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Decentralized variable structure control for mismatched uncertain large-scale systems: a new approach
Systems & Control Letters 43 (2001) 117–125
www.elsevier.com/locate/sysconle
Decentralized variable structure control for mismatched uncertain large-scale systems: a new approach Yao-Wen Tsaia , Kuo-Kai Shyub; ∗ , Kuang-Chiung Changc a Department
of Electronic Engineering, Kuang Wu Institute of Technology, Taipei, Taiwan 112, ROC of Electrical Engineering, National Central University, Chung-Li, Taiwan 320, ROC c Department of Electrical Engineering, Lunghwa Junior College of Technology and Commerce, Taoyuan, Taiwan 300, ROC b Department
Received 12 September 1999; received in revised form 17 November 2000
Abstract In this paper, a decentralized variable structure controller for a class of large-scale systems with mismatched uncertainties is proposed. In every subsystem, two sets of switching surfaces is introduced. New invariance conditions are derived such that the system in the new sliding mode is completely invariant to both matched and mismatched uncertainties. To demonstrate the e4ectiveness of the proposed scheme, a decentralized variable structure controller is synthesized to c 2001 Elsevier perform the new sliding mode. Moreover, the stability analysis of the overall system is also provided. Science B.V. All rights reserved. Keywords: Variable structure systems; Mismatched uncertainty; Sliding mode; Invariance condition
1. Introduction The central subject of variable structure systems (VSS) is the controller design such that the state trajectory is trapped on a switching surface and remains on it thereafter. In this case, the system is called in the sliding mode. In it the system behaves as an equivalent system with the desired dynamics. During the sliding mode, the system behavior has invariance property which is independent of matched uncertainty [2– 4,9–17,19]. If the invariance condition (matching condition) is not satis>ed, however, the mismatched uncertainty will enter into the dynamics of the system in the sliding mode. Thus the system behavior in the sliding mode is not invariant to the mismatched uncertainty. Owing to large dimensions and the e4ects of interconnections, the VSS design for large-scale systems is much more di@cult than that for small-scale systems. For stabilizing large-scale systems, some important studies were carried out by using the VSS design [5 –8,18]. Khurana et al. [5] proposed a hierarchical control method; Matthews and DeCarlo [7,8] inserted an integral term of interconnections to the sliding surface. Xu et al. [18] discussed the stability of the sliding mode. However, these papers considered only the matched uncertain large-scale system. If some mismatched uncertainty exists in the system dynamics, the system ∗ Corresponding author. Tel.: 886-3-4227151 ext. 4463; fax: 886-3-4255830. E-mail address: [email protected] (Kuo-Kai Shyu).
c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 0 8 1 - 0
118
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
performance cannot be assured. Lee and Wang [6] solved some important problems for a class of mismatched uncertain large-scale systems. One of the drawbacks of the method is that the system behavior can still be inKuenced by the mismatched uncertainty. However, extending invariance property of uncertain large-scale systems from matched to mismatched uncertainty is still not a trivial work. In this paper, we consider a class of mismatched uncertain large-scale systems in which the conventional invariance condition (i.e., the matching condition) is not satis>ed. Using two sets of switching surfaces, new invariance conditions are derived such that the system behavior in the new sliding mode is completely independent of both matched and mismatched uncertainties. For designing the sliding phase, a design method of choosing the two set switching surfaces are proposed by using the well known closed loop eigenvalue=eigenvector assignment method. As for the reaching phase design, a decentralized variable structure controller is synthesized. It should be pointed out that this decentralized control law not only asymptotically stabilizes the closed loop system but also establishes the existence and reachability of the new sliding mode. Throughout the work, AT denotes the transpose of matrix A; min (Q) and max (Q) denote, respectively, the minimal and maximal eigenvalue of the symmetric matrix Q; Ir denotes the identity matrix of dimension r × r. · denotes the Euclidean norm for vectors and the induced spectral norm for matrices. 2. Review of VSS theory Consider a class of mismatched uncertain large-scale systems that is decomposed into L subsystems. The state space representation of each subsystem is described by the following: x˜˙i = (A˜ i + QA˜ i )x˜i + B˜ i ui +
L
(A˜ ij + QA˜ ij )x˜j ;
i = 1; 2; : : : ; L;
(1)
j=1 j=i
where x˜i ∈ Rni ; ui ∈ Rmi are the state variable and control input of the subsystem, respectively. A˜ i ; B˜ i ; A˜ ij are constant matrices with appropriate dimensions. The values of the parameters of uncertain matrices QA˜ i and QA˜ ij are unknown, but the upper bound of the values is known. For each subsystem, denote the switching surfaces by ˜i = 0, where the switching functions ˜i = S˜i x˜i ; i = 1; 2; : : : ; L (2) are mi -state vector. The system in sliding mode satis>es ˜i = 0 and ˜˙i = 0. The following theorem shows the well-known invariance condition for the system: Theorem 1. System (1) in the sliding mode is invariant to the uncertainties QA˜ i x˜i and QA˜ ij x˜j if the invariance condition rank(B˜ i ) = rank[B˜ i : QA˜ i : QA˜ ij ] = mi (3) is satis4ed. Remark 1. If the invariance condition (3) is satis>ed, then there exists a matrix T˜ i ∈ Rni ×ni such that 0 0 0 ; T˜ i [B˜ i QA˜ i QA˜ ij ] = B˜ i1 QA˜ i1 QA˜ ij1
