Design of decentralised adaptive sliding mode ... - Semantic Scholar
International Journal of Control Vol. 81, No. 10, October 2008, 1507–1518
Design of decentralised adaptive sliding mode controllers for large-scale systems with mismatched perturbations Chih-Chiang Chenga* and Yaote Changab a
Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC; bDepartment of Electrical Engineering, Kao Yuan University, Kaohsiung County 821, Taiwan, ROC (Received 28 October 2006; final version received 28 September 2007) Based on the Lyapunov stability theorem, a methodology for designing a decentralised adaptive sliding mode control scheme is proposed in this paper. This scheme is implemented for a class of large-scale systems with both matched and mismatched perturbations. The perturbations and the interconnection terms are assumed to be norm bounded under certain mild conditions. The decentralised sliding surfaces with adaptive mechanisms embedded are specially designed for each subsystem, so that when each subsystem enters the sliding mode, the mismatched perturbations and the effects of interconnections can be effectively overcome and achieve asymptotic stability. The decentralised controller with embedded adaptive mechanisms is capable of driving the controlled state trajectories into the designated sliding surface in finite time. This is also achieved without the knowledge of upper bounds of the perturbations except those of the uncertainties in the input channels. A numerical example is included to demonstrate the feasibility of the proposed control scheme. Keywords: Lyapunov stability; adaptive sliding mode control; mismatched perturbations; asymptotical stability
1. Introduction It is generally considered that large-scale and complex systems are very difficult to stabilise with a single controller. This is due to computational complexity caused by large dimensions and effects of interconnections (Richter, Lefebvres, and DeCarlo 1982). Therefore, for designing a large-scale control system, the researchers in this field often divided the entire system into several subsystems, and utilised the decentralised controller to stabilise each subsystem. When the system contains only matched perturbations, the asymptotical stability can be obtained using the sliding mode control (SMC) technique (Chou and Cheng 2003, Shyu, Liu, and Hsu 2005), adaptive and neural techniques (Ge, Hang, and Zhang 2000), or adaptive and fuzzy methods (Tong, Li, and Chen 2004). It is noted that except Chou and Cheng (2003), the other three methodologies require the knowledge of the least upper bounds of perturbations. When the large-scale system contains mismatched perturbations, most researchers need the information of the upper bound of perturbations in order to achieve asymptotical stability. These include Mahmoud and Bingulac (1998), Lu, Mei, Song, Goto, and Konishi (2000), Xie and Xie (2000a,b), Shyu, Tsai, Yu, and Lai (2000), Tsai, Shyu and Chang (2001), Guang-Di and Guang-Da (2001), Liu and Huang (2001), Akar and *Corresponding author. Email: [email protected] ISSN 0020–7179 print/ISSN 1366–5820 online 2008 Taylor & Francis DOI: 10.1080/00207170701713788 http://www.informaworld.com
_ O¨zguner (2002), Guan, Chen, Yu, and Qin (2002), Jiang (2002), Park and Lee (2002), Wu, Xie, and Zhang (2004), Yan, Edwards, and Spurgeon (2004a), Yan, Edwards, Spurgeon, and Bleijs (2004b), Zhu and Pagilla (2007). As for controlling the perturbed largescale system without requiring the upperbound of perturbations, Chiang and Kuo (2002) designed an adaptive fuzzy sliding mode controller for a class of large-scale systems whose subsystems, however, all belong to single-input and single-output (SISO) systems. It is also observed that only some works (Xie and Xie 2000b, Guan et al. 2002, Park and Lee 2002, Wu et al. 2004) considered the uncertainties in the input channels. Chou and Cheng (2003) also proposed adaptive variable structure controllers for a class of large-scale time-varying delay systems. The controller achieved asymptotical stability without the knowledge of upper bounds of perturbations. However, this control scheme can only be applied to systems with matched perturbations. In this paper we propose another adaptive sliding mode control scheme for a class of large-scale MIMO systems with matched, mismatched perturbations and the uncertainties in the input channels to solve the regulation problems. First of all, we specifically design decentralised sliding surfaces for each subsystem to be controlled.
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C.-C. Cheng and Y. Chang
Thus when each subsystem enters the decentralised sliding surface, the trajectories of the state possess the property of asymptotical stability. The next step is to apply adaptive technique to the design of decentralised sliding surface and controller, so that not only the mismatched perturbations are effectively suppressed and thereby driving the state trajectories into the sliding surface within a finite time, but also the knowledge of the upper bounds of perturbations is not required except those of input uncertainties. The proposed technique can also be viewed as an extension of the work of Chou and Cheng (2003). Finally, a numerical example is demonstrated to show the applicability of the proposed design technique.
2. System descriptions and problem formulations Consider a class of MIMO mismatched uncertain large-scale systems consisting of L subsystems. The dynamic equations of each subsystem are governed by x_ i1 ¼ ½Fi1 þ Fi1 ðt, xi Þxi1 þ ½Gi ðt, xi Þ þ Gi ðt, xi Þxi2 þ
L X
F ij1 ðt, xi , xj Þxj1
ð1aÞ
j¼1 j6¼i
x_ i2 ¼ fi2 ðt, xi Þ þ fi2 ðt, xi Þ þ ½Bi ðt, xi Þ þ Bi ðt, xi Þui þ
L X
½F ij2 þ F ij2 ðt, xi , xj Þxj , xTi2 T
ni
where 1 i L, xi ¼ 2 R represents measurable state vector of the ith subsystem, xi1 2 Rni mi , xi2 2 Rmi , and mi 5 ni 2mi. The matrices Fi1 , Gi ðt, xi Þ ¼ G0i ðt, xi1 Þ G00i ðt, xi Þ 2 Rðni mi Þmi , G0i 2 00 ðni mi Þðni mi Þ ðni mi Þð2mi ni Þ mi mi R , Gi 2 R , Bi ðt, xi Þ 2 R are known, and ui ðtÞ 2 Rmi , 1 i L, is the control input of the ith subsystem, which is to be designed to regulate the whole perturbed large-scale system. Note that Fi1 is a constant matrix, the time-varying matrix Gi(t, xi) depends on state xi, and the time-varying matrix G0i ðt, xi1 Þ partitioned from Gi(t, xi) depends on substate vector xi1. The unknown matrices Fi1 2 Rðni mi Þðni mi Þ , Gi ðt,xi Þ ¼ ½G0i ðt,xi ÞG00i ðt,xi Þ 0 ðni mi Þmi ðni mi Þðni mi Þ 2R , Gi 2 R ,G00i 2Rðni mi Þð2mi ni Þ mi and vector fi2 2R denote the model uncertainty of the ith subsystem, and Bi is the uncertainty of the input channel. The interconnection terms L X j¼1 j6¼i
F ij1 xj1
and
L X j¼1 j6¼i
A1:
A2:
Both Bi(t, xi) and G0i ðt, xi1 Þ are non-singular matrices for all t and xi, and each entry of 0 G Þ is continuous in xi1. i ðt, 0xi101 G G r < 1, where r is an unknown i i positive constant. Note that in this paper kxk stands for the Euclidean norm of a vector x, kMk is the induced two-norm of a matrix M, and the index numbers i, j, ‘ are all non-negative integers.
In general, there are two important steps when utilising the SMC technique (Hung, Gao, and Hung 1993). The first step is to design a sliding surface function r, and then analyse the stability of the system when the dynamics of controlled system is in the sliding mode, i.e., r ¼ 0 (this step is presented in x 3). The second step is to design a control input function so that the reaching condition exists, i.e., r will approach zero in a finite time (this is addressed in x 4).
