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Adaptive Fuzzy Output-Feedback Controller Design for Nonlinear Systems via Backstepping and Small-Gain Approach Zhi Liu, Fang Wang, Yun Zhang, Xin Chen, and C. L. Philip Chen

Abstract—This paper focuses on an input-to-state practical stability (ISpS) problem of nonlinear systems which possess unmodeled dynamics in the presence of unstructured uncertainties and dynamic disturbances. The dynamic disturbances depend on the states and the measured output of the system, and its assumption conditions are relaxed compared with the common restrictions. Based on an input-driven filter, fuzzy logic systems are directly used to approximate the unknown and desired control signals instead of the unknown nonlinear functions, and an integrated backstepping technique is used to design an adaptive output-feedback controller that ensures robustness with respect to unknown parameters and uncertain nonlinearities. This paper, by applying the ISpS theory and the generalized small-gain approach, shows that the proposed adaptive fuzzy controller guarantees the closed-loop system being semi-globally uniformly ultimately bounded. A main advantage of the proposed controller is that it contains only three adaptive parameters that need to be updated online, no matter how many states there are in the systems. Finally, the effectiveness of the proposed approach is illustrated by two simulation examples. Index Terms—Adaptive fuzzy control, backsteppng technique, output-feedback, small-gain approach.

I. Introduction

O

VER THE PAST two decades, much progress has been made in the adaptive control of nonlinear systems

Manuscript received April 22, 2013; revised October 2, 2013; accepted November 14, 2013. Date of publication December 3, 2013; date of current version September 12, 2014. This work was supported in part by the National Natural Science Foundation of China under Projects 60974047 and U1134004, in part by the Natural Science Foundation of Guangdong Province under Grant S2012010008967, in part by the Science Fund for Distinguished Young Scholars under Grant S20120011437, in part by the 2011 Zhujiang New Star, in part by the Ministry of Education of New Century Excellent Talent under Grant NCET-12-0637, in part by the 973 Program of China under Grant 2011CB013104, and in part by the Doctoral Fund of the Ministry of Education of China under Grant 20124420130001. This paper was recommended by Associate Editor H. Gao. Z. Liu and Y. Zhang are with the Department of Automation, Guangdong University of Technology, Guangdong 510006, China (e-mail: [email protected]; [email protected]). F. Wang is with the Department of Automation, Guangdong University of Technology, Guangdong 510006, China, and also with the College of Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China (e-mail: [email protected]). X. Chen is with the Department of Mechatronics Engineering, Guangdong University of Technology, Guangdong 510006, China (e-mail: [email protected]). C. L. Philip Chen is with the Faculty of Science and Technology, University of Macau, Macau, 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.2013.2292702

[1]–[11]. The early stages of the study were based on the assumption that the uncertain nonlinearities are either to be linearly parameterized or to have a prior knowledge of the bound. To overcome the limitations existing in the abovementioned control approaches, this kind of approximationbased adaptive control, in recent years, has been developed for nonlinear systems with a triangular structure. By combining the backstepping technique and neural network methods, many researchers developed some significant results such as in [12]–[21]. The controllers that were designed in [12]–[21] can not only guarantee all the signals being uniformly ultimately bounded in the closed-loop system, but also achieve a desired control performance. At the same time, for these triangular nonlinear systems, some researchers have combined the backstepping technique with fuzzy logic systems to solve the adaptive controller design problems and have achieved a desired control performance in [22]–[33]. These adaptive control approaches in [12]–[33] can be used to deal with those nonlinear systems without satisfying the matching conditions, and they do not require the unknown nonlinear functions being linearly parameterized. However, these nonlinear systems included only the nonlinear uncertainties, did not contain unmodeled dynamics or dynamical disturbances. In practice, there generally exist the unmodeled dynamics and the dynamic disturbances, and they may result in the instability of the closed-loop system and the severe deterioration of system performances. It is very important to study the nonlinear systems with unmodeled dynamics and dynamical disturbances in the control theory and its applications. To deal with such problems, adaptive backstepping output feedback controllers were proposed in [34] and [35]. In [34] and [35], the authors studied a class of systems which only contained unstructured uncertainties and dynamic disturbances. By combining the backstepping technique with the small-gain approach, the authors used a nonlinear damping technique to cancel the effects of dynamic disturbance and utilized a dynamic bound signal to accommodate the unmodeled dynamics. Recently, the results of [34] and [35] have been extended by [36]–[39] to the systems with unstructured uncertainties, dynamic uncertainties, and dynamic disturbances. Many adaptive fuzzy output feedback control methods were developed. However, these control methods described above have a substantial drawback, that is, the number of adaptation laws depends on the dimension of systems or the number of the fuzzy rule

