Automatica 38 (2002) 1713–1718www.elsevier.com/locate/automatica

Brief Paper

Decentralized sliding mode control design using overlappingdecompositions�

Mehmet Akar a;1; ∗, 'Umit 'Ozg'unerb;1

aDepartment of Electrical Engineering, The University of Southern California, Los Angeles, CA 90089-2565, USAbDepartment of Electrical Engineering, The Ohio State University, Columbus, OH 43210, USA

Received 30 June 1999; received in revised form 24 April 2001; accepted 2 April 2002

Abstract

This paper discusses overlapping decentralized sliding mode controller design for large-scale continuous-time systems. Design issues,like connective reachability of the sliding manifold and the stability of the sliding mode equations in the expanded and original statespaces are examined. Application of the results to automatic generation control is also discussed brie5y.? 2002 Published by Elsevier Science Ltd.

Keywords: Overlapping decompositions; Sliding mode control; Decentralized control

1. Introduction

Sliding mode control is an e9ective way of controllingboth certain and uncertain, linear and nonlinear systems be-cause the technique not only stabilizes the system, but alsoprovides disturbance rejection and low sensitivity to plantparameter variations (Utkin, 1992). Among many variationsin this area, decentralized sliding mode control design hasbeen one of the research topics for many engineers (Richter,Lefebvre, & DeCarlo, 1982; Matthews & DeCarlo, 1985,1988; Khurana, Ahson, & Lamba, 1986; Yasuda, 1996).This paper further contributes to the research in this direc-tion by investigating the use of overlapping decompositionsin the design of decentralized sliding mode control.

An overlapping decomposition of a large-scale system al-lows the subsystems to share some common parts and thusprovides greater 5exibility in the choice of the subsystems(Ikeda & BSiljak, 1980). In some cases overlapping decom-positions have been shown to succeed, while disjoint decom-positions have failed ( BSiljak, 1990; Akar & Sezer, 2001).

� A preliminary version of this paper was presented at the 14th IFACWorld Congress, Beijing, China. This paper was recommended for pub-lication in revised form by the Associate Editor Carlos Canudas de Witunder the direction of editor Hassan Khalil.

∗ Corresponding author.E-mail addresses: akar@usc.edu (M. Akar), umit@ee.eng.ohio-

state.edu ( 'U. 'Ozg'uner).1 Correspondence also to. Tel.: +1-614-2925940; fax: +1-614-2927596

The theory behind the overlapping decompositions is Inclu-sion Principle (Ikeda & BSiljak, 1980), which justiIes theembedding of a Inite dimensional system into a higher di-mensional system. 2 An overlapping decomposition of theoriginal system corresponds to a disjoint decomposition ofthe expanded system. A decentralized solution for the (dis-joint) pieces of the expanded system is then contracted toobtain a solution for the original system.

The concept of overlapping decompositions has beensuccessfully applied to various large-scale problems includ-ing decentralized optimal control (Ikeda & BSiljak, 1980),parallel distributed compensation for fuzzy systems (Akar& 'Ozg'uner, 2000a) and hybrid system control (Iftar &'Ozg'uner, 1998). This paper extends similar ideas in design-ing decentralized sliding-mode controllers for large-scalecontinuous-time systems. As a basis for the design, con-nective reachability problem is deIned and solved in Sec-tion 2 via the contexts of vector Lyapunov functions andM -matrices (Fiedler & Ptak, 1962). Section 2 outlines theprocedure for design of decentralized sliding mode control

2 The notion of embedding a dynamical system into another dynamicalsystem is relevant, because the expanded system may have some specialproperty that the original system does not have. In this paper, the decou-pled nature of the expanded system will be of signiIcance, whereas thenotion of Immersion (Isidori, 1995) is used for the solution of the prob-lem of output regulation in the case of error feedback. In that context,a nonlinear system is immersed into a Inite dimensional and observablelinear system to Ind necessary and suKcient conditions for the erroroutput regulation problem.