ð1bÞ
j¼1 j6¼i
½xTi1
in the dynamic equations of each subsystem reflect the fact that the performance of each subsystem will be affected by other subsystems. Note that F ij2 2 Rmi nj is a known constant matrix, and F ij1 2 Rðni mi Þðnj mj Þ , F ij2 2 Rmi nj are unknown matrices. In order to design the decentralised adaptive sliding mode controllers for regulating the dynamic system (1) in spite of the existence of both matched and mismatched perturbations, the following are assumed to be valid throughout this article:
3. Design of decentralised sliding surface In this section we design the decentralised sliding surface function by treating the state variable xi2 as a pseudo controller through the use of sliding function to stabilise the dynamic equation of xi1 subject to mismatched perturbations and interconnections. In the design of the pseudo controller xi2, we only consider the case that the dimension of xi1 is smaller or equal to that of the dimension xi2 in this paper, i.e., ni 2mi. Under this condition, one can achieve asymptotical stability for xi1 after the controlled system is in the sliding mode. We first decompose the state vector xi2 into x0i2 and T 0 ni mi x00i2 , i.e., xi2 ¼ ½x0T x00T , and i2 i2 , where xi2 2 R x00i2 2 R2mi ni . According to assumption A1, the sliding surface function can be designed as ri ðxÞ ¼ ½rTi1 rTi2 T , where ri1 2 Rni mi , ri2 2 R2mi ni and n h i o 0 ri1 ¼ x0i2 þ Gi1 ðt, xi1 Þ Ki þ k^i1 ðtÞ þ k^i2 ðtÞkxi1 k Ii xi1 ,
ðF ij2 þ F ij2 Þxj
ð2aÞ ri2 ¼
x00i2 ,
ð2bÞ
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International Journal of Control Ki 2 Rðni mi Þðni mi Þ is a matrix designed in such a way that lmax ðFci1 Þ < 0, Fci1 ¼ Fi1 Ki , Fci1 is a symmetry ðni mi Þðni mi Þ matrix, Ii 2 R . The adaptive gains k^i‘ ðtÞ, ‘ ¼ 1, 2, which are used to tackle the mismatched perturbations when system is in the sliding mode, are given by _ k^i‘ ðtÞ ¼ kxi1 k‘ ,
1 ‘ 2, 1 i L
ð3Þ
and k^i‘ ð0Þ ¼ 0. Now the following theorem illustrates that the property of asymptotical stability can indeed be achieved in the sliding mode if the sliding surface functions (2) are employed. Theorem 1: Consider the large-scale system (1) with assumptions A1–A2. If the sliding function ri and adaptive gains k^i‘ ðtÞ, ‘ ¼ 1, 2, of the ith subsystem are designed as (2) and (3) respectively, and if the mismatched perturbations and interconnections in the domain of interest satisfy Fi1 ðt, xi Þ i0 þ i1 kxi1 k, L L X X F ij1 ðt, xi , xj Þ a þ bi1 kxi1 k, i¼1 j6¼i
V_ 1 ¼
L X 1 T c xi1 Fi1 xi1 þxTi1 ðFi1 G0i G01 i Ki Þxi1 k x i1 k i¼1
ðk^i1 þ k^i2 kxi1 kÞxTi1 xi1 ðk^i1 þ k^i2 kxi1 kÞxTi1 G0i G0i 1 xi1 L 2 o X X _ þxTi1 F ij1 xi1 þð1rÞ ð8Þ k~i‘ k~i‘ : j¼1 j6¼i
L L X xTi1 X F ij1 xj1 kxi1 k j¼1 i¼1 j6¼i L X L L X L X X xT F ji1 kxi1 k j1 F ji1 xi1 ¼ xj1 i¼1 j¼1 j6¼i
L X
L X
Substituting (5) and x00i2 ¼ 0 into (1a) yields
ðk^i1 þ k^i2 kxi1 kÞxi1 þ
L X
V_ 1
i kxi1 k þ 12ð1 rÞ k~i12 þ k~i22 ,
L X L X
bj1 xj1 kxi1 k
i¼1 j¼1
akxi1 k þ
ð9Þ
L L X X 1 T C xi1 Fi1 xi1 þ ði0 þ i1 kxi1 k þ rkKi kÞkxi1 k kxi1 k i¼1 i¼1
þ
L X
ð1 rÞ k^i1 þ k^i2 kxi1 k kxi1 k
i¼1
"
L X
! # L X 1 akxi1 k þ Lbi1 þ bj1 kxi1 k2 2 j¼1
L h i X _ _ ð1 rÞ k~i1 k^i1 þ k~i2 k^i2 i¼1
Now we define a Lyapunov function candidate as L h X
akxi1 k þ
Hence, using (4), (9), assumption A2, and noting _ _ that k~i‘ ðtÞ ¼ k^i‘ ðtÞ, ‘ ¼ 1, 2, one can further simplify (8) as
ð6Þ
j¼1 j6¼i
V1 ¼
j¼1
i¼1
F ij1 xj1:
! bj1 xj1 kxi1 k
L X L 2 1X bj1 kxi1 k2 þxj1 2 i¼1 j¼1 i¼1 " ! # L L X X 1 2 akxi1 k þ bj1 kxi1 k : Lbi1 þ ¼ 2 i¼1 j¼1 L X
n h io Fi1 G0i G01 Ki þ k^i1 þ k^i2 kxi1 k Ii xi1 i
L X
i¼1
Proof: When the controlled system is in the sliding mode, ri ¼ 0, from (2) one can see that x00i2 ¼ 0, and h i x0i2 ¼ G01 Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 : ð5Þ i
i¼1 j¼1 j6¼i
aþ
i¼1
ð4Þ
i¼1
‘¼1
According to (4), it is observed that
¼
respectively in the sliding mode, where i0, i1, a, and bi1 are unknown positive constants, then the trajectories of all state variables xi, 1 i L, will reach zero asymptotically when the controlled system is in the sliding mode, and there exist finite constants k^i‘1 such that limt!1 k^i‘ ðtÞ ¼ k^i‘1 , ‘ ¼ 1, 2.
x_ i1 ¼Fci1 xi1 þ
the trajectories of (3) and (6) yields
ð7Þ
i¼1
where k~i‘ ðtÞ ¼ ki‘ k^i‘ ðtÞ, ‘ ¼ 1, 2, denote the adapta tion errors of the two unknown constants ki1 ¼ PL ði0 þ rkKi k þ aÞ=ð1 rÞ, ki2 ¼ ½i1 þ ðLbi1 þ j¼1 bj1 Þ=2= ð1 rÞ respectively. Differentiating (7) for xi1 6¼ 0 along
L L h X X 1 T c xi1 Fi1 xi1 þ ð1 rÞðki1 þ ki2 kxi1 kÞkxi1 k ¼ kxi1 k i¼1 i¼1 ð1 rÞ k^i1 þ k^i2 kxi1 k kxi1 k i ð1 rÞ k~i1 þ k~i2 kxi1 k kxi1 k
¼
L L X X 1 T c lmax ðFci1 Þkxi1 k 0: xi1 Fi1 xi1 kxi1 k i¼1 i¼1
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C.-C. Cheng and Y. Chang
When xi1 ¼ 0, (7) becomes L 1X ð1 rÞ k~2i1 þ k~2i2 : 2 i¼1
V1 ¼
ð10Þ
that the upper bound of F ij1 satisfies the following constraint F ij1 ðt, xi , xj Þ !0 þ !11 kx11 k þ !21 kx21 k þ þ !L1 kxL1 k
Differentiating (10) along the trajectories of (3) yields V_ 1 ¼
L X
_ _ ð1 rÞðk~i1 k~i1 þ k~i2 k~i2 Þ
i¼1
¼
L X
ð1 rÞ k~i1 kxi1 k þ k~i2 kxi1 k2 ¼ 0
i¼1
Z
t
xi1 ðÞ‘ d,
‘ Rt Since 0 xi1 ðÞ d is a non-decreasing function of t, and k^i‘ ðtÞ is also a bounded function, from Tao (2003, Proposition 2.14), there exist finite constants such that limt!1 k^i‘ ðtÞ ¼ k^i‘1 , 1 i L, 1 ‘ 2. œ Remark 1: When the controlled system is in the sliding mode, ri ¼ 0 and x00i2 ¼ 0, according to (5), x0i2 is a function of xi1 (the time-varying matrix G01 also i depends on xi1), therefore, the state vector xi is a function of xi1 and t when the system is in the sliding mode. This is the reason why we can assume that the functions Fi1 and F ij1 are bounded (in the sliding mode) by norm-bounded functions depending only kxi1 k in (4). Remark 2: When the controlled large-scale system is in the sliding mode, the interconnection term F ij1 may still receive influence from the jth subsystem. We have already considered this fact when we made the assumption about the upper bound of F ij1 in (4). This can be seen as follows. When the whole large-scale system is in the sliding mode, we assume
!i1 kxi1 k,
ð11Þ
where !0 4 0 and !i1 4 0 are unknown constants, and xi1, 1 i L, are the state variables from all subsystems. From (11), it can also be seen that L L L X L X X X F ij1 ðt, xi , xj Þ !0 þ !i1 kxi1 k i¼1
j¼1 i¼1
¼aþ
L X
bi1 kxi1 k,
ð12Þ
i¼1
where a ¼ L!0 and bi1 ¼ L!i1 are also unknown constants. From (11) and (12) it is clear that we have considered the effect of other subsystems on the upper bound of F ij1 when system is in the sliding mode. Remark 3: In Theorem 1 one can also assume that the mismatched perturbation and interconnection terms satisfy r1 X Fi1 ðt, xi Þ i‘1 kxi1 k‘1 ,
1 ‘ 2, 1 i L:
0
L X i¼1
i¼1 j6¼i
since xi1 ¼ 0. Hence the value of V1 will decrease until xi1 ¼ 0, which also means that the state variable xi1 will approach zero as t ! 1. The state variable x0i2 will also approach zero as t ! 1 according to (5) and assumption A1. On the other hand, x00i2 ¼ 0 when the controlled system is in the sliding mode. Therefore, the state variable xi will approach zero as t ! 1. Since V1 4 0 and V_ 1 0, V1 is a bounded function. Hence from (7) one can see that k~il and k^il , 1 ‘ 2, are all bounded when the controlled system is in the sliding mode. On the other hand, from (3), one can see that k^i‘ ðtÞ ¼ k^i‘ ð0Þ þ
¼ !0 þ
‘1 ¼0 r2 L X X F ij1 t, xi , xj a þ bi‘2 kxi1 k‘2 ,
L X i¼1 j6¼i
ð13Þ
i¼1 ‘2 ¼1
where a, i‘1 , bi‘2 , 0 ‘1 r1, 1 ‘2 r2 are unknown positive constants. The positive integers r1 and r2 are determined by the designer in accordance with the knowledge about the order of the perturbations. For example, if the perturbations contain a term such as x3i1 , then one may choose r1 ¼ 3 (or r2 ¼ 3). However, if x4i1 exists in the perturbation, then the inequality (13) might not be satisfied for certain domain of xi1 if one still chooses r1 ¼ 3 (or r2 ¼ 3). Therefore, the property of asymptotical stability proved in Theorem 1 is in the local sense. On the other hand, the magnitude of control input in general will become larger when a larger number r1 (or r2) is used. Although in this paper we choose r1 ¼ r2 ¼ 1, the design of sliding surface function and controller for the general case r1 4 1 (or r2 4 1) can be proceeded in a similar way. Remark 4: The reason why x0i2 will approach zero when xi1 reaches zero is explained as follows.
