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Adaptive Output-Feedback Fuzzy Tracking Control for a Class of Nonlinear Systems Qi Zhou, Peng Shi, Senior Member, IEEE, Jinjun Lu, and Shengyuan Xu

Abstract—This paper is concerned with the problem of adaptive fuzzy tracking control via output feedback for a class of uncertain single-input single-output (SISO) strict-feedback nonlinear systems. The dynamic feedback strategy begins with an input-driven filter. By utilizing fuzzy logic systems to approximate unknown and desired control input signals directly instead of the unknown nonlinear functions, an output-feedback fuzzy tracking controller is designed via a backstepping approach. It is shown that the proposed fuzzy adaptive output controller can guarantee that all the signals remain bounded and that the tracking error converges to a small neighborhood of the origin. Simulations results are presented to demonstrate the effectiveness of the proposed methods. Index Terms—Adaptive control, backstepping, fuzzy control, nonlinear systems, strict-feedback systems.

I. INTRODUCTION DAPTIVE control for nonlinear systems with parametric uncertainties has received a lot of attention over the past few decades (see, for example, [1]–[5] and the references therein). However, the early stages of the research were based on the assumption that the uncertain nonlinearities are either to have a prior knowledge of the bound or to be linearly parameterized. These assumptions cannot always be satisfied because in some practical systems, it is inevitable that they will contain some uncertain elements that cannot be modeled; then, the techniques that are developed in [1]–[3] are not applicable. It is well known that neural networks and fuzzy logic systems have been found to be particularly powerful tools to control nonlinear systems because of their universal approximation properties. Over the past few decades, the issues of utilizing neural networks and fuzzy logic systems to approximate un-

A

Manuscript received January 11, 2011; accepted April 25, 2011. Date of publication June 7, 2011; date of current version October 10, 2011. This work was supported by the National Natural Science Foundation of China under Grant 61074043 and Grant 61074008, the Qing Lan Project, the Natural Science Foundation of Jiangsu Province under Grant BK2008047 and Grant BK2008188, the Engineering and Physical Sciences Research Council, U.K., under Grant EP/F029195, and the China Scholarship Council. Q. Zhou is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China, and also with the School of Engineering and Science, Victoria University, Melbourne, Vic. 8001, Australia (e-mail: [email protected]). P. Shi is with the Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 IDL, U.K., and also with the School of Engineering and Science, Victoria University, Melbourne, Vic. 8001, Australia (e-mail: [email protected]). J. Lu is with the School of Electrical Engineering, Nantong University, Nantong 226019, China (e-mail: [email protected]). S. Xu is with the School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2011.2158652

known nonlinearities have been well investigated (see, for example, [6], [7], and the references therein). However, the systems that are considered in these works are restricted to satisfy matching conditions, that is, the unknown functions must appear on the same equation as the control input channel. Therefore, these schemes are not feasible for more general systems such as the strict-feedback or pure-feedback systems. Recently, a backstepping technique has been successfully applied to analyze the problem of stability and to design adaptive controllers for nonlinear systems. Many researchers have focused on the triangular nonlinear systems and developed some significant results such as [8]–[13] via a backstepping technique and neural networks methods. The controllers that are designed in [8]–[13] cannot only achieve good performance but also guarantee that all the signals in the closed-loop system to be uniformly ultimate bounded. In the meantime, many researchers resorted to a backstepping technique and fuzzy logic systems to solve the adaptive controller design problems for these triangular nonlinear systems and to achieve the performance [14]–[21]. At the same time, the backstepping technique has been used to solve a class of nonlinear stochastic systems (see, for example, [22] and [23]). In [14]–[21], it was assumed that the states of the systems are directly measurable. When the state of the nonlinear systems is unmeasured, the adaptive output-feedback control method may be an effective way to control the systems. In [24], the problem of robust adaptive output-feedback control for a class of nonlinear systems with time-varying actuator faults is investigated, and in [25], the problem of adaptive fuzzy output-feedback control for nonlinear systems with unknown sign of high-frequency gain is researched. By exploiting using fuzzy logic systems to approximate the unknown nonlinear functions and combining backstepping approach, a fuzzy adaptive output-feedback controller is designed in [26] for singleinput single-output (SISO) strict-feedback nonlinear systems and in [27] and [28] for multiple-input multiple-output nonlinear systems, respectively. In addition, by combining neural networks with backstepping approach, an adaptive output-feedback controller is designed for a class of large-scale nonlinear systems in [29]. The weakness of the aforementioned adaptive control methods in [8]–[12], [14]– [16], [24], and [26]–[29] is that many adaptive parameters need to be tuned by online learning laws. When the number of neural network nodes or the number of the fuzzy logic rule bases increase, the number of adaptation laws increase accordingly. Therefore, the learning time tends to be large inevitably. More recently, a new adaptive fuzzy control method has been presented in [30] to reduce the adaptive parameters. The work in [31] developed a new direct adaptive fuzzy control for nonlinear strict-feedback

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ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

systems, in which Mamdani-type fuzzy systems are used to directly approximate the desired control input signals instead of the unknown nonlinearities. And the computation burden has been greatly reduced because only one parameter needs to be tune. However, the results in [31] hold only when the state of the nonlinear system is measurable. Motivated by the previous discussion, in this paper, we will develop a direct adaptive output-feedback control approach for strict-feedback nonlinear systems. The dynamic feedback strategy begins with an input-driven filter, which is different from [26]–[29], and by combining fuzzy logic systems with a backstepping technique, fuzzy adaptive output-feedback controller is constructed recursively. It is proved that the controller that is designed in this paper guarantees that all the signals in the closed-loop remain bounded and achieves good tracking performance. In addition, the method of direct adaptive control that is used in this paper can reduce the computation burden; therefore, the algorithm is convenient to realize in engineering. The remaining of this paper is organized as follows. The problem formulation and preliminaries are presented in Section II, and a direct adaptive output-feedback fuzzy tracking controller is designed by a recursive procedure in Section III. Two simulation examples are given in Section IV to demonstrate the effectiveness of the proposed scheme, and the paper is concluded in Section V.

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.. . .

x ˆ n = u − ln x ˆ1

(2)

where x ˆi is the estimate of xi , (i = 1, . . . , n), and li is the design parameter such that the matrix ⎡ ⎤ −l1 ⎢ . ⎥ ⎥ . Ac = ⎢ In −1 ⎣ . ⎦ −ln

0...

0

is a strict Hurwitz matrix, which means for a given matrix Q > 0, there exists a matrix P > 0 that satisfies the following equation: ATc P + P Ac = −Q.