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LIU et al.: ADAPTIVE FUZZY OUTPUT-FEEDBACK CONTROLLER DESIGN FOR NONLINEAR SYSTEMS

bases. With an increase of dimension or fuzzy rules, the number of parameters to be estimated will increase significantly. Consequently, the online learning time becomes prohibitively large. More recently, to reduce the adaptive parameters, a new adaptive fuzzy control method was presented for nonlinear strict-feedback systems in [40]–[42]. In [40]–[42], the computation burden was greatly reduced because only one parameter needs to be tuned. However, the unmodeled dynamics were neglected and the dynamic disturbances were bounded by constants in [40]–[42]. To our best knowledge, there, to date, have been no results of solving the problem of computation explosion phenomenon for the strict-feedback systems with unmodeled dynamics and dynamic disturbances. It will be very difficult in the theoretical analysis to deal with the problem of computation explosion phenomenon for the nonlinear systems with unmodeled dynamics and dynamic disturbances, and this is a challenging issue. Inspired by the previous works, we will focus on the input-to-state practical stability (ISpS) problem of nonlinear systems which possess unmodeled dynamics in the presence of unstructured uncertainties and dynamic disturbances. The main advantages of this paper are as follows. 1) Compared with [34]–[39], the assumption on the external disturbances i (z, y) of the system has been relaxed. The external disturbances depend on the states and the measured output of the system, and the restrictive conditions are more general than those in the existing researches. This paper has extended the results of [34]–[39]. 2) By applying the estimation of the vector norm of unknown parameters, computation explosion phenomenon is avoided. The algorithm proposed in this paper contains only three adaptive parameters that need to be updated online, no matter how many states there are in the system and how many rules in the fuzzy system are used. Meanwhile, the fuzzy logic system is directly used to approximate the unknown control signal, which makes the control law and the parameter adaptive laws more simple. Therefore, the proposed control method reduces the computation burden significantly, which facilitates its implementation in engineering. The rest of this paper is organized as follows. In the next section, the control problem of systems with unmodeled dynamics is formulated and some preliminaries are given. In Section III, a direct adaptive output-feedback fuzzy controller is designed by using the backstepping technique, and the rigorous stability analysis of the closed-loop system is conducted. The simulation examples are presented to demonstrate the effectiveness of the proposed control method in Section IV. Finally, Section V concludes the work.

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A. Nonlinear Control Problem Consider the following uncertain nonlinear systems: z˙ = q(z, y) x˙ i = xi+1 + fi (y) + i (z, y), i = 1, 2, . . . , n − 1 x˙ n = u + fn (y) + n (z, y) (1)

y = x1

where x = [x1 , x2 , ..., xn ]T ∈ Rn is the state vector of the system, x2 , ..., xn are unmeasured portion of the state, u and y are the input and the output of the system, the z−dynamics shown in (1) are referred to as the unmodeled dynamics, z ∈ Rn0 . i (i = 1, 2, . . . , n) are the dynamic disturbances. It is assumed that i and q are uncertain Lipschitz continuous functions. fi (y) (i = 1, 2, . . . , n) are the unknown smooth functions, and the output y is directly measured. Throughout this paper, the following assumptions are imposed on (1). Assumption 1: For each 1 ≤ i ≤ n, there exist unknown positive constants di1 , di2 such that |i | ≤ di1 ψi1 (|y|) + di2 ψi2 (z)

where ψi1 and ψi2 are known nonnegative smooth functions, and ψi2 (0) = 0. Remark 1: Note that the unmodeled dynamics z and the dynamical disturbances i (z, y) were extensively studied by many references such as [34]–[39]. But our restrictive conditions are different from those in such previous articles as [34]–[39], which depend on the states and the measured output of the system. Meanwhile, the coefficients in front of ψi1 and ψi2 are different, i.e., di1 = di2 . Note that if di1 = di2 , Assumption 1 is the same as those in [34]–[39]. Therefore, this paper has extended the results of [34]–[39]. Assumption 2 [35]: The unmodeled dynamics are exponentially input-to-state practically stable (exp-ISpS), i.e., for the system z˙ = q(z, y), there exists an exp-ISpS Lyapunov function V0 (z) such that α1 (z) ≤ V0 (z) ≤ α2 (z) ∂V0 q(z, x) ≤ −α0 (z) + γ0 (|y|) + d0 ∂z

In this section, the control problem of uncertain nonlinear systems is first formulated. Then, to approximate the desired control signals, the fuzzy logic systems are adopted.