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International Journal of Control
0 According to assumption A1, G01 i ðt, xi1 Þ ¼ ½ðg i Þcj , 1 c, j ni mi, is also a non-singular matrix and each entry ðg 0i Þcj is also continuous in xi1. From (5), it can be seen that
x0i2c ¼
nX i mi j¼1
nX i mi
ðg 0i Þcj
nX i mi
The lumped perturbations pi, 1 i L, are partitioned as pi ðt, xi , xj , ui Þ ¼ ½p0i ðt, xi , xj , ui ÞT p00i ðt, xi , xj , ui ÞT T 0 ni mi pi 2 R , p00i 2 R2mi ni where
p0i ¼ f0i2 þ B0i ui þ
kij‘ xi1‘
‘¼1
i þ Hip Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 8 > : 0
1 c ni mi :
j¼1
where x0i2 ¼ ½x0i21 x0i22 x0i2ðni mi Þ T , xi1 ¼ ½xi11 xi12 xi1ðni mi Þ T , Ki ¼ [kij‘]. Therefore, x0i2c will reach zero when xi1j and xi1l approach zero since ðg 0i Þcj , k^il , k^i2 and kxi1 k are continuous in xi1. From the proof of Theorem 1, one can see that the purpose of k^i‘ , ‘ ¼ 1, 2 in (2) is to adapt the unknown constants k^i‘ so that the mismatched uncertainties and interconnections are suppressed in the sliding mode. Note that the limit values of these adaptive gains k^i‘ ðtÞ, i.e., k^i‘1 , may not be equal to their true values of ki‘ respectively, since V_ 1 is seminegative definite function. Although these mismatched uncertainties and interconnections will affect the dynamics of the systems in the sliding mode, as seen from (6), the property of asymptotical stability still can be achieved. 4. Design of decentralised controllers The second phase of designing the SMC scheme is to design the controllers so that the state trajectories can be driven into a designated sliding surface in finite time. Before introducing the proposed controllers, we first partitioned the vector fi2 and the matrix F ij2 as 0 00 00 T T respectively; fi2 ¼ ½ fi2T fi2T T and F ij2 ¼ ½ F 0T ij2 Fij2 0 00 ni mi 2mi ni , fi2 2 R , F 0ij2 2 Rðni mi Þnj , where fi2 2 R 00 ð2m n Þn F ij2 2 R i i j . Let Hin(t, xi) be the normal part of 0 ðd=dtÞ½Gi1 ðt, xi1 Þ, and Hip(t, xi, xj) be the part of 0 ðd=dtÞ½Gi1 ðt, xi1 Þ which contains perturbations. Then 0 the derivative of Gi1 ðt, xi1 Þ can be written as d 0 1 ½G ðt, xi1 Þ ¼ Hin ðt, xi Þ þ Hip ðt, xi , xj Þ, dt i where 0
0
@G 1 @G 1 Hin ðt, xi Þ ¼ i þ i ðFi1 xi1 þ Gi xi2 Þ, @t @xi1 0 @Gi1 ½Fi1 xi1 þ Gi xi2 Hip ðt, xi , xj Þ ¼ @xi1 L X þ F ij1 ðt, xi , xj Þxj1 : j¼1 j6¼i
F 0ij2 xj
j¼1 j6¼i
h
ðg 0i Þcj ðk^i1 þ k^i2 kxi1 kÞxi1j ,
L X
1
L X B C B F x þ G x þ F ij1 xj1 C i1 i1 i i2 @ A j¼1 j6¼i
0 þ
1
9 > > =
L X C k^i2 T B Fi1 xi1 þ Gi xi2 þ F ij1 xj1 C xi1 B @ Axi1 >, kxi1 k > j¼1 ; j6¼i
p00i ¼ f00i2 þ B00i ui þ
L X
F 00ij2 xj ,
j¼1 j6¼i 0
00
Bi ¼ ½ BiT Bi T T , B0i 2 Rðni mi Þmi , B00i 2 ð2mi ni Þmi R . According to assumption A1 and (3), the proposed controllers are designed as ui ¼ uin þ uis þ uip ,
ð14Þ
where T T T uin ¼ B1 ui1 ui2 , i
ui1 ¼ f0 i2 Hin ½Ki þ k^i1 þ k^i2 kxi1 k Ii xi1 ( h i 01 Ki þ k^i1 þ k^i2 kxi1 k Ii ðFi1 xi1 þ Gi xi2 Þ G i
# ) k^i2 T x ðFi1 xi1 þ Gi xi2 Þ xi1 , þ kxi1 k þ kxi1 k þ kxi1 k i1 "
3
ui2 ¼ f00i2 , uis ¼
L X B1 i ri F ji2 kxi k, kri k j¼1 j6¼i
B1 ri , i ðtÞ ¼ g^ i0 ðtÞ þ Lg^i1 ðtÞkxi k, uip ¼ ½i ðtÞ þ i ðtÞ i kri k 1 gi2 kuin þ uis k þ Bi i þ i ðtÞ ¼ , 1 gi2 B1 i the constant gi2 has to satisfy the constraint , gi2 < 1=B1 i
ð15Þ
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C.-C. Cheng and Y. Chang
and " is a designed positive constant. The adaptive gains g^ i‘ ðtÞ, ‘ ¼ 0, 1 are given by ( ‘ 1 ‘ _g^ ðtÞ ¼ L kxi k , if kri k 6¼ 0 , g^ i‘ ð0Þ ¼ 0, ð16Þ i‘ 0, if kri k ¼ 0
Now we define a Lyapunov function candidate as ! L 1 X 1 X 2 kri k þ ð21Þ g~ , V2 ¼ 2 ‘¼0 i‘ i¼1
where is a positive constant. The following theorem proves that the proposed controllers (14) can indeed drive the state trajectories into the designated sliding surface (2) in finite time.
where g~ i‘ ¼ g^ i‘ gi‘ , 0 ‘ 1 are the errors of adaptive gains. By using (14), one can obtain the time derivative of (21) along the trajectories of (20) and (16) for ri 6¼ 0 as
Theorem 2: Consider the large-scale system (1) with the assumptions A1–A2. If the sliding function, the controller, and the adaptive rules are designed as (2), (14) and (16) respectively, and the lumped perturbations pi, 1 i L, in the domain of interest satisfy the constraints L X p ðt, xi , xj , ui Þ gi0 þ gi1 kxi k þ gi2 kui k, i
L 1 X X rTi r_ i þ g~ i‘ g_~ i‘ V_ 2 ¼ k k r i i¼1 ‘¼0
8 >
¼
< L > X
!
2
3
L X rTi 6 6
1 X
9 > > =
7 g~i‘ g_~i‘ : Fij2 xj þpi þBi ðuis þuip Þ7 4 5 þ > > k k r i > > i¼1 : j¼1 ‘¼0 ; j6¼i
ð17Þ
ð22Þ
i¼1
where gi‘, 0 ‘ 1 are unknown positive constants, whereas gi2 is a known positive constant satisfying the inequality (15); then (a) the sliding variable ri will approach zero in a finite time tf, and the adaptive gains g^i‘ ðtÞ, ‘ ¼ 0, 1, are all bounded. (b) there exist finite constants gi‘1 such that limt!1 g^ i‘ ðtÞ ¼ gi‘1 , ‘ ¼ 0, 1. Proof: (a) From (2), one can obtain the time derivative of switching variable ri as i 9 d 0 1 h ½Gi Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 > > > > dt > > h i > > 0 > 1 = þ Gi Ki þ ðk^i1 þ k^i2 kxi1 kÞIi x_ i1 ð18Þ
> > d > _^ _^ > > þ ki1 þ ki2 kxi1 k þ k^i2 kxi1 k xi1 > > dt > > ; 00 r_ i2 ¼ x_ i2 : r_ i1 ¼ x_ 0i2 þ
Substituting (1) and (14) into (18) yields r_ i1 ¼
L X j¼1 j6¼i
r_ i2 ¼
L X
9 > F0 xj þ p0 þ B0 ðuis þ uip Þ > > > ij2 i i > > > = 00 F ij2 xj
j¼1 j6¼i
> > > þ p00i þ B00i ðuis þ uip Þ, > > > > ;
L X j¼1 j6¼i
F ij2 xj þ pi þ Bi ðuis þ uip Þ:
L L L X L rT L X L X X X rTi X F ji2 kxi k: j F ji2 xi F ij2 xj ¼ kri k j¼1 rj i¼1 i¼1 j¼1 i¼1 j¼1 j6¼i
j6¼i
j6¼i
ð23Þ Using (14), (16), (23), and noting that g_~i‘ ¼ g_~ i‘ , 0 ‘ 1, one can further derive (22) as " # L 1 T X X r i V_ 2 ðp þBi uip Þþ g~ i‘ g_~i‘ kri k i i¼1 ‘¼0 ! L 1 X X rTi _ g~i‘ g~ i‘ pi þ Bi uip þ kri k i¼1 ‘¼0 " L L X X gi0 þ g‘1 kx‘ kþgi2 kui kðg^ i0 þLg^ i1 kxi kþi Þ i¼1
‘¼1
#
þðg^ i0 gi0 ÞþLðg^i1 gi1 Þkxi k ¼
L X
ði þgi2 kui kÞ
i¼1
L X
ði þi Þ i þgi2 ½kuin þuis kþB1 i
i¼1
ð19Þ ¼
L X
þgi2 kuin þuis kþ B1 i i 1gi2 B1 i i
i¼1
00T T 0 ðni mi Þmi where Bi ¼ ½B0T , B00i 2 Rð2mi ni Þmi . i B i , Bi 2 R (19) can also be rewritten as
r_ i ¼
It is also observed that
¼
L X
i þgi2 ðkuin þuis k gi2 kuin þuis kþ B1 i
i¼1
i i þ B1 i
ð20Þ ¼
L X i¼1
¼ L < 0, ri 6¼ 0:
1513
International Journal of Control If ri ¼ 0, it is easy to verify that V_ 2 ¼ 0. The preceding analysis indicates that the value of ri will approach zero in a finite time tf, i.e., ri ¼ 0, 8t tf . It can be verified that tf t0 þ V2(t0)/L". Since V2 is a bounded function, the adaptive gains g^i‘ ðtÞ are all bounded. (b) From (a) it is seen that the value of V2 will decrease until ri(tf) ¼ 0, hence V_ 2 is actually a negative semidefinite function. In addition, from (16) it is known that g^ i‘ ðtÞ are monotonically increasing functions, and these functions are all bounded in accordance with (a). Therefore, according to Tao (2003, Proposition 2.14) one can conclude that there exist finite constants gi‘1 such that limt!1 g^ i‘ ðtÞ ¼ gi‘1 . It is also noted that gi‘1 may not necessarily be equal to gi‘. œ Remark 5: One of the key features of this paper is that the perturbations pi(t, xi, xj, ui), 1 i L, contain the input gain uncertainties, i.e., Bi(t, xi). Therefore, the upper bound of the perturbation pi depends on the control ui. In the proposed control scheme we use control effort ui to overcome the perturbation, however, some portion of Biui becomes the input uncertainty, i.e., Biui. It seems that there is a logic cycle with the uncertain bounds when we design the controller. For cutting this logic cycle, one way is to limit the portion of the input uncertainty Biui. This is why in the Theorem 2 the constant gi2 has to satisfy the constraint (15). For easy understanding of the physical meaning of (15), one can assume that Fi1 ¼ 0, Gi ¼ 0, F ij1 ¼ 0, fi2 ¼ 0, F ij2 ¼ 0 for simplicity. This means that the perturbation of each subsystem contains only input uncertainty. Then pi ðt, xi , xj , ui Þ ¼ kBi ui k gi2 kui k: ð24Þ (15) and (24) clearly indicate that the upper bound of perturbation induced by input gain uncertainty Biui is limited to gi2 kui k. Now if kBi k gi2 ,
ð25Þ
then by using (15), one can see that (25) implies kBi k: kBi k gi2 < 1=B1 i The above inequality is quite reasonable since in most physical systems the magnitude of the uncertainty of input gain kBi k is less than that of input gain kBi k. Remark 6: Note that the effect of other subsystems on the upper bound of pi has been considered when we made the assumption (17); the same reason as that explained in Remark 2. On the other hand, one
can also relax the constraints (17) to the following constraints r3 L X X pi ðt, xi , xj , ui Þ g0 þ g0i‘3 kxi k‘3 þgi2 kui k, i0 i¼1 ‘3 ¼1
ð26Þ where g0i‘3 , 1 ‘ r3 are unknown positive constants. The upper bound in (26) is non-linear for all the state variables of the controlled system, and the choice of r3 is similar to the method of choosing r1 described in Remark 3. Although in this paper we choose r3 ¼ 1, the design of controller for the general case r3 4 1 can be proceeded in a similar way. Remark 7: Theorem 2 clearly indicates that gi2 is a known positive constant and has to satisfy the constraint (15). Knowing gi2 one may require extra information about the model uncertainty of input gain, since input uncertainty in general often comes from the model uncertainty of input gains, i.e., Bi(t, xi)ui in this paper. For example, if one knows that in the worst case the uncertainty of input gain is at most 30% of kBi k, then one of the best choices of gi2 is to let gi2 ¼ 0:3kBi k. Note also that the upper bound of gi2 is not necessarily a small number. For example, if Bi ¼ Ii, then one of the choices of gi2 is to let gi2 ¼ 0.9. From the proof of Theorem 2, one can see that the purpose of the adaptive gains g^i‘ ðtÞ, 0 ‘ 1, is used to adapt the unknown constants gi‘ so that the perturbations (both matched and mismatched) can be suppressed during the reaching mode. The trajectories of the controlled system are capable of entering the sliding surface ri ¼ 0 in finite time. On the other hand, according to Theorem 1 the trajectory of state variable xi will approach zero asymptotically when the system is in the sliding mode. Therefore, one can conclude that the proposed control scheme can guarantee the stability of the controlled system for all time.