(3)

Define ei = xi − x ˆi , (i = 1, . . . , n). Then, we have x1 ) + d1 (t) + l1 y e˙ 1 = e2 − l1 e1 + f1 (¯ e˙ 2 = e3 − l2 e1 + f2 (¯ x2 ) + d2 (t) + l2 y .. . e˙ n = −ln e1 + fn (¯ xn ) + dn (t) + ln y which can be rewritten as x) + D (t) e˙ = Ac e + F (¯

(4)

with e = (e1 , e2 , . . . , en )T

II. PROBLEM FORMULATION AND PRELIMINARIES In this section, the nonlinear control problem is first formulated, and then in order to develop an adaptive dynamic feedback control design procedure, an input-driven filter is introduced. Finally, to approximate the desired control signals, the fuzzy logic systems are given.

F (¯ x) = (f1 (¯ x1 ) + l1 y, . . . , fn (¯ xn ) + ln y)T 

= (F1 (¯ x) , . . . , Fn (¯ x))T D (t) = (d1 (t) , . . . , dn (t))T . After combining (1), (2), and (4), the system can be taken as

A. Nonlinear Control Problem

x) + D (t) e˙ = Ac e + F (¯

Consider an SISO nonlinear dynamic system in the following form: xi ) + di (t) , x˙ i = xi+1 + fi (¯

x ˆ2 = x ˆ 3 − l2 x ˆ1

1≤i≤n−1

.. .

x˙ n = u + fn (¯ x) + dn (t) y = x1

y˙ = x ˆ2 + e2 + f1 (x1 ) + d1 (t) .

.

(1)

x ˆ n = u − ln x ˆ1

(5)

where x ¯ = [x1 , x2 , . . . , xn ]T ∈ Rn denotes state vector of the system; u ∈ R and y ∈ R are input and output of the system, respectively. x ¯i = [x1 , x2 , . . . , xi ]T , i = 1, 2 . . . , n − 1. fi (.), i = 1, 2 . . . , n stand for the unknown smooth system functions with fi (0) = 0. di (t), i = 1, 2 . . . , n are the external disturbance uncertainties of the system, which satisfy |di (t)| ≤ d¯i , with d¯i as a constant.

where y and x ˆi , (i = 1, . . . , n) are available for control design. The control objective of this paper is to design a direct adaptive fuzzy controller such that the system output y can track the reference signal yd , while all the signals in the derived closed-loop system remain bounded. The reference signal yd is (i) assumed to be available together with its ith time derivative yd T  (i) (1) (i) for i = 1, 2, . . . , n, y¯d = yd , yd , . . . , yd .

B. Dynamic Feedback Design

C. Fuzzy Logic Systems

The dynamic feedback strategy begins with an input-driven filter [32] as follows:

Construct fuzzy logic systems with the following IF–THEN rules:

.

x ˆ1 = x ˆ 2 − l1 x ˆ1 .

x ˆ2 = x ˆ 3 − l2 x ˆ1

Ri : If x1 is F1i and . . . and xn is Fni . Then y is B i , i = 1, 2, . . . , N.

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

The fuzzy logic systems with the singleton fuzzifier, product inference, and center average defuzzifier can be formulated as

N ¯ n i=1 Φi j =1 μF ji (xj )

y(x) =

N n i (xj ) μ F i=1 j =1 j where x = [x1 , x2 , . . . , xn ]T ∈ Rn , μF ji (xj ) is the membership ¯ i = max μB i (y). Let of F i , Φ j

y ∈R

n

j =1 μF i (xj ) j

pi (x) =

N n i μ (x ) j i=1 j =1 F j

to be semiglobal stable, and for any given scalar ε > 0, the tracking error satisfies lim supt→∞ χ21 ≤ ε2 by appropriately choosing the design parameters. The proof of Theorem 1 is presented in the following with a backstepping approach. Proof: Step 1: For the reference signal yd , introduce the tracking error variable χ1 = y − yd , and choose the following Lyapunov function candidate: 1 1 V1 = eT P e + χ21 + θ˜2 (10) 2 2r ˆ The time derivative of V1 is given by where θ˜ = θ − θ.

¯1, Φ ¯2, . . . , ξ(x) = [p1 (x), p2 (x), . . . , pN (x)]T , and Φ = [Φ T ¯ ΦN ] . Then, the fuzzy logic systems can be rewritten as follows: T

y(x) = Φ ξ(x).

V˙ 1 = −eT Qe + 2eT P (F + D) 1 ˆ. + χ1 (ˆ x2 + e2 + f1 + d1 − y˙ d ) − θ˜θ. r

(6)

Lemma 1: [33] Let f (x) be a continuous function that is defined on a compact set Ω; then, for any given constant ε > 0, there exists a fuzzy logic system y(x) in the form of (6) such that



x) , . . . , Fn (¯ x))T , and Fi (¯ x) , i = 1, 2, . . . , n is As F = (F1 (¯ an unknown function, by Lemma 1, for any given εi0 > 0, there exists a fuzzy logic system ΦTi0 ξ0 (X0 ) such that Fi (X0 ) = ΦTi0 ξ0 (X0 ) + δi0 (X0 )

sup |f (x) − y(x)| ≤ ε.

|δi0 (X0 )| ≤ εi0

x∈Ω

where X0 = x ¯. Therefore

III. FUZZY ADAPTIVE CONTROL DESIGN AND STABILITY ANALYSIS

F (X0 ) = ΦT0 ξ0 (X0 ) + δ0 (X0 )

The backstepping design procedure contains n steps. In each step, a virtual control function α ˆ i should be developed using an appropriate Lyapunov function Vi , and the real tracking control law u will finally be designed. To begin with the backstepping design procedure, let us define a constant   θ = max Φi 2 : i = 0, 1, 2, . . . , n . From the definition of θ, we know that θ is an unknown constant, and we define θˆ as the estimate of θ. The feasible virtual control signal is designed as 1 ˆ T (Xi ) ξi (Xi ) , i = 1, . . . , n (7) αi (Xi ) = − 2 χi θξ i 2ai  T T  − (1) (i) ˆ ˆ , Xi = x1 , x ˆi , θ, y¯d , with where X1 = x1 , θ, y¯d −

x ˆi = (ˆ x1 , . . . , x ˆi )T for i = 2, . . . , n, and χi will be defined at the proof of Theorem 1. Theorem 1: Consider the nonlinear dynamic system in (1), if we choose the input driven filter (2), and a control law as u=−

1 ˆ T (Xn ) ξn (Xn ) χn θξ n 2a2n

(8)

with the intermediate virtual control signals αi that are described as (7), and the adaptive law that is defined as .

