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Observed-Based Adaptive Fuzzy Tracking Control for Switched Nonlinear Systems With Dead-Zone Shaocheng Tong, Shuai Sui, and Yongming Li

Abstract—In this paper, the problem of adaptive fuzzy outputfeedback control is investigated for a class of uncertain switched nonlinear systems in strict-feedback form. The considered switched systems contain unknown nonlinearities, dead-zone, and immeasurable states. Fuzzy logic systems are utilized to approximate the unknown nonlinear functions, a switched fuzzy state observer is designed and thus the immeasurable states are obtained by it. By applying the adaptive backstepping design principle and the average dwell time method, an adaptive fuzzy output-feedback tracking control approach is developed. It is proved that the proposed control approach 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. Index Terms—Average dwell time, backstepping technique, fuzzy adaptive control, fuzzy logic systems, switched nonlinear systems.

I. I NTRODUCTION N THE past decades, many fuzzy or neural adaptive control design problem have been studied for uncertain nonlinear systems in strict-feedback form by combining the adaptive backstepping design technique with fuzzy-logicsystems or neural-networks [1]–[20]. Among them, the works in [2]–[12] consider adaptive fuzzy or neural state feedback control design problem for single-input and single-output (SISO) nonlinear systems, multiple-input, and multiple-output (MIMO) nonlinear systems, respectively. References [13]– [20] investigated adaptive fuzzy or neural output-feedback control design problem for the SISO or MIMO nonlinear systems in which the states are not available for measurement, while the works [12], [19], [20] study robust adaptive fuzzy or neural state feedback (output-feedback) control design problems for the SISO or MIMO nonlinear systems with deadzone input. The above mentioned adaptive fuzzy or neural control approaches can provide an effective methodology to

I

Manuscript received November 7, 2014; revised December 21, 2014; accepted December 22, 2014. This work was supported by the National Natural Science Foundation of China under Grant 61374113 and Grant 61203008. This paper was recommended by Associate Editor M. J. Er. The authors are with the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou 121001, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2014.2386912

control those uncertain nonlinear systems without satisfying the matching condition. Nevertheless, they need not require that the nonlinear functions included in the controlled systems are known or can be linearly parameterized. However, the above mentioned results are only suitable for the nonswitched nonlinear systems in strict-feedback form, instead of the nonswitched nonlinear systems. As stated in [21]–[26], 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. In fact, many real-word systems, such as mechanical systems, power electronic, automotive industry, air traffic control, etc., can be expressed as switched systems. Thus the control design and stability on the switched systems have attracted many researchers a great interest. Recently, some backstepping control design methods have been proposed for several classes of switched nonlinear systems [27]–[34]. Two state feedback control approaches in [27]–[29] have investigated based on the common Lyapunov function method for a class of switched nonlinear systems, but the nonlinear functions of controlled systems in [27]–[29] are required to be known functions. In [30], an adaptive neural network stabilizer and a switching law have been designed for a class of switched nonlinear systems with triangular structure, in which the neural networks are employed to approximate the unknown nonlinear functions in controlled systems. Two adaptive neural control schemes have been proposed for a class of switched nonlinear systems in [31] and [32], and an admissible switching signal with average dwell-time technique has been given. In [33], an adaptive neural network feedback control scheme of nonlinear switched impulsive systems has been given under all admissible switched strategy. The neural networks is used to compensate for the nonlinear uncertainties of switched impulsive systems. In [34], an adaptive neural network control method has been presented 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, the above mentioned adaptive control approaches are limited to the uncertain nonlinear systems whose states are available for feedback, they can thus not control those switched nonlinear systems with immeasurable states. In addition, in the control design, the aforementioned adaptive control schemes do not consider the dead-zone input effect on the control performance. To the best of our knowledge, to date now, there are not results on

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adaptive fuzzy output feedback control available for uncertain switched nonlinear systems with immeasurable states and unknown dead-zone. This paper investigates the adaptive fuzzy control design problem for a class of uncertain switched nonlinear systems with unmeasured states and unknown dead-zone, and proposed an adaptive fuzzy output control design method. Fuzzy logic systems are first utilized to approximate the unknown nonlinear functions, and a fuzzy switched state observer is designed and thus the immeasurable states are obtained via it. By using the parameters of the dead-zone 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 dwell time method. Compared with the existing literature, the main contributions of the proposed control scheme are summarized as follows. 1) This paper first proposed an adaptive fuzzy tracking output feedback control method for a class of switched nonlinear systems in strict-feedback form based on the average dwell time method. The proposed adaptive control method has solved the state unmeasured problem via designing fuzzy switched state observer. Although, References [30]–[34] also addressed the adaptive fuzzy or neural control design for switched nonlinear strictfeedback systems. However, they all require that states must be available for measurement. 2) This paper first investigated the adaptive fuzzy outputfeedback control design problem for uncertain switched nonlinear system with unknown dead-zone. The proposed control scheme cannot only guarantee the stability of the whole switched control system, but also can attenuate the effect of the dead-zone on the control performance by constructing the dead-zone compensation. Note that our previous results in [19] and [20] only focus on the nonswitched nonlinear system. Thus, they cannot be applied to the switched nonlinear system. The remainder of this paper is organized as follows. The problem formulation and preliminaries are described in Section II. The switched fuzzy state observer design is given in Section III. The adaptive fuzzy control design and stability analysis are presented in Section IV. The simulation example is given in Section V. Finally, the conclusion is drawn in Section VI.