5. Numerical example Consider the following large-scale system with the dynamic equations in the form of (1) (L ¼ 2): Subsystem I: n1 ¼ 3, m1 ¼ 2, F11 ¼ 1, G1 ¼ ½1=ð1 þ 0:1 cos x11 Þ x12 T , x13 1 0 B1 ¼ , , f12 ¼ x211 0 1 0 1 3 : F 122 ¼ 1 1 2
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C.-C. Cheng and Y. Chang
Subsystem II: n2 ¼ 3, m2 ¼ 2, ½1=ð1 0:1 cos x21 Þ x22 T , "
B2 ¼
1 0 "
F 212 ¼
0
#
" f22 ¼
2
0
1 1
,
x23 x221
#
F21 ¼ 1,
G2 ¼
0
0
0:1
f22 ¼
,
F 212 ¼
x223 cosð0:5tÞ
#
x13 cos 0:5t 0
F11 ¼ 1 þ x11 sin t, G1 ¼ ½0:1 cos x12 0:2x213 sin2tT ,
F 121 ¼ x23 þ x22 þ x11 , " # 0:1 sin x11 0 , B1 ¼ 0 0:4sin3t cos x213 " # " # x12 sinð0:5tÞ 0 0 x11 f12 ¼ , F 122 ¼ : 0:5x213 x13 x11 sin t 0 Subsystem II: x223 cos 0:5tT ,
F 221 ¼ x13 þ x12 þ x23 ,
0
3.0 2.5
x11
2.0
x12
1.5
x13
1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 0
Figure 1. State trajectories x1.
2
#
0:4x12
The sliding function and controller are designed as (2) and (14), respectively. The design parameters are chosen to be (K1, K2, , ") ¼ (2, 2, 1, 0.1), so that the eigenvalues of Fc11 ¼ Fc21 are all located at 1. The initial condition of each adaptive rules is zero, and the initial condition of subsystem I and subsystem II are x1 ð0Þ ¼ ½2 2 3T , T x2 ð0Þ ¼ ½2 2 4 . The state trajectories of the subsystem I and the subsystem II are shown in Figures 1 and 2, respectively. One can see that the states of each subsystem are all driven to zero. The control inputs are displayed from Figures 3–6, respectively. The chattering phenomenon can be alleviated by replacing the function ri =kri k with saturation function (Hung et al. 1993). From (14), it can be seen that the control input ui depends on the adaptive gains k^1 ðtÞ, k^2 ðtÞ, and the magnitude of kxi1 k3 , hence if the deviation of the state variable xi1 from the origin is large (in our proposed control scheme, this may happen at t ¼ 0), then a large control effort is needed to overcome this uncertainty before the system enters the sliding mode. Nevertheless, the value of state variable xi1 will decrease asymptotically once the system enters
Subsystem I:
0:5x11 x21
: 0:2x13 x23
3
F21 ¼ 1 þ x21 , G2 ¼ ½0:1 sin x21
,
0:1x221 "
# ,
"
#
To demonstrate the robustness of the proposed controller, it is assumed that the model uncertainties, input uncertainties, disturbances and/or nonlinearities are as follows:
0:2 cos x21 cos 0:4t
B2 ¼
: 1 0
"
4
6
8
1515
International Journal of Control 4.5 4.0 3.5 3.0
x21
2.5
x22
2.0
x23
1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 0
2
4
6
8
2
4
6
8
Figure 2. State trajectories x2.
u11
15 10 5 0 −5 −10 −15 0
Figure 3. Control input u11.
u12
10
0
−10
−20
−30 0
Figure 4. Control input u12.
2
4
6
8
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C.-C. Cheng and Y. Chang 15
u21
10
5
0 −5 −10 −15 −20 0
2
4
6
8
2
4
6
8
Figure 5. Control input u21.
20
u22
15 10 5 0 −5 −10 −15 −20 −25 0
Figure 6. Control input u22.
the sliding surface, as proved in Theorem 1. The magnitude of control ui will also decrease rapidly. Figure 7 illustrates that the sliding variable of each subsystem also approaches zero in finite time. The adaptive gains are shown in Figures 8 and 9, where each adaptive gain approaches a finite constant respectively.
6. Conclusions In this paper, a decentralised adaptive sliding mode control scheme is successfully proposed for a class of
mismatch perturbed large-scale systems. The proposed controller is capable of driving each subsystem into a specially designed sliding surface in a finite time without requiring the knowledge of upper bounds of perturbations except that of input uncertainty. Although the mismatched perturbations and interconnections still affect the dynamics of the controlled system in the sliding mode, they are all suppressed effectively, and each subsystem as well as the whole controlled large-scale system can achieve asymptotical stability. For future study, the case of ni 4 2mi is worth consideration.
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International Journal of Control 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5
σ21 σ22 σ11 σ12
0
2
4
6
8
Figure 7. Sliding surface variables r1 and r2.
1.2
1.0
k21
k11
k22
k12
0.8
0.6
0.4
0.2
0.0 0
1
2
3
4
5
6
7
8
Figure 8. Adaptive gains k^11 , k^12 , k^21 and k^22 . 1.8 1.6 1.4
g20
g10
g21
g11
1.2 1.0 0.8 0.6 0.4 0.2 0.0 0
Figure 9. Adaptive gains g^ 10 , g^ 11 , g^20 and g^ 21 .
2
4
6
8
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C.-C. Cheng and Y. Chang
Acknowledgements The author would like to thank the Editor, Associate Editor, and the anonymous reviewers for their many helpful comments and suggestions that have helped to improve the quality of this paper. The authors are also grateful to the National Science Council of R. O. C. for financial support for this research (NSC93-2218-E-110-033).
References Akar, M., and O¨zguner, U¨. (2002), ‘Decentralized Sliding Mode Control Design Using Overlapping Decompositions,’ Automatica, 38, 1713–1718. Chiang, C.-C., and Kuo, Z.-H. (2002), ‘Decentralized Adaptive Fuzzy Controller Design of Large-Scale Nonlinear Systems with Unmatched Uncertainties,’ Proceedings of the 2002 IEEE International Conference, 1, pp. 668–673. Chou, C.-H., and Cheng, C.C. (2003), ‘A Decentralized Model Reference Adaptive Variable Structure Controller for Large-Scale Time-Varying Delay Systems,’ IEEE Transactions on Automatic Control, 48, 1213–1217. Ge, S.S., Hang, C.C., and Zhang, T. (2000), ‘Stable Adaptive Control for Nonlinear Multivariable Systems with a Triangular Control Structure,’ IEEE Transactions on Automatic Control, 45, 1221–1225. Guan, Z.-H., Chen, G., Yu, X., and Qin, Y. (2002), ‘Robust Decentralized Stabilization for a Class of Large-Scale Time-Delay Uncertain Impulsive Dynamical Systems,’ Automatica, 38, 2075–2084. Guang-Di, H., and Guang-Da, H. (2001), ‘Stabilization of an Uncertain Large-Scale Time-Dependent Bilinear Neutral Differential System by Memory Feedback Control,’ IMA Journal of Mathematical Control and Information, 18, 1–18. Hung, J.Y., Gao, W., and Hung, J.C. (1993), ‘Variable Structure Control: A Survey,’ IEEE Transactions on Industrial Electronic, 40, 2–22. Jiang, Z.-P. (2002), ‘Decentralized Disturbance Attenuating Output-Feedback Trackers for Large-Scale Nonlinear Systems,’ Automatica, 38, 1407–1415. Liu, X., and Huang, G. (2001), ‘Global Decentralized Robust Stabilization for Interconnected Uncertain Nonlinear Systems with Multiple Inputs,’ Automatica, 37, 1435–1442. Lu, O., Mei, S., Song, Y.H., Goto, M., and Konishi, H. (2000), ‘Decentralized Nonlinear H1 Excitation Control Based on Regulation Linearization,’ IEEE Proceedings – Generation Transmission & Distribution, 174, 245–251. Mahmoud, M.S., and Bingulac, S. (1998), ‘Robust Design of Stabilizing Controllers for Interconnected Time-Delay Systems,’ Automatica, 34, 795–800.