0005-1098/02/$ - see front matter ? 2002 Published by Elsevier Science Ltd.PII: S 0005 -1098(02)00077 -8

1714 M. Akar, 2U. 2Ozg2uner / Automatica 38 (2002) 1713–1718

by means of overlapping decompositions using also theresults from Section 2. The application of the results to apower control system is brie5y discussed in Section 3.2,and Inally conclusions are given in Section 4.

In the sequel, R denotes the set of real numbers; Rkdenotes the set of k-dimensional real vectors; Ik denotesthe k-dimensional identity matrix; N, and T denote thesets, {1; : : : ; N}, and [0;∞), respectively. Let x be ak-dimensional vector. Then sgn(x) = [sgn(x1); sgn(x2); : : : ;sgn(xk)]T, where sgn(·) is the signum function, and ‖x‖denotes the Euclidean norm of x. For a k × k dimensionalmatrix X , m(X ) and M (X ) denote the minimum and themaximum eigenvalues of X , respectively.

2. Connective reachability of the sliding manifold

Consider an interconnected system S

S : xi(t) = Aixi(t) + Biui(t) + hi(t; x(t)); i∈N; (1)

which is composed of N subsystems

Si : xi(t) = Aixi(t) + Biui(t); i∈N; (2)

where xi(t)∈Rni , ui(t)∈Rmi , are the state and the input ofSi at time t, x(t) = [xT1 (t); x

T2 (t); : : : ; x

TN (t)]

T ∈Rn, u(t) =[uT1 (t); u

T2 (t); : : : ; u

TN (t)]

T ∈Rm, are the state and the input ofthe overall system S; Ai; Bi are constant matrices of dimen-sions ni × ni and ni ×mi, respectively, hi :T×Rn → Rni isthe ith interconnection function. The goal is to control thecomposite system S using local sliding mode control lawsgiven as

ui(xi) =−Mi(xi) sgn(si(xi)); i∈N; (3)

where Mi(xi)∈R is the gain, si(xi) = Cixi = 0 is the slid-ing manifold for the ith subsystem Si, and s = Cx = 0is the sliding manifold for the composite system S withC = blockdiag (C1; C2; : : : ; CN ). The following assumptionis made about the structure of the system given in (1).

Assumption 1. (i) Each xi; i∈N; can be locally ob-served.

(ii) (Ai; Bi); i∈N; is stabilizable.(iii) CiBi; i∈N; is invertible.(iv) The interconnections hi(t; x(t)) satisfy the inequalities

‖Cihi(t; x(t))‖6∑j∈N�ij‖xj‖;

∀t ∈T; ∀x∈Rn; i∈N; (4)

for some nonnegative numbers �ij.

The connective reachability problem that will be dealtwith in this section, is now deIned as follows.

Problem 1. Assuming that the local controls; ui(xi); i∈N;guarantee that the manifolds for the subsystems Si arereached; 9nd conditions on the system parameters underwhich the same set of controls will enforce reachability ofthe sliding manifold for the composite system S.

To obtain a solution for Problem 1, one Irst has to makesure that the local manifolds are reached, and this is estab-lished with the following assumption.

Assumption 2. Assume that the equivalent control

ui; eq =−(CiBi)−1CiAixi; i∈N; (5)

for the ith subsystem satisIes ‖ui; eq‖6Fi(xi) and

m(Di)Mi(xi)=√mi − ‖CiBi‖Fi(xi)¿ �i‖xi‖;

for some positive constants �i and positive deInite matricesDi where

Di = 12 [CiBi + (CiBi)T]; i∈N: (6)

Remark 1. The requirements of Assumption 2 can be ful-Illed by letting Fi(xi) = ‖(CiBi)−1CiAi‖:‖xi‖ and choosingthe gain; Mi(xi); as

Mi(xi) =√mi

m(Di)(Fi(xi)‖CiBi‖+ �i‖xi‖); (7)

for some desired �i ¿ 0; i∈N.

Then the following theorem proposes one solution forProblem 1 in terms of M -matrices (Fiedler & Ptak, 1962).

