Fuzzy adaptive quantized output feedback tracking control for switched nonlinear systems with input quantization

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Fuzzy adaptive quantized output feedback tracking control for switched nonlinear systems with input quantization ✩ Shuai Sui ∗ , Shaocheng Tong Department of Mathematics, Liaoning University of Technology, Jinzhou, Liaoning 121001, China Received 19 January 2015; received in revised form 18 May 2015; accepted 13 July 2015

Abstract In this paper, the problem of adaptive fuzzy quantized output-feedback control is investigated for a class of uncertain switched nonlinear systems in strict feedback form. The considered switched systems contain unknown nonlinearities, hysteretic quantized input and without requiring the system states being available for measurement. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and a switched fuzzy state observer is designed to estimate the unmeasured states. The hysteretic quantized input is implemented to avoid the oscillation caused by logarithmic quantizer and decomposed into two bounded nonlinear functions. Based on the estimated states and using the backstepping design principle, an adaptive fuzzy quantized output feedback control scheme is developed. It is proved that whole adaptive fuzzy quantized control scheme can guarantee that all the variables in the closed-loop system are bounded under a class of switching signals with average dwell time, and also that the system output can track a given reference signal as closely as possible. The simulation results are given to check the effectiveness of the proposed approach. © 2015 Elsevier B.V. All rights reserved.

Keywords: Fuzzy logic systems; Fuzzy control; Adaptive control; Switched systems; Input quantization

1. Introduction In several control engineering problems, the system to be controlled is characterized by a finite set of possible control actions. Such systems are referred to as systems with quantized control input and the possible values of the input represent the levels of quantization. For example, hydraulic systems using on/off valves are systems with quantized input, digital control, hybrid systems, automotive powertrain systems, networked control systems (for information processing of networked systems, all signals must be quantized before data transmission) [3–5,41–44]. Therefore, the control design for quantized control systems is important. ✩

This work was supported by the National Natural Science Foundation of China (Nos. 61374113, 61203008, 61074014).

* Corresponding author. Tel.: +86 416 4199101; fax: +86 416 4199415.

E-mail address: [email protected] (S. Sui). http://dx.doi.org/10.1016/j.fss.2015.07.012 0165-0114/© 2015 Elsevier B.V. All rights reserved.

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Recently quantized feedback control has attracted a great deal of attention. An important aspect is that utilizing quantization schemes can not only have sufficient precision, but also require low communication rate [1,2]. Compared to the traditional logarithmic quantizer [1,2], Hayakawaa et al. [3] extended the results from uncertain linear systems to uncertain nonlinear systems, and first developed a hysteretic quantizer, which can avoid the oscillation caused by logarithmic quantizer. The work in [4] investigated the adaptive quantized backstepping feedback control problems for SISO strict-feedback uncertain nonlinear systems with hysteretic quantized input. However, the choice of quantization parameters depends on controller design parameters and certain system parameters, it is difficult to choose appropriate quantizer parameters for a complex nonlinear system. It is easy to see that the results in [3,4] include only the nonlinear uncertainties in strict-feedback form. In addition, Hayakawaa et al. [3] and Zhou et al. [4] required that the nonlinear functions included in the controlled systems are known or can be linearly parameterized. To overcome this difficult, Liu et al. [5] proposed an adaptive fuzzy quantized control for a class of nonlinear stochastic systems, which don’t need assume the controlled systems are completely known. Although, the results of the above have made some achievement, they all required the states of the controlled systems are measured directly, and did not consider the control design problem for uncertain switched nonlinear systems in strict-feedback form, in which the states are unavailable for measurement. In the past decade, adaptive backstepping fuzzy or neural control design schemes have got great development. The authors in [6–10] studied the adaptive fuzzy or neural control design problems for non-strict-feedback nonlinear systems by combing backstepping technique. The characteristic of the non-strict-feedback nonlinear system is that system functions contain the whole state variables. In [11,12], the adaptive fuzzy state feedback control approaches were proposed for a class of SISO nonlinear systems with unknown dead zone output and unknown virtual control coefficients, respectively. The authors in [13–15] proposed adaptive fuzzy state feedback control design methods for a class of MIMO nonlinear systems with unknown directions and time-varying delays, respectively. Different from the above state feedback control design, the authors in [16–23] proposed adaptive fuzzy output feedback control approaches for a class of uncertain nonlinear systems with input nonlinearities. It should be mentioned that the adaptive fuzzy control design methods on uncertain nonlinear systems in strict-feedback form with input quantization has not been reported yet. And also, although many fuzzy or neural adaptive control design problems have been investigated for uncertain nonlinear systems by combining the adaptive backstepping design technique with fuzzy-logic-systems (FLSs) or neural-networks (NNs) [6–25,45,46], these adaptive fuzzy or neural control schemes are all focused on the non-switched nonlinear systems. As we know, switched systems are an important class of hybrid systems, which can be described by a family of subsystems and a rule that orchestrates the switching between them [26–30]. The control design and stability on the switched systems have attracted many researchers a great interest. Recently, some control design methods have been proposed via the backstepping design technique for several classes of switched nonlinear systems [31–38]. The works [31–34] have investigated for a class of switched nonlinear systems based on the common Lyapunov function method. Two adaptive neural control schemes have been proposed for a class of switched nonlinear systems in [35,36] based on average dwell-time technique. The work [37] proposed an adaptive neural network feedback control scheme for nonlinear switched impulsive systems under all admissible switched strategy. And also, Han et al. [38] proposed an adaptive neural network control method for a class of switched nonlinear systems with switching jumps and uncertainties in both system models and switching signals based on dwell-time property. However, all the above-mentioned adaptive control approaches required the states are measured directly, which limits the applicability of these control schemes in practical industrial systems. In addition, in the control design, the aforementioned adaptive control schemes do not consider the quantized input effect on the control performance. To the best of our knowledge, to date now, there are no results on adaptive fuzzy output feedback control available for uncertain switched nonlinear systems with immeasurable states and input quantization. In this paper, the problem of adaptive fuzzy quantized output-feedback control is investigated for a class of uncertain switched nonlinear systems in strict-feedback form. The considered switched systems contain unknown nonlinearities, hysteretic quantized input and without requiring the system states being available for measurement. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, and a switched fuzzy state observer is designed to estimate the unmeasured states. By using the nonlinear decomposition of input quantization and in the framework of adaptive backstepping technique, a robust adaptive fuzzy output-feedback tracking control approach is developed. The stability of the closed-loop system is proved based on Lyapunov function method and the average