θˆ =

n  r 2 T ˆ 2 χi ξi (Xi ) ξi (Xi ) − k0 θ 2a i i=1

(11)

(9)

where positive constants ai (i = 1, . . . , n), r, and k0 are design parameters so that the closed-loop system can be guaranteed

δ0 (X0 ) ≤ ε0 where Φ0 = (Φ10 , . . . , Φn 0 ) δ0 (X0 ) = (δ10 (X0 ) , . . . , δn 0 (X0 ))T . As ξ0T ξ0 ≤ 1, and according to the definition of θ, we know Φ0 2 ≤ θ. Therefore, the following inequality holds:   2eT P F = 2eT P ΦT0 ξ0 (X0 ) + δ0 (X0 ) ≤ e2 + P 2 θ + P 2 ε20 .

(12)

By the definition of D, we have

 2 ¯ 2eT P D ≤ e2 + P 2 D T  ¯ = d¯1 , d¯2 , . . . , d¯n . where D Similarly

(13)

1 −2 2 1 2 ¯2 ρ χ1 + ρ d 1 (14) 2 2 1 χ1 e2 ≤ e22 + χ21 (15) 4 where ρ is a positive constant. Substituting (3) and the inequalities (12)–(15) into (11), the time derivative of V1 is rewritten as χ1 d 1 ≤

V˙ 1 ≤ − [λm in (Q) − 3] e2   1 1 + χ1 x ˆ2 + f1 + ρ−2 χ1 + χ1 − y˙ d 2 4 + P 2 θ + P 2 ε20

ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

 ∂α1 ˙ = V 1 + χ2 x ˆ1 − (ˆ x2 + e2 + f1 + d1 ) ˆ 3 − l2 x ∂x1  ∂α1 ˆ. ∂α1 ∂α1 (2) − y˙ d − y θ− . (23) ∂yd ∂ y˙ d d ∂ θˆ

 2 1 2 2 1 . ¯  + ρ d¯1 − θ˜θˆ + P  D 2 r   2 ˆ2 + f¯1 = − [λm in (Q) − 3] e + χ1 x  2 ¯ + P 2 θ + P 2 ε20 + P 2 D 2

1 1 1 . + ρ2 d¯21 − χ21 − θ˜θˆ 2 2 r

(16)

where 1 1 1 f¯1 (X1 ) = f1 + ρ−2 χ1 + χ1 − y˙ d + χ1 . 2 4 2 Now, take the intermediate control signal α ˆ 1 (X1 ) as   ¯ α ˆ 1 (X1 ) = − k1 χ1 + f1

Using a similar way as (14) and (15) in Step 1, the following inequalities can be obtained: 2  ∂α1 1 ∂α1 2 −χ2 e2 ≤ e2 + χ22 (24) ∂x1 4 ∂x1  2 ∂α1 1 −2 ∂α1 1 d1 ≤ ρ χ22 + ρ2 d¯12 . (25) −χ2 ∂x1 2 ∂x1 2 Then, the substitution of (24) and (25) into (23) gives

where k1 > 0. Then, we have

V˙ 2 ≤ − [λm in (Q) − 4] e2 − k1 χ21 + χ1 χ2  . r 2 T 1 ˆ + Δ1 + θ˜ χ ξ ξ − θ 1 r 2a21 1 1  1 2 ¯2 + ρ d 1 + χ2 x ˆ1 ˆ 3 − l2 x 2 2  ∂α1 1 ∂α1 − (ˆ x2 + f1 ) + χ2 ∂x1 4 ∂x1  2 ∂α1 1 ∂α1 ˆ. + ρ−2 χ2 − θ 2 ∂x1 ∂ θˆ  ∂α1 (2) ∂α1 y˙ d − y − ∂yd ∂ y˙ d d

2

V˙ 1 ≤ − [λm in (Q) − 3] e + χ1 (ˆ x2 − α ˆ1 )  2 ¯ − k1 χ2 + P 2 θ + P 2 ε2 + P 2 D 1

0

.

1 1 1 ˆ + ρ2 d¯12 − χ21 − θ˜θ. 2 2 r

(17)

However, α ˆ 1 (X1 ) is an unknown nonlinear function as it contains f1 (x1 ), which cannot be implemented in practice. Therefore, according to Lemma 1, for any given constant ε1 > 0, there exists a fuzzy logic system ΦT1 ξ1 (X1 ) such that α ˆ 1 (X1 ) = ΦT1 ξ1 (X1 ) + δ1 (X1 ) |δ1 (X1 )| ≤ ε1 .

(18)

From the definitions of θ and α1 , we have ˆ 1 = −χ1 −χ1 α ≤

ΦT1 Φ1 

= − [λm in (Q) − 4] e2 − k1 χ21 + χ1 χ2  . r 2 T 1 ˆ + θ˜ χ ξ ξ − θ + Δ1 1 r 2a21 1 1   1 ˆ3 + f¯2 + ρ2 d¯21 + χ2 x 2   ∂α1 ˆ. 1 + χ2 ϕ2 (X2 ) − θ − χ22 ˆ 2 ∂θ

ξ1 Φ1  − χ1 δ1

1 2 T 1 1 1 χ1 θξ1 ξ1 + a21 + χ21 + ε21 2 2a1 2 2 2

(19)

1 2ˆ T χ θξ ξ1 . 2a21 1 1

(20)

χ1 α1 = −

where

Then, the substitution of (19) and (20) into (17) yields

∂α1 f¯2 (X2 ) = −l2 x ˆ1 − (ˆ x2 + f1 ) ∂x1 2  2  ∂α1 1 ∂α1 1 + χ2 + ρ−2 χ2 4 ∂x1 2 ∂x1

V˙ 1 ≤ − [λm in (Q) − 3] e2 + χ1 (ˆ x2 − α1 ) − k1 χ21  . r 2 T 1 ˆ + θ˜ χ ξ ξ − (21) θ + Δ1 1 r 2a21 1 1 where

 2 ¯ Δ1 = P 2 θ + P 2 ε20 + P 2 D



1 1 1 + ρ2 d¯12 + a21 + ε21 . 2 2 2 ˆ2 − α1 , and consider the Step 2: Define the variable χ2 = x following Lyapunov function: 1 V2 = V1 + χ22 . 2 The time derivative of V2 is V˙ 2 = V˙ 1 + χ2 χ˙ 2

