Decentralized Control of Linear Switched Nested Systems With...

420 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 2, NO. 4, DECEMBER 2015

Decentralized Control of Linear Switched NestedSystems With �2-Induced Norm Performance

Anshuman Mishra, Cédric Langbort, and Geir E. Dullerud, Fellow, IEEE

Abstract—This paper considers a decentralized switched con-trol problem where exact conditions for controller synthesis areobtained in the form of semidefinite programming (SDP). Theformulation involves a discrete-time switched linear plant that hasa nested structure, and whose system matrices switch betweena finite number of values according to finite-state automation.The goal of this paper is to synthesize a commensurately nestedswitched controller to achieve a desired level of �2-induced normperformance. The nested structures of both plant and controllerare characterized by block lower-triangular system matrices. Forthis setup, exact conditions are provided for the existence of afinite path-dependent synthesis. These include conditions for thecompletion of scaling matrices obtained through an extendedmatrix completion lemma. When individual controller dimensionsare chosen at least as large as the plant, these conditions reduceto a set of linear matrix inequalities. The completion lemma alsoprovides an algorithm to complete closed-loop scaling matrices,leading to inequalities for controller synthesis that are solvableeither algebraically or numerically through SDP.

Index Terms—Decentralized control, H∞ control, linear matrixinequalities (LMIs), switched systems.

I. INTRODUCTION

IN THIS paper, we are interested in decentralized controlof nested systems with switched dynamics as depicted in

Fig. 1. The system matrices of the linear plant switch within afinite set, with the switching being governed by a parameter θ(t)generated by finite-state automation. The controller has accessto recent values of this parameter with finite memory. Further,the plant and controller dynamics are restricted to be nested,representing a hierarchy of subsystems with a unidirectionalflow of information amongst them. Such a nested structurealso corresponds to the system matrices having a block lowertriangular sparsity structure, which further translates to aninput–output mapping of the same sparsity structure as depictedin blocks of Fig. 1. For this setup, our goal is to stabilize theclosed-loop system while achieving a contractive-induced �2norm performance. This architecture of hierarchical systemswith underlying switched dynamics can be encountered, forinstance, in networked control systems, power systems, inter-connected vehicle formations, and economic theory [1]–[8].

Past studies on the control of discrete time switched systemsare quite diverse but mostly limited to centralized control.

Manuscript received June 13, 2014; revised December 14, 2014; acceptedJanuary 17, 2015. Date of publication April 28, 2015; date of current versionDecember 14, 2015. This work was supported by AFOSR under FA9550-10-1-0573. Recommended by Senior Editor Jeff S. Shamma and Associate EditorPablo A. Iglesias.

The authors are with Coordinated Science Laboratory, University of Illinois,Urbana, IL 61801 USA (e-mail: [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TCNS.2015.2426753

Fig. 1. Interconnection diagram showing the interaction of a controller with aplant. It does not show the disturbance input and performance output intercon-nections, and corresponding parts of the plant.

These include [9]–[13], which vary in terms of the type ofparameters switching dynamics, parameter availability to con-trollers, and performance criteria. While much of the earlierliterature primarily considers mode-dependent control wherecontroller dynamics depend only on the current mode, theauthors in [11]–[13] generalize this idea to allow the controllerto depend on a finite path of switched parameters. In [12],synthesis conditions for a finite-path-dependent controller areprovided to achieve exponential stability and disturbance atten-uation, and it forms the centralized version of the results pre-sented in this paper. The results in [12] were further extended in[13] to also allow controller access to a finite number of futureparameters as well.

In this paper, we will use the H∞-type cost criteria ofinduced �2 norm, which is also referred to as disturbanceattenuation [12] or root-mean square (rms) gain [14] in theliterature. In general, decentralized control of systems underthe H∞-type cost criterion has been a challenging problemwith a few notable results even for the nonswitched setup.Some of the prior work includes [15]–[17] where authorshave considered systems distributed over lattices/graphs andsynthesized controllers which assume the same topology as theplant. To extend the centralized synthesis scheme, these studiesrestrict the scaling matrices to be of block-diagonal structurewith separate blocks corresponding to time and spatial updates.This, however, leads only to sufficient conditions for the exis-tence of controllers. Recently, [18] considered the decentralizedcontrol of continuous-time linear time-invariant (LTI) systemswith a nested interconnection structure. Although the intercon-nection topology is more restrictive than those considered in[15]–[17], the conditions for existence of controllers are tight.

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MISHRA et al.: DECENTRALIZED CONTROL OF LINEAR SWITCHED NESTED SYSTEMS 421

A discrete-time Linear time-varying (LTV) version of theproblem considered in [18] was solved in [19] using similarideas and operator theory. Corresponding controller synthe-sis conditions are presented as infinite dimensional operatorinequalities which can be reduced to tractable LMIs in spe-cial cases, such as LTI and periodic systems. In the con-trol of such LTV systems, the controller is assumed tohave knowledge of the entire switching path. In contrast,this paper considers the control of switched systems wherethe controller has knowledge of only a finite portion ofthe switching path. Other recent studies, which considerdecentralized control of nested systems, include [20]–[23].In particular, [21]–[23] consider a H2 cost criteria and provideanalytical solutions to the problem.

This paper is organized as follows. In Section II, we providesome preliminaries describing the notation used in this paper.In Section III, we describe the switched problem under consid-eration while laying out necessary background and results re-garding switched systems. In Section IV, necessary conditionsfor the existence of path-dependent controllers are developedwhich, upon use of a new result on the completion of scalingmatrices in Section V, leads to the exact conditions presented inSection VI. In Section VII, the controller synthesis procedure isdescribed, followed by a discussion on computational complex-ity in Section VIII. Finally, a numerical example is providedin Section IX.

II. PRELIMINARIES

In this section, we describe useful notations and backgroundmaterial (derived from [12]) for switched systems. For the classof systems encountered here, existing analysis results in theform of conditions for achieving stability and performance arealso provided.

A. Notational Preliminaries

We denote the set of real numbers, non-negative, and positiveintegers by R, N0, and Z+, respectively. The n-dimensionalEuclidean space is denoted by R

n with the corresponding2-norm being | · |2. The space of n×m dimensional real valuedmatrices is denoted by R

n×m. The spaces of n-dimensionalsymmetric, positive-definite, and positive-semidefinite matricesare denoted by S

n, Sn+, and Sn+, respectively. Elements (say X)

of Sn+ and Sn+ are also often indicated by X � 0 and X � 0,

respectively. For a matrix W : WT , W †, rank(W ), Im(W ), andKer(W ) represent its transpose, pseudo-inverse, rank, imagespace, and kernel space, respectively. As a shorthand notation,we represent a block-diagonal matrix by diag(D1, . . . , Dk)

with {Di}ki=1 being its diagonal blocks. An identity matrix ofdimension n is denoted by In or simply I . For a matrix W , weuse W⊥ and W‖ to denote full-column rank matrices, satisfyingIm(W⊥) = Ker(W ), WT

⊥ W⊥ = I , Im(W‖) = Ker(W )⊥, theorthogonal complement of Ker(W ), and WT

‖ W‖ = I .We denote �n to be the space of infinite-indexed sequence of

elements as

x = (x(0), x(1), x(2), . . .) with x(t) ∈ Rn for t ∈ N0. (1)

When the dimension n is clear from context, this space issimply denoted as �. A subspace of � is the Hilbert space�n2 (or simply �2) which is equipped with the inner-product〈x, y〉 :=

∑∞t=0 x(t)

T y(t) and with norm∑∞

t=0 |x(t)|22 < ∞.For the �2 Hilbert space, we denote its norm by ‖ · ‖. The spaceof linear-bounded operators mapping Hilbert spaces �k2 to �m2is denoted by L(�k2 , �m2 ) (or simply L(�k2) when the two spacesare the same). The induced norm corresponding to this space isdenoted by ‖ · ‖.

The Schur complement formula for positive-definite matricesdescribes the following equivalence:⎡

⎣X11 X12 X13

XT12 X22 X23

XT13 XT

23 X33

⎤⎦ � 0 ⇔ X22 � 0 and

[X11 X13

XT13 X33

]−[X12

XT23

]X−1

22 [XT12 X23 ] � 0.

