Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer

Communicated by Jianbin Qiu

Accepted Manuscript

Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer Yang Cui, Huaguang Zhang, Qiuxia Qu, Chaomin Luo PII: DOI: Reference:

S0925-2312(17)30610-0 10.1016/j.neucom.2017.03.064 NEUCOM 18297

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

28 December 2016 7 March 2017 31 March 2017

Please cite this article as: Yang Cui, Huaguang Zhang, Qiuxia Qu, Chaomin Luo, Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer, Neurocomputing (2017), doi: 10.1016/j.neucom.2017.03.064

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Synthetic adaptive fuzzy tracking control for MIMO uncertain nonlinear systems with disturbance observer College of Information Science and Engineering, Northeastern University, Shenyang, 110819, China b

Department of Electrical and Computer Engineering, University of Detroit Mercy, Michigan, USA

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Yang Cuia , Huaguang Zhanga,∗ , Qiuxia Qua , Chaomin Luob

Abstract

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In this paper, a synthetic adaptive fuzzy tracking control method is studied for a class of multi-input multi-output (MIMO) uncertain nonlinear systems with time-varying disturbances. The unknown nonlinear functions are approximated by employing generalized fuzzy hyperbolic model. A synthetic adaptive fuzzy control is designed by dynamic surface control, serial-parallel estimation model and the disturbance observer. Then by using Lyapunov stability theory, it is guaranteed that all the variables of the closed-loop systems are semi-globally uniformly ultimately bounded (SGUUB). Finally, a satisfactory tracking performance with faster and higher accuracy can be obtained by adjusting the parameters appropriately. A practical example simulation can demonstrate the effectiveness and applicability of the proposed control approach.

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Keywords: Dynamic surface control (DSC), generalized fuzzy hyperbolic model (GFHM), nonlinear systems, serial-parallel estimation model, disturbance observer

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Corresponding author Email addresses: [email protected] (Yang Cuia ), [email protected] (Huaguang Zhanga,∗ ), [email protected] (Qiuxia Qua ), [email protected] (Chaomin Luob )

Preprint submitted to Neurocomputing

April 3, 2017

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1. Introduction

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Since the 20th century, industrial development is getting more and more rapidly, such as aircraft flight control system [1], autonomous underwater vehicle system [2]-[3], power system control [4], and marine vessels control [5]. Almost all the actual industrial system can be described as a nonlinear system. The establishment of the system model [6], the controller design [7][8] and stability analysis [9]-[10] are three important roles for the nonlinear systems, especially the controller design. As well known that the nonlinear uncertainties always exist in industrial systems, it is the difficulty of control design for the nonlinear systems. With the advent of fuzzy logic systems (FLSs) [11]-[17] and Neural network (NN) [19]-[21], the problem of nonlinear uncertainties can be solved. An arbitrary unknown nonlinear function can be approximated by FLSs or NN with any accuracy, it has been already confirmed. In the 90s, Kokotovic et al. [22] proposed a new kind of adaptive control scheme, i.e., adaptive backstepping control. The adaptive backstepping control showed great potential for improving the quality of transition process, especially in the field of aeronautics and astronautics. Due to the successful applied in aircraft and missiles, adaptive backstepping control approach for the uncertain nonlinear systems has got great development [23]-[29]. An adaptive neural decentralized control approach was proposed for a class of MIMO uncertain stochastic nonlinear strong interconnected systems [24]. However, for the traditional backstepping control method, the main shortcoming is that the control design is too complexity, that is “explosion of complexity” phenomenon. With the system order increasing, the complexity of the designed control grows drastically. The control design is too complicated on account of repeated derivations of nonlinear function. Nevertheless, the “explosion of complexity” phenomenon can be avoided by added low-order filter into traditional backstepping control method, i.e., dynamic surface control (DSC) [30]-[36]. It has been known that most of the control approaches were focus on tracking performance and stability of the systems. Few results were focus on the precision and speed of the tracking. For the designed control in [37]-[39], a serial-parallel estimation model was confirmed for the sake of the precision of the tracking. It is well known that the time-varying uncertain disturbance always inevitably exists in many practical industrial areas. For example, the unmanned aerial vehicles are affected by the ocean currents and wind, and 2

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maritime transport are affected by the waves and tides. For this time-varying disturbance problem, a general approach is to design a robust control making the system run normally. Nevertheless, the drawback of robust control may cause unsatisfactory tracking accuracy and energy waste. In [40], a composite fuzzy control was investigated for nonlinear systems with dead zone based on disturbance observer. In [41], a neural adaptive output feedback control method was studied for nonlinear systems with unmeasured states, external disturbances and unknown hysteresis by designing disturbance observer. So, an adaptive control by designing the disturbance observer is an ideal control method to eliminate above problem. Until now, it has been an open problem for designing a tracking controller guaranteeing a good tracking performance for a class of uncertain MIMO nonlinear systems with unknown time-varying disturbance. This is our research motivation. A fuzzy adaptive tracking control is designed for uncertain MIMO nonlinear systems with time-varying disturbance in this paper. The approximator GFHM is applied to approximate an unknown nonlinear function. The disturbance observer is designed by GFHM approximator and disturbance estimation. By designing the serial-parallel estimation model and prediction error, a synthetic fuzzy adaptive tracking control is designed on the strength of the backstepping DSC technique, which can obtain a better tracking performance. And all the signals of the closed-loop systems are bounded. The outputs of systems can track the given bounded signals by adjusting parameters appropriately. The major contributions are as follows.

