Data Driven Robust Superstable Control of Switched Systems⁎

9th IFAC Symposium on Robust Control Design 9th IFAC Symposium on Robust Control Design Florianopolis, Brazil, September 3-5, 2018Design 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018 Available online at www.sciencedirect.com 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018Design Florianopolis, Brazil, September 3-5, 2018

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IFAC PapersOnLine 51-25 (2018) 402–408

Data Data Data Data

Driven Driven Robust Robust Superstable Superstable DrivenSwitched Robust Superstable  Systems DrivenSwitched Robust Superstable Systems Switched Systems  Switched Systems ∗∗ Tianyu Dai ∗∗∗ Mario Sznaier ∗∗ ∗∗

Control Control Control Control

of of of of

Tianyu Dai ∗∗ Mario Sznaier ∗∗ Tianyu Dai Mario Sznaier ∗∗ ∗ ∗∗ ∗ Tianyu Dai Mario Boston, SznaierMA, Dept, Northeastern University, 02115(e-mail: ∗ ∗ ECE ECE Dept, Boston, MA, 02115(e-mail: Northeastern University, ∗ ∗ ECE Dept, Northeastern University, Boston, MA, 02115(e-mail: [email protected]). [email protected]). ∗ ∗∗ ECE Dept, University, Dept, Northeastern Northeastern University, Boston, Boston, MA, MA, 02115(e-mail: 02115 (e-mail: ∗∗ [email protected]). ∗∗ ECE University, Boston, MA, 02115 (e-mail: ∗∗ ECE Dept, Northeastern ∗∗ [email protected]). [email protected]). ECE Dept, Northeastern University, Boston, MA, 02115 (e-mail: [email protected]). ∗∗ ECE Dept, Northeastern University, Boston, MA, 02115 (e-mail: [email protected]). [email protected]). Abstract: This paper considers the problem of data driven control of switched linear MIMO Abstract: This paper considers the of data control switched linear Abstract: This paper considers the problem problem data driven driven control of ofoperating switched points linear MIMO MIMO systems. Our goal is, given experimental dataofcollected at different and asystems. Our goal is, given experimental data collected at different operating points and aAbstract: This paper considers thedesign problem ofcollected data driven control ofoperating switched linear MIMO priori model structure, to directly a controller that super-stabilizes all plants in systems. Our goal is, given experimental data at different points and apriori model structure, to directly design data a controller that super-stabilizes all points plants and in the the systems. Our goal experimental collected at different operating aconsistency set, for is, anygiven arbitrary switching sequence, without explicitly identifying the in plant. priori model structure, to directly design a controller that super-stabilizes all plants the consistency set, for any to arbitrary switching sequence, without explicitly identifying the in plant. priori model structure, directly design a controller that super-stabilizes all plants the Our main result shows that this problem cansequence, be recast without into a convex optimization problem and consistency set, for anythat arbitrary switching explicitly identifying the plant. Our main result shows this problem can be recast into optimization problem and consistency set, for anythat arbitrary switching without explicitly identifying the plant. Our main solved, result shows this problem cansequence, bewith recast into aa convex convex optimization problem and efficiently leading to aa robust controller guaranteed ��∞ worse-case performance for efficiently solved, leading to robust controller with guaranteed worse-case performance for ∞ optimization problem and ∞ Our main solved, result shows that this problem canhave bewith recast into athe convex any switching amongst all plants that could generated observed experimental data. efficiently leading to a robust controller guaranteed � worse-case performance for ∞ ∞ any switching amongst all plants that could have generated the observed experimental data. efficiently solved, leading a robust with guaranteed performance ∞ worse-case any switching amongst alltoplants thatcontroller could have generated the �observed experimental data.for © 2018, IFAC (International Federation of Automatic Hosting Elsevier experimental Ltd. All rights reserved. any switching amongst all plants that could haveControl) generated theby observed data. Keywords: Keywords: Data Data Driven Driven MIMO MIMO Control, Control, Superstable Superstable Control, Control, Convex Convex Optimization Optimization Keywords: Data Driven MIMO Control, Superstable Control, Convex Optimization Keywords: Data Driven MIMO Control, Superstable Control, Convex Optimization 1. INTRODUCTION polyhedral polyhedral Lyapunov Lyapunov function function based based conditions conditions given given in in 1. 