975

(22)

∂α1 ∂α1 (2) 1 y˙ d − y + χ2 − ϕ2 (X2 ) ∂yd ∂ y˙ d d 2

with r ∂α1 ϕ2 (X2 ) = −k0 θˆ − χ2 2 ˆ 2a ∂θ 2 ∂α1 r 2 T + 2 χ1 ξ1 ξ1 . ∂ θˆ 2a1

   ∂α1   χ2  ∂ θˆ 

Now, take the intermediate control signal α ˆ 2 (X2 ) as   α ˆ 2 (X2 ) = − k2 χ2 + χ1 + f¯2

(26)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

 m ∂αm −1 ˆ.  ∂αm −1 (i) y θ− . − (i−1) d ∂ θˆ ∂y

where k2 > 0; then, addition and subtraction of α ˆ 2 (X2 ) in (26) yield V˙ 2 ≤ − [λm in (Q) − 4] e2 −

2 

i=1

ki χ2i

Using a similar way as in Step 1 and Step 2, the following inequalities can be obtained: 2  ∂αm −1 1 ∂αm −1 2 e2 ≤ e2 + χ2m (34) −χm ∂x1 4 ∂x1  2 ∂αm −1 1 −2 ∂αm −1 1 d1 ≤ ρ χ2m + ρ2 d¯21 . (35) −χm ∂x1 2 ∂x1 2

i=1

 . r 2 T 1 ˆ + χ2 (ˆ x3 − α ˆ 2 ) + θ˜ χ ξ ξ − θ 1 r 2a21 1 1 1 + Δ1 + ρ2 d¯21 2   ∂α1 ˆ. 1 + χ2 ϕ2 (X2 ) − θ − χ22 . ˆ 2 ∂θ

(27)

Since α ˆ 2 (X2 ) is also an unknown nonlinear function, the fuzzy logic system ΦT2 ξ2 (X2 ) is now employed to approximate it. For any given constant ε2 > 0, there exists a fuzzy logic system ΦT2 ξ2 (X2 ) such that

Substituting (34) and (35) into (33), we get V˙ m ≤ − [λm in (Q) − (2 + m)] e2 −

(28)

1 2 T 1 1 1 χ θξ ξ2 + a22 + χ22 + ε22 2a22 2 2 2 2 2

(29)

1 2ˆ T χ θξ ξ2 . 2a22 2 2

(30)

χ2 α2 = −

  ∂αi−1 ˆ. χi ϕi (Xi ) − θ ∂ θˆ i=2  ∂αm −1 + χm x ˆ1 − (ˆ x2 + f1 ) ˆm +1 − lm x ∂x1 +

Then, by substituting (29) and (30) into (27), we have V˙ 2 ≤ − [λm in (Q) − 4] e2 −

2 



i=1

1 + 4

(31)



m −1  i=1

∂αm −1 (ˆ xi+1 − li x ˆ1 ) ∂x ˆi

∂αm −1 ∂x1

2

d

1 χm + ρ−2 2



∂αm −1 ∂x1 m −1 



2 χm

ki χ2i

i=1

1 + χm −1 χm + Δm −1 + ρ2 d¯21 2  m −1 . 1˜  r 2 T + θ χ ξ ξi − θˆ r 2a2i i i i=1

Step m: Similarly, for each step m (3 ≤ m ≤ n − 1), define χm = x ˆm − αm −1 , and choose the Lyapunov function candidate 1 Vm = Vm −1 + χ2m . (32) 2 The time derivative of Vm is ∂αm −1 (ˆ x2 + e2 + f1 + d1 ) ∂x1



= − [λm in (Q) − (2 + m)] e2 −

2 2 2  i 2 ¯2 1  2 1  2 ρ d1 + + ai + ε . 2 2 i=1 2 i=1 i i=1



∂αm −1 (ˆ xi+1 − li x ˆ1 ) ∂x ˆi m

 2 ¯ Δ2 = P 2 θ + P 2 ε20 + P 2 D

V˙ m = V˙ m −1 + χm [ˆ xm +1 − lm x ˆ1

m −1 

∂αm −1 ˆ.  ∂αm −1 (i) y θ− − (i−1) d ∂ θˆ ∂y

i=1

where

m −1 

i=1

ki χ2i + χ2 (ˆ x3 − α2 )

  2 . 1˜  r 2 T ˆ + θ χ ξ ξi − θ r 2a2i i i i=1   ∂α1 ˆ. + χ2 ϕ2 (X2 ) − θ + Δ2 ∂ θˆ

ki χ2i

i=1

Using a similar procedure as in (19) and (20), we can get ˆ2 ≤ −χ2 α

m −1 

1 + χm −1 χm + Δm −1 + ρ2 d¯21 2 m −1  . 1˜  r 2 T ˆ + θ χ ξ ξi − θ r 2a2i i i i=1

α ˆ 2 (X2 ) = ΦT2 ξ2 (X2 ) + δ2 (X2 ) |δ2 (X2 )| ≤ ε2 .

(33)

d

+

  ∂αi−1 ˆ. χi ϕi (Xi ) − θ ∂ θˆ i=2

m 

  1 + χm x ˆm +1 + f¯m − χ2m 2 where ∂αm −1 ˆ1 − (ˆ x2 + f1 ) f¯m (Xm ) = −lm x ∂x1 −

m −1  i=1

∂αm −1 (ˆ xi+1 − li x ˆ1 ) ∂x ˆi

(36)

ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS



m  ∂αm −1

(i)

y + (i−1) d

i=1

∂yd 

1 + ρ−2 2

∂αm −1 ∂x1

1 4



∂αm −1 ∂x1

2

where χm ∂αi−1 − ϕi (Xi ) = −k0 θˆ ∂ θˆ

2 χm

∂αm −1 ϕm (Xm ) = −k0 θˆ ∂ θˆ +

m −1 

Step n: Define the variable as χn = x ˆn − αn −1 , and consider the following Lyapunov function: 1 Vn = Vn −1 + χ2n . 2

ˆ m (Xm ) in where km > 0; then, addition and subtraction of α (36) yield V˙ m ≤ − [λm in (Q) − (2 + m)] e2 m 



ki χ2i + χm (ˆ xm +1 − α ˆm )



  ∂αi−1 ˆ. χi ϕi (Xi ) − θ ∂ θˆ i=2

m 

1 1 + Δm −1 + ρ2 d¯21 − χ2m . (37) 2 2 Similarly, α ˆ m (Xm ) can be approximated by the fuzzy logic system ΦTm ξm (Xm ) as α ˆ m (Xm ) = ΦTm ξm (Xm ) + δm (Xm ) |δm (Xm )| ≤ εm .