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1. By introducing low-order filter in traditional backstepping control method, the “explosion of complexity” problem can be avoided which exists in [23]. 2. Comparing with existing control methods, a satisfactory tracking performance with faster and higher accuracy can be obtained by designing the serial-parallel estimation model, prediction error and disturbance observer. 3. The problem of fuzzy adaptive tracking control is investigated for MIMO nonlinear systems with time-delay disturbance. By constructing the disturbance observer, the problem of uncertain nonlinear disturbance can be solved.

The structure of this paper is organized as follows. Section 2 describes problem formulation and preliminaries. A synthetic fuzzy adaptive control 3

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2. Problem formulation and preliminaries

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with disturbance observer is given in Section 3. Section 4 analyzes the systems stability. Section 5 gives two simulation examples. Section 6 is the conclusion of this paper.

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2.1. System description The class of MIMO uncertain nonlinear systems with disturbance is described as   x˙ l,1 = fl,1 (xl,1 ) + xl,2 + dl,1 (t)     x˙ l,2 = fl,2 (xl,2 ) + xl,3 + dl,2 (t)    .. . (1)  x ˙ = f (x + d (t) ) + x  l,b −1 l,b −1 l,b −1 l,b l,bl −1 l l l l     (x, u = f x ˙ (t), ) + u + d  l,bl l,bl l l,bl l−1   y = x . l l,1

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where xl,pl = [xl,1 , . . . , xl,pl ]T ∈ Rpl (l = 1, 2, . . . , n; pl = 1, 2, . . . , bl ) is the system state vector, ul is the system control input with ul = [u1 , . . . , ul ]T , and yl is the system output. fl,pl (·) is an unknown smooth nonlinear function. dl,pl (t) is an unknown time-varying disturbance. x = [xT1 , . . . , xTn ]T with xl = [xl,1 , . . . , xl,bl ]T . Assumption 1 [42]: For the positive constant d¯l,pl and d∗l,pl , the external unknown disturbance dl,pl satisfy |dl,pl | ≤ d¯l,pl and its slope satisfy |d˙l,pl | ≤ d∗l,pl . The constants d¯l,pl and d∗l,pl will be used in the stability analysis and do not need to be known for the controller design. Control objectives: For the given reference signals yl,r , we design a synthetic fuzzy adaptive tracking control and adaptive laws θl,pl such that all the signals of the closed-loop system are SGUUB, and the output of the systems can track a given reference signal. 2.2. The generalized fuzzy hyperbolic model In the following control design and stability analysis procedure, generalized fuzzy hyperbolic model is often used as an approximator for nonlinear functions by reason of their well-pleasing approximation performance which is proved by many literatures.

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For the generalized fuzzy hyperbolic model [43], we design the membership functions of Px and Nx as follows 1

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µNx (xz ) = e− 2 (xz +kz ) ,

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µPx (xz ) = e− 2 (xz −kz ) ,

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where Px represent a positive fuzzy set, Nx represent a negative fuzzy set, and kz is positive constant. Definition 1 [43]: Let x = [x1 (t), x2 (t), . . . , xn (t)]T is an n dimension input variable, and y(t) is an output variable. x¯ = [¯ x1 (t), . . . , x¯mP (t)]T is designed as the generalized input variable, where x¯i = xz − dzj , m = ni=1 ωi is the number of generalized input variables, ωz (z = 1, . . . , n) are the number to be transformed about xz , dzj (z = 1, . . . , n, j = 1, . . . , ωz ) are constants where xz is transformed. If the following conditions are satisfied for the fuzzy rule base, then it is called generalized fuzzy hyperbolic rule base: 1) The fuzzy rules can be described as IF (x1 − d11 ) is Fx11 and . . . and (x1 − d1ω1 ) is Fx1ω1 and (x2 − d21 ) is Fx21 and . . . and (x2 − dω2 ) is Fxn1 and . . . and (xn − dnωn ) is Fxnωn THEN y = cF11 + · · · + cF1ω1 + cF21 + · · · + cFn1 + · · · + cFnωn ,

(3)

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where Fxzj are fuzzy sets corresponding to xz − dzj , which contain Px (Positive) and Nx (Negative) subsets. cFzj are constants corresponding to Fxzj . 2) The constants cFzj (z = 1, . . . , n, j = 1, . . . , ωz ) in the “THEN” part correspond to Fxzj in the “IF” part, that is, if there is Fxzj in the “IF” part, cFzj must appear in the “THEN” part. Otherwise, cFzj does not appear in the “THEN” part. P 3) There are 2m fuzzy rules in the rule base, where m = ni=1 ωi ; that is, all the possible Px and Nx combinations of input variables in the “IF” part and all the linear combinations of constants in the “THEN” part. Lemma 1 [43]: Let x and y(t) are the input and the output of the GFHM, respectively. If the membership functions of the generalized input variables Px and Nx are designed as (2), the generalized input variables and generalized fuzzy hyperbolic rule base are designed as definition 1, then the following