(4)
where B˜ i1 ∈ Rmi ×mi ; QA˜ i1 ∈ Rmi ×ni and QA˜ ij1 ∈ Rmi ×ni . 3. System formulation and new switching surfaces We are now in a position to present a new design method for system (1). Here we make the following assumptions:
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
119
Assumption 1. rank(B˜ i ) = mi . Assumption 2. All the pairs (A˜ i ; B˜ i ) are completely controllable. Assumption 3. ni ¿2mi . Assumption 4. For each subsystem, suppose mi = rank(B˜ i )6rank[B˜ i : QA˜ i : QA˜ ij ]62mi ;
(5)
ni ×ni
such that i.e. there exists a matrix Ti ∈ R 0 0 Ti B˜ i = 0 , Bi ; Ti QA˜ i Ti−1 = QAi2 , QAi Bi3 QAi3 and
0 Ti QA˜ ij Tj−1 = QAij2 , QAij ; QAij3
(6)
(7)
where QAi2 ; QAi3 ; QAij2 and QAij3 ∈ Rmi ×ni . Based on Assumption 4 and using xi = Ti x˜i , we can rewrite (1) by x˙i = (Ai + QAi )xi + Bi ui +
L
(Aij + QAij )xj ;
i = 1; 2; : : : ; L;
(8)
j=1 j=i
where Bi = Ti B˜ i ;
Ai = Ti A˜ i Ti−1 ;
Aij = Ti A˜ ij Tj−1 :
In the following, the analysis and design methods are on the basis of the transformed system (8). Remark 2. Comparing (3) and (5), we can see that Eq. (3) is a special case of (5). In other words, the considered system (1) includes mismatched uncertainties. Remark 3. Following Assumption 4, we have mi = rank(Bi )6rank[Bi : QAi : QAij ]62mi . Now, for each subsystem, de>ne two sets of switching functions as Ri = SRi xi ;
ˆi = Sˆi xi ;
(9)
where Ri ∈ Rmi and ˆi ∈ Rmi and the matrices SRi and Sˆi are full rank. Hence, we have two-set switching surfaces Ri = 0 and ˆi = 0. The new sliding mode under two sets of switching functions (9) is de>ned by the following two equations: Ri = 0
and
ˆi = 0
and
R˙i = 0; ˆ˙i = 0:
(10)
4. Switching surfaces design In this section, we present the design of the switching surfaces Ri = 0 and ˆi = 0. The design is based on the method developed by El-Ghezawi et al. [1].
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
Consider the well-known fact about the eigenvalue=eigenvector assignment problem of the linear feedback systems (x˙ = Ax + Bu), (A + BK)V1 = V1 J;
(11)
where V1 ∈ Rn×(n−m) is the eigenvector matrix, K ∈ Rm×n is the feedback matrix and J ∈ R(n−m)×(n−m) is a freely chosen Jordan matrix which determines the system performances. If rank(K) = m, then (11) implies that Col(AV1 − V1 J ) ∈ Range(B). This method can be used to design the switching surfaces of the large-scale system (8). For each subsystem, we can use Col(Ai Vi − Vi Ji ) ∈ Range(Bi ), and we suppose the eigenvector matrix and Jordan matrix can be partitioned by Vi = [Ni
Wi ]
and
Ji =
Ji1 0
i Ji2
;
(12)
where Ni ∈ Rni ×(ni −2mi ) and Wi ∈ Rni ×mi . The matrix Ji is a Jordan matrix which has (ni − mi ) real negative eigenvalues and is partitioned by Ji1 ∈ R(ni −2mi )×(ni −2mi ) ; Ji2 ∈ Rmi ×mi and i is compatibly dimensioned. It can be seen that, from (12), Col(Ai Ni − Ni Ji1 ) ∈ Range(Bi ); Col(Ai Wi − Ni i − Wi Ji2 ) ∈ Range(Bi ):
(13)
In the eigenvalue=eigenvector assignment method, it also requires Range(Vi ) ∩ Range(Bi ) = {0}. Hence, [Vi Bi ] = [Ni Wi Bi ] is invertible. The inverse of [Ni Wi Bi ] must have the form
Nig
g Wi ; Big where Nig ; Wig and Big denote generalized inverse of Ni ; Wi and Bi , respectively. If we let Sˆi = Big and SRi − Sˆi = Wig , then the following matrix:
Nig
Mi = SRi − Sˆi Sˆi
(14)
is with the inverse form Mi−1 = [Ni
Wi
Bi ]:
(15)
Hence, the switching matrices SRi and Sˆi have been designed and we get Nig Bi =0; (SRi − Sˆi )Bi =0 and Sˆi Bi =Imi . Introduce the new coordinates
zi
Ri − ˆi = Mi xi ; ˆi
(16)
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
121
where Mi is de>ned in (14). Then system (8) in the new coordinates (16) has the form z˙i Nig zi zj zi L ˙ AR ij Rj − ˆj + SRi − Sˆi QAi Mi−1 Ri − ˆi Ri − ˆ˙i = AR i Ri − ˆi + ˆ˙i
j=i
ˆi
Nig
+ SRi − Sˆi Sˆi where
L j=i
Nig Ai Ni AR i = Mi Ai Mi−1 = (SRi − Sˆi )Ai Ni Sˆi Ai Ni
Nig Aij Nj AR ij = Mi Aij Mj−1 = (SRi − Sˆi )Aij Nj Sˆi Aij Nj
Sˆi
ˆi
0 QAij Mj−1 Rj − ˆj + 0 ui ; I mi ˆj
and
ˆj zj
Nig Ai Wi (SRi − Sˆi )Ai Wi
Nig Ai Bi (SRi − Sˆi )Ai Bi
Sˆi Ai Wi
Sˆi Ai Bi
Nig Aij Wj (SRi − Sˆi )Aij Wj Sˆi Aij Wj
(17)
Nig Aij Bj (SRi − Sˆi )Aij Bj : Sˆi Aij Bj
According to (17), it can be seen that the system dynamics in the new sliding mode (10) still depends on the mismatched uncertainty. 5. New invariance condition In this section, we will propose a new invariance condition which extends the conventional invariance condition (i.e., the matching conditions) from the matched system to the mismatched uncertain variable structure systems (8). This new invariance condition will show that the system behavior in the new sliding mode (10) is completely invariant to the matched and mismatched uncertainties. Theorem 2. Partition the matrix Mi−1 into Ni1 Wi1 0 Mi−1 = [Ni Wi Bi ] = Ni2 Wi2 0 ; Ni3 Wi3 Bi3
(18)
where Bi3 is de4ned in (6); Ni1 ∈ R(ni −2mi )×(ni −2mi ) ; Wi2 ∈ Rmi ×mi and the other matrices are compatibly dimensioned. Let Hi1 = (Wi2 − Ni2 Ni1−1 Wi1 )−1 and Hi2 = (Ni1 − Wi1 Wi2−1 Ni2 )−1 . Then in the new sliding mode (10); the following two statements are satis4ed: (i) If Ni1 is invertible; then Ni1−1 Wi1 Hi1 [QAi2
QAij2 ] = 0
(19)
is an invariance condition with respect to the mismatched uncertainty; where QAi2 and QAij2 are de4ned in (6) and (7); respectively. (ii) If Wi2 is invertible; then Hi2 Wi1 Wi2−1 [QAi2
QAij2 ] = 0
is an invariance condition with respect to the mismatched uncertainty.
