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Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems
Author's Accepted Manuscript
Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems Zhaoke Ning, Jinyong Yu, Xing Xing, Huanqing Wang
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)00990-X http://dx.doi.org/10.1016/j.neucom.2015.06.075 NEUCOM15766
To appear in:
Neurocomputing
Received date: 30 January 2015 Revised date: 3 May 2015 Accepted date: 29 June 2015 Cite this article as: Zhaoke Ning, Jinyong Yu, Xing Xing, Huanqing Wang, Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems, Neurocomputing, http://dx.doi.org/10.1016/j. neucom.2015.06.075 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems
∗
Zhaoke Ninga, Jinyong Yua†, Xing Xingb and Huanqing Wangc‡ a
Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, China, b
College of Information Science and Technogy, Bohai University, China, c
School of Mathematics and Physics, Bohai University, China.
Abstract: In this paper, the problem of approximation-based adaptive decentralized control is considered for a class of noaffine stochastic nonlinear interconnected large-scale systems. Based on backstepping design, an adaptive state-feedback controller is proposed, which can guarantees that all the signals in the closed-loop systems remain fourth-moment bounded. During the controller design, fuzzy logic systems are utilized to model unknown nonlinear functions. The main contributions of the developed control scheme lie in that the proposed control scheme is independent of the information of basis functions of fuzzy systems and only one adaptive parameter needs to be estimated online for each subsystem. Simulation results are provided to illustrate the effectiveness of the proposed approach. Keywords: Pure-feedback stochastic nonlinear interconnected systems, Fuzzy logic systems, Adaptive decentralized control, Backstepping. ∗
This work is partially supported by the Natural Science Foundation of China (61304002 and 61304003). E-mail address: [email protected] (J. Yu). ‡ E-mail address: [email protected] (H. Wang). †
1
1
Introduction
During the past decades, the controller design and stability analysis for highly uncertain nonlinear dynamical systems has been one of the most research topics both in the control theory and practical applications, and many interesting results have been reported in [1–20] and the reference therein. Backstepping-based adaptive control was initially proposed in [1] for a class of nonlinear strict-feedback systems without satisfying matching condition. Then, adaptive backstepping technique was extended to control strict-feedback nonlinear systems with uncertain nonlinearities by using the capability of universal approximation of radial basis function (RBF) neural networks or fuzzy logical systems. So far, much significant progress have been achieved, for example, see [21–32] for deterministic nonlinear systems and in [33–39] for stochastic cases. Compared with strict-feedback nonlinear systems, nonaffine pure-feedback systems, a more general class of lower-triangular systems, have no affine appearance of the variables to be used as virtual control signals, which means that the controller design of pure-feedback nonlinear systems is a difficult and challenging work. By combining the backstepping methodology with adaptive neural control, in [40, 41], two special classes of pure-feedback systems were considered, where the last equation of the controlled system is in affine form in order to avoid the algebraic loop problem. In [42], an “ISS-modular” approach combined with the small-gain theorem was presented for adaptive neural control of completely non-affine pure-feedback systems. Since then, some researchers further studied other kinds of nonlinear systems in non-affine form [43–46]. Large-scale systems which are composed of interconnected subsystems often exist in practical systems, such as power systems, digital communication networks and multi-agent systems. Decentralized control and centralized control are two main control approaches for controlling large-scale nonlinear interconnected subsystems. Because of the complexity of the control synthesis and physical restrictions on information exchange among subsystems, designing a centralized controller is a technically difficult task [47]. Therefore, decentralized control approach in which each subsystem is controlled independently of its locally available information is often required to be designed for achieving a control objective of the overall large-scale system. Earlier works on decentralized control were mainly considered the linear systems or nonlinear systems with the so-called strict matching conditions [48]. In [49], Wen first presented an adaptive backstepping decentralized control scheme for a class of 2
large-scale systems without satisfying the matching condition. By combining adaptive backstepping control technique with fuzzy logic systems or neural networks, much research work has focused on the control design of large-scale systems with unknown continuous nonlinear functions, for example, see [50–54]. However, though the aforementioned results focus on the decentralized control of nonlinear interconnected systems, most of them are in affine structure or the effect of stochastic disturbance is ignored. So far, to the best of the authors’ knowledge, the adaptive fuzzy control for pure-feedback stochastic nonlinear interconnected systems merely appeared in [55] where the diffusion terms are free of interconnections and the number of adaption laws depends on the number of the fuzzy rules bases. Obviously, while the order of the considered systems increases, the number of adaptive parameters will increase correspondingly, and as a result, the online learning time may be very large. Therefore, it is a meaningful issue to design an approximation-based adaptive controller containing fewer adaptive parameters for pure-feedback stochastic nonlinear interconnected systems with a general form. Motivated by the aforementioned observations, the problem of approximation-based decentralized adaptive control of pure-feedback stochastic nonlinear interconnected large-scale systems is further considered in this paper. During the controller design, fuzzy logic systems are used to model the packaged unknown nonlinearities, and then an decentralized adaptive controller is systematically constructed via backstepping. The developed state-feedback controller ensures that all the signals in the closed-loop systems are semi-globally uniformly ultimately bounded in the sense of fourth-moment. The main contributions of this paper are summarized as follows: (1) both the drift and diffusion terms are related to the output variable of each subsystems, (2) the proposed control scheme is independent of the information of basis functions of fuzzy systems, and (3) only one adaptive parameter needs to be estimated online for each subsystem. The simulation results validate the effectiveness of the proposed control approach. The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. An adaptive control scheme is presented in Section 3. The simulation example is given in section 4, and followed by Section 5 which concludes the work.
3
2
Problem Formulation And Preliminaries
Before presenting the main results, let us first introduce some necessary definitions and lemmas, consider the following stochastic system: dx = f (x)dt + h(x)dw, ∀x ∈ Rn ,
(1)
where w is an r-dimensional independent standard Brownian motion defined on the complete probability space (Ω, F, {Ft }t≥0 , P ) with Ω being a sample space, F being a σ−field, {Ft }t≥0 being a filtration, and P being a probability measure; x ∈ Rn is the state, f : Rn → Rn and h : Rn → Rn×r are locally Lipschitz functions in x and satisfy f (0) = 0 and h(0) = 0. Definition 1. For any given V (x) ∈ C 2 , associated with the stochastic differential equation (1), define the differential operator L as follows: 2 ∂V 1 T∂ V h}, LV = f + T r{h ∂x 2 ∂x2
(2)
where T r(A) is the trace of A. 2
o correction Remark 1. As commented in [50], the term 12 T r{hT ∂∂xV2 h} in (2) is called Itˆ term where the second-order differential
∂2V ∂x2
makes the controller design much more difficult
than that of the deterministic system. Definition 2. [35] The trajectory {x(t), t ≥ 0} of the stochastic system (1) is said to be semi-globally uniformly ultimately bounded in the pth moment, if for some compact set Ω ∈ Rn and any initial state x0 = x(t0 ), there exist a constant ε > 0 and a time constant T = T (ε, x0 ) such that E(|x(t)|p ) < ε for all t > t0 + T. Especially, when p = 2, it is usually called semi-globally uniformly ultimately bounded in mean square. Lemma 1. [35] Suppose that there exists a C 2 function V (x) : Rn → R+ , two constants ¯ 1 and α ¯ 2 such that c1 > 0 and c2 > 0, class K∞ −functions α ⎧ ⎨ α ¯ 2 (|x|), ¯ 1 (|x|) ≤ V (x) ≤ α ⎩ LV (x) ≤ −c V (x) + c , 1
2
for all x ∈ Rn and t > t0 . Then, there is an unique strong solution of system (1) for each x0 ∈ Rn and it satisfies E[V (x)] ≤ V (x0 )e−c1 t + 4
c2 , ∀t > t0 . c1
In this paper, we consider a class of pure-feedback interconnected large-scale stochastic nonlinear systems composed of N subsystems. The ith (i = 1, 2, . . . , N) subsystem is described by ⎧ ⎪ ⎪ dx = (fi,j (¯ xi,j , xi,j+1 ) + ρi,j (¯ y ))dt + hTi,j (¯ y )dwi , 1 ≤ j ≤ ni − 1, ⎪ ⎨ i,j xi,ni , ui ) + ρi,ni (¯ y ))dt + hTi,ni (¯ y )dwi , dxi,ni = (fi,ni (¯ ⎪ ⎪ ⎪ ⎩ y =x , i i,1
(3)
where x¯i,j = [xi,1 , xi,2 , . . . , xi,j ]T and y¯ = [y1 , y2 , . . . , yN ]T . xi = [xi,1 , xi,2 , . . . , xi,ni ]T ∈ Rni , ui ∈ R and yi ∈ R are the state, the scalar control input and the scalar output of the ith nonlinear subsystem, respectively; wi (i = 1, 2, . . . , N) are ri -dimensional standard Brownian motions defined on a probability space (Ω, F, P ) with Ω being a sample space, F being a σ−field, and P being a probability measure; fi,j (·) : Rj+1 → R, (j = 1, 2, . . . , ni ) are unknown smooth nonlinear functions, ρi,j (·) : RN → R, hi,j (·) : RN → Rri , (j = 1, 2, . . . , ni ) are unknown smooth interconnections between the ith subsystem and other subsystems, with fi,j (0) = ρi,j (0) = 0, hi,j (0) = 0. By applying mean value theorem [56], fi,j (.) in (5) can be expressed as fi,j (¯ xi,j , xi,j+1 ) = fi,j (¯ xi,j , x0i,j+1)+gμi,j (xi,j+1 −x0i,j+1 ), xi,ni , ui ) = fi,ni (¯ xi,ni , u0i )+gμi,ni (ui −u0i ), fi,ni (¯
(4)
xi,j , xi,j+1 ) and fi,j (¯ xi,j , x0i,j+1), where smooth function fi,j (·) is explicitly analyzed between fi,j (¯ xi,j , xμi,j ) = gμi,j : = gi,j (¯
∂fi,j (¯ xi,j ,xi,j+1 ) |xi,j+1 =xµi,j , xi,ni +1 ∂xi,j+1
= ui , xμi,j = μi,j xi,j+1 + (1 −
μi,j )x0i,j+1 , 0 < μi,j < 1, i = 1, 2, . . . , N, j = 1, 2, . . . , ni . Furthermore, by substituting (4) into (3) and choosing x0i,j+1 = 0, u0i = 0, it follows ⎧ ⎪ ⎪ xi,j , 0) + ρi,j (¯ y ))dt + hTi,j (¯ y )dwi , 1 ≤ j ≤ ni − 1, ⎪ dxi,j = (gμi,j xi,j+1 + fi,j (¯ ⎨ dxi,ni = (gμi,ni ui + fi,ni (¯ xi,ni , 0) + ρi,ni (¯ y ))dt + hTi,ni (¯ y )dwi , ⎪ ⎪ ⎪ ⎩ y =x , i i,1
(5)
Assumption 1. For 1 ≤ i ≤ N, 1 ≤ j ≤ ni , the signs of gμi,j do not change and without loss of generality, it is supposed that there are unknown constants bm and bM such that 0 < bm ≤ |gμi,j | ≤ bM < ∞.