(3) (4)

where functions α0 , α1 , α2 , and γ0 are of class κ∞ -functions, and d0 is a known nonnegative constant. Moreover, γ0 is a known function.

B. ISpS and Small-Gain Theorem Lemma 1 (Small-Gain Theorem [43]): Given the interconnected systems x˙ 1 = f1 (x1 , x2 , u1 ) x˙ 2 = f2 (x1 , x2 , u2 )

II. Problem Formulation and Some Preliminaries

(2)

(5) (6)

where xi ∈ Rni , ui ∈ Rmi , and fi : Rn1 × Rn2 × Rmi → Rni (i = 1, 2) is locally Lipschitz. For i = 1, 2, it is assumed that there exists an ISpS-Lyapunov function Vi for the xi subsystems such that we have the following.

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1) There exist κ∞ -functions ρi1 and ρi2 so that ρi1 (xi ) ≤ Vi (xi ) ≤ ρi2 (xi )

∀xi ∈ R . (7) ni



2) There exist κ∞ -functions αi and κ-functions χi , γi and some constants ci ≥ 0 such that if V1 (x1 ) ≥ max{χ1 (V2 (x2 )), γ1 (u1 ) + c1 )}

III. Adaptive Fuzzy Control Design and ISpS Analysis In this section, to handle unmeasured states, an inputdriven filter is first introduced; then, an adaptive fuzzy outputfeedback control scheme is constructed. Finally, the analysis of input-to-state practical stability is conducted. A. Dynamic Feedback Design

then

∇V1 (x1 )f1 (x1 , x2 , u1 ) ≤ −α1 (V1 ).

(8)

The dynamic feedback strategy begins with an input-driven filter as follows: xˆ˙ i = xˆ i+1 − li xˆ 1 xˆ˙ n = u − ln xˆ 1

If V2 (x2 ) ≥ max{χ2 (V1 (x1 )), γ2 (u2 ) + c2 )} then

∇V2 (x2 )f2 (x1 , x2 , u2 ) ≤ −α2 (V2 ).

(9)

Furthermore, if there exists c0 ≥ 0 such that χ1 ◦ χ2 (s) < s

∀s > c0

(10)

where xˆ i is the estimate of xi and li (i = 1, 2, . . . , n) is the design parameter such that the matrix ⎡ ⎤ −l1 ⎢ ⎥ A = ⎣ ... ⎦ In−1 −ln . . . 0

AT P + PA = −Q.

C. Fuzzy Logic Systems In this paper, a fuzzy logic system will be used to approximate a continuous function f (x) defined on some compact set

. Adopt the singleton fuzzifier, the product inference, and the center-average defuzzifier to deduce the following fuzzy rules: Rl : If x1 is F1l and . . . and xn is Fnl , Then y is Gl , l = 1, 2, . . . , N where x = [x1 , x2 , ..., xn ]T ∈ Rn and y ∈ R are the input and the output of the fuzzy system, respectively, Fil and Gl are fuzzy sets in R, and N is the number of the rules. Through the singleton function, the center average defuzzification, and the product inference [44], the output of the fuzzy system is N n l=1 l i=1 μFil (xi ) y(x) = N n l=1 [ i=1 μFil (xi )] where l = max μGl (y),  = (1 , 2 , . . . , N )T . y∈R

n

i=1 μF l (xi ) ξl (x) = N n i l=1 [ i=1 μFil (xi )]

and ξ(x) = (ξ1 (x), ξ2 (x), . . . , ξN (x))T . The fuzzy logic system can be rewritten as y(x) = T ξ(x).

(11)

Lemma 2 [44]: Let f (x) be a continuous function defined on a compact set . Then, for ∀ε > 0, there exists a fuzzy logic system (11) such that sup |f (x) − T ξ(x)| ≤ ε. x∈

(13)

is a strict Hurwitz matrix, that is, for a given matrix Q > 0, there exists a matrix P > 0 such that the following equation holds:

then the interconnected systems (5) and (6) are ISpS.