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model can be obtained: m X cPxi ekxi x¯i + cNxi e−kxi x¯i y= ekxi x¯i + e−kxi x¯i i=1 m X

ai +

i=1

m X ekxi x¯i − e−kxi x¯i bi kx x¯i e i + e−kxi x¯i i=1

=% + ϑT tanh(Φ¯ x) = G(x),

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=

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Pm c −c c +c T where ai = Pi 2 Ni , bi = Pi 2 Ni , % = i=1 ai , ϑ = [b1 , · · · , bm ] , Φ = diag[kx1 , · · · , kxm ], and tanh(Φ¯ x) is designed as tanh(Φ¯ x) = [tanh(kx1 x¯1 ), · · · , tanh(kxm x¯m )]T . Then (4) is called GFHM. Remark 1 [44]: The GFHM can be further written as a extension form such as y = θT ϕ(x)

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where θ = [%, ϑT ]T , ϕ(x) = [1, tanh(kx1 x¯1 ), · · · , tanh(kxm x¯m )]T . Lemma 2 [43]: For an arbitrary continuous function f (x) which is defined on the compact set U , there exists a GFHM such that sup |f (x) − θT ϕ(x)| < ε.

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x∈U

where ε is any positive constant.

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3. Synthetic control design

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In this section, by using backstepping DSC approach, a fuzzy adaptive control method will be studied by introducing serial-parallel estimation model. The control design is divided into n steps. In the first n − 1 steps, the virtual controllers are designed. In the last step, the actual controller ul is designed. For convenience, we first design Rl,pl = xl,pl , Rl,bl = [x, ul−1 ]T . By Lemma 2, the GFHM is a universal approximator, i.e., it can approximate any a smooth function on a compact space, thus we can assume that the nonlinear unknown function fl,pl (Rl,pl ) in (1) can be approximated by the following GFHM T fˆl,pl (Rl,pl |θl,pl ) = θl,p ϕ (Rl,pl ) l l,pl

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∗ According to [31], [27], the optimal parameter vectors θl,p is defined as l

min [

sup

θl,pl ∈Ωl,pl Rl,p ∈Ul,p l l

|fˆl,pl (Rl,pl |θl,pl ) − fl,pl (Rl,pl )|]

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∗ = arg θl,p l

where Ωl,pl and Ul,pl are bounding compact regions for θl,pl and Rl,pl , respectively. Moreover, the approximation error εl,pl is designed as ∗ εl,pl = fl,pl (Rl,pl ) − fˆl,pl (Rl,pl |θl,p ) l

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where εl,pl satisfy |εl,pl | ≤ ε¯l,pl , ε¯l,pl is a positive constant. step l, 1: The first error surface is defined as sl,1 = yl − yl,r .

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We design Dl,1 = dl,1 + εl,1 , and choose the first virtual control xdl,2 as T ˆ l,1 xdl,2 = −θl,1 ϕl,1 (Rl,1 ) − cl,1 sl,1 + y˙ l,r − D

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ˆ l,1 is the estimation of Dl,1 . where cl,1 > 0 is the design parameter, D c A new state variable xl,2 is designed by making xdl,2 get through a firstorder filter as κl,2 x˙ cl,2 + xcl,2 = xdl,2 , xcl,2 (0) = xdl,2 (0). (12)

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where κl,2 is a positive constant. Define sl,2 = xl,2 − xcl,2 . Then, we obtain T ˜ l,1 − cl,1 sl,1 + xcl,2 − xdl,2 . s˙ l,1 = θ˜l,1 ϕl,1 (Rl,1 ) + sl,2 + D

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∗ ˜ l,1 = Dl,1 − D ˆ l,1 . where θ˜l,1 = θl,1 − θl,1 and D The compensating signal is designed with the purpose of eliminating the influence on the error xcl,2 − xdl,2 as

z˙l,1 = −cl,1 zl,1 + zl,2 + xcl,2 − xdl,2 ,

zl,1 (0) = 0

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where zl,2 will be defined next. The compensated tracking error signals are designed gl,1 = sl,1 − zl,1 ,

gl,2 = sl,2 − zl,2 . 7

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The derivative of compensated tracking error can be obtained as

We design the serial-parallel estimation model

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T ˜ l,1 − cl,1 gl,1 . g˙ l,1 = θ˜l,1 ϕl,1 (Rl,1 ) + gl,2 + D

T ˆ l,1 xˆ˙ l,1 = θl,1 ϕl,1 (Rl,1 ) + xl,2 + βl,1 ql,1 + D

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where βl,1 is a positive constant, ql,1 is the prediction error, and

Define the adaptive law θl,1 as

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ql,1 = xl,1 − xˆl,1 .

θ˙l,1 = ηl,1 [(gl,1 + γl,1 ql,1 )ϕl,1 (Rl,1 ) − σl,1 θl,1 ]

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where ηl,1 , γl,1 and σl,1 are design constants. The disturbance observer is designed by using GFHM as follows

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ˆ l,1 = xl,1 − ξl,1 D

T ˆ l,1 + gl,1 + γl,1 ql,1 ξ˙l,1 = θl,1 ϕl,1 (Rl,1 ) + xl,2 + D

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ˆ l,1 can be obtained as According to (1) and (21), the derivative of D T ˆ˙ l,1 = θ˜l,1 ˜ l,1 − gl,1 − γl,1 ql,1 D ϕl,1 (Rl,1 ) + D

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˜ l,1 can be written as Then the derivative of D

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˜˙ l,1 = D˙ l,1 − θ˜T ϕl,1 (Rl,1 ) − D ˜ l,1 + gl,1 + γl,1 ql,1 D l,1

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step l, pl (pl = 2, . . . , bl − 1): Define the pl th error surface sl,pl as sl,pl = xl,pl − xcl,pl

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where xcl,pl will be defined in the following procedure (26). We design Dl,pl = dl,pl + εl,pl , and choose the pl th virtual control xdl,pl +1 as T ˆ l,p xdl,pl +1 = −θl,p ϕ (Rl,pl ) − cl,pl sl,pl − sl,pl −1 + x˙ cl,pl − D (25) l l l,pl ˆ l,p is the estimation of Dl,p . where cl,pl > 0 is the design parameter, D l l 8

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A new state variable xcl,pl +1 is designed by making xdl,pl +1 get through a first-order filter as xcl,pl +1 (0) = xdl,pl +1 (0).