We will encounter several inequalities of the formWTHW � 0. To save space, we will sometimes write suchinequalities as [•]THW � 0. Also, for partitioned symmetric

matrices, say

[X1 X2

XT2 X3

], we occasionally suppress repeated

sub-blocks as

[X1 X2

· X3

].

Throughout this paper, we use M to denote the number ofdecentralized subsystems in the nested setup. We define thespace of block-lower triangular matrices of the form⎡

⎢⎢⎣H11 0 . . . 0H21 H22 0

.... . .

...HM1 HM2 . . . HMM

⎤⎥⎥⎦

by S((m1, . . . ,mM ), (k1, . . . , kM )) so that Hij ∈ Rmi×kj and

Hij = 0 for i < j. We also use the notation J = {1, . . . ,M}and J = {0, . . . ,M}.

B. Mode-Dependent Switched Systems

Let us consider a switched system

x(t+ 1) = Aθ(t)x(t) +Bθ(t)w(t)

z(t) = Cθ(t)x(t) +Dθ(t)w(t) (2)

where the system matrices depend on switching parametersθ(t) sequenced in time. Such a system whose dynamics de-pend only on the current value of the switched parameter iscalled a mode-dependent system. We assume that the switchingparameters take values from a finite set N = {1, . . . , ns} andthe switching between these values in time is governed by afinite-state automata. The parameter sequences generated bysuch an automation will be referred to as admissible sequences.We denote the set of admissible sequences of length r ∈ N0

as Ar. As an example, consider the case where switchingparameter θ(t) is governed by an automation with three statesN = {1, 2, 3} as shown in Fig. 2(a). Here, the directed edges


Fig. 2. (a) Example of switching automata. (b) Corresponding induced au-tomata for L = 1.

indicate allowed switching transitions which occur exactly onceevery time step. Thus

A1 = N , A2 = {12, 13, 23, 33, 31}A3 = {123, 131, 133, 231, 233, 312, 313, 331, 333}.

We denote a sequence of zero length as ∅ and adopt theconvention A0 = {∅}.

C. Path-Dependent Systems and Induced Switching Sequence

Consider the switched system

x(t+ 1) = AΩ(t)x(t) +BΩ(t)w(t)

z(t) = CΩ(t)x(t) +DΩ(t)w(t) (3)

whose system matrices at time t depend on a switching pathΩ(t) = (θ(t− L), . . . , θ(t)) ∈ AL+1 consisting of L+ 1 re-cent values of the switching parameters. Such a system isreferred to as a finite-path-dependent system with memory oflength L. We can view such systems as mode-dependent sys-tems by using switching paths of length L as the modes of thesystem. To describe the corresponding switching behavior, weintroduce an induced automata (as previously suggested in [11]and [13]) with state-space N := AL+1 and whose transitionsare governed by the original automata. To explain this further,consider states αa = (βa0, . . . , βaL) and αb = (βb0, . . . , βbL)of the induced automata where βai, βbi ∈ N for i = 0, . . . , L.Then, a directed edge from state αa to αb in the induced au-tomata is same as saying that βai = βb(i−1) for i = 1, . . . , L−1 and (βa0, . . . , βaL, βbL) ∈ AL+2 is a sequence generated bythe original automata.1 Admissible sequences of length r in theinduced automata are denoted by AL

r . It is not hard to verify thatelements in AL

r are equivalent to those of Ar+L for r > 0. Toexplain this, we consider a finite-path-dependent system withmemory 1, governed by the same switching automation as inFig. 2(a). The induced automation shown in Fig. 2(b) has fivestates N = A2, so the set containing admissible sequences oflength 2 is given by

A12 = {(12, 23), (13, 31), (13, 33), (23, 31), (23, 33),

(31, 12), (31, 13), (33, 31), (33, 33)} .

This is equivalent to A3 (denoted as A12 � A3).

For a sequence Φ = (α0, . . . , αr) ∈ ALr+1, there exists an

equivalent sequence (β0, . . . , βr+L) ∈ Ar+L+1. Correspond-

1This induced automata graph is also known as the De Bruijn graph and isdiscussed extensively in [24].

ingly, for r > 0, we define Φ, Φ ∈ Ar+L � ALr , Φ† ∈ N =

AL+1, and Φ� ∈ N as

Φ := (β1, . . . , βr+L) � (α1, . . . , αr)

Φ := (β0, . . . , βr+L−1) � (α0, . . . , αr−1)

Φ† := (βr, . . . , βr+L) = αr, Φ� := βr+L.

For r = 0, these definitions reduce to

Φ := (β1, . . . , βL), Φ := (β0, . . . , βL−1)

Φ† := (β0, . . . , βL), Φ� := βL.

However, unlike r > 0, in this case, AL is not equivalent toAL

0 = {∅}.When memory L = 0, which also corresponds to mode-

dependent systems, A0r coincides with Ar. For a sequence

Φ = (β0, . . . , βr) ∈ Ar+1 with r > 0, earlier definitions give

Φ := (β1, . . . , βr), Φ := (β0, . . . , βr−1), Φ† = Φ� = βr.

For r = 0, Φ = Φ = ∅ and Φ† = Φ� = β0.The above definitions depend on the type of the sequence,

determined by length r and memory L. To keep the notationsimple, symbols used for sequences (e.g., Φ, Ψ used later) donot carry this information. However, the exact set on which theyare defined will be clearly specified.

D. System Analysis

Consider the following linear time-varying (LTV) systemdynamics

x(t+ 1) = At x(t) +Bt w(t)

z(t) = Ct x(t) +Dt w(t) (4)

with x(0) = 0, x(t) ∈ Rn, w(t) ∈ R

nw, z(t) ∈ R

nz, and At,

Bt, Ct, Dt are bounded matrices of commensurate dimensionsfor t ∈ N0. Note that given w ∈ �, there is a unique solutionx ∈ �. The input to output mapping from �n

wto �n

zis denoted

by w → z. The system (4) is said to be exponentially stable, iffor w ≡ 0 and x(0) �= 0, there exist constants α > 0 and 0 <β < 1 such that |x(t)|2 ≤ αβt|x(0)|2 holds for all t ∈ N0. Inthis paper, we consider the performance criteria of a contractiveinduced �2 norm or ‖w → z‖ < 1. In this regard, we recall theLTV version of the well-known Kalman–Yakubovich–Popov(KYP) lemma (see [25]).

Lemma 1: The system (4) is exponentially stable and sat-isfies the performance criteria of ‖w → z‖ < 1 if and only ifthere exist positive constants a, b, and ε, and positive definitematrices {Xt}t∈N0

satisfying aI � Xt � bI and[Xt 00 I

]−[At Bt

Ct Dt

]T [Xt+1 00 I

] [At Bt

Ct Dt

]� εI

for each t ∈ N0. �Since switched systems that were introduced earlier are

special cases of LTV systems, the definitions of stability andperformance criteria defined before apply to such systemsas well. Analogous to the aforementioned KYP lemma, the


next lemma presents conditions for stability and performancefor switched systems; it was proved in [12, Theor. 3.3] andextended to incorporate a look-ahead horizon in [13, Theor. 26].

Lemma 2: The mode-dependent system (2) is exponentiallystable and satisfies ‖w → z‖ < 1 if and only if there existsan r ∈ N0 and a set of positive-definite matrices {XΨ}Ψ∈Ar

satisfying[XΦ 00 I

]−[AΦ�

BΦ�

CΦ�DΦ�

]T [XΦ 00 I

] [AΦ�

BΦ�

CΦ�DΦ�

]� 0

for all Φ ∈ Ar+1. �For the case of r = 0, the notation of Φ = Φ = ∅ implies

that XΦ = XΦ = X∅, or that the scaling matrices take the samevalue X∅ for any choice of switching path Φ ∈ A1. Also notethat the number of inequalities in the above lemma is finite,unlike in Lemma 1; hence, we do not need to explicitly specifythe uniform bounds for the inequalities. The following lemmais an extension of the above lemma to path-dependent systems.