Park, J.H., and Lee, S.G. (2002), ‘Robust Decentralized Stabilization of Uncertain Large-Scale Discrete-Time Systems,’ International Journal of Systems Science, 33, 649–654. Richter, R., Lefebvres, S., and DeCarlo, R. (1982), ‘Control of a Class of Nonlinear Systems by Decentralized Control,’ IEEE Transactions on Automatic Control, 27, 492–494. Shyu, K.-K., Liu, W.-J., and Hsu, K.-C. (2005), ‘Design of Large-Scale Time-Delayed Systems with Dead-Zone Input via Variable Structure Control,’ Automatica, 41, 1239–1246. Shyu, K.-K., Tsai, Y.-W., Yu, Y., and Lai, C.-K. (2000), ‘Decentralized Dynamic Output Feedback Controllers for Interconnected Systems with Mismatched Uncertainties Using New Sliding Mode Control,’ International Journal of Systems Science, 31, 1011–1020. Tao, G. (2003), Adaptive Control Design and Analysis, New Jersey: John Wiley & Sons. Tong, S., Li, H.-X., and Chen, G. (2004), ‘Adaptive Fuzzy Decentralized Control for a Class of Large-Scale Nonlinear Systems,’ IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 34, 770–775. Tsai, Y.W., Shyu, K.K., and Chang, K.C. (2001), ‘Decentralized Variable Structure Control for Mismatched Uncertain Large-Scale Systems: A New Approach,’ Systems & Control Letters, 43, 117–125. Wu, Z.-J., Xie, V., and Zhang, S.-Y. (2004), ‘Robust Decentralized Adaptive Stabilization for a Class of Interconnected Systems with Unmodeled Dynamics,’ International Journal of Systems Science, 35, 389–404. Xie, S., and Xie, L. (2000a), ‘Decentralized Global Disturbance Attenuation for a Class of Large-Scale Uncertain Nonlinear Systems,’ International Journal of Systems Science, 31, 1285–1297. Xie, S., and Xie, L. (2000b), ‘Stabilization of a Class of Uncertain Large-scale Stochastic Systems with Time Delays,’ Automatica, 36, 161–167. Yan, X.-G., Edwards, C., and Spurgeon, S.K. (2004a), ‘Decentralized Robust Sliding Mode Control for a Class of Nonlinear Interconnected Systems by Static Output Feedback,’ Automatica, 40, 613–620. Yan, X.-G., Edwards, C., Spurgeon, S.K., and Bleijs, J.A.M. (2004b), ‘‘Decentralized Sliding-Mode Control for Multimachine Power Systems Using only Output Information,’’ IEEE Proceedings – Control Theory and Applications, 151, 627–635. Zhu, Y., and Pagilla, P.R. (2007), ‘Decentralized Output Feedback Control for a Class of Large-Scale Interconnected Systems,’ IMA Journal of Mathematical Control and Information, 24, 57–69.
Design of decentralised adaptive sliding mode controllers for large-scale systems with mismatched perturbations Chih-Chiang Chenga* and Yaote Changab a
Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, ROC; bDepartment of Electrical Engineering, Kao Yuan University, Kaohsiung County 821, Taiwan, ROC (Received 28 October 2006; final version received 28 September 2007) Based on the Lyapunov stability theorem, a methodology for designing a decentralised adaptive sliding mode control scheme is proposed in this paper. This scheme is implemented for a class of large-scale systems with both matched and mismatched perturbations. The perturbations and the interconnection terms are assumed to be norm bounded under certain mild conditions. The decentralised sliding surfaces with adaptive mechanisms embedded are specially designed for each subsystem, so that when each subsystem enters the sliding mode, the mismatched perturbations and the effects of interconnections can be effectively overcome and achieve asymptotic stability. The decentralised controller with embedded adaptive mechanisms is capable of driving the controlled state trajectories into the designated sliding surface in finite time. This is also achieved without the knowledge of upper bounds of the perturbations except those of the uncertainties in the input channels. A numerical example is included to demonstrate the feasibility of the proposed control scheme. Keywords: Lyapunov stability; adaptive sliding mode control; mismatched perturbations; asymptotical stability
1. Introduction It is generally considered that large-scale and complex systems are very difficult to stabilise with a single controller. This is due to computational complexity caused by large dimensions and effects of interconnections (Richter, Lefebvres, and DeCarlo 1982). Therefore, for designing a large-scale control system, the researchers in this field often divided the entire system into several subsystems, and utilised the decentralised controller to stabilise each subsystem. When the system contains only matched perturbations, the asymptotical stability can be obtained using the sliding mode control (SMC) technique (Chou and Cheng 2003, Shyu, Liu, and Hsu 2005), adaptive and neural techniques (Ge, Hang, and Zhang 2000), or adaptive and fuzzy methods (Tong, Li, and Chen 2004). It is noted that except Chou and Cheng (2003), the other three methodologies require the knowledge of the least upper bounds of perturbations. When the large-scale system contains mismatched perturbations, most researchers need the information of the upper bound of perturbations in order to achieve asymptotical stability. These include Mahmoud and Bingulac (1998), Lu, Mei, Song, Goto, and Konishi (2000), Xie and Xie (2000a,b), Shyu, Tsai, Yu, and Lai (2000), Tsai, Shyu and Chang (2001), Guang-Di and Guang-Da (2001), Liu and Huang (2001), Akar and *Corresponding author. Email: [email protected] ISSN 0020–7179 print/ISSN 1366–5820 online 2008 Taylor & Francis DOI: 10.1080/00207170701713788 http://www.informaworld.com
_ O¨zguner (2002), Guan, Chen, Yu, and Qin (2002), Jiang (2002), Park and Lee (2002), Wu, Xie, and Zhang (2004), Yan, Edwards, and Spurgeon (2004a), Yan, Edwards, Spurgeon, and Bleijs (2004b), Zhu and Pagilla (2007). As for controlling the perturbed largescale system without requiring the upperbound of perturbations, Chiang and Kuo (2002) designed an adaptive fuzzy sliding mode controller for a class of large-scale systems whose subsystems, however, all belong to single-input and single-output (SISO) systems. It is also observed that only some works (Xie and Xie 2000b, Guan et al. 2002, Park and Lee 2002, Wu et al. 2004) considered the uncertainties in the input channels. Chou and Cheng (2003) also proposed adaptive variable structure controllers for a class of large-scale time-varying delay systems. The controller achieved asymptotical stability without the knowledge of upper bounds of perturbations. However, this control scheme can only be applied to systems with matched perturbations. In this paper we propose another adaptive sliding mode control scheme for a class of large-scale MIMO systems with matched, mismatched perturbations and the uncertainties in the input channels to solve the regulation problems. First of all, we specifically design decentralised sliding surfaces for each subsystem to be controlled.
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C.-C. Cheng and Y. Chang
Thus when each subsystem enters the decentralised sliding surface, the trajectories of the state possess the property of asymptotical stability. The next step is to apply adaptive technique to the design of decentralised sliding surface and controller, so that not only the mismatched perturbations are effectively suppressed and thereby driving the state trajectories into the sliding surface within a finite time, but also the knowledge of the upper bounds of perturbations is not required except those of input uncertainties. The proposed technique can also be viewed as an extension of the work of Chou and Cheng (2003). Finally, a numerical example is demonstrated to show the applicability of the proposed design technique.
2. System descriptions and problem formulations Consider a class of MIMO mismatched uncertain large-scale systems consisting of L subsystems. The dynamic equations of each subsystem are governed by x_ i1 ¼ ½Fi1 þ Fi1 ðt, xi Þxi1 þ ½Gi ðt, xi Þ þ Gi ðt, xi Þxi2 þ
L X
F ij1 ðt, xi , xj Þxj1
ð1aÞ
j¼1 j6¼i
x_ i2 ¼ fi2 ðt, xi Þ þ fi2 ðt, xi Þ þ ½Bi ðt, xi Þ þ Bi ðt, xi Þui þ
L X
½F ij2 þ F ij2 ðt, xi , xj Þxj , xTi2 T
ni
where 1 i L, xi ¼ 2 R represents measurable state vector of the ith subsystem, xi1 2 Rni mi , xi2 2 Rmi , and mi 5 ni 2mi. The matrices Fi1 , Gi ðt, xi Þ ¼ G0i ðt, xi1 Þ G00i ðt, xi Þ 2 Rðni mi Þmi , G0i 2 00 ðni mi Þðni mi Þ ðni mi Þð2mi ni Þ mi mi R , Gi 2 R , Bi ðt, xi Þ 2 R are known, and ui ðtÞ 2 Rmi , 1 i L, is the control input of the ith subsystem, which is to be designed to regulate the whole perturbed large-scale system. Note that Fi1 is a constant matrix, the time-varying matrix Gi(t, xi) depends on state xi, and the time-varying matrix G0i ðt, xi1 Þ partitioned from Gi(t, xi) depends on substate vector xi1. The unknown matrices Fi1 2 Rðni mi Þðni mi Þ , Gi ðt,xi Þ ¼ ½G0i ðt,xi ÞG00i ðt,xi Þ 0 ðni mi Þmi ðni mi Þðni mi Þ 2R , Gi 2 R ,G00i 2Rðni mi Þð2mi ni Þ mi and vector fi2 2R denote the model uncertainty of the ith subsystem, and Bi is the uncertainty of the input channel. The interconnection terms L X j¼1 j6¼i
F ij1 xj1
and
L X j¼1 j6¼i
A1:
A2:
Both Bi(t, xi) and G0i ðt, xi1 Þ are non-singular matrices for all t and xi, and each entry of 0 G Þ is continuous in xi1. i ðt, 0xi101 G G r < 1, where r is an unknown i i positive constant. Note that in this paper kxk stands for the Euclidean norm of a vector x, kMk is the induced two-norm of a matrix M, and the index numbers i, j, ‘ are all non-negative integers.
In general, there are two important steps when utilising the SMC technique (Hung, Gao, and Hung 1993). The first step is to design a sliding surface function r, and then analyse the stability of the system when the dynamics of controlled system is in the sliding mode, i.e., r ¼ 0 (this step is presented in x 3). The second step is to design a control input function so that the reaching condition exists, i.e., r will approach zero in a finite time (this is addressed in x 4).