5
Assumption 2 [54]. There are unknown smooth functions ρi,j,l (yl ) and hi,j,l (yl ) with ρi,j,l (0) = 0 and hi,j,l (0) = 0 such that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni , y )| |ρi,j (¯
2
N
≤
l=1 N
y ) 2 ≤ hi,j (¯
ρ2i,j,l (yl ),
(6)
h2i,j,l (yl )
(7)
l=1
with ρi,j,l (0) = hi,j,l (0) = 0, l = 1, 2, . . . , N. Remark 2. Since ρi,j,l (yl ) and hi,j,l (yl ) are smooth functions and satisfy ρi,j,l (0) = 0 and ¯ i,j,l (yl ) such that hi,j,l (0) = 0, there exist unknown smooth functions ρ¯i,j,l (yl ) and h y )|2 ≤ |ρi,j (¯ y ) 2 ≤ hi,j (¯
N l=1 N
yl2 ρ¯2i,j,l (yl ),
(8)
¯ 2 (yl ). yl2 h i,j,l
(9)
l=1
During the controller design, fuzzy logic systems will be used to approximate nonlinear functions. The following lemma has been proved in [57], which means that fuzzy logic systems could be applied as an approximator of nonlinear functions. Lemma 2 [57]. Let f (x) be a continuous function defined on a compact set Ω. Then for any given constant ε > 0, there exists a fuzzy logic system W T S(x) such that sup | f (x) − W T ψ(x) |≤ ε. x∈Ω
where W = [w1 , w2 , . . . , wN ]T is the ideal constant weight vector, ψ(x) = [ψ1 (x), . . . , ψN (x) ]T / N j=1 ψj (x) is the basis function vector, N > 1 is the number of the fuzzy rules and si (x) are chosen as Gaussian functions, that is, −(x − μi )T (x − μi ) ψj (x) = exp , i = 1, 2, . . . , N ηi2 with μi = [μi1 , μi2 , . . . , μin ]T being the center vector and ηi the width of the Gaussian function. Lemma 3(Young’s inequality [2]). For ∀(x, y) ∈ R2 , the following inequality holds: xy ≤
1 εp p |x| + q |y|q , p qε
where ε > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1. 6
3
Adaptive fuzzy control design
In this section, a backstepping-based adaptive control design procedure is proposed. During T ψi,j (Zi,j ) will be utilized to approxthe controller design process, the fuzzy logic system Wi,j
imate the packaged unknown function f¯i,j (Zi,j ) at step j. Both the virtual control laws and adaption laws will be designed as: αi,j (Zi,j ) = −ki,j ei,j −
1 3 ˆ e θi , 2c2i,j i,j
ni λi 6 ˙ ˆ θi = ei,j − γi θˆi , 2 2c i,j j=1
(10) (11)
where i = 1, 2, . . . , N, j = 1, 2, . . . , ni , ki,j , ai,j , λi and γi are positive design parameters, xTi,j , θˆi ]T , (j = 2, . . . , ni ), and ei,j is defined by the following variable Zi,1 = xi,1 , Zi,j = [¯ transformation: ei,j = xi,j − αi,j−1
(12)
with αi,0 = 0. θˆi is the estimation of an unknown constant θi which will be specified as θi = max{
1 Wi,j 2 ; j = 1, 2, · · · , ni }, bm
(13)
where bm has been given in Assumption 1, and Wi,j is the norm of the ideal weight vector of fuzzy logic systems, which will be defined at Step j. Specifically, αi,ni is the real controller ui . In the following, time variable t and state vector x¯i,j will be omitted for simplicity from the corresponding functions and define ψi,j (Zi,j ) = ψi,j . Step 1. Based on (12), we have dei,1 = (gμi,1 ei,2 + gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + ρi,1 (¯ y ))dt + hTi,1 (¯ y )dwi .
(14)
Choose a Lyapunov function as 1 1 Vi,1 = e4i,1 = yi4 . 4 4
(15)
With the help of (2) and (14), we have 3 xi,1 , 0) + ρi,1 (¯ y )) + yi2hTi,1 (¯ y )hi,1 (¯ y ). LVi,1 = yi3 (gμi,1 ei,2 + gμi,1 αi,1 + fi,1 (¯ 2 7
(16)
By taking (8), (9) and Young’s inequality into account, one has 3 1 gμi,1 yi4 + gμi,1 e4i,2 , 4 4 3 4 1 4 3 yi ρi,1 (¯ y) ≤ y) y + ρ (¯ 4 i 4 i,1 N 3 4 1 4 4 ≤ y + N y ρ¯ (yl ), 4 i 4 l=1 l i,1,l
yi3 gμi,1 ei,2 ≤
3 2 T 3 4 y )hi,1 (¯ y) ≤ yi hi,1 (¯ y + 2 4 i 3 4 y + ≤ 4 i
(17)
(18)
3 y ) 4 hi,1 (¯ 4 N 3 4¯ 4 N yl hi,1,l (yl ). 4
(19)
l=1
Substituting (17)-(19) into (16) results in 3 3 1 LVi,1 = yi3(gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + yi + gμi,1 yi ) + gμi,1 e4i,2 2 4 4 N
1 3 ¯ 4 (yl ) . Nyl4 ρ¯4i,1,l (yl ) + Nyl4 h + i,1,l 4 4 l=1
(20)
Step j(2 ≤ j ≤ ni − 1). Reusing the Itˆ o formula and (12) shows xi,j , 0) + ρi,j (¯ y ) − αi,j−1)dt dei,j = (gμi,j ei,j+1 + gμi,j αi,j + fi,j (¯ +(hi,j (¯ y) −
j−1 ∂αi,j−1 k=1
∂xi,k
hi,j−1 (¯ y ))T dwi ,
(21)
where αi,j−1 =
j−1 ∂αi,j−1 k=1
∂xi,k
(fi,k (¯ xi,k+1 ) + ρi,k (¯ y )) +
∂αi,j−1 ˆ˙ θi ∂ θˆi
j−1 1 ∂ 2 αi,j−1 T + h (¯ y )hi,q (¯ y ). 2 p,q=1 ∂xi,p ∂xi,q i,p
(22)
Taking a Lyapunov function candidate as Vi = 14 e4i,j . And following (2), one has LVi,j =
e3i,j
xi,j , 0) − gμi,j ei,j+1 + gμi,j αi,j + fi,j (¯
j−1 ∂αi,j−1 k=1
∂xi,k
fi,k (¯ xi,k+1) −
∂αi,j−1 ˆ˙ θi ∂ θˆi
j−1 1 ∂ 2 αi,j−1 T +ρi,j (¯ y) − ρi,k (¯ y) − hi,p (¯ y )hi,q (¯ y) ∂x 2 ∂x ∂x i,k i,p i,q p,q=1 k=1
T
j−1 j−1 ∂αi,j−1 ∂αi,j−1 3 2 hi,j (¯ y) − hi,k (¯ y) y) − hi,k (¯ y) . + ei,j hi,j (¯ 2 ∂x ∂x i,k i,k k=1 k=1 j−1 ∂αi,j−1
8
(23)
Furthermore, following the same procesures from (17) to (19), the following inequalities hold 3 1 (24) gμi,j e4i,j + gμi,j e4i,j+1, 4 4 j−1 N j−1 j−1 ∂αi,j−1 3 4 ∂αi,j−1 4 1 4 4 3 ei,j −ei,j ρi,k (¯ y) ≤ ( )3 + N yl ρ¯i,k,l (yl ),(25) ∂x 4 ∂x 4 i,k i,k k=1 k=1 k=1 l=1 e3i,j gμi,j ei,j+1 ≤
y) e3i,j ρi,j (¯
3 4 1 4 4 ei,j + N ≤ y ρ¯ (yl ), 4 4 l=1 l i,j,l N
(26)
j−1 j−1 j−1 ∂ 2 αi,j−1 T 1 3 ∂ 2 αi,j−1 T 1 |ei,j | hi,p (¯ y )hi,q (¯ y) ≤ | |hi,p (¯ y )hi,p (¯ y) − e3i,j 2 ∂x ∂x 2 ∂x ∂x i,p i,q i,p i,q p,q=1 p=1 q=1 2 j−1 j−1 2 1 6 ∂ αi,j−1 ≤ e 4 i,j p=1 q=1 ∂xi,p ∂xi,q j−1
j−1
∂αi,j−1 3 2 y) − hi,k (¯ y ) 2 ei,j hi,j (¯ 2 ∂x i,k k=1
1 ¯ 4 (yl ), + (j − 1)N yl4 h i,p,l 4 p=1 l=1
j−1 3 4 ∂αi,j−1 4 1+ ≤ ( ) je 4 i,j ∂x i,k k=1 j
N
3 ¯ 4 (yl ). yl4h + jN i,k,l 4 k=1 l=1
(27)
N
(28)
Thus, the inequalities (23)-(28) imply that j−1 ∂αi,j−1 3 3 3 xi,j , 0) − (fi,k (¯ xi,k+1 )) + gμi,j ei,j + ei,j LVi,j ≤ ei,j gμi,j αi,j + fi,j (¯ ∂xi,k 4 4 k=1 2 j−1 j−1
j−1 j−1 3 ∂αi,j−1 4 1 3 ∂ 2 αi,j−1 3 ∂αi,j−1 4 3 + ei,j ( ) + ei,j + jei,j ( ) 4 ∂xi,k 4 ∂xi,p ∂xi,q 4 ∂xi,k p=1 q=1 k=1 k=1 j N 1 3 ∂αi,j−1 ˆ˙ 1 4 4 4 + jei,j − y ρ¯ (yl ) θi + gμi,j ei,j+1 + N 4 4 4 k=1 l=1 l i,k,l ∂ θˆi j−1
j
1 ¯ 4 (yl ) + 3 jN ¯ 4 (yl ). yl4 h yl4h + (j − 1)N i,p,l i,k,l 4 4 p=1 l=1 k=1 l=1 N