Let

1≤i≤n−1

(12)

(14)

Define estimate error as e = x − xˆ where xˆ = (ˆx1 , . . . , xˆ n )T . From (1) and (13), the estimate error system is modeled as e˙ = Ae + [F (y) + ]

(15)

where F (y) = (f1 (y) + l1 y, . . . , fn (y) + ln y)T = (F1 (y), . . . , Fn (y))T ,  = (1 , . . . , n )T . B. Adaptive Fuzzy Output-Feedback Controller Design The backstepping design procedure consists of n steps. At Step i (1 ≤ i ≤ n − 1), the desired, but unknown, control signal πˆ i is first considered to stabilize the first i subsystems theoretically. Then, the fuzzy logic system will be employed to approximate this unknown control signal, and consequently, to construct the virtual control signal. The real regulating control law u will be designed at the last step. To develop the backstepping-based design procedure, we first define the following constants:

 θ = max i 2 , i = 0, 1, 2, . . . , n 2 d1 = max{|di1 |, di1 , i = 1, 2, . . . , n} 2 d2 = max{|di2 |, di2 , i = 1, 2, . . . , n}.

Obviously, θ, di (i = 1, 2) are unknown positive constants because i  and di1 are unknown. Define θˆ as the estimate ˆ Define dˆ 1 as the estimate of d1 and d˜ 1 = of θ and θ˜ = θ − θ. ˆ ˆ d1 − d1 . Define d2 as the estimate of d2 and d˜ 2 = d2 − dˆ 2 . At

LIU et al.: ADAPTIVE FUZZY OUTPUT-FEEDBACK CONTROLLER DESIGN FOR NONLINEAR SYSTEMS

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Step i (1 ≤ i ≤ n − 1), the feasible virtual control signal is constructed as θˆ πi (Xi ) = − 2 x¯ i ξiT (Xi )ξi (Xi ) (16) 2ai

By using Assumption 1 and Young’s inequality, we obtain

ˆ y)T , x¯ i = xˆ i − πi−1 (2 ≤ i ≤ n), where Xi = (ˆx1 , . . . , xˆ i , θ, x¯ 1 = y.

≤ 2e2 + d1 P2 +d2 P2

Theorem 1: For system (1) with unmodeled dynamics, if we choose the input driven filter (13), employ a control law as θˆ u = − 2 x¯ n ξnT (Xn )ξn (Xn ) (17) 2an with the intermediate virtual control signals defined by and use the following adaptive laws: n r1 2 T ˆθ˙ = r1 P2 + x¯ ξ (Xi )ξi (Xi ) − k0 θˆ 2 i i 2a i i=1 n

2 n − 1 2 y 2 + P2 ψi1 (|y|) + dˆ˙1 = r2 ψ11 (|y|) 4 4 i=1 n

2 n − 1 y2 2 dˆ˙2 = + P2 ψi2 (z) + (z) r3 ψ12 4 4 i=1

y(e2 + 1 )

(16)

(18)

V˙ 1

(19)

i=1 n

i=1

eP|i | 2

ψi1 (|y|)

2

ψi2 (z)

(25)

i=1 2

y d1 2 (|y|) + y2 + ψ11 4 4 d2 2 2 +ψ12 (z) + y . 4 ≤ e2 +

(26)

≤ −eT Qe + 5e2 + y[ˆx2 + f1 (y)] + P2 θ y 2 d1 2 d2 2 2 +P2 ε20 + + y + y + ψ11 (|y|) 4 4 4 n n ψi1 (|y|))2 + d2 P2 ( ψi2 (z))2 +d1 P2 ( i=1

(20)

i=1

1 1 1 − θ˜ θˆ˙ − d˜ 1 dˆ˙1 − d˜ 2 dˆ˙2 r1 r2  r 3 y ≤ −(λmin (Q) − 5)e2 + y xˆ 2 + f¯ 1 (y) − 2 ˜ ˜ θ˜ d d 1 2 ˆ˙ + (w1 − dˆ˙1 ) + (τ1 − dˆ˙2 ) − (r1 P2 − θ) r1 r2 r3 n 2 (z) + dˆ 2 P2 ( (27) ψi2 (z))2 +σ0 + ψ12 2 +ψ12 (z)

i=1

where

Differentiating V1 yields = eT (AT P + PA)e + 2eT P(F (y) + ) 1 +y[ˆx2 + e2 + f1 (y) + 1 ] − θ˜ θˆ˙ r1 1 1 − d˜ 1 dˆ˙1 − d˜ 2 dˆ˙2 . r2 r3

n



n

Substituting (14) and inequalities (24)–(26) into (22) yields

where the constants ai (i = 1, 2, . . . , n), ri (i = 1, 2, 3), and k0 are positive design parameters; then, the closed-loop system consisting of (1) is ISpS. Proof: Step 1: Consider the following Lyapunov function candidate: x¯ 2 d˜ 2 d˜ 2 θ˜ 2 V1 = eT Pe + 1 + + 1 + 2. (21) 2 2r1 2r2 2r3 V˙ 1