Consider the following uncertain switched nonlinear system in strict-feedback form: x˙ i = xi+1 + fi,σ (t) (xi ) + i,σ (t) (t) i = 1, . . . , n − 1 x˙ n = uσ (t) + fn,σ (t) (xn ) + n,σ (t) (t) (1)

where xi = [x1 , x2 , · · · , xi ∈ i = 1, 2, . . . , n(x = xn ) are the states, y ∈  is the output. 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. ]T

i ,

In (2), vσ (t) ∈  is the input to the dead zone; mσ (t) stands for the slope of the dead-zone characteristic; br,σ (t) and bl,σ (t) represent the breakpoints of the input nonlinearity. In this paper, it is assumed that mσ (t) is an unknown constant and only the output y is available for measurement. Remark 1: If for t ∈ [0, ∞), the switching signal σ (t) = k, then (1) represents a class of nonswitched nonlinear systems in strict-feedback form studied extensively (see [13]–[16], [19]). If the functions fi,σ (t) (xi ) are known or unknown, the states are available for measurement and without considering the dead-zone in (1), then the switched system (1) is the plants investigated in [27]–[34]. According to [35], we can express the dead-zone (2) as the following form: uσ (t) = mσ (t) vσ (t) (t) + hσ (t) (t)

(3)

where

⎧ −mσ (t) br,σ (t) , vσ (t) ≥ br,σ (t) ⎪ ⎪ ⎪ ⎨ −mσ (t) vσ (t) (t) hσ (t) (t) = −bl,σ (t) < vσ (t) < br,σ (t) ⎪ ⎪ ⎪ ⎩ mσ (t) bl,σ (t) , vσ (t) ≤ −bl,σ (t) .

(4)

From (4), we have   hσ (t)  ≤ h∗ , h∗ = max {mσ (t) bl,σ (t) , mσ (t) br,σ (t) }. (5) σ (t) σ (t) σ (t)∈M

II. P ROBLEM F ORMULATIONS AND P RELIMINARIES

y = x1

Moreover, σ (t) = k implies that the kth subsystem is active. fi,k (xi ),(i = 1, 2, . . . , n, k ∈ M) are unknown smooth nonlinear functions. i,k (t) (i = 1, 2, . . . , n, k ∈ M) are the dynamic dis¯ i,k with  ¯ i,k being known turbances and satisfy i,k (t) ≤  constants. 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 [22], [25]–[26]. uσ (t) ∈  is the output of the dead-zone, which is described by ⎧ mσ (t) (vσ (t) − br,σ (t) ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if vσ (t) ≥ br,σ (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 0, if  uσ (t) = Dσ (t) (vσ (t) ) = (2) −bl,σ (t) < vσ (t) < br,σ (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ mσ (t) (vσ (t) + bl,σ (t) ) ⎪ ⎪ ⎪ ⎩ if vσ (t) ≤ −bl,σ (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 subsystem is active. In this paper, we assume kj = kj+1 for all j. Lemma 1 [24], [25]: A switched nonlinear 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 , ∀T ≥ t ≥ 0 τa

(6)

where Nσ (T, t) is the number of switches occurring in the interval [t, T).

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Lemma 2 [36]: Let f (x) be a continuous function defined on a compact set . Then for any constant ε > 0, there exists a fuzzy logic system fˆ (x|θ ) = θ T ϕ(x) such as     (7) sup  f (x) − θ ∗ T ϕ(x) ≤ ε x∈

ϕ T (x)

where =  N =n [ϕ1 (x), . . . , ϕN (x)] and ϕl n μ (x )/ ( μ (x )) are fuzzy basis funcl l i i i=1 Fi l=1 i=1 Fi tions. μFl (xi ) is a fuzzy membership function of the inference i antecedent variable xi in IF-then rule, which is usually chosen as Gaussian type-function. θ = [θ1 , θ2 , . . . , θN ]T is the adaptive parameter vector, and θi is the inference consequent variable corresponding to the ith IF-then rule. The optimal parameter vector θ ∗ for θ is a quantity only for analytical purposes, which is typically chosen as the value of θ minimizing f (x) − θ ∗T ϕ(x) over , that is

  ∗ ∗T   (8) θ = arg minn sup f (x) − θ ϕ(x) ≤ ε . θ∈R

By substituting (9) into (1), the switched system (1) can be expressed as   ∗ x˙ i = xi+1 + fˆi,σ (t) xˆ i |θi,σ (t) + εi,σ (t) + i,σ (t) (t) i = 1, . . . , n − 1

(10)   ∗ x˙ n = uσ (t) + fˆn,σ (t) xˆ n |θn,σ + ε +  (t) n,σ (t) n,σ (t) (t) y = x1 . Rewrite (10) as x˙ = Aσ (t) x + Lσ (t) y +

III. S WITCHED F UZZY S TATE O BSERVER D ESIGN Since the states x2 , x3 , . . . , xn−1 , and xn in the switched nonlinear system (1) are not available for feedback, this section gives the design of a switched fuzzy state observer to estimate the system states. Note that the functions fi,σ (t) (xi ) are unknown. Thus the fuzzy logic systems are utilized to approximate them. By Lemma 2, we assume that   T fi,σ (t) xi |θi,σ (t) = θi,σ (t) ϕi,σ (t) xi    fˆi,σ (t) xˆ θi,σ (t) = θ T ϕi,σ (t) xˆ i,σ (t)

i

i

n  i=1

 ∗T ϕ ˆi Bi θi,σ (t) i,σ (t) x (11)

+ εσ (t) + σ (t) + Buσ (t) y = Cx ⎤

⎡

where Aσ (t)

x∈

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) (xi ), unmeasured states xi , i = 2, . . . , n and unknown dead-zone.

3

σ (t)

=

−l1,σ (t) ⎢ .. = ⎣ .