N
(29)
o formula, we have Step ni . Applying (2) and Itˆ xi,ni , 0) + ρi,ni (¯ y ) − αi,ni −1 )dt dei,ni = (gμi,ni ui + fi,ni (¯ y) − +(hi,ni (¯
n i −1 k=1
∂αi,ni −1 hi,k (¯ y ))T dwi , ∂xi,k
(30)
where αi,ni −1 is given in (22) with j = ni . Construct a stochastic Lyapunov function as 1 bm ˜2 θ , Vi,ni = e4i,ni + 4 2λi i 9
(31)
where θ˜i = θi − θˆi denotes the parameter estimation error and λi is a positive design constant. Then, with the help of (2), the time derivative of Vi,ni is expressed as n i −1 ∂αi,ni −1 ∂αi,ni −1 ˆ˙ 3 xi,ni , 0) − fi,k (¯ xi,k+1) − LVi,ni = ei,ni gμi,ni ui + fi,ni (¯ θi ∂xi,k ∂ θˆi k=1
n ni −1 i −1 ∂αi,ni −1 ∂ 2 αi,ni −1 T 1 +ρi,ni (¯ y) − ρi,k (¯ y) − h (¯ y )hi,q (¯ y) ∂xi,k 2 p,q=1 ∂xi,p ∂xi,q i,p k=1 i ∂αi,ni −1 bm ˜ ˆ˙ 3 y) − hi,k (¯ y ) 2 − + e2i,ni hi,ni (¯ θi θi . 2 ∂x λ i,k i k=1
n −1
(32)
Repeating the same derivations as (25)-(29) yields n i −1 3 ∂αi,ni −1 3 xi,ni , 0) − fi,k (¯ xi,k+1) + ei,ni LVi,ni ≤ ei,ni gμi,ni ui + fi,ni (¯ ∂xi,k 4 k=1 2 n ni −1 n i −1 i −1
∂ 2 αi,ni −1 3 ∂αi,ni −1 4 1 3 3 + ei,ni ( ) + ei,ni 4 ∂xi,k 4 ∂xi,p ∂xi,q p=1 q=1 k=1
n i −1 3 ∂αi,ni −1 4 3 ∂αi,ni −1 ˆ˙ bm ˜ ˆ˙ + ni ei,ni ( ) + ni ei,ni − θi θi θi − ˆi 4 ∂x 4 λ i,k i ∂ θ k=1 ni n N N i −1 1 1 4 4 ¯ 4 (yl ) + N yl ρ¯i,k,l (yl ) + (ni − 1)N yl4 h i,p,l 4 k=1 l=1 4 p=1 l=1 i 3 ¯ 4 (yl ). + ni N yl4 h i,k,l 4 k=1 l=1
n
N
(33)
Now, choose the Lyapunov function candidate for the whole system as V =
ni N i=1 j=1
Vi,j =
ni N 1 1 bm ˜2 θi ). ( yi4 + e4i,j + 4 4 2λ i i=1 j=2
Subsequently, combining the inequalities (20), (29) and (33), we have nl N N s 3 3 1 3 LV ≤ yi gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + yi + gμi,1 yi + Nyi ρ¯4l,k,i (yi ) 2 4 4 i=1 l=1 s=1 k=1 n n −1 N N s s l l 1 3 ¯ 4 (yi ) + Nyi ¯ 4 (yi ) h h s s + Nyi l,k,i l,p,i 4 4 p=1 l=1 s=1 k=1 l=1 s=1 j−1 N n i −1 ∂αi,j−1 3 ei,j gμi,j αi,j + fi,j (¯ xi,j , 0) − fi,k (¯ xi,k+1 ) + ∂xi,k i=1 j=2 k=1 2 j−1 j−1
j−1 3 ∂αi,j−1 4 1 3 ∂ 2 αi,j−1 3 ( ) 3 + ei,j + gμi,j ei,j + ei,j 4 4 ∂x 4 ∂xi,p ∂xi,q i,k p=1 q=1 k=1 10
(34)
j−1 3 ∂αi,j−1 4 3 1 + jei,j ( ) + (j + 1)ei,j + gμi,j−1 ei,j 4 ∂xi,k 4 4 k=1 n N i −1 ∂αi,ni −1 3 ei,ni gμi,ni ui + fi,ni (¯ xi,ni , 0) − fi,k (¯ xi,k+1 ) + ∂xi,k i=1 k=1 2 n ni −1 n i −1 i −1
∂ 2 αi,ni−1 3 ∂αi,ni −1 4 1 3 + ei,ni ( ) 3 + ei,ni 4 ∂x 4 ∂xi,p ∂xi,q i,k p=1 q=1 k=1 n i −1 3 ∂αi,ni −1 4 3 1 + ni ei,ni ( ) + (ni + 1)ei,ni + gμi,ni −1 ei,ni 4 ∂x 4 4 i,k k=1 −
ni N
∂αi,j−1 ˆ˙ e3i,j θi ∂ θˆi i=1 j=2
−
N bm i=1
λi
˙ θ˜i θˆi ,
(35)
where the following results have been utilized in (35). n
i l 1 1 4 4 N N yl ρ¯i,k,l (yl ) = yi4 ρ¯4l,k,i (yi ), 4 i=1 s=1 k=1 l=1 4 i=1 l=1 s=1 k=1
(36)
nl ni N s N s N N 3 3 4¯ 4 ¯ 4 (yi ), s y h (yl ) = s y 4h N N 4 i=1 s=1 k=1 l=1 l i,k,l 4 i=1 l=1 s=1 k=1 i l,k,i
(37)
N ni −1 s N s N nl −1 N 1 1 4¯ 4 ¯ 4 (yi ). s y h (yl ) = s y4h N N 4 i=1 s=1 p=1 l=1 l i,p,l 4 i=1 l=1 s=1 p=1 i l,p,i
(38)
N
n
s
N
N
N
s
By (11) and rearranging sequence, it follows that −
ni N
e3i,j
i=1 j=2
=−
ni N i=1 j=2
=
∂αi,j−1 e3i,j ∂ θˆi
ni N
∂αi,j−1 ˆ e3i,j γi θi ∂ θˆi i=1 j=2
−
ni N i=1 j=2
≤
∂αi,j−1 ˆ˙ θi ∂ θˆi
e3i,j
ni λi 6 ei,k − γi θˆi 2 2c i,k k=1
−
j−1 3 ∂αi,j−1 ei,j ∂ θˆi k=1 i=1 j=2
ni N
λi 6 e 2c2i,k i,k
ni ∂αi,j−1 λi 6 ei,k 2 ∂ θˆi k=j 2ci,k
ni N
∂αi,j−1 ˆ e3i,j γθ ˆi i i ∂ θ i=1 j=2
−
j−1 3 ∂αi,j−1 ei,j ∂ θˆi k=1 i=1 j=2
ni N
λi 6 e 2c2i,k i,k
j ni N λi 6 3 ∂αi,k−1 + e ( |e |). 2c2i,j i,j k=2 i,k ∂ θˆi i=1 j=2
Substituting (39) into (35) produces LV
≤
N
yi3
N n i −1 ¯ gμi,1 αi,1 + fi,1 (Zi,1) + e3i,j gμi,j αi,j + f¯i,j (Zi,j )
i=1
i=1 j=2
11
(39)
+
N i=1
e3i,ni
ni N N 3 bm ˜ ˆ˙ 4 ¯ gμi,ni ui + fi,ni (Zi,ni ) − ei,j − θi θi , 4 i=1 j=1 λi i=1
(40)
where the functions f¯i,j (Zi,j ), i = 1, 2, . . . , N are defined as n
l 9 3 1 ¯ xi,1 , 0) + yi + gμi,1 yi + Nyi ρ¯4l,k,i (yi ) fi,1 (Zi,1 ) = fi,1 (¯ 4 4 4 l=1 s=1 k=1
N
s
nl N N n s s l −1 1 3 4 ¯ ¯ 4 (yi ), hl,k,i (yi ) + Nyi h s s + Nyi l,p,i 4 4 p=1 l=1 s=1 k=1 l=1 s=1
(41)
j−1
j−1 ∂αi,j−1
∂αi,j−1 4 3 3 xi,j , 0) − fi,k (¯ xi,k+1 ) + gμi,j ei,j + ei,j ( )3 f¯i,j (Zi,j ) = fi,j (¯ ∂x 4 4 ∂x i,k i,k k=1 k=1 j−1 j−1 2 j−1 2 ∂ αi,j−1 ∂αi,j−1 1 3 3 + e3i,j + jei,j ( )4 + (j + 2)ei,j 4 ∂xi,p ∂xi,q 4 ∂xi,k 4 p=1 q=1 k=1
j−1
1 ∂αi,j−1 ˆ ∂αi,j−1 λi 6 + gμi,j−1 ei,j + γi θi − ei,k 2 4 ∂ θˆi ∂ θˆi k=1 2ci,k +
j λi 3 3 ∂αi,k−1 e ( |e |), j = 2, . . . , ni − 1, 2c2i,j i,j k=2 i,k ∂ θˆi
(42)
n i −1 3 ∂αi,ni −1 4 ∂αi,ni −1 fi,k (¯ xi,k+1) + ei,ni ( )3 ∂x 4 ∂x i,k i,k k=1 k=1
n −1 −1 n −1 n 2 i i i ∂ 2 αi,n −1 ∂αi,n −1 1 3 i i + e3i,ni + ni ei,ni ( )4 4 ∂x ∂x 4 ∂x i,p i,q i,k p=1 q=1 k=1
f¯i,ni (Zi,ni ) = fi,ni (¯ xi,ni , 0) −
n i −1
3 1 ∂αi,ni −1 ˆ + (ni + 2)ei,ni + gμi,ni −1 ei,ni + γi θi 4 4 ∂ θˆi ni −1 ni λi 6 ∂αi,k−1 ∂αi,ni −1 λi 3 − e + e ( |e3i,k |). i,k i,ni 2 2 ˆi 2ci,ni ∂ θˆi k=1 2ci,k ∂ θ k=2
(43)
¯ l,k,i are unknown, f¯i,j (Zi,j ), i = 1, 2, . . . , N, j = As smooth functions fi,j , gμi,j , ρ¯l,k,i and h 1, 2, . . . , ni cannot be directly employed to construct virtual control signal αi,j or the actual control input ui . According to Lemma 2, for any given εi,j > 0, there exists a fuzzy logic T ψi,j (Zi,j ) such that system Wi,j T ψi,j (Zi,j ) + δi,j (Zi,j ), f¯i,j (Zi,j ) = Wi,j
(44)
where δi,j (Zi,j ) denotes the approximation error and satisfies |δi,j (Zi,j )| < εi,j . Furthermore, T by using Young’s inequality and the fact of ψi,j ψi,j ≤ 1, one has
e3i,j f¯i,j (Zi,j )
=
e3i,j
T Wi,j ψi,j Wi,j + e3i,j δi,j (Zi,j ) Wi,j
12
≤
bm 6 1 3 1 ei,j θi + c2i,j + e4i,j + ε4i,j , 2 2ci,j 2 4 4
(45)
where i = 1, 2, . . . , N, j = 1, 2, . . . , ni and the unknown constant θi = max
1 Wi,j 2 ; bm
j=
1, 2, · · · , ni }. Substituting (44) into (40) and using (45) produces LV
N n i −1 bm 3 bm 6 3 ei,j gμi,j αi,j + 2 ei,j θi ≤ gμi,1 αi,1 + 2 ei,1 θi + 2ai,1 2ci,j i=1 i=1 j=2
ni
N N 1 2 bm 3 1 2 3 ci,j + εi,j ei,ni gμi,ni ui + 2 ei,ni θi + + 2a 2 4 i,n i i=1 i=1 j=1 N
−
e3i,1
N bm i=1
λi
˙ θ˜i θˆi .