≤ 2eP ≤ 2

2eT P

3 dˆ 1 dˆ 2 = f1 + y + y + y + 11 (y) 4 4  ⎧ 4  2 n 2 2 ⎨ dˆ 1 P ψ (|y|) +ψ11 (|y|) i=1 i1 11 (y) = , y = 0 y ⎩ 0, y = 0 n

2 2 y 2 w1 = + P ψi1 (|y|) 4 i=1 n

2 y2 τ1 = + P2 ψi2 (z) 4 i=1 f¯ 1 (y)

(22)

As F (y) = (F1 (y), . . . , Fn (y))T and Fi (y) (i = 1, 2, . . . , n) is an unknown function, according to Lemma 2, for any given εi0 > 0, there exists a fuzzy logic system Ti0 ξ0 (y) such that Fi (y) = Ti0 ξ0 (y) + δi0 (y), |δi0 (y)| ≤ εi0 .

σ0

Therefore, the following equality holds:

= P2 θˆ + P2 ε20 .

To stabilize this system, take the intermediate control signal πˆ 1 (X1 ) as

F (y) = T0 ξ0 (y) + δ0 (y), δ0 (y) ≤ ε0 where

πˆ 1 (X1 ) = −(k1 x¯ 1 + f¯ 1 ) 0 = (10 , . . . , n0 ), δ0 (y) = (δ10 (y), . . . , δn0 (y))T ,  ε0 = ε210 + . . . + ε2n0 . (23)

As ξ0T ξ0 0 2 ≤

≤ 1, and from the definition of θ, we can obtain θ. Consequently, the following inequality holds: 2eT PF

= 2eT P[T0 ξ0 (y) + δ0 (y)] ≤ 2e2 + P2 θ + P2 ε20 .

(24)

where k1 > 0. Then, the following inequality holds: V˙ 1

≤ −(λmin (Q) − 5)e2 + x¯ 1 [ˆx2 − πˆ 1 (X1 )] − k1 x¯ 12 ˜ ˜ θ˜ ˆ˙ + d1 (w1 − dˆ˙1 ) + d2 (τ1 − dˆ˙2 ) + σ0 − (r1 P2 − θ) r1 r2 r3 n x¯ 2 2 +ψ12 (28) (z) + dˆ 2 P2 ( ψi2 (z))2 − 1 . 2 i=1

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IEEE TRANSACTIONS ON CYBERNETICS, VOL. 44, NO. 10, OCTOBER 2014

τm−1

φi (Xi )

i=1

a12 ε21 a 2 ε2 + = P2 θˆ + P2 ε20 + 1 + 1 2 2 2 2 r1 = r1 P2 + 2 x¯ 12 ξ1T ξ1 . 2a1 = σ0 +

Step m (2 ≤ m ≤ n − 1): It is assumed that the Lyapunov function 1 2 Vm−1 = Vm−2 + x¯ m−1 (32) 2 satisfies the following inequality: V˙ m−1

≤ −(λmin (Q) − (m + 3))e2 −

m−1

+

r3

d˜ 2 + r3

x¯ i+1

i=1

i=2

∂πi  (τm−1 − dˆ˙2 ) ∂dˆ 2

∂πi−1 ˆ˙  x¯ i φi (Xi ) − θ ∂θˆ

(33)