ð1bÞ
j¼1 j6¼i
½xTi1
in the dynamic equations of each subsystem reflect the fact that the performance of each subsystem will be affected by other subsystems. Note that F ij2 2 Rmi nj is a known constant matrix, and F ij1 2 Rðni mi Þðnj mj Þ , F ij2 2 Rmi nj are unknown matrices. In order to design the decentralised adaptive sliding mode controllers for regulating the dynamic system (1) in spite of the existence of both matched and mismatched perturbations, the following are assumed to be valid throughout this article:
3. Design of decentralised sliding surface In this section we design the decentralised sliding surface function by treating the state variable xi2 as a pseudo controller through the use of sliding function to stabilise the dynamic equation of xi1 subject to mismatched perturbations and interconnections. In the design of the pseudo controller xi2, we only consider the case that the dimension of xi1 is smaller or equal to that of the dimension xi2 in this paper, i.e., ni 2mi. Under this condition, one can achieve asymptotical stability for xi1 after the controlled system is in the sliding mode. We first decompose the state vector xi2 into x0i2 and T 0 ni mi x00i2 , i.e., xi2 ¼ ½x0T x00T , and i2 i2 , where xi2 2 R x00i2 2 R2mi ni . According to assumption A1, the sliding surface function can be designed as ri ðxÞ ¼ ½rTi1 rTi2 T , where ri1 2 Rni mi , ri2 2 R2mi ni and n h i o 0 ri1 ¼ x0i2 þ Gi1 ðt, xi1 Þ Ki þ k^i1 ðtÞ þ k^i2 ðtÞkxi1 k Ii xi1 ,
ðF ij2 þ F ij2 Þxj
ð2aÞ ri2 ¼
x00i2 ,
ð2bÞ
1509
International Journal of Control Ki 2 Rðni mi Þðni mi Þ is a matrix designed in such a way that lmax ðFci1 Þ < 0, Fci1 ¼ Fi1 Ki , Fci1 is a symmetry ðni mi Þðni mi Þ matrix, Ii 2 R . The adaptive gains k^i‘ ðtÞ, ‘ ¼ 1, 2, which are used to tackle the mismatched perturbations when system is in the sliding mode, are given by _ k^i‘ ðtÞ ¼ kxi1 k‘ ,
1 ‘ 2, 1 i L
ð3Þ
and k^i‘ ð0Þ ¼ 0. Now the following theorem illustrates that the property of asymptotical stability can indeed be achieved in the sliding mode if the sliding surface functions (2) are employed. Theorem 1: Consider the large-scale system (1) with assumptions A1–A2. If the sliding function ri and adaptive gains k^i‘ ðtÞ, ‘ ¼ 1, 2, of the ith subsystem are designed as (2) and (3) respectively, and if the mismatched perturbations and interconnections in the domain of interest satisfy Fi1 ðt, xi Þ i0 þ i1 kxi1 k, L L X X F ij1 ðt, xi , xj Þ a þ bi1 kxi1 k, i¼1 j6¼i
V_ 1 ¼
L X 1 T c xi1 Fi1 xi1 þxTi1 ðFi1 G0i G01 i Ki Þxi1 k x i1 k i¼1
ðk^i1 þ k^i2 kxi1 kÞxTi1 xi1 ðk^i1 þ k^i2 kxi1 kÞxTi1 G0i G0i 1 xi1 L 2 o X X _ þxTi1 F ij1 xi1 þð1rÞ ð8Þ k~i‘ k~i‘ : j¼1 j6¼i
L L X xTi1 X F ij1 xj1 kxi1 k j¼1 i¼1 j6¼i L X L L X L X X xT F ji1 kxi1 k j1 F ji1 xi1 ¼ xj1 i¼1 j¼1 j6¼i
L X
L X
Substituting (5) and x00i2 ¼ 0 into (1a) yields
ðk^i1 þ k^i2 kxi1 kÞxi1 þ
L X
V_ 1
i kxi1 k þ 12ð1 rÞ k~i12 þ k~i22 ,
L X L X
bj1 xj1 kxi1 k
i¼1 j¼1
akxi1 k þ
ð9Þ
L L X X 1 T C xi1 Fi1 xi1 þ ði0 þ i1 kxi1 k þ rkKi kÞkxi1 k kxi1 k i¼1 i¼1
þ
L X
ð1 rÞ k^i1 þ k^i2 kxi1 k kxi1 k
i¼1
"
L X
! # L X 1 akxi1 k þ Lbi1 þ bj1 kxi1 k2 2 j¼1
L h i X _ _ ð1 rÞ k~i1 k^i1 þ k~i2 k^i2 i¼1
Now we define a Lyapunov function candidate as L h X
akxi1 k þ
Hence, using (4), (9), assumption A2, and noting _ _ that k~i‘ ðtÞ ¼ k^i‘ ðtÞ, ‘ ¼ 1, 2, one can further simplify (8) as
ð6Þ
j¼1 j6¼i
V1 ¼
j¼1
i¼1
F ij1 xj1:
! bj1 xj1 kxi1 k
L X L 2 1X bj1 kxi1 k2 þxj1 2 i¼1 j¼1 i¼1 " ! # L L X X 1 2 akxi1 k þ bj1 kxi1 k : Lbi1 þ ¼ 2 i¼1 j¼1 L X
n h io Fi1 G0i G01 Ki þ k^i1 þ k^i2 kxi1 k Ii xi1 i
L X
i¼1
Proof: When the controlled system is in the sliding mode, ri ¼ 0, from (2) one can see that x00i2 ¼ 0, and h i x0i2 ¼ G01 Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 : ð5Þ i
i¼1 j¼1 j6¼i
aþ
i¼1
ð4Þ
i¼1
‘¼1
According to (4), it is observed that
¼
respectively in the sliding mode, where i0, i1, a, and bi1 are unknown positive constants, then the trajectories of all state variables xi, 1 i L, will reach zero asymptotically when the controlled system is in the sliding mode, and there exist finite constants k^i‘1 such that limt!1 k^i‘ ðtÞ ¼ k^i‘1 , ‘ ¼ 1, 2.
x_ i1 ¼Fci1 xi1 þ
the trajectories of (3) and (6) yields
ð7Þ
i¼1
where k~i‘ ðtÞ ¼ ki‘ k^i‘ ðtÞ, ‘ ¼ 1, 2, denote the adapta tion errors of the two unknown constants ki1 ¼ PL ði0 þ rkKi k þ aÞ=ð1 rÞ, ki2 ¼ ½i1 þ ðLbi1 þ j¼1 bj1 Þ=2= ð1 rÞ respectively. Differentiating (7) for xi1 6¼ 0 along
L L h X X 1 T c xi1 Fi1 xi1 þ ð1 rÞðki1 þ ki2 kxi1 kÞkxi1 k ¼ kxi1 k i¼1 i¼1 ð1 rÞ k^i1 þ k^i2 kxi1 k kxi1 k i ð1 rÞ k~i1 þ k~i2 kxi1 k kxi1 k
¼
L L X X 1 T c lmax ðFci1 Þkxi1 k 0: xi1 Fi1 xi1 kxi1 k i¼1 i¼1
1510
C.-C. Cheng and Y. Chang
When xi1 ¼ 0, (7) becomes L 1X ð1 rÞ k~2i1 þ k~2i2 : 2 i¼1
V1 ¼
ð10Þ
that the upper bound of F ij1 satisfies the following constraint F ij1 ðt, xi , xj Þ !0 þ !11 kx11 k þ !21 kx21 k þ þ !L1 kxL1 k
Differentiating (10) along the trajectories of (3) yields V_ 1 ¼
L X
_ _ ð1 rÞðk~i1 k~i1 þ k~i2 k~i2 Þ
i¼1
¼
L X
ð1 rÞ k~i1 kxi1 k þ k~i2 kxi1 k2 ¼ 0
i¼1
Z
t
xi1 ðÞ‘ d,
‘ Rt Since 0 xi1 ðÞ d is a non-decreasing function of t, and k^i‘ ðtÞ is also a bounded function, from Tao (2003, Proposition 2.14), there exist finite constants such that limt!1 k^i‘ ðtÞ ¼ k^i‘1 , 1 i L, 1 ‘ 2. œ Remark 1: When the controlled system is in the sliding mode, ri ¼ 0 and x00i2 ¼ 0, according to (5), x0i2 is a function of xi1 (the time-varying matrix G01 also i depends on xi1), therefore, the state vector xi is a function of xi1 and t when the system is in the sliding mode. This is the reason why we can assume that the functions Fi1 and F ij1 are bounded (in the sliding mode) by norm-bounded functions depending only kxi1 k in (4). Remark 2: When the controlled large-scale system is in the sliding mode, the interconnection term F ij1 may still receive influence from the jth subsystem. We have already considered this fact when we made the assumption about the upper bound of F ij1 in (4). This can be seen as follows. When the whole large-scale system is in the sliding mode, we assume
!i1 kxi1 k,
ð11Þ
where !0 4 0 and !i1 4 0 are unknown constants, and xi1, 1 i L, are the state variables from all subsystems. From (11), it can also be seen that L L L X L X X X F ij1 ðt, xi , xj Þ !0 þ !i1 kxi1 k i¼1
j¼1 i¼1
¼aþ
L X
bi1 kxi1 k,
ð12Þ
i¼1
where a ¼ L!0 and bi1 ¼ L!i1 are also unknown constants. From (11) and (12) it is clear that we have considered the effect of other subsystems on the upper bound of F ij1 when system is in the sliding mode. Remark 3: In Theorem 1 one can also assume that the mismatched perturbation and interconnection terms satisfy r1 X Fi1 ðt, xi Þ i‘1 kxi1 k‘1 ,
1 ‘ 2, 1 i L:
0
L X i¼1
i¼1 j6¼i
since xi1 ¼ 0. Hence the value of V1 will decrease until xi1 ¼ 0, which also means that the state variable xi1 will approach zero as t ! 1. The state variable x0i2 will also approach zero as t ! 1 according to (5) and assumption A1. On the other hand, x00i2 ¼ 0 when the controlled system is in the sliding mode. Therefore, the state variable xi will approach zero as t ! 1. Since V1 4 0 and V_ 1 0, V1 is a bounded function. Hence from (7) one can see that k~il and k^il , 1 ‘ 2, are all bounded when the controlled system is in the sliding mode. On the other hand, from (3), one can see that k^i‘ ðtÞ ¼ k^i‘ ð0Þ þ
¼ !0 þ
‘1 ¼0 r2 L X X F ij1 t, xi , xj a þ bi‘2 kxi1 k‘2 ,
L X i¼1 j6¼i
ð13Þ
i¼1 ‘2 ¼1
where a, i‘1 , bi‘2 , 0 ‘1 r1, 1 ‘2 r2 are unknown positive constants. The positive integers r1 and r2 are determined by the designer in accordance with the knowledge about the order of the perturbations. For example, if the perturbations contain a term such as x3i1 , then one may choose r1 ¼ 3 (or r2 ¼ 3). However, if x4i1 exists in the perturbation, then the inequality (13) might not be satisfied for certain domain of xi1 if one still chooses r1 ¼ 3 (or r2 ¼ 3). Therefore, the property of asymptotical stability proved in Theorem 1 is in the local sense. On the other hand, the magnitude of control input in general will become larger when a larger number r1 (or r2) is used. Although in this paper we choose r1 ¼ r2 ¼ 1, the design of sliding surface function and controller for the general case r1 4 1 (or r2 4 1) can be proceeded in a similar way. Remark 4: The reason why x0i2 will approach zero when xi1 reaches zero is explained as follows.