(46)
Furthermore, by taking (10) and (11) into account and using the following inequality: e3i,j gμi,j αi,j ≤ −ki,j bm e4i,j −
bm 6 ˆ e θi , i = 1, 2, . . . , N, j = 1, 2, . . . , ni , 2c2i,j i,j
(46) can be rewritten as LV
≤ −
ni N i=1 j=1
≤ −
ki,j bm e4i,j
n N i i=1
+
N γi bm i=1
λi
θ˜i θˆi +
γi bm ˜2 θ ki,j bm e4i,j + 2λi i j=1
ni
N 1
2
i=1 j=1
+
ni
N 1 i=1 j=1
2
c2i,j
c2i,j
1 + ε2i,j 4
1 2 γi bm 2 + εi,j + θ , 4 2λi i
(47)
2 where the inequality θ˜i θˆi ≤ − 12 θ˜i + 12 θi2 has been used in the above inequality.
At the present stage, we summarize our main result in the following theorem. Theorem 1. Consider the large-scale stochastic nonlinear systems (5) consisting of Assumptions 1-2 and apply the above design procedures, the controller (10), and adaptive law (11). Assume that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni , the packaged unknown functions f¯i,j (Zi,j ) can T be well approximated by the fuzzy logic systems Wi,j ψi,j (Zi,j ) in the sense that the approxi-
mation errors δi,j (Zi,j ) are bounded, then for bounded initial conditions [ziT (0), θˆi (0)]T ∈ Ω0 , all involved signals in the closed-loop system remain bounded in the sense of fourth-moment, and the error signals ei,j and θ˜i eventually converge to the compact set Ωs defined by ni N 2λi b0 b0 E[|ei,j |4 ] ≤ 4 , |θ˜i | ≤ ,1 ≤ i ≤ N . (48) Ωs = ei,j , θ˜i a0 bm a0 i=1 j=1 13
Proof: Let a0 = min{4ki,j bm , γi , i = 1, 2, . . . , N, j = 1, 2, . . . , ni }, ni N 1 1 γi bm 2 b0 = ( c2i,j + ε2i,j + θi ), 2 4 2λ i i=1 j=1 then we can rewrite (47) as LV ≤ −a0 V + b0 , t ≥ 0.
(49)
Therefore, it can be concluded from Lemma 1 and (34) that all the signals in the closed-loop system (5) remain fourth-moment bounded. Then, from (49), the following inequality can be reached directly dE[V (t)] ≤ −a0 E[V (t)] + b0 , dt
(50)
which implies that E[V (t)] ≤ e−a0 t [V (0)] +
b0 , ∀t > 0. a0
(51)
Then, it is easily obtained that E[V (t)] ≤
b0 , t → +∞. a0
(52)
Therefore, based on the definition of V in (34), the error signals ei,j and θ˜i eventually converge to the compact set Ωs specified in (48).
4
Simulation Example
Example 1. To illustrate the effectiveness of our result, consider the following the following pure-feedback interconnected stochastic nonlinear system: ⎧ y1 y2 3 ⎪ ⎪ dx dt + 1+y = (2 + sin(x ))x + x + y y 2 dw1 , 1,1 1,1 1,2 1 2 1,2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ dx1,2 = ((2 + cos(x1,1 x1,2 ))u1 + 0.1 sin(u1 ) + y12y2 ) dt + ln(1 + y12 )y2 dw1, ⎪ ⎪ ⎪ ⎪ ⎨ y =x , 1 1,1 ⎪ ⎪ dx2,1 = ((3 + sin(x2,1 ))x2,2 + 2x32,2 + y1 y22)dt + y1 sin(y22)dw2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dx2,2 = ((1 + x22,1 )u2 + 0.2u52 + y1 y2 )dt + y22 cos (y1 )dw2 , ⎪ ⎪ ⎪ ⎪ ⎩ y =x , 2 2,1 14
(53)
where x1,1 , x1,2 , x2,1 and x2,2 are the system states, u1 and u2 denote the system input signals. For simplicity, w˙ is chosen as an one-dimensional Gaussian white noise with zero mean and variance 1. It can be easily verified that Assumptions 1 and 2 are satisfied, and the existing approximation-based adaptive control schemes are not suitable for controlling this system. The control objective is to design an adaptive fuzzy controller such that all the signals in the closed-loop system are bounded. By applying Theorem 1, the virtual control inputs, actual controller and adaptive laws are constructed as ⎧ ⎪ ⎪ αi,1 = −ki,1 ei,1 − 2c12 e3i,1 θˆi , i = 1, 2, ⎪ ⎪ i,1 ⎨ 1 ui = −ki,2 ei,2 − 2c2 e3i,2 θˆi , i = 1, 2, i,2 ⎪ ⎪ 2 ⎪ ˙ ⎪ λ 6 i ˆ ⎩ θi = j=1 2 ei,j − γi θˆi , i = 1, 2, 2c
(54)
i,j
where ei,1 = xi,1 , ei,2 = xi,2 − αi,1 , i = 1, 2. The simulation is carried out under the initial conditions [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.6, 0.2, 0.5, 0.2]T , and [θˆ1 (0), θˆ2 (0)]T = [0, 0]T . In the simulation, the design parameters are chosen as: k1,1 = k1,2 = k2,1 = k2,2 = 2.5, c1,1 = c1,2 = c2,1 = c2,2 = 1, γ1 = γ2 = 1, and λ1 = λ2 = 2. Then we get the simulation results, which are shown by Figures 1-4. Fig. 1 displays the state variables y1 and y2 and Fig. 2 shows the state variables x2,1 and x2,2 . Figs. 3 displays the control input signals u1 and u2 , and Fig. 4 illustrates the response curve of the adaptive parameters θˆ1 and θˆ2 . Simulation results show that good convergence performances are achieved and all the signals in the closed-loop system are bounded.
5
Conclusion
In this paper, an approximation-based adaptive decentralized control scheme has been proposed for a class of large-scale pure-feedback stochastic nonlinear interconnected systems. In the controller design, fuzzy logic systems were used to model unknown nonlinearities and the decentralized adaptive controller is constructed based on backstepping. The proposed decentralized adaptive controllers guarantees that all the signals within the closed-loop systems remain bounded in the sense of fourth-moment. A simulation example is used to validity the effectiveness of the proposed approach.
15
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21
0.7 y1 0.6
y
2
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2
0
1
2
3
4 Time(Sec)
5
6
7
8
Fig. 1. States of closed-loop system y1 and y2 . 0.4 x12 x22
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
1
2
3
4 Time(Sec)
5
6
7
Fig. 2. States of closed-loop system x2,1 and x2,2 .
22
8
2 u1 u
2
1
0
−1
−2
−3
−4
−5
0
1
2
3
4 Time(Sec)
5
6
7
8
Fig. 3. The control input signals u1 and u2 . 0.7 θˆ1 θˆ2
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4 Time(Sec)
5
6
Fig. 4. Adaptive laws θˆ1 and θˆ2 .