Lemma 3: The finite-path-dependent system (3) with a mem-ory L∈N0 is exponentially stable and satisfies ‖w → z‖<1if and only if there exists an r ∈ N0 and a set of positive-definitematrices {XΨ}Ψ∈Ar+L

satisfying[XΦ 00 I

]−[AΦ† BΦ†

CΦ† DΦ†

]T [XΦ 00 I

] [AΦ† BΦ†

CΦ† DΦ†

]� 0

(5)

for all Φ ∈ ALr+1. �

Proof of the above lemma can be found in [12, Lemma 3.5].Note that the above lemma is immediate from Lemma 2 forr > 0, through the use of induced switching automata. For thecase of r = 0, the sufficiency part of the proof requires retracingthe proof of Lemma 2 (see [12] and [13]). For the necessitypart, one can increase r to be greater than 0, so that the aboveinequalities are satisfied for a large enough r.

Remark 4: Finite-path-dependent systems with memoryL1 ∈ N0 are also contained in the set of finite-path-dependentsystems with memory L2 > L1. Also, suppose the system in(3) with a memory L1 has positive-definite scaling matrices{XΨ}Ψ∈Ar1+L1

satisfying (5) for some r1 > 0. Then, we canalternatively choose a memory L2 = L1 + r′ and r2 = r1 − r′

for some non-negative integer r′ ≤ r1 and use the same scalingmatrices {XΨ}Ψ∈Ar2+L2

to describe the same set of inequalitiesand, hence, the same stability and performance properties.

III. SWITCHED DECENTRALIZED CONTROL PROBLEM

We now introduce the decentralized switched problem underconsideration in this paper.

A. Plant Description

For the decentralized control problem, we consider the fol-lowing mode-dependent switched plant

x(t+ 1) = Aθ(t)x(t) +Bwθ(t)w(t) +Bu

θ(t)u(t)

z(t) = Czθ(t)x(t) +Dzw

θ(t)w(t) +Duwθ(t)u(t) (6)

y(t) = Cyθ(t)x(t) +Dyw

θ(t)w(t)

with x(0) = 0. Here, w(t) ∈ Rnw

is the disturbance input,z(t) ∈ R

nzis the performance output, u(t) ∈ R

nuis the con-

trol input, and y(t) ∈ Rny

is the measurement available tothe controller. These vectors, indexed by t, further define thecorresponding elements in � similar to (1) and are denoted withthe same name x, w and z. The states, inputs, and outputs arepartitioned as

x(t) =

⎡⎢⎣

x1(t)...

xM (t)

⎤⎥⎦ , u(t) =

⎡⎢⎣

u1(t)...

uM (t)

⎤⎥⎦ , y(t) =

⎡⎢⎣

y1(t)...

yM (t)

⎤⎥⎦

where xi(t) ∈ Rni , ui(t) ∈ R

nui , and yi(t) ∈ R

nyi . Corre-

sponding sequences xi, ui, yi in � for i ∈ J are also de-fined. The dimensions satisfy n =

∑Mi=1 ni, nu =

∑Mi=1 n

ui ,

and ny =∑M

i=1 nyi . We introduce the tuple n = (n1, . . . , nM )

and similarly define nu and ny . As described in Section II-B,the switching sequence (θ(0), θ(1), . . .) is governed by a finite-state automation with θ(t) taking values in a finite set N .For the system (6), we have the following assumption whichenforces the nested structure.

Assumption 5: We assume that Aφ ∈ S(n, n), Bwφ ∈

S(n, nu), and Cyφ ∈ S(ny, n) for all φ ∈ N .

As a result, it is clear that the mappings uj → xi, xj → yi,and uj → yi are all zero operators for i < j and i, j ∈ J .

B. Synthesis Problem

For the plant (6), our goal is to design finite-dimensionalfinite-path-dependent linear controller with a block lower tri-angular sparsity structure in order to stabilize the closed-loopsystem and achieve a performance of contractive induced �2norm from disturbance w to performance output z. We use thefollowing state-space representation for a finite-path-dependentcontroller

xK(t+ 1) = AKΩ(t)x

K(t) +BKΩ(t)y(t)

u(t) = CKΩ(t)x

K(t) +DKΩ(t)y(t). (7)

For a controller with memory L, the aforementioned systemmatrices at time t depend on a switching path given by Ω(t) =(θ(t− L), . . . , θ(t)) ∈ AL+1 consisting of L+ 1 recent valuesof the plant’s switching parameter. The controller state xK(t) ∈R

nKis partitioned as [(xK

1 (t))T. . . (xK

M (t))T]T

with xKi (t) ∈

RnKi , thus satisfying nK = nK

1 + . . .+ nKM . For given con-

troller dimensions {nKi }Mi=1, our objective is to design the

aforementioned controller by determining a finite memory Land associated structured controller matrices

AKΨ ∈ S(nK , nK), BK

Ψ ∈ S(nK , ny)

CKΨ ∈ S(nu, nK), DK

Ψ ∈ S(nu, ny) (8)

for every admissible sequence Ψ ∈ AL+1. Here, we have usednK = (nK

1 , . . . , nKM ). The resulting controller has a y to u

mapping with a lower triangular sparsity structure as depictedin Fig. 1.


IV. NECESSARY CONDITIONS FOR THE

EXISTENCE OF CONTROLLER SYNTHESIS

This section is devoted to developing necessary conditionsfor the existence of finite-path-dependent synthesis. But firstwe describe the closed-loop system and define notations asso-ciated with it. We also present a lemma useful for eliminatingcontroller matrices from the closed-loop KYP inequality.

A. Closed-Loop System

While using a path-dependent controller of memory L (asdescribed in (7)) with system (6), it is clear that the closed-loopsystem is also path dependent with memory L. In particular, theclosed loop has the following dynamics:

xC(t+1) = AC

Ω(t)xC(t) +BC

Ω(t)w(t)

z(t) = CCΩ(t)x

C(t) +DCΩ(t)w(t) (9)

with xC(t) =

[x(t)xK(t)

]. At time t, the closed-loop system

matrices ACΩ(t), B

CΩ(t), C

CΩ(t) and DC

Ω(t) depend on the sameswitching sequence Ω(t) = (θ(t− L), . . . , θ(t)) ∈ AL+1 asthe controller in (7). For all Ψ ∈ AL+1, we can write the closed-loop system matrices as an affine combination of the controllermatrices as QC

Ψ :=

It is well-known that the above can be written as

QCΨ = RΨ�

+(UCΨ�

)TQK

ΨV CΨ�

(10)

with QKΨ =

[AK

Ψ BKΨ

CKΨ DK

Ψ

]representing the unknown controller

matrices, and the following defined for φ ∈ N :

The matrix QKΨ being structured, can be written as a linear

combination of unstructured ones as described by the followingrelation2

QKΨ =

M∑i=1

[EK

i−1

0 Eui−1

]QK

i,Ψ

[EK

i 00 Ey

i

]T(11)

where QKi,Ψ for each i ∈ J is an unstructured matrix of

dimension((nKi + nu

i ) + · · ·+ (nKM + nu

M ))× ((nK1 + ny

1) +

2Note that the decomposition (11) is not unique. However, the existence ofQK

Ψ implies the existence of {QKi,Ψ}i∈J and vice-versa.

· · ·+ (nKi + ny

i )). Matrices E•i and E•

i are defined below fori ∈ J

E•i =

[In•

1+...+n•i

0

], E•

i =

[0

In•i+1

+...+n•M

]

with • denoting one of K, u or y; the row dimension ofthe aforementioned matrices is n•. Note that these satisfy(E•T

i )⊥ = E•i and (E•T

i )⊥ = E•i . We can thus write (10) as

QCΨ = RΨ�

+

M∑i=1

(UCi,Ψ�

)TQK

i,ΨVCi,Ψ�

(12)

with

for φ ∈ N and i ∈ J . Note that UC1,φ = UC

φ and V CM,φ = V C

φ .The following matrices (through their image spaces) describe

the kernels of the above matrices:

which further rely on the following definitions:

Nyi,φ =

[Ny,x

i,φ

Ny,wi,φ

]=

[(Ey

i )TCy

φ (Eyi )

TDyw

φ

]⊥

Nui,φ =

[Nu,x

i,φ

Nu,zi,φ

]=

[ (Eu

i

)T (Bu

φ

)T (Eu

i

)T (Dzu

φ

)T ]⊥

with the row-dimensions of Ny,xi,φ , Ny,w

i,φ , Nu,xi,φ , Nu,z

i,φ , Nyi,φ,

and Nui,φ being n, nw, n, nz , n+ nw, and n+ nz , respectively.

