f − k =0 ) of thesystem output over a future horizon given the inputs u =col( { u k } T f − k =0 ) . There are infinitely many output trajectories y that satisfy (2), corresponding to different initial states x .The authors of [19] propose to implicitly fix the initial state byusing recent input-output samples u ini = col( { u k } − k = − T ini ) , We call { u ini , y ini } recent data . Fig. 1. Chronological order of data used in data-driven prediction y ini = col( { y k } − k = − T ini ) , which are available at time (seeFig. 1). More precisely, let U = (cid:20) U p U f (cid:21) (cid:44) H T ini + T f (¯ u ) , Y = (cid:20) Y p Y f (cid:21) (cid:44) H T ini + T f (¯ y ) , where U p and Y p consist of the first T ini block rows of U and Y , respectively; while U f and Y f consist of the last T f block rows of U and Y , respectively. The following lemmasummarizes the prediction algorithm. Lemma 2 ([19]) . Suppose that ¯ u is persistently exciting oforder T ini + T f + n , and T ini ≥ l ( G ) . Then, for a given recentsystem trajectory ( u ini , y ini ) and the input sequence u , there exists at least one vector g verifying U p Y p U f g = u ini y ini u , (3)2) the output prediction y is unique and given by y = Y f g, (4) for any g satisfying (3) . Note that, collectively (3) and (4) are equivalent to (2),which can be rewritten as U p Y p U f Y f g = u ini y ini uy . Throughout the paper, we assume that ¯ u and T ini verify theconditions in Lemma 2, which implies that the matrices U , U p and U f have full row rank.III. P ROBLEM F ORMULATION
In view of the prediction algorithm described in Lemma 2,consider the following data-driven linear-quadratic trackingproblem min u,y,g T f − (cid:88) k =0 (cid:16) (cid:107) y k − r k (cid:107) Q + (cid:107) u k (cid:107) R (cid:17) s.t. (3) , (4) (5)where { r k } T f − k =0 is the tracking reference and Q ≥ , R > are weight matrices. In this paper, we assume that the noisyrecent outputs y ini verify y ini = ˇ y ini + w, where ˇ y ini represents the noiseless output and w denotes themeasurement noise. Besides, as in [12] and [10], we assumethat w satisfies the following quadratic constraint (cid:20) w (cid:21) (cid:62) (cid:20) Φ Φ Φ (cid:62) Φ (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Φ (cid:20) w (cid:21) ≥ , (6)where Φ = Φ (cid:62) < . Remark 3.
The negative definiteness of Φ ensures that (cid:107) w (cid:107) is bounded. In the special case that Φ = and Φ = − I , (6) reduces to w (cid:62) w = (cid:88) k w (cid:62) k w k ≤ Φ , which, as highlighted in [12], has the interpretation ofbounded noise energy. We are interested in designing a control input u thatminimizes the worst-case quadratic tracking error among allfeasible noise trajectories, which are defined as follows. Definition 4.
For recent data ( u ini , y ini ) , a noise trajectory w is called feasible if it verifies (6) and ( u ini , y ini − w ) is atrajectory of G . Next, we provide a robust formulation of the trackingproblem (5) based on the linear quadratic tracking errorLQTE ( u, y, w ) (cid:44) T f − (cid:88) k =0 (cid:16) (cid:107) y k − r k (cid:107) Q + (cid:107) u k (cid:107) R (cid:17) . Problem P1 : Find the input sequence u solving min u,y,g max w LQTE ( u, y, w ) (7a)subject to U p Y p U f Y f g = u ini y ini uy − w , (7b) w is a feasible noise trajectory . (7c)The constraint (7c) makes the min-max optimization prob-lem difficult to solve. However, as we show in the nextsection, this issue can be circumvented by using a suitableparameterization of feasible noise trajectories.IV. R OBUST C ONTROLLER D ESIGN
Problem P1 can be reformulated as follows min u,γ,g,y γ s.t. , LQTE ( u, y, w ) ≤ γ, ∀ w satisfying (7c) . (8)For notational simplicity, we have omitted the dependence ofthe problem on r . In the sequel, we will derive a tractablereformulation of (8). We first show in Section IV-A thatany noise trajectory w fulfilling (7c) can be expressed as anaffine function of a vector g w satisfying a quadratic constraint.In Section IV-B, we show that the output y is completelydetermined by the input u and the vector g w , which allowsone to express the tracking error constraint as a quadratic constraint on g w . In light of these results, in Section IV-C, weprove that (8) is equivalent to a minimization problem with alinear cost and LMI constraints. Finally, in Section IV-D, weshow how to reduce the size of the LMI constraints to reducethe computational burden. A. Feasible Noise Parameterization
Since ¯ u is persistently exciting of order T ini + T f + n ,it is also persistently exciting of order T ini + n . In view ofLemma 1, ( u ini , y ini − w ) is a trajectory of G if and only ifthere exists g ini ∈ R T d − T ini − T f +1 such that (cid:20) u ini y ini − w (cid:21) = (cid:20) U p Y p (cid:21) g ini . (9)Consider the solution g ∗ ini = U (cid:62) p (cid:0) U p U (cid:62) p (cid:1) − u ini to the firstequation in (9), i.e., U p g ∗ ini = u ini . Any other solution g ini verifying U p g ini = u ini can be written as g ini = g ∗ ini + M g w for some g w ∈ R T d − ( m +1) T ini − T f +1 , where M = N ( U p ) .Furthermore, from the second block row of (9), any w thatmakes ( u ini , y ini − w ) a trajectory of G can be written as w = − Y p M g w + ( − Y p g ∗ ini + y ini ) (cid:124) (cid:123)(cid:122) (cid:125) w (10)for some g w .In view of the above results, the feasible noise trajectoriescan be explicitly parameterized as follows. Lemma 5.