(ˆ xi+1 − li x ˆ1 ) −

∂x ˆi

∂αn −1 ˆ. θ ∂ θˆ



n  ∂αn −1

(i) y (i−1) d ∂y i=1 d

.

(43)

Similar to the aforementioned steps, we can obtain the following inequalities: 2  ∂αn −1 1 ∂αn −1 e2 ≤ e22 + χ2n (44) −χn ∂x1 4 ∂x1  2 ∂αn −1 ∂αn −1 1 1 d1 ≤ ρ−2 χ2n + ρ2 d¯21 . (45) −χn ∂x1 2 ∂x1 2 Substituting (44) and (45) into (43), we get

(38) V˙ n ≤ − [λm in (Q) − (2 + n)] e2 −

We can also get −χm α ˆm ≤

n −1  ∂αn −1 i=1

m −1  . 1˜  r 2 T + θ χ ξ ξi − θˆ r 2a2i i i i=1

1 2 T 1 1 1 χ θξ ξm + a2m + χ2m + ε2m (39) 2a2m m m 2 2 2

χm αm = −

1 2 ˆT χ θξ ξm . 2a2m m m

(40)

Then, by substituting (39) and (40) into (37), we have V˙ m ≤ − [λm in (Q) − (2 + m)] e2 −

m  i=1

+

i=2

+ Δm

χi

ki χ2i

1 + χn −1 χn + Δn −1 + ρ2 d¯21 2  n −1 . 1˜  r 2 T + θ χ ξ ξi − θˆ r 2a2i i i i=1 n −1  i=2

 χi

 ∂αi−1 ˆ. θ ϕi (Xi ) − ∂ θˆ

 ∂αn −1 + χn u − ln x ˆ1 − (ˆ x2 + f1 ) ∂x1

 m . 1˜  r 2 T ˆ + θ 2 χi ξi ξi − θ r 2a i i=1 

n −1  i=1

+ ki χ2i + χm (ˆ xm +1 − αm )



m 

(42)

The time derivative of Vn is given by  ∂αn −1 V˙ n = V˙ n −1 + χn u − ln x ˆ1 − (ˆ x2 + e2 + f1 + d1 ) ∂x1

i=1

+

   ∂αj −1    χj  ∂ θˆ 

m m m  i 2 ¯2 1  2 1  2 ρ d1 + ai + ε . 2 2 i=1 2 i=1 i i=1

+

Take the intermediate control signal α ˆ m (Xm ) as   α ˆ m = − km χm + χm −1 + f¯m



r χi 2 2ai j =2

 2 ¯ Δm = P 2 θ + P 2 ε20 + P 2 D

  m  r  ∂αj −1  − χm 2  χ j 2am ∂ θˆ  j =2

∂αm −1 r 2 T 2 χj ξj ξj . ∂ θˆ 2aj

j =1

i 

i−1  ∂αi−1 r 2 T 2 χj ξj ξj ∂ θˆ 2aj j =1

+

1 + χm − ϕm (Xm ) 2 with

977



 ∂αi−1 ˆ. θ ϕi (Xi ) − ∂ θˆ

n −1  ∂αn −1 i=1

∂x ˆi

(ˆ xi+1 − li x ˆ1 )

∂αn −1 ˆ.  ∂αn −1 (i) y θ− (i−1) d ∂ θˆ ∂y n

(41)



i=1

d

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

1 + 4



∂αn −1 ∂x1

2

1 χn + ρ−2 2



∂αn −1 ∂x1

= − [λm in (Q) − (2 + n)] e2 −

n −1 



2

Following the similar procedure and by the definition of u, we have 1 1 1 1 ˆ n ≤ 2 χ2n θξnT ξn + a2n + χ2n + ε2n (49) −χn α 2an 2 2 2

χn

ki χ2i

χn u = −

i=1

1 + χn −1 χn + Δn −1 + ρ2 d¯21 2  n −1 . 1˜  r 2 T + θ χ ξ ξi − θˆ r 2a2i i i i=1 +

V˙ n ≤ − [λm in (Q) − (2 + n)] e2 −   n . 1˜  r 2 T ˆ + θ 2 χi ξi ξi − θ r 2a i i=1

  ∂αi−1 ˆ. χi ϕi (Xi ) − θ ∂ θˆ i=2

  1 + χn u + f¯n − χ2n 2

(46)

+ where

∂αn −1 f¯n (Xn ) = −ln x ˆ1 − (ˆ x2 + f1 ) ∂x1 ∂x ˆi

i=1



(ˆ xi+1 − li x ˆ1 ) (i)

y + (i−1) d

∂yd 

1 + ρ−2 2

∂αn −1 ∂x1

2



χi

1 4



 ∂αi−1 ˆ. θ + Δn ϕi (Xi ) − ∂ θˆ

n n n  i 2 ¯2 1  2 1  2 ρ d1 + ai + ε . 2 2 i=1 2 i=1 i i=1

2 χn

V˙ n ≤ − [λm in (Q) − (2 + n)] e2 −

+ Δn +

n 

 χi

i=2

Take the intermediate control signal α ˆ n (Xn ) as 

¯n + +Δ

i=1

 n −1 . 1˜  r 2 T ˆ + θ χ ξ ξi − θ r 2a2i i i i=1 +

n 

χi

i=2

+ Δn −1

n 

ki χ2i −

i=1

k0 ˜2 θ 2r

 ∂αi−1 ˆ. θ ϕi (Xi ) − ∂ θˆ

V˙ n ≤ − [λm in (Q) − (2 + n)] e2 −

n 

ki χ2i −

i=1

Denote (47)

shown

that

∂ θˆ

i=2

 ∂αi−1 ˆ. θ ϕi (Xi ) − ∂ θˆ

1 1 + ρ2 d¯21 − χ2n . 2 2

χi

i=2

k0 ˜ˆ θθ r

 ∂αi−1 ˆ. θ ϕi (Xi ) − ∂ θˆ

¯ n = Δn + k0 θ2 . Δ 2r From the work in [31], it can be . n

χi ϕi (Xi ) − ∂ α i −1 θˆ ≤ 0; therefore

ki χ2i + χn (u − α ˆn )





ki χ2i +

where

V˙ n ≤ − [λm in (Q) − (2 + n)] e2 −

n 

n  i=1

≤ − [λm in (Q) − (2 + n)] e2 −

where kn > 0; then, addition and subtraction of α ˆ n (Xn ) in (46) yield

n 

i=1

ˆ we can get By the definition of θ,

1 χn + χn − ϕn (Xn ) . 2

α ˆ n = − kn χn + χn −1 + f¯n

ki χ2i

.