Theorem 1. Under Assumptions 1 and 2; the slidingmanifold; s = 0; is reached for the composite system S ifthe N × N matrix W is an M -matrix where

wij =

{�i − �ii; i = j;

−�ij; i �= j:(8)

Remark 2. Theorem 1 simply says that the connectivereachability is guaranteed once �i; i∈N; are high enoughto beat the interconnections; but as pointed out in Remark 1this can always be achieved by choosing high gains Mi(xi);i∈N.

Proof. For the composite system S; let a Lyapunov-likefunction be

v(x) =∑i∈N

di sgn(si)Tsi; (9)

where di ¿ 0; i∈N are to be chosen. Then

v(x)(S) =∑i∈N

di sgn(si)T[CiBi(ui − ui; eq) + Cihi]

6−∑i∈N

√midi

�i‖xi‖ −∑

j∈N�ij‖xj‖

=−dT

Wz;

M. Akar, 2U. 2Ozg2uner / Automatica 38 (2002) 1713–1718 1715

where d = [√m1d1;

√m2d2; : : : ;

√mNdN ]T and z =

[‖x1‖; ‖x2‖; : : : ; ‖xN‖]T. Since W is an M -matrix; there

exists a d¿ 0 such that dTW ¿ 0. Thus it follows that

connective reachability is achieved in Inite time (Utkin;1992).

Remark 3. Theorem 1 applies to the class of systemsconsidered by Yasuda (1996); since the interconnectionhi(t; x(t)) can simply be taken as

hi(t; x(t)) = Fi∑j∈N

Gij(t; x)Ejxj; i∈N;

where Ei; Fi are full rank; constant matrices; and Gij(t; x)is bounded; i.e.; ‖Gij(t; x)‖6 gij; for some nonnegativenumbers gij.

A similar solution to the one given in Theorem 1 can beobtained by modifying the sliding mode control law (3) asin the following assumption.

Assumption 3. Assume that the equivalent control in (5)satisIes ‖ui; eq‖6Fi(xi) and m(Di)[Mi(xi) − Fi(xi)]¿ #i‖xi‖ for some positive numbers #i and positive deInite ma-trices Di given in (6); where ui(xi) in (3) is updated as

ui(xi)=−Mi(xi) sgn(s∗i ); (s∗i )T = sTi CiBi; i∈N: (10)

Theorem 2. Under Assumptions 1 and 3; the slidingmanifold; s = 0; is reached for the composite system S ifthe N × N matrix W is an M -matrix where

wij =

{#i − �ii; i = j;

−�ij; i �= j:(11)

Proof. For the composite system; let a Lyapunov-like func-tion be

v(x) =∑i∈N

di(sTi si)1=2; (12)

where di ¿ 0; i∈N. Then

v(x)(S) =∑i∈N

di(sTi si)−1=2sTi [CiBi(ui − ui; eq) + Cihi]

6−∑i∈N

di

#i‖xi‖ −∑

j∈N�ij‖xj‖

=−dTWz;

where d=[d1; d2; : : : ; dN ]T; and z=[‖x1‖; ‖x2‖; : : : ; ‖xN‖]T.Since W is an M -matrix; it follows that connective reacha-bility is assured in Inite time (Utkin; 1992).

Similar to Remark 2, once the gains Mi(xi), i∈N, arehigh enough, Theorem 2 says that connective reachabilityis assured. Another point that has to be emphasized is that

Theorem 1 is more conservative than Theorem 2. This istrue, because the inequalities

m(Di)Mi(xi)=√mi6 m(Di)Mi(xi);

‖CiBi‖Fi(xi)¿ m(Di)Fi(xi); i∈N;

imply that �i6 #i for Ixed gains Mi(xi). However, the twotheorems perform the same when all the subsystems aresingle input and the individual manifolds are designed sothat ‖CiBi‖= m(Di), i∈N.

3. Overlapping decentralized design

In the previous section, conditions for connective reach-ability were derived based on a disjoint decompositionof the composite system. In this section, those resultswill be used to design decentralized sliding-mode con-trol laws using overlapping decompositions. For clarityof the presentation, the design will be carried out for lin-ear systems which have two subsystems overlapping. Theextension to nonlinear systems will be discussed in theconclusion.

