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Adaptive fuzzy output constrained decentralized control for switched nonlinear large-scale systems with unknown dead zones
Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Adaptive fuzzy output constrained decentralized control for switched nonlinear large-scale systems with unknown dead zones Lili Zhang a,b,∗ , Guang-Hong Yang a,b a
College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China
b
State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, 110819, PR China
article
info
Article history: Received 13 December 2015 Accepted 21 July 2016
Keywords: Switched nonlinear large-scale systems Decentralized state observer Fuzzy logic systems Prescribed performance theory Unknown dead zones
abstract In this paper, an adaptive fuzzy output-feedback control design with output constrained is investigated for a class of switched uncertain nonlinear large-scale systems with unknown dead zones and immeasurable states. Based on dynamic surface backstepping control design technique and incorporated by the average dwell time method and the prescribed performance theory, a new adaptive fuzzy output-feedback control method is developed. It is strictly proved that the resulting closed-loop system is stable in the sense of uniformly ultimately boundedness and both transient and steady-state performances of the outputs are preserved. Comparison simulation studies illustrate the effectiveness of the proposed approach. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction A large-scale system is often considered as a set of interconnected subsystems, such as power systems, computer and telecommunications networks, economic systems and multiagent systems. Owing to the complexity of control synthesis and the physical restrictions on information exchange among subsystems, it is often required to design for each subsystem a decentralized controller depending only on local measurements, even if to achieve an objective for the whole large-scale system [1]. Since fuzzy logic system and neural networks are the universal approximator, and can approximate any nonlinear function with arbitrary precision [2]. During the past two decades, various adaptive decentralized control schemes have been developed for large-scale nonlinear systems [2]. In [3–7], adaptive neural network and fuzzy decentralized state feedback control designs were proposed for large-scale nonlinear systems in strict feedback form. In [8,9], adaptive neural network and fuzzy decentralized output feedback control methods were studied for large-scale nonlinear systems by designing a state observer for estimating the unmeasured states. The above mentioned adaptive fuzzy or neural decentralized control approaches can provide an effective methodology to control those uncertain nonlinear large-scale systems. Nevertheless, they need not require that the nonlinear functions included in the controlled systems be known or be linearly parameterized. However, the above mentioned results are only suitable for the non-switched nonlinear large-scale systems, instead of the switched nonlinear large-scale systems. Recently, some control design methods have been proposed for several classes of switched nonlinear systems in [10–16]. Two state feedback control approaches in [10,11] have proposed for a class of switched single-input and singleoutput (SISO) nonlinear systems based on the common Lyapunov function method. By applying average dwell-time
∗
Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China. E-mail addresses: [email protected] (L. Zhang), [email protected] (G.-H. Yang).
http://dx.doi.org/10.1016/j.nahs.2016.07.002 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
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technique, works in [12,13] have been investigated switched adaptive control schemes for a class of switched nonlinear systems with time-varying delay, and [14] firstly solved the problem of output feedback control design for stochastic nonlinear switched systems with unmodeled dynamics. In [15], a novel adaptive neural network control method has been presented for a class of SISO switched nonlinear systems with switching jumps. An adaptive fuzzy output tracking control problem has been investigated in [16] for a class of switched uncertain nonlinear large-scale systems under arbitrary switchings. Obviously, all the above mentioned adaptive control approaches can ensure that all the signals of the resulting closed-loop system are bounded. But all the existing switched control methods are not investigated in the problem of prescribed performance control (PPC). It is well known that the PPC demands the convergence rate no less than a prescribed value, exhibiting a maximum overshoot less than a sufficiently small constant and the output or tracking error is confined within the prescribed performance bounds for all times. The robust adaptive control for SISO strict feedback nonlinear system and feedback linearizable multiple-input and multiple-output (MIMO) nonlinear systems with PPC were investigated in [17,18]. A multiple constraints control was considered in [19,20] with PPC approach. The dead zone issues have been skillfully addressed with the prescribed performance theory for a class of non-switched systems in [21,22]. However, the prescribed performance design methodology still not be applied to switched nonlinear systems and the output feedback design for switched nonlinear large-scale systems with unknown dead zone is still a challenge. In this paper, an adaptive fuzzy output feedback control design with prescribed performance is developed for a class of Switched nonlinear large-scale systems with unknown dead zone. The main contributions of the proposed control scheme are as follows: (i) by introducing the prescribed performance control technique, both transient and steady-state performances of the hybrid switched large-scale systems with unknown dead zone and immeasurable states are ensured; (ii)parameter adaptive method is adopted to deal with unknown dead-zone issues, which makes the controller has better robustness. The remainder of this paper is organized as follows. The problem statement and preliminaries are given in Section 2. The decentralized switched fuzzy state observer design is given in Section 3. Adaptive controller design and stability analysis are in Section 4. The simulation studies are given in Section 5, and followed by Section 6 which concludes the work. 2. Problem statement and preliminaries 2.1. Systems descriptions and assumptions Consider a class of uncertain switched nonlinear large-scale systems that is composed of N subsystems interconnected by their outputs. The ith switched subsystem can be described by switched nonlinear systems:
σ (t ) σ (t ) σ (t ) x˙ i,1 = xi,2 + fi,1 (xi,1 ) + ∆i,1 (¯y) + di,1 σ (t ) σ (t ) σ (t ) x˙ i,2 = xi,3 + fi,2 (xi,2 ) + ∆i,2 (¯y) + di,2 . .. σ (t ) σ (t ) σ (t ) x˙ i,ni −1 = xi,ni + fi,ni −1 (xi,ni −1 ) + ∆i,ni −1 (¯y) + di,n−1 σ (t ) σ (t ) σ (t ) σ (t ) σ (t ) x˙ i,ni = Di (ui ) + fi,ni (xi,ni ) + ∆i,ni (¯y) + di,n yi = xi,1
(1)
where xi,j = [xi,1 , xi,2 , . . . , xi,j ]T ∈ ℜi , i = 1, 2, . . . , N ; j = 1, 2, . . . ni are the states, yi ∈ ℜ is the output. The function σ (t ) : [0, ∞) → M = {1, 2, . . . , m} is switching signal which is assumed to be a piecewise continuous (from σ (t ) the right) function of time. Moreover σ (t ) = k implies that the kth switched large-scale system is active. fi,j (xi,j ), are σ (t )
unknown smooth nonlinear functions. ∆i,j (¯y), (¯y = [y1 , y2 , . . . , yN ]) are unknown smooth functions representing the σ (t ) σ (t ) interconnection effects between the ith subsystem and the other subsystems. Di (ui ) ∈ ℜ is the output of the deadzones. In addition, we assume that the state of the system (1) does not jump at the switching instants, i.e., the solution is everywhere continuous, which is a standard assumption in the switched system literature [16,23]. Now we recall the definition of average dwell time, which plays a key role in research of switched nonlinear control. A switched nonlinear large-scale system (1) is called to have a switching signal σ (t ) with average dwell time τa , if there exist two positive numbers N0 and τa such that Nσ (T , t ) ≤ N0 +
T −t
τa
∀T ≥t ≥0
(2)
where Nσ (T , t ) is the number of switches occurring in the interval [t , T ). Let T > 0 be an arbitrary time. Denote by t1 , . . . , tNσ (T ,0) the switching times on the interval (0, T ) (t0 = 0). When t ∈ [tj , tj+1 ), σ (t ) = kj , that is, the kj th large-scale system is active. In this study, we assume kj ̸= kj+1 for all j.
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63
Remark 1. It should be mentioned that if for t ∈ [0, ∞), the switching signal σ (t ) = k, then the system (1) represents a class of nonswitched nonlinear large-scale systems in strict feedback form, which is investigated commonly in the literatures, for example [3,4]. σ (t )
Similar to [24], the output of the dead zone Di σ (t )
Di
σ (t )
(ui
),
σ (t ) σ (t ) σ (t ) mi,r (ui − di,r ) 0
if ui
i ,l
i
≥ dσi,r(t ) < uσi (t ) < dσi,r(t ) ≤ −dσi,l(t ) .
− dσi,l(t ) σ (t )
if
mσ (t ) (uσ (t ) + dσ (t ) ) i,l
σ (t )
σ (t )
if ui
(uσi (t ) ) is described by: (3)
∈ ℜ is the input to the dead zone; mσi,r(t ) and mσi,l(t ) denote the slope of the dead zone and dσi,r(t ) and dσi,l(t ) represent
In (2), ui
σ (t )
the dead zone width parameters. The dead zone outputs are not available for measurement. The dead zone parameters mi,r , σ (t )
σ (t )
σ (t )
mi,l , di,r
and di,l
σ (t )
are unknown, but their signs are known (mi,r
> 0, mσi,l(t ) > 0, dσi,r(t ) ≥ 0 and dσi,l(t ) ≥ 0, respectively). σ (t )
σ (t )
σ (t )
σ (t )
Assumption 1 ([24]). Dead zone slopes are bounded by known constants mi,rmin , mi,rmax , mi,lmin and mi,lmax such that 0<
σ (t ) mi,rmin
≤
σ (t ) mi,r
≤
σ (t ) mi,rmax
and 0 <
σ (t ) mi,lmin
≤
σ (t ) mi,l
≤
σ (t ) mi,lmax .
σ (t )
The dead zone inverse technique is useful for compensating the dead zone effect. Setting ui,d as the control input from σ (t )
the controller to achieve the control objective for the plant without a dead zone, the following control signal ui according to the certainty equivalence dead zone inverse: σ (t )
= (Diσ (t ) (uσi,d(t ) ))−1 =
ui
σ (t )
σ (t )
σ (t )
σ (t )
σ (t )
σ (t )
ui,d + dˆ i,r ,m σ (t )
δi
σ (t )
ˆ i,r m
σ (t )
ui,d + dˆ i,l,m
+
σ (t )
ˆ i ,l m
σ (t )
σ (t )
σ (t )
(1 − δiσ (t ) )
σ (t ) σ (t )
ˆ i ,r , m ˆ i,l , dˆ i,r ,m and dˆ i,l,m are estimates of mi,r , mi,l , mi,r di,r where m
σ (t )
δi
=
1 0
Di
(ui
σ (t )
˜ i,l where m
σ (t ) σ (t )
and mi,l di,l , respectively, and
σ (t )
(5)
σ (t )
if ui,d < 0. σ (t )
σ (t )
(4)
if ui,d ≥ 0
The resulting errors between ui σ (t )
is generated
)−
(uσi,d(t ) )
=
σ (t ) d˜ i,r ,m
σ (t )
and ui,d are given by σ (t )
−
σ (t )
ui,d + dˆ i,r ,m
ˆ σi,r(t ) m
˜ σi,r(t ) m
σ (t )
δi
+
σ (t ) d˜ i,l,m
σ (t )
−
σ (t )
ui,d + dˆ i,l,m
ˆ σi,l(t ) m
˜ σi,l(t ) m
(1 − δiσ (t ) ) + εiσ,d(t )
(6)
ˆ σi,l(t ) − mσi,l(t ) , m ˜ σi,r(t ) = m ˆ σi,r(t ) − mσi,r(t ) , d˜ σi,l(,tm) = dˆ σi,l(,tm) − dσi,l(,tm) and d˜ σi,r(,tm) = dˆ σi,r(,tm) − dσi,r(,tm) are parameter errors. =m
εiσ,d(t ) can be expressed as
εiσ,d(t ) = −mσi,r(t ) χiσ,r(t ) (uσi (t ) − dσi,r(t ) ) − mσi,m(t ) χiσ,l(t ) (uσi (t ) − dσi,r(t ) ) where
εiσ,d(t )
(7)
is bounded with
χiσ,r(t ) =
1, 0,
σ (t )
σ (t )
if 0 ≤ ui < di,r otherwise ;
χiσ,l(t ) =
1, 0,
σ (t )
σ (t )
if di,l < ui otherwise.
<0
σ (t )
Assumption 2 ([25]). For the nonlinear functions ∆i,j (¯y) satisfy N σ (t ) σ (t ) σ (t ) qi,l βl (|yl |) ∆i,j (¯y) ≤
(8)
l=1
σ (t )
σ (t )
where qi,l (i = 1, 2, . . . , N , l = 1, 2, · · · N ) is unknown constant denoting the strength of the interaction; βl is a known nonlinear smooth function.
(|yl |) ≥ 0
σ (t )
Thus, according to [25], there exists smooth non-negative function h¯ i,l (yl ) such that
N l=1
σ (t ) σ (t ) qi,l βl
2 (|yl |)
≤
N l =1
σ (t ) h¯ i,l y2l + 2
N
σ (t )
βl
2 (0)
.
(9)
l =1
Lemma 1 ([26]). For any constant η > 0 and any a variable ϖ ∈ R, limϖ →0 ϖ1 tanh2 ( ϖ ) = 0. η Lemma 2 ([26]). Consider the bounded set given by Ωη := {η ||η| < 0.8814τ }, τ is the positive constant. Then, ϖ ̸∈ Ωϖ , the inequality 1 − 16 tanh2 (τ /η) ≤ 0 holds.