23
7
8
Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems Zhaoke Ning, Jinyong Yu, Xing Xing, Huanqing Wang
www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(15)00990-X http://dx.doi.org/10.1016/j.neucom.2015.06.075 NEUCOM15766
To appear in:
Neurocomputing
Received date: 30 January 2015 Revised date: 3 May 2015 Accepted date: 29 June 2015 Cite this article as: Zhaoke Ning, Jinyong Yu, Xing Xing, Huanqing Wang, Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems, Neurocomputing, http://dx.doi.org/10.1016/j. neucom.2015.06.075 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Robust adaptive decentralized control for a class of noaffine stochastic nonlinear interconnected systems
∗
Zhaoke Ninga, Jinyong Yua†, Xing Xingb and Huanqing Wangc‡ a
Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, China, b
College of Information Science and Technogy, Bohai University, China, c
School of Mathematics and Physics, Bohai University, China.
Abstract: In this paper, the problem of approximation-based adaptive decentralized control is considered for a class of noaffine stochastic nonlinear interconnected large-scale systems. Based on backstepping design, an adaptive state-feedback controller is proposed, which can guarantees that all the signals in the closed-loop systems remain fourth-moment bounded. During the controller design, fuzzy logic systems are utilized to model unknown nonlinear functions. The main contributions of the developed control scheme lie in that the proposed control scheme is independent of the information of basis functions of fuzzy systems and only one adaptive parameter needs to be estimated online for each subsystem. Simulation results are provided to illustrate the effectiveness of the proposed approach. Keywords: Pure-feedback stochastic nonlinear interconnected systems, Fuzzy logic systems, Adaptive decentralized control, Backstepping. ∗
This work is partially supported by the Natural Science Foundation of China (61304002 and 61304003). E-mail address: [email protected] (J. Yu). ‡ E-mail address: [email protected] (H. Wang). †
1
1
Introduction
During the past decades, the controller design and stability analysis for highly uncertain nonlinear dynamical systems has been one of the most research topics both in the control theory and practical applications, and many interesting results have been reported in [1–20] and the reference therein. Backstepping-based adaptive control was initially proposed in [1] for a class of nonlinear strict-feedback systems without satisfying matching condition. Then, adaptive backstepping technique was extended to control strict-feedback nonlinear systems with uncertain nonlinearities by using the capability of universal approximation of radial basis function (RBF) neural networks or fuzzy logical systems. So far, much significant progress have been achieved, for example, see [21–32] for deterministic nonlinear systems and in [33–39] for stochastic cases. Compared with strict-feedback nonlinear systems, nonaffine pure-feedback systems, a more general class of lower-triangular systems, have no affine appearance of the variables to be used as virtual control signals, which means that the controller design of pure-feedback nonlinear systems is a difficult and challenging work. By combining the backstepping methodology with adaptive neural control, in [40, 41], two special classes of pure-feedback systems were considered, where the last equation of the controlled system is in affine form in order to avoid the algebraic loop problem. In [42], an “ISS-modular” approach combined with the small-gain theorem was presented for adaptive neural control of completely non-affine pure-feedback systems. Since then, some researchers further studied other kinds of nonlinear systems in non-affine form [43–46]. Large-scale systems which are composed of interconnected subsystems often exist in practical systems, such as power systems, digital communication networks and multi-agent systems. Decentralized control and centralized control are two main control approaches for controlling large-scale nonlinear interconnected subsystems. Because of the complexity of the control synthesis and physical restrictions on information exchange among subsystems, designing a centralized controller is a technically difficult task [47]. Therefore, decentralized control approach in which each subsystem is controlled independently of its locally available information is often required to be designed for achieving a control objective of the overall large-scale system. Earlier works on decentralized control were mainly considered the linear systems or nonlinear systems with the so-called strict matching conditions [48]. In [49], Wen first presented an adaptive backstepping decentralized control scheme for a class of 2
large-scale systems without satisfying the matching condition. By combining adaptive backstepping control technique with fuzzy logic systems or neural networks, much research work has focused on the control design of large-scale systems with unknown continuous nonlinear functions, for example, see [50–54]. However, though the aforementioned results focus on the decentralized control of nonlinear interconnected systems, most of them are in affine structure or the effect of stochastic disturbance is ignored. So far, to the best of the authors’ knowledge, the adaptive fuzzy control for pure-feedback stochastic nonlinear interconnected systems merely appeared in [55] where the diffusion terms are free of interconnections and the number of adaption laws depends on the number of the fuzzy rules bases. Obviously, while the order of the considered systems increases, the number of adaptive parameters will increase correspondingly, and as a result, the online learning time may be very large. Therefore, it is a meaningful issue to design an approximation-based adaptive controller containing fewer adaptive parameters for pure-feedback stochastic nonlinear interconnected systems with a general form. Motivated by the aforementioned observations, the problem of approximation-based decentralized adaptive control of pure-feedback stochastic nonlinear interconnected large-scale systems is further considered in this paper. During the controller design, fuzzy logic systems are used to model the packaged unknown nonlinearities, and then an decentralized adaptive controller is systematically constructed via backstepping. The developed state-feedback controller ensures that all the signals in the closed-loop systems are semi-globally uniformly ultimately bounded in the sense of fourth-moment. The main contributions of this paper are summarized as follows: (1) both the drift and diffusion terms are related to the output variable of each subsystems, (2) the proposed control scheme is independent of the information of basis functions of fuzzy systems, and (3) only one adaptive parameter needs to be estimated online for each subsystem. The simulation results validate the effectiveness of the proposed control approach. The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. An adaptive control scheme is presented in Section 3. The simulation example is given in section 4, and followed by Section 5 which concludes the work.
3
2
Problem Formulation And Preliminaries
Before presenting the main results, let us first introduce some necessary definitions and lemmas, consider the following stochastic system: dx = f (x)dt + h(x)dw, ∀x ∈ Rn ,
(1)
where w is an r-dimensional independent standard Brownian motion defined on the complete probability space (Ω, F, {Ft }t≥0 , P ) with Ω being a sample space, F being a σ−field, {Ft }t≥0 being a filtration, and P being a probability measure; x ∈ Rn is the state, f : Rn → Rn and h : Rn → Rn×r are locally Lipschitz functions in x and satisfy f (0) = 0 and h(0) = 0. Definition 1. For any given V (x) ∈ C 2 , associated with the stochastic differential equation (1), define the differential operator L as follows: 2 ∂V 1 T∂ V h}, LV = f + T r{h ∂x 2 ∂x2
(2)
where T r(A) is the trace of A. 2
o correction Remark 1. As commented in [50], the term 12 T r{hT ∂∂xV2 h} in (2) is called Itˆ term where the second-order differential
∂2V ∂x2
makes the controller design much more difficult
than that of the deterministic system. Definition 2. [35] The trajectory {x(t), t ≥ 0} of the stochastic system (1) is said to be semi-globally uniformly ultimately bounded in the pth moment, if for some compact set Ω ∈ Rn and any initial state x0 = x(t0 ), there exist a constant ε > 0 and a time constant T = T (ε, x0 ) such that E(|x(t)|p ) < ε for all t > t0 + T. Especially, when p = 2, it is usually called semi-globally uniformly ultimately bounded in mean square. Lemma 1. [35] Suppose that there exists a C 2 function V (x) : Rn → R+ , two constants ¯ 1 and α ¯ 2 such that c1 > 0 and c2 > 0, class K∞ −functions α ⎧ ⎨ α ¯ 2 (|x|), ¯ 1 (|x|) ≤ V (x) ≤ α ⎩ LV (x) ≤ −c V (x) + c , 1
2
for all x ∈ Rn and t > t0 . Then, there is an unique strong solution of system (1) for each x0 ∈ Rn and it satisfies E[V (x)] ≤ V (x0 )e−c1 t + 4
c2 , ∀t > t0 . c1
In this paper, we consider a class of pure-feedback interconnected large-scale stochastic nonlinear systems composed of N subsystems. The ith (i = 1, 2, . . . , N) subsystem is described by ⎧ ⎪ ⎪ dx = (fi,j (¯ xi,j , xi,j+1 ) + ρi,j (¯ y ))dt + hTi,j (¯ y )dwi , 1 ≤ j ≤ ni − 1, ⎪ ⎨ i,j xi,ni , ui ) + ρi,ni (¯ y ))dt + hTi,ni (¯ y )dwi , dxi,ni = (fi,ni (¯ ⎪ ⎪ ⎪ ⎩ y =x , i i,1
(3)
where x¯i,j = [xi,1 , xi,2 , . . . , xi,j ]T and y¯ = [y1 , y2 , . . . , yN ]T . xi = [xi,1 , xi,2 , . . . , xi,ni ]T ∈ Rni , ui ∈ R and yi ∈ R are the state, the scalar control input and the scalar output of the ith nonlinear subsystem, respectively; wi (i = 1, 2, . . . , N) are ri -dimensional standard Brownian motions defined on a probability space (Ω, F, P ) with Ω being a sample space, F being a σ−field, and P being a probability measure; fi,j (·) : Rj+1 → R, (j = 1, 2, . . . , ni ) are unknown smooth nonlinear functions, ρi,j (·) : RN → R, hi,j (·) : RN → Rri , (j = 1, 2, . . . , ni ) are unknown smooth interconnections between the ith subsystem and other subsystems, with fi,j (0) = ρi,j (0) = 0, hi,j (0) = 0. By applying mean value theorem [56], fi,j (.) in (5) can be expressed as fi,j (¯ xi,j , xi,j+1 ) = fi,j (¯ xi,j , x0i,j+1)+gμi,j (xi,j+1 −x0i,j+1 ), xi,ni , ui ) = fi,ni (¯ xi,ni , u0i )+gμi,ni (ui −u0i ), fi,ni (¯
(4)
xi,j , xi,j+1 ) and fi,j (¯ xi,j , x0i,j+1), where smooth function fi,j (·) is explicitly analyzed between fi,j (¯ xi,j , xμi,j ) = gμi,j : = gi,j (¯
∂fi,j (¯ xi,j ,xi,j+1 ) |xi,j+1 =xµi,j , xi,ni +1 ∂xi,j+1
= ui , xμi,j = μi,j xi,j+1 + (1 −
μi,j )x0i,j+1 , 0 < μi,j < 1, i = 1, 2, . . . , N, j = 1, 2, . . . , ni . Furthermore, by substituting (4) into (3) and choosing x0i,j+1 = 0, u0i = 0, it follows ⎧ ⎪ ⎪ xi,j , 0) + ρi,j (¯ y ))dt + hTi,j (¯ y )dwi , 1 ≤ j ≤ ni − 1, ⎪ dxi,j = (gμi,j xi,j+1 + fi,j (¯ ⎨ dxi,ni = (gμi,ni ui + fi,ni (¯ xi,ni , 0) + ρi,ni (¯ y ))dt + hTi,ni (¯ y )dwi , ⎪ ⎪ ⎪ ⎩ y =x , i i,1
(5)
Assumption 1. For 1 ≤ i ≤ N, 1 ≤ j ≤ ni , the signs of gμi,j do not change and without loss of generality, it is supposed that there are unknown constants bm and bM such that 0 < bm ≤ |gμi,j | ≤ bM < ∞.