Consider a continuous-time system S described by

S:

{y(t) = Ay(t) + Bu(t);

s(y(t)) = Cy(t);(13)

where y(t)∈Rn, u(t)∈Rm are the state and the input of thesystem S at time t; A; B; C are constant matrices of dimen-sions, n × n, n × m, m × n, respectively, and s(y(t)) = 0is the sliding manifold. Suppose that the system matrices in(13) are partitioned as

(14)

(15)

where the dotted lines show two subsystems sharing infor-mation through another subsystem, i.e., the overlapping sub-system. Without loss of generality, the above partitioning isassumed to be satisfying the conditions dim(A11)=dim(B1)and dim(A55) = dim(B2). As for the results in Section 2,n1 = dim(A11) + dim(A22), m1 = dim(B1), n2 = dim(A33),m2 = 0, n3 = dim(A44) + dim(A55), and m3 = dim(B2).

1716 M. Akar, 2U. 2Ozg2uner / Automatica 38 (2002) 1713–1718

The state transformation y = Ty where

T =

In1 0 0

0 In2 0

0 In2 0

0 0 In3

(16)

deInes an expanded system S described by

S:

{ ˙y(t) = Ay(t) + Bu(t);

s(y(t)) = Cy(t);(17)

where y(t)∈Rn, u(t)∈Rm are the state and the input ofthe system S at time t; A; B; C are constant matrices of di-mensions, n× n, n×m, m× n, respectively, and s(y(t))=0is the sliding manifold. It can be shown that (Akar,1999) if

AT = TA; B= TB; CT = C

hold, then y(t; Ty0)=Ty(t; y0) and s(y(t; Ty0))=s(y(t; y0))for all t¿ 0 and y0 ∈Rn. Moreover, any sliding mode con-trol law designed for the expanded system S can be suit-ably transformed back to the original state space for im-plementation on S. In the expanded state space, S can beviewed as

S:

˙x1(t) = A1x1(t) + B1u1(t) + A12x2(t);

s1(t) = C1x1(t) + C12x2(t);

˙x2(t) = A2x2(t) + B2u2(t) + A21x1(t);

s2(t) = C2x2(t) + C21x1(t)

(18)

which is an interconnection of two subsystems

S1:

{ ˙x1(t) = A1x1(t) + B1u1(t);

s1(t) = C1x1(t);(19)

S2:

{ ˙x2(t) = A2x2(t) + B2u2(t);

s2(t) = C2x2(t);(20)

where

A1 =

A11 A12 A13

A21 A22 A23

A31 A32 A33

; A12 =

0 A14 A15

0 A24 A25

0 A34 A35

;

B1 =

B1

0

0

; (21)

A2 =

A33 A34 A35

A43 A44 A45

A53 A54 A55

; A21 =

A31 A32 0

A41 A42 0

A51 A52 0

;

B2 =

0

0

B2

; (22)

C1 = [Im1 C12 C13]; C12 = [0 C14 C15];

C2 = [C23 C24 Im2 ]; C21 = [C21 C22 0]: (23)

It is clear that if A12 = 0, A21 = 0, C12 = 0, and C21 = 0,one has a totally decoupled system in the expanded statespace. However, this is a severe restriction on the originalsystem. Instead, it will be assumed that sliding modes can beenforced for the overall system with the constraints C12 = 0and C21 = 0.

3.1. Design for individual subsystems

For designing appropriate sliding mode control laws forthe subsystems, reconsider the expanded system (18) withthe additional conditions, C12 = 0 and C21 = 0, as an inter-connected system

S:

˙x11 = A1;11x11 + A1;12x12 + B1u1 + A12;11x21

+ A12;12x22;

˙x12 = A1;21x11 + A1;22x12 + A12;21x21

+ A12;22x22;

s(x1) = x11 + C1;12x12;

˙x21 = A2;11x21 + A2;12x22 + A21;11x11

+ A21;12x12;