= P2 θˆ + P2 ε20 +

m−1 2 a i

i=1

μm−1 wm−1

2

+

m−1 r1 2 T x¯ ξ ξ 2 i i i 2a i i=1 n

2 y2 + P2 = ψi1 (|y|) 4 i=1

= r1 P2 +

+

m−2 2 (|y|) r2 ψ11 4

+

ki x¯ i2

i

2

1˜ d2 (τm−1 − dˆ˙2 ) + r3

i=1 m−2 i=1

x¯ i+1

∂πi (τm−1 − dˆ˙2 ) ∂dˆ 2 n

m−1 2 ε i=1

m−1

2 m−2ˆ 2 ψi2 (z) d2 ψ12 (z) + dˆ 2 P2 + 4 i=1 +

where σm−1

≤ −(λmin (Q) − (m + 4))e2 −

i=1

n

2 +ψ12 (z) +

∂πm−1 (ˆx2 + e2 + f1 (y) + 1 ) ∂y ∂πm−1 ˆ˙ ∂πm−1 ˆ˙ ∂πm−1 ˆ˙  θ− (35) d1 − − d2 . ∂θˆ ∂dˆ 1 ∂dˆ 2 Using a similar way as in Step 1, we can get the following inequalities: 1 ∂πm−1 2 2 −¯xm ∂π∂ym−1 e2 ≤ |e2 |2 + (36) x¯ m 4 ∂y

∂π  d1 2 d2 2 m−1 2 2 −¯xm ∂π∂ym−1 1 ≤ 2 x¯ m + ψ11 (|y|) + ψ12 (z). ∂y 4 4 (37)

θ˜ ˆ˙ +σm−1 + x¯ m−1 x¯ m − (μm−1 − θ) r1 m−2 1 ∂πi + d˜ 1 (wm−1 − dˆ˙1 ) + x¯ i+1 (wm−1 − dˆ˙1 ) r2 ∂dˆ 1

2 m−2ˆ 2 + d2 ψ12 (z) + dˆ 2 P2 ψi2 (z) 4 i=1 m−1

Choose the Lyapunov function candidate 1 2 Vm = Vm−1 + x¯ m . (34) 2 Then, the time derivative of Vm is m−1

∂πm−1 V˙ m = V˙ m−1 + x¯ m xˆ m+1 − (ˆxi+1 − li xˆ 1 ) ∂ˆxi i=1

V˙ m

θ˜ ˆ˙ +¯xm−1 x¯ m + σm−1 − (μm−1 − θ) r1 m−2 1 ˜ ∂πi  d1 + r 2 + (wm−1 − dˆ˙1 ) x¯ i+1 r2 ∂dˆ 1 i=1 m−2

j=1

∂πi−1 r1 2 T x¯ j ξj ξj . ∂θˆ 2aj2

Based on (33) and (35)–(37), we conclude

ki x¯ i2

i=1

1

i−1

−lm xˆ 1 −

where

μ1

+

(30)

≤ −(λmin (Q) − 5)e2 + x¯ 1 [ˆx2 − π1 ] − k1 x¯ 12 ˜ ˜ θ˜ ˆ˙ + d1 (w1 − dˆ˙1 ) + d2 (τ1 − dˆ˙2 ) + (μ1 − θ) r1 r2 r3 n 2 (31) (z) + dˆ 2 P2 ( +σ1 + ψ12 ψi2 (z))2

σ1

m−2 2 r3 ψ12 (z) 4 i ∂πi−1 r1  ∂πj−1  − = −k0 θˆ x¯ i 2 ¯xj  2ai ∂θˆ ∂θˆ j=2

(29)

Now, substituting (29)–(30) into (28) yields V˙ 1

= +

πˆ 1 (X1 ) = T1 ξ1 (X1 ) + δ1 (X1 ), |δ1 (X1 )| ≤ ε1 . According to the definitions of θ and π1 , we can get θ a2 x¯ 2 ε2 −¯x1 πˆ 1 (X1 ) ≤ 2 x¯ 12 ξ1T ξ1 + 1 + 1 + 1 2 2 2 2a1 ˆθ x¯ 1 π1 = − 2 x¯ 12 ξ1T ξ1 . 2a1

2 y2 + P2 ψi2 (z) 4 i=1 n

As πˆ 1 (X1 ) contains f1 (y), it is an unknown nonlinear function. From Lemma 2, for any given constant ε1 > 0, there exists a fuzzy logic system T1 ξ1 (X1 ) such that

m−1

∂πi−1 ˆ˙  x¯ i φi (Xi ) − θ + x¯ m xˆ m+1 + f¯ m ∂θˆ i=1

+wm

m−2 ∂πm−1 r2 ∂πi 2 + x¯ i+1 ψ11 (|y|) + φm (Xm ) ˆ 4 i=1 ∂ d1 ∂dˆ 1

m−2 ∂πi 2 ∂πm−1 r3 x¯ i+1 ψ12 (z) + τm 4 i=1 ∂dˆ 2 ∂dˆ 2 2 ∂πm−1 ˆ˙ ∂πm−1 ˆ˙ ∂πm−1 ˆ˙  x¯ m θ− − d1 − d2 − 2 ∂θˆ ∂dˆ 1 ∂dˆ 2 ˜1 d d 2 2 2 2 +ψ12 (z) + ψ11 (|y|) + ψ12 (z) 4 4