5
Assumption 2 [54]. There are unknown smooth functions ρi,j,l (yl ) and hi,j,l (yl ) with ρi,j,l (0) = 0 and hi,j,l (0) = 0 such that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni , y )| |ρi,j (¯
2
N
≤
l=1 N
y ) 2 ≤ hi,j (¯
ρ2i,j,l (yl ),
(6)
h2i,j,l (yl )
(7)
l=1
with ρi,j,l (0) = hi,j,l (0) = 0, l = 1, 2, . . . , N. Remark 2. Since ρi,j,l (yl ) and hi,j,l (yl ) are smooth functions and satisfy ρi,j,l (0) = 0 and ¯ i,j,l (yl ) such that hi,j,l (0) = 0, there exist unknown smooth functions ρ¯i,j,l (yl ) and h y )|2 ≤ |ρi,j (¯ y ) 2 ≤ hi,j (¯
N l=1 N
yl2 ρ¯2i,j,l (yl ),
(8)
¯ 2 (yl ). yl2 h i,j,l
(9)
l=1
During the controller design, fuzzy logic systems will be used to approximate nonlinear functions. The following lemma has been proved in [57], which means that fuzzy logic systems could be applied as an approximator of nonlinear functions. Lemma 2 [57]. Let f (x) be a continuous function defined on a compact set Ω. Then for any given constant ε > 0, there exists a fuzzy logic system W T S(x) such that sup | f (x) − W T ψ(x) |≤ ε. x∈Ω
where W = [w1 , w2 , . . . , wN ]T is the ideal constant weight vector, ψ(x) = [ψ1 (x), . . . , ψN (x) ]T / N j=1 ψj (x) is the basis function vector, N > 1 is the number of the fuzzy rules and si (x) are chosen as Gaussian functions, that is, −(x − μi )T (x − μi ) ψj (x) = exp , i = 1, 2, . . . , N ηi2 with μi = [μi1 , μi2 , . . . , μin ]T being the center vector and ηi the width of the Gaussian function. Lemma 3(Young’s inequality [2]). For ∀(x, y) ∈ R2 , the following inequality holds: xy ≤
1 εp p |x| + q |y|q , p qε
where ε > 0, p > 1, q > 1, and (p − 1)(q − 1) = 1. 6
3
Adaptive fuzzy control design
In this section, a backstepping-based adaptive control design procedure is proposed. During T ψi,j (Zi,j ) will be utilized to approxthe controller design process, the fuzzy logic system Wi,j
imate the packaged unknown function f¯i,j (Zi,j ) at step j. Both the virtual control laws and adaption laws will be designed as: αi,j (Zi,j ) = −ki,j ei,j −
1 3 ˆ e θi , 2c2i,j i,j
ni λi 6 ˙ ˆ θi = ei,j − γi θˆi , 2 2c i,j j=1
(10) (11)
where i = 1, 2, . . . , N, j = 1, 2, . . . , ni , ki,j , ai,j , λi and γi are positive design parameters, xTi,j , θˆi ]T , (j = 2, . . . , ni ), and ei,j is defined by the following variable Zi,1 = xi,1 , Zi,j = [¯ transformation: ei,j = xi,j − αi,j−1
(12)
with αi,0 = 0. θˆi is the estimation of an unknown constant θi which will be specified as θi = max{
1 Wi,j 2 ; j = 1, 2, · · · , ni }, bm
(13)
where bm has been given in Assumption 1, and Wi,j is the norm of the ideal weight vector of fuzzy logic systems, which will be defined at Step j. Specifically, αi,ni is the real controller ui . In the following, time variable t and state vector x¯i,j will be omitted for simplicity from the corresponding functions and define ψi,j (Zi,j ) = ψi,j . Step 1. Based on (12), we have dei,1 = (gμi,1 ei,2 + gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + ρi,1 (¯ y ))dt + hTi,1 (¯ y )dwi .
(14)
Choose a Lyapunov function as 1 1 Vi,1 = e4i,1 = yi4 . 4 4
(15)
With the help of (2) and (14), we have 3 xi,1 , 0) + ρi,1 (¯ y )) + yi2hTi,1 (¯ y )hi,1 (¯ y ). LVi,1 = yi3 (gμi,1 ei,2 + gμi,1 αi,1 + fi,1 (¯ 2 7
(16)
By taking (8), (9) and Young’s inequality into account, one has 3 1 gμi,1 yi4 + gμi,1 e4i,2 , 4 4 3 4 1 4 3 yi ρi,1 (¯ y) ≤ y) y + ρ (¯ 4 i 4 i,1 N 3 4 1 4 4 ≤ y + N y ρ¯ (yl ), 4 i 4 l=1 l i,1,l
yi3 gμi,1 ei,2 ≤
3 2 T 3 4 y )hi,1 (¯ y) ≤ yi hi,1 (¯ y + 2 4 i 3 4 y + ≤ 4 i
(17)
(18)
3 y ) 4 hi,1 (¯ 4 N 3 4¯ 4 N yl hi,1,l (yl ). 4
(19)
l=1
Substituting (17)-(19) into (16) results in 3 3 1 LVi,1 = yi3(gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + yi + gμi,1 yi ) + gμi,1 e4i,2 2 4 4 N
1 3 ¯ 4 (yl ) . Nyl4 ρ¯4i,1,l (yl ) + Nyl4 h + i,1,l 4 4 l=1
(20)
Step j(2 ≤ j ≤ ni − 1). Reusing the Itˆ o formula and (12) shows xi,j , 0) + ρi,j (¯ y ) − αi,j−1)dt dei,j = (gμi,j ei,j+1 + gμi,j αi,j + fi,j (¯ +(hi,j (¯ y) −
j−1 ∂αi,j−1 k=1
∂xi,k
hi,j−1 (¯ y ))T dwi ,
(21)
where αi,j−1 =
j−1 ∂αi,j−1 k=1
∂xi,k
(fi,k (¯ xi,k+1 ) + ρi,k (¯ y )) +
∂αi,j−1 ˆ˙ θi ∂ θˆi
j−1 1 ∂ 2 αi,j−1 T + h (¯ y )hi,q (¯ y ). 2 p,q=1 ∂xi,p ∂xi,q i,p
(22)
Taking a Lyapunov function candidate as Vi = 14 e4i,j . And following (2), one has LVi,j =
e3i,j
xi,j , 0) − gμi,j ei,j+1 + gμi,j αi,j + fi,j (¯
j−1 ∂αi,j−1 k=1
∂xi,k
fi,k (¯ xi,k+1) −
∂αi,j−1 ˆ˙ θi ∂ θˆi
j−1 1 ∂ 2 αi,j−1 T +ρi,j (¯ y) − ρi,k (¯ y) − hi,p (¯ y )hi,q (¯ y) ∂x 2 ∂x ∂x i,k i,p i,q p,q=1 k=1
T
j−1 j−1 ∂αi,j−1 ∂αi,j−1 3 2 hi,j (¯ y) − hi,k (¯ y) y) − hi,k (¯ y) . + ei,j hi,j (¯ 2 ∂x ∂x i,k i,k k=1 k=1 j−1 ∂αi,j−1
8
(23)
Furthermore, following the same procesures from (17) to (19), the following inequalities hold 3 1 (24) gμi,j e4i,j + gμi,j e4i,j+1, 4 4 j−1 N j−1 j−1 ∂αi,j−1 3 4 ∂αi,j−1 4 1 4 4 3 ei,j −ei,j ρi,k (¯ y) ≤ ( )3 + N yl ρ¯i,k,l (yl ),(25) ∂x 4 ∂x 4 i,k i,k k=1 k=1 k=1 l=1 e3i,j gμi,j ei,j+1 ≤
y) e3i,j ρi,j (¯
3 4 1 4 4 ei,j + N ≤ y ρ¯ (yl ), 4 4 l=1 l i,j,l N
(26)
j−1 j−1 j−1 ∂ 2 αi,j−1 T 1 3 ∂ 2 αi,j−1 T 1 |ei,j | hi,p (¯ y )hi,q (¯ y) ≤ | |hi,p (¯ y )hi,p (¯ y) − e3i,j 2 ∂x ∂x 2 ∂x ∂x i,p i,q i,p i,q p,q=1 p=1 q=1 2 j−1 j−1 2 1 6 ∂ αi,j−1 ≤ e 4 i,j p=1 q=1 ∂xi,p ∂xi,q j−1
j−1
∂αi,j−1 3 2 y) − hi,k (¯ y ) 2 ei,j hi,j (¯ 2 ∂x i,k k=1
1 ¯ 4 (yl ), + (j − 1)N yl4 h i,p,l 4 p=1 l=1
j−1 3 4 ∂αi,j−1 4 1+ ≤ ( ) je 4 i,j ∂x i,k k=1 j
N
3 ¯ 4 (yl ). yl4h + jN i,k,l 4 k=1 l=1
(27)
N
(28)
Thus, the inequalities (23)-(28) imply that j−1 ∂αi,j−1 3 3 3 xi,j , 0) − (fi,k (¯ xi,k+1 )) + gμi,j ei,j + ei,j LVi,j ≤ ei,j gμi,j αi,j + fi,j (¯ ∂xi,k 4 4 k=1 2 j−1 j−1
j−1 j−1 3 ∂αi,j−1 4 1 3 ∂ 2 αi,j−1 3 ∂αi,j−1 4 3 + ei,j ( ) + ei,j + jei,j ( ) 4 ∂xi,k 4 ∂xi,p ∂xi,q 4 ∂xi,k p=1 q=1 k=1 k=1 j N 1 3 ∂αi,j−1 ˆ˙ 1 4 4 4 + jei,j − y ρ¯ (yl ) θi + gμi,j ei,j+1 + N 4 4 4 k=1 l=1 l i,k,l ∂ θˆi j−1