The noise trajectory w is feasible if and only ifthere exists g w satisfying (10) and (cid:20) g w (cid:21) (cid:62) (cid:20) [ A w ] [ A w ] [ A w ] (cid:62) [ A w ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) A w (cid:20) g w (cid:21) ≥ , (11) where [ A w ] = Φ + w (cid:62) Φ (cid:62) + Φ w + w (cid:62) Φ w , [ A w ] = − Φ Y p M − w (cid:62) Φ Y p M, [ A w ] = M (cid:62) Y (cid:62) p Φ Y p M. (12) Proof.
In addition to making ( u ini , y ini − w ) a trajectoryof G , a feasible w should also satisfy the constraint (6).Substituting (10) into (6), we write the constraint on g w as (cid:20) w (cid:21) (cid:62) Φ (cid:20) w (cid:21) = Φ + ( − Y p M g w + w ) (cid:62) Φ (cid:62) + Φ ( − Y p M g w + w )+ ( − Y p M g w + w ) (cid:62) Φ ( − Y p M g w + w )= (cid:20) g w (cid:21) (cid:62) A w (cid:20) g w (cid:21) , with A w defined in (11), (12). Remark 6.
We note that, from (10) , for a given vector g ∗ ini ,there might be multiple g w parameterizing the same noisetrajectory w . In Section IV-D, we show that this redundancy in-creases the computational complexity of the proposed method,and provide a method to overcome this problem. B. Reformulation of the Tracking Error Constraint In this section, we show that for a feasible noise trajectory w , the tracking error constraint LQTE ( u, y, w ) ≤ γ can bereformulated as a quadratic constraint on the parameter vector g w . We achieve this goal by first writing the output y asan affine function of u and g w , and then substituting theexpression of y into the tracking error constraint.To express y in terms of u and g w , we compute g from thefirst three block rows of (7b), and substitute it into the lastblock row of (7b). First of all, we show how to construct asolution g from U p Y p U f g = u ini y ini − wu . (13)Since ( u ini , y ini − w ) is a feasible system trajectory, in viewof Lemma 2, for any given input u , there exists a (possiblynonunique) vector g verifying (13). Any solution g to (13) canbe decomposed as g (cid:44) g ini + g u , where g ini verifies (9) and g u solves U p Y p U f g u = u − U f g ini . (14)Therefore, if we can find a solution g u to (14), we can obtaina solution g to (13).Before showing how to construct g u in Lemma 7, thefollowing results are needed. In view of Theorem 2 of [23],the matrix [ U (cid:62) p , Y (cid:62) p , U (cid:62) f ] (cid:62) does not always have full rowrank, even though [ U (cid:62) p , U (cid:62) f ] (cid:62) does. Therefore, there existsa row permutation matrix P Y transforming Y p as P Y Y p =[ Y (cid:62) p , Y (cid:62) p ] (cid:62) such that Λ (cid:44) [ U (cid:62) p , Y (cid:62) p , U (cid:62) f ] (cid:62) has full row rankand rank (Λ) = rank U p Y p U f . Lemma 7.
A solution to (14) is given by g u = Λ (cid:62) (ΛΛ (cid:62) ) − u − U f g ini . (15) Proof.