(20)
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
Proof. Observing (17), it is easy to get that if Nig [QAi
QAij ] = 0;
(21)
then the system dynamics in the new sliding mode (10) are independent of the mismatched uncertainty. If the matrix Mi−1 is partitioned by (18), then we have
Nig
Ni1
Wi1
−1
Ni2 Wi2 Mi = SRi − Sˆi = Ni1 −1 Sˆi −Bi3 [Ni3 Wi3 ] Ni2
0 Wi1 Wi2
−1
: −1
(22)
Bi3
(i) Suppose Ni1 is invertible. Let Nig = [Ni1g Ni2g Ni3g ], where Ni1g ∈ R(ni −2mi )×(ni −2mi ) ; Ni2g ∈ R(ni −2mi )×mi and Ni3g ∈ R(ni −2mi )×mi . Using (6) and (7), we have the invariance condition (21) that can be rewritten by Nig [QAi QAij ] = Ni2g [QAi2 QAij2 ] = 0. It follows from (22) that Ni2g = −Ni1−1 Wi1 Hi1 . Thus, we have (19) as an invariance condition with respect to the mismatched uncertainty. (ii) Similar conclusion can be drawn. Remark 4. Consider system (17). If we use only one switching surface ˆi = 0, then the sliding mode satis>es ˆi = 0 and ˆ˙i = 0. By (17), we can show that if Nig [QAi
QAij ] = 0;
(SRi − Sˆi )[QAi
QAij ] = 0;
(23)
then the system dynamics are independent of the mismatched uncertainties. On the other hand, the new approach uses two switching surfaces Ri = 0 and ˆi = 0. Using (17), it is easy to see that we only need to satisfy the >rst condition (Nig [QAi QAij ] = 0) of (23) in this new approach. Theorem 2 shows that if Eqs. (19) or (20) is satis>ed, then the condition Nig [QAi QAij ] = 0 holds. Thus, the system is invariant to the mismatched uncertainties. Therefore, the approach renders robustness to a wide class of the systems. We now discuss the system behavior in the new sliding mode (10) if one of the new invariance conditions in Theorem 2 is satis>ed. The following lemma is needed: Lemma. If (13) is satis4ed; then Ni Ai Nig = Ji1 ; N i Ai Wi = i ; (SRi − Sˆi )Ai Nig = 0; (SRi − Sˆi )Ai Wi = Ji2 :
(24)
Proof. Using Mi Mi−1 = Ini , we have Ni Nig = Ini −2mi ; Ni Wi = 0; Ni Bi = 0; (SRi − Sˆi )Nig = 0; (SRi − Sˆi )Wi = Imi and (SRi − Sˆi )Bi = 0. Let (13) hold, since Ni Nig = Ini −2mi and Ni Bi = 0, we have Ni Ai Nig = Ji1 . Also, since (SRi − Sˆi )Nig = 0 and (SRi − Sˆi )Bi = 0, we can see that (SRi − Sˆi )Ai Nig = 0. Consider (13), since Ni Nig = Ini −2mi ; Ni Wi = 0 and Ni Bi = 0, we get Ni Ai Wi = i . Also, using (SRi − Sˆi )Nig = 0; (SRi − Sˆi )Wi = Imi and (SRi − Sˆi )Bi = 0, we have (SRi − Sˆi )Ai Wi = Ji2 .
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
123
From Theorem 2, if one of the new invariance conditions is satis>ed, we then have Nig [QAi QAij ] = 0. Thus, by using the Lemma and Eq. (17), the system equation in the new sliding mode (10) can be written as z˙i = Ji1 zi + Nig
L
Aij Nj zj ;
j=i
R˙i = 0; ˆ˙i = 0:
(25)
It is noted that the negative eigenvalues of the diagonal matrix Ji1 are determined in advance. Thus, the desired performance can be obtained. 6. The reaching phase design The goal of the reaching phase is to determine a decentralized variable structure controller such that the state trajectory of each subsystem will move to the sliding surfaces Ri = 0 and ˆi = 0. The decentralized control is selected to be ui = − i ˆi xi =ˆi − ˆi (ˆi + Ri − ˆi )=ˆi ;
(26)
where i and are positive values and will be designed later. Theorem 3. Suppose there exist known nonnegative constants Let i ¿ Sˆi Ai + +
L
ˆ
i S i
+ (SRi − Sˆi )Ai +
i
and !ij such that QAi 6
i
and QAij 6!ij .