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2.2. Performance function and error transformation According to [18], the prescribed performance is achieved by ensuring that each output yi (t ) (i = 1, 2, . . . , N ) evolves strictly within prescribed decaying bounds as follows:
− δi,min µi (t ) < yi (t ) < δi,max µi (t ) ,
∀t ≥0
(10)
where δi,min and δi,max are design constants, and the performance functions µi (t ) are bounded and strictly positive decreasing smooth functions with the property limt →∞ µi (t ) = µi,∞ ; µi,∞ > 0 is a constant. In this paper, the performance functions are chosen as the exponential form µi (t ) = (µi,0 − µi,∞ )e−ai t + µi,∞ , where ai , µi,0 and µi,∞ are strictly positive constants, µi,0 > µi,∞ , and µi,0 = µi (0) is selected such that −δi,min µi (0) < yi (0) < δi,max µi (0) is satisfied. Furthermore, the maximum overshoot of yi (t ) is prescribed less than max{δi,min µi (0), δi,max µi (0)}. Therefore, choosing the performance function µi (t ) and the constants δi,min , δi,max appropriately determines the performance bounds of the output yi (t ). To represent (10) by an equality form, we set yi (t ) = µi (t ) Φi (ζi (t )), where Φi (ζi ) =
∀t ≥0
δi,max eζi −δi,min e−ζi eζi +e−ζi
(11)
.
Since the function Φi (ζi ) is strictly monotonic increasing, its inverse function can be expressed as
ζi (t ) = Φ −1
yi (t )
=
µi (t )
1 2
ln
Φi − δimin δimax − Φi
(12)
and
µ ˙ y ˙ζi (t ) = ρi y˙ i − i i µi 1 with ρi = 21µ Φ +δ1 − . Φi −δi,max i i i,min
(13)
For control design of the nonlinear system, we design the following errors transformation zi,1 (t ) = ζi (t ) −
1 2
ln
δi,min . δi,max
(14)
And the transformation state dynamics is
z˙i,1 (t ) = ρi
µ ˙ i yi y˙ i − µi
.
(15)
Lemma 3 ([18]). Consider output yi (t ) and transformed errors zi,1 (t ) defined in (14). If zi,1 (t ) is bounded, prescribed performance of yi (t ) is satisfied for all t ≥ 0, i.e., (10) is satisfied. 2.3. Fuzzy logic systems A fuzzy logic system (FLS) consists of four parts: a knowledge base, a fuzzifier, a fuzzy inference engine with fuzzy rules, and a defuzzifier. The knowledge base for FLS is comprised of a collection of fuzzy If-then rules as following: Rl : If x1 is F1l and x2 is F2l and . . . and xn is Fnl , then y is Gl , l = 1, 2, . . . , N where x = [x1 , . . . , xn ]T and y are the FLS input and output, respectively. Fil and Gl are fuzzy sets with membership functions µF l (xi ) and µGl (y), respectively. N is the number of rules. i
Through singleton fuzzifier, center average defuzzification and product inference Wang2016, the fuzzy logic system can be expressed as N
n
µF l ( x i ) i y(x) = n N µF l (xi ) y¯ l
l=1
i=1
l=1
(16)
i
i =1
where y¯ l = maxy∈R µGl (y). Define the fuzzy basis functions as n
ϕl =
µF l ( x i )
i=1 N
n
l=1
i=1
i
. µF l (xi ) i
(17)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
65
Denoting θ T = [¯y1 , y¯ 2 , . . . , y¯ N ] = [θ1 , θ2 , . . . , θN ] and ϕ(x) = [ϕ1 (x), . . . , ϕN (x)]T , then fuzzy logic system (16) can be rewritten as y(x) = θ T ϕ(x).
(18)
Lemma 4 ([16,27]). Let f (x) be a continuous function defined on a compact set Ω . Then for any constant ε > 0, there exists a fuzzy logic system fˆ (x |θ ) = θ T ϕ(x) such as sup f (x) − (θ ∗ )T ϕ(x) ≤ ε.
(19)
x∈Ω
3. Decentralized switched fuzzy state observer design Since the states xi,2 , xi,3 , . . . , xi,ni −1 and xni in the switched nonlinear large-scale system (1) are not available for feedback, this section aims to design a switched decentralized state observer to estimate the unmeasured states in (1). σ (t ) Note that the functions fi,j (xi,j ) are unknown. Thus the fuzzy logic systems are utilized to approximate them. By Lemma 4, we assume that σ (t )
fi,j
(xi,j |θiσ,j(t ) ) = θiσ,j(t )T ϕiσ,j(t ) (xi,j ); fˆi,σj (t ) (ˆxi,j |θiσ,j(t ) ) = θiσ,j(t )T ϕiσ,j(t ) (ˆxi,j ),
(20)
where xˆ i,j = [ˆxi,1 , xˆ i,2 , . . . , xˆ i,ni ] are the estimations of xi,j = [xi,1 , xi,2 , . . . , xi,ni ] . T
T
∗σ (t )
According to definition of the optimal parameter vectors θi,j (t ) θi∗σ ,j
= arg min σ (t )
sup
σ (t )
(xi,j ׈xi,j )∈Ui,j,1 ×Ui,j,2
θi,j ∈Ωi,j
in [16], let
ˆ σ (t ) σ (t ) σ (t ) fi,j (ˆxi,j |θi,j ) − fi,j (xi,j ) ,
σ (t )
(21)
σ (t )
where Ωi,j , Ui,j,1 and Ui,j,2 are bounded compact regions for θi,j , xi,j and xˆ i,j , respectively. In addition, the approximation σ (t )
error εi,j
is defined as
σ (t )
(t ) (xi,j ) = fˆi,σj (t ) (ˆxi,j |θi∗σ ) + εiσ,j(t ) , i = 1, 2, . . . , N , j = 1, 2, . . . , ni ,j σ (t ) ∗σ (t ) ∗σ (t ) σ (t ) being an unknown constant. where εi,j satisfies εi,j ≤ εi,j , with εi,j
fi,j
(22)
By substituting (21) into (1), the switched system (1) can be expressed as σ (t )
x˙ i,j = xi,j+1 + fˆi,j σ (t )
x˙ i,ni = Di
(t ) (ˆxi,j |θi∗σ ) + εiσ,j(t ) + ∆σi,j(t ) (t ) ,j
(t ) σ (t ) (uiσ (t ) ) + fˆi,σn(it ) (ˆxi,ni |θi∗σ )εi,ni + ∆σi,n(ti ) (t ) ,j
(23)
yi = xi,1 i = 1, 2, . . . , N , j = 1, 2, . . . , ni − 1. Rewrite (23) as σ (t )
x˙ i,ni = Ai
σ (t )
xi,ni + Li
yi +
ni
∗σ (t )T
Bi,j θi,j
ϕiσ,j(t ) (ˆxi,j ) + εiσ (t ) + ∆σi (t ) + Bi,ni Dσi (t ) (uσi (t ) )
j =1
(24)
yi = Ci xi,ni where σ (t )
Ai
σ (t ) −li,1 = ... −lσi,n(ti )
Ini −1 0
···
σ (t )
,
∆i
σ (t ) ∆i,1 (¯y) = ... ,
σ (t ) σ (t )
Li
σ (t )
σ (t )
∆i,ni (¯y)
0
εiσ (t ) = [εiσ,1(t ) , εiσ,2(t ) , . . . , εiσ,n(it ) ]T ,
Ci = [1, . . . , 0],
li,1
= ... , li,ni
Bi,j = [0, . . . , 0, 1, 0, . . . , 0]T .
j
To estimate the states of the system, the following switched fuzzy decentralized state observer is designed σ (t )
x˙ˆ i,ni = Ai
σ (t )
xˆ i,ni + Li
yi +
ni
σ (t )
Bi,j fˆi,j
(ˆxi,j |θiσ,j(t ) ) + Bi,ni Dσi (t ) (uσi (t ) )
j =1
(25)
yˆ i = Ci xˆ i,ni . Define observation error vector ei as ei = [ei,1 , ei,2 , . . . , ei,ni ]T = xi,ni − xˆ i,ni .
(26)
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From (24)–(26), the observation error can be expressed as σ (t )
e˙ i = Ai
ni
ei +
σ (t )
σ (t )
σ (t )
Bi,j (θ˜i,j )T ϕi,j (ˆxi,j ) + εi
+ ∆σi (t )
(27)
j =1
σ (t )
∗σ (t )
σ (t )
where θ˜i,j = θi,j − θi,j , i = 1, 2, . . . , N , j = 1, 2, . . . ni are the adaptive parameter vector errors. For the switching signal σ (t ) = k, k = 1, 2, . . . , m; the kj th switched large-scale system is active and the remaining large-scale systems are inactive. Choose vector Lki such that matrix Aki is a strict Hurwitz matrix, therefore, for any given matrix Qik = (Qik )T > 0, there exists a positive definite matrix Pik = (Pik )T > 0 such that
(Aki )T Pik + Pik Aki = −Qik .
(28)
To evaluate the property of the state observer (25), we consider the following Lyapunov candidate V0k =
N
Vik,0 =
N
i =1
ei T Pik ei .
V0k
for (28) as (29)
i=1
Then the time derivative of V0k along with the solutions of (26) is V0k
˙ ≤
N
(
k min Qi
−λ
) ∥ei ∥ + 2
ε +
2eTi Pik
k i
∆ki
i=1
+
ni
Bi,j (θ˜ ) ϕ (ˆxi,j ) k T i,j
k i,j
(30)
j =1
where λmin (Qi ) is the smallest eigenvalue of matrix Qi . Note that βm (|ym |) , m = 1, 2, . . . , N is a known non-linear smooth function, thus, according to [28], there exists smooth non-negative function lki,m (ym ) such that
N
2 qki,m βm (|ym |)
≤
m=1
N
lki,m y2m + 2
m=1
N
2 βm (0)
.
(31)
m=1
By completing the squares and Assumption 2, the fact that ϕik,j (ˆxi,j )T ϕik,j (ˆxi,j ) ≤ 1, we can obtain N
2eTi Pik ∆ki ≤
N
2 N N 2 ∥ei ∥2 + ni Pik 2 βm (0) lki,m y2m + 2ni Pik
i =1
i=1
≤
N
m=1
i =1 ni
m=1
2 N N N N k 2 k 2 2 k 2 ∥ei ∥ + ni Pi lm,i yi + 2ni Pi βm (0) m=1 i=1
(32)
i=1 m=1
ni 2 (θ˜ ki,j )T θ˜ik,j
Bi,j (θ˜ik,j )T ϕik,j (ˆxi,j ) ≤ ni ∥ei ∥2 + ni Pik
(33)
j =1
j =1
∗ 2
2eTi Pik εik ≤ ∥ei ∥2 + Pik ε ki .
(34)
Substituting (32)–(34) into (30) results in V0k
˙ ≤
N
ni N N k 2 2 k T −λ ∥ei ∥ + Pi (θ˜ i,j ) θ˜i,j + Di,0 + ni Pik lkm,i y2i k i,0
2
i=1
j =1
2 N N
where Dki,0 = 2ni Pik (
i =1
m=1
(35)
m=1 i=1
∗ 2 βm (0))2 + Pik ε ki and λki,0 = λmin (Qi ) − 2 − ni .
4. Adaptive controller design and stability analysis This section will give the adaptive fuzzy output-feedback control design procedure based on dynamic surface backstepping design technique in [29] and the fuzzy switched decentralized state observer designed in the above section. The stability of the closed-loop system will be proved by using Lyapunov function and average dwell time method. When t ∈ [tj , tj+1 ) , and σ (t ) = k, that is, the ni th switched large-scale system is active. For the kth switched largescale system, the ni step adaptive fuzzy backstepping output feedback control design is based on the following change of coordinates: Si,1 = zi,1 ,
Si,j = xˆ i,j − qi,j ;
χi,j = qi,j − αik,j−1 ,
i = 2, . . . , N ; j = 2, . . . , ni
(36)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
67
where Si,j is called error surface, qi,j is a state variable, which is obtained through a first-order filter on intermediate control function αi,j−1 and χi,j is called the output error of the first-order filter. Step i, 1: Expressing xi,2 in terms of its estimate as xi,2 = xˆ i,2 + ei,2 , we obtain S˙i,1 = ρi
y˙ i −
µ ˙ i yi µi
= ρi Si,2 + χi,2 + αik,1 + ei,2 + (θik,1 )T ϕik,1 (ˆxi,1 ) µ ˙ i yi + εik,1 . µi
+ (θ˜ik,1 )T ϕik,1 (ˆxi,1 ) + ∆ki,1 (¯y) −
(37)
Choose the Lyapunov function candidate as N
Vik,1
=
V0k
i =1
+
N 1 2
i =1
Si2,1
1
+
2δik,1
(θ˜ ) θ˜
k T k i ,1 i ,1
(38)
where δik,1 > 0 is design parameter. From (1), and (36)–(37), the time derivative of Vik,1 satisfies V˙ ik,1 ≤ V˙ ik,0 + Si,1 ρi
Si,2 + χi,2 + αik,1 + εik,1 + (θik,1 )T ϕik,1 (ˆxi,1 )
+ ei,2 + (θ˜ik,1 )T ϕik,1 (ˆxi,1 ) + ∆ki,1 (¯y) −
µ ˙ i yi µi
+
1
δik,1
k
(θ˜ik,1 )T θ˙˜ i,1 .