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κl,pl +1 x˙ cl,pl +1 + xcl,pl +1 = xdl,pl +1 ,

where κl,pl +1 is a positive constant. Define sl,pl +1 = xl,pl +1 − xcl,pl +1 . Then we can obtain

T ϕ (Rl,pl ) + sl,pl +1 − cl,pl sl,pl s˙ l,pl =θ˜l,p l l,pl ˜ l,p + xcl,p +1 − xdl,p +1 . − sl,p −1 + D l

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z˙l,pl = −cl,pl zl,pl − zl,pl −1 + zl,pl +1 + xcl,pl +1 − xdl,pl +1 ,

zl,pl (0) = 0.

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where zl,pl +1 will be defined next. The compensated tracking error signal is designed as gl,pl = sl,pl − zl,pl

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The derivative of compensated tracking error can be obtained as

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T ˜ l,p . g˙ l,pl = θ˜l,p ϕ (Rl,pl ) + gl,pl +1 − cl,pl gl,pl − gl,pl −1 + D l l l,pl

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We design the serial-parallel estimation model giving by

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T ˆ l,p xˆ˙ l,pl = θl,p ϕ (Rl,pl ) + xl,pl +1 + βl,pl ql,pl + D l l l,pl

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where βl,pl is a positive constant, ql,pl is the prediction error, and ql,pl = xl,pl − xˆl,pl .

(32)

Define the adaptive law θl,pl as θ˙l,pl = ηl,pl [(gl,pl + γl,pl ql,pl )ϕl,pl (Rl,pl ) − σl,pl θl,pl ]

where ηl,pl , γl,pl and σl,pl are design constants. 9

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The disturbance observer is designed by using GFHM as follows (34)

T ˆ l,p + gl,p + γl,p ql,p ξ˙l,pl = θl,p ϕ (Rl,pl ) + xl,pl +1 + D l l l l l l,pl

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ˆ l,p = xl,p − ξl,p D l l l ˆ l,p can be obtained as According to (1) and (35), the derivative of D l ˆ˙ l,p = θ˜T ϕl,p (Rl,p ) + D ˜ l,p − gl,p − γl,p ql,p D l,pl l l l l l l l ˜ l,p can be written as Then the derivative of D l

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˜ l,p + gl,p + γl,p ql,p ˜˙ l,p = D˙ l,p − θ˜T ϕl,p (Rl,p ) − D D l,pl l l l l l l l l

(36)

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step l, bl : Define the bl th error surface sl,bl as sl,bl = xl,bl − xcl,bl .

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T ˆ l,b ul = −θl,b ϕ (Rl,bl ) − cl,bl sl,bl − sl,bl −1 + x˙ cl,bl − D l l l,bl

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ˆ l,b is the estimation of Dl,b . where cl,bl > 0 is the design constant, D l l The derivative of sl,bl can be written as (40)

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T ˜ l,b . s˙ l,bl = θ˜l,b ϕ (Rl,bl ) − cl,bl sl,bl − sl,bl −1 + D l l l,bl

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∗ ˜ l,b = Dl,b − D ˆ l,b . where θ˜l,bl = θl,b − θl,bl and D l l l l The compensating signal is designed as

z˙l,bl = −cl,bl zl,bl − zl,bl −1 ,

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The compensated tracking error signal is designed as gl,bl = sl,bl − zl,bl .

(42)

T ˜ l,b . g˙ l,bl = θ˜l,b ϕ (Rl,bl ) − cl,bl gl,bl − gl,bl −1 + D l l l,bl

(43)

Then we have

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We design the serial-parallel estimation model giving by T ˆ l,b xˆ˙ l,bl = θl,b ϕ (Rl,bl ) + ul + βl,bl ql,bl + D l l l,bl

(44)

ql,bl = xl,bl − xˆl,bl . Define the adaptive law θl,bl as

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θ˙l,bl = ηl,bl [(gl,bl + γl,bl ql,bl )ϕl,bl (Rl,bl ) − σl,bl θl,bl ]

(45)

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where ηl,bl , γl,bl and σl,bl are design constants. The disturbance observer is designed by using GFHM as follows ˆ l,b = xl,b − ξl,b D l l l

(47)

T ˆ l,b + gl,b + γl,b ql,b ξ˙l,bl = θl,b ϕ (Rl,bl ) + ul + D l l l l l l,bl

(48)

ˆ l,b can be obtained as According to (1) and (48), the derivative of D l

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˜ l,b can be written as Then the derivative of D l T ˜˙ l,b = D˙ l,b − θ˜l,b ˜ l,b + gl,b + γl,b ql,b D ϕ (Rl,bl ) − D l l l l l l l l,bl

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Remark 2 [40]: The compounded disturbance Dl,pl (t) is composed of the effect of fuzzy approximation error and time-varying disturbance. There exists an unknown positive constant Dl,pl ,max such that |Dl,pl (t)| ≤ Dl,pl ,max . Remark 3 [40]: Following the boundedness of |ε˙l,pl | ≤ ε∗l,pl (ε∗l,pl is an unknown constant) and according to Assumption 1, we known that there exists an unknown positive constant ζl,pl such that |D˙ l,pl | ≤ ζl,pl .