∂αn −1 ∂x1



n 

 2 ¯ Δn = P 2 θ + P 2 ε20 + P 2 D +

n  ∂αn −1 i=1

n  i=2

where

n −1  ∂αn −1

(50)

Then, by substituting (49) and (50) into (47), we have

n 



1 2ˆ T χ θξ ξn . 2a2n n n



c = min

k0 ˜2 ¯ θ + Δn . 2r

 λm in (Q) − (2 + n) , 2ki , k0 ; i = 1, . . . , n . λm ax (P )

Then, we get Similar to the aforementioned steps, α ˆ n (Xn ) can be approximated by the fuzzy logic system ΦTn ξn (Xn ) as α ˆ n (Xn ) = ΦTn ξn (Xn ) + δn (Xn ) |δn (Xn )| ≤ εn .

(48)

¯ n. V˙ n ≤ −cVn + Δ Therefore, the following inequality holds:  ¯n  ¯n Δ Δ , Vn (t) ≤ Vn (0) − e−ct + c c

t ≥ 0.

(51)

ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

979

Choose λm in (Q) − (3 + n) > 0; then, (51) implies that the sigˆ αi , and u are bounded. Therefore, for any ˆi , χi , θ, nals xi , x given ε > 0, by appropriately choosing the design parameters ki and k0 and choosing parameters ai , εi , and ρ to be sufficiently small, as well as r to be sufficiently large, it is possible to make ¯ n /c) ≤ (ε2 /2). Therefore, from (51), the following holds: (Δ  ¯n  ¯n Δ Δ χ21 ≤ 2 Vn (0) − e−ct + 2 c c which means lim sup χ21 ≤ 2 t→∞

¯n Δ ≤ ε2 . c

Remark 1: It should be pointed out that the fuzzy logic system is directly used to approximate the intermediate control signal α ˆ i (i = 1, 2, . . . , n) rather than the unknown nonlinear function fi (i = 1, 2, . . . , n). In addition, if the design parameters ai , k0 , ki , and r are chosen appropriately, the tracking error can converge to a small neighborhood of the origin. Remark 2: The main contribution of this paper is utilizing an input-driven filter to design a fuzzy adaptive output-feedback controller for SISO strict-feedback nonlinear systems. In addition, a direct adaptive fuzzy control method is used, which can reduce the computation burden, because only one parameter needs to be estimated online.

Fig. 1.

System output y (“-”) and reference signal y d (“-.-”) of Example 1.

Fig. 2.

Trajectory of the tracking error y − y d of Example 1.

IV. SIMULATION In this section, two examples are presented to demonstrate the effectiveness of our main results. Example 1 [26]: Consider the following system: x˙ 1 = x2

The adaptive law is given as

x˙ 2 = −0.1x2 − x31 + 12 cos (t) + u y = x1 .

.

(52)

The reference signal is given as yd = sin (t). Choose the following fuzzy membership functions:     (x + 1.5)2 −(x + 1)2 μF i1 (x) = exp , μF i2 (x) = exp 2 2     −(x + 0.5)2 −x2 μF i3 (x) = exp , μF i4 (x) = exp 2 2     2 −(x − 0.5) −(x − 1)2 μF i5 (x) = exp , μF i6 (x) = exp 2 2   2 −(x − 1.5) μF i7 (x) = exp . (53) 2 According to Theorem 1, the virtual control function α1 and the true control law u are chosen, respectively, as α1 = −

1 ˆ T ξ1 , χ1 θξ 1 2a21

u=−

ˆ2 − α1 . where χ1 = y − yd , and χ2 = x

1 ˆ T ξ2 χ2 θξ 2 2a22

(54)

θˆ =

2  r 2 T ˆ 2 χi ξi (Xi ) ξi (Xi ) − k0 θ. 2a i i=1

(55)

In the simulation, we choose design parameters l1 = 144, l2 = 24, a1 = 0.305, a2 = 0.215, r = 8.5, k0 = 0.005, and ˆ1 (0), x ˆ2 (0)]T = the same initial conditions [x1 (0), x2 (0), x T [0.2, 0.2, 1.5, 1.5] as in [26, Ex. 1] and θˆ (0) = 0. The simulation results are shown by Figs. 1– 4, respectively. Fig. 1 shows the system output y and reference signal yd . Fig. 2 depicts the trajectory of the tracking error y − yd . Fig. 3 plots the trajectory of input u. Fig. 4 illustrates the trajectory of adaptive ˆ By comparing the simulation results with the ones parameter θ. in [26], it can be seen that the method in this paper cannot only achieve the tracking performance but requires a smaller control gain as well. In addition, the number of adaptive parameter is only one in our paper, while the number is 7 in [26]. Example 2 [26]: To further show the effectiveness of our result, we consider the following nonlinear dynamic system: x˙ 1 = x2 + x1 e−0.5x 1   x˙ 2 = u + x1 sin x22 y = x1 .

(56)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

Fig. 3.

Control law u of Example 1.

Fig. 6.

Trajectories of the tracking errors y − y d of Example 2.

Fig. 4.

Adaptive parameter θˆ of Example 1.

Fig. 7.

Control law u of Example 2.

Fig. 5.

System output y (“-”) and reference signal y d (“-.-”) of Example 2.

Fig. 8.

Adaptive parameter θˆ of Example 2.

The tracking reference signal is yd = (1/2) sin(t). For this system, we still consider the fuzzy membership functions that are defined in (53). Similarly, the direct adaptive fuzzy controller for system (56) is designed by Theorem 1. The virtual control function α1 , the true control law u, and the adaptive law θˆ are chosen as (54) and (55). In the simulation, the design parameters are chosen as l1 = 5, l2 = 1, a1 = 0.298, a2 = 0.215,

r = 8.6, and k0 = 0.005. The initial conditions are chosen as ˆ1 (0), x ˆ2 (0)]T = [0, −0.2, 0, 0.3]T , which is the [x1 (0), x2 (0), x same as [26, Ex. 2]. The simulation results are shown by Figs. 5– 8, from which we can see that the tracking performance is achieved well as in [26]. However, the computation burden is reduced, because in our paper, the adaptive parameter is only one, while the number is 10 in [26].