1. INTRODUCTION INTRODUCTION polyhedral Lyapunov function based conditions giventhe in Blanchini et al. [2009] are necessary and sufficient, Blanchini et al. [2009] are necessary and sufficient, the 1. INTRODUCTION polyhedralstability Lyapunov function based conditions giventhe in quadratic counterparts are only sufficient. Blanchini et al. [2009] are necessary and sufficient, During the past decade, a large research effort has been destability counterparts are only sufficient. During the past decade, a large research effort has been de- quadratic Blanchini et al. [2009] are necessary and sufficient, the During the past decade, a large research effort has been dequadratic stability counterparts are only sufficient. voted to the problem of designing controllers for switched voted tothe the problem of designing controllers for switched data control methods quadratic stability counterparts only (DDC) sufficient. Duringto decade, large research effort has been de- As As an an alternative, alternative, data driven driven are control (DDC) methods voted thepast problem of adesigning controllers for switched linear systems, described by a model that switches among seek to avoid the plant identification step by synthesizlinear systems, described by a model that switches among As an alternative, data driven control (DDC) methods voted to the problem of designing controllers for switched seek to avoid the plant identification step by synthesizlinear described by a to model that switches severalsystems, submodels according a switching signal.among This As an alternative, data from driven control (DDC) methods ing a controller directly the experimental data. several submodels according to a switching signal. This seek to avoid the plant identification step by synthesizlinear systems, described by aof model that switches among a to controller directly from the experimental data. For For several submodels according to a switching signal. This ing interest is due to the ability these systems to provide seek avoid the plant identification step by synthesizinterest is due to the ability of these systems to provide ing a controller directly from the experimental data. iterative feedback tuning (Hjalmarsson et For al. several submodels according to a switching signal. This instance, instance, iterative feedback tuning (Hjalmarsson et al. interest is due to the ability of these systems to provide tractable approximations to problems in a wide range a controller directly from the experimental data. tractable approximations to of problems in a wide range ing instance, iterative feedback (Hjalmarsson et For al. [1998]), aims to minimize aa tuning composite performance criinterest is due to the ability these systems to provide [1998]), aims to minimize composite performance critractable approximations problems a widebiology range instance, iterative feedback tuning (Hjalmarsson et al. of scenarios, ranging from to instance frominsystems terion by iteratively computing the gradient of the input of scenarios, ranging from instance from systems biology [1998]), aims to minimize a composite performance critractable approximations to problems insystems aetwide range terion byaims iteratively computing the gradient of the input of scenarios, rangingtofrom instance biology (Liang et al. [2010]) dynamic visionfrom (Vidal al. [2007]). [1998]), to respect minimize a composite performance criand output with to the controller parameters. Fre(Liang et al. [2010]) dynamic vision (Vidal et al. [2007]). terion by iteratively computing the gradient of the input of scenarios, rangingto from instance from systems biology and output with respect to the controller parameters. Fre(Liang et al. [2010]) to dynamic vision (Vidal et al. [2007]). Under the assumption that the switching signal is meaterion by iteratively computing the gradient of the input Under the assumption that the switching signal is meaand output with respect to the controller parameters. Frequency domain tuning (Kammer et is on (Liang et al.assumption [2010])controllers to dynamic vision (Vidal et al. [2007]). quency domain (Kammer et al. al. [2000]) [2000]) is based basedFreon Under the that the switching is measurable, feedback guaranteed to stabilize the output withtuning respect to the controller parameters. surable, feedback controllers guaranteed tosignal stabilize the and quency domain tuning (Kammer al. [2000]) is criterion. based on spectral analysis and minimizes aa et noise rejection Under the assumption that the switching signal is measpectral analysis and minimizes noise rejection criterion. surable, feedback guaranteed to stabilize the quency domain tuning (Kammer et al. [2000]) is based on closed loop systemcontrollers under arbitrary switching have been Correlation basedand tuning (Karimi et al. [2002]) criterion. seeks to closed systemcontrollers under arbitrary switching have been analysis minimizes a noise rejection surable,loop feedback guaranteed to stabilize the spectral Correlation basedand tuning (Karimi et al. [2002]) criterion. seeks to closed loop under arbitrary switching have[2006], been proposed forsystem instance in Geromel and Colaneri spectral analysis minimizes aunder noise rejection minimize a reference criterion the assumption of proposed for instance in Geromel and Colaneri [2006], Correlation based tuning (Karimi et al. [2002]) seeks to closed loop system under arbitrary switching have[2006], been minimize a reference criterion under the assumption of proposed for instance in Geromel and Colaneri Lin and Antsaklis [2009], Blanchini et al. [2009], LiberCorrelation based tuning (Karimi et al. [2002]) seeks to Lin and Antsaklis [2009], Blanchini et al. [2009], Liberminimize a reference criterion under the assumption of independence of the reference and noise signals. Similarly, proposed forFiacchini instance in Geromel and Colaneri [2006], of the reference and noise signals. Similarly, Lin and Antsaklis [2009], Blanchini et al. [2009], Liber- independence zon [2012], and Jungers [2014] and references minimize a reference criterion under the assumption of virtual reference feedback tuning (Campi et al. [2002]) also zon [2012], Fiacchini and Jungers [2014] and references independence of the reference and noise signals. Similarly, Lin and Antsaklis [2009], et al. and [2009], Liberreference tuning (Campi et al. [2002]) also zon [2012], Fiacchini and Blanchini Jungers references therein). While very effective, all of[2014] these techniques re- virtual independence of feedback the reference and noiseJsignals. Similarly, aims to minimize a reference criterion by introducing therein). While very effective, all of these techniques revirtual reference feedback tuning (Campi et al. [2002]) also y zon [2012], Fiacchini and Jungers [2014] and references toreference minimizefeedback a reference criterion Jyyetby therein). While effective, all ofin these techniques re- aims quire a model ofvery the system. Thus, practical scenarios, VR virtual tuning (Campi al.Jintroducing [2002]) also quire a model of the system. Thus, practical scenarios, aims to minimize a reference criterion Jyy by an equivalent virtual reference cost function V VR R .. VRFT While very effective, all ofin these techniques rean equivalent virtual reference cost function Jintroducing quire a model of the system. Thus, in practical scenarios, atherein). multi-step procedure is required: (i) Firstly, identify a VR R VRFT aims to minimize a reference criterion J by introducing V y aquire multi-step procedure is required: (i) Firstly, identify a are they do require an equivalent virtual since reference function J Viterations. . VRFT a model of the system. Thus, in scenarios, methods are popular popular since theycost do not not require asuitable multi-step is required: (i)practical Firstly, identify plantprocedure and associated worst-case identification er-a methods R an equivalent virtual reference cost function J iterations. .systems VRFT methods are popular since they do not require iterations. suitable plant and associated worst-case identification erFurther, they can be extended to handle MIMO a multi-step procedure is required: (i) Firstly, identify a they can be since extended to handle MIMO systems suitable plant identification er- Further, ror bounds, (ii)and use associated additional worst-case data to refine these bounds, methods are popular they do not require iterations. Further, they can be extended to handle MIMO systems ror bounds, (ii) use additional data to refine these bounds, (Formentin et al. [2012]) and even nonlinear plants (Campi suitable plant associated worst-case identification er- (Formentin et al. [2012]) and even nonlinear plants (Campi ror bounds, (ii)and additional datathat to refine theseall bounds, and (iii) design ause robust controller stabilizes plants Further, they [2006]). can[2012]) be extended to handle MIMO and (iii) design robust controller stabilizes plants and Savaresi While these methods work systems well in (Formentin al. and even ror bounds, (ii) a use additional datathat toand refine theseall bounds, and (iii) design a robust controller that stabilizes all plants consistent with the nominal model bounds. Unforand Savaresiet [2006]). While thesenonlinear methodsplants work (Campi well in (Formentin et al. [2012]) and even nonlinear plants (Campi consistent with the nominal model and bounds. Unformany scenarios, there are still open issues that need to and Savaresi [2006]). While these methods work well in and (iii) design a robust controller that stabilizes all plants scenarios, there are still open issues that need to be be consistent with the nominal model and bounds. tunately, the identification/validation steps are farUnforfrom many and Savaresi [2006]). While these methods work well in many scenarios, there are still open issues that need to be tunately, the identification/validation steps are far from addressed. Firstly, VRFT requires an appropriate selection consistent with the nominal model and bounds. UnforFirstly, VRFT requires anissues appropriate selection tunately, identification/validation steps are far et from trivial for the switched systems (see for instance Ozay al. addressed. many scenarios, there are still open that need to be trivial for switched systems (see for instance Ozay et al. of both the reference model and the family of controllers, addressed. Firstly, VRFT requires an appropriate selection tunately, the identification/validation stepscan arebe far et from reference model and the of controllers, trivial for switched systems this (see approach for instance al. of both theFirstly, [2014, 2015]). In requires an family appropriate selection [2014, 2015]). In addition, addition, this approach canOzay be overly overly which itself is aVRFT nontrivial Secondly, in the of bothin the reference model andproblem. the