Left multiply both sides of (14) with (cid:104)
I 0 00 P Y
00 0 I (cid:105) toobtain U p Y p Y p U f g u = u − U f g ini . (16)By definition, the rows of Y p can be written as linear combi-nations of the rows of [ U (cid:62) p , Y (cid:62) p , U (cid:62) f ] (cid:62) . Therefore there existsan ordered sequence of elementary row operations { E k } ek =1 captured by the matrix E (cid:44) E e E e − . . . E such that E U p Y p Y p U f = U p Y p U f . (17) We next show by contradiction that the rows of Y p can alsobe written as linear combinations of the rows of [ U (cid:62) p , Y (cid:62) p ] (cid:62) .Suppose that the rows of Y p cannot be written as linear com-binations of the rows of [ U (cid:62) p , Y (cid:62) p ] (cid:62) . Then, left multiplying E to both sides of (16), we obtain U p Y p U f g u = linear combination of rows of u − U f g ini u − U f g ini . (18)Since [ (cid:62) , (cid:62) ] (cid:62) is a feasible system trajectory, in view ofLemma 2, there always exists a g u solving (14). There-fore, (14) and further (16), (18) should always be compatiblefor any u − U f g ini . However, it is clear from the third block rowof (18) that, (18) is not always compatible for any u − U f g ini .This makes a contradiction. Therefore, the rows of Y p can bewritten as linear combinations of the rows of [ U (cid:62) p , Y (cid:62) p ] (cid:62) . Asa result, the matrix E can be constructed such that E u − U f g ini = u − U f g ini , (19)that is, the matrix E only applies elementary row operationson the first three block rows of [ (cid:62) , (cid:62) , (cid:62) , ( u − U f g ini ) (cid:62) ] (cid:62) . The vector g u in (15) satisfies U p Y p U f g u = u − U f g ini . Then, we have U p Y p U f g u = u − U f g ini . (20)Left multiplying both sides of (20) by E − , in view of (17)and (19), one obtains U p Y p Y p U f g u = u − U f g ini . Furthermore, from the definition of P Y , we have U p Y p U f = I 0 00 P − Y
00 0 I U p (cid:20) Y p Y p (cid:21) U f . Therefore, we conclude the proof by showing that U p Y p U f g u = I 0 00 P − Y
00 0 I (cid:20) (cid:21) u − U f g ini = (cid:20) (cid:21) u − U f g ini . A solution g = g ini + g u to (13) can be obtained from a g ini verifying (9) and the g u in Lemma 7. We next show that y can be expressed as an affine function of u and g w , and furtherreformulate the tracking error constraint in terms of u and g w . Lemma 8.
Given a feasible noise trajectory w and a controlsequence u , the unique output y satisfying (7b) is given by y = B u u + B w g w + y , (21) where g w parameterizes w through (10) , y = B ini g ∗ ini , B w = B ini M , B ini = Y f I + Λ (cid:62) (ΛΛ (cid:62) ) − − U f ,B u = Y f Λ (cid:62) (ΛΛ (cid:62) ) − . Moreover, the tracking error constraint
LQTE( u, y, w ) ≤ γ can be equivalently expressed as (cid:20) g w (cid:21) (cid:62) (cid:20) [ Q g ] [ Q g ] [ Q g ] (cid:62) [ Q g ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Q g ( u,γ ) (cid:20) g w (cid:21) ≥ , (22) where ¯ R = I ⊗ R, ¯ Q = I ⊗ Q, [ Q g ] = γ − u (cid:62) ¯ Ru − ( B u u + y − r ) (cid:62) ¯ Q ( B u u + y − r ) , [ Q g ] = − ( B u u + y − r ) (cid:62) ¯ QB w , [ Q g ] = − B (cid:62) w ¯ QB w . Proof.
Substituting g = g ini + g u into the fourth block rowof (7b), one obtains y = Y f g ini + Y f g u = Y f g ini + Y f Λ (cid:62) (ΛΛ (cid:62) ) − u + − U f g ini = B ini g ini + B u u = B ini ( g ∗ ini + M g w ) + B u u, which proves (21). We then have the following equivalentconditionsLQTE ( u, y, w ) ≤ γ ⇔ γ − u (cid:62) ¯ Ru − ( y − r ) (cid:62) ¯ Q ( y − r ) ≥ , ⇔ γ − u (cid:62) ¯ Ru − ( y + B w g w + B u u − r ) (cid:62) ¯ Q ( y + B w g w + B u u − r ) ≥ , ⇔ (22) . The proof is complete.
C. Main Result
The following theorem leverages the results obtained inLemma 5 and Lemma 8 to show that (8) and, hence, P1 , areequivalent to a minimization problem with a linear cost andLMI constraints. Theorem 9.