R − Sˆi
i S i
{Sˆj Aji + !ji Sˆj + (SRj − Sˆj )Aji + !ji SRj − Sˆj };
j=i
¿ 0:
(27)
Then the state trajectory of system (8) asymptotically moves towards the switching surfaces Ri = 0 and ˆi = 0. Proof. Choose a Lyapunov function candidate V=
L
(ˆi + Ri − ˆi ):
(28)
i=1
It follows from (8) and (28) that
L T T T ˆ − ˆ R i i i ˙ ˙ V˙ = ˆi + (R˙i − ˆi ) ˆi Ri − ˆi i=1
=
L
ˆTi Sˆi (Ai + QAi )xi =ˆi +
i=1
+
L L i=1 j=i
L
ˆTi Sˆi Bi ui =ˆi
i=1
ˆTi Sˆi (Aij + QAij )xj =ˆi
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
+
L
(RTi − ˆTi )(SRi − Sˆi )(Ai + QAi )xi =Ri − ˆi
i=1
+
L
(RTi − ˆTi )(SRi − Sˆi )Bi ui =Ri − ˆi
i=1
+
L L
(RTi − ˆTi )(SRi − Sˆi )(Aij + QAij )xj =Ri − ˆi :
(29)
i=1 j=i
Using QAij 6!ij , we have L L
ˆTi Sˆi (Aij + QAij )xj =ˆi
i=1 j=i
=
L L
ˆTj Sˆj (Aji + QAji )xi =ˆj 6
L L
i=1 j=i
{Sˆj Aji + !ji Sˆj }xi
(30)
i=1 j=i
and, we also have L L
(RTi − ˆTi )(SRi − Sˆi )(Aij + QAij )xj =Ri − ˆi
i=1 j=i
6
L L
{(SRj − Sˆj )Aji + !ji SRj − Sˆj }xi :
(31)
i=1 j=i
By Mi Mi−1 = Ini , it can be seen that Sˆi Bi = Imi and (SRi − Sˆi )Bi = 0. Now, by (26), (29) – (31) and QAi 6 i , we have V˙ 6
L
{Sˆi Ai +
ˆ
i S i }xi
+
i=1
+
L L
{Sˆj Aji + !ji Sˆj }xi
i=1 j=i
L
{(SRi − Sˆi )Ai +
R − Sˆi }xi
i S i
i=1
+
L L
{(SRj − Sˆj )Aji + !ji SRj − Sˆj }xi
i=1 j=i
−
L
{ i xi + (ˆi + Ri − ˆi )}
i=1
¡ −
L
(ˆi + Ri − ˆi ):
i=1
Hence V˙ ¡ − V . That is the switching functions Ri and ˆi will asymptotically converge to zero. Remark 5. The reachability result shows that the system slides asymptotically, it is only when sliding that the robustness holds. In the new sliding mode (10), if one of the invariance conditions in Theorem 2 is satis>ed, the system is invariant to the mismatched uncertainties.
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
125
7. Conclusions A new decentralized VSS design method has been developed in detail for a class of mismatched uncertain large-scale systems. In the sliding phase, we propose two sets of switching surfaces. New invariance conditions have been derived to guarantee that the system dynamic is completely independent of both matched uncertainty and mismatched uncertainty. In the reaching phase, a decentralized variable structure controller is introduced to guarantee the existence of the new sliding motion and to stabilize the overall system. References [1] O.M.E. El-Ghezawi, A.S.I. Zinober, S.A. Billings, Analysis and design of variable structure systems using a geometric approach, Internat. J. Control 38 (1983) 657–671. [2] W. Gao, J.-C. Hung, Variable structure control of nonlinear systems: a new approach, IEEE Trans. Ind. Electron. IE-40 (1993) 45–55. ˙ [3] S. Hui, S.H. Zak, Robust control synthesis for uncertain=nonlinear dynamical systems, Automatica 28 (1992) 289–298. [4] J.-Y. Hung, W. Gao, J.-C. Hung, Variable structure control: a survey, IEEE Trans. Ind. Electron. IE-40 (1993) 2–22. [5] H. Khurana, S.I. Ahson, S.S. Lamba, On stabilization of large-scale control systems using variable structure systems theory, IEEE Trans. Automat. Control AC-31 (1986) 176–178. [6] J.-L. Lee, W.-J. Wang, Robust decentralized stabilization via sliding mode control, Control-Theory Adv. Technol. 9 (1993) 721–731. [7] G.P. Matthews, R.A. DeCarlo, Decentralized variable structure control of interconnected multiinput=multioutput nonlinear systems, Circuits Systems Signal Process. 6 (1987) 363–387. [8] G.P. Matthews, R.A. DeCarlo, Decentralized tracking for a class of interconnected nonlinear systems using variable structure control, Automatica 24 (1988) 187–193. [9] E.P. Ryan, A variable structure approach to feedback regulation of uncertain dynamical systems, Internat. J. Control 38 (1983) 1121–1134. [10] K.-K. Shyu, H.-J. Shieh, A new switching surface sliding mode speed control for induction motor drive systems, IEEE Trans. Power Electron. PE-11 (1996) 660–667. [11] K.-K. Shyu, Y.-W. Tsai, C.-F. Yung, A modi>ed variable structure controller, Automatica 28 (1992) 1209–1213. [12] K.-K. Shyu, J.-J. Yan, Robust stability of uncertain time-delay systems and its stabilization by variable structure control, Internat. J. Control 57 (1993) 237–246. [13] J.J. Slotine, Sliding controller design for non-linear systems, Internat. J. Control 40 (1984) 421–434. [14] S.K. Spurgeon, Choice of discontinuous control component for robust sliding mode performance, Internat. J. Control 53 (1991) 163–179. [15] V.I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Control AC-22 (1977) 212–222. [16] V.I. Utkin, Sliding Modes and Their Application in Variable Structure Systems, MIR Publishers, Moscow, 1978. [17] V.I. Utkin, Variable structure systems, Automatica 19 (1983) 5–25. [18] X. Xu, Y. Wu, W. Huang, Variable structure control approach of decentralized model-reference adaptive systems, IEE Proc. D 137 (1990) 302–306. ˙ [19] S.H. Zak, S. Hui, Output feedback variable structure controllers and state estimators for uncertain=nonlinear dynamic systems, IEE Proc. D 140 (1993) 41–50.