⎤ l1,σ (t) ⎥ ⎢ = ⎣ ... ⎦ ⎡

⎥ In−1 ⎦, Lσ (t) −ln,σ⎤(t) 0 . . . 0 ln,σ (t) ⎡ 1,σ (t) (t) ⎥ ⎢ .. ⎦, C = [1, . . . , 0], εσ (t) = ⎣ .

n,σ (t) (t) [ε1,σ (t) , ε2,σ (t) . . . , εn,σ (t) ]T , Bi = [0 · · · 1 · · · 0]T B = [0, . . . , 1]. To estimate the states of the system, a switched fuzzy state observer is designed as x˙ˆ = Aσ (t) xˆ + Lσ (t) y +

n 

  Bi fˆi,σ (t) xˆ i θi,σ (t) + Buσ (t)

i=1

yˆ = Cˆx.

(12)

Define observation error vector e as e = [e1 , e2 , . . . , en ]T = x − xˆ .

(13)

From (11)–(13), the dynamics of the observer error is e˙ = Aσ (t) e +

n 

 T Bi θ˜i,σ ˆ i + εσ (t) + σ (t) (t) ϕi,σ (t) x

(14)

i=1

where xˆ i = [ˆx1 , xˆ 2 , . . . , xˆ i ]T are the actual estimations of xi = [x1 , x2 , . . . , xi ]T , i = 1, 2, . . . , n. According to definition of ∗ the optimal parameter vectors θi,σ (t) in [36]–[39], let

∗ θi,σ (t)

= arg min

θi,σ (t) ∈i,σ (t)

    fˆi,σ (t) xˆ θi, σ (t) sup i (xi ׈xi )∈U1i ×Ui2    − fi,σ (t) xi  , 1 ≤ i ≤ n

where i,σ (t) , U1i , and U2i are bounded compact regions for θi,σ (t) , xi , and xˆ i , respectively. In addition, the approximation error εi,σ (t) is defined as    ∗ fi,σ (t) xi = fˆi,σ (t) xˆ i |θi,σ (9) (t) + εi,σ (t) , 1 ≤ i ≤ n ∗ ∗ where εi,σ (t) satisfies |εi,σ (t) | ≤ εi,σ (t) , with εi,σ (t) being an unknown constant.

∗ where θ˜i,σ (t) = θi,σ (t) − θi,σ (t) is the adaptive parameter vector error. Choose vector Lσ (t) such that matrix Aσ (t) is a strict Hurwitz matrix, therefore, for any a given matrix Qσ (t) = QTσ (t) > 0, there exists a positive definite matrix Pσ (t) = PTσ (t) > 0 such that

ATσ (t) Pσ (t) + Pσ (t) Aσ (t) = −Qσ (t) .

(15)

Remark 2: If t ∈ [tj , tj+1 ), and σ (t) = k, that is, the kth subsystem is active and the remaining subsystems are inactive. = 0, the dynamics of the In addition, assuming that i,k (t) T ϕ (ˆ observer error becomes e˙ = Ak e + ni=1 Bi θ˜i,k i,k xi ) + εk . For this case, choose the Lyapunov function as V0,k = eT Pk e, then from (14) and (15), we can obtain that V˙ 0,k ≤ −λ0,k e 2 +

n  j=1

 2 T ˜ θ˜j,k θj,k + εk∗ 

(16)

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with λ0,k = λmin (Qk ) − (n + 1) Pk 2 and εk∗ = ∗ , ε ∗ . . . , ε ∗ ]T . Since the fuzzy logic system fˆ (ˆ [ε1,k i,k xi |θi,k ) 2,k n,k can well approximate the unknown function fi,k (xi ), εk∗ is bounded. On the otherhand, from (16), it is clear that if T θ˜ λ0,k > 0 and the term nj=1 θ˜j,k j,k is bounded, then the state observer (12) is asymptotically stable and the observer errors can be made small. Thus, the designed state observer (12) is reasonable. Next section will how to design an adaptive  study T θ˜ control scheme to ensure nj=1 θ˜j,k j,k bounded.

3 2 1 2 1 1 ∗2 z1 + z2 + e 2 + ε1,k 2 2 2 2  1 2 1 2  ¯ . |z1 | 1,k (t) ≤ z1 +  2 2 1,k Substituting (21)–(24) into (20) results in z1 (z2 + ε1,k + e2 ) ≤

1 V˙ 1,k ≤ −λ1,k e 2 + z22 2   T + z1 α1,k + 2z1 + θ1,k ϕ1,k xˆ 1 − y˙ r   n   1 T T ˜ ϕ1,k xˆ 1 z1 − θ˙1,k + Pk 2 θ˜j,k θj,k + θ˜1,k δ1,k + C1,k

To realize the control objective, this section will give the adaptive fuzzy output-feedback control design procedure based on the backstepping design technique and the switched fuzzy 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. In the sequel, the n-step adaptive backstepping design procedures [38], [40], [41] will be developed for the kth activated subsystem. Let z1 = y − yr , zi = xˆ i − αi−1,k

(17)

where αi−1,k is intermediate control function. Step 1: Since x2 = xˆ 2 + e2 , we obtain  z˙1 = x2 + f1,k x1 + 1,k (t) − y˙ r

where  λ1,k = λmin (Qk ) − n − (5/2), and C1,k = Pk 2 εk∗ 2 + n ∗2 2 ¯2 ¯2 Pk i=1 i,k + (1/2)ε1,k + (1/2)1,k . Design the intermediate control function α1,k , the parameter adaptation function θ1,k as  T α1,k = −β1,k z1 − 2z1 − θ1,k ϕ1,k xˆ 1 + y˙ r (26) ˙θ1,k = δ1,k ϕ1,k (ˆx1 )z1 − τ1,k θ1,k (27) where β1,k > 0 and τ1,k > 0 are design parameters. From (26) and (27), it follows that: 1 τ1,k T θ˜ θ1,k V˙ 1,k ≤ −λ1,k e 2 + z22 − β1,k z21 + 2 δ1,k 1,k n  T ˜ + Pk 2 θ˜j,k θj,k + C1,k . (28) Step i(2 ≤ i ≤ n): Since zi = xˆ i − αi−1,k , similar to step 1, the derivative time of zi along with (12) and (17) is