(39)
By completing the squares, by Assumption 2, we can obtain Si,1 ρi (Si,2 + χi,2 ) ≤ Si2,1 ρi2 + Si,1 ρi (εik,∗2 + ei,2 ) ≤ Si2,1 ρi2 + N
ρ
Si,1 i ∆ki,1
(¯y) ≤
i=1
N 1 i =1
2
1 2 1
Si2,2 +
1 2
χi2,2
(40)
1
∥ei ∥2 + εik∗2
2
2
ρ +
Si2,1 i2
N N i=1 m=1
lkm,i y2i
+
N N
2 βm (0)
(41)
i=1 m=1
where εik∗ = [εik,∗1 , εik,∗2 , . . . , εik,∗ni ]T . Substituting (40)–(41) into (39) results in
Γk ˙ i yi ˙Vik,1 ≤ −λki,1 ∥ei ∥2 + 1 Si2,2 + Si,1 ρi αik,1 + i,1 + 2Si,1 ρi + (θik,1 )T ϕik,1 (ˆxi,1 ) − µ 2 Si,1 ρi µi ni 2 1 1 (θ˜ik,1 )T θ˜ik,j + Dki,1 + χi2,2 + (θ˜ik,1 )T Si,1 ρi ϕik,1 (ˆxi,1 ) − k θ˙ik,1 + ni Pik 2 δi,1 j =1 where λki,1 = λki,0 + 12 , Dki,1 = Dki,0 + (
N
m=1
Note that when Si,1 ρi = 0, functions 2 Si,1 ρi
tanh ( τ ) is introduced to deal with i 1
k
2 βm (0))2 and Γi,k1 = (ni Pik + 1) Nm=1 lkm,i .
Γi,1 Si,1 ρi
are discontinuous function. In order to overcome this problem, the function
1 Γk . Si,1 ρi i,1
Then (42) becomes
1
χi2,2 + Si,1 ρi αik,1 + (θik,1 )T ϕik,1 (ˆxi,1 ) + 2Si,1 ρi 2 2 16 Si,1 ρi µ ˙ i yi Si,1 ρi 2 tanh Γi,k1 − + 1 − tanh2 Γi,k1 + Si,1 ρi τi µi τi ni 1 k k T k k k T k + (θ˜i,j ) θ˜i,j + Di,1 + (θ˜i,1 ) ϕi,1 (ˆxi,1 )Si,1 ρi − k θ˙i,1 . δi,1 j=1
V˙ ik,1 ≤ −λki,1 ∥ei ∥2 +
Si2,2 +
(42)
(43)
Design the intermediate control function αik,1 , the adaptation function θik,1 as
αik,1 = −βik,1 Si,1 −
16 Si,1 ρi
tanh2
θ˙ik,1 = δik,1 ϕik.1 (ˆxi,1 )Si,1 ρi − τik,1 θik,1
Si,1 ρi
τi
µ ˙ i yi Γi,k1 − (θik,1 )T ϕik.1 xˆ i,1 − Si,1 ρi + µi
(44) (45)
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
where βik,1 > 0, τik,1 > 0 are design parameters. From Lemma 1, one knows that limSi,1 ρi →0 According to Lemma 2 and (43)–(45), it follows that V˙ ik,1 ≤ −λi,1 ∥ei ∥2 +
1 2
Si2,2 − βik,1 Si2,1 ρi2 +
1 2
τik,1
χi2,2 +
δ
k i,1
(θ˜ik,1 )T θik,1 +
16 Si,1 ρi
S
ρ
tanh2 ( i,τ1 i )Γi,k1 exists. i
ni (θ˜ik,j )T θik,j + Dki,1 .
(46)
j =1
Step i, j(2 ≤ j ≤ ni ): Since Si,j = xˆ i,j − qki,j , similar to Step i, 1, the derivative time of Si,j along with (25) and (36) is S˙i,j = x˙ˆ i,j − q˙ i,j = Si+1 + χi,j+1 + αik,j + lki,j ei,1 + θiT,k ϕi,k (ˆxi,j ) − q˙ i,j .
(47)
To avoid repeatedly differentiating αik,j in the traditional backstepping design, which leads to the so called the problem
‘‘explosion of complexity’’ as mentioned in [29], we introduce a new state variable qi,j+1 and let αik,j pass through a first-order filter with a constant ςi,j+1 to obtain qi,j+1 , i.e.,
ςi,j+1 q˙ i,j+1 + qi,j+1 = αik,j ,
qi,j+1 (0) = αik,j (0).
(48)
By definition of χi,j+1 = qi,j+1 − αik,j , it yields q˙ i,j+1 =
χ − ςii,,jj++11
and
χi,j+1 ςi,j+1
χ˙ i,j+1 = q˙ i,j+1 − α˙ ik,j = Hi,j+1 (·) −
(49)
where Hi,j+1 (ˆxi,j , yi , S2 , . . . , Si,j+1 , χ2 , . . . , χi,j+1 , θi,1 , . . . , θi,j ) is a continuous function. Consider the following Lyapunov function candidate N
Vik,j =
i=1
N
Vik,j−1 +
i=1
1 2
zi2,j +
1
1
2δik,j
(θ˜ik,j )T θ˜ik,j + χi2,j
(50)
2
where δik,j > 0 is the design parameter. From (49) and (50), we can obtain V˙ ik,j ≤ V˙ ik,j−1 + Si,j (Si,j+1 + χi,j+1 + αik,j + lki,j ei,1 − q˙ i,j + (θik,j )T ϕik,j (ˆxi,j ) − (θ˜ik,j )T ϕik,j (ˆxi,j ))
− χi,j
χi,j Hi,j − ςi,j
+ (θ˜ )
Si,j ϕ (ˆxi,j ) −
k T i,j
k i,j
1
δik,j
θ˙
k i,j
.
(51)
By completing the squares, we have Si,j (Si,j+1 + χi,j+1 + lki,j ei,1 ) ≤
−Si,j (θ˜ik,j )T ϕik,j (ˆxi,j ) ≤
1
3 2
Si2,j +
1 2
Si2,j+1 +
1 2
1
χi2,j+1 + (lki,j )2 ∥ei ∥2
1
(θ˜ik,j )T θ˜ik,j + Si2,j .
2 Substituting (52)–(53) into (51) yields
(53)
2
V˙ ik,j ≤ V˙ ik,j−1 + Si,j (2Si,j + αik,j + (θik,j )T ϕik,j (ˆxi,j ) − q˙ i,j ) +
+ (θ˜ )
k T i,j
(52)
2
Si,j ϕ (ˆxi,j ) − k i,j
1
δik,j
θ˙
k i,j
1 2
Si2,j+1 +
1 2
χi2,j+1
χi,j 1 − χi,j − + Hi,j + (lki,j )2 ∥ei ∥2 . ςi,j 2
(54)
Design the intermediate control function αik,j , the adaptation function θik,j as
αik,j = −βik,j Si,j − 2Si,j − (θik,j )T ϕik,j (ˆxi,j ) + q˙ i,j
(55)
θ˙ = δ
(56)
k i,j
k i,j Si,j
ϕ (ˆxi,j ) − τ θ k i,j
k k i,j i,j
where βik,j > 0 and τik,j > 0 are design parameters. Substituting (55)–(56) into (54) yields V˙ ik,j ≤ −λki,j ∥ei ∥2 +
j 1
2 p=1
ni
+
χi2,p+1 −
j
βi,p Si2,p + Dki,1 − βi,1 Si2,1 ρi +
p=2 j
p=1 j
1 k T k χi,p (θ˜ik,j )T θ˜ik,j + (θ˜i,p ) θ˜i,p − χi,p Hi,p − 2 ςi,p j=1 p=2 p=2
where λki,j = λki,1 −
j −1 2
(lki,j )2 .
j τik,p
1
δik,p
+ Si2,j+1 2
(θ˜ik,p )T θik,p
(57)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
69
Step i, ni : In the final design step, the dead-zone input uki,d will be obtained. From (3), (5) and (36), the time derivative of Si,ni is S˙i,ni = x˙ˆ i,ni − α˙ ik,ni −1 = Dki (uki ) + lki,ni ei,1 + (θik,ni )T ϕik,ni (ˆxi,ni ) − q˙ i,ni
=
uki,d
dki,m,r
+ ˜
d˜ ki,m,l −
+
−
uki,d + dˆ ki,m,r
ˆ ki,r m
uki,d + dˆ ki,m,l
ˆ ki,l m
δik + lki,ni ei,1 + (θik,ni )T ϕik,ni (ˆxi,ni )
mki,r
˜
(1 − δik ) + εik,d − q˙ i,ni .
mki,l
˜
(58)
Consider the following Lyapunov function candidate Vik,ni = Vik,ni +
+
1 2
1
Si2,ni +
1 2δ
1
˜ ki,r )2 + (m
2ηik,2
1
1
2
2ηik,1
(θ˜iT,ni )T θ˜i,ni + χi2,ni +
k i,ni
2ηik,3
(d˜ ki,m,l )2 +
1 2ηik,4
˜ ki,l )2 (m
(d˜ ki,m,r )2
(59)
k
where δi,ini > 0 and ηik,p (p = 1, 2, 3, 4) are design parameters. V˙ ik,ni = V˙ ik,ni −1 + Si,ni (uki,d + 2Si,ni + εik,d + (θik,ni )T ϕik,ni (ˆxi,ni ) − q˙ i,ni )
k 1 k ˙ k T k ˜ + (Si,ni η δ + dˆ i,m,r ) + (θ˜i,ni ) Si,ni φi,ni (ˆxi,ni ) − k θ˙i,ni η δi,ni k ˆk 1 k ˙ˆ k − ui,d + di,m,l Si,n ηk (1 − δ k ) ˜ i ,l m + k m i ,l i i i,1 k ˆ i,l m ηi,1 uki,d + dˆ ki,m,r k 1 k 1 ˙k k k ˙ ˜ i ,r m ˆ i,r − Si,ni ηi,2 δi + k d˜ ki,m,l (Si,ni ηik,3 (1 − δik ) + dˆ i,m,l ). + k m k ˆ i ,r m ηi,2 ηi,3 1
dki,m,r k i ,4
k k i,4 i
(60)
Similar to the procedures in step i, j, design the input of the dead zones uki,d and parameter adaptation functions as follows: uki,d = −βik,ni Si,ni − 2Si,ni − (θik,ni )T ϕik,ni (ˆxi,ni ) + q˙ i,ni
(61)
θ˙ik,ni = δik,ni Si,ni ϕik,ni (ˆxi,ni ) − τik,ni θik,ni
(62)
where β > 0 and τ > 0 are design parameters. Similar to the results in literature [21,22], the unknown deadzone parameter adaptation functions are omitted here for brevity. Substituting (61)–(62) and dead-zone parameter adaptation laws into (60) yields k i,ni
k i,ni
V˙ k ≤ −λki,ni ∥ei ∥2 −
ni
βik,j Si2,j − βik,j Si2,j ρi +
j =2
ni τik,j j =1
δik,j
(θ˜ik,j )T θik,j
ni ni ni k 2 χi,j k T ˜k ˜ + Pi (θi,j ) θi,j + χi,j − + Hi,j + χi2,j + Dki,ni ςi,j j=1 j =2 i=2 +
τik,ni η
k i,1
˜ ki,l m ˆ ki,l + m
τik,ni η
k i,2
˜ ki,r m ˆ ki,r + m
τik,ni η
k i,3
d˜ ki,l,m dˆ ki,l,m +
τik,ni ηik,4
d˜ ki,r ,m dˆ ki,r ,m
(63)
where λki,ni = λki,ni −1 − 21 , Dki,ni = Dki,1 + 21 (¯εik,d )2 , ε¯ ik,d ≤ ε¯ i,d and ε¯ i,d is a constant.