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dwell time method. Compared with the existing literature, the main contributions of this paper are summarized as follows. (i) This paper proposed an adaptive fuzzy tracking output feedback control method for a class of switched nonlinear systems in strict-feedback form. The proposed adaptive control method has solved the state unmeasured problem via designing fuzzy switched state observer. Although the previous results in [31–38] also studied the design problem for switched nonlinear systems, they all require that the states must be available for measurement. In addition, the control methods in [31–38] do not consider the problem of quantized input. (ii) This paper first investigated the adaptive fuzzy output-feedback quantized control design problem for uncertain switched nonlinear system with hysteretic quantized input. The proposed control scheme can not only guarantee the stability of the whole switched control system, but also can attenuate the effect of the hysteretic quantizer on the control performance. Note that recent results in [3–5] only focus on the non-switched nonlinear systems. In addition, the controlled systems in [3–5] all require that the states must be available for measurement. Thus they cannot be applied to the switched nonlinear system under consideration in this study. 2. Problem formulations and preliminaries 2.1. System descriptions and assumptions Consider the following uncertain switched nonlinear system in strict-feedback form: ⎧ x˙i = xi+1 + fi,σ (t) (x i ) + di,σ (t) (t), ⎪ ⎪ ⎨ i = 1, · · · , n − 1, x ˙ = qσ (t) (uσ (t) ) + fn,σ (t) (x) + dn,σ (t) (t), ⎪ n ⎪ ⎩ y = x1

(1)

where x i = [x1 , x2 , · · · , xi ]T ∈ i , i = 1, 2, · · · , n() are the states, y ∈  is the output of system. The function σ (t) : [0, ∞) → M = {1, 2, · · · , m} is a switching signal which is assumed to be a piecewise continuous (from the right) function of time. Moreover, σ (t) = k implies that the kth subsystem is active. fi,σ (t) (x), (i = 1, 2, · ·· , n, ) are unknown smooth nonlinear functions. di,σ (t) (t)(i = 1, 2, · · · , n) are the dynamic disturbances and satisfy di,k (t) ≤ d¯i,k with d¯i,k being known constants. qσ (t) (uσ (t) ) ∈  is the output of the hysteresis quantizater. For this class of nonlinear systems (1), we assume that the existence and uniqueness of solution are satisfied. 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 [27], and [29,30]. Let T > 0 be an arbitrary time. Denote t1 , . . . , tNσ (T ,0) as the switching times on the interval (0, T )(t0 = 0). When t ∈ [tj , tj +1 ), σ (t) = kj , that is, the kj th subsystem is active. In this study, we assume kj = kj +1 for all j . In this paper, we use hysteresis quantizer to avoid chattering. qσ (t)(uσ (t) ) is a quantized input signal, which is expressed as following form similar to [3–5]:   ⎧ ui,σ (t) uσ (t)  ≤ ui,σ (t) , u˙ σ (t) < 0, or sgn(u ), if < u ⎪ i,σ (t) σ (t) 1+δ 1−δσ (t) ⎪ σ (t) ⎪ ⎪   ⎪ u i,σ ⎪ ui,σ (t) < uσ (t)  ≤ 1−δσ(t)(t) , u˙ σ (t) > 0 ⎪ ⎪ ⎪ ⎪   ⎪ ui,σ (t) ⎪ ⎪ u (1 + δσ (t) )sgn(uσ (t) ), if ui,σ (t) < uσ (t)  ≤ 1−δ , u˙ σ (t) < 0, or ⎪ σ (t) ⎨ i,σ (t)    ui,σ (t) (2) qσ (t) (uσ (t) ) =  ui,σ (t) (1+δσ (t) ) , u˙ σ (t) > 0  ⎪ 1−δσ (t) ≤ uσ (t) ≤ 1−δσ (t) ⎪ ⎪   ⎪ umin ⎪ ⎪ , u˙ σ (t) < 0, or 0, if 0 ≤ uσ (t)  < 1+δ ⎪ σ (t) ⎪ ⎪   ⎪ u ⎪ min   ⎪ ˙ σ (t) > 0 ⎪ 1+δσ (t) ≤ uσ (t) ≤ umin , u ⎪ ⎩ − othercase qσ (t) (uσ (t) (t )), where uσ (t) = ρ (1−i) u0,σ (t) with integer and parameters u0,σ (t) > 0 and 0 < ρ < 1, δσ (t) = u0,σ (t) 1+δσ (t)

1−ρσ (t) 1+ρσ (t) .qσ (t) (uσ (t) )

is the

determines the size of the dead-zone for qσ (t) (uσ (t) ). The map set U = {0, ±ui,σ (t) , ±ui,σ (t) (1 + δσ (t) )}.umin = of the hysteretic quantizer qσ (t) (uσ (t) ) for uσ (t) > 0 is shown in Fig. 1.

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Fig. 1. Map of qσ (t) (uσ (t) ) for uσ (t) > 0.

Remark 1. The parameter ρσ (t) is considered as a measure of quantization density. The smaller ρσ (t) is, the coarser the quantizer is. When ρσ (t) approaches to zero, δσ (t) approaches to 1, then qσ (t) (uσ (t) ) will have fewer quantization levels as uσ (t) ranges over that interval. Remark 2. The control action for the hysteretic quantizer (2) should be satisfied in terms of existence and uniqueness of solution of the closed-loop systems. Since the system (1) is uncertain so the parameter ρσ (t) of the hysteretic quantizer is not given a prior. Instead, it should be chosen based on a guideline that ensures the stability of the closed loop system. In order to propose a suitable control scheme, we decompose the hysteretic quantizer qσ (t)(uσ (t) ) into the following form: qσ (t) (uσ (t) ) = Dσ (t) (uσ (t) )uσ (t) + sσ (t) (t)

(3)

where Dσ (t) (uσ (t) ) and sσ (t) (t) are nonlinear functions. Regarding the nonlinearity Dσ (t) (uσ (t) ) and sσ (t) (t), we have the following lemma. Lemma 1. The nonlinearities Dσ (t) (uσ (t) ) and sσ (t) (t) satisfy   sσ (t) (t) ≤ umin 1 − δσ (t) ≤ Dσ (t) (uσ (t) ) ≤ 1 + δσ (t) ,

(4)

  Proof. From Fig. 1 and using sector bound property, we can get that for uσ (t)  ≥ umin 1 − δσ (t) ≤ sσ (t) (uσ (t) ) ≤ 1 + δσ (t) q

(u

(5)

)

where sσ (t) (uσ (t) ) = σ (t)uσ (t)σ (t) .   For uσ (t)  ≤ uσ (t) min , qσ (t) (uσ (t) ) = 0 from the definition (3), we have 0 = Dσ (t) (uσ (t) )uσ (t) + sσ (t) (t) Define