j
1 ¯ 4 (yl ) + 3 jN ¯ 4 (yl ). yl4 h yl4h + (j − 1)N i,p,l i,k,l 4 4 p=1 l=1 k=1 l=1 N
N
(29)
o formula, we have Step ni . Applying (2) and Itˆ xi,ni , 0) + ρi,ni (¯ y ) − αi,ni −1 )dt dei,ni = (gμi,ni ui + fi,ni (¯ y) − +(hi,ni (¯
n i −1 k=1
∂αi,ni −1 hi,k (¯ y ))T dwi , ∂xi,k
(30)
where αi,ni −1 is given in (22) with j = ni . Construct a stochastic Lyapunov function as 1 bm ˜2 θ , Vi,ni = e4i,ni + 4 2λi i 9
(31)
where θ˜i = θi − θˆi denotes the parameter estimation error and λi is a positive design constant. Then, with the help of (2), the time derivative of Vi,ni is expressed as n i −1 ∂αi,ni −1 ∂αi,ni −1 ˆ˙ 3 xi,ni , 0) − fi,k (¯ xi,k+1) − LVi,ni = ei,ni gμi,ni ui + fi,ni (¯ θi ∂xi,k ∂ θˆi k=1
n ni −1 i −1 ∂αi,ni −1 ∂ 2 αi,ni −1 T 1 +ρi,ni (¯ y) − ρi,k (¯ y) − h (¯ y )hi,q (¯ y) ∂xi,k 2 p,q=1 ∂xi,p ∂xi,q i,p k=1 i ∂αi,ni −1 bm ˜ ˆ˙ 3 y) − hi,k (¯ y ) 2 − + e2i,ni hi,ni (¯ θi θi . 2 ∂x λ i,k i k=1
n −1
(32)
Repeating the same derivations as (25)-(29) yields n i −1 3 ∂αi,ni −1 3 xi,ni , 0) − fi,k (¯ xi,k+1) + ei,ni LVi,ni ≤ ei,ni gμi,ni ui + fi,ni (¯ ∂xi,k 4 k=1 2 n ni −1 n i −1 i −1
∂ 2 αi,ni −1 3 ∂αi,ni −1 4 1 3 3 + ei,ni ( ) + ei,ni 4 ∂xi,k 4 ∂xi,p ∂xi,q p=1 q=1 k=1
n i −1 3 ∂αi,ni −1 4 3 ∂αi,ni −1 ˆ˙ bm ˜ ˆ˙ + ni ei,ni ( ) + ni ei,ni − θi θi θi − ˆi 4 ∂x 4 λ i,k i ∂ θ k=1 ni n N N i −1 1 1 4 4 ¯ 4 (yl ) + N yl ρ¯i,k,l (yl ) + (ni − 1)N yl4 h i,p,l 4 k=1 l=1 4 p=1 l=1 i 3 ¯ 4 (yl ). + ni N yl4 h i,k,l 4 k=1 l=1
n
N
(33)
Now, choose the Lyapunov function candidate for the whole system as V =
ni N i=1 j=1
Vi,j =
ni N 1 1 bm ˜2 θi ). ( yi4 + e4i,j + 4 4 2λ i i=1 j=2
Subsequently, combining the inequalities (20), (29) and (33), we have nl N N s 3 3 1 3 LV ≤ yi gμi,1 αi,1 + fi,1 (¯ xi,1 , 0) + yi + gμi,1 yi + Nyi ρ¯4l,k,i (yi ) 2 4 4 i=1 l=1 s=1 k=1 n n −1 N N s s l l 1 3 ¯ 4 (yi ) + Nyi ¯ 4 (yi ) h h s s + Nyi l,k,i l,p,i 4 4 p=1 l=1 s=1 k=1 l=1 s=1 j−1 N n i −1 ∂αi,j−1 3 ei,j gμi,j αi,j + fi,j (¯ xi,j , 0) − fi,k (¯ xi,k+1 ) + ∂xi,k i=1 j=2 k=1 2 j−1 j−1
j−1 3 ∂αi,j−1 4 1 3 ∂ 2 αi,j−1 3 ( ) 3 + ei,j + gμi,j ei,j + ei,j 4 4 ∂x 4 ∂xi,p ∂xi,q i,k p=1 q=1 k=1 10
(34)
j−1 3 ∂αi,j−1 4 3 1 + jei,j ( ) + (j + 1)ei,j + gμi,j−1 ei,j 4 ∂xi,k 4 4 k=1 n N i −1 ∂αi,ni −1 3 ei,ni gμi,ni ui + fi,ni (¯ xi,ni , 0) − fi,k (¯ xi,k+1 ) + ∂xi,k i=1 k=1 2 n ni −1 n i −1 i −1
∂ 2 αi,ni−1 3 ∂αi,ni −1 4 1 3 + ei,ni ( ) 3 + ei,ni 4 ∂x 4 ∂xi,p ∂xi,q i,k p=1 q=1 k=1 n i −1 3 ∂αi,ni −1 4 3 1 + ni ei,ni ( ) + (ni + 1)ei,ni + gμi,ni −1 ei,ni 4 ∂x 4 4 i,k k=1 −
ni N
∂αi,j−1 ˆ˙ e3i,j θi ∂ θˆi i=1 j=2
−
N bm i=1
λi
˙ θ˜i θˆi ,
(35)
where the following results have been utilized in (35). n
i l 1 1 4 4 N N yl ρ¯i,k,l (yl ) = yi4 ρ¯4l,k,i (yi ), 4 i=1 s=1 k=1 l=1 4 i=1 l=1 s=1 k=1
(36)
nl ni N s N s N N 3 3 4¯ 4 ¯ 4 (yi ), s y h (yl ) = s y 4h N N 4 i=1 s=1 k=1 l=1 l i,k,l 4 i=1 l=1 s=1 k=1 i l,k,i
(37)
N ni −1 s N s N nl −1 N 1 1 4¯ 4 ¯ 4 (yi ). s y h (yl ) = s y4h N N 4 i=1 s=1 p=1 l=1 l i,p,l 4 i=1 l=1 s=1 p=1 i l,p,i
(38)
N
n
s
N
N
N
s
By (11) and rearranging sequence, it follows that −
ni N
e3i,j
i=1 j=2
=−
ni N i=1 j=2
=
∂αi,j−1 e3i,j ∂ θˆi
ni N
∂αi,j−1 ˆ e3i,j γi θi ∂ θˆi i=1 j=2
−
ni N i=1 j=2
≤
∂αi,j−1 ˆ˙ θi ∂ θˆi
e3i,j
ni λi 6 ei,k − γi θˆi 2 2c i,k k=1
−
j−1 3 ∂αi,j−1 ei,j ∂ θˆi k=1 i=1 j=2
ni N
λi 6 e 2c2i,k i,k
ni ∂αi,j−1 λi 6 ei,k 2 ∂ θˆi k=j 2ci,k
ni N
∂αi,j−1 ˆ e3i,j γθ ˆi i i ∂ θ i=1 j=2
−
j−1 3 ∂αi,j−1 ei,j ∂ θˆi k=1 i=1 j=2
ni N
λi 6 e 2c2i,k i,k
j ni N λi 6 3 ∂αi,k−1 + e ( |e |). 2c2i,j i,j k=2 i,k ∂ θˆi i=1 j=2
Substituting (39) into (35) produces LV
≤
N
yi3
N n i −1 ¯ gμi,1 αi,1 + fi,1 (Zi,1) + e3i,j gμi,j αi,j + f¯i,j (Zi,j )
i=1
i=1 j=2
11
(39)
+
N i=1
e3i,ni
ni N N 3 bm ˜ ˆ˙ 4 ¯ gμi,ni ui + fi,ni (Zi,ni ) − ei,j − θi θi , 4 i=1 j=1 λi i=1
(40)
where the functions f¯i,j (Zi,j ), i = 1, 2, . . . , N are defined as n
l 9 3 1 ¯ xi,1 , 0) + yi + gμi,1 yi + Nyi ρ¯4l,k,i (yi ) fi,1 (Zi,1 ) = fi,1 (¯ 4 4 4 l=1 s=1 k=1
N
s
nl N N n s s l −1 1 3 4 ¯ ¯ 4 (yi ), hl,k,i (yi ) + Nyi h s s + Nyi l,p,i 4 4 p=1 l=1 s=1 k=1 l=1 s=1
(41)
j−1
j−1 ∂αi,j−1
∂αi,j−1 4 3 3 xi,j , 0) − fi,k (¯ xi,k+1 ) + gμi,j ei,j + ei,j ( )3 f¯i,j (Zi,j ) = fi,j (¯ ∂x 4 4 ∂x i,k i,k k=1 k=1 j−1 j−1 2 j−1 2 ∂ αi,j−1 ∂αi,j−1 1 3 3 + e3i,j + jei,j ( )4 + (j + 2)ei,j 4 ∂xi,p ∂xi,q 4 ∂xi,k 4 p=1 q=1 k=1
j−1
1 ∂αi,j−1 ˆ ∂αi,j−1 λi 6 + gμi,j−1 ei,j + γi θi − ei,k 2 4 ∂ θˆi ∂ θˆi k=1 2ci,k +
j λi 3 3 ∂αi,k−1 e ( |e |), j = 2, . . . , ni − 1, 2c2i,j i,j k=2 i,k ∂ θˆi
(42)
n i −1 3 ∂αi,ni −1 4 ∂αi,ni −1 fi,k (¯ xi,k+1) + ei,ni ( )3 ∂x 4 ∂x i,k i,k k=1 k=1
n −1 −1 n −1 n 2 i i i ∂ 2 αi,n −1 ∂αi,n −1 1 3 i i + e3i,ni + ni ei,ni ( )4 4 ∂x ∂x 4 ∂x i,p i,q i,k p=1 q=1 k=1
f¯i,ni (Zi,ni ) = fi,ni (¯ xi,ni , 0) −
n i −1
3 1 ∂αi,ni −1 ˆ + (ni + 2)ei,ni + gμi,ni −1 ei,ni + γi θi 4 4 ∂ θˆi ni −1 ni λi 6 ∂αi,k−1 ∂αi,ni −1 λi 3 − e + e ( |e3i,k |). i,k i,ni 2 2 ˆi 2ci,ni ∂ θˆi k=1 2ci,k ∂ θ k=2
(43)
¯ l,k,i are unknown, f¯i,j (Zi,j ), i = 1, 2, . . . , N, j = As smooth functions fi,j , gμi,j , ρ¯l,k,i and h 1, 2, . . . , ni cannot be directly employed to construct virtual control signal αi,j or the actual control input ui . According to Lemma 2, for any given εi,j > 0, there exists a fuzzy logic T ψi,j (Zi,j ) such that system Wi,j T ψi,j (Zi,j ) + δi,j (Zi,j ), f¯i,j (Zi,j ) = Wi,j
(44)
where δi,j (Zi,j ) denotes the approximation error and satisfies |δi,j (Zi,j )| < εi,j . Furthermore, T by using Young’s inequality and the fact of ψi,j ψi,j ≤ 1, one has
e3i,j f¯i,j (Zi,j )
=
e3i,j
T Wi,j ψi,j Wi,j + e3i,j δi,j (Zi,j ) Wi,j
12
≤
bm 6 1 3 1 ei,j θi + c2i,j + e4i,j + ε4i,j , 2 2ci,j 2 4 4
(45)
where i = 1, 2, . . . , N, j = 1, 2, . . . , ni and the unknown constant θi = max
1 Wi,j 2 ; bm
j=
1, 2, · · · , ni }. Substituting (44) into (40) and using (45) produces LV