(
ni 1
1 T k i =1 2 e i P i e i
+
2
j =1
(Si2,j +
1
δik,j
(θ˜ik,j )k θ˜ik,j ) +
ni
χi2,j ) ≤ M}, where M > 0 is a known constant. Since Θ is a compact set and Hi,j is a continuous function, there exists a positive constant Mi,j such that Hi,j ≤ Mi,j on Θ , therefore, we Let Θ = {
N
1 2
j=2
have
χi,j Hi ≤ 1 χ 2 M 2 + 1 . i,j i,j
2 2 By completing the square for each parameter estimate: 1
1
2
2
(θ˜ik,j )T θik,j ≤ − (θ˜ik,j )T θ˜ik,j + (θik,j )∗T (θik,j )∗ .
(64)
(65)
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
Substituting the above inequalities into (63) yields k
V˙ ≤
N
−λki,ni ∥ei ∥2 −
ni
ni
τik,j 2δik,j
j=2
−
ni
τik,ni 2ηik,1
1
−
˜ ki,l )2 − (m
2 − Pik (θ˜ik,j )T θ˜ik,j −
2
τik,ni ηik,2
˜ ki,r )2 − (m
τik,ni
τik,j 2δik,j
(d˜ ki,l,m )2 −
ηik,3
1
− 1 − Hi2,j
ςi,j k 2 − Pi (θ˜ik,1 )T θ˜ik,1
τik,ni
k
(d˜ ki,r ,m )2 + Dki
ηik,4
τik,n
τi,j j=1 2δ k i,j
ni
χi2,j
j =2
j =2
i =1
−
βik,j Si2,j − βik,1 Si2,1 ρi +
τik,n
(66) τik,n
τik,n
(θik,j )∗T + 2ηk i (mki,l )2 + ηk i (mki,r )2 + ηk i (dki,l,m )2 + ηk i (dki,r ,m )2 + 12 . Let Ci k = i,1 i,2 i,3 i,4 2 2 min λki,ni /λi,max (Pik ), 2βi,j , τik,j − 1 − Pik δik,j , τik,1 − 2 Pik , τik,ni ηik,l (i = 1, . . . , N ; j = 1, . . . , ni ; l = 1, 2, 3, 4).
where Dki
= Dki,ni −1 +
Then (66) can be rewritten as V˙ k ≤ −CV k + D
(67)
where C = mink∈M {C1 , C2 , . . . , CN } and D = { , , . . . , }. The aforementioned control design and analysis are summarized in the following theorem. k
k
k
maxk∈M Dk1
Dk2
DkN
Theorem 1. For switched uncertain nonlinear system (1) with unmeasured states, for every switching signal σ (t ), if the average ln µ dwell time satisfies τa > C , then the controller (61) and fuzzy state observer (25), together with the intermediate control functions (44) and (55), parameter adaptation functions (45), (56) and (62), can guarantee that all the signals in the closed-loop system are bounded. Moreover, each output is confined within the prescribed performance bounds for all t ≥ 0. Remark 1. Notice that the PPC scheme can get better control performance than the ones without PPC. However, failures can cause performance deterioration of control systems. In the past decades, a great deal of results of fault-tolerant control (see [30–32] and the reference therein) has been received. Therefore, the fault-tolerant control with prescribed performance will be the future study. It is easy to see that the function W (t ) = eCt V σ (t ) (x(t )) is piecewise differentiable along solutions of the system (1). In view of (67), on each interval [tj , tj+1 ), one has
˙ (t ) = CeCt Vσ (t ) (x(t )) + eCt V˙ σ (t ) (x(t )) ≤ DeCt , W
t ∈ [tj , tj+1 ).
(68)
With the results in [10,33], we have V (x(t )) ≤ µV (x(t ))(µ > 1, k, l ∈ M ), which implies that k
l
W (tj+1 ) = µeCtj+1 V˙ σ (tj+1 ) (x(tj+1 )) ≤ µeCtj+1 V˙ σ (t ) (x(tj+1 ))
=
µW (tj−+1 )
≤ µ W (tj ) +
tj+1
De dt . Ct
(69)
tj
For an arbitrary T > t0 = 0, after iterating the inequality (69) from j = 0 to j = Nσ (T , 0) − 1, we obtain that W (T − ) ≤ W (tNσ (T , 0)) +
T
DeCt dt
tNσ (T ,0)
≤ µ W (tNσ (T ,0)−1 ) +
tNσ (T ,0) tNσ (T ,0)−1
De dt + µ Ct
−1
T
Ct
De dt tNσ (T ,0)
≤ ··· ≤µ
−Nσ (T ,0)
W (0) +
Nσ (T ,0)
j =0
µ
−j
tj+1
De dt + µ
tj
Ct
−Nσ (T ,0)
T
eD dt . Ct
tNσ (T ,0)
(70)
Since τa > ln µ/C , for any δ ∈ (0, C − ln µ/τa ), one has τa > ln µ/(C − δ). By (2), it holds that Nσ (T , t ) ≤ N0 +
(C − δ)(T − t ) , ln µ
∀ T ≥ t ≥ 0.
(71)
In addition, it is clear that Nσ (T , 0) − j ≤ 1 + Nσ (T , tj+1 ), j = 0, 1, . . . , Nσ (T , 0), one has
µNσ (T ,0)−j ≤ µ1+N0 e(C −δ)(T −tj+1 ) .
(72)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
71
In addition, since δ < C and tj+1
DeCt dt ≤ e(C −δ)tj+1
tj+1
e(C −δ)t dt .
(73)
tj
tj
From (72) and (73), it then follows that W (T − ) ≤ µNσ (T ,0) W (0) + µ1+N0 e(C −δ)T
T
Deδ t dt .
(74)
0
According to [33], there exist two class κ functions α (|x|) and α¯ (|x|), which satisfy α (|x|) ≤ V k (x) ≤ α¯ (|x|). It indicates that
α (∥x(T )∥) ≤ V σ (T ) (x(T − )) −
≤ eN0 ln µ e
ln µ τa −C T
≤ eN0 ln µ e
ln µ τa −C T
D
α¯ (∥x(0)∥) + µ1+N0 (1 − e−δT ) δ D
α¯ (∥x(0)∥) + µ1+N0 , δ
∀ T > 0.
(75)
We conclude that, by (75) and δ > 0, if τa > (ln µ/C ), then for bounded initial conditions, ei,j , Si,j and θ˜ik,j i = 1, 2, . . . , N , j = 1, 2, . . . , ni are bounded. Since (θik,j )∗ are constants, θik,j are bounded. Further, it is easy to obtain that uki , xˆ i,j , xi,j , zi,1 are bounded. Hence, for bounded initial conditions, all the signals in the closed-loop system (1) are bounded for switching signal with average dwell time. 5. Simulation studies Within this section, we will present illustrative examples. While the approach could be applied to inter-vehicle spacing regulation in a platoon of an automated highway system, since that system fits the assumptions of our framework, instead we study the control of two inverted pendulums connected by a spring. Each pendulum may be positioned by a torque input σ (t ) σ (t ) Di (ui ) applied by a servomotor at its base. It is assumed that both θ1 = x1,1 and θ2 = x2,1 (angular position and rate) are available to the ith controller for i = 1, 2; σ (t ) = k ∈ {1, 2}. The equations which describe the motion of the pendulums are defined by
x˙ i,1 = xi,2 σ (t ) σ (t ) Di (ui ) mi g Kr 2 kr (l − b) x˙ i,2 = + − sin(xi,1 ) + + 1fi,σ2(t ) (xi,2 ) + ∆σi,2(t ) (¯y) J J 4J 2J i i i i yi = xi,1 x1,1 sin(x1,1 x1,2 ) , m1 x2,1 sin(x2,1 x2,2 ) 1 , ∆2,2 y m2
where 1f11,2 (x1,1 , x1,2 ) =
x1,2 cos(x1,1 ) , m1 Kr 2 2 ∆2,2 y 4J2
1f12,2 (x1,1 , x1,2 ) = Kr 2 4J2
∆11,2 (¯y) =
Kr 2 4J1
sin(x2,1 ), ∆11,2 (¯y) =
(76)
Kr 2 4J1
sin(x1,1 x2,1 ),
(¯ ) = sin(x1,1 ), (¯ ) = sin(x1,1 x2,1 ). (x2,1 , x2,2 ) = The parameters m1 = 0.2 kg and m2 = 0.2 kg are the pendulum end masses, J1 = 0.05 kg and J2 = 0.0625 kg are the moments of inertia, K = 10 N/m is the spring constant of the connecting spring, r = 0.5 m is the pendulum height, l = 0.5 m is the natural length of the spring, and g = 9.81 m/s2 is gravitational acceleration. The distance between the pendulum hinges is defined as b = 0.4 m, where, in this example b < l so that the pendulums repel one another when both are in the upright position. In the simulation study, eleven fuzzy sets are defined over interval [−2k, 2k], and by choosing partitioning points as −2k, −k, 0, k, 2k (k ∈ {1, 2}) their fuzzy membership functions are given as follows: 1f21,2
µF 1 (ˆxi,j ) = exp(−(ˆxi,j − 2k)2 ), µF 2 (ˆxi,j ) = exp(−(ˆxi,j − k)2 ), µF 3 (ˆxi,j ) = exp(−(ˆxi,j )2 ), i,j,k
i,j,k
i,j,k
µF 4 (ˆxi,j ) = exp(−(ˆxi,j + k)2 ), µF 5 (ˆxi,j ) = exp(−(ˆxi,j + 2k)2 ), i,j,k
i,j,k
i = 1, 2; j = 1, 2.
The FLSs can be expressed in the form: fˆi,kj xˆ i,j θik,j = (θik,j )T ϕik,j (ˆxi,j ), i = 1, 2; j = 1, 2; k ∈ {1, 2}. The performance functions are chosen as: µ1 (t ) = 0.137e−0.5t + 0.013 and µ2 (t ) = 0.136e−0.4t + 0.014. For the switched system 1: l11,1 = l11,2 = l12,1 = l12,2 = 5.
For the switched system 2: l21,1 = l21,2 = l22,1 = l22,2 = 30.
1
In addition, in (23), by selecting Q1k = Q2k = 10I, we can obtain positive definite symmetric matrices P1k = −0.1
0.1667
P2k = −0.0028
−0.0028 . 0.1668
−0.1 , 1.02
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
Fig. 1. The trajectories of y1 .
Fig. 2. The trajectories of y2 .
ˆ ki,r , m ˆ ki,l , dˆ ki,r ,m , The design parameters for the intermediate control function αik,1 , the parameter adaptive laws θik,1 , θik,2 , m
dˆ ki,l,m and the actual control Dki (uki ) are given as βik,j = 5, γik,j = 0.08, τik,j = 0.01 and ηik,l = 0.01 (i = 1, 2; j = 1, 2; l = 1, 2, 3, 4).
The initial conditions are chosen as x1,1 (0) = x2,1 (0) = xˆ 2,2 (0) = 0.1, xˆ 1,1 (0) = −0.1, xˆ 1,2 (0) = 0.2, θ11,1 (0) = θ21,1 =
ˆ ki,r (0) = m ˆ ki,l (0) = dˆ ki,l,m (0) = dˆ ki,r ,m (0) = 1 the [0.1, 0.3, 0.5, 0.7, 0.9], θ12,1 (0) = θ22,1 = [−0.1, −0.3, −0.5, −0.7, −0.9], m
other initial values are chosen as zeros.
The average dwell time is chosen as τa = 11.1, and let µ = 0.18, then, we can obtain τa = 11.1 > ln 6.7755/0.18. Thus, the adaptive fuzzy output-feedback control problem of the resulting closed-loop system (76) is solvable under every switching signal k ∈ {1, 2}. The simulation results are shown by Figs. 1–6, where Figs. 1 and 2 show the trajectories of the outputs; Figs. 3 and 4 show the trajectories of the states and their estimates; Fig. 5 shows the trajectories of Dki (uki ) and uki,d (i = 1, 2); Fig. 6 shows switching signal.
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
73
Fig. 3. The trajectories of x1,2 and xˆ 1,2 .
Fig. 4. The trajectories of x2,2 and xˆ 2,2 .
6. Conclusions For a class of uncertain nonlinear switched uncertain nonlinear large-scale systems with unknown dead-zones and unmeasured states, the adaptive fuzzy output-feedback decentralized control problem has been considered. The fuzzy switched decentralized state observer has been designed to estimate the unmeasured states and the controller has been obtained based on dynamic surface backstepping technique, the average dwell time method and the prescribed performance theory. It is proved that the proposed control approach can guarantee that all the signals of the closed-loop system are bounded. Comparison simulation studies illustrate that both transient and steady-state performances of the outputs are better than without prescribed performance. In the future study, we will concentrate on the adaptive fault tolerant control design for the uncertain nonlinear systems with prescribed performances based on the results of this paper. Acknowledgments This work was supported in part by the Funds of National Science of China (Grant Nos. 61273148, 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2013ZCX01).
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
Fig. 5. The trajectories of Dki (uki ) and uki,d , i = 1, 2; k = 1, 2.
Fig. 6. Switching signal σ (t ).