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4. Stability analysis

Theorem 1: For the uncertain nonlinear system (1), if the actual controller (39), the virtual controllers (11), (25), the adaptive laws (19), (33), (46), and the disturbance observers (20), (34), (47) exist, then all the signals in system (1) are bounded, and the output of the system output can track the given reference signals. 11

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Proof: The Lyapunov function candidate Vl is considered as b

b

b

b

l l l l X 1 ˜T ˜ 1X 1X 1X 2 2 2 ˜ + + + gl,p γ q λ D θl,pl θl,pl . l,pl l,pl l,pl l,pl l 2 p =1 2 p =1 2 p =1 2η l,p l p =1 l

l

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By using (17),(28),(44), the derivative of prediction error is obtained as T ˜ l,p − βl,p ql,p . ϕ (Rl,pl ) + D q˙l,pl = θ˜l,p l l l l l,pl

The time derivative of Vl is obtained as gl,pl g˙ l,pl +

pl =1

γl,pl ql,pl q˙l,pl +

pl =1

pl =1

+

bl X

bl X

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pl =1

+

b

l X 1 ˜T ˙ ˜ l,p D ˜˙ l,p − λl,pl D θl,pl θl,pl l l η l,p l p =1 p =1 l

˜ l,p ] + gl,pl [−cl,pl gl,pl + D l

bl X

bl X

l

˜ l,p − βl,p ql,p ] γl,pl ql,pl [D l l l

˜ l,p [D˙ l,p − θ˜T ϕl,p (Rl,p ) − D ˜ l,p + gl,p + γl,p ql,p ] λl,pl D l,pl l l l l l l l l T σl,pl θ˜l,p θ . l l,pl

pl =1

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bl X

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bl X

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V˙ l =

(52)

(53)

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By using Assumption 1, Youngs inequality and the fact ϕl,pl (Rl,pl )T ϕl,pl (Rl,pl ) ≤ $l,pl , we obtain the following inequalities: ˜ l,p ≤ 1 g 2 + 1 D ˜2 gl,pl D l,pl l 2 2 l,pl

γ2 ˜2 ˜ l,p ≤ 1 q 2 + l,pl D γl,pl ql,pl D l 2 l,pl 2 l,pl λ2 ˜ l,p D˙ l,p ≤ 1 D ˜ 2 + l,pl ζ 2 λl,pl D l l 2 l,pl 2 l,pl λ2l,pl T $l,pl ˜ 2 T ˜ ˜ −λl,pl Dl,pl θl,pl ϕl,pl ≤ Dl,pl + θ˜ θ˜l,p 2 2 l,pl l λ2 ˜ l,p gl,p ≤ 1 D ˜ 2 + l,pl g 2 λl,pl D l l 2 l,pl 2 l,pl 12

(54) (55) (56) (57) (58)

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2 γl,p λ2 2 l l,pl 2 ˜ l,p γl,p ql,p ≤ 1 D ˜ l,p ql,pl λl,pl D + l l l l 2 2

(59)

Then, substituting (54)-(59) into (53), the derivative V˙ l becomes

+

(cl,pl

pl =1 bl X

bl 2 X λ2l,pl γl,p λ2 1 2 1 2 l l,pl − (γl,pl βl,pl − − )gl,pl − − )ql,p l 2 2 2 2 p =1 l

(λl,pl −

2 γl,p l

2

pl =1 bl X

b

l X λ2l,pl T $l,pl 2 ˜ l,p + − 2)D θ˜l,bl θ˜l,bl l 2 2 p =1

−

l

T σl,pl θ˜l,p θ + φl,1 l l,pl

pl =1

where φl,1 = Since

Pbl

λ2l,p

pl =1 [ 2

l

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−

bl X

2 ]. ζl,p l

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V˙ l ≤ −

(60) can be rewritten as (cl,pl −

pl =1 bl X

(λl,pl

bl 2 X γl,p λ2 λ2l,pl 1 2 1 2 l l,pl − )gl,p − (γ β − − )ql,p l,pl l,pl l l 2 2 2 2 p =1

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−

bl X

l

bl X λ2l,pl $l,pl 1 T ˜ 2 ˜ − − − 2)Dl,pl − (σl,pl − − )θ˜l,p θ + φl l l,pl 2 2 2 2 p =1 2 γl,p l

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V˙ l ≤ −

pl =1

(61)

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2 σl,p 1 ˜T ˜ T ∗ T ˜ l σl,pl θ˜l,p θ θ + θ ≤ kθl,p k2 − σl,pl θ˜l,p θ l,p l,p l l l l l,pl 2 l,pl l 2

(60)

l

(62)

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P σ2 ∗ l where φl = φl,1 + bpll =1 l,p kθl,p k2 . 2 l P We design the whole Lyapunov function as V = nl=1 Vl . From (62), we