family of controllers, trivial for switched systemsthe (see for instance Ozay et al. addressed. [2014, 2015]). In addition, this approach can be overly conservative, since typically identification error bounds which in itself is a nontrivial problem. Secondly, in the of both the reference model and the family of controllers, conservative, since typically the identification error bounds original formulation, closed loop internal stability is which in itself is a nontrivial problem. Secondly, inguarthe [2014, 2015]). In addition, this approach can be overly conservative, sincestep typically thetight. identification errorthat bounds provided by each are not Finally, note the original formulation, closed loop internal stability is which in itself is a nontrivial problem. Secondly, inguarthe original formulation, closed loop internal stability is guarprovided by each step are not tight. Finally, note that the anteed only for minimum phase plants. Campestrini et al. conservative, since typically the identification error bounds anteed only for minimum phase plants. Campestrini et al. provided by each stepprovided are not tight. Finally, note switched that the original formulation, closed loop internal stability is guarstability guarantees by most existing stability guarantees provided by most existing switched [2011] address the case of non minimum phase plants anteed only for minimum phase plants. Campestrini et al. provideddesign by each stepprovided arehold not only tight. Finally, note that the the [2011] address the case of non minimum phase plants stability guarantees by most existing switched control methods for the case where anteed only for minimum phase plants. Campestrini et al. control design methods hold only for the case where the through the use of tunable parameters, but only local [2011] address the case of non minimum phase plants stabilityswitches guarantees provided by most existing switched the usetheof case tunable parameters, but onlyplants local control design methods hold only formodels. the caseIn where the through system amongst the nominal principle, [2011] address of non minimum phase system switches amongst the nominal models. In principle, convergence guaranteed. note perthe is of tunableFinally, parameters, but actual only local controlmethods design methods only to formodels. the caseIn where the through convergence isuse guaranteed. Finally, note that that system switches amongst the nominal principle, these can be hold modified handle uncertainty, VperR through the use ofthetunable parameters, butJactual only local convergence is guaranteed. Finally, note that actual perthese methods can be modified to handle uncertainty, formance matches virtual reference (e.g. = J V R y V R ), system switches amongst the nominal models. In principle, formance matches the virtual reference (e.g. J = J these methods can be modified handle but uncertainty, for instance by enforcing quadratictostability, this can convergence is guaranteed. Finally, note that actual y VperR ), y V R for instance by enforcing quadratic stability, but this can and hence closed loop stability is guaranteed, only when formance matches the virtual reference (e.g. J = J ), y yonly when these methods can be modified tostability, handle uncertainty, and hence closed loop stability is guaranteed, for instance by enforcing quadratic but this can lead to very conservative results. For instance, while the VR formance matches the virtual reference (e.g. J = J ), lead to very conservative results. For instance, while the y an infinite number of data points is considered. The pracand hence closed loop stability is guaranteed, only when for instance enforcing quadratic stability, butwhile this can infinite number of data pointsisisguaranteed, considered. only The when praclead to very by conservative results. For instance, the an and hence closed loop stability  This work was supported in part by NSF grants ECCS–1404163, an infinite number of data points is considered. The pracrelevant issue guaranteeing stability only aa  lead verywas conservative For grants instance, while the tically tically relevant issueofof ofdata guaranteeing stability using using Thistowork supported in results. part by NSF ECCS–1404163, an infinite number points is considered. Theonly prac This work was tically relevant issue of guaranteeing stability using only a CNS–1646121 and CMMI–1638234; grant FA9550-12-1finite, noisy data record is still wide open. supported in part byAFOSR NSF grants ECCS–1404163, CNS–1646121 and CMMI–1638234; AFOSR grant FA9550-12-1finite, noisy data record is still wide open.  tically noisy relevant issue of guaranteeing stability using only a 0271; thewas Alert DHS Center of Excellence under Award Number Thisand work supported in part byAFOSR NSF grants ECCS–1404163, CNS–1646121 and CMMI–1638234; grant FA9550-12-1finite, data record is still wide open. 0271; and the Alert DHS Center of Excellence under Award Number 2008-ST-061-ED0001. CNS–1646121 and DHS CMMI–1638234; AFOSRunder grantAward FA9550-12-1finite, noisy data record is still wide open. 0271; and the Alert Center of Excellence Number 2008-ST-061-ED0001.