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Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Adaptive fuzzy output constrained decentralized control for switched nonlinear large-scale systems with unknown dead zones Lili Zhang a,b,∗ , Guang-Hong Yang a,b a
College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China
b
State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang, 110819, PR China
article
info
Article history: Received 13 December 2015 Accepted 21 July 2016
Keywords: Switched nonlinear large-scale systems Decentralized state observer Fuzzy logic systems Prescribed performance theory Unknown dead zones
abstract In this paper, an adaptive fuzzy output-feedback control design with output constrained is investigated for a class of switched uncertain nonlinear large-scale systems with unknown dead zones and immeasurable states. Based on dynamic surface backstepping control design technique and incorporated by the average dwell time method and the prescribed performance theory, a new adaptive fuzzy output-feedback control method is developed. It is strictly proved that the resulting closed-loop system is stable in the sense of uniformly ultimately boundedness and both transient and steady-state performances of the outputs are preserved. Comparison simulation studies illustrate the effectiveness of the proposed approach. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction A large-scale system is often considered as a set of interconnected subsystems, such as power systems, computer and telecommunications networks, economic systems and multiagent systems. Owing to the complexity of control synthesis and the physical restrictions on information exchange among subsystems, it is often required to design for each subsystem a decentralized controller depending only on local measurements, even if to achieve an objective for the whole large-scale system [1]. Since fuzzy logic system and neural networks are the universal approximator, and can approximate any nonlinear function with arbitrary precision [2]. During the past two decades, various adaptive decentralized control schemes have been developed for large-scale nonlinear systems [2]. In [3–7], adaptive neural network and fuzzy decentralized state feedback control designs were proposed for large-scale nonlinear systems in strict feedback form. In [8,9], adaptive neural network and fuzzy decentralized output feedback control methods were studied for large-scale nonlinear systems by designing a state observer for estimating the unmeasured states. The above mentioned adaptive fuzzy or neural decentralized control approaches can provide an effective methodology to control those uncertain nonlinear large-scale systems. Nevertheless, they need not require that the nonlinear functions included in the controlled systems be known or be linearly parameterized. However, the above mentioned results are only suitable for the non-switched nonlinear large-scale systems, instead of the switched nonlinear large-scale systems. Recently, some control design methods have been proposed for several classes of switched nonlinear systems in [10–16]. Two state feedback control approaches in [10,11] have proposed for a class of switched single-input and singleoutput (SISO) nonlinear systems based on the common Lyapunov function method. By applying average dwell-time
∗
Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang, 110819, PR China. E-mail addresses: [email protected] (L. Zhang), [email protected] (G.-H. Yang).
http://dx.doi.org/10.1016/j.nahs.2016.07.002 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
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technique, works in [12,13] have been investigated switched adaptive control schemes for a class of switched nonlinear systems with time-varying delay, and [14] firstly solved the problem of output feedback control design for stochastic nonlinear switched systems with unmodeled dynamics. In [15], a novel adaptive neural network control method has been presented for a class of SISO switched nonlinear systems with switching jumps. An adaptive fuzzy output tracking control problem has been investigated in [16] for a class of switched uncertain nonlinear large-scale systems under arbitrary switchings. Obviously, all the above mentioned adaptive control approaches can ensure that all the signals of the resulting closed-loop system are bounded. But all the existing switched control methods are not investigated in the problem of prescribed performance control (PPC). It is well known that the PPC demands the convergence rate no less than a prescribed value, exhibiting a maximum overshoot less than a sufficiently small constant and the output or tracking error is confined within the prescribed performance bounds for all times. The robust adaptive control for SISO strict feedback nonlinear system and feedback linearizable multiple-input and multiple-output (MIMO) nonlinear systems with PPC were investigated in [17,18]. A multiple constraints control was considered in [19,20] with PPC approach. The dead zone issues have been skillfully addressed with the prescribed performance theory for a class of non-switched systems in [21,22]. However, the prescribed performance design methodology still not be applied to switched nonlinear systems and the output feedback design for switched nonlinear large-scale systems with unknown dead zone is still a challenge. In this paper, an adaptive fuzzy output feedback control design with prescribed performance is developed for a class of Switched nonlinear large-scale systems with unknown dead zone. The main contributions of the proposed control scheme are as follows: (i) by introducing the prescribed performance control technique, both transient and steady-state performances of the hybrid switched large-scale systems with unknown dead zone and immeasurable states are ensured; (ii)parameter adaptive method is adopted to deal with unknown dead-zone issues, which makes the controller has better robustness. The remainder of this paper is organized as follows. The problem statement and preliminaries are given in Section 2. The decentralized switched fuzzy state observer design is given in Section 3. Adaptive controller design and stability analysis are in Section 4. The simulation studies are given in Section 5, and followed by Section 6 which concludes the work. 2. Problem statement and preliminaries 2.1. Systems descriptions and assumptions Consider a class of uncertain switched nonlinear large-scale systems that is composed of N subsystems interconnected by their outputs. The ith switched subsystem can be described by switched nonlinear systems:
σ (t ) σ (t ) σ (t ) x˙ i,1 = xi,2 + fi,1 (xi,1 ) + ∆i,1 (¯y) + di,1 σ (t ) σ (t ) σ (t ) x˙ i,2 = xi,3 + fi,2 (xi,2 ) + ∆i,2 (¯y) + di,2 . .. σ (t ) σ (t ) σ (t ) x˙ i,ni −1 = xi,ni + fi,ni −1 (xi,ni −1 ) + ∆i,ni −1 (¯y) + di,n−1 σ (t ) σ (t ) σ (t ) σ (t ) σ (t ) x˙ i,ni = Di (ui ) + fi,ni (xi,ni ) + ∆i,ni (¯y) + di,n yi = xi,1
(1)
where xi,j = [xi,1 , xi,2 , . . . , xi,j ]T ∈ ℜi , i = 1, 2, . . . , N ; j = 1, 2, . . . ni are the states, yi ∈ ℜ is the output. The function σ (t ) : [0, ∞) → M = {1, 2, . . . , m} is switching signal which is assumed to be a piecewise continuous (from σ (t ) the right) function of time. Moreover σ (t ) = k implies that the kth switched large-scale system is active. fi,j (xi,j ), are σ (t )
unknown smooth nonlinear functions. ∆i,j (¯y), (¯y = [y1 , y2 , . . . , yN ]) are unknown smooth functions representing the σ (t ) σ (t ) interconnection effects between the ith subsystem and the other subsystems. Di (ui ) ∈ ℜ is the output of the deadzones. In addition, we assume that the state of the system (1) does not jump at the switching instants, i.e., the solution is everywhere continuous, which is a standard assumption in the switched system literature [16,23]. Now we recall the definition of average dwell time, which plays a key role in research of switched nonlinear control. A switched nonlinear large-scale system (1) is called to have a switching signal σ (t ) with average dwell time τa , if there exist two positive numbers N0 and τa such that Nσ (T , t ) ≤ N0 +
T −t
τa
∀T ≥t ≥0
(2)
where Nσ (T , t ) is the number of switches occurring in the interval [t , T ). Let T > 0 be an arbitrary time. Denote by t1 , . . . , tNσ (T ,0) the switching times on the interval (0, T ) (t0 = 0). When t ∈ [tj , tj+1 ), σ (t ) = kj , that is, the kj th large-scale system is active. In this study, we assume kj ̸= kj+1 for all j.
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63
Remark 1. It should be mentioned that if for t ∈ [0, ∞), the switching signal σ (t ) = k, then the system (1) represents a class of nonswitched nonlinear large-scale systems in strict feedback form, which is investigated commonly in the literatures, for example [3,4]. σ (t )
Similar to [24], the output of the dead zone Di σ (t )
Di
σ (t )
(ui
),
σ (t ) σ (t ) σ (t ) mi,r (ui − di,r ) 0
if ui
i ,l
i
≥ dσi,r(t ) < uσi (t ) < dσi,r(t ) ≤ −dσi,l(t ) .
− dσi,l(t ) σ (t )
if
mσ (t ) (uσ (t ) + dσ (t ) ) i,l
σ (t )
σ (t )
if ui
(uσi (t ) ) is described by: (3)
∈ ℜ is the input to the dead zone; mσi,r(t ) and mσi,l(t ) denote the slope of the dead zone and dσi,r(t ) and dσi,l(t ) represent
In (2), ui
σ (t )
the dead zone width parameters. The dead zone outputs are not available for measurement. The dead zone parameters mi,r , σ (t )
σ (t )
σ (t )
mi,l , di,r
and di,l
σ (t )
are unknown, but their signs are known (mi,r
> 0, mσi,l(t ) > 0, dσi,r(t ) ≥ 0 and dσi,l(t ) ≥ 0, respectively). σ (t )
σ (t )
σ (t )
σ (t )
Assumption 1 ([24]). Dead zone slopes are bounded by known constants mi,rmin , mi,rmax , mi,lmin and mi,lmax such that 0<
σ (t ) mi,rmin
≤
σ (t ) mi,r
≤
σ (t ) mi,rmax
and 0 <
σ (t ) mi,lmin
≤
σ (t ) mi,l
≤
σ (t ) mi,lmax .
σ (t )
The dead zone inverse technique is useful for compensating the dead zone effect. Setting ui,d as the control input from σ (t )
the controller to achieve the control objective for the plant without a dead zone, the following control signal ui according to the certainty equivalence dead zone inverse: σ (t )
= (Diσ (t ) (uσi,d(t ) ))−1 =
ui
σ (t )
σ (t )
σ (t )
σ (t )
σ (t )
σ (t )
ui,d + dˆ i,r ,m σ (t )
δi
σ (t )
ˆ i,r m
σ (t )
ui,d + dˆ i,l,m
+
σ (t )
ˆ i ,l m
σ (t )
σ (t )
σ (t )
(1 − δiσ (t ) )
σ (t ) σ (t )
ˆ i ,r , m ˆ i,l , dˆ i,r ,m and dˆ i,l,m are estimates of mi,r , mi,l , mi,r di,r where m
σ (t )
δi
=
1 0
Di
(ui
σ (t )
˜ i,l where m
σ (t ) σ (t )
and mi,l di,l , respectively, and
σ (t )
(5)
σ (t )
if ui,d < 0. σ (t )
σ (t )
(4)
if ui,d ≥ 0
The resulting errors between ui σ (t )
is generated
)−
(uσi,d(t ) )
=
σ (t ) d˜ i,r ,m
σ (t )
and ui,d are given by σ (t )
−
σ (t )
ui,d + dˆ i,r ,m
ˆ σi,r(t ) m
˜ σi,r(t ) m
σ (t )
δi
+
σ (t ) d˜ i,l,m
σ (t )
−
σ (t )
ui,d + dˆ i,l,m
ˆ σi,l(t ) m
˜ σi,l(t ) m
(1 − δiσ (t ) ) + εiσ,d(t )
(6)
ˆ σi,l(t ) − mσi,l(t ) , m ˜ σi,r(t ) = m ˆ σi,r(t ) − mσi,r(t ) , d˜ σi,l(,tm) = dˆ σi,l(,tm) − dσi,l(,tm) and d˜ σi,r(,tm) = dˆ σi,r(,tm) − dσi,r(,tm) are parameter errors. =m
εiσ,d(t ) can be expressed as
εiσ,d(t ) = −mσi,r(t ) χiσ,r(t ) (uσi (t ) − dσi,r(t ) ) − mσi,m(t ) χiσ,l(t ) (uσi (t ) − dσi,r(t ) ) where
εiσ,d(t )
(7)
is bounded with
χiσ,r(t ) =
1, 0,
σ (t )
σ (t )
if 0 ≤ ui < di,r otherwise ;
χiσ,l(t ) =
1, 0,
σ (t )
σ (t )
if di,l < ui otherwise.
<0
σ (t )
Assumption 2 ([25]). For the nonlinear functions ∆i,j (¯y) satisfy N σ (t ) σ (t ) σ (t ) qi,l βl (|yl |) ∆i,j (¯y) ≤
(8)
l=1
σ (t )
σ (t )
where qi,l (i = 1, 2, . . . , N , l = 1, 2, · · · N ) is unknown constant denoting the strength of the interaction; βl is a known nonlinear smooth function.
(|yl |) ≥ 0
σ (t )
Thus, according to [25], there exists smooth non-negative function h¯ i,l (yl ) such that
N l=1
σ (t ) σ (t ) qi,l βl
2 (|yl |)
≤
N l =1
σ (t ) h¯ i,l y2l + 2
N
σ (t )
βl
2 (0)
.
(9)
l =1
Lemma 1 ([26]). For any constant η > 0 and any a variable ϖ ∈ R, limϖ →0 ϖ1 tanh2 ( ϖ ) = 0. η Lemma 2 ([26]). Consider the bounded set given by Ωη := {η ||η| < 0.8814τ }, τ is the positive constant. Then, ϖ ̸∈ Ωϖ , the inequality 1 − 16 tanh2 (τ /η) ≤ 0 holds.