ZHOU et al.: ADAPTIVE OUTPUT-FEEDBACK FUZZY TRACKING CONTROL FOR A CLASS OF NONLINEAR SYSTEMS

From the two examples, we can see that the direct adaptive fuzzy control method that is proposed in this paper can achieve the tracking performance well. It is worth mentioning that the adaption law that is proposed by this method is only one. Therefore, the computation burden can be significantly reduced. V. CONCLUSION By using Fuzzy logic systems and backstepping approach, a direct adaptive fuzzy tracking control scheme has been proposed in this paper. Additionally, by introducing the input-driven filter, a fuzzy adaptive output-feedback controller has been constructed. The method that has been proposed in this paper can be applied to a class of systems with unmeasurable states. In addition, as the number of the online adaptive parameter is only one, the computation burden can be reduced accordingly; therefore, it is convenient to implement this algorithm in practical systems. Finally, two simulation examples have been presented to illustrate the effectiveness of the method that is proposed in this paper. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the reviewers for their very constructive comments and suggestions that have helped improve the quality and presentation of this paper. REFERENCES [1] S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Control, vol. 34, no. 11, pp. 1123–1131, May 1989. [2] Z. Jiang and D. Hill, “A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics,” IEEE Trans. Automat. Control, vol. 44, no. 9, pp. 1705–1711, Sep. 1999. [3] W. Lin and C. Qian, “Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback,” Automatica, vol. 37, no. 7, pp. 1093–1101, 2001. [4] B. Chen, Y. Yang, B. Lee, and T. Lee, “Fuzzy adaptive predictive flow control of ATM network traffic,” IEEE Trans. Fuzzy Syst., vol. 11, no. 4, pp. 568–581, Aug. 2003. [5] B. Chen, B. Lee, and S. Chen, “Adaptive power control of cellular CDMA systems via the optimal predictive model,” IEEE Trans. Wireless Commun., vol. 4, no. 4, pp. 1914–1927, Jul. 2005. [6] J. Spooner and K. Passino, “Stable adaptive control of a class of nonlinear systems and neural network,” IEEE Trans. Fuzzy Syst., vol. 4, no. 3, pp. 339–359, Aug. 1996. [7] J. Park, S. Seo, and G. Park, “Robust adaptive fuzzy controller for nonlinear system using estimation of bounds for approximation errors,” Fuzzy Sets Syst., vol. 133, no. 1, pp. 19–36, 2003. [8] S. Ge and C. Wang, “Direct adaptive NN control of a class of nonlinear systems,” IEEE Trans. Neural Netw., vol. 13, no. 1, pp. 214–221, Jan. 2002. [9] Y. Zhang, P. Peng, and Z. Jiang, “Stable neural controller design for unknown nonlinear systems using backstepping,” IEEE Trans. Neural Netw., vol. 11, no. 6, pp. 1347–1360, Nov. 2000. [10] C. Kwan and F. Lewis, “Robust backstepping control of nonlinear systems using neural networks,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 30, no. 6, pp. 753–766, Nov. 2000. [11] T. Zhang, S. Ge, and C. Hang, “Adaptive neural network control for strictfeedback nonlinear systems using backstepping design,” Automatica, vol. 36, no. 12, pp. 1835–1846, 2000. [12] M. Wang, B. Chen, and P. Shi, “Adaptive neural control for a class of perturbed strict-feedback nonlinear time-delay systems,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 3, pp. 721–730, Jun. 2008. [13] C. Hua, X. Guan, and P. Shi, “Robust output feedback tracking control for time-delay nonlinear systems using neural network,” IEEE Trans. Neural Netw., vol. 18, no. 2, pp. 495–505, Mar. 2007.

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[14] B. Chen and X. Liu, “Fuzzy approximate disturbance decoupling of MIMO nonlinear systems by backstepping and application to chemical processes,” IEEE Trans. Fuzzy Syst., vol. 13, no. 6, pp. 832–847, Dec. 2005. [15] S. Zhou, G. Feng, and C. Feng, “Robust control for a class of uncertain nonlinear systems: Adaptive fuzzy approach based on backstepping,” Fuzzy Sets Syst., vol. 151, no. 1, pp. 1–20, 2005. [16] M. Wang, B. Chen, X. Liu, and P. Shi, “Adaptive fuzzy tracking control for a class of perturbed strict-feedback nonlinear time-delay systems,” Fuzzy Sets Syst., vol. 159, no. 8, pp. 949–967, 2008. [17] S. Tong, Y. Li, and P. Shi, “Fuzzy adaptive backstepping robust control for SISO nonlinear system with dynamic uncertainties,” Inf. Sci., vol. 179, no. 9, pp. 1319–1332, 2009. [18] T. Wang, S. Tong, and Y. Li, “Robust adaptive fuzzy control for nonlinear system with dynamic uncertainties based on backstepping,” Int. J. Innovat. Comput., Inf. Control, vol. 5, no. 9, pp. 2675–2688, 2009. [19] S. Tong, Y. Li, and T. Wang, “Adaptive fuzzy backstepping fault-tolerant control for uncertain nonlinear systems based on dynamic surface,” Int. J. Innovat. Comput., Inf. Control, vol. 5, no. 10, pp. 3249–3261, 2009. [20] H. Han, “Adaptive fuzzy control for a class of uncertain nonlinear systems via LMI approach,” Int. J. Innovat. Comput., Inf. Control, vol. 6, no. 1, pp. 275–286, 2010. [21] H. Jiang, C. Zhou, and J. Yu, “Robust fuzzy control of nonlinear discrete fuzzy impulsive delayed systems,” ICIC Exp. Lett., vol. 4, no. 3(B), pp. 973–978, 2010. [22] Z. Wu, X. Xie, P. Shi, and Y. Xia, “Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching,” Automatica, vol. 45, no. 4, pp. 997–1004, 2009. [23] Y. Xia, M. Fu, P. Shi, Z. Wu, and J. Zhang, “Adaptive backstepping controller design for stochastic jump systems,” IEEE Trans. Automat. Control, vol. 54, no. 12, pp. 2853–2859, Dec. 2009. [24] Z. Zhang, S. Xu, Y. Guo, and Y. Chu, “Robust adaptive output-feedback control for a class of nonlinear systems with time-varying actuator faults,” Int. J. Adaptive Control Signal Process., DOI: 10.1002/acs.1165, 2011. [25] C. Liu, S. Tong, and Y. Li, “Adaptive fuzzy backstepping output feedback control of nonlinear systems with unknown sign of high-frequency gain,” ICIC Exp. Lett., vol. 4, no. 5(A), pp. 1689–1694, 2010. [26] S. Tong and Y. Li, “Observer-based fuzzy adaptive control for strictfeedback nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 12, pp. 1749– 1764, 2009. [27] S. Tong, C. Li, and Y. Li, “Fuzzy adaptive observer backstepping control for MIMO nonlinear systems,” Fuzzy Sets Syst., vol. 160, no. 19, pp. 2755– 2775, 2009. [28] B. Chen, X. Liu, and S. Tong, “Adaptive fuzzy output tracking control of MIMO nonlinear uncertain systems,” IEEE Trans. Fuzzy Syst., vol. 15, no. 2, pp. 287–300, Apr. 2007. [29] W. Chen and J. Li, “Decentralized output-feedback neural control for systems with unknown interconnections,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 38, no. 1, pp. 258–266, Feb. 2008. [30] Y. Yang and C. Zhou, “Adaptive fuzzy H ∞ stabilization for strict-feedback canonical nonlinear systems via backstepping and small-gain approach,” IEEE Trans. Fuzzy Syst., vol. 13, no. 1, pp. 104–114, Feb. 2005. [31] B. Chen, X. Liu, K. Liu, and C. Lin, “Direct adaptive fuzzy control of nonlinear strict-feedback systems,” Automatica, vol. 45, no. 6, pp. 1530– 1535, 2009. [32] Z. Jiang, I. Mareels, D. Hill, and J. Huang, “A unifying framework for global regulation via nonlinear output feedback: From ISS to iISS,” IEEE Trans. Automat. Control, vol. 49, no. 4, pp. 549–562, Apr. 2004. [33] H. Lee and M. Tomizuka, “Robust adaptive control using a universal approximator for SISO nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 1, pp. 95–106, Feb. 2000.