2008-ST-061-ED0001. 0271; and the Alert DHS Center of Excellence under Award Number 2008-ST-061-ED0001. 2008-ST-061-ED0001. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 592 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 IFAC 592 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 592 10.1016/j.ifacol.2018.11.140 Copyright © 2018 IFAC 592

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In order to address the issue of guaranteed stability using a finite data record, Cheng et al. [2015] recently proposed a robust optimization based approach to DDC of SISO systems. This approach is guaranteed to stabilize all systems that could have generated the observed experimental data. However at the present time, it is restricted to the SISO case, and extensions to MIMO do not seem to be trivial, due to minimality issues when considering MIMO matrix fractional descriptions. However, this method, as all the methods described above, strongly depends on the timeinvariance assumption. To the best of our knowledge, at the present time there are no DDC methods for switched systems, capable of guaranteeing closed loop stability. Motivated by these open questions, in this paper we take initial steps towards designing a robust switched DDC framework, capable of handling uncertainty while providing stability certificates. This will be accomplished by exploiting the concept of superstability (Blanchini and Sznaier [2000], Polyak and Halpern [1999], Sznaier et al. [2002]), combined with tools from robust optimization. While requiring closed loop super-stability is a stronger requirement than Schur stability, the advantage of this approach is that it allows to reduce the problem to a tractable, Linear Programming Problem. The resulting controllers are guaranteed to robustly stabilize, under arbitrary switching, all plants compatible with the observed data. In addition, due to superstability, these controllers also guarantee worst case �∞ -induced performance. The paper is organized as follows. In Section 2 we recall some background results and introduce the problem under consideration. Section 3 shows that this problem can be reduced to a Linear Program. These results are illustrated with an example in Section 4. Finally, Section 5 points out to directions for further research. 2. PRELIMINARIES 2.1 Notation and background results R, N x, X |x| x(i) X(i, j) X(:, j) X≥0 �X�∞

set of real number, set of nonnegative integers a vector in Rn , a matrix in Rm×n a vector with elements |xi | the i-th entry of x the (i, j)-th entry of X the j-th column of X X is element-wise non-negative X(i, j) ≥ 0) �∞ -norm of the matrix X ∈ Rm×n n  . �X�∞ = sup |X(i, j)| i

�x�∞

mat(x)

Next, for ease of reference, we recall a result that plays a key role in recasting the switched DDC problem into a tractable Linear Programming form. Lemma 1 (Extended Farkas Lemma, Henrion et al. . [1999]). Consider two polyhedrons of the form PN = . {x : Nx ≤ ν} and PM = {x : Mx ≤ μ}, where N, M are suitable matrices and ν, μ are column vectors. Then PN ⊆ PM if and only if there exists a non-negative matrix Y such that YN = M (1) Yν ≤ μ 2.2 Superstability Definition 1. Given a full column rank matrix T, a switched state space model of the form (2) xk+1 = Ai xk , i = 1, ..s where Ai denotes the dynamics of each subsystem is said to be superstable with respect to T if there exists some δ < 1 such that �TAi T† �∞ ≤ δ < 1, i = 1, . . . , s (3) † where T denotes a left inverse of T. The importance of superstability in the context of this paper is given by the following result Lemma 2. The origin is an asymptotically stable equilibrium point of a superstable system under any arbitrary switching sequence. Proof. Follows from noting that if (3) holds, then, regardless of the switching sequence, �Txk+1 �∞ ≤ δ�Txk �∞ < �Txk �∞ Hence �Tx�∞ is a Lyapunov function for the system. Remark 1. In the sequel, for simplicity, we will assume that T = I, in which case Def. 1 coincides with the definition in (Sznaier et al. [2002]). However, any full rank fixed T can be accommodated by the proposed technique with straightforward modifications. Note that in principle, conservatism can be reduced by optimizing over T. However, in this case, the resulting problem is no longer convex. While in principle it can be convexified using polynomial optimization techniques, at the price of increased computational complexity, such an approach is outside the scope of the present paper.