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2.2. Performance function and error transformation According to [18], the prescribed performance is achieved by ensuring that each output yi (t ) (i = 1, 2, . . . , N ) evolves strictly within prescribed decaying bounds as follows:
− δi,min µi (t ) < yi (t ) < δi,max µi (t ) ,
∀t ≥0
(10)
where δi,min and δi,max are design constants, and the performance functions µi (t ) are bounded and strictly positive decreasing smooth functions with the property limt →∞ µi (t ) = µi,∞ ; µi,∞ > 0 is a constant. In this paper, the performance functions are chosen as the exponential form µi (t ) = (µi,0 − µi,∞ )e−ai t + µi,∞ , where ai , µi,0 and µi,∞ are strictly positive constants, µi,0 > µi,∞ , and µi,0 = µi (0) is selected such that −δi,min µi (0) < yi (0) < δi,max µi (0) is satisfied. Furthermore, the maximum overshoot of yi (t ) is prescribed less than max{δi,min µi (0), δi,max µi (0)}. Therefore, choosing the performance function µi (t ) and the constants δi,min , δi,max appropriately determines the performance bounds of the output yi (t ). To represent (10) by an equality form, we set yi (t ) = µi (t ) Φi (ζi (t )), where Φi (ζi ) =
∀t ≥0
δi,max eζi −δi,min e−ζi eζi +e−ζi
(11)
.
Since the function Φi (ζi ) is strictly monotonic increasing, its inverse function can be expressed as
ζi (t ) = Φ −1
yi (t )
=
µi (t )
1 2
ln
Φi − δimin δimax − Φi
(12)
and
µ ˙ y ˙ζi (t ) = ρi y˙ i − i i µi 1 with ρi = 21µ Φ +δ1 − . Φi −δi,max i i i,min
(13)
For control design of the nonlinear system, we design the following errors transformation zi,1 (t ) = ζi (t ) −
1 2
ln
δi,min . δi,max
(14)
And the transformation state dynamics is
z˙i,1 (t ) = ρi
µ ˙ i yi y˙ i − µi
.
(15)
Lemma 3 ([18]). Consider output yi (t ) and transformed errors zi,1 (t ) defined in (14). If zi,1 (t ) is bounded, prescribed performance of yi (t ) is satisfied for all t ≥ 0, i.e., (10) is satisfied. 2.3. Fuzzy logic systems A fuzzy logic system (FLS) consists of four parts: a knowledge base, a fuzzifier, a fuzzy inference engine with fuzzy rules, and a defuzzifier. The knowledge base for FLS is comprised of a collection of fuzzy If-then rules as following: Rl : If x1 is F1l and x2 is F2l and . . . and xn is Fnl , then y is Gl , l = 1, 2, . . . , N where x = [x1 , . . . , xn ]T and y are the FLS input and output, respectively. Fil and Gl are fuzzy sets with membership functions µF l (xi ) and µGl (y), respectively. N is the number of rules. i
Through singleton fuzzifier, center average defuzzification and product inference Wang2016, the fuzzy logic system can be expressed as N
n
µF l ( x i ) i y(x) = n N µF l (xi ) y¯ l
l=1
i=1
l=1
(16)
i
i =1
where y¯ l = maxy∈R µGl (y). Define the fuzzy basis functions as n
ϕl =
µF l ( x i )
i=1 N
n
l=1
i=1
i
. µF l (xi ) i
(17)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
65
Denoting θ T = [¯y1 , y¯ 2 , . . . , y¯ N ] = [θ1 , θ2 , . . . , θN ] and ϕ(x) = [ϕ1 (x), . . . , ϕN (x)]T , then fuzzy logic system (16) can be rewritten as y(x) = θ T ϕ(x).
(18)
Lemma 4 ([16,27]). Let f (x) be a continuous function defined on a compact set Ω . Then for any constant ε > 0, there exists a fuzzy logic system fˆ (x |θ ) = θ T ϕ(x) such as sup f (x) − (θ ∗ )T ϕ(x) ≤ ε.
(19)
x∈Ω
3. Decentralized switched fuzzy state observer design Since the states xi,2 , xi,3 , . . . , xi,ni −1 and xni in the switched nonlinear large-scale system (1) are not available for feedback, this section aims to design a switched decentralized state observer to estimate the unmeasured states in (1). σ (t ) Note that the functions fi,j (xi,j ) are unknown. Thus the fuzzy logic systems are utilized to approximate them. By Lemma 4, we assume that σ (t )
fi,j
(xi,j |θiσ,j(t ) ) = θiσ,j(t )T ϕiσ,j(t ) (xi,j ); fˆi,σj (t ) (ˆxi,j |θiσ,j(t ) ) = θiσ,j(t )T ϕiσ,j(t ) (ˆxi,j ),
(20)
where xˆ i,j = [ˆxi,1 , xˆ i,2 , . . . , xˆ i,ni ] are the estimations of xi,j = [xi,1 , xi,2 , . . . , xi,ni ] . T
T
∗σ (t )
According to definition of the optimal parameter vectors θi,j (t ) θi∗σ ,j
= arg min σ (t )
sup
σ (t )
(xi,j ׈xi,j )∈Ui,j,1 ×Ui,j,2
θi,j ∈Ωi,j
in [16], let
ˆ σ (t ) σ (t ) σ (t ) fi,j (ˆxi,j |θi,j ) − fi,j (xi,j ) ,
σ (t )
(21)
σ (t )
where Ωi,j , Ui,j,1 and Ui,j,2 are bounded compact regions for θi,j , xi,j and xˆ i,j , respectively. In addition, the approximation σ (t )
error εi,j
is defined as
σ (t )
(t ) (xi,j ) = fˆi,σj (t ) (ˆxi,j |θi∗σ ) + εiσ,j(t ) , i = 1, 2, . . . , N , j = 1, 2, . . . , ni ,j σ (t ) ∗σ (t ) ∗σ (t ) σ (t ) being an unknown constant. where εi,j satisfies εi,j ≤ εi,j , with εi,j
fi,j
(22)
By substituting (21) into (1), the switched system (1) can be expressed as σ (t )
x˙ i,j = xi,j+1 + fˆi,j σ (t )
x˙ i,ni = Di
(t ) (ˆxi,j |θi∗σ ) + εiσ,j(t ) + ∆σi,j(t ) (t ) ,j
(t ) σ (t ) (uiσ (t ) ) + fˆi,σn(it ) (ˆxi,ni |θi∗σ )εi,ni + ∆σi,n(ti ) (t ) ,j
(23)
yi = xi,1 i = 1, 2, . . . , N , j = 1, 2, . . . , ni − 1. Rewrite (23) as σ (t )
x˙ i,ni = Ai
σ (t )
xi,ni + Li
yi +
ni
∗σ (t )T
Bi,j θi,j
ϕiσ,j(t ) (ˆxi,j ) + εiσ (t ) + ∆σi (t ) + Bi,ni Dσi (t ) (uσi (t ) )
j =1
(24)
yi = Ci xi,ni where σ (t )
Ai
σ (t ) −li,1 = ... −lσi,n(ti )
Ini −1 0
···
σ (t )
,
∆i
σ (t ) ∆i,1 (¯y) = ... ,
σ (t ) σ (t )
Li
σ (t )
σ (t )
∆i,ni (¯y)
0
εiσ (t ) = [εiσ,1(t ) , εiσ,2(t ) , . . . , εiσ,n(it ) ]T ,
Ci = [1, . . . , 0],
li,1
= ... , li,ni
Bi,j = [0, . . . , 0, 1, 0, . . . , 0]T .
j
To estimate the states of the system, the following switched fuzzy decentralized state observer is designed σ (t )
x˙ˆ i,ni = Ai
σ (t )
xˆ i,ni + Li
yi +
ni
σ (t )
Bi,j fˆi,j
(ˆxi,j |θiσ,j(t ) ) + Bi,ni Dσi (t ) (uσi (t ) )
j =1
(25)
yˆ i = Ci xˆ i,ni . Define observation error vector ei as ei = [ei,1 , ei,2 , . . . , ei,ni ]T = xi,ni − xˆ i,ni .
(26)
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
From (24)–(26), the observation error can be expressed as σ (t )
e˙ i = Ai
ni
ei +
σ (t )
σ (t )
σ (t )
Bi,j (θ˜i,j )T ϕi,j (ˆxi,j ) + εi
+ ∆σi (t )
(27)
j =1
σ (t )
∗σ (t )
σ (t )
where θ˜i,j = θi,j − θi,j , i = 1, 2, . . . , N , j = 1, 2, . . . ni are the adaptive parameter vector errors. For the switching signal σ (t ) = k, k = 1, 2, . . . , m; the kj th switched large-scale system is active and the remaining large-scale systems are inactive. Choose vector Lki such that matrix Aki is a strict Hurwitz matrix, therefore, for any given matrix Qik = (Qik )T > 0, there exists a positive definite matrix Pik = (Pik )T > 0 such that
(Aki )T Pik + Pik Aki = −Qik .
(28)
To evaluate the property of the state observer (25), we consider the following Lyapunov candidate V0k =
N
Vik,0 =
N
i =1
ei T Pik ei .
V0k
for (28) as (29)
i=1
Then the time derivative of V0k along with the solutions of (26) is V0k
˙ ≤
N
(
k min Qi
−λ
) ∥ei ∥ + 2
ε +
2eTi Pik
k i
∆ki
i=1
+
ni
Bi,j (θ˜ ) ϕ (ˆxi,j ) k T i,j
k i,j
(30)
j =1
where λmin (Qi ) is the smallest eigenvalue of matrix Qi . Note that βm (|ym |) , m = 1, 2, . . . , N is a known non-linear smooth function, thus, according to [28], there exists smooth non-negative function lki,m (ym ) such that
N
2 qki,m βm (|ym |)
≤
m=1
N
lki,m y2m + 2
m=1
N
2 βm (0)
.
(31)
m=1
By completing the squares and Assumption 2, the fact that ϕik,j (ˆxi,j )T ϕik,j (ˆxi,j ) ≤ 1, we can obtain N
2eTi Pik ∆ki ≤
N
2 N N 2 ∥ei ∥2 + ni Pik 2 βm (0) lki,m y2m + 2ni Pik
i =1
i=1
≤
N
m=1
i =1 ni
m=1
2 N N N N k 2 k 2 2 k 2 ∥ei ∥ + ni Pi lm,i yi + 2ni Pi βm (0) m=1 i=1
(32)
i=1 m=1
ni 2 (θ˜ ki,j )T θ˜ik,j
Bi,j (θ˜ik,j )T ϕik,j (ˆxi,j ) ≤ ni ∥ei ∥2 + ni Pik
(33)
j =1
j =1
∗ 2
2eTi Pik εik ≤ ∥ei ∥2 + Pik ε ki .
(34)
Substituting (32)–(34) into (30) results in V0k
˙ ≤
N
ni N N k 2 2 k T −λ ∥ei ∥ + Pi (θ˜ i,j ) θ˜i,j + Di,0 + ni Pik lkm,i y2i k i,0
2
i=1
j =1
2 N N
where Dki,0 = 2ni Pik (
i =1
m=1
(35)
m=1 i=1
∗ 2 βm (0))2 + Pik ε ki and λki,0 = λmin (Qi ) − 2 − ni .
4. Adaptive controller design and stability analysis This section will give the adaptive fuzzy output-feedback control design procedure based on dynamic surface backstepping design technique in [29] and the fuzzy switched decentralized state observer designed in the above section. The stability of the closed-loop system will be proved by using Lyapunov function and average dwell time method. When t ∈ [tj , tj+1 ) , and σ (t ) = k, that is, the ni th switched large-scale system is active. For the kth switched largescale system, the ni step adaptive fuzzy backstepping output feedback control design is based on the following change of coordinates: Si,1 = zi,1 ,
Si,j = xˆ i,j − qi,j ;
χi,j = qi,j − αik,j−1 ,
i = 2, . . . , N ; j = 2, . . . , ni
(36)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
67
where Si,j is called error surface, qi,j is a state variable, which is obtained through a first-order filter on intermediate control function αi,j−1 and χi,j is called the output error of the first-order filter. Step i, 1: Expressing xi,2 in terms of its estimate as xi,2 = xˆ i,2 + ei,2 , we obtain S˙i,1 = ρi
y˙ i −
µ ˙ i yi µi
= ρi Si,2 + χi,2 + αik,1 + ei,2 + (θik,1 )T ϕik,1 (ˆxi,1 ) µ ˙ i yi + εik,1 . µi
+ (θ˜ik,1 )T ϕik,1 (ˆxi,1 ) + ∆ki,1 (¯y) −
(37)
Choose the Lyapunov function candidate as N
Vik,1
=
V0k
i =1
+
N 1 2
i =1
Si2,1
1
+
2δik,1
(θ˜ ) θ˜
k T k i ,1 i ,1
(38)
where δik,1 > 0 is design parameter. From (1), and (36)–(37), the time derivative of Vik,1 satisfies V˙ ik,1 ≤ V˙ ik,0 + Si,1 ρi
Si,2 + χi,2 + αik,1 + εik,1 + (θik,1 )T ϕik,1 (ˆxi,1 )
+ ei,2 + (θ˜ik,1 )T ϕik,1 (ˆxi,1 ) + ∆ki,1 (¯y) −
µ ˙ i yi µi
+
1
δik,1
k
(θ˜ik,1 )T θ˙˜ i,1 .