Also, Ny0,φ = I and Nu

M,φ = I .With respect to Lemma 3, the closed-loop scaling matrices

are denoted by XCΨ ∈ S

n+nK

+ , defined for each Ψ ∈ Ar+L

and some appropriately chosen r ∈ N0. These matrices arepartitioned into plant and controller sections as

XCΨ =

[XΨ XGK

Ψ(XGK

Ψ

)TXK

Ψ

],(XC

Ψ

)−1=

[YΨ Y GK

Ψ(Y GKΨ

)TY KΨ

](13)

with XΨ, YΨ∈Sn+, XGK

Ψ , Y GKΨ ∈R

n×nK, and XK

Ψ , Y KΨ ∈S

nK

+ .We further define the following for i ∈ J :

Zi,Ψ :=

{XΨ−XGK

Ψ EKi

((EK

i

)TXK

ΨEKi

)−1(XGK

Ψ EKi

)T}−1

= YΨ−Y GKΨ EK

i

((EK

i

)TY KΨ EK

i

)−1(Y GKΨ EK

i

)T(14)

while noting that Z0,Ψ = YΨ and ZM,Ψ = X−1Ψ . Note that Zi,Ψ

is the (1, 1) block of the inverse of XCi,Ψ or, alternatively, Z−1

i,Ψ


is the (1, 1) block of the inverse of Y Ci,Ψ, which are defined as

XCi,Ψ : =

[XΨ XGK

Ψ EKi

·(EK

i

)TXK

Ψ EKi

]

Y Ci,Ψ : =

[YΨ Y GK

Ψ EKi

·(EK

i

)TY KΨ EK

i

]. (15)

We point out that the conditions for the existence of the con-troller presented later use {Zi,Ψ}i∈J as constituent variables. Asimilar choice was also made in [18] for the LTI problem.

B. Elimination Lemma

The following lemma is useful for obtaining the necessaryconditions for synthesis and for controller synthesis itself.

Lemma 6: Consider W ∈ Sn, matrices S0 = 0, PM+1 = 0,

{Pi}Mi=1, and {Si}Mi=1 each with column dimension n, satisfying

Ker(P1) ⊂ Ker(P2) ⊂ · · · ⊂ Ker(PM )

and full-column rank matrices {Ni}Mi=0 satisfying Im(Ni) =Ker(S0) ∩ · · · ∩Ker(Si−1) for i∈J . Then the following hold.

i) The inequality

W +

M∑i=1

(PTi QiSi + ST

i QTi Pi

)� 0 (16)

in the unstructured variables {Qi}Mi=1 has a solution if andonly if W is positive-definite on the subspaces Ker(S0) ∩· · · ∩Ker(Si) ∩Ker(Pi+1) for i ∈ J .

ii) Further, if a solution exists, {Qi}Mi=1 can be constructedby solving the following inequalities:

NTi

⎛⎝W +

M∑j=i

(PTj QjSj + ST

j QTj Pj

)⎞⎠Ni � 0 (17)

in the order i = M, . . . , 1. �The above lemma is from [26, Theor. 2], which was further

used in [18] to solve a decentralized control problem in continu-ous time. For the discrete-time setting, we present the followinglemma while making use of Lemma 6(i).

Lemma 7: Given Z ∈ Sn+, Z ∈ S

m+ , R ∈ R

n×m, and matri-ces {Ui}Mi=1 and {Vi}Mi=1 with column dimensions n and m,respectively, satisfying

Ker(U1) ⊂ Ker(U2) ⊂ · · · ⊂ Ker(UM )and Ker(V1) ⊃ Ker(V2) ⊃ · · · ⊃ Ker(VM )

the following inequality in the unstructured variables {Qi}Mi=1

Z−(R+

M∑i=1

UTi QiVi

)T

Z

(R+

M∑i=1

UTi QiVi

)�0 (18)

has a solution if and only if the following hold:[Ui+⊥ 00 Vi⊥

]T[Z−1 RRT Z

][Ui+1⊥ 0

0 Vi⊥

]�0 for i∈J . (19)

Here, we have used V0⊥ = I and UM+⊥ = I .

Proof: Using the Schur complement formula, we canwrite (18) equivalently in the form of (16) with

W =

[Z−1 RRT Z

], Pi = [Ui 0 ] and Si = [ 0 Vi ].

Further, Pi⊥ =

[Ui⊥ 00 I

], Si⊥ =

[I 00 Vi⊥

]and[

Pi+1

Si

]⊥=

[Ui+1 00 Vi

]⊥=

[Ui+⊥ 00 Vi⊥

]

whose columns also form a basis of the space Ker(Si) ∩Ker(Pi+1). Having the above definitions in place, we can useLemma 6 to show the equivalence between (18) and (19). �

C. Necessary Conditions

In this section, we develop conditions in the form of LMIswhich are necessary for the existence of a decentralized con-troller.

Lemma 8: Consider the system (6) along with the struc-tural description in Assumption 5. There exists a finite-path-dependent controller (7) structured as (8) which stabilizes thissystem and achieves performance ‖w → z‖ < 1 if and only ifthere exists an L ∈ N0 and positive-definite {XC

Ψ}Ψ∈ALsuch

that the corresponding {Zi,Ψ}i∈J ,Ψ∈AL(defined by (13) and

(14)) satisfy

Proof: (=⇒) Given a finite-path-dependent controller

with memory L′ as QKΨ =

[AK

Ψ BKΨ

CKΨ DK

Ψ

]for Ψ ∈ AL′+1 that

stabilizes the plant and achieves a contractive-induced �2 normperformance, we can construct the closed-loop system withmemory L′ using (9) and (10). Since the closed loop is stableand satisfies ‖w → z‖ < 1, using Lemma 3, we know thatthere exist an r ∈ N0 and positive-definite scaling matrices{XC

Ψ}Ψ∈Ar+L′ satisfying[XC

Φ 00 I

]−(QC

Φ†

)T[XC

Φ0

0 I

]QC

Φ† � 0 (21)

for all Φ ∈ AL′r+1. Substituting expansion (12) into the above,

we obtain a set of inequalities in unstructured controller vari-ables {QK

i,Ψ}i∈J ,Ψ∈AL′+1

in addition to the scaling matrices.

We next eliminate these controller matrices from the aboveinequalities using Lemma 7. This, however, can be done onlyfor the case of r = 0; this is because the unknown matrix QK

i,Ψ

for each Ψ ∈ AL′+1 appears in multiple inequalities of (21),which makes direct elimination using Lemma 7 not feasiblefor r > 0. So we extend the controller/closed-loop memory toL = L′ + r (see Remark 4). Now applying Lemma 7, we knowthat (21) implies the existence of L ∈ N0 and positive-definite


{XCΨ}Ψ∈AL

such that the following is satisfied for all i ∈ J and

Φ ∈ AL1 � AL+1

Applying permutations to the above inequality, it can be shownto be equivalent to

[•]T

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

YΦ 0 Y GKΦ

AΦ�Bw

Φ�0

0 I 0 CzΦ�

DzwΦ�

0(Y GKΦ

)T0 Y K

Φ0 0 0

(AΦ�)T

(Cz

Φ�

)T0 XΦ 0 XGK

Φ(Bw

Φ�

)T (Dzw

Φ�

)T0 0 I 0

0 0 0(XGK

Φ

)T

0 XKΦ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎣

Nu,xi,Φ�

0 0 0

Nu,zi,Φ�

0 0 0

0 EKi 0 0

0 0 Ny,xi,Φ�

0

0 0 Ny,wi,Φ�

0

0 0 0 EKi

⎤⎥⎥⎥⎥⎥⎥⎦� 0.