www.elsevier.com/locate/sysconle
Decentralized variable structure control for mismatched uncertain large-scale systems: a new approach Yao-Wen Tsaia , Kuo-Kai Shyub; ∗ , Kuang-Chiung Changc a Department
of Electronic Engineering, Kuang Wu Institute of Technology, Taipei, Taiwan 112, ROC of Electrical Engineering, National Central University, Chung-Li, Taiwan 320, ROC c Department of Electrical Engineering, Lunghwa Junior College of Technology and Commerce, Taoyuan, Taiwan 300, ROC b Department
Received 12 September 1999; received in revised form 17 November 2000
Abstract In this paper, a decentralized variable structure controller for a class of large-scale systems with mismatched uncertainties is proposed. In every subsystem, two sets of switching surfaces is introduced. New invariance conditions are derived such that the system in the new sliding mode is completely invariant to both matched and mismatched uncertainties. To demonstrate the e4ectiveness of the proposed scheme, a decentralized variable structure controller is synthesized to c 2001 Elsevier perform the new sliding mode. Moreover, the stability analysis of the overall system is also provided. Science B.V. All rights reserved. Keywords: Variable structure systems; Mismatched uncertainty; Sliding mode; Invariance condition
1. Introduction The central subject of variable structure systems (VSS) is the controller design such that the state trajectory is trapped on a switching surface and remains on it thereafter. In this case, the system is called in the sliding mode. In it the system behaves as an equivalent system with the desired dynamics. During the sliding mode, the system behavior has invariance property which is independent of matched uncertainty [2– 4,9–17,19]. If the invariance condition (matching condition) is not satis>ed, however, the mismatched uncertainty will enter into the dynamics of the system in the sliding mode. Thus the system behavior in the sliding mode is not invariant to the mismatched uncertainty. Owing to large dimensions and the e4ects of interconnections, the VSS design for large-scale systems is much more di@cult than that for small-scale systems. For stabilizing large-scale systems, some important studies were carried out by using the VSS design [5 –8,18]. Khurana et al. [5] proposed a hierarchical control method; Matthews and DeCarlo [7,8] inserted an integral term of interconnections to the sliding surface. Xu et al. [18] discussed the stability of the sliding mode. However, these papers considered only the matched uncertain large-scale system. If some mismatched uncertainty exists in the system dynamics, the system ∗ Corresponding author. Tel.: 886-3-4227151 ext. 4463; fax: 886-3-4255830. E-mail address: [email protected] (Kuo-Kai Shyu).
c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 0 8 1 - 0
118
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
performance cannot be assured. Lee and Wang [6] solved some important problems for a class of mismatched uncertain large-scale systems. One of the drawbacks of the method is that the system behavior can still be inKuenced by the mismatched uncertainty. However, extending invariance property of uncertain large-scale systems from matched to mismatched uncertainty is still not a trivial work. In this paper, we consider a class of mismatched uncertain large-scale systems in which the conventional invariance condition (i.e., the matching condition) is not satis>ed. Using two sets of switching surfaces, new invariance conditions are derived such that the system behavior in the new sliding mode is completely independent of both matched and mismatched uncertainties. For designing the sliding phase, a design method of choosing the two set switching surfaces are proposed by using the well known closed loop eigenvalue=eigenvector assignment method. As for the reaching phase design, a decentralized variable structure controller is synthesized. It should be pointed out that this decentralized control law not only asymptotically stabilizes the closed loop system but also establishes the existence and reachability of the new sliding mode. Throughout the work, AT denotes the transpose of matrix A; min (Q) and max (Q) denote, respectively, the minimal and maximal eigenvalue of the symmetric matrix Q; Ir denotes the identity matrix of dimension r × r. · denotes the Euclidean norm for vectors and the induced spectral norm for matrices. 2. Review of VSS theory Consider a class of mismatched uncertain large-scale systems that is decomposed into L subsystems. The state space representation of each subsystem is described by the following: x˜˙i = (A˜ i + QA˜ i )x˜i + B˜ i ui +
L
(A˜ ij + QA˜ ij )x˜j ;
i = 1; 2; : : : ; L;
(1)
j=1 j=i
where x˜i ∈ Rni ; ui ∈ Rmi are the state variable and control input of the subsystem, respectively. A˜ i ; B˜ i ; A˜ ij are constant matrices with appropriate dimensions. The values of the parameters of uncertain matrices QA˜ i and QA˜ ij are unknown, but the upper bound of the values is known. For each subsystem, denote the switching surfaces by ˜i = 0, where the switching functions ˜i = S˜i x˜i ; i = 1; 2; : : : ; L (2) are mi -state vector. The system in sliding mode satis>es ˜i = 0 and ˜˙i = 0. The following theorem shows the well-known invariance condition for the system: Theorem 1. System (1) in the sliding mode is invariant to the uncertainties QA˜ i x˜i and QA˜ ij x˜j if the invariance condition rank(B˜ i ) = rank[B˜ i : QA˜ i : QA˜ ij ] = mi (3) is satis4ed. Remark 1. If the invariance condition (3) is satis>ed, then there exists a matrix T˜ i ∈ Rni ×ni such that 0 0 0 ; T˜ i [B˜ i QA˜ i QA˜ ij ] = B˜ i1 QA˜ i1 QA˜ ij1