13

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have

− − Pn

where φ =

Choose cl,pl −

and σl,pl − 2 λ2 γl,p l,p

l

2

l=1 pl =1

bl n X X

λ2l,p

l

2

−

1 2

λ2l,pl 1 2 − )gl,p l 2 2

(γl,pl βl,pl −

l=1 pl =1

bl n X X

(λl,pl

l=1 pl =1

bl n X X

2 γl,p λ2 1 2 l l,pl − )ql,p l 2 2

l=1 pl =1

λ2l,pl 1 T ˜ − )θ˜l,p θ +φ l l,pl 2 2

− 12 > 0, γl,pl βl,pl −

2 λ2 γl,p l,p l

2

l

− 12 > 0, λl,pl −

> 0. Let ζl = min{2(cl,pl −

1 2 ), λl,p (λl,pl 2 l

2 γl,p

$l,pl 2

λ2l,p

l

2

− 2 − − − 2), 2ηl,pl (σl,pl − min{ζ1 , ζ2 , . . . , ζn }. Then, (63) can be further rewritten as l

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2

l

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l

(63)

2 γl,p $l,pl l ˜2 − − − 2)D l,pl 2 2

(σl,pl −

φl .

l=1

λ2l,p

(cl,pl −

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−

bl n X X

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V˙ ≤ −

V˙ ≤ −ζV + φ

2 γl,p

l

2

−

$l,pl −2 2

>0

2 − 12 ), γl,p (γl,pl βl,pl −

λ2l,p 2

l

l

−

1 )} 2

and ζ =

(64)

V (t) ≤ V (0)e−ζt +

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The solution of (64) can be written as φ . ζ

(65)

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ˆ l,p , and θl,p According to (65), it can be shown that the signals gl,pl , ql,pl , D l l are all bounded. From Refs. [35] [39], it is concludedp that all the signals of the closed-loop system are SGUUB. Meanwhile, |gl,1 | ≤ 2(V (0) exp(−ζt) + φ/ζ) p and |ql,1 | ≤ 2(V (0) exp(−ζt) + φ/ζ)/γl,1 . Since lim exp(−ζt) = 0, it folt→∞ p p lows that |gl,1 | ≤ 2φ/ζ) and |ql,1 | ≤ 2φ/(ζγl,1 ). From [35] and [44, Lemma 3], we can obtain zl,pl (t) (l = 1, 2, . . . , n; pl = 1, 2, . . . , bl ) are bounded, thus we can further get that all the signals in the closed-loop systems are bounded. 14

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5. Simulation

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In this part, we will give two simulation examples to illustrate the availability of the proposed method in this paper.

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5.1. Example one: The MIMO uncertain nonlinear systems are considered in the following   x˙ l,1 = fl,1 (xl,1 ) + xl,2 + dl,1 (t), (66) x˙ l,2 = fl,2 (x) + ul + dl,2 (t),   yl = xl,1 , l = 1, 2 where f1,1 (x1,1 ) = 0.5x21,1 , f1,2 (x) = x1,1 x1,2 , f2,1 (x2,1 ) = −x2,1 sin2 (x2,1 ), f2,2 (x) = x2,2 x1,1 sin(x1,2 ), d1,1 (t) = 0.1 cos(t), d1,2 (t) = 2 sin(0.1t), d2,1 (t) = 0.1 sin(3t), d2,2 (t) = cos(0.5t). The reference signals are given as y1,r = sin(t) and y2,r = 2 sin(t) + sin(0.5t). The virtual control function xdl,2 , the actual control ul , the adaptive laws θ˙l,1 and θ˙l,1 are chosen as

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T ˆ l,1 xdl,2 = −θl,1 ϕl,1 (Rl,1 ) − cl,1 sl,1 + y˙ l,r − D

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T ˆ l,2 ul = −θl,2 ϕl,2 (Rl,2 ) − cl,2 sl,2 − sl,1 + x˙ cl,2 + D

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θ˙l,1 = ηl,1 [(gl,1 + γl,1 ql,1 )ϕl,1 (Rl,1 ) − σl,1 θl,1 ] θ˙l,2 = ηl,2 [(gl,2 + γl,2 ql,2 )ϕl,2 (Rl,2 ) − σl,2 θl,2 ]

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The initial state of the system is [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.5, 0.5, 0.5, 0.5]T . The parameters chosen for simulation are: c1,1 = 228, c1,2 = 180, c2,1 = 280, c2,2 = 174, κ1,2 = κ2,2 = 0.01, γ1,1 = 1.7, γ1,2 = 1.3, γ2,1 = 3, γ2,2 = 1.5. The simulation results are demonstrated in Figure 1-4. In order to further illustrate the superiority of the disturbance observer based synthetic fuzzy control (DOB-SFC) in this paper, we make a comparison with adaptive tracking control (ATC) in [46]. From Fig.1(B) and Fig.2(B), we can see that DOB-SFC has higher precision. From Fig.1(C) and Fig.2(C), the tracking speed is more faster by using DOB-SFC. Figs.5 and Figs.6 are the control trajectories of ul (l = 1, 2).