(18)

Choose the Lyapunov function candidate as 1 1 T θ˜ θ˜1,k V1,k = eT Pk e + z21 + 2 2δ1,k 1,k

(19)

z˙i = x˙ˆ i − α˙ i−1,k i−1   ∂αi−1,k ˙ T = zi+1 + αi,k + li,k e1 + θi,k ϕi,k xˆ i − xˆ j,k ∂ xˆ j,k j=1

V˙ 1,k ≤ −λmin (Qk ) e 2

n   T + 2eT Pk Bi θ˜i,k ϕi,k (ˆxi ) + εk + k

1 T ˙ θ˜ θ˜ 1,k . δ1,k 1,k

(20) the

fact

i=1

(21)

i=1

T Bi θ˜i,k ϕi,k (ˆxi ) ≤ n e 2 + Pk 2

n  i=1

T ˜ θ˜i,k θi,k

(22)

(29)

where i−1 i−1    ∂αi−1,k ˙ ∂αi−1,k T Hi,k = li,k e1 + θi,k θ˙j,k ϕi,k xˆ i − xˆ j − ∂ xˆ j ∂θj,k j=1

n   2 ¯ 2i,k  2eT Pk (k + εk ) ≤ 2 e 2 + Pk 2 εk∗  + Pk 2

2eT Pk

−

∂αi−1,k = zi+1 + αi,k + Hi,k − ∂x1    T × θ˜1,k ϕ1,k xˆ 1 + ε1,k + e2 + 1,k (t)

i=1

n 

i−1 

+ ε1,k + 1,k (t)

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

using

i−1 

∂αi−1,k ∂αi−1,k ( j) θ˙j,k − y ( j−1) r ∂θj,k j=1 j=1 ∂yr   ∂αi−1,k  T T xˆ 2 + e2 + θ1,k − ϕ1,k xˆ 1 + θ˜1,k ϕ1,k xˆ 1 ∂x1 

where δ1,k > 0 is a design parameter. From (1), (14), and (17)–(19), the time derivative of V1,k satisfies

and

(25)

j=1

 T = z2 + α1,k + e2 + ε1,k + θ1,k ϕ1,k xˆ 1  T + θ˜1,k ϕ1,k xˆ 1 + 1,k (t) − y˙ r .

By completely the squares T (ˆ ϕi,k xi )ϕi,k (ˆxi ) ≤ 1, we can obtain

(24)

j=1

IV. A DAPTIVE F UZZY C ONTROL D ESIGN AND S TABILITY A NALYSIS

+

(23)

−

i−1  ∂αi−1,k

y( j) ( j−1) r ∂y r j=1

−

j=1

  ∂αi−1,k  T ϕ1,k xˆ 1 . xˆ 2 + θ1,k ∂x1

Consider the following Lyapunov function candidate: 1 1 T Vi,k = Vi−1,k + z2i + θ˜ θ˜i,k 2 2δi,k i,k

(30)

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where δi,k > 0 is a design parameter. From (29) and (30), we can obtain 1 T ˙ θ˜ θ˜ i,k V˙ i,k ≤ V˙ i−1,k + zi z˙i + δi,k i,k ≤ −λi−1,k e 2 + Ci−1,k  + zi zi+1 + αi,k + Hi,k  ∂αi−1,k  T θ˜1,k ϕ1,k (ˆx1 ) + e2 + 1,k (t) + ε1,k − ∂x1    1 T zi ϕi,k xˆ i − θ˙i,k + θ˜i,k δi,k i−1   T − βj,k z2j − zi θ˜i,k ϕi,k xˆ i j=1

+

i−1  τj,k j=1

+

1 2

δj,k

i−1  j=2

T θ˜j,k θj,k + Pk 2

n 

T ˜ θ˜j,k θj,k

j=1

1 i−2 T T ˜ θ˜j,k θj,k + z2i + θ˜ θ˜1,k . 2 2 1,k

(31)

By completing the squares, we have ∂αi−1,k (εi,k + e2 + 1,k (t)) ∂x1   1 2 3 ∂αi−1,k 2 2 1 ∗2 1 2 ¯ zi ≤ εi,k + e + 1,k + 2 2 2 2 ∂x1 (32) 2  ∂αi−1,k T 1 T 1 ∂αi−1,k θ˜ ϕ1,k (ˆx1 ) ≤ θ˜1,k θ˜1,k + −zi z2i (33) ∂x1 1,k 2 2 ∂x1 1 1 T T θ˜i,k . zi (zi+1 − θ˜i,k ϕi,k (ˆxi )) ≤ z2i + z2i+1 + θ˜i,k (34) 2 2 Substituting (32)–(34) into (31) yields −zi

V˙ i,k ≤ −λi,k e 2 + Ci,k

   3 ∂αi−1,k 2 + zi αi,k + zi + 2 zi + Hi,k 2 ∂x1    i−1  1 2 1 T ˜ ˙ + zi+1 + θi,k zi ϕi,k xˆ i − βj,k z2j θi,k − 2 δi,k j=1

+

i−1  j=1

τj,k T θ˜ θj,k + Pk 2 δj,k j,k

n 

T ˜ θ˜j,k θj,k

j=1

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

+

(35)

j=2

∗2 + 1  ¯2 where λi,k = λi−1,k − 12 , and Ci,k = Ci−1,k + 12 εi,k 2 1,k . Design the intermediate control function αi,k , the parameter adaptation function θi,k as   3 ∂αi−1,k 2 αi,k = −βi,k zi − zi − 2 zi − Hi,k (36) 2 ∂x1  θ˙i,k = δi,k zi ϕi,k xˆ i − τi,k θi,k (37)

where βi,k > 0 and τi,k > 0 are design parameters.