(39)
By completing the squares, by Assumption 2, we can obtain Si,1 ρi (Si,2 + χi,2 ) ≤ Si2,1 ρi2 + Si,1 ρi (εik,∗2 + ei,2 ) ≤ Si2,1 ρi2 + N
ρ
Si,1 i ∆ki,1
(¯y) ≤
i=1
N 1 i =1
2
1 2 1
Si2,2 +
1 2
χi2,2
(40)
1
∥ei ∥2 + εik∗2
2
2
ρ +
Si2,1 i2
N N i=1 m=1
lkm,i y2i
+
N N
2 βm (0)
(41)
i=1 m=1
where εik∗ = [εik,∗1 , εik,∗2 , . . . , εik,∗ni ]T . Substituting (40)–(41) into (39) results in
Γk ˙ i yi ˙Vik,1 ≤ −λki,1 ∥ei ∥2 + 1 Si2,2 + Si,1 ρi αik,1 + i,1 + 2Si,1 ρi + (θik,1 )T ϕik,1 (ˆxi,1 ) − µ 2 Si,1 ρi µi ni 2 1 1 (θ˜ik,1 )T θ˜ik,j + Dki,1 + χi2,2 + (θ˜ik,1 )T Si,1 ρi ϕik,1 (ˆxi,1 ) − k θ˙ik,1 + ni Pik 2 δi,1 j =1 where λki,1 = λki,0 + 12 , Dki,1 = Dki,0 + (
N
m=1
Note that when Si,1 ρi = 0, functions 2 Si,1 ρi
tanh ( τ ) is introduced to deal with i 1
k
2 βm (0))2 and Γi,k1 = (ni Pik + 1) Nm=1 lkm,i .
Γi,1 Si,1 ρi
are discontinuous function. In order to overcome this problem, the function
1 Γk . Si,1 ρi i,1
Then (42) becomes
1
χi2,2 + Si,1 ρi αik,1 + (θik,1 )T ϕik,1 (ˆxi,1 ) + 2Si,1 ρi 2 2 16 Si,1 ρi µ ˙ i yi Si,1 ρi 2 tanh Γi,k1 − + 1 − tanh2 Γi,k1 + Si,1 ρi τi µi τi ni 1 k k T k k k T k + (θ˜i,j ) θ˜i,j + Di,1 + (θ˜i,1 ) ϕi,1 (ˆxi,1 )Si,1 ρi − k θ˙i,1 . δi,1 j=1
V˙ ik,1 ≤ −λki,1 ∥ei ∥2 +
Si2,2 +
(42)
(43)
Design the intermediate control function αik,1 , the adaptation function θik,1 as
αik,1 = −βik,1 Si,1 −
16 Si,1 ρi
tanh2
θ˙ik,1 = δik,1 ϕik.1 (ˆxi,1 )Si,1 ρi − τik,1 θik,1
Si,1 ρi
τi
µ ˙ i yi Γi,k1 − (θik,1 )T ϕik.1 xˆ i,1 − Si,1 ρi + µi
(44) (45)
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
where βik,1 > 0, τik,1 > 0 are design parameters. From Lemma 1, one knows that limSi,1 ρi →0 According to Lemma 2 and (43)–(45), it follows that V˙ ik,1 ≤ −λi,1 ∥ei ∥2 +
1 2
Si2,2 − βik,1 Si2,1 ρi2 +
1 2
τik,1
χi2,2 +
δ
k i,1
(θ˜ik,1 )T θik,1 +
16 Si,1 ρi
S
ρ
tanh2 ( i,τ1 i )Γi,k1 exists. i
ni (θ˜ik,j )T θik,j + Dki,1 .
(46)
j =1
Step i, j(2 ≤ j ≤ ni ): Since Si,j = xˆ i,j − qki,j , similar to Step i, 1, the derivative time of Si,j along with (25) and (36) is S˙i,j = x˙ˆ i,j − q˙ i,j = Si+1 + χi,j+1 + αik,j + lki,j ei,1 + θiT,k ϕi,k (ˆxi,j ) − q˙ i,j .
(47)
To avoid repeatedly differentiating αik,j in the traditional backstepping design, which leads to the so called the problem
‘‘explosion of complexity’’ as mentioned in [29], we introduce a new state variable qi,j+1 and let αik,j pass through a first-order filter with a constant ςi,j+1 to obtain qi,j+1 , i.e.,
ςi,j+1 q˙ i,j+1 + qi,j+1 = αik,j ,
qi,j+1 (0) = αik,j (0).
(48)
By definition of χi,j+1 = qi,j+1 − αik,j , it yields q˙ i,j+1 =
χ − ςii,,jj++11
and
χi,j+1 ςi,j+1
χ˙ i,j+1 = q˙ i,j+1 − α˙ ik,j = Hi,j+1 (·) −
(49)
where Hi,j+1 (ˆxi,j , yi , S2 , . . . , Si,j+1 , χ2 , . . . , χi,j+1 , θi,1 , . . . , θi,j ) is a continuous function. Consider the following Lyapunov function candidate N
Vik,j =
i=1
N
Vik,j−1 +
i=1
1 2
zi2,j +
1
1
2δik,j
(θ˜ik,j )T θ˜ik,j + χi2,j
(50)
2
where δik,j > 0 is the design parameter. From (49) and (50), we can obtain V˙ ik,j ≤ V˙ ik,j−1 + Si,j (Si,j+1 + χi,j+1 + αik,j + lki,j ei,1 − q˙ i,j + (θik,j )T ϕik,j (ˆxi,j ) − (θ˜ik,j )T ϕik,j (ˆxi,j ))
− χi,j
χi,j Hi,j − ςi,j
+ (θ˜ )
Si,j ϕ (ˆxi,j ) −
k T i,j
k i,j
1
δik,j
θ˙
k i,j
.
(51)
By completing the squares, we have Si,j (Si,j+1 + χi,j+1 + lki,j ei,1 ) ≤
−Si,j (θ˜ik,j )T ϕik,j (ˆxi,j ) ≤
1
3 2
Si2,j +
1 2
Si2,j+1 +
1 2
1
χi2,j+1 + (lki,j )2 ∥ei ∥2
1
(θ˜ik,j )T θ˜ik,j + Si2,j .
2 Substituting (52)–(53) into (51) yields
(53)
2
V˙ ik,j ≤ V˙ ik,j−1 + Si,j (2Si,j + αik,j + (θik,j )T ϕik,j (ˆxi,j ) − q˙ i,j ) +
+ (θ˜ )
k T i,j
(52)
2
Si,j ϕ (ˆxi,j ) − k i,j
1
δik,j
θ˙
k i,j
1 2
Si2,j+1 +
1 2
χi2,j+1
χi,j 1 − χi,j − + Hi,j + (lki,j )2 ∥ei ∥2 . ςi,j 2
(54)
Design the intermediate control function αik,j , the adaptation function θik,j as
αik,j = −βik,j Si,j − 2Si,j − (θik,j )T ϕik,j (ˆxi,j ) + q˙ i,j
(55)
θ˙ = δ
(56)
k i,j
k i,j Si,j
ϕ (ˆxi,j ) − τ θ k i,j
k k i,j i,j
where βik,j > 0 and τik,j > 0 are design parameters. Substituting (55)–(56) into (54) yields V˙ ik,j ≤ −λki,j ∥ei ∥2 +
j 1
2 p=1
ni
+
χi2,p+1 −
j
βi,p Si2,p + Dki,1 − βi,1 Si2,1 ρi +
p=2 j
p=1 j
1 k T k χi,p (θ˜ik,j )T θ˜ik,j + (θ˜i,p ) θ˜i,p − χi,p Hi,p − 2 ςi,p j=1 p=2 p=2
where λki,j = λki,1 −
j −1 2
(lki,j )2 .
j τik,p
1
δik,p
+ Si2,j+1 2
(θ˜ik,p )T θik,p
(57)
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
69
Step i, ni : In the final design step, the dead-zone input uki,d will be obtained. From (3), (5) and (36), the time derivative of Si,ni is S˙i,ni = x˙ˆ i,ni − α˙ ik,ni −1 = Dki (uki ) + lki,ni ei,1 + (θik,ni )T ϕik,ni (ˆxi,ni ) − q˙ i,ni
=
uki,d
dki,m,r
+ ˜
d˜ ki,m,l −
+
−
uki,d + dˆ ki,m,r
ˆ ki,r m
uki,d + dˆ ki,m,l
ˆ ki,l m
δik + lki,ni ei,1 + (θik,ni )T ϕik,ni (ˆxi,ni )
mki,r
˜
(1 − δik ) + εik,d − q˙ i,ni .
mki,l
˜
(58)
Consider the following Lyapunov function candidate Vik,ni = Vik,ni +
+
1 2
1
Si2,ni +
1 2δ
1
˜ ki,r )2 + (m
2ηik,2
1
1
2
2ηik,1
(θ˜iT,ni )T θ˜i,ni + χi2,ni +
k i,ni
2ηik,3
(d˜ ki,m,l )2 +
1 2ηik,4
˜ ki,l )2 (m
(d˜ ki,m,r )2
(59)
k
where δi,ini > 0 and ηik,p (p = 1, 2, 3, 4) are design parameters. V˙ ik,ni = V˙ ik,ni −1 + Si,ni (uki,d + 2Si,ni + εik,d + (θik,ni )T ϕik,ni (ˆxi,ni ) − q˙ i,ni )
k 1 k ˙ k T k ˜ + (Si,ni η δ + dˆ i,m,r ) + (θ˜i,ni ) Si,ni φi,ni (ˆxi,ni ) − k θ˙i,ni η δi,ni k ˆk 1 k ˙ˆ k − ui,d + di,m,l Si,n ηk (1 − δ k ) ˜ i ,l m + k m i ,l i i i,1 k ˆ i,l m ηi,1 uki,d + dˆ ki,m,r k 1 k 1 ˙k k k ˙ ˜ i ,r m ˆ i,r − Si,ni ηi,2 δi + k d˜ ki,m,l (Si,ni ηik,3 (1 − δik ) + dˆ i,m,l ). + k m k ˆ i ,r m ηi,2 ηi,3 1
dki,m,r k i ,4
k k i,4 i
(60)
Similar to the procedures in step i, j, design the input of the dead zones uki,d and parameter adaptation functions as follows: uki,d = −βik,ni Si,ni − 2Si,ni − (θik,ni )T ϕik,ni (ˆxi,ni ) + q˙ i,ni
(61)
θ˙ik,ni = δik,ni Si,ni ϕik,ni (ˆxi,ni ) − τik,ni θik,ni
(62)
where β > 0 and τ > 0 are design parameters. Similar to the results in literature [21,22], the unknown deadzone parameter adaptation functions are omitted here for brevity. Substituting (61)–(62) and dead-zone parameter adaptation laws into (60) yields k i,ni
k i,ni
V˙ k ≤ −λki,ni ∥ei ∥2 −
ni
βik,j Si2,j − βik,j Si2,j ρi +
j =2
ni τik,j j =1
δik,j
(θ˜ik,j )T θik,j
ni ni ni k 2 χi,j k T ˜k ˜ + Pi (θi,j ) θi,j + χi,j − + Hi,j + χi2,j + Dki,ni ςi,j j=1 j =2 i=2 +
τik,ni η
k i,1
˜ ki,l m ˆ ki,l + m
τik,ni η
k i,2
˜ ki,r m ˆ ki,r + m
τik,ni η
k i,3
d˜ ki,l,m dˆ ki,l,m +
τik,ni ηik,4
d˜ ki,r ,m dˆ ki,r ,m
(63)
where λki,ni = λki,ni −1 − 21 , Dki,ni = Dki,1 + 21 (¯εik,d )2 , ε¯ ik,d ≤ ε¯ i,d and ε¯ i,d is a constant.
(
ni 1
1 T k i =1 2 e i P i e i
+
2
j =1
(Si2,j +
1
δik,j
(θ˜ik,j )k θ˜ik,j ) +
ni
χi2,j ) ≤ M}, where M > 0 is a known constant. Since Θ is a compact set and Hi,j is a continuous function, there exists a positive constant Mi,j such that Hi,j ≤ Mi,j on Θ , therefore, we Let Θ = {
N
1 2
j=2
have
χi,j Hi ≤ 1 χ 2 M 2 + 1 . i,j i,j
2 2 By completing the square for each parameter estimate: 1
1
2
2
(θ˜ik,j )T θik,j ≤ − (θ˜ik,j )T θ˜ik,j + (θik,j )∗T (θik,j )∗ .