(e.g. 2.3 The Consistency Set Consider a multiple input multiple output (MIMO) switched plant described by the model (4) xik+1 = Ai xik + Bi uk + wk , i ∈ {1, . . . , s}

j=1

�∞ -norm of the vector x ∈ Rn . �x�∞ = sup |x(i)|

where xik ∈ Rn , uk ∈ Rm , and wk ∈ Rn , denote the state, input, and process noise (or equivalenty, modelling error), respectively, the superscript i indicates that the ith system was active at time k, and the switching sequence is arbitrary. Given labeled experimental data {uk , xik , xik+1 }Tk=0 , i = 1, . . . , s, and a bound � on �w�, the norm of the noise, the consistency set is defined as the set of all plants of the form (4) that could have generated the

i

vec(X)

403

matrix vectorizing operation  T vec(X) = X(:, 1)T , . . . X(:, n)T vector to matrix operation

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observed data. In the case of �∞ bounded noise, this set is a collection of polyhedrons of the form: . Pi = {Ai , Bi :�xtik +1 − Ai xtik − Bi utik �∞ (5) ≤ � for all tik ∈ Ki }

where Ki denotes the set of all times where the ith system is active. 2.4 Statement of the Problem

Consider the setup shown in Figure 1, where each partition represents a region in state space where a given subsystem is active. The goal of this paper is to design a state feedback controller that renders the origin an asymptotically stable equilibrium point for any arbitrary switching, based only on experimental input/output data and some minimal a-priori information about the system. Formally, this leads to the following problem:

?

?

c(P2)

c(P1)

⎡

⎢ . ⎢ Uk = ⎢ ⎣

01×n 01×n . . . (xik )T uTk 01×n 01×n uTk .. .. . . 01×n 01×n

... ...

0 0 .. .

... . . . uTk

and define ⎡

⎤ ⎥ ⎥ ⎥ ⎦

(8)

⎤ ⎤ ⎤ ⎡ ⎡ Xiti xtik +1 Uti1 1 . ⎢ .. ⎥ . ⎥ i . ⎢ i . ⎢ . ⎥ ⎥ Xi = ⎢ ⎣ . ⎦ , U = ⎣ .. ⎦ , ξ = ⎣ .. ⎦ Utim xtim +1 Xitim i i i

where tik ∈ Ki , k = 1, . . . , mi and mi = card(Ki ). Then (5) can be rewritten as: � � � i i � ai . n2 nm Pi = {ai ∈ R , bi ∈ R : � X U bi (9) i − ξ �∞ ≤ �} 2

? c(Pn)

Fig. 1.

by rewriting each of the polyhedrons in the consistency . . set in terms of ai = vec(ATi ) and bi = vec(BTi ). Let ⎤ ⎡ i T (xk ) 01×n . . . 0 i T ⎥ ⎢ . ⎢ 01×n (xk ) . . . 0 ⎥ (7) Xik = ⎢ . . . .. .. ⎥ ⎦ ⎣ .. ...

Setup for Switched Data Driven Control Synthesis.

Problem 1. Assume that the system under consideration consists of s LTI subsystems, each described by a model of the form (4). Given labeled experimental data {uk , xik , xik+1 }Tk=0 , i = 1, . . . , s, and a bound � on �∞ norm of the noise, find a switched state feedback gain Fi such that the resulting closed loop matrix Aσt + Bσt Fσt is asymptotically stable for any switching sequence σt ∈ {1, . . . , s}, for all pairs (Ai , Bi ) in the consistency set.

Note that Problem 1 is generically non-convex, and thus challenging to solve. In the sequel, we propose a relaxation where rather than stability, we will enforce superstability of the switched system, in the sense of Definition 1. Thus, formally we are interested in solving: Problem 2. Find a switched state feedback law Fi such that for all pairs (Ai , Bi ) consistent with the experimental data and a-priori information, the closed loop system is superstable. Remark 2. Note that from Definition 1 and Lemma 2 it follows that the problem above is equivalent to finding a set of gains Fi such that (6) �Ai + Bi Fi �∞ ≤ δ < 1, i = 1, . . . , s for all pairs (Ai , Bi ) consistent with the experimental data and a-priori information. 3. MAIN RESULTS This section contains the main result of the paper: a convex reformulation of Problem 2. To this effect, begin 594

Theorem 1. Given a matrix S ∈ Rr×n and vector d ∈ Rr , dj ≥ 0, there exist a switched feedback gain Fi such that Svec[(Ai + Bi Fi )T ] ≤ d for all pairs (Ai , Bi ) in the consistency set (5) if and only if there exist s matrices 2 Yi ∈ Rr×2nmi , Yi ≥ 0 and F i ∈ Rn ×nm such that � i � � � X Ui i Yi (10) i i = S SF −X −U and � i � ξ + �1 i Y ≤d (11) −ξ i + �1