The robust tracking control problem P1 is equiv-alent to solving min u,γ,α ≥ γ (23a) s . t ., ( ¯ R + B (cid:62) u ¯ QB u ) − (cid:2) u (cid:3)(cid:20) u (cid:62) (cid:21) Q ag ( u, γ ) − αA w ≥ , (23b) where Q ag ( u, γ ) = Q g ( u, γ ) + (cid:20) u (cid:62) ( R + B (cid:62) u ¯ QB u ) u
00 0 (cid:21) , (24) A w is defined in Lemma 5, and ¯ Q, ¯ R, Q g , and B u are definedin Lemma 8.Proof. Based on Lemma 5 and Lemma 8, the minimizationproblem (8) is equivalent to min u,γ γ s.t. , (22) holds ∀ g w satisfying (11) . In view of the S-lemma [21], the constraint of this minimiza-tion problem holds if and only if there exist u and α ≥ suchthat Q g ( u, γ ) − αA w ≥ . Using Schur complement [24], the above matrix inequalitycan be transformed into the LMI in (23b). Note that thequadratic term u (cid:62) ( R + B (cid:62) u ¯ QB u ) u in the right hand sideof (24) cancels out with the quadratic term of u in Q g ( u, γ ) ,therefore making Q ag ( u, γ ) a linear function of u and γ .Minimizing the performance index γ further gives the solutionof (8) and hence P1 . D. Implementation Aspects: Dimension Reduction for Com-putational Efficiency
In view of the analysis in Section IV-A, the sequence w makes ( u ini , y ini − w ) a trajectory of G if and only if thereexists g w , such that w = − Y p N ( U p ) g w + w . (25)However, if g w is mapped into w through (25), any g w + v ,where v ∈ ker( Y p N ( U p )) , is also mapped into the same w .This is especially true when the length T d of historical datais large, i.e., T d (cid:29) mT ini and T d (cid:29) pT ini , which makes ker( U p ) ∩ ker( Y p ) , and therefore ker( Y p N ( U p )) , nonempty.As a result, any parameterization of a subspace through g w is redundant. Redundancy affects also the constraint (23b).Indeed, if the length of the vector g w is unnecessarily large,so are the sizes of the matrices A w and Q g ( u, γ ) in (11), (22),as well as the LMI constraint in (23b), making the optimizationproblem (23) inefficient.More formally, denote W as the set of noise trajectories w that make ( u ini , y ini − w ) a trajectory of G . We have, from (25), W = range( Y p N ( U p )) + w , where we represent the vectorspace range( Y p N ( U p )) as { Y p N ( U p ) g w | g w ∈ R T d − ( m +1) T ini − T f +1 } . (26)The cause of redundancy is that the dimension of the freevector g w in (26) can be much larger than the dimension of range( Y p N ( U p )) . In the following theorem, we address thisissue to present a non-redundant representation of W . Theorem 10.
The vector w belongs to W if and only if thereexists g w ∈ R ¯ n w such that w = − Y p N ( U p ) R ( N ( U p ) (cid:62) Y (cid:62) p ) g w + w . (27) where ¯ n w = rank( Y p N ( U p )) . Moreover, the above mappingfrom R ¯ n w to W is bijective.Proof. Since two vector spaces are isomorphic if and onlyif they have the same dimension, to eliminate the redundantrepresentation problem, we introduce an isomorphism from R ¯ n w to range( Y p N ( U p )) and represent range( Y p N ( U p )) interms of this isomorphism. Notice that range( Y p N ( U p )) = { Y p N ( U p ) g | g ∈ R T d − T ini +1 − T ini m } ( a ) = { Y p N ( U p )( g + g ) | g ∈ ker( Y p N ( U p )) ,g ∈ range( N ( U p ) (cid:62) Y (cid:62) p ) } = { Y p N ( U p ) g | g ∈ range( N ( U p ) (cid:62) Y (cid:62) p ) } = { Y p N ( U p ) R ( N ( U p ) (cid:62) Y (cid:62) p ) g w | g w ∈ R ¯ n w } where ( a ) follows from the fact that ker( Y p N ( U p )) ⊥ range( N ( U p ) (cid:62) Y (cid:62) p ) . Therefore an isomorphism from R ¯ n w to range( Y p N ( U p )) isgiven by the matrix Y p N ( U p ) R ( N ( U p ) (cid:62) Y (cid:62) p ) . Furthermore,since W is range( Y p N ( U p )) shifted by w , the mapping from R ¯ n w to W given by (27) is bijective.In view of the above theorem, the representation of W through (27) using g w ∈ R ¯ n w is non-redundant. To applythe above result in the implementation of (23), we only needto replace the matrix M = N ( U p ) in the derivations ofSection IV-A–IV-C with M = N ( U p ) R ( N ( U p ) (cid:62) Y (cid:62) p ) . Remark 11.
Since ¯ n w = rank (cid:18)(cid:20) U p Y p (cid:21)(cid:19) − T ini m ( a ) = n, where ( a ) follows from Theorem 2 of [23], the length of thevector g w in (27) is equal to n . This guarantees that the sizeof the matrix in the LMI in (23b) scales with n + T f m . On thecontrary, the length of the vector g w in (10) scales with T d .As T d is usually significantly larger than n , the non-redundantparameterization shown in this section can reduce the size ofthe LMI constraint (23b) considerably. V. G
ENERALIZATIONS
In this section, we provide several extensions to the robustcontrol design method described in the previous section. First,in Section V-A, we show how to add input and outputconstraints to the controller. In Section V-B, we show howto take into account actuator disturbances, before presentingthe overall LMI optimization problem incorporating bothextensions in Section V-C.