N n i −1 bm 3 bm 6 3 ei,j gμi,j αi,j + 2 ei,j θi ≤ gμi,1 αi,1 + 2 ei,1 θi + 2ai,1 2ci,j i=1 i=1 j=2
ni
N N 1 2 bm 3 1 2 3 ci,j + εi,j ei,ni gμi,ni ui + 2 ei,ni θi + + 2a 2 4 i,n i i=1 i=1 j=1 N
−
e3i,1
N bm i=1
λi
˙ θ˜i θˆi .
(46)
Furthermore, by taking (10) and (11) into account and using the following inequality: e3i,j gμi,j αi,j ≤ −ki,j bm e4i,j −
bm 6 ˆ e θi , i = 1, 2, . . . , N, j = 1, 2, . . . , ni , 2c2i,j i,j
(46) can be rewritten as LV
≤ −
ni N i=1 j=1
≤ −
ki,j bm e4i,j
n N i i=1
+
N γi bm i=1
λi
θ˜i θˆi +
γi bm ˜2 θ ki,j bm e4i,j + 2λi i j=1
ni
N 1
2
i=1 j=1
+
ni
N 1 i=1 j=1
2
c2i,j
c2i,j
1 + ε2i,j 4
1 2 γi bm 2 + εi,j + θ , 4 2λi i
(47)
2 where the inequality θ˜i θˆi ≤ − 12 θ˜i + 12 θi2 has been used in the above inequality.
At the present stage, we summarize our main result in the following theorem. Theorem 1. Consider the large-scale stochastic nonlinear systems (5) consisting of Assumptions 1-2 and apply the above design procedures, the controller (10), and adaptive law (11). Assume that for 1 ≤ i ≤ N, 1 ≤ j ≤ ni , the packaged unknown functions f¯i,j (Zi,j ) can T be well approximated by the fuzzy logic systems Wi,j ψi,j (Zi,j ) in the sense that the approxi-
mation errors δi,j (Zi,j ) are bounded, then for bounded initial conditions [ziT (0), θˆi (0)]T ∈ Ω0 , all involved signals in the closed-loop system remain bounded in the sense of fourth-moment, and the error signals ei,j and θ˜i eventually converge to the compact set Ωs defined by ni N 2λi b0 b0 E[|ei,j |4 ] ≤ 4 , |θ˜i | ≤ ,1 ≤ i ≤ N . (48) Ωs = ei,j , θ˜i a0 bm a0 i=1 j=1 13
Proof: Let a0 = min{4ki,j bm , γi , i = 1, 2, . . . , N, j = 1, 2, . . . , ni }, ni N 1 1 γi bm 2 b0 = ( c2i,j + ε2i,j + θi ), 2 4 2λ i i=1 j=1 then we can rewrite (47) as LV ≤ −a0 V + b0 , t ≥ 0.
(49)
Therefore, it can be concluded from Lemma 1 and (34) that all the signals in the closed-loop system (5) remain fourth-moment bounded. Then, from (49), the following inequality can be reached directly dE[V (t)] ≤ −a0 E[V (t)] + b0 , dt
(50)
which implies that E[V (t)] ≤ e−a0 t [V (0)] +
b0 , ∀t > 0. a0
(51)
Then, it is easily obtained that E[V (t)] ≤
b0 , t → +∞. a0
(52)
Therefore, based on the definition of V in (34), the error signals ei,j and θ˜i eventually converge to the compact set Ωs specified in (48).
4
Simulation Example
Example 1. To illustrate the effectiveness of our result, consider the following the following pure-feedback interconnected stochastic nonlinear system: ⎧ y1 y2 3 ⎪ ⎪ dx dt + 1+y = (2 + sin(x ))x + x + y y 2 dw1 , 1,1 1,1 1,2 1 2 1,2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ dx1,2 = ((2 + cos(x1,1 x1,2 ))u1 + 0.1 sin(u1 ) + y12y2 ) dt + ln(1 + y12 )y2 dw1, ⎪ ⎪ ⎪ ⎪ ⎨ y =x , 1 1,1 ⎪ ⎪ dx2,1 = ((3 + sin(x2,1 ))x2,2 + 2x32,2 + y1 y22)dt + y1 sin(y22)dw2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dx2,2 = ((1 + x22,1 )u2 + 0.2u52 + y1 y2 )dt + y22 cos (y1 )dw2 , ⎪ ⎪ ⎪ ⎪ ⎩ y =x , 2 2,1 14
(53)
where x1,1 , x1,2 , x2,1 and x2,2 are the system states, u1 and u2 denote the system input signals. For simplicity, w˙ is chosen as an one-dimensional Gaussian white noise with zero mean and variance 1. It can be easily verified that Assumptions 1 and 2 are satisfied, and the existing approximation-based adaptive control schemes are not suitable for controlling this system. The control objective is to design an adaptive fuzzy controller such that all the signals in the closed-loop system are bounded. By applying Theorem 1, the virtual control inputs, actual controller and adaptive laws are constructed as ⎧ ⎪ ⎪ αi,1 = −ki,1 ei,1 − 2c12 e3i,1 θˆi , i = 1, 2, ⎪ ⎪ i,1 ⎨ 1 ui = −ki,2 ei,2 − 2c2 e3i,2 θˆi , i = 1, 2, i,2 ⎪ ⎪ 2 ⎪ ˙ ⎪ λ 6 i ˆ ⎩ θi = j=1 2 ei,j − γi θˆi , i = 1, 2, 2c
(54)
i,j
where ei,1 = xi,1 , ei,2 = xi,2 − αi,1 , i = 1, 2. The simulation is carried out under the initial conditions [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.6, 0.2, 0.5, 0.2]T , and [θˆ1 (0), θˆ2 (0)]T = [0, 0]T . In the simulation, the design parameters are chosen as: k1,1 = k1,2 = k2,1 = k2,2 = 2.5, c1,1 = c1,2 = c2,1 = c2,2 = 1, γ1 = γ2 = 1, and λ1 = λ2 = 2. Then we get the simulation results, which are shown by Figures 1-4. Fig. 1 displays the state variables y1 and y2 and Fig. 2 shows the state variables x2,1 and x2,2 . Figs. 3 displays the control input signals u1 and u2 , and Fig. 4 illustrates the response curve of the adaptive parameters θˆ1 and θˆ2 . Simulation results show that good convergence performances are achieved and all the signals in the closed-loop system are bounded.
5
Conclusion
In this paper, an approximation-based adaptive decentralized control scheme has been proposed for a class of large-scale pure-feedback stochastic nonlinear interconnected systems. In the controller design, fuzzy logic systems were used to model unknown nonlinearities and the decentralized adaptive controller is constructed based on backstepping. The proposed decentralized adaptive controllers guarantees that all the signals within the closed-loop systems remain bounded in the sense of fourth-moment. A simulation example is used to validity the effectiveness of the proposed approach.
15
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21
0.7 y1 0.6
y
2
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2
0
1
2
3
4 Time(Sec)
5
6
7
8
Fig. 1. States of closed-loop system y1 and y2 . 0.4 x12 x22
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
1
2
3
4 Time(Sec)
5
6
7
Fig. 2. States of closed-loop system x2,1 and x2,2 .
22
8
2 u1 u
2
1
0
−1
−2
−3
−4
−5
0
1
2
3
4 Time(Sec)
5
6
7
8
Fig. 3. The control input signals u1 and u2 . 0.7 θˆ1 θˆ2
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4 Time(Sec)
5
6
Fig. 4. Adaptive laws θˆ1 and θˆ2 .
23
7
8