5

From (36) and (37), it follows that: V˙ i,k ≤ −λi,k e 2 −

i  j=1

+

i  j=1

+

1 2

1 βj,k z2j + z2i+1 2

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

j=1

i 

T ˜ θ˜j,k θj,k +

j=2

i−1 T θ˜ θ˜1,k + Ci,k . 2 1,k

(38)

Step n: In the final design step, the dead-zone input vk will be obtained. Define the change of coordinate as zn = xˆ n − αn−1,k .

(39)

The time derivative of zn is  T ϕn,k xˆ n + Hn,k z˙n = xˆ˙ n − α˙ n−1,k = uk + θn,k   ∂αn−1,k  T θ˜1,k ϕ1,k xˆ 1 + ε1,k + e2 + 1,k (t) − ∂x1 ˆ k vk (t) + hk + Hn,k =m ˜ k vk (t) + m   ∂αn−1,k  T − θ˜1,k ϕ1,k xˆ 1 + ε1,k + e2 + 1,k (t) ∂x1 where n−1   ∂αn−1,k ˙ T ϕn,k xˆ n − Hn,k = ln,k e1 + θn,k xˆ j ∂ xˆ j

(40)

j=1

−

n−1  j=1

∂αn−1,k θ˙j,k − ∂θj,k

n−1 

∂αn−1,k

y( j) ( j−1) r j=1 ∂yr

  ∂αn−1,k  T − ϕ1,k xˆ 1 . xˆ 2 + θ1,k ∂x1 Consider the following Lyapunov function candidate: 1 1 T 1 2 1 ˜2 Vk = Vn−1,k + z2n + h (41) m ˜ + θ˜ θ˜n,k + 2 2δn,k n,k 2κk k 2υk k where δn,k > 0, κk > 0, and υk > 0 are the design parameters; ˆ k and h˜ k = h∗k − hˆ k are parameter errors. m ˜ k = mk − m From (40) and (41), we can obtain V˙ k ≤ V˙ n−1,k + zn z˙n +

1

T ˙˜ θ n,k + θ˜n,k

1 ˙˜ k + 1 h˜ h˙˜ m ˜ km κk υk k k

δn,k ≤ −λn−1,k e 2 + Cn−1,k   T + zn m ϕn,k xˆ n + Hn,k ˆ k vk (t) + θn,k

  ∂αn−1,k  T θ˜1,k ϕ1,k xˆ 1 + ε1,k + e2 + 1,k (t) − ∂x1   1 T + θ˜n,k zn ϕn,k (ˆxn ) − θ˙n + |zn | hˆ k δn,k    n−1  1 ˙ T + h˜ k |zn | − hˆ k − βj,k z2j − zn θ˜n,k ϕn,k xˆ n υk j=1

  n−1  τj,k T 1 ˙ n−2 T θ˜ θj,k ˆk + θ˜1,k θ˜1,k + +m ˜ k zn vk (t) − m κk 2 δj,k j,k j=1

+ Pk 2

n  j=1

T ˜ θ˜j,k θj,k +

1 2

n−1  j=2

1 T ˜ θ˜j,k θj,k + z2n . 2

(42)

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Design the input of the dead-zone vk and parameter tation functions as follows: !   1 ∂αn−1,k 2 vk = zn −βn,k zn − zn − 2 m ˆk ∂x1  − sgn(zn )hˆ k − Hn,k  θ˙n,k = δn,k zn ϕn,k xˆ n − τn,k θn,k ⎧ κk zn vk − ηk m ˆk ⎪ ⎪     ⎪ ⎪ ⎪ ˆ k  < Mmk or m ˆ k  = Mmk if m ⎪ ⎪ ⎨ ˙ˆ k = and m ˜ k zn vk (t) ≤ 0 m ⎪ ⎪ ⎪ κ m ˆ z v m ˆ ⎪ κk zn vk − ηk m ˆ k − k k n 2k k ⎪ ⎪ m ˆ ⎪ | | k   ⎩ ˆ k  = Mmk and m if m ˜ k zn vk (t) > 0   ˙ˆ hk = υk zn  − ρk hˆ k

adap-

(43) (44)

(45)

(46)

where βn,k > 0, τn,k > 0, ηk > 0, ρk > 0, and Mmk > 0 are design parameters. Similar to the derivations in step i, we have V˙ k ≤ −λn,k e 2 −

n 

βj,k z2j +

j=1

+ Pk 2

n 

T ˜ θ˜j,k θj,k +

j=1

+

1 2

n  τj,k

j=1 n 

δj,k

T θj,k θ˜j,k

j=2

(47)

∗2 + where λn,k = λn−1,k − (1/2), Cn,k = Cn−1,k + (1/2)εi,k 2 ¯ (1/2)1,k . By completing the square

1 T 1 ∗  T 2 θ˜i,k + θi,k θ˜i,k θi,k ≤ − θ˜i,k 2 2 1 2 1 2 ˜ + m m ˜ km ˆk ≤ − m 2 k 2 k 1 1 h˜ k hˆ k ≤ − h˜ 2k + h2k . 2 2

(48)

V˙ k ≤ −λn,k e 2 −

Therefore, (51) can be rewritten as V˙ k ≤ −CVk + D.