Dσ (t) (uσ (t) ) = and

 sσ (t) (t) =

sσ (t) (uσ (t) ), 1,

(6)   uσ (t)  > umin   uσ (t)  ≤ umin

  uσ (t)  > umin   −uσ (t) , uσ (t)  ≤ umin 0,

(7)

(8)

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Then qσ (t) (uσ (t) ) = Dσ (t) (uσ (t) )uσ (t) + sσ (t) (t) holds, where Dσ (t) (uσ (t) ) and sσ (t) (t) satisfy (4). The proof is completed. 2 Remark 3. If the states are available for measurement and without considering the quantized input signal in system (1), then the subsystems in (1) represents a class of non-switched nonlinear systems in strict-feedback form, because the controlled systems under this study contain the unmeasured states, quantized input signal and the switching signals, the control design and stability analysis in this paper are more difficult and challenging than those in [6–22]. Assumption 1. (See [22].) There exists a known constant liσ (t) such that     fi,σ (t) (x i ) − fi,σ (t) (xˆ i ) ≤ li,σ (t) x i − xˆ i  , i = 1, 2, . . . , n

(9)   where xˆ i = [xˆ1 , xˆ2 , . . . , xˆi ]T is the estimate of x i = [x1 , x2 , . . . , xi ]T ; where x i  denotes the 2- norm of a vector x i . Lemma 2. (See [28,29].) A switched nonlinear system 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

(10)

where Nσ (T , t) is the number of switches occurring in the interval [t, T ). 2.2. Fuzzy logic systems We introduce the fuzzy logic systems [39]. A fuzzy logic system (FLS) consists of four parts, that is, the knowledge base, the fuzzifier, the fuzzy inference engine and the defuzzifier. The knowledge base for FLS is composed of a series of fuzzy If–Then inference rules of the following form: R l : 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 fuzzy logic system input and output, respectively. Fuzzy sets Fil and Gl , μF l (xi ) i

and μGl (y) are the fuzzy functions of Fil and Gl , respectively. N is the rule number. Through singleton function, center average defuzzification and product inference [39], a fuzzy logic system can be expressed as

n N l=1 y¯l i=1 μFil (xi ) y(x) = N n (11) l=1 [ i=1 μF l (xi )] i

where y¯l = maxy∈R μGl (y). Define the fuzzy basis functions as

n i=1 μF l (xi ) ϕl = N n i l=1 ( i=1 μF l (xi ))

(12)

i

Denote θ = [y¯1 , y¯2 , . . . , y¯N ]T = [θ1 , θ2 , . . . , θN ]T and φ T (x) = [φ1 (x), · · · , φN (x)], then the fuzzy logic system (11) can be rewritten as y(x) = θ T φ(x)

(13)

Lemma 3. (See [39].) Let f (x) be a continuous function defined on a compact set . Then for any constant ε > 0, there exist an FLS (11) such as     sup f (x) − θ T φ(x) ≤ ε (14) x∈

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Control objective. The control objective of this paper is to design an adaptive fuzzy output feedback control scheme such that all the variables in the switched closed-loop system are bounded and the system output y can track the given reference signal yr (t) in presence of the unknown function fi,σ (t) (x i ), unmeasured states xi , i = 2, · · · , n and quantized input signal qσ (t) (uσ (t) ). 3. Switched fuzzy state observer design In this section, a switched fuzzy state observer is proposed for estimating the immeasurable states xi , i = 2, · · · , n in the switched strict-feedback nonlinear system (1). For the kth subsystem in (1), it can be written as ⎧ x˙1 = f1,k (xˆ1 ) + x2 + f1,k + d1,k ⎪ ⎪ ⎪ ⎪ x ˙2 = f2,k (xˆ 2 ) + x3 + f2,k + d2,k ⎪ ⎪ ⎪ ⎨ .. . (15) ⎪ x˙n−1 = fn−1,k (xˆ n−1 ) + xn + fn−1,k + dn−1,k ⎪ ⎪ ⎪ ⎪ ˆ + qk (uk ) + fn,k + dn,k ⎪ x˙n = fn,k (x) ⎪ ⎩ y = x1 where fi,k = fi,k (x i ) − fi,k (xˆ i ), i = 1, . . . , n; k ∈ {1, 2, · · · , m}. By Lemma 2, we can assume that the nonlinear function fi,k (x i ) in (12) 1 ≤ i ≤ n; k ∈ {1, 2, · · · , m} can be approximated by the following fuzzy logic systems  T φi,k (xˆ i ) (16) fˆi,k (xˆ i θi,k ) = θi,k The optimal parameter vector θi∗ is defined as      θi∗ = arg min [ sup fˆi,m (xˆ i θi,k ) − fi,k (xˆ i )] θi,k ∈ i,k xˆ ∈U i

where i,k and U are compact regions for θi,k and xˆ i , respectively. Also the fuzzy minimum approximation errors εi,k (xˆ i ) are defined as  ∗ εi,k (xˆ i ) = fi,k (xˆ i ) − fˆi,k (xˆ i θi,k ) (17)   ∗ , and ε ∗ is an unknown constant. Here εi,k satisfies that εi,k  ≤ εi,k i,k By substituting (17) into (15), the kth switched subsystem (15) can be expressed as x˙ = Ak x + Lk y +

n 

∗T Bi θi,k φi,k (xˆ i ) + εk + k + Fk + Bn qk (uk )

i=1

¯ y = Cx where

(18)

⎡

⎤

−L1,k ⎢ .. Ak = ⎣ . −Ln,k

0

In−1 ...

⎥ ⎦,

⎡

⎤ L1,k ⎢ ⎥ Lk = ⎣ ... ⎦ ,

0

⎤ f1,k ⎢ ⎥ Fk = ⎣ ... ⎦ ,

Ln,k

k = [d1,k , · · · , dn,k ] , εk = [ε1,k , · · · , εn,k ] , Bi = [ 0 T

⎡

T

C¯ = [1, · · · , 0],

fn,k ···

1

···

0 ]T .