Using the Schur complement formula twice, followed by rela-tions defined in (14), the above inequality (hence also (22)) canbe shown to be equivalent to (20).(⇐=) The proof follows the same steps as in the converse

direction but in the reverse order. Note that the step involvingthe use of elimination lemma leading to inequality (21) provesthe existence of a controller of memory L. �

The above lemma, provides the exact condition for theexistence of the controller in terms of the existence of anL and {XC

Ψ}Ψ∈ALsatisfying (20). However, inequalities (20)

in {Zi,Ψ}i∈J ,Ψ∈ALby themselves present only a necessary

condition for the existence of the controller, because the ex-istence of {Zi,Ψ}i∈J does not directly imply the existence ofa XC

Ψ . Additional conditions that ensure sufficiency will bedeveloped in the next section. Also, the inequalities (20) are notlinear in {Zi,Ψ}i∈J ,Ψ∈AL

. Toward the goal of obtaining linearinequalities, the next lemma defines a factorization which wasoriginally performed in [18] for a similar context.

Lemma 9: For a symmetric matrix X =

[X1 X2

XT2 X3

]with

invertible X1 ∈ Sm1 , X2 ∈ R

m1×m2 and X3 ∈ Sm2 , we can

define the triple {Za, Zb, Zc} with Za ∈ Sm1 , Zb ∈ R

m1×m2 ,and Zc ∈ S

m2 , related to X by the following bijective mapping:

Za = X−11 , Zb = −X−1

1 X2, Zc = X3 −XT2 X

−11 X2.

The triple then defines the following unique factorization:

X =

[I 0

−(Zb)T

Zc

] [Za Zb

0 I

]−1

. (23)

Further, X � 0 if and only if Za � 0 and Zc � 0. �

In view of this lemma, for positive-definite {Zi,Ψ}i∈J ,Ψ∈AL,

we define the following associated matrices:

Zai,Ψ :=

(ET

i Zi,ΨEi

)−1, Zb

i,Ψ := −Zai,Ψ

(ET

i Zi,ΨEi

),

Zci,Ψ := ET

i Zi,ΨEi−(ET

i Zi,ΨEi

)T(ET

i Zi,ΨEi

)−1ET

i Zi,ΨEi

(24)

for i ∈ J and Ψ ∈ AL. Note that Zc0,Ψ = YΨ and Za

M,Ψ =

XΨ, while Za0,Ψ, Zb

0,Ψ, ZbM,Ψ, and Zc

M,Ψ have at least one oftheir dimensions as zero. Since Zi,Ψ ∈ S

n+, using the above

relations, it can be verified that Zai,Ψ ∈ S

n1+...+ni+ , Zb

i,Ψ ∈R

(n1+...+ni)×(ni+1+...+nM ), and Zci,Ψ ∈ S

ni+1+...+nM

+ for i ∈J . These matrices define the following factorization similarto (23):

Zi,Ψ =Zli,Ψ

(Zui,Ψ

)−1=

(Zui,Ψ

)−T (Zli,Ψ

)T(25)

with Zli,Ψ =

[I 0

−(Zbi,Ψ

)TZci,Ψ

]and Zu

i,Ψ=

[Zai,Ψ Zb

i,Ψ

0 I

]. (26)

Note that Zli,Ψ and Zu

i,Ψ are invertible due to the positive-definiteness of Zc

i,Ψ and Zai,Ψ, respectively.

We now use the factorization in (25) and correspondingchange of variables to convert the inequalities in Lemma 8 tobe linear in the new variables.

Lemma 10: Given positive-definite matrices {XCΨ}Ψ∈AL

, de-fine associated{Zi,Ψ}i∈J ,Ψ∈AL

and{Zai,Ψ, Z

bi,Ψ, Z

ci,Ψ}i∈J ,Ψ∈AL

using (14) and (24). Then, the inequality (20) is equivalentto the following inequalities linear in variables {Za

i,Ψ, Zbi,Ψ,

Zci,Ψ}i∈J ,Ψ∈AL

:

�

Remark 11: The above inequalities are linear in the variablesdue to the following simplifications: (Zu

i,Ψ)TZl

i,Ψ = (Zli,Ψ)

T

Zui,Ψ =

[Zai,Ψ 00 Zc

i,Ψ

]for Ψ ∈ AL and (Zu

i,Φ)TAΦ�

Zli,Φ =[

Zai,Φ

A11i,Φ�

0

(Zbi,Φ

)TA11

i,Φ�+A21

i,Φ�−A22

i,Φ�(Zb

i,Φ)T

A22i,Φ�

Zci,Φ

]for Φ ∈

AL+1 with A11i,Φ�

= ETi AΦ�

Ei, A21i,Φ�

= ETi AΦ�

Ei, and

A22i,Φ�

= ETi AΦ�

Ei.Proof of Lemma 10: Using Lemma 20 in the Appendix,

we have the inequality (20), which is equivalent to the followingfor all i ∈ J and Φ ∈ AL+1

((Wi,Φ�

Si,Φ)⊥)T

STi,ΦHi,ΦSi,Φ (Wi,Φ�

Si,Φ)⊥ � 0


with , Si,Φ=diag(Zui,Φ, I,Zl

i,Φ, I)

and Wi,Φ� =

[(Eu

i )T(Bu

Φ�)T (Eu

i )T(Dzu

Φ�)T 0 0

0 0 (Eyi )

TCyΦ�

(Eyi )

TDywΦ�

].

Using the relations (Eyi )

TCy

Φ�Zli,Φ = (Ey

i )TCy

Φ�and (Eu

i )T

(BuΦ�

)TZui,Φ

= (Eui )

TBu T

Φ�, we have Wi,Φ�

Si,Φ = Wi,Φ�.

Further, along with (25), the above inequality leads to in-equality (27). �

V. COMPLETION OF SCALING MATRICES

In this section, we discuss conditions required for com-pletion of the closed-loop scaling matrices {XC

Ψ}Ψ∈ALusing

{Zi,Ψ}i∈J ,Ψ∈AL, which in the next section will enable us to

ensure sufficiency for existence of controller synthesis. Further,in the process we also provide steps involved in efficientlyperforming such a completion, which is helpful for controllersynthesis presented in Section VII. But first we present thefollowing extension of a well known matrix completion lemma(see, for example, [27, Lemma 7.9]), when the dimensions ofthe know submatrices need not be the same.

Lemma 12: Given matrices R11 ∈ Sn+ and S1 =[

S11 S12

ST12 S22

]∈ S

n+m+ such that S11 ∈ S

n+. Then, for a positive

integer nK , there exists matrices R12 ∈ Rn×m, R22 ∈ S

m+ ,

R13, S13 ∈ Rn×nK

, R23, S23 ∈ Rm×nK

, and R33, S33 ∈ SnK

+

satisfying

R :=

⎡⎣R11 R12 R13

RT12 R22 R23

RT13 RT

23 R33

⎤⎦ � 0

and S := R−1 =

⎡⎣S11 S12 S13

ST12 S22 S23

ST13 ST

23 S33

⎤⎦

if and only if[R11 II S−1

11

]� 0 and rank

[R11 II S−1

11

]≤ n+ nK (28)

with S11 = (S11 − S12S−122 S

T12)

−1. The above rank condition is

always satisfied for nK ≥ n.Proof: Let us define the matrices

R1 =

[R11 R12

RT12 R22

], R2 =

[R13

R23

], R3 = R33

S12 ∈ Rn×m and S22 ∈ S

m+ so that

[S11 S12

ST12 S22

]= S−1

1 .

Since S1 � 0 and due to the following relation:[R1 II S1

]=

[I S−1

1

0 I

] [R1 − S−1

1 00 S1

] [I 0

S−11 I

]

we have [R1 II S1

]� 0 ⇔ R1 − S−1

1 � 0 (29)

and

rank

[R1 II S1

]= n+m+ rank

(R1 − S−1

1

). (30)

Also note the following expansion:

R1−S−11 =

[R11−

(S11−S12S

−122 S

T12

)−1R12−S12

RT12 − ST

12 R22 − S22

]. (31)

We now provide the main arguments of the proof.(=⇒) From Lemma [27, Lemma 7.9], we know that R �

0 and S � 0 implies

[R1 II S1

]� 0 and rank

[R1 II S1

]≤

n+m+ nK . Using (29) and (31), this further implies

R1 − S−11 � 0 ⇒ R11 −

(S11 − S12S

−122 S

T12

)−1 � 0.