˙x22 = A2;21x21 + A2;22x22 + B2u2 + A21;21x11

+ A21;22x12;

s(x2) = x22 + C2;21x21

(24)

which is composed of two subsystems

S1:

˙x11 = A1;11x11 + A1;12x12 + B1u1;

˙x12 = A1;21x11 + A1;22x12;

s(x1) = x11 + C1;12x12

(25)

and

S2:

˙x21 = A2;11x21 + A2;12x22;

˙x22 = A2;21x21 + A2;22x22 + B2u2;

s(x2) = x22 + C2;21x21;

(26)

M. Akar, 2U. 2Ozg2uner / Automatica 38 (2002) 1713–1718 1717

where xT1 = [xT11, xT12], x

T2 = [xT21, x

T22], x11 ∈Rm1 , x12 ∈

Rn1+n2−m1 , x21 ∈Rn2+n3−m3 , x22 ∈Rm3 and the system ma-trices in (24) are expressed in terms of the matrices in (21)–(23). Note that the sliding mode equations for the systemsS1, S2 and S are ˙x12 = Asm;11x12, ˙x21 = Asm;22x21 and˙xsm = Asm xsm, respectively, where xTsm = [xT12; x

T21] and

Asm =

[Asm;11 Asm;12

Asm;21 Asm;22

]

=

[A1;22 − A1;21C1;12 A12;21 − A12;22C2;21

A21;12 − A21;11C1;12 A2;11 − A2;12C2;21

]:

Two things are important in designing appropriate slidingmode control laws for the subsystems S1 and S2. First,reachability of the sliding manifold for the individual slid-ing manifolds should imply the connective reachability ofthe overall sliding manifold. Second, sliding mode equationin the expanded space should be designed so that the slidingmode equation in the original state space is stable. Assum-ing that stabilizing decentralized sliding mode control lawsui(xi)=−M i(xi) sgn(si); i=1; 2 which enforce stable slid-ing motion on the manifolds si(xi) = 0 have been designed,it is clear from the development in Section 2, that connec-tive reachability is assured if the gains ‖M i(xi)‖ are highenough.

The second issue in designing an appropriate sliding modecontrol in the expanded system is related to the sliding modeequation. If the sliding mode equation in the expanded spaceis stable, so is the one in the original state space. Moreover,it can be shown (Akar, 1999) that the eigenvalues of Asm

include the eigenvalues of A33 and Asm where xsm =Asmxsmis the sliding motion equation for the original system. Thisimplies that the sliding mode equation in the expanded spacemay be unstable, even if the sliding motion equation in theoriginal space is stable. Still, for this case, one should alsobe aware of the fact that the sliding motion in the expandedspace is stable for those initial conditions satisfying x0=Tx0.

In case A33 is stable, the sliding mode equation for theexpanded system will be stable if the following condition issatisIed

‖Asm;12‖‖Asm;21‖6 14m(Q1)M (P1)

m(Q2)M (P2)

; (27)

where sliding motion equation matrix Asm; ii for each sub-system satisIes the matrix Lyapunov equation

ATsm; iiPi + PiAsm; ii =−Qi; i = 1; 2 (28)

for some symmetric positive deInite matrices Pi and Qi.This condition is easily obtained by manipulating the Lya-punov function v(xsm) = d1x

T12P1x12 + d2x

T21P2x21 for prop-

erly chosen positive constants d1 and d2.

Remark 4. For simplicity; sliding mode control design bymeans of overlapping decompositions has been presentedfor two subsystems overlapping. If there were N subsystemsoverlapping; then connective reachability would be assuredprovided that the control gains for subsystems were highenough to satisfy the condition in Theorem 2. As for the sta-bility of the motion equation; assuming that all self matricesof the overlapping states (e.g.; A33 for the two subsystemcase above) are stable; then interconnected stability is as-sured if an N ×N test matrix is an M -matrix ( BSiljak; 1978).