+

(38)

LIU et al.: ADAPTIVE FUZZY OUTPUT-FEEDBACK CONTROLLER DESIGN FOR NONLINEAR SYSTEMS

where = −lm xˆ 1 −

f¯ m

m−1 ∂πm−1

∂ˆxi

Then, substituting (40)–(41) into (39), we obtain m ki x¯ i2 V˙ m ≤ −(λmin (Q) − (m + 4))e2 −

(ˆxi+1 − li xˆ 1 )

i=1

θ˜ ˆ˙ +¯xm (ˆxm+1 − πm ) + σm − (μm − θ) r1 m−1 1 ˜ ∂πi  d1 + r2 + (wm − dˆ˙1 ) x¯ i+1 ˆ1 r2 ∂ d i=1

i=1 2 (|y|) dˆ 1 ψ11

∂πm−1 x¯ m − + (ˆx2 + f1 (y)) 2 4¯xm ∂y ∂πm−1 9 ∂πm−1 2 + x¯ m − wm 4 ∂y ∂dˆ 1 m−2 ∂πi 2 r2 ∂πm−1 − x¯ i+1 ψ11 (|y|) − τm 4 ∂dˆ 1 ∂dˆ 2 +

−

r3 4

i=1 m−2

x¯ i+1

i=1

1 ˜ ∂πi  d2 + r3 (τ − dˆ˙2 ) x¯ i+1 ˆ2 m r3 ∂ d i=1 n

2 m−1ˆ 2 2 ˆ + d2 ψ12 (z) + d2 P ψi2 (z) 4 i=1 m

∂πi−1 ˆ˙  2 + x¯ i φi (Xi ) − θ + ψ12 (z) (42) ˆ ∂ θ i=2 m−1

+

∂πi 2 ψ12 (z) − φm (Xm ) ∂dˆ 2

r2 2 ψ (|y|) 4 11 r3 2 = τm−1 + ψ12 (z). 4 = wm−1 +

wm τm

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where

Now, take the intermediate control signal πˆ m (Xm ) as

m a2

σm

= P2 θˆ + P2 ε20 +

μm

m r1 2 T = r1 P2 + x¯ ξ ξi . 2ai2 i i i=1

i

i=1

πˆ m (Xm ) = −(¯xm−1 + km x¯ m + f¯ m ), km > 0. Then, we have V˙ m

≤ −(λmin (Q) − (m + 4))e2 −

m

ki x¯ i2

i=1

i

i=1

2

∂πn−1 (ˆx2 + e2 + f1 (y) + 1 ) ∂y ∂πn−1 ˆ˙  ∂πn−1 ˆ˙ ∂πn−1 ˆ˙ d1 − d2 . θ− − ∂θˆ ∂dˆ 1 ∂dˆ 2

∂πi  1 ˜ 2 d2 + r3 (τ − dˆ˙2 ) + ψ12 x¯ i+1 (z) ˆ2 m r3 ∂ d i=1 n

2 m−1ˆ 2 d2 ψ12 (z) + dˆ 2 P2 + ψi2 (z) 4 i=1 m−1

(39)

Similarly, for any given positive constant εm , the fuzzy logic system Tm ξm (Xm ) is utilized to approximate the unknown πˆ m (Xm ) such that πˆ m (Xm ) = Tm ξm (Xm ) + δm (Xm ), |δm (Xm )| ≤ εm .

≤ −(λmin (Q) − (n + 4))e2 −

n−1

ki x¯ i2

i=1

θ 2 T a2 x¯ 2 ε2 x¯ m ξm ξm + m + m + m 2 2am 2 2 2

(40)

θ˜ ˆ˙ +σn−1 + x¯ n−1 x¯ n − (μn−1 − θ) r1 n−2 ∂πi 1 x¯ i+1 (wn−1 − dˆ˙1 ) + d˜ 1 (wn−1 − dˆ˙1 ) + r2 ∂dˆ 1 i=1

x¯ m πm = −

1 2ˆ T x¯ θξ ξ . 2 m m m 2am

(41)

(44)