A. Input and Output Constraints
In this section, we show how to add quadratic input andoutput constraints to problem P1 . Since constraints on theinput can be directly incorporated into (23), hereafter we focuson constraints on the output y only and in the form θ ( y ) = (cid:20) y (cid:21) (cid:62) (cid:20) Θ Θ Θ (cid:62) Θ (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Θ (cid:20) y (cid:21) ≥ . (28)When Θ = Θ (cid:62) < and Θ = 0 , the above constraintimposes an upper bound on (cid:107) y (cid:107) .Since the output is related to the input u and the noisetrajectory w via (21), the output constraint (28) can be writtenas Θ + Θ ( y + B u u + B w g w ) + ( y + B u u + B w g w ) (cid:62) Θ (cid:62) + ( y + B u u + B w g w ) (cid:62) Θ ( y + B u u + B w g w )= (cid:20) g w (cid:21) (cid:62) (cid:20) [Θ g ] [Θ g ] [Θ g ] (cid:62) [Θ g ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Θ g (cid:20) g w (cid:21) ≥ , where [Θ g ] = B (cid:62) w Θ B w , [Θ g ] = Θ + Θ ( y + B u u ) + ( y + B u u ) (cid:62) Θ (cid:62) + ( y + B u u ) (cid:62) Θ ( y + B u u ) , [Θ g ] = Θ B w + ( y + B u u ) (cid:62) Θ B w . In principle, we want the constraint (28) to hold for all feasiblenoise trajectories. Similarly to the proof of Theorem 9, thisrequirement is equivalent to the existence of α y ≥ suchthat Θ g − α y A w ≥ , which can be reformulated as an LMIconstraint and added to the optimization problem (23). B. Actuator Disturbances
In this section, we show how to consider actuator distur-bances. Suppose the actuation input ˇ u ini to the system G togenerate the recent data is also noisy, i.e., ˇ u ini = u ini − d ini , where u ini is nominal control input and d ini is the actuatordisturbance. Moreover, we also consider a disturbance d actingon the computed control input u , i.e., ˇ u = u − d . Therefore,the data-dependent relation (7b) becomes U p Y p U f Y f g = u ini y ini uy − d ini wd . (29)We assume that the actuator disturbance ¯ d (cid:44) [ d (cid:62) ini , d (cid:62) ] (cid:62) satisfies the quadratic constraint (cid:20) d (cid:21) (cid:62) (cid:20) Φ d, Φ d, Φ (cid:62) d, Φ d, (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Φ d (cid:20) d (cid:21) ≥ , (30)where Φ d, < . Our goal is to solve a min-max robust control problemsimilar to P1 . Due to the existence of actuator disturbances, wereplace (cid:107) u (cid:107) R with the true system input (cid:107) ˇ u (cid:107) R in the cost (7a),replace (7b) with (29), and optimize over all feasible noise anddisturbance trajectories [ d (cid:62) ini , w (cid:62) , d (cid:62) ] (cid:62) . We first characterizefeasible trajectories [ d (cid:62) ini , w (cid:62) ] (cid:62) such that ( u ini − d ini , y ini − w ) is a trajectory of G . Similarly to the argument in Section IV-A, [ d (cid:62) ini , w (cid:62) ] (cid:62) satisfies the above requirement if and only if thereexists g ini such that (cid:20) d ini w (cid:21) = − (cid:20) U p Y p (cid:21) g ini + (cid:20) u ini y ini (cid:21) . (31)Therefore, the set of noise and disturbance trajectories [ d (cid:62) ini , w (cid:62) ] (cid:62) that make ( u ini − d ini , y ini − w ) a trajectory of G is ˜ W = range( (cid:20) U p Y p (cid:21) ) + (cid:20) u ini y ini (cid:21) . Let ¯ n d = rank( (cid:104) U p Y p (cid:105) ) = ¯ n w + mT ini . Similarly to the analysisin Section IV-D, [ d (cid:62) ini , w (cid:62) ] (cid:62) ∈ ˜ W if and only if there exists g w ∈ R ¯ n d such that (cid:20) d ini w (cid:21) = − (cid:20) U p Y p (cid:21) R ( (cid:20) U p Y p (cid:21) (cid:62) ) g w + (cid:20) u ini y ini (cid:21) . (32)Moreover, the above mapping from R ¯ n d to ˜ W is bijective.Therefore, from (32), one gets (cid:20) d ini w (cid:21) d = − (cid:20) U p Y p (cid:21) R ( (cid:20) U p Y p (cid:21) (cid:62) )