1511
International Journal of Control
0 According to assumption A1, G01 i ðt, xi1 Þ ¼ ½ðg i Þcj , 1 c, j ni mi, is also a non-singular matrix and each entry ðg 0i Þcj is also continuous in xi1. From (5), it can be seen that
x0i2c ¼
nX i mi j¼1
nX i mi
ðg 0i Þcj
nX i mi
The lumped perturbations pi, 1 i L, are partitioned as pi ðt, xi , xj , ui Þ ¼ ½p0i ðt, xi , xj , ui ÞT p00i ðt, xi , xj , ui ÞT T 0 ni mi pi 2 R , p00i 2 R2mi ni where
p0i ¼ f0i2 þ B0i ui þ
kij‘ xi1‘
‘¼1
i þ Hip Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 8 > : 0
1 c ni mi :
j¼1
where x0i2 ¼ ½x0i21 x0i22 x0i2ðni mi Þ T , xi1 ¼ ½xi11 xi12 xi1ðni mi Þ T , Ki ¼ [kij‘]. Therefore, x0i2c will reach zero when xi1j and xi1l approach zero since ðg 0i Þcj , k^il , k^i2 and kxi1 k are continuous in xi1. From the proof of Theorem 1, one can see that the purpose of k^i‘ , ‘ ¼ 1, 2 in (2) is to adapt the unknown constants k^i‘ so that the mismatched uncertainties and interconnections are suppressed in the sliding mode. Note that the limit values of these adaptive gains k^i‘ ðtÞ, i.e., k^i‘1 , may not be equal to their true values of ki‘ respectively, since V_ 1 is seminegative definite function. Although these mismatched uncertainties and interconnections will affect the dynamics of the systems in the sliding mode, as seen from (6), the property of asymptotical stability still can be achieved. 4. Design of decentralised controllers The second phase of designing the SMC scheme is to design the controllers so that the state trajectories can be driven into a designated sliding surface in finite time. Before introducing the proposed controllers, we first partitioned the vector fi2 and the matrix F ij2 as 0 00 00 T T respectively; fi2 ¼ ½ fi2T fi2T T and F ij2 ¼ ½ F 0T ij2 Fij2 0 00 ni mi 2mi ni , fi2 2 R , F 0ij2 2 Rðni mi Þnj , where fi2 2 R 00 ð2m n Þn F ij2 2 R i i j . Let Hin(t, xi) be the normal part of 0 ðd=dtÞ½Gi1 ðt, xi1 Þ, and Hip(t, xi, xj) be the part of 0 ðd=dtÞ½Gi1 ðt, xi1 Þ which contains perturbations. Then 0 the derivative of Gi1 ðt, xi1 Þ can be written as d 0 1 ½G ðt, xi1 Þ ¼ Hin ðt, xi Þ þ Hip ðt, xi , xj Þ, dt i where 0
0
@G 1 @G 1 Hin ðt, xi Þ ¼ i þ i ðFi1 xi1 þ Gi xi2 Þ, @t @xi1 0 @Gi1 ½Fi1 xi1 þ Gi xi2 Hip ðt, xi , xj Þ ¼ @xi1 L X þ F ij1 ðt, xi , xj Þxj1 : j¼1 j6¼i
F 0ij2 xj
j¼1 j6¼i
h
ðg 0i Þcj ðk^i1 þ k^i2 kxi1 kÞxi1j ,
L X
1
L X B C B F x þ G x þ F ij1 xj1 C i1 i1 i i2 @ A j¼1 j6¼i
0 þ
1
9 > > =
L X C k^i2 T B Fi1 xi1 þ Gi xi2 þ F ij1 xj1 C xi1 B @ Axi1 >, kxi1 k > j¼1 ; j6¼i
p00i ¼ f00i2 þ B00i ui þ
L X
F 00ij2 xj ,
j¼1 j6¼i 0
00
Bi ¼ ½ BiT Bi T T , B0i 2 Rðni mi Þmi , B00i 2 ð2mi ni Þmi R . According to assumption A1 and (3), the proposed controllers are designed as ui ¼ uin þ uis þ uip ,
ð14Þ
where T T T uin ¼ B1 ui1 ui2 , i
ui1 ¼ f0 i2 Hin ½Ki þ k^i1 þ k^i2 kxi1 k Ii xi1 ( h i 01 Ki þ k^i1 þ k^i2 kxi1 k Ii ðFi1 xi1 þ Gi xi2 Þ G i
# ) k^i2 T x ðFi1 xi1 þ Gi xi2 Þ xi1 , þ kxi1 k þ kxi1 k þ kxi1 k i1 "
3
ui2 ¼ f00i2 , uis ¼
L X B1 i ri F ji2 kxi k, kri k j¼1 j6¼i
B1 ri , i ðtÞ ¼ g^ i0 ðtÞ þ Lg^i1 ðtÞkxi k, uip ¼ ½i ðtÞ þ i ðtÞ i kri k 1 gi2 kuin þ uis k þ Bi i þ i ðtÞ ¼ , 1 gi2 B1 i the constant gi2 has to satisfy the constraint , gi2 < 1=B1 i
ð15Þ
1512
C.-C. Cheng and Y. Chang
and " is a designed positive constant. The adaptive gains g^ i‘ ðtÞ, ‘ ¼ 0, 1 are given by ( ‘ 1 ‘ _g^ ðtÞ ¼ L kxi k , if kri k 6¼ 0 , g^ i‘ ð0Þ ¼ 0, ð16Þ i‘ 0, if kri k ¼ 0
Now we define a Lyapunov function candidate as ! L 1 X 1 X 2 kri k þ ð21Þ g~ , V2 ¼ 2 ‘¼0 i‘ i¼1
where is a positive constant. The following theorem proves that the proposed controllers (14) can indeed drive the state trajectories into the designated sliding surface (2) in finite time.
where g~ i‘ ¼ g^ i‘ gi‘ , 0 ‘ 1 are the errors of adaptive gains. By using (14), one can obtain the time derivative of (21) along the trajectories of (20) and (16) for ri 6¼ 0 as
Theorem 2: Consider the large-scale system (1) with the assumptions A1–A2. If the sliding function, the controller, and the adaptive rules are designed as (2), (14) and (16) respectively, and the lumped perturbations pi, 1 i L, in the domain of interest satisfy the constraints L X p ðt, xi , xj , ui Þ gi0 þ gi1 kxi k þ gi2 kui k, i
L 1 X X rTi r_ i þ g~ i‘ g_~ i‘ V_ 2 ¼ k k r i i¼1 ‘¼0
8 >
¼
< L > X
!