For the kth subsystem, a switched fuzzy state observer is designed as x˙ˆ = Ak xˆ + Lk y +

n 

 Bi fˆi,k (xˆ i θi,k ) + Bn qk (uk )

i=1

yˆ = C¯ xˆ Define observation error vector e as

(19)

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e = [e1 , e2 , · · · , en ]T = x − xˆ

7

(20)

From (18) and (19), the dynamics of the observer error is e˙ = Ak e +

n 

T Bi θ˜i,k φi,k (xˆ i ) + εk + k + Fk

(21)

i=1 ∗ −θ where θ˜i,k = θi,k i,k is the adaptive parameter vector error. Choose vector Lk such that matrix Ak is a strict Hurwitz matrix. Therefore, for any a given matrix Qk = QTk > 0, there exists a positive definite matrix Pk = PkT > 0 such that

ATk Pk + Pk Ak = −Qk

(22)

Remark 4. For the kth subsystem and assume that di,k (t) = 0. Choose the Lyapunov function as V0,k = eT Pk e, then from (21) and (22), it follows that V˙0,k ≤ −λ0,k e2 +

n 

 2 T ˜ θ˜j,k θj,k + εk∗ 

(23)

j =1

2 ) − (n + 1) P 2 and ε ∗ = [ε ∗ , ε ∗ · · · , ε ∗ ]T . Since the fuzzy logic with λ0,k = λmin (Qk ) − Pk 2 ( nj=1 lj,k k k n,k  1,k 2,k system fˆi,k (x|θ ˆ i,k ) can well approximate the unknown function fi,k (x), εk∗  is bounded. Moreover, if λ0,k > 0 and T θ˜ the term nj=1 θ˜j,k j,k is bounded, then from (21) and (22), we can know that the state observer (19) is asymptotically stable in the sense of the Lyapunov function. Thus, the designed fuzzy state observer (19) is reasonable. Next section T θ˜ will study how to design an adaptive control scheme to ensure that the term nj=1 θ˜j,k j,k is bounded. Furthermore, that the error dynamics (21) is asymptotically stable is concluded. Remark 5. If fˆi,k (x|θ ˆ i,k ) = 0, then the fuzzy state observer (19) in this study will be reduced to a linear-reduced state observer in [10,17–20]. It is worth noting that since the linear-reduced state observer in [10,17–20] is independent of the controlled nonlinear systems, it cannot obtain the good estimations of the unmeasured states when fi,k (x i ) = 0. Moreover, it is only suitable for the stabilization control design problem, not for the tracking control design problem discussed in this paper. 4. Adaptive fuzzy control design and stability analysis To realize the control objective, this section will give the adaptive fuzzy output-feedback control design via backstepping design technique and the designed fuzzy state observer in the last section. The stability of the switched closed-loop system will be proved by using Lyapunov function and average dwell time method. In the sequel, the n-step adaptive backstepping design procedures will be developed for the kth subsystem. Let z1 = y − yr ,

zi = xˆi − αi−1,k ,

2 ≤ i ≤ n; k ∈ {1, 2, · · · , m}

(24)

where αi−1,k is a virtual controller. Step 1: From (24) and x2 = xˆ2 + e2 , we have z˙ 1 = x2 + f1,k (x1 ) + d1,k (t) − y˙r T T = z2 + α1,k + e2 + ε1,k + θ1,k φ1,k (xˆ1 ) + θ˜1,k φ1,k (xˆ1 ) + d1,k (t) − y˙r + f1,k

(25)

Choose the Lyapunov function candidate as 1 1 T V1,k = eT Pk e + z12 + θ˜ θ˜1,k 2 2r1,k 1,k where r1,k > 0 is a design parameter. From (21), (22) and (24)–(26), the time derivative of V1,k satisfies

(26)

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V˙1,k ≤ −λmin (Qk ) e2 + 2eT Pk [εk + k + Fk +

n 

T Bi θ˜i,k φi,k (xˆ i )]

i=1 T φ1,k (xˆ i ) + e2 + z1 (z2 + α1,k + ε1,k + θ1,k T + θ˜1,k φ1,k (xˆ1 ) + d1,k (t) − y˙r + f1,k ) +

1 r1,k

T ˙˜ θ˜1,k θ 1,k

(27)

T (xˆ )φ (xˆ ) ≤ 1, we can obtain By completely the squares, Assumption 1 and the fact φi,k i i,k i n   2 2 2eT Pk ( k + εk ) ≤ 2 e2 + Pk 2 εk∗  + Pk 2 d¯i,k

(28)

i=1

2eT Pk Fk ≤ e2 + Pk 2 Fk 2 ≤ e2 + Pk 2 (|f1 |2 + · · · + |fn |2 ) n  2 e2 ) lj,k ≤ e2 + P 2 ( j =1 n  T T ˜ 2eT Pk Bi θ˜i,k φi,k (xˆ i ) ≤ n e2 + Pk 2 θ˜i,k θi,k i=1 i=1 n 

(29)

(30)

3 1 1 ∗2 1 2 + d¯1,k z1 (ε1,k + e2 + d1,k (t)) ≤ z12 + e2 + ε1,k 2 2 2 2   1 2 1 2 2 z1 f1,k  ≤ z1 + l1,k e 2 2 Substituting (28)–(32) into (27) results in

(31) (32)

T φ1,k (xˆ1 ) − y˙r ) V˙1,k ≤ −λ1,k e2 + z1 (α1,k + 2z1 + z2 + θ1,k T + θ˜1,k (φ1,k (xˆ1 )z1 −

1 r1,k

θ˙1,k ) + Pk 2

n 

T ˜ θ˜j,k θj,k + M1,k

(33)

j =1

2   2  ∗ 2 2 ) − n − 7 − l1,k and M 2 ) + 1 (ε ∗2 + d¯ 2 ). where λ1,k = λmin (Qk ) − P 2 ( nj=1 lj,k + ni=1 d¯i,k 1,k = Pk  ( εk 1,k 2 2 2 1,k Design the virtual controller α1,k and the parameter adaptation function θ1,k as

T α1,k = −β1,k z1 − 2z1 − θ1,k φ1,k (xˆ1 ) + y˙r ˙θ1,k = r1,k φ1,k (xˆ1 )z1 − τ1,k θ1,k

(34) (35)

where β1,k > 0 and τ1,k > 0 are design parameters. From (34)–(35), it follows that  τ1,k T T ˜ V˙1,k ≤ −λ1,k e2 + z1 z2 − c1,k z12 + θ˜1,k θ1,k + Pk 2 θ˜j,k θj,k + M1,k r1,k n

(36)

j =1

Step i(2 ≤ i ≤ n): Since zi = xˆi − αi−1,k , the derivative time of zi along with (19) and (24) is z˙ i = x˙ˆ i − α˙ i−1,k T = zi+1 + αi,k + Li,k e1 + θi,k φi,k (xˆ i ) −

i−1  ∂αi−1,k l=1

−

i−1  ∂αi−1,k

(j ) y (j −1) r j =1 ∂yr

−

∂ xˆl

x˙ˆ l −

i−1  ∂αi−1,k j =1

∂θj,k

θ˙j,k

∂αi−1,k T [xˆ2 + e2 + θ1,k φ1,k (xˆ1 ) ∂x1

T + θ˜1,k φ1,k (xˆ1 ) + ε1,k + d1,k (t) + f1,k ] ∂αi−1,k ˜ T (θ1,k φ1,k (xˆ1 ) + ε1,k + e2 + d1,k (t) + f1,k ) = zi+1 + αi,k + Hi,k − ∂x1