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1.5 y1r

1

1.005

DOB−SFC ATC

0.5

1 0.995

0

0.985

−1

0.98

−1.5

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0.99

−0.5

y1r

DOB−SFC ATC

0.975 0

5

10 Time(sec) (A)

15

20

7.6

0.3

0.01

DOB−SFC ATC

0.2

7.7

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0

7.9

DOB−SFC ATC

0.005

0.1

7.8 Time(sec) (B)

0

−0.1

−0.005

−0.2 5

10 Time(sec) (C)

15

−0.01

20

3

4

5

6 7 Time(sec) (D)

8

9

10

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0

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Figure 1: Tracking performance for Example 1:(A) tracking performance of two methods;(B) tracking detail over a period of time;(C) tracking errors of two methods;(D) tracking errors in time period [2.5sec,10sec].

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5.2. Example two: In this part, two inverted pendulums are considered as follows   x˙ l,1 = fl,1 (xl,1 ) + xl,2 + dl,1 (t), x˙ l,2 = fl,2 (x) + ul + dl,2 (t),   yl = xl,1 2

2

(67)

AC

kr kr where f1,1 (x1,1 ) = 0, f1,2 (x) = ( mJ11gr − 4J )sin(x1,1 )+ 4J sin(x2,1 ), f2,1 (x2,1 ) = 1 1 m2 gr kr2 kr2 0, f2,2 (x) = ( J2 − 4J2 )sin(x2,1 ) + 4J2 sin(x1,2 ), d1,1 (t) = 0, d1,2 (t) = sin(t) + kr kr (L − b), d2,1 (t) = 0, d2,2 = cos(t) + 2J (L − b). 2J1 2 As well as [14] [46], x1,1 = θ1 , x2,1 = θ2 are angular positions. And the system parameters m1 = m2 = 2 kg are the pendulum end masses, k = 10 N/m is the connecting spring constant, J1 = J2 = 1 kg are the moments of inertia, natural length of the spring is defined as L = 0.5 m, r = 0.1 m is the

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3

−2.58

y2,r

y2,r

DOB−SFC ATC

1

DOB−SFC ATC

−2.59

0

−2.6

−1 −2.61

−2 −3

0

5

10 Time(sec) (A)

15

20

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0.01

0 −0.1

10.47 10.48 Time(sec) (B)

10.49

10.5

DOB−SFC ATC

0.02

0.1

0

−0.2

−0.01

−0.3 −0.4

10.46

0.03

DOB−SFC ATC

0.2

−2.62 10.45

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2

0

5

10 Time(sec) (C)

15

20

−0.02

8

9

10 Time(sec) (D)

11

12

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Figure 2: Tracking performance for Example 1:(A) tracking performance of two methods;(B) tracking detail over a period of time;(C) tracking errors of two methods;(D) tracking errors in time period [8sec,12sec]. 100

80 60

PT

40 20

0

AC

CE

−20 −40 −60 −80

−100

0

2

4

6

8

10 Time(sec)

12

14

16

18

20

Figure 3: The trajectories of control u1 .

pendulum height, and g = 9.81m/s2 is the gravitational acceleration. b = 0.4 m is The distance between the pendulum hinges. Two inverted pendulums 17

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100 80 60 40

0 −20 −40 −60 −80 −100

0

2

4

6

8

10 Time(sec)

12

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18

20

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Figure 4: The trajectories of control u2 .

Figure 5: Two inverted pendulums.

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connected by a spring and a damper is showed as Fig.5. The initial state of the system is [x1,1 (0), x1,2 (0), x2,1 (0), x2,2 (0)]T = [0.5, 0.5, 0.5, 0.5]T . The parameters chosen for simulation are: c1,1 = 78, c1,2 = 75, c2,1 = 76, c2,2 = 77, κ1,2 = κ2,2 = 0.01, γ1,1 = 1.5, γ1,2 = 1.2, γ2,1 = 2.4, γ2,2 = 0.5. The reference signals are given as y1,r = 5 sin(t) + 2 sin(0.5t) and y2,r = 3 cos(2t) + cos(t). The simulation results are demonstrated in Figure 6-9. From Fig.6 and Fig.7, we can further illustrate that the better tracking performance can be achieved by DOB-SFC in this paper.

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10

−1.88 y1,r

y1,r

−1.9

DOB−SFC ATC

5

DOB−SFC ATC

−1.92 −1.94

−5 −10

−1.96 0

5

10 Time(sec) (A)

15

20

0.2

3.985 3.99 Time(sec) (B)

3.995

4

DOB−SFC ATC

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0.005

3.98

0

0

−0.005

−0.1 −0.2

0.01

DOB−SFC ATC

0.1

−1.98 3.975

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0

−0.01

0

5

10 Time(sec) (C)

15

20

8.5

9

9.5

10

Time(sec) (D)

6. Conclusion

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Figure 6: Tracking performance for Example 2: (A) tracking performance of two methods; (B) tracking detail over a period of time; (C) tracking errors of two methods; (D) tracking errors in time period [8.5sec,10sec].

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A synthetic fuzzy adaptive tracking control based on GFHM has been investigated for MIMO nonlinear systems with unknown time-varying disturbance in this paper. GFHM has been considered as universal approximator to approximated an unknown nonlinear function. Based on the the design of serial-parallel estimation model and prediction error, a novel fuzzy adaptive tracking control approach has been developed by using backstepping DSC technique. Not only the “explosion of complexity” can be avoided, but also the satisfactory tracking performance can be obtained. It has been guaranteed that all the signals of the closed-loop systems were SUUB, and the output of the systems can track the given reference signals. In my future work, an adaptive tracking control method will be studied for uncertain nonlinear systems with unmeasured states.