(4)
where B˜ i1 ∈ Rmi ×mi ; QA˜ i1 ∈ Rmi ×ni and QA˜ ij1 ∈ Rmi ×ni . 3. System formulation and new switching surfaces We are now in a position to present a new design method for system (1). Here we make the following assumptions:
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119
Assumption 1. rank(B˜ i ) = mi . Assumption 2. All the pairs (A˜ i ; B˜ i ) are completely controllable. Assumption 3. ni ¿2mi . Assumption 4. For each subsystem, suppose mi = rank(B˜ i )6rank[B˜ i : QA˜ i : QA˜ ij ]62mi ;
(5)
ni ×ni
such that i.e. there exists a matrix Ti ∈ R 0 0 Ti B˜ i = 0 , Bi ; Ti QA˜ i Ti−1 = QAi2 , QAi Bi3 QAi3 and
0 Ti QA˜ ij Tj−1 = QAij2 , QAij ; QAij3
(6)
(7)
where QAi2 ; QAi3 ; QAij2 and QAij3 ∈ Rmi ×ni . Based on Assumption 4 and using xi = Ti x˜i , we can rewrite (1) by x˙i = (Ai + QAi )xi + Bi ui +
L
(Aij + QAij )xj ;
i = 1; 2; : : : ; L;
(8)
j=1 j=i
where Bi = Ti B˜ i ;
Ai = Ti A˜ i Ti−1 ;
Aij = Ti A˜ ij Tj−1 :
In the following, the analysis and design methods are on the basis of the transformed system (8). Remark 2. Comparing (3) and (5), we can see that Eq. (3) is a special case of (5). In other words, the considered system (1) includes mismatched uncertainties. Remark 3. Following Assumption 4, we have mi = rank(Bi )6rank[Bi : QAi : QAij ]62mi . Now, for each subsystem, de>ne two sets of switching functions as Ri = SRi xi ;
ˆi = Sˆi xi ;
(9)
where Ri ∈ Rmi and ˆi ∈ Rmi and the matrices SRi and Sˆi are full rank. Hence, we have two-set switching surfaces Ri = 0 and ˆi = 0. The new sliding mode under two sets of switching functions (9) is de>ned by the following two equations: Ri = 0
and
ˆi = 0
and
R˙i = 0; ˆ˙i = 0:
(10)
4. Switching surfaces design In this section, we present the design of the switching surfaces Ri = 0 and ˆi = 0. The design is based on the method developed by El-Ghezawi et al. [1].
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
Consider the well-known fact about the eigenvalue=eigenvector assignment problem of the linear feedback systems (x˙ = Ax + Bu), (A + BK)V1 = V1 J;
(11)
where V1 ∈ Rn×(n−m) is the eigenvector matrix, K ∈ Rm×n is the feedback matrix and J ∈ R(n−m)×(n−m) is a freely chosen Jordan matrix which determines the system performances. If rank(K) = m, then (11) implies that Col(AV1 − V1 J ) ∈ Range(B). This method can be used to design the switching surfaces of the large-scale system (8). For each subsystem, we can use Col(Ai Vi − Vi Ji ) ∈ Range(Bi ), and we suppose the eigenvector matrix and Jordan matrix can be partitioned by Vi = [Ni
Wi ]
and
Ji =
Ji1 0
i Ji2
;
(12)
where Ni ∈ Rni ×(ni −2mi ) and Wi ∈ Rni ×mi . The matrix Ji is a Jordan matrix which has (ni − mi ) real negative eigenvalues and is partitioned by Ji1 ∈ R(ni −2mi )×(ni −2mi ) ; Ji2 ∈ Rmi ×mi and i is compatibly dimensioned. It can be seen that, from (12), Col(Ai Ni − Ni Ji1 ) ∈ Range(Bi ); Col(Ai Wi − Ni i − Wi Ji2 ) ∈ Range(Bi ):
(13)
In the eigenvalue=eigenvector assignment method, it also requires Range(Vi ) ∩ Range(Bi ) = {0}. Hence, [Vi Bi ] = [Ni Wi Bi ] is invertible. The inverse of [Ni Wi Bi ] must have the form
Nig
g Wi ; Big where Nig ; Wig and Big denote generalized inverse of Ni ; Wi and Bi , respectively. If we let Sˆi = Big and SRi − Sˆi = Wig , then the following matrix:
Nig
Mi = SRi − Sˆi Sˆi
(14)
is with the inverse form Mi−1 = [Ni
Wi
Bi ]:
(15)
Hence, the switching matrices SRi and Sˆi have been designed and we get Nig Bi =0; (SRi − Sˆi )Bi =0 and Sˆi Bi =Imi . Introduce the new coordinates
zi
Ri − ˆi = Mi xi ; ˆi
(16)
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
121
where Mi is de>ned in (14). Then system (8) in the new coordinates (16) has the form z˙i Nig zi zj zi L ˙ AR ij Rj − ˆj + SRi − Sˆi QAi Mi−1 Ri − ˆi Ri − ˆ˙i = AR i Ri − ˆi + ˆ˙i
j=i
ˆi
Nig
+ SRi − Sˆi Sˆi where
L j=i
Nig Ai Ni AR i = Mi Ai Mi−1 = (SRi − Sˆi )Ai Ni Sˆi Ai Ni
Nig Aij Nj AR ij = Mi Aij Mj−1 = (SRi − Sˆi )Aij Nj Sˆi Aij Nj
Sˆi
ˆi
0 QAij Mj−1 Rj − ˆj + 0 ui ; I mi ˆj
and
ˆj zj
Nig Ai Wi (SRi − Sˆi )Ai Wi
Nig Ai Bi (SRi − Sˆi )Ai Bi
Sˆi Ai Wi
Sˆi Ai Bi
Nig Aij Wj (SRi − Sˆi )Aij Wj Sˆi Aij Wj
(17)
Nig Aij Bj (SRi − Sˆi )Aij Bj : Sˆi Aij Bj
According to (17), it can be seen that the system dynamics in the new sliding mode (10) still depends on the mismatched uncertainty. 5. New invariance condition In this section, we will propose a new invariance condition which extends the conventional invariance condition (i.e., the matching conditions) from the matched system to the mismatched uncertain variable structure systems (8). This new invariance condition will show that the system behavior in the new sliding mode (10) is completely invariant to the matched and mismatched uncertainties. Theorem 2. Partition the matrix Mi−1 into Ni1 Wi1 0 Mi−1 = [Ni Wi Bi ] = Ni2 Wi2 0 ; Ni3 Wi3 Bi3
(18)
where Bi3 is de4ned in (6); Ni1 ∈ R(ni −2mi )×(ni −2mi ) ; Wi2 ∈ Rmi ×mi and the other matrices are compatibly dimensioned. Let Hi1 = (Wi2 − Ni2 Ni1−1 Wi1 )−1 and Hi2 = (Ni1 − Wi1 Wi2−1 Ni2 )−1 . Then in the new sliding mode (10); the following two statements are satis4ed: (i) If Ni1 is invertible; then Ni1−1 Wi1 Hi1 [QAi2
QAij2 ] = 0
(19)
is an invariance condition with respect to the mismatched uncertainty; where QAi2 and QAij2 are de4ned in (6) and (7); respectively. (ii) If Wi2 is invertible; then Hi2 Wi1 Wi2−1 [QAi2
QAij2 ] = 0
is an invariance condition with respect to the mismatched uncertainty.