Qi Zhou received the B.S. and M.S. degrees in mathematics from Bohai University, Jinzhou, China, in 2006 and 2009, respectively. She is currently working toward the Ph.D. degree with the School of Automation, Nanjing University of Science and Technology, Nanjing, China. She is a Visiting Student with Victoria University, Melbourne, Australia. Her research interest includes fuzzy control, stochastic control, and robust control.

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Peng Shi (SM’97) received the B.Sc. degree in mathematics from the Harbin Institute of Technology, Harbin, China, in 1982, the M.E. degree in systems engineering from Harbin Engineering University in 1985, the Ph.D. degree in electrical engineering from the University of Newcastle, Newcastle, Australia, in 1994, the Ph.D. degree in mathematics from the University of South Australia, Mount Gambier, Australia, in 1998, and the D.Sc. degree from the University of Glamorgan, Pontypridd, U.K., in 2006. He was a Lecturer with Heilongjiang University, Harbin, during 1985–1989 and with the University of South Australia during 1997–1999, and he was a Senior Scientist with Defence Science and Technology Organization, Department of Defence, Australia, during 1999–2005. He joined the University of Glamorgan as a Professor in 2004. Since 2008, he has also been a Professor with Victoria University, Melbourne, Australia. In addition, he is a coauthor of three research monographs: Analysis and Synthesis of Systems with Time-Delays (Berlin, Germany: Springer, 2009), Fuzzy Control and Filtering Design for Uncertain Fuzzy Systems (Berlin, Germany: Springer, 2006), and Methodologies for Control of Jump Time-Delay Systems (Boston, MA: Kluwer, 2003). His research interests include control system design, fault detection techniques, Markov decision processes, and operational research. He has published a number of papers in these areas. Dr. Shi is the Editor-in-Chief of the International Journal of Innovative Computing, Information and Control. He is also an Advisory Board Member, an Associate Editor, and an Editorial Board Member for a number of international journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON SYSTEMS, MAN AND CYBERNETICS–PART B, the IEEE TRANSACTIONS ON FUZZY SYSTEMS, Information Sciences, and the International Journal of Systems Science. He is the recipient of the Most Cited Paper Award of Signal Processing in 2009. He is a Fellow of the Institute of Engineering and Technology (U.K.) and the Institute of Mathematics and its Applications (U.K.).

Jinjun Lu was born in 1964. He received the B.S. and M.S. degrees in mathematics from Nanjing University of Technology, Nanjing, China, and Hohai University, respectively. He is currently a Professor with Nantong Vocational College and College of Electrical Engineering, Nantong University, Nantong, China. His current research interests include congestion control, robust control, and fault-tolerant control.

Shengyuan Xu received the B.Sc. degree from the Hangzhou Normal University, Hangzhou, China, in 1990, the M.Sc. degree from the Qufu Normal University, Qufu, China, in 1996, and the Ph.D. degree from the Nanjing University of Science and Technology, Nanjing, China, in 1999. From 1999 to 2000, he was a Research Associate with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong. From December 2000 to November 2001 and December 2001 to September 2002, he was a Postdoctoral Researcher with the Center for Systems Engineering and Applied Mechanics, Catholic University of Louvain, Louvain-la-Neuve, Belgium, and the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, respectively. From September 2002 to September 2003 and September 2003 to September 2004, he was a William Mong Young Researcher and an Honorary Associate Professor, respectively, with the Department of Mechanical Engineering, The University of Hong Kong, Hong Kong. Since November 2002, he has been a Professor with the School of Automation, Nanjing University of Science and Technology. His research interests include robust filtering and control, singular systems, time-delay systems, neural networks, and multidimensional and nonlinear systems. Dr. Xu was a recipient of the National Excellent Doctoral Dissertation Award in 2002 from the Ministry of Education of China. He obtained a grant from the National Science Foundation for Distinguished Young Scholars of China in 2006. He received a Cheung Kong Professorship in 2008 from the Ministry of Education of China. He is a member of the Editorial Boards of Multidimensional Systems and Signal Processing and Circuits, Systems, and Signal Processing.

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IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 34, NO. 3, JUNE 2006. 1013. Fuzzy Logic and Support Vector Machine Approaches to Regime ...

Adaptive Air-to-Ground Secure Communication System ... - IEEE Xplore
Corresponding author, e-mail: [email protected]. Abstract—A novel ... hardware setup for the ADS-B based ATG system is analytically established and ...

Distributed Adaptive Learning of Graph Signals - IEEE Xplore
Abstract—The aim of this paper is to propose distributed strate- gies for adaptive learning of signals defined over graphs. Assuming the graph signal to be ...

Control Design for Unmanned Sea Surface Vehicles ... - IEEE Xplore
Nov 2, 2007 - the USSV, and the actual hardware and software components used for control of ... the control design problem was developed in our previous.