(49) (50)

Next, we will give the stability proof of the closed-loop system based on the above controller design and Lemma 1. Theorem 1: For switched uncertain nonlinear system (1), for every switching signal σ (t), if the average dwell time satisfies τa > (ln μ/C), then the controller (43) and fuzzy state observer (12), together with the intermediate control functions (26) and (36), parameter adaptation functions (27), (37), and (44)–(46), 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. 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 (52), on each interval [tj , tj+1 ), one has (53)

As the same proof in [22] and [42], 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 ⎥ ⎢  DeCt dt⎦. (54) ≤ μ ⎣ W tj + Pick an arbitrary T > t0 = 0. Iterating the inequality (54) from j = 0 to j = Nσ (T, 0) − 1, we obtain that %T − DeCt dt W(T ) ≤ W(tNσ (T,0) ) + tNσ (T,0) tN% σ (T,0)

βi,k z2i

≤ μ[W(tNσ (T,0)−1 ) +

i=1

 n  1  τi,k 2 T ˜ θi,k − − 2 Pk − 1 θ˜i,k 2 δi,k i=2   1 τ1,k T ˜ − − n + 1 − 2 Pk 2 θ˜1,k θ1,k 2 δ1,k ηk 2 ρk ˜ 2 h + Dk − m ˜ − 2κk k 2υk k

(52)

tj

Substituting the above inequalities into (47) yields n 

j = 1, . . . , n.

˙ W(t) = CeCt Vσ (t) (x(t)) + eCt V˙ σ (t) (x(t))   ≤ DeCt , t ∈ tj , tj+1 .

T ˜ θ˜j,k θj,k + Cn,k

n−1 T ηk ρk θ˜1,k θ˜1,k + m ˜ km ˆ k + h˜ k hˆ k 2 κk υk

(ρk /υk ) > 0(k ∈ M), C = min Ck , D = max Dk , and k∈M k∈M " # Ck = min λn,k λmax (Pk ), 2βj,k , τj,k − 2 Pk 2 δj,k − δj,k $ τ1,k − nδ1,k + δ1,k − 2 Pk 2 δ1,k , ηk , ρk

+ μ−1 ≤ ··· (51)

 ∗T θ ∗ + (η /2κ )m2 + where Dk = Cn,k + ni=1 (τi,k /2δi,k )θi,k k k i,k k 2 (ρk /2υk )hk . Let βi,k > 0(i = 1, . . . , n), (τ1,k /δ1,k ) − n + 1 − 2 Pk 2 (τi,k /δi,k ) − 2 Pk 2 − 1 > 0(i = 2, . . . , n), (ηk /κk ) > 0

tNσ (T,0)−1

%T

DeCt dt]

tNσ (T,0)

⎡

Nσ (T,0) ⎢

≤μ

DeCt dt

⎣W(0) +

Nσ (T,0)−1 

−j

μ

j=0

+ μ−Nσ (T,0)

%tj+1 DeCt dt tj

%T

tNσ (T,0)

⎤

⎥ DeCt dtS⎦. (55)

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Since τa > ln μ/C, for any δ ∈ (0, C − ln μ/τa ), one has τa > ln μ/(C − δ). By (6), it holds that Nσ (T, t) ≤ N0 + ((C − δ)(T − t)/ln μ),

∀T ≥ t ≥ 0.

In addition, it is clear that Nσ (T, 0) − j Nσ (T, tj+1 ), j = 0, 1, . . . , Nσ (T, 0), one has μNσ (T,0)−j ≤ μ1+N0 e(C−δ)(T−tj+1 ).

≤

(56) 1 + (57)

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

(58)

tj

It then follows from (57) and (58), one has: %T − Nσ (T,0) 1+N0 (C−δ)T W(T ) ≤ μ W(0) + μ e Deδt dt.

(59)

0

According to [34], there exist two class κ functions α(|x|) ¯ and α(|x|), ¯ which satisfy α(|x|) ≤ Vk (x) ≤ α(|x|). It indicates that α( x(T) ) ≤ Vσ (T − ) (x(T − )) 

≤ eN0 ln μ e

ln μ τa −C

 T

α( x(0) ) ¯  D + μ1+N0 1 − e−δT   δ

≤ eN0 ln μ e

ln μ τa −C

T

α( x(0) ) ¯

1+N0 D

, ∀T > 0. (60) δ # We conclude that, by (60) and δ > 0, if τa > (ln μ C), then for bounded initial conditions, e, zi , θ˜i,k , m ˜ k , and h˜ k , i = ∗ ∗ 1, 2, . . . , n are bounded. Since θi,k , mk and hi , i = 1, 2, . . . , n, ˆ k , and hˆ i , i = 1, 2, . . . , n, are bounded. are constants, θi,k , m Further, it is easy to obtain that xˆ i , xi , and uσ (t) 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 τa > (ln μ/C). On the other hand, for any given constant ξ > 0 the inequality μ1+N0 (D/δ) ≤ (1/2)ξ 2 can be obtained by appropriately choosing the matrices Qk , k ∈ M, and the design parameters βi and τi,k , and choosing δi,k sufficiently large. In addition, from (60), it follows that:   ln μ 1 2 D N0 ln μ τa −C T e α( x(0) ) ¯ + μ1+N0 1 − e−δT z ≤e 2 1 δ # which, together with τa > (ln μ C), implies that D lim z21 (t) = lim |y(t) − yr (t)|2 ≤ 2μ1+N0 ≤ ξ 2 . t→∞ t→∞ δ This completes the proof of Theorem 1. Remark 3: Note that from (59), we can only conclude that the state observer error and tracking   error √ √ satisfying that e ≤ max 2D/(Cλmin (Pk )) and z1  ≤ 2D/C, we can+μ

k∈M

not conclude that the state observer error and tracking error converges to zero. However, According to [13]–[20], we can make both the state observer error and tracking error to be small by increasing the design parameters li,k , λmin (Qk ), βi,k and κk , δi,k , or decreasing τi,k and ηk (i = 1, . . . , n).