(20)
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
Proof. Observing (17), it is easy to get that if Nig [QAi
QAij ] = 0;
(21)
then the system dynamics in the new sliding mode (10) are independent of the mismatched uncertainty. If the matrix Mi−1 is partitioned by (18), then we have
Nig
Ni1
Wi1
−1
Ni2 Wi2 Mi = SRi − Sˆi = Ni1 −1 Sˆi −Bi3 [Ni3 Wi3 ] Ni2
0 Wi1 Wi2
−1
: −1
(22)
Bi3
(i) Suppose Ni1 is invertible. Let Nig = [Ni1g Ni2g Ni3g ], where Ni1g ∈ R(ni −2mi )×(ni −2mi ) ; Ni2g ∈ R(ni −2mi )×mi and Ni3g ∈ R(ni −2mi )×mi . Using (6) and (7), we have the invariance condition (21) that can be rewritten by Nig [QAi QAij ] = Ni2g [QAi2 QAij2 ] = 0. It follows from (22) that Ni2g = −Ni1−1 Wi1 Hi1 . Thus, we have (19) as an invariance condition with respect to the mismatched uncertainty. (ii) Similar conclusion can be drawn. Remark 4. Consider system (17). If we use only one switching surface ˆi = 0, then the sliding mode satis>es ˆi = 0 and ˆ˙i = 0. By (17), we can show that if Nig [QAi
QAij ] = 0;
(SRi − Sˆi )[QAi
QAij ] = 0;
(23)
then the system dynamics are independent of the mismatched uncertainties. On the other hand, the new approach uses two switching surfaces Ri = 0 and ˆi = 0. Using (17), it is easy to see that we only need to satisfy the >rst condition (Nig [QAi QAij ] = 0) of (23) in this new approach. Theorem 2 shows that if Eqs. (19) or (20) is satis>ed, then the condition Nig [QAi QAij ] = 0 holds. Thus, the system is invariant to the mismatched uncertainties. Therefore, the approach renders robustness to a wide class of the systems. We now discuss the system behavior in the new sliding mode (10) if one of the new invariance conditions in Theorem 2 is satis>ed. The following lemma is needed: Lemma. If (13) is satis4ed; then Ni Ai Nig = Ji1 ; N i Ai Wi = i ; (SRi − Sˆi )Ai Nig = 0; (SRi − Sˆi )Ai Wi = Ji2 :
(24)
Proof. Using Mi Mi−1 = Ini , we have Ni Nig = Ini −2mi ; Ni Wi = 0; Ni Bi = 0; (SRi − Sˆi )Nig = 0; (SRi − Sˆi )Wi = Imi and (SRi − Sˆi )Bi = 0. Let (13) hold, since Ni Nig = Ini −2mi and Ni Bi = 0, we have Ni Ai Nig = Ji1 . Also, since (SRi − Sˆi )Nig = 0 and (SRi − Sˆi )Bi = 0, we can see that (SRi − Sˆi )Ai Nig = 0. Consider (13), since Ni Nig = Ini −2mi ; Ni Wi = 0 and Ni Bi = 0, we get Ni Ai Wi = i . Also, using (SRi − Sˆi )Nig = 0; (SRi − Sˆi )Wi = Imi and (SRi − Sˆi )Bi = 0, we have (SRi − Sˆi )Ai Wi = Ji2 .
Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
123
From Theorem 2, if one of the new invariance conditions is satis>ed, we then have Nig [QAi QAij ] = 0. Thus, by using the Lemma and Eq. (17), the system equation in the new sliding mode (10) can be written as z˙i = Ji1 zi + Nig
L
Aij Nj zj ;
j=i
R˙i = 0; ˆ˙i = 0:
(25)
It is noted that the negative eigenvalues of the diagonal matrix Ji1 are determined in advance. Thus, the desired performance can be obtained. 6. The reaching phase design The goal of the reaching phase is to determine a decentralized variable structure controller such that the state trajectory of each subsystem will move to the sliding surfaces Ri = 0 and ˆi = 0. The decentralized control is selected to be ui = − i ˆi xi =ˆi − ˆi (ˆi + Ri − ˆi )=ˆi ;
(26)
where i and are positive values and will be designed later. Theorem 3. Suppose there exist known nonnegative constants Let i ¿ Sˆi Ai + +
L
ˆ
i S i
+ (SRi − Sˆi )Ai +
i
and !ij such that QAi 6
i
and QAij 6!ij .