Proof. Consider all the time instants that the ith system is active. The corresponding consistency set (9) can be rewritten as � �� � � i � X i Ui ai ξ + �1 ≤ (Pi ) −X i −U i bi −ξi + �1 Given a matrix Fi with columns fji , define ⎤ ⎡ i T (f1 ) 01×n . . . 0 ⎢0 (f1i )T . . . 0 ⎥ ⎥ i . ⎢ 1×n F =⎢ . .. .. ⎥ ⎣ .. . ... . ⎦

(12)

01×n 01×n . . . (fni )T It can be easily seen that vec((Ai + Bi Fi )T ) = ai + F i bi

Given a non-negative vector d, consider now the polyhedron obtained by imposing bounds on each element Svec[(Ai + Bi Fi )T ], that is: � � Svec[(Ai + Bi Fi )T ] j ≤ dj

(13)

In order for (13) to hold for all (ai , bi ) in the consistency set, the polyhedron Pi must be included in the polyhedron � � � � � ai � . i ≤d (14) Pcl,i = (ai , bi ) : S SF bi

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The proof follows now from Lemma 1 (Farka’s Lemma) with � i � � � X Ui N= , M = S SF i , −X i −U i � � i ξ + �1 ,μ = d ν= −ξ i + �1 Theorem 2. Problem 2 is solvable iff there exist s ma2 trices Yi ∈ Rn×2nmi ≥ 0 and F i ∈ Rn ×nm and a vector n d ∈ R , 1 > d ≥ 0 such that (10)-(11) hold for all S 2 matrices in Rn×n of the form: ⎤ ⎡ T 01×n . . . 0 s1 T ⎢ ... 0 ⎥ . ⎢ 01×n s2 ⎥ (15) S=⎢ . . .. ⎥ .. ⎣ .. ... . ⎦ 01×n 01×n . . . sTn where sj ∈ Rn is a vector with elements sj,� = ±1. Proof. For a given pair (Ai , Bi ), let acl j,� denote the ele. ments of Acl = Ai + Bi Fi . Note that � � |acl sj,� acl j,� | < 1 ⇐⇒ j,� ≤ dj < 1∀sj,� = ±1 � � (16) � � ⇐⇒ Svec[(Ai + Bi Fi )T ] j ≤ dj

405

set) Dj,� on each element of |(Ai + Bi Fi )|, consistent numerical experience shows that the relaxation works well, specially in cases where �, the bound on the process noise, is relatively small. Finally, in the next result we consider the effect of external disturbances and show that the switched controller Fi provides guaranteed worse case �∞ disturbance rejection. Theorem 4. Consider now the case where the system is subject to �∞ bounded disturbances, that is xk+1 = Ai xk + Bi uk + Ei wk (19) Then, the switched controller Fi obtained from Theorem 3 guarantees that, for every trajectory starting from xo = 0, and for any arbitrary switching sequence, �xk � ≤ . maxi �Ei �∞ �w�∞ = μ, where δ ∗ indicates the optimal value 1−δ ∗ in (18) . Proof. Define the set Sδ = {x : �x�∞ ≤ μ}. From (19) it follows that if xk ∈ Sδ , then �xk+1 �∞ ≤ δ ∗ �xk �∞ + max �Ei �∞ �w�∞ i

δ∗ ≤( + 1)max �Ei �∞ �w�∞ = μ i 1 − δ∗

Hence the set Sδ is positively invariant, so trajectories starting in its interior never leave.

The proof follows now from Theorem 1.

Remark 3. The result above shows that indeed one can design super-stabilizing data driven controllers by solving a Linear Programming problem. However, this result is mostly of theoretical interest, except for low order systems, 2 since the resulting problem has roughly O(2n ) constraints due to the need to consider all possible sign combinations when forming S. This issue is addressed in the next result, where we introduce a low complexity relaxation that works well in practice. Theorem 3. Problem 2 is solvable if there exist s matrices 2 2 Y i ∈ R2n ×2nmi , Yi ≥ 0, F i ∈ Rn ×nm and D ∈ Rn×n ≥ 0 such D1 < 1 and (10)-(11) hold with � � � � In2 ×n2 vec(DT ) S= and d = (17) −In2 ×n2 vec(DT ) Proof. If (10)-(11) hold, then from Theorem 1 it follows cl that each element acl j,� of (Ai + Bi Fi ) satisfies |aj,� | < Dj,� . Hence |(Ai + Bi Fi )|1 ≤ D1 < 1 which implies superstability. Note that conditions in the Theorem above are linear in the variables Yi , Fi , D and therefore Problem 2 reduces to the linear programming problem: min δ subject to (10),(11), (17) and: Yi ≥ 0, D ≥ 0 D1 ≤ δ1