00 I (cid:20) g w d (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ¯ g + (cid:20) u ini y ini (cid:21) , (33)i.e., the vector ¯ g ∈ R ¯ n d + mT f parameterizes all feasible noiseand disturbance trajectories. By following the arguments usedin the proof of Lemma 5, the quadratic constraints on w and ¯ d can be transformed into quadratic constraints on ¯ g as (cid:20) w (cid:21) (cid:62) Φ (cid:20) w (cid:21) ≥ ⇐⇒ (cid:20) g (cid:21) (cid:62) ¯Φ w (cid:20) g (cid:21) ≥ , (34) (cid:20) d (cid:21) (cid:62) Φ d (cid:20) d (cid:21) ≥ ⇐⇒ (cid:20) g (cid:21) (cid:62) ¯Φ d (cid:20) g (cid:21) ≥ , (35)where the matrices ¯Φ w and ¯Φ d directly follow from (6), (30),and (33), and their expressions are omitted for brevity.Since every [ d (cid:62) ini , w (cid:62) ] (cid:62) ∈ ˜ W can be written as (32), bysubstituting this representation into (31), we obtain that fora given [ d (cid:62) ini , w (cid:62) ] (cid:62) , the solution g ini to (31) is given by g ini = M d g w , where M d = R ( (cid:104) U p Y p (cid:105) (cid:62) ) . We can followthe procedure in Section IV-B to derive the solution g = g ini + g u to the first three equations in (29), where g u =Λ (cid:62) (ΛΛ (cid:62) ) − (cid:2) (ˇ u − U f g ini ) (cid:62) (cid:3) (cid:62) verifies (14) with thenoisy control input ˇ u instead of u . Then, since y = Y f g ,the following holds with the matrices B ini and B u defined inLemma 8 y = B ini g ini + B u ˇ u = B ini M d g w − B u d + B u u = (cid:2) B ini M d − B u (cid:3)(cid:124) (cid:123)(cid:122) (cid:125) ¯ B g ¯ g + B u u. (36) Since ˇ u = u − Ξ¯ g, where Ξ = [ , I ] , the performanceconstraint can be rewritten as γ − T f − (cid:88) k =0 (cid:16) (cid:107) y k − r t + k (cid:107) Q + (cid:107) ˇ u k (cid:107) R (cid:17) = γ − ˇ u (cid:62) ¯ R ˇ u − ( y − r ) (cid:62) ¯ Q ( y − r )= γ − ( u − Ξ¯ g ) (cid:62) ¯ R ( u − Ξ¯ g ) − ( ¯ B g ¯ g + B u u − r ) (cid:62) ¯ Q ( ¯ B g ¯ g + B u u − r )= (cid:20) g (cid:21) (cid:62) (cid:20) [ ¯ Q g ] [ ¯ Q g ] [ ¯ Q g ] (cid:62) [ ¯ Q g ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ¯ Q g ( u,γ ) (cid:20) g (cid:21) ≥ , where [ ¯ Q g ] = γ − u (cid:62) ¯ Ru − ( B u u − r ) (cid:62) ¯ Q ( B u u − r ) , [ ¯ Q g ] = u (cid:62) ¯ R Ξ − ( B u u − r ) (cid:62) ¯ Q ¯ B g , [ ¯ Q g ] = − Ξ (cid:62) ¯ R Ξ − ¯ B (cid:62) g ¯ Q ¯ B g . As such, the overall data-driven robust control objective isto find u and γ such that (cid:20) g (cid:21) (cid:62) ¯ Q g ( u, γ ) (cid:20) g (cid:21) ≥ holds for all feasible noise and disturbance trajectories param-eterized by ¯ g satisfying quadratic constraints (34), (35). Usingthe S-lemma for multiple quadratic inequalities [21], this istrue if there exist u , α w ≥ , and α d ≥ such that ¯ Q g ( u, γ ) − α w ¯Φ w − α d ¯Φ d ≥ . (37)We can further convert the above inequality into an LMIthrough the Schur complement. Therefore, the problem P1 with input disturbances is solved if the following optimizationproblem is solved min u,γ,α w ≥ ,α d ≥ γ s.t. , (37) . C. Co-existence of Quadratic Input/Output Constraints andActuator Disturbance
In this section, we use the results in Sections V-A and V-Bfor dealing simultaneously with the quadratic input/output con-straints and actuator disturbances. The overall robust controlproblem is given by min u max w,d ini ,d T f − (cid:88) k =0 (cid:16) (cid:107) y k − r k (cid:107) Q + (cid:107) u k − d k (cid:107) R (cid:17) (38a)subject to U p Y p U f Y f g = u ini y ini uy − d ini wd , (38b) (cid:20) u − d (cid:21) (cid:62) (cid:20) Ψ Ψ Ψ (cid:62) Ψ (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) Ψ (cid:20) u − d (cid:21) ≥ , (38c)output quadratic constraints (28) . (38d) The following theorem provides an LMI-based representationof (38) . Theorem 12.