(37)

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where T Hi,k = Li,k e1 + θi,k φi,k (x) ˆ −

i−1  ∂αi−1,k

∂ xˆl

l=1

−

i−1  ∂αi−1,k

(j ) y (j −1) r j =1 ∂yr

−

x˙ˆ l −

i−1  ∂αi−1,k

∂θj,k

j =1

θ˙j,k

∂αi−1,k T [xˆ2 + θ1,k φ1,k (x)]. ˆ ∂x1

Consider the following Lyapunov function candidate 1 1 T Vi,k = Vi−1,k + zi2 + θ˜ θ˜i,k 2 2ri,k i,k

(38)

where ri,k > 0 is a design parameter. From (37) and (38), we can obtain 1 T ˙ V˙i,k ≤ V˙i−1,k + zi z˙ i + θ˜ θ˜ i,k δi,k i,k ∂αi−1,k ˜ T (θ1,k φ1,k (xˆ1 ) + e2 ∂x1 i−1  1 T T + d1,k (t) + ε1,k + f1,k )] + θ˜i,k (zi φi,k (xˆ i ) − cj,k zj2 − zi θ˜i,k φi,k (xˆ i ) θ˙i,k ) − ri,k

≤ −λi−1,k e2 + Mi−1,k + zi [zi−1 + zi+1 + αi,k + Hi,k −

j =1

+

i−1  j =1

τj,k T θ˜ θj,k + Pk 2 rj,k j,k ∗2 − 2)( 12 ε1,k

where Mi−1,k = M1,k + (i + By completing the squares, we have

n 

T ˜ θ˜j,k θj,k +

j =1

i−1 

1 2

T ˜ θ˜j,k θj,k +

j =2

i −2 T θ˜ θ˜1,k 2 1,k

(39)

1 ¯2 2 d1,k ).

∂αi−1,k 1 ∗2 1 1 2 3 ∂αi−1,k 2 2 (ε1,k + e2 + d1,k (t)) ≤ ε1,k + e2 + d¯1,k + ( ) zi ∂x1 2 2 2 2 ∂x1 ∂αi−1,k T 1 T 1 2 ∂αi−1,k 2 2 e2 + ( (θ˜1,k φ1,k (xˆ1 ) + f1,k ) ≤ θ˜1,k ) zi −zi θ˜1,k + l1,k ∂x1 2 2 ∂x1 1 1 T T φi,k (xˆ i ) ≤ zi2 + θ˜i,k −zi θ˜i,k θ˜i,k 2 2 Substituting (40)–(42) into (39) yields −zi

(40) (41) (42)

5 ∂αi−1,k 2 ) zi + Hi,k ] V˙i,k ≤ −λi,k e2 + Mi,k + zi [zi−1 + αi,k + zi + zi+1 + ( 2 ∂x1 i−1 i−1   τj,k T 1 T + θ˜i,k (zi φi,k (xˆ i ) − cj,k zj2 + θ˙i,k ) − θ˜ θj,k ri,k rj,k j,k j =1

+ Pk 2

n 

T ˜ θ˜j,k θj,k +

j =1

1 2

i 

T ˜ θ˜j,k θj,k +

j =2

j =1

i −1 T θ˜ θ˜1,k 2 1,k

(43)

∗2 + 1 d¯ 2 . where λi,k = λi−1,k − − Mi,k = Mi−1,k + 12 ε1,k 2 1,k Design the virtual controller αi,k and the parameter adaptation function θi,k as 1 2

1 2 2 l1,k ,

5 ∂αi−1,k 2 αi,k = −ci,k zi − zi−1 − zi − ( ) zi − Hi,k 2 ∂x1 θ˙i,k = ri,k zi φi,k (xˆ i ) − τi,k θi,k where ci,k > 0 and τi,k > 0 are design parameters. From (44)–(45), it follows that

(44) (45)

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V˙i,k ≤ −λi,k e2 −

i 

cj,k zj2 + zi zi+1 +

j =1

i  τj,k j =1

rj,k

T θj,k + Pk 2 θ˜j,k

n 

T ˜ θ˜j,k θj,k

j =1

1 T i −1 T θ˜ θ˜1,k + Mi,k θ˜j,k θ˜j,k + 2 2 1,k i

+

(46)

j =2

Step n: In the final design step, the quantized input uk will be obtained. From (3), (4) and (24), the time derivative of zn is ∂αn−1,k T z˙ n = x˙ˆ n − α˙ n−1,k = qk (uk ) + Hn,k − (θ˜1,k φ1,k (xˆ1 ) + ε1,k + e2 ∂x1 + d1,k (t) + f1,k ) ∂αn−1,k T = Gk (uk )uk + sk (t) + Hn,k − (θ˜1,k φ1,k (xˆ1 ) + ε1,k ∂x1 + e2 + d1,k (t) + f1,k )

(47)

where T Hn,k = Ln,k e1 + θn,k φn,k (x) ˆ −

n−1  ∂αi−1,k l=1

−

n−1 

∂αn−1,k

(j ) y (j −1) r j =1 ∂yr

−

∂ xˆl

x˙ˆ l −

n−1  ∂αn−1,k j =1

∂θj,k

θ˙j,k

∂αn−1,k T [xˆ2 + θ1,k φ1,k (x)]. ˆ ∂x1

Consider the following Lyapunov function candidate 1 1 T Vk = Vn−1,k + zn2 + θ˜ θ˜n,k 2 2rn,k n,k

(48)

where rr,k > 0 is the design parameter. Applying (3), (47) and (48), we can obtain 1 T ˙ V˙k = V˙n−1,k + zn z˙ n + θ˜ θ˜ n,k rn,k n,k ≤ −λn−1,k e2 + Mn−1,k + zn [Dk (uk )uk + sk (t) + zn−1 + Hn,k ∂αn−1,k T − (θ˜1,k φ1,k (xˆ1 ) + ε1,k + e2 + d1,k (t)) + f1,k ] ∂x1 T (zn φn,k (x) ˆ − + θ˜n,k

T − zn θ˜n,k φn,k (x) ˆ +

1 rn,k

n−1  τj,k j =1

+

1 2

n−1 

θ˙n ) −

rj,k

n−1  j =1

cj,k zj2 +

n−2 T θ˜ θ˜1,k 2 1,k

T θj,k + Pk 2 θ˜j,k

n 

T ˜ θ˜j,k θj,k

j =1

T ˜ θ˜j,k θj,k

(49)

j =2

By completing the squares and combine (4), we have 1 1 zn sk (t) ≤ zn2 + u2min 2 2 Similar to the inequalities (40)–(42) in step i, then (49) can be rewritten in the following form:

(50)

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1 T ˙ V˙k = V˙n−1,k + zn z˙ n + θ˜ θ˜ n,k rn,k n,k 3 5 ∂αn−1,k 2 ≤ −λn−1,k e2 + Mn,k + zn [Dk (uk )uk + zn + zn−1 + ( ) zn + Hn,k ] 2 2 ∂x1 n−1  1 n−2 T T ˜ ˙ + θn,k (zn φn,k (x) ˆ − cj,k zj2 + θ˜ θ˜1,k θn ) − rn,k 2 1,k j =1

T − zn θ˜n,k φn,k (x) ˆ +

n−1  j =1

 τj,k T 1 T T ˜ θ˜j,k θj,k + Pk 2 θ˜j,k θj,k + θ˜j,k θ˜j,k rj,k 2 n

n−1

j =1

j =2

(51)

Design the input of the quantized input uk and parameter adaptation functions as follows: 1 3 5 ∂αn−1,k 2 (−cn,k zn − zn − zn−1 − ( ) zn − Hn,k ) 1 − δk 2 2 ∂x1 ˆ − τn,k θn,k θ˙n,k = rn,k zn φn,k (x)

uk =

(52) (53)

where cn,k > 0 and τn,k > 0 are design parameters. Note that, from (4) and (52), and by completing the square, we can obtain 3 5 ∂αn−1,k 2 Dk (uk )uk ≤ −cn,k zn − zn − zn−1 − ( ) zn − Hn,k 2 2 ∂x1 1 T 1 ∗T ∗ T θj,k ≤ − θ˜j,k θi,k θ˜j,k θ˜j,k + θi,k 2 2 Substituting (52)–(55) into (51) yields V˙k ≤ −λn,k e2 −

n 

cj,k zj2 +

j =1

+

1 2

n 

n  τj,k j =1

T ˜ θ˜j,k θj,k +

j =2

rj,k

T θj,k + Pk 2 θ˜j,k

n  j =1

n−1 T θ˜ θ˜1,k + Mk 2 1,k

2

n 

(55)

T ˜ θ˜j,k θj,k

l 1 ∗2 1 ¯2 1 2 where λn,k = λn−1,k − 12 − 1,k 2 , Mk = Mn−1,k + 2 ε1,k + 2 d1,k + 2 umin + By completing the square, (56) can be rewritten as

V˙k ≤ −λn,k e2 −

(54)

(56) n

τi,k ∗T ∗ i=1 2ri,k θi,k θi,k .

1  τi,k T ˜ − ( − 2 Pk 2 − 1)θ˜i,k θi,k 2 ri,k n

ci,k zi2

i=1

i=2

1 τ1,k T ˜ − ( − n + 1 − 2 Pk 2 )θ˜1,k θ1,k + Mk 2 r1,k Let C = mink∈M Ck , D = maxk∈M Mk and Ck = min{λn,k /λmax (Pk ), 2ci,k , τ1,k − (n − 1)r1,k − 2 Pk 2 r1,k , τl,k − 2 Pk 2 rl,k − rl,k , },

(57)

i = 1, · · · , n,

l = 2, · · · , n − 1. Therefore, (57) can be rewritten as V˙k ≤ −CVk + D

(58)

Next, we will give the stability proof of the closed-loop system based on the above controller design and Lemma 1. Theorem 1. For uncertain switched strict-feedback nonlinear system (1), if Assumption 1 is satisfied, and the average dwell time of the switching signal σ (t) satisfies the condition τa > lnCμ , then the designed controller (52) and fuzzy state observer (19), the virtual controllers (34) and (44), together with parameter adaptation functions (35), (45) and (53), can guarantee that all the variables in the closed-loop system are bounded. Moreover, the observer and the tracking errors can be made to converge to a small neighborhood of zero by suitably choosing the design parameters.

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12

Proof. It is easy to see that the function W (t) = eCt Vσ (t) (x(t)) is piecewise differentiable along solutions of the system (1). In view of (58), on each interval [tj , tj +1 ), one has W˙ (t) = CeCt Vσ (t) (x(t)) + eCt V˙σ (t) (x(t)) ≤ DeCt ,

t ∈ [tj , tj +1 )

(59)

As the same proof in [27], we can have that Vk (x(t)) ≤ μVl (x(t))(μ > 1, k, l ∈ M), implies that W (tj +1 ) ≤ eCtj +1 Vσ (tj +1 ) (x(tj +1 )) ≤ μeCtj +1 Vσ (tj ) (x(tj +1 )) = μW (tj−+1 ) tj +1 ≤ μ[W (tj ) + DeCt dt]

(60)

tj

Pick an arbitrary T > t0 = 0. Iterating the inequality (60) from j = 0 to j = Nσ (T , 0) − 1, we obtain that T

W (T − ) ≤ W (tNσ (T ,0) ) +

DeCt dt

tNσ (T ,0) tN σ (T ,0)

≤ μ[W (tNσ (T ,0)−1 ) +

De dt + μ Ct

−1

tNσ (T ,0)−1

T DeCt dt]

tNσ (T ,0)

≤ ··· ≤μ

Nσ (T ,0)

[W (0) +

Nσ (T ,0)−1 

μ

−j

j =0

tj +1 DeCt dt tj

T

+ μ−Nσ (T ,0)

DeCt dt]

(61)

tNσ (T ,0)

Since τa > ln μ/C, for any δ ∈ (0, C − ln μ/τa ), one has τa > ln μ/(C − δ). By (10), it holds that Nσ (T , t) ≤ N0 +

(C − δ)(T − t) , ln μ

∀T ≥ t ≥ 0

(62)

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 )

(63)

In addition, since δ < C and tj +1 tj +1 Ct (C−δ)tj +1 De dt ≤ e e(C−δ)t dt tj

(64)

tj

It then follows from (63) and (64), one has −

W (T ) ≤ μ

T Nσ (T ,0)

W (0) + μ

1+N0 (C−δ)T

Deδt dt

e

0

According to [40], there exist two κ functions α(|x|) and α(|x|), ¯ which satisfy α(|x|) ≤ Vk (x) ≤ α(|x|). ¯

(65)

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13

It indicates that α(x(T )) ≤ Vσ (T − ) (x(T − )) ≤ eN0 ln μ e(

ln μ τa −C)T

≤ eN0 ln μ e(

ln μ τa −C)T

α(x(0)) ¯ + μ1+N0

D (1 − e−δT ) δ

α(x(0)) ¯ + μ1+N0

D , δ

∀T > 0

(66)