Using a Schur complement argument again, the above is thesame as the matrix inequality in (28). Using (30) and (31), wehave rank(R11−(S11−S12S

−122 S

T12)

−1)≤ rank(R1−S−11 )≤nK.

Using a property similar to (30), we can arrive at the rankcondition in (28) from above.(⇐=) This part of the proof is constructive. If we assume

that R1 is known completely, then by combining Lemma [27,Lemma 7.9] with (29)–(31) we know that the matrices R and Scan be completed if and only if

R1 − S−11 � 0 and rank

(R1 − S−1

1

)≤ nK .

So we can instead focus on the problem of completing the ma-trix R1 which satisfies the above conditions. The expansion (31)suggests that choosing R12 = S12 and R22 = S22 would resultin R1−S−1

1 =diag(R11−S11, 0). From (28), it is clear that theabove conditions are satisfied. Thereafter, we can complete theremaining blocks ofR by followingthe steps in [27, Lemma 7.9]:choose R3 = I and R2 such that R1 − S−1

1 = R2RT2 . This is

the same as setting R33 = I , R23 = 0 and choosing R13 suchthat R11 − S11 = R13R

T13. Finally, obtain the unknown blocks

of S by inverting the constructed R. �Lemma 13: Given positive-definite matrices{Zi,Ψ}i∈J ,Ψ∈AL

,

we can construct positive-definite {XCΨ}Ψ∈AL

satisfying (13)and (14) if and only if[

Z−1i,Ψ II Zi−1,Ψ

]�0, rank

[Z−1i,Ψ II Zi−1,Ψ

]≤n+nK

i (32)

for all i ∈ J and Ψ ∈ AL. Further, the above rank conditionsare always satisfied for nK

i ≥ n.Proof: We will use matrices {Zi,Ψ}i∈J to construct XC

Ψ

for each Ψ ∈ AL, as shown in the following steps

• First, we construct Y C1,Ψ (defined in (15)) using Z0,Ψ =

YΨ and Z1,Ψ. We do this pointwise using Lemma[27, Lemma 7.9], which yields the following condition forcompletion:[Z−11,Ψ II Z0,Ψ

]� 0, rank

[Z−11,Ψ II Z0,Ψ

]≤ n+ nK

1

for all Ψ ∈ AL. A possible construction can also be foundin [27, Lemma 7.9].


• We construct Y Ci,Ψ in a recursive manner in the order i =

2, . . . ,M . For this, we use Lemma 12 with R11 = Z−1i,Ψ

and S1 = Y Ci−1,Ψ to complete the matrix S = Y C

i,Ψ � 0.This can be done if and only if the following conditionsare satisfied:[Z−1i,Ψ II Zi−1,Ψ

]� 0, rank

[Z−1i,Ψ II Zi−1,Ψ

]≤ n+ nK

i .

Note the use of relation (14) for index i− 1 while usingLemma 12. A possible construction is given in the proofof Lemma 12.

After performing the above steps, we are left with Y CM,Ψ

which is the same as Y CΨ for Ψ ∈ AL. Since the above steps

use “if and only if” arguments, the converse direction of theproof also holds. �

VI. EXACT CONDITIONS FOR THE EXISTENCE OF

CONTROLLER SYNTHESIS

We now present the main existence result of this paper.Theorem 14: Consider the mode-dependent system (6) along

with the structural description in Assumption 5. There exists asynthesis of a finite-path-dependent controller (7) which

i) is structured as (8);ii) has dimensions {nK

i }Mi=1;iii) stabilizes the plant;iv) achieves closed-loop performance ‖w → z‖ < 1

if and only if there exists an L ∈ N0 and matrices{Za

i,Ψ, Zbi,Ψ, Z


satisfying the following:

Zai,Ψ � 0, Zc

i,Ψ � 0 for all i ∈ J ,Ψ ∈ AL (33a)

[ (Zui,Ψ

)TZli,Ψ

(Zli,Ψ

)TZui−1,Ψ(

Zui−1,Ψ

)TZli,Ψ

(Zui−1,Ψ

)TZli−1,Ψ

]� 0 (33c)

and rank

[ (Zui,Ψ

)TZli,Ψ

(Zli,Ψ

)TZui−1,Ψ(

Zui−1,Ψ

)TZli,Ψ

(Zui−1,Ψ

)TZli−1,Ψ

]≤n+ nK

i

for all Ψ ∈ AL and i ∈ J (33d)

where Zli,Ψ and Zu

i,Ψ are defined using Zai,Ψ, Zb

i,Ψ, and Zci,Ψ as

in (26). Further, the rank conditions above are always satisfiedwhen nK

i ≥ n, leaving us with LMIs (33a)–(33c). �We have already verified in Remark 11 that all elements in

the above inequalities are affine in constituent variables. Theonly additional term encountered here is(Zli,Ψ

)TZui−1,Ψ = diag

(Zai−1,Ψ, Ini

, Zci,Ψ

)+

[0 Zb

i−1,Ψ

0 0

]−[0 Zb

i,Ψ

0 0

]

which is also affine.Proof of Theorem 14: (⇐=) Let there be {Za

i,Ψ, Zbi,Ψ,

Zci,Ψ}i∈J ,Ψ∈AL

satisfying (33a)–(33d). We can then userelations in (24) to obtain corresponding positive-definite

matrices {Zi,Ψ}i∈J ,Ψ∈AL. Now using definitions (26) and the

transformation

[Zli,Ψ 00 Zu

i−1,Ψ

]T [Z−1i,Ψ II Zi−1,Ψ

] [Zli,Ψ 00 Zu

i−1,Ψ

]

=

[ (Zui,Ψ

)TZli,Ψ

(Zli,Ψ

)TZui−1,Ψ(

Zui−1,Ψ

)TZli,Ψ

(Zui−1,Ψ

)TZli−1,Ψ

]

it is clear that (33c) and (33d) imply that Zi,Ψ and Zi−1,Ψ

satisfy (32) for each i ∈ J and Ψ ∈ AL. Thus, Lemma 13 canbe applied to construct XC

Ψ satisfying (13) and (14). Now usingLemma 10 along with (33b), we know that the inequalities in(20) are satisfied. Finally, we can apply Lemma 8 to argue theexistence of a desired controller.(=⇒) This part of the proof retraces the above steps in

the backwards direction. However, Lemma 13 is not applieddirectly, but through steps contained in it. �

VII. CONTROLLER SYNTHESIS

In this section, we discuss the decentralized controller syn-thesis using the scaling matrices obtained earlier. We first startwith the following lemma motivated by [28], [29, Lemma 5.2],applicable for centralized controller synthesis.

Lemma 15: Given Z ∈ Sn+, Z ∈ S

m+ , R ∈ R

n×m, and matri-ces U and V with column dimensions n and m, respectively,satisfying

V T⊥ (Z −RT ZR)V⊥ � 0 and

UT⊥ (Z−1 −RZ−1RT )U⊥ � 0 (34)

we can construct Q, satisfying the inequality

[Z−1 R+ UTQV· Z

]� 0 (35)

as

Q =(UT‖ UT

)† (−WT

23 +WT13W

−111 W12

)(V V‖)

† (36)

where Wij := HTi WHj for i, j ∈ {1, 2, 3} with

W =

[Z−1 RRT Z

], H1 =

[U⊥ 00 V⊥

], H2 =

[0V‖

],

H3 =

[U‖0

]

being commensurately partitioned matrices. �Proof of the above lemma uses ideas similar to Lemma 3.1.

Note that in the above lemma, the matrices U⊥, U‖, V⊥, V‖ canbe computed using the singular value decompositions (SVD) ofU and V . In this case, the pseudo-inverses in (36) can be writtendirectly by inspection.