Denote α (cid:44) [ α w , α d , α u,w , α u,d , α y,w , α y,d ] (cid:62) .The optimization problem (38) is solved when the followingminimization problem is solved min u,γ,α ≥ γ s.t. , (37) , (42) , (45) . (39) where constraints (42) and (45) are given in the proof.Proof. Similarly to the proof of Theorem 9, we aim tominimize γ , subject to the tracking error constraint and theconstraint that the input/output constraints hold for all feasiblenoise and disturbance trajectories. The tracking error constraintcan be shown to be given as (37). In the following, we showhow to characterize the constraint that the input/output con-straints hold for all feasible noise and disturbance trajectories.In light of ˇ u = u − Ξ¯ g , one sees that the input constraintin (38c) is equivalent to ¯ ψ (¯ g ) = (cid:20) g (cid:21) (cid:62) (cid:20) [ ¯Ψ] [ ¯Ψ] [ ¯Ψ] (cid:62) [ ¯Ψ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ¯Ψ (cid:20) g (cid:21) ≥ , (40)where [ ¯Ψ] = Ψ + u (cid:62) Ψ u + Ψ u + u (cid:62) Ψ (cid:62) , [ ¯Ψ] = − Ψ Ξ − u (cid:62) Ψ Ξ , [ ¯Ψ] = Ξ (cid:62) Ψ Ξ . (41)We need to ensure that (40) holds for all ¯ g satisfying thequadratic constraints (34), (35). In view of the S-lemma, thisis possible if there exist u , α u,w ≥ , and α u,d ≥ such that ¯Ψ − α u,w ¯Φ w − α u,d ¯Φ d ≥ , (42)which can be converted to an LMI using the Schur com-plement. Similarly, considering y = ¯ B g ¯ g + B u u , the outputconstraint in (28) is equivalent to ¯ θ (¯ g ) = (cid:20) g (cid:21) (cid:62) (cid:20) [ ¯Θ] [ ¯Θ] [ ¯Θ] (cid:62) [ ¯Θ] (cid:21)(cid:124) (cid:123)(cid:122) (cid:125) ¯Θ (cid:20) g (cid:21) ≥ , (43)where [ ¯Θ] = Θ + u (cid:62) B (cid:62) u Θ B u u + Θ B u u + u (cid:62) B (cid:62) u Θ (cid:62) , [ ¯Θ] = Ψ ¯ B g + u (cid:62) B (cid:62) u Θ ¯ B g , [ ¯Θ] = ¯ B (cid:62) g Θ ¯ B g . (44)Following the same procedure as for input constraint, it canbe shown that (43) is satisfied if there exist u , α y,w ≥ , and α y,d ≥ such that the following is satisfied ¯Θ − α y,w ¯Φ w − α y,d ¯Φ d ≥ , (45)which can be converted to an LMI using the Schur comple-ment. Combining the above results, we get (39). Even though the matrix inequalities in the theorem and proof are not LMIs,they can be transformed to LMIs using Schur complement in a similar way tothe proof of Theorem 9. To save space, the resulting LMIs are not displayed.Furthermore, we refer to these matrix inequalities as LMIs without ambiguity.
Remark 13.
The extensions presented in this section in-volve the use of the S-lemma with multiple quadratic con-straints [21] and Schur complement with semidefinite ma-trices [24], which are only sufficient. Therefore, the controldesign procedure in (39) is conservative, i.e., it may have nosolution even though a control input u solving the min-maxcontrol problem (38) exists. Remark 14.
The proposed control design can be easilyapplied in a receding horizon fashion, in order to implementa data-driven predictive controller. In doing so, at each timeinstance, one needs to update the output reference r as wellas recent input and output data u ini and y ini with the onlinedata, solve (39) , and apply only the first control input fromthe computed optimal control sequence u . Moreover, it can beshown that the resulting controller is equivalent to a robustmodel predictive controller (MPC) with bounded uncertaintyon the initial state. As such, the stability of the resulting closed-loop system can be studied using the existing results on robustMPC. Such a discussion is omitted so as to emphasize therobust data-driven nature of the proposed controller, which isthe main contribution of this paper. VI. S
IMULATIONS
We illustrate the performance of the proposed controllerthrough numerical simulations. We consider an unstable LTIsystem (1) with randomly selected system matrices A = . − . − . . . . . . . . − . . − . . . . ,B = − . − . − . . − . − . . . − . − . . − . ,C = (cid:20) . . . − . . − . − . . (cid:21) , D = . By choosing a random initial condition, historical input-outputdata of length T d = 110 are collected with inputs generatedfrom a uniform distribution in the interval [ − , . We assumethat the exact order n = 4 of the system is unknown andonly the upper bound ¯ n = 6 is available. Consequently, recent input-output data of length T ini = ¯ n = 6 ≥ l ( G ) arecollected with inputs from the uniform distribution in [ − , .Moreover, recent data is corrupted by input disturbances andoutput noises as in Section V-C, where the trajectories w and ¯ d are selected to satisfy quadratic constraints (6) and (30),respectively, with Φ = T ini p(cid:15), Φ = , Φ = − I , Φ d, = ( T ini + T f ) m(cid:15), Φ d, = , Φ d, = − I , (46)and (cid:15) = 0 . . We are interested in solving the min-maxproblem (38). We select T f = 20 , r = , Q = I , and R = I to robustly regulate the output of the system to zero withina horizon of length . Moreover, we seek to do so while Fig. 2. Output responses with the designed robust control under differentfeasible noise and disturbance trajectories. ensuring that ˇ u and y satisfy quadratic constraints (38c), (28)with Ψ = T f (cid:15) u , Ψ = , Ψ = − I , Θ = T f (cid:15) y , Θ = , Θ = − I , and (cid:15) u = (cid:15) y = 0 . .As shown in Theorem 12, it is possible to convert thisrobust control input design problem into the minimizationproblem (39). This problem is then solved using Yalmip [25]on Matlab with MOSEK [26] specified as the solver, whichreturns the optimal control sequence u . This control sequenceis tested with multiple compatible realizations of noise tra-jectories. In particular, we randomly select vectors ¯ g that satisfy the quadratic constraints (34) and (35), which,in view of (33), parameterize feasible sequences of w and ¯ d . As shown in Figure 2, output trajectories are quicklybrought around zero for all noise and disturbance realizations.Moreover, Figure 3 displays the robustness of the closed-loop system. Specifically, the first plot in Figure 3 shows that γ ≤ γ ∗ for all selected noise and disturbance realizations,where each blue circle corresponds to a specific realizationand γ ∗ is the optimal value of (39). Moreover, the secondand third plots show that the input and output constraints aresatisfied for all selected noise and disturbance trajectories, i.e., ψ (ˇ u ) ≥ and θ ( y ) ≥ , respectively.When parameterizing the noise trajectories as in (33), theparameterization methods proposed in Sections IV-D and V-Ballow for a significant reduction in the sizes of the LMIconditions (37), (42), and (45). In particular, the size of eachLMI condition is reduced from to .It is seen from Figure 3 that the obtained γ values are not ashigh as the optimal value γ ∗ . Moreover, the input and outputconstraints are not active in any of the different simulationscenarios, i.e., ψ (ˇ u ) (cid:54) = 0 and θ ( y ) (cid:54) = 0 . These limitationsare due to the fact that the version of S-lemma for multiplequadratic inequalities and the semidefinite version of Schurcomplement used in Section V-C are conservative. In order todemonstrate that they are the only sources of conservativity,we run another simulation with the same LTI system, in which
10 20 30 40 50 60 70 80 90 100101520 10 20 30 40 50 60 70 80 90 1001.61.71.8 10 20 30 40 50 60 70 80 90 10088.59
Fig. 3. Robustness against different noise and disturbance trajectories. Top:Tracking errors γ = y (cid:62) Qy + ˇ u (cid:62) R ˇ u (blue circles), and the optimal value γ ∗ (red line) computed from (39). Middle: Values of the input constraint ψ (ˇ u ) computed as in (38c). Bottom: Values of the output constraint θ ( y ) computedas in (28).
10 20 30 40 50 60 70 80 90 10077.17.27.37.4
Fig. 4. Tracking errors γ = y (cid:62) Qy + u (cid:62) Ru (blue circles), and the optimalvalue γ ∗ (red line) computed from (23). we do not consider actuator disturbances and input/outputconstraints. Specifically, the same historical data as in theprevious simulation are used to construct the Hankel matrices U p , U f , Y p , and Y f . Moreover, the same input sequence u ini is used to generate the recent trajectory. Differently tothe previous case, only the recent output trajectory y ini isaffected by measurement noise w . This noise is chosen tosatisfy (6) with the matrix Φ defined by (46). The T f , r , Q ,and R of the previous example are chosen to ensure robustregulation of system output to zero. By utilizing the resultsof Section IV, we solve the problem (23) with the matrix M defined in (27). Similarly to the previous simulation, thecalculated control input u is used to control the system with different realizations of the vector g w parameterizingdifferent feasible noise trajectories w . The results of thissimulation are presented in Figure 4, where one sees that thetracking costs γ in blue circles are smaller than the robustoptimal tracking cost γ ∗ computed from (23) for all feasiblenoise trajectories. It can also be seen from this figure that some γ values get quite close to γ ∗ , hence supporting the claim thatTheorem 9 in Section IV is not conservative.VII. C ONCLUSIONS
Willems’ Fundamental Lemma shows that finite-length per-sistently exciting data can characterize the behaviors of linear systems, which enables data-driven simulation and control. Inthis paper, we build on this data-dependent representation toconsider the case that the recent output data are noisy andsolve worst-case robust optimal tracking control problems ina data-driven fashion. The key ingredient of our approach isa suitable parameterization of the feasible noise trajectoriesand the performance specification, which allows one to ex-press them as quadratic constraints. Then, by applying theS-lemma, we show that the worst-case robust control problemis equivalent to a minimization problem with a linear costand LMI constraints. Moreover, by carefully selecting thenoise parameterization, we can show that the dimension ofthe LMI optimization problem does not scale with the lengthof historical data. Our method can also easily incorporate inputand output constraints, as well as actuator disturbances.At present, the proposed method assumes that noise affectsonly recent data. Future work will be devoted to generaliza-tions accounting for noise also in historical data.R
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