∗ , i = 1, 2, . . . , n We conclude that, by (66) and δ > 0, if τa > (ln μ/C), then for bounded initial conditions, ei , zi , θ˜i,k ∗ are bounded. Since θi,k , i = 1, 2, . . . , n, are constants, θi,k , i = 1, 2, . . . , n, are bounded. Further, it is easy to obtain that xˆi , xi , uk and qk (uk ), i = 1, 2, . . . , n, are also bounded. Hence, for bounded initial conditions, all the signals in the closed-loop system (1) are bounded for switching signal σ (t) with average dwell time satisfying that τa > (ln μ/C). From the previous discussions, the control design procedures and the guideline of the parameter selections are given as follows:

Step 1: Define the fuzzy IF–THEN rules and the membership functions, construct the fuzzy basis functions, and obtain the FLS (14). Step 2: specify the vector Lk such that matrix Ak is a strict Hurwitz matrix. Step 3: specify positive-definite matrices Qk and by solving the Lyapunov equation (22), positive definite matrices Pk are obtained. Step 4: select appropriately design parameters such that ci,k > 0 and τi,k > 0, and the determine intermediate control functions α1,k (34), αi,k (44), and the parameters adaptation laws θ1,k (35), θi,k (45), i = 1, . . . , n − 1; k ∈ M. Step 5: select appropriately design parameters cn,k > 0 and τn,k > 0, actual controller uk , and adaptive update law θn,k . 2 Remark 6. Note that that the state observer error and tracking error satisfying √ from (66), we can only conclude √ that e ≤ maxk∈M 2D/(Cλmin (Pk )) and |z1 | ≤ 2D/C, we cannot conclude that the state observer error and tracking error asymptotically converges to zero. However, according to [13–22], we can make both the state observer error and tracking error to be small by increasing the design parameters Li,k , λmin (Qk ), ci,k and ri,k or decreasing τi,k (i = 1, · · · , n). 5. Simulation studies In this section, a simulation example is given to illustrate the effectiveness of the proposed adaptive fuzzy output feedback control approach. Example 1 (A numerical example). Consider the following uncertain switched nonlinear system: x˙1 = x2 + f1,σ (t) (x1 ) + d1,σ (t) (t) x˙2 = qσ (t) (uσ (t) ) + f2,σ (t) (x1 , x2 ) + d2,σ (t) (t) y = x1

(67)

where f1,1 (x1 , x2 ) = x1 sin(x12 ), f2,1 (x1 , x2 ) = x2 /(1 + x12 ), d1,1 (t) = 0.1 sin(t), d2,1 (t) = 0.1 cos(t), f1,2 (x1 , x2 ) = 0.2 cos(x1 ), f2,2 (x1 , x2 ) = 0.2x1 x22 , d1,2 (t) = cos(t), 1,2 (t) = 0.1 sin(t), d2,2 (t) = 0.1 cos(t). The output is assumed to be a Gaussian white noise with zero mean and variance 1.0, and the given reference tracking signals is yr (t) = sin(0.2t). The parameters in the hysteretic quantized input (2) are selected as δ1 = 0.5, δ2 = 0.1, umin = 0.2. The fuzzy membership functions are defined as

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14

Fig. 2. The trajectories of y “blue line” and yr “red line”.

(xˆ1 − 3 + l)2 ], 1,k 2 The fuzzy basis functions are μF l (xˆ1 ) = exp[−

] exp[− (xˆ1 −3+l) 2

μF l (xˆ2 ) = exp[− 2,k

(xˆ2 − 3 + l)2 ], l = 1, · · · , 5, k ∈ {1, 2} 2

2

φ1,l,k (xˆ1 ) = 5 φ2,l,k (xˆ1 , xˆ2 ) =

,

l = 1, · · · , 5,

(xˆ1 −3+n)2 ] 2 2 2 ] × exp[− (xˆ2 −3+l) ] exp[− (xˆ1 −3+l) 2 2 , 5 (xˆ1 −3+n)2 (xˆ2 −3+n)2 ] × exp[− ] n=1 exp[− 2 2

n=1 exp[−

k ∈ {1, 2} l = 1, · · · , 5, k ∈ {1, 2}.

Therefore, the FLSs can be expressed in the form 5   T T φi,k (xˆ i ) = θi,j,k φi,j,k (xˆ i ), fˆi,k (xˆ i θi,k ) = θi,k

i = 1, 2, k = 1, 2.

j =1

Setting the parameters L1,1 = 6, L2,1 = 8, and L1,2 = 8, L2,2 = 10 the switched fuzzy state observer (19) is  ⎧ ⎨ x˙ˆ 1 = xˆ2 + fˆ1,k (xˆ1 θ1,k ) + L1,k (x1 − xˆ1 ) k = 1, 2 x˙ˆ = qk (uk ) + fˆ2,k (xˆ1 , xˆ2 θ2,k ) + L2,k (x1 − xˆ1 ) , ⎩ 2 yˆ = xˆ1

(68)

In addition, in (22), by select Q1 = 3I and Q2 = 3I , we can obtain two positive-definite symmetric matrices     2.2500 −1.5000 2.0625 −1.5000 P1 = , and P2 = . −1.5000 1.4062 −1.5000 1.4063 The design parameters in virtual control α1,k , the control input of quantizer uk , and the parameter adaptive laws θ1,k , θ2,k , are chosen as ci,1 = 0.1, ci,2 = 0.5, ri,k = 0.1, τi,k = 1.5, i = 1, 2; k = 1, 2. The initial conditions are chosen as x1 (0) = 0.5, x2 (0) = −0.5, xˆ1 (0) = 0.1, xˆ2 (0) = −0.1, and the others initial values are chosen zeros. The average dwell time is chosen as τa = 10, and let μ = 1.1, then, we can obtain τa = 10 > ln 1.1/0.01. Thus, the adaptive fuzzy output-feedback control problem of the resulting closed-loop system (64) is solvable under every switching signal k ∈ {1, 2}. The simulation results are shown in Figs. 2–9, where Fig. 2 expresses the trajectories of the system output y and tracking signal yr ; Fig. 3 and Fig. 4 show the trajectories of xi (i = 1, 2) and their estimates xˆi ,

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15

Fig. 3. The trajectories of x1 “blue line” and xˆ1 “red line”.