3.2. Application: automatic generation control

The design technique discussed in the previous sectioncan be employed on the two-area model described in ( BSiljak,1978) by the equation

(29)

where the dashed lines identify areas; Rf1 ∈R8 andRf2 ∈R8 are the deviations of states from their nominalvalues; Rv1 ∈R and Rv2 ∈R are the variables introducedto achieve integral control; Rpe ∈R is the variation ofthe total power exchange between the areas; u1 and u2 arescalar inputs. The actual design and simulation results canbe found in the technical report by Akar and 'Ozg'uner,(2000b). In the report, it is also shown by simulations thatthe modiIed decentralized sliding mode control law hasdisturbance rejection capability.

4. Conclusions

This paper has studied the use of overlapping decom-positions in designing decentralized sliding mode controllaws for large-scale continuous-time systems. The connec-tive reachability problem has been deIned and solved inSection 2. The solution obtained can be easily extended tocover systems composed of nonlinear subsystems. In Sec-tion 3, the design procedure has been outlined only for linear

1718 M. Akar, 2U. 2Ozg2uner / Automatica 38 (2002) 1713–1718

systems, however the ideas can be generalized to nonlinearsystems that are aKne in the control with some additionalconstraints (by combining the techniques used in Section 3and the results in BSiljak, 1990). Another possible extensionis to apply similar ideas for designing decentralized vari-able structure controllers for a class of switching systems(Akar, 1999).

Acknowledgements

This research was supported by the Center of Intelli-gent Transportation Research of The Ohio State Universitythrough a graduate fellowship. The authors would like tothank the associate editor and the anonymous reviewers fortheir comments and suggestions which improved the pre-sentation of the paper.

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Mehmet Akar was born in G'ulnar, Turkeyin 1973. He received his B.S. and M.S.degrees from Bilkent University, Ankara in1994 and 1996, respectively, and his Ph.D.degree from the Ohio State University,Columbus, Ohio in 1999, all in electricalengineering. Between January 2000 andJuly 2001, he was a postdoctoral researchassociate at the Center for System Scienceat Yale University. Since July 2001, he hasheld a similar position within the Commu-nication Sciences Institute at the University

of Southern California. His current research interests include controlof communication systems in general, resource allocation for wirelessnetworks, and stability and control of hybrid systems.

Umit &Ozg&uner received his Ph.D. from theUniversity of Illinois in 1975. He has heldresearch and teaching positions at I.B.M.T.J. Watson Research Center, University ofToronto and Istanbul Technical Universityand worked at Ohio Aerospace Institute andFord Motor Co. during a sabbatical year.He has been with the Ohio State Univer-sity since 1981 where he is presently Pro-fessor of Electrical Engineering and holdsthe T.R.C. Inc. Chair on Intelligent Trans-portation Research. His areas of research

interest are in intelligent transportation systems, decentralization and au-tonomy issues in large systems and applied automotive control. Professor'Ozg'uner represented the Control Society in the IEEE TAB IntelligentTransportation Systems Committee, which he chaired in 1998 during itstransition to Council status. He was the Irst President of the IEEE ITSCouncil (1999 and 2000). He has also served the Control Society in manypositions, and was an elected member of its Board of Governors (1999–2001). He participated in the organization of many conferences (includ-ing the IFAC Congress) and has been the Technical Program Co-Chairof the 1993, and General Chair of the 1994, and co-chair of the 1997International Symposium on Intelligent Control. He was the ProgramChair for the 1997 IEEE ITS Conference, which he helped initiate. He isthe General Chair for the 2002 CDC. Professor 'Ozg'uner’s research hasbeen supported by NSF, AFOSR, NASA LeRC and LaRC, Ford, Vis-teon, LLNL, Sandia Labs, NAHSC, Delco-Delphi, OKI and Honda R&D, lately concentrating on automated vehicles. The team he coordinatedparticipated successfully in the 1997 Automated Highway Demonstrationin San Diego (Demo’97), where they demonstrated 3 fully automated carsdoing lane-keeping, convoying and passing using radar and vision basedguidance. In Demo’99 in Ohio, they also demonstrated a vision-based“electronic tow-bar”, and GPS and map based automated speed adjust-ment. He is the author of over 250 publications which have appeared injournals, as book chapters and in conference proceedings.