The next theorem presents an algebraic method for theconstruction of a finite-path-dependent controller with memoryL and dimensions nK

i = n for i ∈ J . But first let us considerthe following alternative expansion of the closed-loop matricessimilar to (10):

QCΨ = RΨ�

+

M∑i=1

(UCi,Ψ�

)TQK

i,ΨVCi,Ψ�

(37)

with

Note that QKi,Ψ consists of the ith block columns of lower-

triangular parts of AKΨ , BK

Ψ , CKΨ , and DK

Ψ . In (10), the

controller was decomposed into {QKi,Ψ}

M

i=1containing redun-

dancies, which did not affect the existence conditions as weeliminated these controller matrices. However, for synthesis,we choose the above decomposition, which eliminates suchredundancies and keeps the number of variables to a minimum.For V C defined above, we can verify the following for i ∈ J :

Ker(V C1,Ψ�

)∩ · · · ∩Ker

(V Ci,Ψ�

)= Ker

(V Ci,Ψ�

). (38)

Theorem 16: Given matrices {Zai,Ψ, Z

bi,Ψ, Z


,which satisfy existence conditions (33a)–(33c), corresponding{XC

Ψ}Ψ∈ALrelated to (13), (14) and (24) can be used to obtain

the following LMI in (39), shown at bottom of the page, invariable QK

i,Φ for each Φ ∈ AL+1, and solved in the order i =M, . . . , 1. Further, this can be accomplished point-wise for eachi ∈ J and Φ ∈ AL+1 using (36) in Lemma 15 by choosing

Q = QKi,Φ, U = UC

i,Φ�, V = V C

i,Φ�

(V Ci−1,Φ�

)⊥

Z=diag(XC

Φ , I), Z=

(V Ci−1,Φ�

)T⊥ diag

(XC

Φ , I)(

V Ci−1,Φ�

)⊥

R =

⎛⎝RΦ�

+

M∑j=i+1

(UCj,Φ�

)TQK

j,ΦVCj,Φ�

⎞⎠(

V Ci−1,Φ�

)⊥ . (40)

Proof: With {Zai,Ψ, Z

bi,Ψ, Z


, we can constructscaling matrices {XC

Ψ}Ψ∈ALusing steps in the proof of

Theorem 14. Now use Lemma 6 with the following choice:

W =

⎡⎣ diag

((XC

Φ

)−1, I)

RΦ�

RTΦ�

diag(XC

Φ , I)⎤⎦

Qi = QKi,Φ, Pi = [UC

i,Φ�0 ] and Si = [ 0 V C

i,Φ�] .

so that corresponding inequality (16) is the same as the KYP-type inequality (5) for the closed loop and is already known tohold for some choice of the controller from Theorem 14. Thus,Lemma 6(i), along with (38), implies that (22) holds. The abovedefinitions along with (38) yield

Ni =

⎡⎣S1

...Si

⎤⎦⊥

=

⎡⎢⎣0 V C

1,Φ�

......

0 V Ci,Φ�

⎤⎥⎦⊥

=

[I 00

(V Ci,Φ�

)⊥

].

corresponding to Lemma 6, which further leads to (39) usinginequality (17) in Lemma 6(ii).

The use of Lemma 15 with the choice (40) to solve for QKi,Φ

in (39) requires us to show that corresponding inequalities (34)are satisfied. This is indeed true, because the inequalities in (34)for i = M correspond to (22) with i = M and i = M − 1. Forany other i = k, inequalities in (34) correspond to (22) withi = k − 1 and (39) with i = k + 1. Note that in intermediatesteps, we use the Schur complement formula, relation (38), andthe following property obtained using Lemma 19

Im(V Ck−1,Φ�

(V Ck,Φ�

(V Ck−1,Φ�

)⊥

)⊥

)= Im

((V Ck,Φ�

)⊥

).

�The above theorem provides the steps involved in obtaining

the controller {QKi,Φ}i∈J ,Φ∈AL+1

from the solution of the exis-

tence LMIs in (33). The controller in the originally described

form QKΦ =

[AK

Φ BKΦ

CKΦ DK

Φ

]structured as (8) can then be recov-

ered as follows for all Φ ∈ AL+1:

QKΦ =

M∑i=1

[EK

i−1

0 Eui−1

]QK

i,Φ

[eKi 00 eyi

]T.

Remark 17: An alternative to the synthesis procedure de-scribed before is to solve the following LMI in the structuredcontroller matrices QK

Φ point-wise for Φ ∈ AL+1

⎡⎣ diag

((XC

Φ

)−1, I)

RΦ�+(UCΦ�

)TQK

Φ V CΦ�(

RΦ�+(UCΦ�

)TQK

Φ V CΦ�

)T

diag(XC

Φ , I)

⎤⎦�0.

(41)

⎡⎣ diag

((XC

Φ

)−1, I) (

RΦ�+∑M

j=i

(UCj,Φ�

)TQK

j,ΦVCj,Φ�

) (V Ci−1,Φ�

)⊥

·(V Ci−1,Φ�

)T⊥ diag

(XC

Φ , I) (

V Ci−1,Φ�

)⊥

⎤⎦ � 0 (39)


Remark 18: If a closed-loop performance of ‖w → z‖ < γis sought, Theorem 14 can be updated to have Cz

φ, Cyφ, Dzw

φ ,Dzu

φ and Dywφ scaled by (1/γ) for all φ ∈ N . The controller

obtained for this modified system using the procedure above,can be used to find the desired controller by scaling BK

Ψ andDK

Ψ with (1/γ) for all Ψ.

VIII. COMPUTATIONAL COMPLEXITY

For simplicity, let us assume that all subsystems have thesame dimension ni = a for all i ∈ J , and that all inputs andoutputs have a single channel, that is, nu

i = nwi = ny

i = nzi = 1

for all i ∈ J . This results in nKi = n = Ma for i ∈ J . In this

case, we can make the following remarks regarding complexityof the various LMIs derived so far.

Existence LMIs: The coupled LMIs in (33) have a total of|AL|(M + 1)(Ma+ 1)(Ma+ 2)/2 variables and combinedLMI row dimension of, at most, 2|AL+1|(Ma+ 1)(M + 1) +|AL|Ma(3M + 1).

Synthesis LMIs (39) Solved Numerically: There are Mstages, with stage k having |AL+1| uncoupled LMIs, each withk(Ma+ 1)2 variables and LMI row dimension of, at most,M2a+ (kM +M + k)a+ 1.

Synthesis LMIs (41) Solved Numerically: Single stage with|AL+1| uncoupled LMIs, each with M(M + 1)(Ma+ 1)2/2variables and LMI row dimension of, at most,2(M2a+Ma+1).

The variable and constraint dimensions of the aforemen-tioned LMIs depend on |AL|, which itself can grow combina-torially with L, depending on the structure of the underlyingautomation. However, for the synthesis LMIs (39), Theorem 16provides a roundabout around such computational complexi-ties, by providing an algebraic scheme whose complexity (as-sociated with SVD computation, matrix inversion, etc.) is onlya fraction of the cost involved in solving the LMIs numerically.

IX. EXAMPLE

Let us consider a two-player example with dynamics (6)and 3-mode switching automation as shown in Fig. 2(a). Thecorresponding system matrices are chosen as

A1 = A2 =

[1.4 00.2 1.4

], A3 =

[0.7 00.2 0.7

]

Bu1 = Bu

2 =

[0 00 1

], Bu

3 =

[1 00 0

]

Cy1 = Cy

2 = diag(1, 0), Cy3 = I2

Dzu1 = Dzu

2 = [ 0 1 ] , Dzu3 = [ 4 0 ]

and the following defined for φ ∈ {1, 2, 3}:

Bwφ =

[11

], Dyw

φ =

[01

], Cz

φ = [ 0.5 2 ] , Dzwφ = 0.5.

Here, we have chosen dimensions n1 = n2 = nu1 = nu

2 =ny1 = ny

2 = nz = nw = 1.