(64)
(65)
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L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
Substituting the above inequalities into (63) yields k
V˙ ≤
N
−λki,ni ∥ei ∥2 −
ni
ni
τik,j 2δik,j
j=2
−
ni
τik,ni 2ηik,1
1
−
˜ ki,l )2 − (m
2 − Pik (θ˜ik,j )T θ˜ik,j −
2
τik,ni ηik,2
˜ ki,r )2 − (m
τik,ni
τik,j 2δik,j
(d˜ ki,l,m )2 −
ηik,3
1
− 1 − Hi2,j
ςi,j k 2 − Pi (θ˜ik,1 )T θ˜ik,1
τik,ni
k
(d˜ ki,r ,m )2 + Dki
ηik,4
τik,n
τi,j j=1 2δ k i,j
ni
χi2,j
j =2
j =2
i =1
−
βik,j Si2,j − βik,1 Si2,1 ρi +
τik,n
(66) τik,n
τik,n
(θik,j )∗T + 2ηk i (mki,l )2 + ηk i (mki,r )2 + ηk i (dki,l,m )2 + ηk i (dki,r ,m )2 + 12 . Let Ci k = i,1 i,2 i,3 i,4 2 2 min λki,ni /λi,max (Pik ), 2βi,j , τik,j − 1 − Pik δik,j , τik,1 − 2 Pik , τik,ni ηik,l (i = 1, . . . , N ; j = 1, . . . , ni ; l = 1, 2, 3, 4).
where Dki
= Dki,ni −1 +
Then (66) can be rewritten as V˙ k ≤ −CV k + D
(67)
where C = mink∈M {C1 , C2 , . . . , CN } and D = { , , . . . , }. The aforementioned control design and analysis are summarized in the following theorem. k
k
k
maxk∈M Dk1
Dk2
DkN
Theorem 1. For switched uncertain nonlinear system (1) with unmeasured states, for every switching signal σ (t ), if the average ln µ dwell time satisfies τa > C , then the controller (61) and fuzzy state observer (25), together with the intermediate control functions (44) and (55), parameter adaptation functions (45), (56) and (62), can guarantee that all the signals in the closed-loop system are bounded. Moreover, each output is confined within the prescribed performance bounds for all t ≥ 0. Remark 1. Notice that the PPC scheme can get better control performance than the ones without PPC. However, failures can cause performance deterioration of control systems. In the past decades, a great deal of results of fault-tolerant control (see [30–32] and the reference therein) has been received. Therefore, the fault-tolerant control with prescribed performance will be the future study. It is easy to see that the function W (t ) = eCt V σ (t ) (x(t )) is piecewise differentiable along solutions of the system (1). In view of (67), on each interval [tj , tj+1 ), one has
˙ (t ) = CeCt Vσ (t ) (x(t )) + eCt V˙ σ (t ) (x(t )) ≤ DeCt , W
t ∈ [tj , tj+1 ).
(68)
With the results in [10,33], we have V (x(t )) ≤ µV (x(t ))(µ > 1, k, l ∈ M ), which implies that k
l
W (tj+1 ) = µeCtj+1 V˙ σ (tj+1 ) (x(tj+1 )) ≤ µeCtj+1 V˙ σ (t ) (x(tj+1 ))
=
µW (tj−+1 )
≤ µ W (tj ) +
tj+1
De dt . Ct
(69)
tj
For an arbitrary T > t0 = 0, after iterating the inequality (69) from j = 0 to j = Nσ (T , 0) − 1, we obtain that W (T − ) ≤ W (tNσ (T , 0)) +
T
DeCt dt
tNσ (T ,0)
≤ µ W (tNσ (T ,0)−1 ) +
tNσ (T ,0) tNσ (T ,0)−1
De dt + µ Ct
−1
T
Ct
De dt tNσ (T ,0)
≤ ··· ≤µ
−Nσ (T ,0)
W (0) +
Nσ (T ,0)
j =0
µ
−j
tj+1
De dt + µ
tj
Ct
−Nσ (T ,0)
T
eD dt . Ct
tNσ (T ,0)
(70)
Since τa > ln µ/C , for any δ ∈ (0, C − ln µ/τa ), one has τa > ln µ/(C − δ). By (2), it holds that Nσ (T , t ) ≤ N0 +
(C − δ)(T − t ) , ln µ
∀ T ≥ t ≥ 0.
(71)
In addition, it is clear that Nσ (T , 0) − j ≤ 1 + Nσ (T , tj+1 ), j = 0, 1, . . . , Nσ (T , 0), one has
µNσ (T ,0)−j ≤ µ1+N0 e(C −δ)(T −tj+1 ) .
(72)
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In addition, since δ < C and tj+1
DeCt dt ≤ e(C −δ)tj+1
tj+1
e(C −δ)t dt .
(73)
tj
tj
From (72) and (73), it then follows that W (T − ) ≤ µNσ (T ,0) W (0) + µ1+N0 e(C −δ)T
T
Deδ t dt .
(74)
0
According to [33], there exist two class κ functions α (|x|) and α¯ (|x|), which satisfy α (|x|) ≤ V k (x) ≤ α¯ (|x|). It indicates that
α (∥x(T )∥) ≤ V σ (T ) (x(T − )) −
≤ eN0 ln µ e
ln µ τa −C T
≤ eN0 ln µ e
ln µ τa −C T
D
α¯ (∥x(0)∥) + µ1+N0 (1 − e−δT ) δ D
α¯ (∥x(0)∥) + µ1+N0 , δ
∀ T > 0.
(75)
We conclude that, by (75) and δ > 0, if τa > (ln µ/C ), then for bounded initial conditions, ei,j , Si,j and θ˜ik,j i = 1, 2, . . . , N , j = 1, 2, . . . , ni are bounded. Since (θik,j )∗ are constants, θik,j are bounded. Further, it is easy to obtain that uki , xˆ i,j , xi,j , zi,1 are bounded. Hence, for bounded initial conditions, all the signals in the closed-loop system (1) are bounded for switching signal with average dwell time. 5. Simulation studies Within this section, we will present illustrative examples. While the approach could be applied to inter-vehicle spacing regulation in a platoon of an automated highway system, since that system fits the assumptions of our framework, instead we study the control of two inverted pendulums connected by a spring. Each pendulum may be positioned by a torque input σ (t ) σ (t ) Di (ui ) applied by a servomotor at its base. It is assumed that both θ1 = x1,1 and θ2 = x2,1 (angular position and rate) are available to the ith controller for i = 1, 2; σ (t ) = k ∈ {1, 2}. The equations which describe the motion of the pendulums are defined by
x˙ i,1 = xi,2 σ (t ) σ (t ) Di (ui ) mi g Kr 2 kr (l − b) x˙ i,2 = + − sin(xi,1 ) + + 1fi,σ2(t ) (xi,2 ) + ∆σi,2(t ) (¯y) J J 4J 2J i i i i yi = xi,1 x1,1 sin(x1,1 x1,2 ) , m1 x2,1 sin(x2,1 x2,2 ) 1 , ∆2,2 y m2
where 1f11,2 (x1,1 , x1,2 ) =
x1,2 cos(x1,1 ) , m1 Kr 2 2 ∆2,2 y 4J2
1f12,2 (x1,1 , x1,2 ) = Kr 2 4J2
∆11,2 (¯y) =
Kr 2 4J1
sin(x2,1 ), ∆11,2 (¯y) =
(76)
Kr 2 4J1
sin(x1,1 x2,1 ),
(¯ ) = sin(x1,1 ), (¯ ) = sin(x1,1 x2,1 ). (x2,1 , x2,2 ) = The parameters m1 = 0.2 kg and m2 = 0.2 kg are the pendulum end masses, J1 = 0.05 kg and J2 = 0.0625 kg are the moments of inertia, K = 10 N/m is the spring constant of the connecting spring, r = 0.5 m is the pendulum height, l = 0.5 m is the natural length of the spring, and g = 9.81 m/s2 is gravitational acceleration. The distance between the pendulum hinges is defined as b = 0.4 m, where, in this example b < l so that the pendulums repel one another when both are in the upright position. In the simulation study, eleven fuzzy sets are defined over interval [−2k, 2k], and by choosing partitioning points as −2k, −k, 0, k, 2k (k ∈ {1, 2}) their fuzzy membership functions are given as follows: 1f21,2
µF 1 (ˆxi,j ) = exp(−(ˆxi,j − 2k)2 ), µF 2 (ˆxi,j ) = exp(−(ˆxi,j − k)2 ), µF 3 (ˆxi,j ) = exp(−(ˆxi,j )2 ), i,j,k
i,j,k
i,j,k
µF 4 (ˆxi,j ) = exp(−(ˆxi,j + k)2 ), µF 5 (ˆxi,j ) = exp(−(ˆxi,j + 2k)2 ), i,j,k
i,j,k
i = 1, 2; j = 1, 2.
The FLSs can be expressed in the form: fˆi,kj xˆ i,j θik,j = (θik,j )T ϕik,j (ˆxi,j ), i = 1, 2; j = 1, 2; k ∈ {1, 2}. The performance functions are chosen as: µ1 (t ) = 0.137e−0.5t + 0.013 and µ2 (t ) = 0.136e−0.4t + 0.014. For the switched system 1: l11,1 = l11,2 = l12,1 = l12,2 = 5.
For the switched system 2: l21,1 = l21,2 = l22,1 = l22,2 = 30.
1
In addition, in (23), by selecting Q1k = Q2k = 10I, we can obtain positive definite symmetric matrices P1k = −0.1
0.1667
P2k = −0.0028
−0.0028 . 0.1668
−0.1 , 1.02
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Fig. 1. The trajectories of y1 .
Fig. 2. The trajectories of y2 .
ˆ ki,r , m ˆ ki,l , dˆ ki,r ,m , The design parameters for the intermediate control function αik,1 , the parameter adaptive laws θik,1 , θik,2 , m
dˆ ki,l,m and the actual control Dki (uki ) are given as βik,j = 5, γik,j = 0.08, τik,j = 0.01 and ηik,l = 0.01 (i = 1, 2; j = 1, 2; l = 1, 2, 3, 4).
The initial conditions are chosen as x1,1 (0) = x2,1 (0) = xˆ 2,2 (0) = 0.1, xˆ 1,1 (0) = −0.1, xˆ 1,2 (0) = 0.2, θ11,1 (0) = θ21,1 =
ˆ ki,r (0) = m ˆ ki,l (0) = dˆ ki,l,m (0) = dˆ ki,r ,m (0) = 1 the [0.1, 0.3, 0.5, 0.7, 0.9], θ12,1 (0) = θ22,1 = [−0.1, −0.3, −0.5, −0.7, −0.9], m
other initial values are chosen as zeros.
The average dwell time is chosen as τa = 11.1, and let µ = 0.18, then, we can obtain τa = 11.1 > ln 6.7755/0.18. Thus, the adaptive fuzzy output-feedback control problem of the resulting closed-loop system (76) is solvable under every switching signal k ∈ {1, 2}. The simulation results are shown by Figs. 1–6, where Figs. 1 and 2 show the trajectories of the outputs; Figs. 3 and 4 show the trajectories of the states and their estimates; Fig. 5 shows the trajectories of Dki (uki ) and uki,d (i = 1, 2); Fig. 6 shows switching signal.
L. Zhang, G.-H. Yang / Nonlinear Analysis: Hybrid Systems 23 (2017) 61–75
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Fig. 3. The trajectories of x1,2 and xˆ 1,2 .
Fig. 4. The trajectories of x2,2 and xˆ 2,2 .
6. Conclusions For a class of uncertain nonlinear switched uncertain nonlinear large-scale systems with unknown dead-zones and unmeasured states, the adaptive fuzzy output-feedback decentralized control problem has been considered. The fuzzy switched decentralized state observer has been designed to estimate the unmeasured states and the controller has been obtained based on dynamic surface backstepping technique, the average dwell time method and the prescribed performance theory. It is proved that the proposed control approach can guarantee that all the signals of the closed-loop system are bounded. Comparison simulation studies illustrate that both transient and steady-state performances of the outputs are better than without prescribed performance. In the future study, we will concentrate on the adaptive fault tolerant control design for the uncertain nonlinear systems with prescribed performances based on the results of this paper. Acknowledgments This work was supported in part by the Funds of National Science of China (Grant Nos. 61273148, 61420106016), and the Research Fund of State Key Laboratory of Synthetical Automation for Process Industries (Grant No. 2013ZCX01).
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Fig. 5. The trajectories of Dki (uki ) and uki,d , i = 1, 2; k = 1, 2.
Fig. 6. Switching signal σ (t ).
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