Fig. 4. The trajectories of x2 “blue line” and xˆ2 “red line”.

respectively; Fig. 5 and Fig. 6 show the control input uk and the quantized input signal qk (uk ), k = 1, 2; Fig. 7 and  2 Fig. 8 show the trajectories of θi,k  (i = 1, 2, k = 1, 2); Fig. 9 shows the switching signal σ (t). Example 2. [One-link robot system] In this example, we consider a one-link manipulator with the inclusion of motor dynamics. The dynamic equation of such system is given by [7,18]. D q¨ + B q˙ + N sin(q) = τ + τd M τ˙ + H τ = u − Km q˙

(69)

where q, q, ˙ q¨ denote the link position, velocity and acceleration, respectively. τ is the torque produced by the electrical subsystem and τd represents the torque disturbance. u is the control input used to represent the electromechanical torque. D = 1 kg m2 is the mechanical inertia, B = 1 Nm s/rad is the coefficient of viscous friction at the joint,

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16

Fig. 5. The trajectories of u1 “blue line” and q1 (u1 ) “red line”.

Fig. 6. The trajectories of u2 “blue line” and q2 (u2 ) “red line”.

N = 10 is a positive constant related to the mass of the load and the coefficient of gravity, M = 0.05H is the armature inductance, H = 0.5 is the armature resistance, Km = 10 Nm/A is the back-emf coefficient. By introducing the variable changes x1 = q, x3 , x3 = τ , and assume that the switched functions are unknown and hysteretic quantizer existing in system (69), the dynamics given by (69) is rewritten in the following form x˙1 = x2 + f1,σ (t) (x1 ) + d1,σ (t) (t) x˙2 = x3 + f2,σ (t) (x1 , x2 ) + d2,σ (t) (t) x˙3 = qσ (t) (uσ (t) ) + f3,σ (t) (x1 , x2 , x3 ) + d3,σ (t) (t) y = x1

(70)

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17

2  Fig. 7. The trajectories of θ1,k  , k ∈ {1, 2}.

2  Fig. 8. The trajectories of θ2,k  , k ∈ {1, 2}.

B where f1,1 (x1 ) = f1,2 (x1 ) = 0, f2,1 (x1 , x2 ) = − D x2 − N D sin x1 , f2,2 (x1 , x2 ) = −x2 + sin x1 x2 , f3,1 (x1 , x2 , x3 ) = x1 Km Km H H 1 − M x2 − M x3 , f3,2 (x1 , x2 , x3 ) = − M x2 − M x3 + 4 , d2,1 (t) = d2,2 (t) = D τd , d1,1 (t) = d1,2 (t) = d3,1 (t) = 1+x3

d3,2 (t) = 0. The control objective is to maintain the systems to track the desired angle trajectory yr = π/5(sin(0.5t) + 0.5 sin(0.2t)). The parameters in the hysteretic quantized input (2) are selected as δ1 = 0.2, δ2 = 0.1, umin = 0.2. The FLSs are similar to Example 1. Setting observer gains Li,k in the fuzzy switched state observer (19) as L1,1 = L2,1 = L3,1 = 2, and L1,2 = L2,2 = L3,2 = 4. In addition, by selecting Q1 = I and Q2 = I , and by solving (22), two positive-definite symmetric matrices are obtained as follows

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Fig. 9. Switching signal σ (t).

Fig. 10. The trajectories of y “blue line” and yr “red line”.

⎡

3.5000 −0.5000 2.7500 P1 = ⎣ −0.5000 −2.75000 −0.5000

⎤ −2.7500 −0.5000 ⎦ 3.0000

⎡

⎤ 2.1667 −0.5000 −1.5417 and P2 = ⎣ −0.5000 1.5417 −0.5000 ⎦ . −1.5417 −0.5000 1.9167

Choosing the design parameters in virtual controllers α1,k , α2,k , the control input of quantizer uk , and the parameters adaptive laws of θ1,k , θ2,k , θ3,k (k = 1, 2) as ci,1 = 2, ci,2 = 3, τi,k = 0.01, i = 1, 2, 3; k = 1, 2. The initial conditions are chosen as x1 (0) = 0.5, x2 (0) = −0.5, x3 (0) = 0.5, xˆ1 (0) = 0.1, xˆ2 (0) = −0.1, xˆ3 (0) = 0.1, and the others initial values are chosen zeros. The average dwell time is chosen as τa = 8.889, and let μ = 3.5, then, we can obtain τa = 8.889 > ln 3.5/0.15. Thus, the adaptive fuzzy output-feedback control problem of the resulting closed-loop system (70) is solvable under every switching signal k ∈ {1, 2}. The simulation results are shown in Figs. 10–16, where Fig. 10 expresses the trajectories of the system output y and tracking signal yr ; Figs. 11–13 show the trajectories of xi (i = 1, 2, 3) and

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Fig. 11. The trajectories of x1 “blue line” and xˆ1 “red line”.

Fig. 12. The trajectories of x2 “blue line” and xˆ2 “red line”.

their estimates xˆi , respectively; Fig. 14 and Fig. 15 show the control input uk and the quantized input signal qk (uk ), k = 1, 2; Fig. 16 shows the switching signal σ (t). From the above simulation results, it is clear that even though the switched nonlinear systems under study contain the uncertain nonlinearities, hysteretic quantized input and without requiring the states being available for measurement, the proposed adaptive fuzzy output feedback control scheme is able to guarantee the stability of the switched control systems and the tracking and observer errors can be made as small as possible by appropriate choice of the design parameters. 6. Conclusions This paper has studied the tracking control design problem for a class of switched nonlinear systems in strictfeedback form. The considered switched systems have completely unknown nonlinear functions, hysteretic quantized

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Fig. 13. The trajectories of x3 “blue line” and xˆ3 “red line”.

Fig. 14. The trajectories of u1 “blue line” and q1 (u1 ) “red line”.

input and without direct requirement of the states measurement. The hysteretic quantized input is decomposed by two bounded nonlinear functions, fuzzy logic systems are utilized to model the switched nonlinear systems and a switched fuzzy state observer has been established for estimating the unmeasured states. A robust adaptive fuzzy output feedback control scheme has been constructed via backstepping design for each subsystem. The stability of the whole switched control system has been proved by the Lyapunov function theory and the average dwell time method. The proposed adaptive fuzzy control method has not only solved the adaptive backstepping tracking control design problem of switched strict feedback nonlinear system with unmeasured states, but also solved the hysteretic quantized input problem. Therefore, the proposed adaptive fuzzy control method in this paper has extended the results of the previous literature. Further research will concentrate on adaptive fuzzy control for MIMO or large-scale switched nonlinear systems based on this paper.

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Fig. 15. The trajectories of u2 “blue line” and q2 (u2 ) “red line”.

Fig. 16. Switching signal σ (t).

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