7

V. S IMULATION E XAMPLE In this section, a simulation example is given to illustrate the effectiveness of the proposed adaptive fuzzy output feedback control approach. Example: Consider the following switched uncertain nonlinear system: x˙ 1 = x2 + f1,σ (t) (x1 ) + 1,σ (t) (t) x˙ 2 = uσ (t) + f2,σ (t) (x2 ) + 2,σ (t) (t) y = x1

(61)

where f1,1 (x1 ) = −x1 e−0.5x1 , f2,1 (x1 , x2 ) = x1 sin(x22 ), 1,1 (t) = cos(t), 2,1 (t) = cos(2t), f1,2 (x1 ) = −x12 , f2,2 (x1 , x2 ) = x1 x22 , 1,2 (t) = cos(t2 ), and 2,1 (x) = sin(t). The given reference tracking signals is yr (t) = sin(0.5t). The parameters in the dead-zone model (2) are selected br,2 = 0.3, bl,1 = 0.5, bl,2 = 2.5, and br,1 = 0.1. The parameters m1 and m2 will be obtained via parameter adaptive laws (45). In the simulation studies, the IF-then rules are chosen as follows. 1 and x 1 , then y is G1 . ˆ 2 is F2,k 1) R1k : If xˆ 1 is F1,k k 1 and x 1 , then y is G2 . 2) R2k : If xˆ 1 is F2,k ˆ 2 is F2,k k 1 and x 1 , then y is G3 . ˆ 2 is F3,k 3) R3k : If xˆ 1 is F3,k k 1 and x 1 , then y is G4 . 4) R4k : If xˆ 1 is F4,k ˆ 2 is F4,k k 1 and x 1 , then y is G5 . ˆ 2 is F2,k 5) R5k : If xˆ 1 is F5,k k 1 = (NL), F 1 = (NL), Where fuzzy sets are chosen as F1,k 2,k 3 3 2 2 = (NS), F2,k = (NS), F1,k = (ZE), F2,k = (ZE), F1,k 5 5 4 4 = (PS), F2,k = (PS), F1,k = (PL), F2,k = (PL), F1,k k ∈ {1, 2}, which are defined over the interval [−2,2] for variables xˆ 1 and xˆ 2 , respectively. NL, NS, ZO, PS, and PL denote negative large, negative small, zero, positive small, positive large, respectively. Their center points are selected as −2, −1, 0, 1, 2, respectively. The corresponding fuzzy membership functions are given by

  (ˆx1 − 3 + l)2 μFl xˆ 1 = exp − 1,k 2

 2   xˆ 1 − 3 + l μFl xˆ 1 , xˆ 2 = exp − 1,k 2

 2  xˆ 2 − 3 + l × exp − 2 l = 1, . . . , 5, k ∈ {1, 2}. They are shown in Fig. 1. We obtain fuzzy basis functions as follows:  2 xˆ −3+l exp − ( 1 2 )   ϕ1,l,k xˆ 1 = 5 (xˆ 1 −3+n)2 exp − n=1 2   2 2 xˆ 1 −3+l) xˆ −3+l ( × exp − ( 2 2 ) exp − 2   ϕ2,l,k (ˆx1 , xˆ 2 ) = 5 (xˆ 1 −3+n)2 × exp − (xˆ 2 −3+n)2 exp − n=1 2 2 l = 1, . . . , 5, k ∈ {1, 2}.

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Fig. 1.

Fig. 3.

Fuzzy membership functions.

Trajectories of x1 (black line) and xˆ 1 (red line).

According to the intermediate function, the dead-zone input and parameters designs in the last section, vk , α1,k , and ˆ k , hˆ k , k ∈ {1, 2} are chosen as θ1,k , θ2,k , m  T ϕ1,k xˆ 1 + 0.5 cos(0.5t) α1,k = −z1 − 2z1 − θ1,k  θ˙1,k = 0.001ϕ1,k xˆ 1 z1 − 0.05θ1,k !   1 ∂αn−1,k 2 vk = z2 −20z2 − z2 − 2 m ˆk ∂x1 & − sgn(z2 )hˆ − H2,k k

Fig. 2.

θ˙2,k = 0.001z2 ϕ2,k (ˆx2 ) − 0.05θ2,k ⎧   ˆ k < 4 ˆ k , if m ⎪ ⎪ 0.01z2 − 0.5m ⎪   ⎪ ⎪ ⎪ ˆ k  = 4 and m or m ˜ k z2 vk ≤ 0 ⎨ m ˆ˙ k = 0.1m ˆ k z2 vk m ˆk ⎪ ˆk − ⎪ 0.01z2 − 0.5m 2 ⎪ m ˆ k| | ⎪   ⎪ ⎪ ⎩ ˆ k  = 4 and m ˜ k z2 vk > 0 if m

Trajectories of y (black line) and yr (red line).

The fuzzy logic systems can be expressed in the form

  ˙ hˆ k = 0.1 z2  − 0.5hˆ k .