For different memory lengths, the above system was ex-amined with nK

1 = nK2 = 2, that is, the total controller state

dimension of nK = 4. Using a bisection algorithm, the smallestperformance level γ (see Remark 18) satisfying conditions inin Theorem 14 was found. The numerical solution of existenceLMIs was obtained using the CVX toolbox [30] with SDPT3solver, while the synthesis LMIs were solved algebraically. Thevalues thus obtained are tabulated below along with the cor-responding performance obtained for a centralized controller.The centralized controller was synthesized following [12] witha controller state dimension of 2.

Memory 0 1 2 3 4 5

Decentralized ∞ 5.468 3.663 3.606 3.604 3.604Centralized ∞ 5.461 3.634 3.561 3.561 3.561

For zero memory length, the system is not stabilizable, resultingin infinite induced norm. For the above example, there is verylittle difference between the performance of centralized anddecentralized control.

Here, changing Cy3 to diag(0, 1) does not affect the cen-

tralized performance. However, the decentralized performancegain increases to 13.278 for L = 1, . . . , 5.

Sample impulse response trajectories for L = 2 are shown inFig. 3.

Note that the existing LMIs have 18 |AL| variables and aconstraint dimension of 18 |AL+1|+ 14 |AL|, with |A1| = 3,|A2| = 5, |A3| = 9, |A4| = 17, |A5| = 31, and |A6| = 57.

X. POSSIBLE VARIATIONS AND CONCLUSION

A. Nested LTI Systems

For a linear time invariant (LTI) formulation, we can obtainexistence and synthesis results similar to Theorem 14 andTheorem 16 by simply choosing an automation with one ele-ment (ns = 1) having a self-loop. As a result, for any memorylength L, there exists only one sequence in AL+1 implyingthat the controller is time-invariant and there is a single scalingmatrix (in fact we can ignore the switching subscripts Ψ,Φaltogether). Since the size L doesn’t play any role, we cansimply choose L = 0 in Theorem 14 and adopt the sameconditions.

Note that suchanLTIresultwaspresented in [19, Theorem 10],however it was obtained from the LTV version of the resultsdescribed here.

B. Extensions

We note that the synthesis conditions and procedure pre-sented here can be extended to the more general setting ofnonregular switching sequences, and also systems with finitelook-ahead horizon (i.e., controller has knowledge of futuremodes for a predefined horizon). For background on thesetopics, see [13] in the context of centralized control.

Extensions to control of Markovian jump linear systemswhere the switching sequence is generated by a Markov chain


Fig. 3. Plot on the left shows a sample switching sequence. Plot on the right shows corresponding impulse response performance output z(t) for the decentralizedcontroller when Cy

3 = I2 (solid), centralized controller when Cy3 = I2 (dashed), and decentralized controller when Cy

3 = diag(0, 1) (dot-dashed).

instead of an automation, to achieve almost sure performance,can also be achieved (see [12, Sec. 4]).

C. Conclusion

We presented the exact conditions for synthesis of nestedfinite-path-dependent controllers for a nested mode-dependentsystem under �2-induced norm performance criteria. To ourknowledge, this is the first such exploration involving de-centralized control of a switched system under an H∞-typeperformance criteria. It can be noted that solving a decentral-ized control problem with controllers having access to nestedparameters, instead of a common parameter as studied here,appears challenging and is a direction of future study.

APPENDIX

We present a couple of useful lemmas here.Lemma 19: Consider matrices W and P with identical

column dimensions. Define V such that Im(V ) = Ker(W ) ∩Ker(P ). Then Im(V ) = Im(P⊥(WP⊥)⊥).

Proof: First we prove Im(V ) ⊂ Im(P⊥(WP⊥)⊥). Con-sider nonzero x ∈ Im(V ), this implies x ∈ Ker(W ) and x ∈Ker(P ) = Im(P⊥). Thus there exists a nonzero z such that x =P⊥z. Since Wx = WP⊥z = 0 we must have z ∈ Ker(WP⊥).Thus x ∈ Im(P⊥(WP⊥)⊥).

Now we prove Im(P⊥(WP⊥)⊥)⊂ Im(V ). Consider nonzerox ∈ Im(P⊥(WP⊥)⊥). Clearly x ∈ Im(P⊥) = Ker(P ). Alsothere exists nonzero z such that x = P⊥(WP⊥)⊥z. ClearlyWx = WP⊥(WP⊥)⊥z = 0. We have thus proved that x isan element of both Ker(W ) and Ker(P ) implying thatx ∈ Im(V ). �

Lemma 20: Consider W ∈ Rm×k, H ∈ R

k×k and S ∈R

k×k with H being symmetric and S being invertible. Then,we have

WT⊥ HW⊥ � 0 if and only if (WS)T⊥(S

THS)(WS)⊥ � 0.

�The proof is immediate from Lemma 19 by setting P = 0,

V = W⊥ and P⊥ = S.

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Anshuman Mishra received the B.Tech. degree from the Department of Elec-trical Engineering, Indian Institute of Technology, Kharagpur, India, in 2007,the M.S. degree in electrical engineering from the Northeastern University,Boston, MA, USA, in 2009, and the Ph.D. degree in mechanical engineeringfrom University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 2014.

Currently, he is an Algorithm Engineer at Apple Inc., Cupertino, CA, USA.His interests broadly include control theory, dynamical modeling, game theory,and signal processing.

Cédric Langbort received the Ph.D. degree in Theoretical & Applied Mechan-ics from Cornell University, Ithaca, NY, USA, in 2005.

Currently, is an Associate Professor of Aerospace Engineering at the Univer-sity of Illinois at Urbana-Champaign (UIUC), Champaign, IL, USA, where heis also affiliated with the Decision & Control Group at the Coordinated ScienceLaboratory (CSL). Prior to joining UIUC in 2006, he studied at the ÉcoleNationale Supérieure de l’Aéronautique et de l’Espace-Supaero, Toulouse,France; the Institut Non-Linéaire, Nice, France; and Cornell University. He alsospent a year-and-a-half as a Postdoctoral Scholar with the Center for the Math-ematics of Information at the California Institute of Technology, Pasadena, CA,USA. He works on applications of control, game, and optimization theory to avariety of fields, and, most recently, to “smart infrastructures” problems withinthe Center for People & Infrastructures which he co-founded and co-directsat CSL.

Prof. Langbort is a recipient of the National Science Foundation CAREERAward and was a subject editor for the Journal of Optimal Control Applicationsand Methods (OCAM).

Geir E. Dullerud (F’08) was born in Oslo, Norway, in 1966. He received theB.A.Sc. degree in engineering science and the M.A.Sc. degree in electricalengineering from the University of Toronto, Toronto, ON, Canada, in 1988 and1990, respectively, and the Ph.D. degree in engineering from the University ofCambridge, Cambridge, U.K., in 1994.

Since 1998, he has been a faculty member in Mechanical Science andEngineering at the University of Illinois, Urbana-Champaign, IL, USA, wherehe is currently a Professor. He is a member of the Coordinated ScienceLaboratory, where he is Director of the Decision and Control Laboratory. Hehas held visiting positions at KTH, Stockholm, Sweden, in 2013 in electricalengineering, and Stanford University, Stanford, CA, from 2005 to 2006 inaeronautics and astronautics. From 1996 to 1998, he was an Assistant Professorin Applied Mathematics at the University of Waterloo, Waterloo, ON, Canada.He was a Research Fellow and Lecturer at the California Institute of Technol-ogy, Pasadena, CA, USA, in the Control and Dynamical Systems Departmentin 1994 and 1995. He has published two books: A Course in Robust ControlTheory (with F. Paganini) (Springer 2000) and Control of Uncertain Sampled-Data Systems (Birkhauser, 1996). He is an Associate Editor with IEEE TRANS-ACTIONS ON AUTOMATIC CONTROL. His current research interests includegames and networked control, robotic vehicles, hybrid dynamical systems, andcyberphysical systems security.

Dr. Dullerud received the National Science Foundation CAREER Award in1999, and in 2005, the Xerox Faculty Research Award at UIUC. He becameASME Fellow in 2011. Currently, he is an Associate Editor with the SIAMJournal on Control and Optimization and Automatica.

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