2
3
L X rTi 6 6
1 X
9 > > =
7 g~i‘ g_~i‘ : Fij2 xj þpi þBi ðuis þuip Þ7 4 5 þ > > k k r i > > i¼1 : j¼1 ‘¼0 ; j6¼i
ð17Þ
ð22Þ
i¼1
where gi‘, 0 ‘ 1 are unknown positive constants, whereas gi2 is a known positive constant satisfying the inequality (15); then (a) the sliding variable ri will approach zero in a finite time tf, and the adaptive gains g^i‘ ðtÞ, ‘ ¼ 0, 1, are all bounded. (b) there exist finite constants gi‘1 such that limt!1 g^ i‘ ðtÞ ¼ gi‘1 , ‘ ¼ 0, 1. Proof: (a) From (2), one can obtain the time derivative of switching variable ri as i 9 d 0 1 h ½Gi Ki þ ðk^i1 þ k^i2 kxi1 kÞIi xi1 > > > > dt > > h i > > 0 > 1 = þ Gi Ki þ ðk^i1 þ k^i2 kxi1 kÞIi x_ i1 ð18Þ
> > d > _^ _^ > > þ ki1 þ ki2 kxi1 k þ k^i2 kxi1 k xi1 > > dt > > ; 00 r_ i2 ¼ x_ i2 : r_ i1 ¼ x_ 0i2 þ
Substituting (1) and (14) into (18) yields r_ i1 ¼
L X j¼1 j6¼i
r_ i2 ¼
L X
9 > F0 xj þ p0 þ B0 ðuis þ uip Þ > > > ij2 i i > > > = 00 F ij2 xj
j¼1 j6¼i
> > > þ p00i þ B00i ðuis þ uip Þ, > > > > ;
L X j¼1 j6¼i
F ij2 xj þ pi þ Bi ðuis þ uip Þ:
L L L X L rT L X L X X X rTi X F ji2 kxi k: j F ji2 xi F ij2 xj ¼ kri k j¼1 rj i¼1 i¼1 j¼1 i¼1 j¼1 j6¼i
j6¼i
j6¼i
ð23Þ Using (14), (16), (23), and noting that g_~i‘ ¼ g_~ i‘ , 0 ‘ 1, one can further derive (22) as " # L 1 T X X r i V_ 2 ðp þBi uip Þþ g~ i‘ g_~i‘ kri k i i¼1 ‘¼0 ! L 1 X X rTi _ g~i‘ g~ i‘ pi þ Bi uip þ kri k i¼1 ‘¼0 " L L X X gi0 þ g‘1 kx‘ kþgi2 kui kðg^ i0 þLg^ i1 kxi kþi Þ i¼1
‘¼1
#
þðg^ i0 gi0 ÞþLðg^i1 gi1 Þkxi k ¼
L X
ði þgi2 kui kÞ
i¼1
L X
ði þi Þ i þgi2 ½kuin þuis kþB1 i
i¼1
ð19Þ ¼
L X
þgi2 kuin þuis kþ B1 i i 1gi2 B1 i i
i¼1
00T T 0 ðni mi Þmi where Bi ¼ ½B0T , B00i 2 Rð2mi ni Þmi . i B i , Bi 2 R (19) can also be rewritten as
r_ i ¼
It is also observed that
¼
L X
i þgi2 ðkuin þuis k gi2 kuin þuis kþ B1 i
i¼1
i i þ B1 i
ð20Þ ¼
L X i¼1
¼ L < 0, ri 6¼ 0:
1513
International Journal of Control If ri ¼ 0, it is easy to verify that V_ 2 ¼ 0. The preceding analysis indicates that the value of ri will approach zero in a finite time tf, i.e., ri ¼ 0, 8t tf . It can be verified that tf t0 þ V2(t0)/L". Since V2 is a bounded function, the adaptive gains g^i‘ ðtÞ are all bounded. (b) From (a) it is seen that the value of V2 will decrease until ri(tf) ¼ 0, hence V_ 2 is actually a negative semidefinite function. In addition, from (16) it is known that g^ i‘ ðtÞ are monotonically increasing functions, and these functions are all bounded in accordance with (a). Therefore, according to Tao (2003, Proposition 2.14) one can conclude that there exist finite constants gi‘1 such that limt!1 g^ i‘ ðtÞ ¼ gi‘1 . It is also noted that gi‘1 may not necessarily be equal to gi‘. œ Remark 5: One of the key features of this paper is that the perturbations pi(t, xi, xj, ui), 1 i L, contain the input gain uncertainties, i.e., Bi(t, xi). Therefore, the upper bound of the perturbation pi depends on the control ui. In the proposed control scheme we use control effort ui to overcome the perturbation, however, some portion of Biui becomes the input uncertainty, i.e., Biui. It seems that there is a logic cycle with the uncertain bounds when we design the controller. For cutting this logic cycle, one way is to limit the portion of the input uncertainty Biui. This is why in the Theorem 2 the constant gi2 has to satisfy the constraint (15). For easy understanding of the physical meaning of (15), one can assume that Fi1 ¼ 0, Gi ¼ 0, F ij1 ¼ 0, fi2 ¼ 0, F ij2 ¼ 0 for simplicity. This means that the perturbation of each subsystem contains only input uncertainty. Then pi ðt, xi , xj , ui Þ ¼ kBi ui k gi2 kui k: ð24Þ (15) and (24) clearly indicate that the upper bound of perturbation induced by input gain uncertainty Biui is limited to gi2 kui k. Now if kBi k gi2 ,
ð25Þ
then by using (15), one can see that (25) implies kBi k: kBi k gi2 < 1=B1 i The above inequality is quite reasonable since in most physical systems the magnitude of the uncertainty of input gain kBi k is less than that of input gain kBi k. Remark 6: Note that the effect of other subsystems on the upper bound of pi has been considered when we made the assumption (17); the same reason as that explained in Remark 2. On the other hand, one
can also relax the constraints (17) to the following constraints r3 L X X pi ðt, xi , xj , ui Þ g0 þ g0i‘3 kxi k‘3 þgi2 kui k, i0 i¼1 ‘3 ¼1
ð26Þ where g0i‘3 , 1 ‘ r3 are unknown positive constants. The upper bound in (26) is non-linear for all the state variables of the controlled system, and the choice of r3 is similar to the method of choosing r1 described in Remark 3. Although in this paper we choose r3 ¼ 1, the design of controller for the general case r3 4 1 can be proceeded in a similar way. Remark 7: Theorem 2 clearly indicates that gi2 is a known positive constant and has to satisfy the constraint (15). Knowing gi2 one may require extra information about the model uncertainty of input gain, since input uncertainty in general often comes from the model uncertainty of input gains, i.e., Bi(t, xi)ui in this paper. For example, if one knows that in the worst case the uncertainty of input gain is at most 30% of kBi k, then one of the best choices of gi2 is to let gi2 ¼ 0:3kBi k. Note also that the upper bound of gi2 is not necessarily a small number. For example, if Bi ¼ Ii, then one of the choices of gi2 is to let gi2 ¼ 0.9. From the proof of Theorem 2, one can see that the purpose of the adaptive gains g^i‘ ðtÞ, 0 ‘ 1, is used to adapt the unknown constants gi‘ so that the perturbations (both matched and mismatched) can be suppressed during the reaching mode. The trajectories of the controlled system are capable of entering the sliding surface ri ¼ 0 in finite time. On the other hand, according to Theorem 1 the trajectory of state variable xi will approach zero asymptotically when the system is in the sliding mode. Therefore, one can conclude that the proposed control scheme can guarantee the stability of the controlled system for all time.
5. Numerical example Consider the following large-scale system with the dynamic equations in the form of (1) (L ¼ 2): Subsystem I: n1 ¼ 3, m1 ¼ 2, F11 ¼ 1, G1 ¼ ½1=ð1 þ 0:1 cos x11 Þ x12 T , x13 1 0 B1 ¼ , , f12 ¼ x211 0 1 0 1 3 : F 122 ¼ 1 1 2
1514
C.-C. Cheng and Y. Chang
Subsystem II: n2 ¼ 3, m2 ¼ 2, ½1=ð1 0:1 cos x21 Þ x22 T , "
B2 ¼
1 0 "
F 212 ¼
0
#
" f22 ¼
2
0
1 1
,
x23 x221
#
F21 ¼ 1,
G2 ¼
0
0
0:1
f22 ¼
,
F 212 ¼
x223 cosð0:5tÞ
#
x13 cos 0:5t 0
F11 ¼ 1 þ x11 sin t, G1 ¼ ½0:1 cos x12 0:2x213 sin2tT ,
F 121 ¼ x23 þ x22 þ x11 , " # 0:1 sin x11 0 , B1 ¼ 0 0:4sin3t cos x213 " # " # x12 sinð0:5tÞ 0 0 x11 f12 ¼ , F 122 ¼ : 0:5x213 x13 x11 sin t 0 Subsystem II: x223 cos 0:5tT ,
F 221 ¼ x13 þ x12 þ x23 ,
0
3.0 2.5
x11
2.0
x12
1.5
x13
1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 0
Figure 1. State trajectories x1.
2
#
0:4x12
The sliding function and controller are designed as (2) and (14), respectively. The design parameters are chosen to be (K1, K2, , ") ¼ (2, 2, 1, 0.1), so that the eigenvalues of Fc11 ¼ Fc21 are all located at 1. The initial condition of each adaptive rules is zero, and the initial condition of subsystem I and subsystem II are x1 ð0Þ ¼ ½2 2 3T , T x2 ð0Þ ¼ ½2 2 4 . The state trajectories of the subsystem I and the subsystem II are shown in Figures 1 and 2, respectively. One can see that the states of each subsystem are all driven to zero. The control inputs are displayed from Figures 3–6, respectively. The chattering phenomenon can be alleviated by replacing the function ri =kri k with saturation function (Hung et al. 1993). From (14), it can be seen that the control input ui depends on the adaptive gains k^1 ðtÞ, k^2 ðtÞ, and the magnitude of kxi1 k3 , hence if the deviation of the state variable xi1 from the origin is large (in our proposed control scheme, this may happen at t ¼ 0), then a large control effort is needed to overcome this uncertainty before the system enters the sliding mode. Nevertheless, the value of state variable xi1 will decrease asymptotically once the system enters
Subsystem I:
0:5x11 x21
: 0:2x13 x23
3
F21 ¼ 1 þ x21 , G2 ¼ ½0:1 sin x21
,
0:1x221 "
# ,
"
#
To demonstrate the robustness of the proposed controller, it is assumed that the model uncertainties, input uncertainties, disturbances and/or nonlinearities are as follows:
0:2 cos x21 cos 0:4t
B2 ¼
: 1 0
"
4
6
8
1515
International Journal of Control 4.5 4.0 3.5 3.0
x21
2.5
x22
2.0
x23
1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 0
2
4
6
8
2
4
6
8
Figure 2. State trajectories x2.
u11
15 10 5 0 −5 −10 −15 0
Figure 3. Control input u11.
u12
10
0
−10
−20
−30 0
Figure 4. Control input u12.
2
4
6
8
1516
C.-C. Cheng and Y. Chang 15
u21
10
5
0 −5 −10 −15 −20 0
2
4
6
8
2
4
6
8
Figure 5. Control input u21.
20
u22
15 10 5 0 −5 −10 −15 −20 −25 0
Figure 6. Control input u22.
the sliding surface, as proved in Theorem 1. The magnitude of control ui will also decrease rapidly. Figure 7 illustrates that the sliding variable of each subsystem also approaches zero in finite time. The adaptive gains are shown in Figures 8 and 9, where each adaptive gain approaches a finite constant respectively.
6. Conclusions In this paper, a decentralised adaptive sliding mode control scheme is successfully proposed for a class of
mismatch perturbed large-scale systems. The proposed controller is capable of driving each subsystem into a specially designed sliding surface in a finite time without requiring the knowledge of upper bounds of perturbations except that of input uncertainty. Although the mismatched perturbations and interconnections still affect the dynamics of the controlled system in the sliding mode, they are all suppressed effectively, and each subsystem as well as the whole controlled large-scale system can achieve asymptotical stability. For future study, the case of ni 4 2mi is worth consideration.
1517
International Journal of Control 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −3.0 −3.5
σ21 σ22 σ11 σ12
0
2
4
6
8
Figure 7. Sliding surface variables r1 and r2.
1.2
1.0
k21
k11
k22
k12
0.8
0.6
0.4
0.2
0.0 0
1
2
3
4
5
6
7
8
Figure 8. Adaptive gains k^11 , k^12 , k^21 and k^22 . 1.8 1.6 1.4
g20
g10
g21
g11
1.2 1.0 0.8 0.6 0.4 0.2 0.0 0
Figure 9. Adaptive gains g^ 10 , g^ 11 , g^20 and g^ 21 .
2
4
6
8
1518
C.-C. Cheng and Y. Chang
Acknowledgements The author would like to thank the Editor, Associate Editor, and the anonymous reviewers for their many helpful comments and suggestions that have helped to improve the quality of this paper. The authors are also grateful to the National Science Council of R. O. C. for financial support for this research (NSC93-2218-E-110-033).
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