R − Sˆi
i S i
{Sˆj Aji + !ji Sˆj + (SRj − Sˆj )Aji + !ji SRj − Sˆj };
j=i
¿ 0:
(27)
Then the state trajectory of system (8) asymptotically moves towards the switching surfaces Ri = 0 and ˆi = 0. Proof. Choose a Lyapunov function candidate V=
L
(ˆi + Ri − ˆi ):
(28)
i=1
It follows from (8) and (28) that
L T T T ˆ − ˆ R i i i ˙ ˙ V˙ = ˆi + (R˙i − ˆi ) ˆi Ri − ˆi i=1
=
L
ˆTi Sˆi (Ai + QAi )xi =ˆi +
i=1
+
L L i=1 j=i
L
ˆTi Sˆi Bi ui =ˆi
i=1
ˆTi Sˆi (Aij + QAij )xj =ˆi
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Y.-W. Tsai et al. / Systems & Control Letters 43 (2001) 117–125
+
L
(RTi − ˆTi )(SRi − Sˆi )(Ai + QAi )xi =Ri − ˆi
i=1
+
L
(RTi − ˆTi )(SRi − Sˆi )Bi ui =Ri − ˆi
i=1
+
L L
(RTi − ˆTi )(SRi − Sˆi )(Aij + QAij )xj =Ri − ˆi :
(29)
i=1 j=i
Using QAij 6!ij , we have L L
ˆTi Sˆi (Aij + QAij )xj =ˆi
i=1 j=i
=
L L
ˆTj Sˆj (Aji + QAji )xi =ˆj 6
L L
i=1 j=i
{Sˆj Aji + !ji Sˆj }xi
(30)
i=1 j=i
and, we also have L L
(RTi − ˆTi )(SRi − Sˆi )(Aij + QAij )xj =Ri − ˆi
i=1 j=i
6
L L
{(SRj − Sˆj )Aji + !ji SRj − Sˆj }xi :
(31)
i=1 j=i
By Mi Mi−1 = Ini , it can be seen that Sˆi Bi = Imi and (SRi − Sˆi )Bi = 0. Now, by (26), (29) – (31) and QAi 6 i , we have V˙ 6
L
{Sˆi Ai +
ˆ
i S i }xi
+
i=1
+
L L
{Sˆj Aji + !ji Sˆj }xi
i=1 j=i
L
{(SRi − Sˆi )Ai +
R − Sˆi }xi
i S i
i=1
+
L L
{(SRj − Sˆj )Aji + !ji SRj − Sˆj }xi
i=1 j=i
−
L
{ i xi + (ˆi + Ri − ˆi )}
i=1
¡ −
L
(ˆi + Ri − ˆi ):
i=1
Hence V˙ ¡ − V . That is the switching functions Ri and ˆi will asymptotically converge to zero. Remark 5. The reachability result shows that the system slides asymptotically, it is only when sliding that the robustness holds. In the new sliding mode (10), if one of the invariance conditions in Theorem 2 is satis>ed, the system is invariant to the mismatched uncertainties.
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7. Conclusions A new decentralized VSS design method has been developed in detail for a class of mismatched uncertain large-scale systems. In the sliding phase, we propose two sets of switching surfaces. New invariance conditions have been derived to guarantee that the system dynamic is completely independent of both matched uncertainty and mismatched uncertainty. In the reaching phase, a decentralized variable structure controller is introduced to guarantee the existence of the new sliding motion and to stabilize the overall system. References [1] O.M.E. El-Ghezawi, A.S.I. Zinober, S.A. Billings, Analysis and design of variable structure systems using a geometric approach, Internat. J. Control 38 (1983) 657–671. [2] W. Gao, J.-C. Hung, Variable structure control of nonlinear systems: a new approach, IEEE Trans. Ind. Electron. IE-40 (1993) 45–55. ˙ [3] S. Hui, S.H. Zak, Robust control synthesis for uncertain=nonlinear dynamical systems, Automatica 28 (1992) 289–298. [4] J.-Y. Hung, W. Gao, J.-C. Hung, Variable structure control: a survey, IEEE Trans. Ind. Electron. IE-40 (1993) 2–22. [5] H. Khurana, S.I. Ahson, S.S. Lamba, On stabilization of large-scale control systems using variable structure systems theory, IEEE Trans. Automat. Control AC-31 (1986) 176–178. [6] J.-L. Lee, W.-J. Wang, Robust decentralized stabilization via sliding mode control, Control-Theory Adv. Technol. 9 (1993) 721–731. [7] G.P. Matthews, R.A. DeCarlo, Decentralized variable structure control of interconnected multiinput=multioutput nonlinear systems, Circuits Systems Signal Process. 6 (1987) 363–387. [8] G.P. Matthews, R.A. DeCarlo, Decentralized tracking for a class of interconnected nonlinear systems using variable structure control, Automatica 24 (1988) 187–193. [9] E.P. Ryan, A variable structure approach to feedback regulation of uncertain dynamical systems, Internat. J. Control 38 (1983) 1121–1134. [10] K.-K. Shyu, H.-J. Shieh, A new switching surface sliding mode speed control for induction motor drive systems, IEEE Trans. Power Electron. PE-11 (1996) 660–667. [11] K.-K. Shyu, Y.-W. Tsai, C.-F. Yung, A modi>ed variable structure controller, Automatica 28 (1992) 1209–1213. [12] K.-K. Shyu, J.-J. Yan, Robust stability of uncertain time-delay systems and its stabilization by variable structure control, Internat. J. Control 57 (1993) 237–246. [13] J.J. Slotine, Sliding controller design for non-linear systems, Internat. J. Control 40 (1984) 421–434. [14] S.K. Spurgeon, Choice of discontinuous control component for robust sliding mode performance, Internat. J. Control 53 (1991) 163–179. [15] V.I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Control AC-22 (1977) 212–222. [16] V.I. Utkin, Sliding Modes and Their Application in Variable Structure Systems, MIR Publishers, Moscow, 1978. [17] V.I. Utkin, Variable structure systems, Automatica 19 (1983) 5–25. [18] X. Xu, Y. Wu, W. Huang, Variable structure control approach of decentralized model-reference adaptive systems, IEE Proc. D 137 (1990) 302–306. ˙ [19] S.H. Zak, S. Hui, Output feedback variable structure controllers and state estimators for uncertain=nonlinear dynamic systems, IEE Proc. D 140 (1993) 41–50.