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6

3.1 y2,r

4

DOB−SFC ATC

2

3

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2.9 0

0

5

10 Time(sec) (A)

15

2.7 5.82

20

DOB−SFC ATC

1

DOB−SFC ATC

5.84

0.06

5.88

5.9

DOB−SFC ATC

0.04

0

5.86 Time(sec) (B)

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−4

y2,r

2.8

−2

0.02

−1

0

−0.02

−2 0

5

10 Time(sec) (C)

15

20

−0.04

5.5

6

6.5 Time(sec) (D)

7

7.5

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Figure 7: Tracking performance for Example 2: (A) tracking performance of two methods; (B) tracking detail over a period of time; (C) tracking errors of two methods; (D) tracking errors in time period [5sec, 7.5sec]. 150

100

PT

50

0

AC

CE

−50

−100

−150

0

2

4

6

8

10 Time(sec)

12

14

16

18

20

Figure 8: The trajectories of control u1 .

Acknowledgement This work was supported by the National Natural Science Foundation of China (6143300461627809), and IAPI Fundamental Research Funds 2013ZCX14. 20

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200

150

100

0

−50

−100

−150

−200

0

2

4

6

8

10 Time(sec)

12

14

16

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50

18

20

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Figure 9: The trajectories of control u2 .

This work was supported also by the development project of key laboratory of Liaoning province. References

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[44] M.J. Zhang, C. Shao, S.K. Li, Robust indirect adaptive fuzzy control based on the GFHM, System Science and Engineering (2012) 248-252.

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[45] W.J. Dong, J.A. Farrell, M. Polycarpou, V. Djapic, M. Sharma, Command filtered adaptive backstepping, IEEE Trans. Contr. Syst. Technol. 20 (2012) 566-580. [46] Y. Cui, H.G. Zhang, Y.C. Wang, Adaptive tracking control of uncertain MIMO nonlinear systems based on generalized fuzzy hyperbolic model, Fuzzy Sets Sys. 306 (2017) 105-117.

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Yang Cui received the B.S. degree in information and computing science and the M.S. degree in applied mathematics form Liaoning University of Technology, Jinzhou, China, in 2009 and 2012, respectively. She is currently working toward the Ph.D. degree in control theory and control engineering, Northeastern University. Her research interests include dynamic surface control, neural networks and adaptive control.

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Huaguang Zhang (M’03, SM’04, F’14) received the B.S. degree and the M.S. degree in control engineering from Northeast Dianli University of China, Jilin City, China, in 1982 and 1985, respectively. He received the Ph.D. degree in thermal power engineering and automation from Southeast University, Nanjing, China,in 1991. He joined the Department of Automatic Control, Northeastern University, Shenyang, China, in 1992, as a Postdoctoral Fellow for two years. Since 1994, he has been a Professor and Head of the Institute of Electric Automation, School of InformationScience and Engineering, Northeastern University, Shenyang, China. His main research interests are fuzzy control, stochastic system control, neural networks based control, nonlinear control, and their 26

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applications. He has authored and coauthored over 280 journal and conference papers, six monographs and co-invented 90 patents. Dr. Zhang is the fellow of IEEE, the E-letter Chair of IEEE CIS Society, the former Chair of the Adaptive Dynamic Programming & Reinforcement Learning Technical Committee on IEEE Computational Intelligence Society. He is an Associate Editor of AUTOMATICA, IEEE TRANSACTIONS ONNEURAL NETWORKS, IEEE TRANSACTIONS ON CYBERNETICS, and NEUROCOMPUTING, respectively. He was an Associate Editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS (2008-2013). He was awarded the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named the Cheung Kong Scholar by the Education Ministry of China in 2005. He is a recipient of the IEEE Transactions on Neural Networks 2012Outstanding Paper Award. He is also a recipient of Andrew P. Sage Best Transactions Paper Award 2015.

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QiuxiaQu received the M.S. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2010, where she is currently pursuing the Ph.D. degree in control theory and control engineering. Her current research interests include adaptive dynamic programming, neural network, optimal control, and their industrial applications.

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ChaominLuo (S’ 01–M’ 08) received the B.Eng. degree in radio engineering from Southeast University, Nanjing, China, the M.Sc. degree in engineering systems and computing from the University of Guelph, Guelph, ON, Canada, and the Ph.D. degree in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada, in 2008. He is currently an Associate Professor with the Advanced Mobility Laboratory, Department of Electrical and Computer Engineering, University of Detroit Mercy, Detroit, MI, USA. His current research interests include control and automation, computational intelligence, intelligent controls and robotics, and embedded systems. Dr. Luo was a recipient of the NSERC Postgraduate Award in Canada and President’s Graduate Awards from the University of Waterloo, and the Best Student Paper Presentation Award at 2007 SWORD. He also received Faculty Research Awards from the University of Detroit Mercy in 2009, 2015, and 2016. He serves as the Editorial Board Member of the International Journal of Complex Systems-Computing, Sensing and Control and an Associative Editor of the International Journal of Robotics and Automation. He was the general Co-chair in the IEEE International Workshop on Computational Intelligence in Smart Technologies, and Journal Special Issues Chair, IEEE 2016 International Conference on Smart Technologies, Cleveland, OH, USA. He has organized and chaired several special sessions on the topics of intelligent vehicle systems and bio-inspired intelligence in the IEEE reputed international conferences, such as the IJCNN and the IEEE-SSCI and WCCI.

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