(18)

This linear program has far fewer variables than the one obtained from Theorem 2 since it uses a single, fixed matrix S. While in principle these results are conservative, since they impose a uniform bound (over the consistency 595

4. ILLUSTRATIVE EXAMPLE In this section, we illustrate the proposed method using an example involving a MIMO system that arbitrarily switches between two modes, one stable an the second unstable. The two systems are given by the following dynamics: � � � � 0.61 0.8 0 0.0479 0.1349 A1 = −0.8 0.61 0 , B1 = 0.8509 0.9376 0 0 0.2 0.4780 0.6490 (System 1) � 0.2 0.4 0 0.9192 −0.0469 A2 = −0.4 0.2 0 , B2 = 0.2927 0.8238 0 0 0.6 −0.2409 −0.9703 (System 2) To highlight the ability of the method to handle subsystem dependent noise levels, we selected a 20% noise level for System1 and 50% for System 2. The open loop responses of each of these systems are shown in Fig. 2; �

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� −0.1063 −0.3001 0.0755 , F2 = 0.3154 −0.1284 0.6062

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with optimal value δ ∗ = 0.9282 < 1, which certifies super-stability of the closed loop system regardless of 1 With 150 experimental data points the algorithm took 8.25 seconds on an 3.4 GHz iMac, equipped with a 4 core i7 processor.

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Fig. 2. Open loop system responses. Top: System 1; Bottom: System 2 the switching sequence. The closed loop norms obtained when closing the loop with these gains around the actual plants used to generate the data are �A1 + B1 F1 �∞ = 0.8282 and �A1 + B1 F1 �∞ = 0.6992. As expected, these values are smaller than δ, since the later is the worse case value over all plants in the consistency set, not just the ground truth. Simulations showing the performance of the proposed controller are shown in figures 3-5. For illustration purposes the first two figures consider a nonswitching scenario where either system 1 or system 2 is always active. The third plot shows the response of the closed loop system, starting from a random initial condition with �x0 �∞ ≤ 10, driven by the switching sequence i = [21112112121122221112]. As shown in the plot, as expected, the states of the system converge to zero exponentially. with �xk �∞ ≤ 0.928k �x0 �∞ . Figure 6 shows the value of δ ∗ as a function of the number of experimental data points. It is worth noting that the curve shows a sharp drop at the beginning, since

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Fig. 4. Closed loop step response, System 2 as more points are added, the consistency set (and hence uncertainty) decrease. The leveling off after around 200 points indicates that at that point, uncertainty in driven by the noise and that the diameter of the consistency set cannot longer be reduced by simply considering more (noisy) data points. Finally, the disturbance rejection properties of the proposed controller are illustrated in Fig. 7. 5. CONCLUSIONS This paper considers the problem of synthesizing robust data driven controllers for switched discrete time systems. Its main result shows that if the problem is modified to enforce closed loop superstability (a stronger concept than stability), then it can be reduced to a robust optimization. Finally, exploiting Farkas’ lemma leads to a computationally efficient algorithm. To the best of our knowledge this is the first data driven paper capable of handling MIMO switching systems, while guaranteeing exponential

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Fig. 6. Optimal value versus experiment size. After approximately 200 data points, the uncertainty in the plant is dominated by the noise. closed loop stability, with a known convergence rate, for all plants compatible with the observed data and a-priori information. Further, by making the closed loop system superstable, this approach has guaranteed worse case ∞ disturbance rejection properties. Perhaps the most serious limitation of the formalism in its present form is that these results depend on the specific coordinate system chosen for the state space representation. This is due to the fact that Definition 1 is not invariant under similarity transformations. Research is currently under way to remove this limitation and to extend the techniques proposed here to the design of stabilizing (rather than super-stabilizing) controllers. Preliminary results, beyond the scope of this paper, indicate that this extension can be achieved by recasting the problem into a robust polynomial optimization form. However, in this case computational complexity becomes an issue. Finding computationally tractable alternatives remains an open problem. REFERENCES Blanchini, F., Miani, S., and Mesquine, F. (2009). A separation principle for linear switching systems and 597

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