5      T T fˆ1,k xˆ 1 θ1,k = θ1,k ϕ1,k xˆ 1 = θ1,j,k ϕ1,j,k xˆ 1 j=1

fˆ2,k



5   T T xˆ 2 θ2,k = θ2,k ϕ2,k (ˆx1 , xˆ 2 ) = θ2,j,k ϕ2,j,k (ˆx1 , xˆ 2 ) j=1

k ∈ {1, 2}. Setting the parameters l1,1 = 4, l2,1 = 12, l1,2 = 8, and l2,2 = 12, the fuzzy switched state observer (12) is    x˙ˆ 1 = xˆ 2 + fˆ1,k xˆ 1 θ1,k + l1,k x1 − xˆ 1    xˆ˙ 2 = uk + fˆ2,k xˆ 1 , xˆ 2 θ2,k + l2,k x1 − xˆ 1 yˆ = xˆ 1 ,

k ∈ {1, 2}.

(62)

In addition, in (15), by select Q1 = Q2 = 6I, we can obtain two positive-definite P1 =

symmetric matrices   4.8750 − 3 9.7500 − 3 , and P2 = −3 2.4062 . −3 1.8125

= The initial conditions are chosen as x1 (0) 0.1, x2 (0) = 0.1, xˆ 1 (0) = xˆ 2 (0) = 0, ˆ 1 (0) = m ˆ 2 (0) = 3.5 hˆ 1 (0) = hˆ 2 (0) = 0, m θ1,k (0) = [θ1,1,k (0), θ1,2,k (0), θ1,3,k (0), θ1,4,k (0), θ1,5,k (0)] = [−0.1, −0.3, −0.5, −0.7, −0.9], θ2,k (0) = [θ2,1,k (0), = [0.1, 0.3, 0.5, θ2,2,k (0), θ2,3,k (0), θ2,4,k (0), θ2,5,k (0)] 0.7, 0.9], k ∈ {1, 2}. The average dwell time is chosen as τa = 11.0294, and let μ = 1.2, then, we can obtain τa = 11.0294 > ln 1.2/0.02. Thus, the adaptive fuzzy output-feedback control problem of the resulting closed-loop system (61) is solvable under every switching signal k ∈ {1, 2}. The simulation results are shown in Figs. 2–10, where Fig. 2 expresses the trajectories of the system output y and tracking signal yr . Figs. 3 and 4 show the trajectories of xi , (i = 1, 2) and their estimates xˆ i , respectively. Fig. 5 shows the trajectories of uk ; Fig. 6 shows the switching

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Fig. 4.

Trajectories of x2 (black line) and xˆ 2 (red line).

Fig. 7.

 2 Trajectories of θ1,k  , k ∈ {1, 2}.

Fig. 5.

Trajectories of uk , k ∈ {1, 2}.

Fig. 8.

 2 Trajectories of θ2,k  , k ∈ {1, 2}.

Fig. 6.

Switching signal σ (t).

Fig. 9.

Trajectories of m ˆ k , k ∈ {1, 2}.

signals σ (t). Figs. 7 and 8 show the trajectories of θi,k 2 , (i = 1, 2, k = 1, 2). Figs. 9 and 10 show the trajectories of mk and hˆ k , (k = 1, 2), respectively.

9

To verify the effectiveness of the proposed control method, we select the average dwell time τa = 8.5, then τa = 8.5 < ln 1.2/0.02 (does not satisfy the condition of Theorem 1).

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VI. C ONCLUSION

Fig. 10.

Trajectories of hˆ k , k ∈ {1, 2}.

This paper has studied the tracking control design problem for a class of switched nonlinear systems in strict-feedback form. The considered switched systems have completely unknown nonlinear functions, unknown dead-zone and without direct requirement of the states measurement. 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 by utilizing the parameters of the dead-zone and in the framework of the backstepping design technique for each subsystem. The stability of the whole switched control system has been proved via Lyapunov function and the average dwell time method. The proposed adaptive fuzzy control method has not only solved the adaptive backstepping control design problem for a class of switched nonlinear system with unmeasured states, but also solved the dead-zone problem. Therefore, the proposed adaptive fuzzy control method of this paper has extended the results of the previous literature. Future research will concentrate on the adaptive fuzzy control methods for uncertain MIMO switched nonlinear systems with unknown control direction based on the results of this paper. R EFERENCES

Fig. 11.

Trajectories of y (black line) and yr (red line).

Fig. 12.

Trajectories of uk , k ∈ {1, 2}.

For this case, we apply the above control method to (61), the simulation results shown in Figs. 11 and 12 indicate that the control system is unstable.

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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TONG et al.: OBSERVED-BASED ADAPTIVE FUZZY TRACKING CONTROL FOR SWITCHED NONLINEAR SYSTEMS WITH DEAD-ZONE

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Shaocheng Tong received the B.S. degree in mathematics from Jinzhou Normal College, Jinzhou, China, the M.S. degree in fuzzy mathematics from Dalian Marine University, Dalian, China, and the Ph.D. degree in fuzzy control from the Northeastern University, Shenyang, China, in 1982, 1988, and 1997, respectively. He is currently a Professor with the Department of Basic Mathematics, Liaoning University of Technology, Jinzhou. His current research interests include fuzzy control theory, nonlinear adaptive control, and intelligent control.

Shuai Sui received the B.S. and M.S. degrees in applied mathematics from the Liaoning University of Technology, Jinzhou, China, in 2011 and 2014, respectively. He is currently an Assistant with the Department of Basic Mathematics, Liaoning University of Technology. His current research interests include fuzzy control, adaptive control, and stochastic control.

Yongming Li received the B.S. and M.S. degrees in applied mathematics from the Liaoning University of Technology, Jinzhou, China, in 2004 and 2007, respectively, and the Ph.D. degree in transportation information engineering and control from Dalian Maritime University, Dalian, China, in 2014. He is currently an Associate Professor with the Department of Basic Mathematics, Liaoning University of Technology. His current research interests include adaptive control, fuzzy control, and neural networks control for nonlinear systems.