Similar to the aforementioned steps, the following inequalities hold: 1 ∂πn−1 2 2 −¯xn ∂π∂yn−1 e2 ≤ |e2 |2 + (45) x¯ n 4 ∂y

∂π 2 d1 2 n−1 −¯xn ∂π∂yn−1 1 ≤ 2 x¯ n2 + ψ11 (|y|) ∂y 4 d2 2 + ψ12 (z). (46) 4 Substituting (45)–(46) into (44), we can obtain V˙ n

Using Young’s inequality and the definition of πm yields −¯xm πˆ m (Xm ) ≤

(43)

−ln xˆ 1 −

+

m−1 2 ∂πi−1 ˆ˙  x¯ m x¯ i φi (Xi ) − θ − . 2 ∂θˆ i=2

m ε2

Step n: Choose the Lyapunov function candidate as 1 Vn = Vn−1 + x¯ n2 . 2 The time derivative of Vn is n−1

∂πn−1 V˙ n = V˙ n−1 + x¯ n u − (ˆxi+1 − li xˆ 1 ) ∂ˆxi i=1

θ˜ ˆ˙ +¯xm (ˆxm+1 − πˆ m ) + σm−1 − (μm−1 − θ) r1 m−1 ∂πi  1 ˜ d 1 + r2 (wm − dˆ˙1 ) x¯ i+1 + ˆ r2 ∂ d1 i=1

+

2

+

+

1˜ d2 (τn−1 − dˆ˙2 ) + r3

n−2 i=1

x¯ i+1

∂πi (τn−1 − dˆ˙2 ) ∂dˆ 2

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2 n−2ˆ 2 d2 ψ12 (z) + dˆ 2 P2 ψi2 (z) 4 i=1 n

+

n−1

∂πi−1 ˆ˙  + x¯ i φi (Xi ) − θ + x¯ n u + f¯ n ∂θˆ i=1

+wn

n−1ˆ 2 d2 ψ12 (z) 4 n

2 ψi2 (z) +dˆ 2 P2 2 (z) + +σn + ψ12

n−2 ∂πn−1 r2 ∂πi 2 + x¯ i+1 ψ11 (|y|) + φn (Xn ) ˆ 4 i=1 ∂ d1 ∂dˆ 1

n−2 r3 ∂πi 2 ∂πn−1 + x¯ i+1 ψ12 (z) + τn ˆ 4 i=1 ∂ d2 ∂dˆ 2 ∂πn−1 ˆ˙ ∂πn−1 ˆ˙ ∂πn−1 ˆ˙  x¯ n2 θ− − d1 − d2 − 2 ∂θˆ ∂dˆ 1 ∂dˆ 2 ˜d1 d2 2 2 2 (z) + ψ11 +ψ12 (|y|) + ψ12 (z) 4 4

= −ln xˆ 1 −

n−1 ∂πn−1

∂ˆxi

where = P2 θˆ + P2 ε20 +

μn

n r1 2 T = r1 P2 + x¯ ξ ξi . 2ai2 i i i=1

(47)

(ˆxi+1 − li xˆ 1 )

2

+

n ε2 i

i=1

2

ˆ˙ dˆ˙1 , and dˆ˙2 , we Furthermore, according to the definitions of θ, have n V˙ n ≤ −(λmin (Q) − (n + 4))e2 − ki x¯ i2 i=1

k0 ∂πi−1 ˆ˙  + θ˜ θˆ + x¯ i φi (Xi ) − θ r1 ∂θˆ i=2 n−1ˆ 2 2 (z) + d2 ψ12 (z) +σn + ψ12 4 n

2 +dˆ 2 P2 ψi2 (z) . n

n−2 ∂πn−1 r3 ∂πi 2 − x¯ i+1 ψ12 (z) ˆ 4 i=1 ∂ d1 ∂dˆ 2 ∂πn−1 − φn (Xn ) −τn ∂dˆ 2 r2 2 (|y|) wn = wn−1 + ψ11 4 r3 2 τn = τn−1 + ψ12 (z). 4 Now, take the intermediate control signal πˆ n (Xn ) as

−wn

From [41], the following equality holds: n

∂πi−1 ˆ˙  θ ≤ 0. x¯ i φi (Xi ) − ∂θˆ i=2 ˜ − θ) ˜ ≤ − 1 θ˜ 2 + 1 θ 2 . θ˜ θˆ = θ(θ 2 2 Substituting (52)–(53) into (51), we obtain V˙ n

≤ −(λmin (Q) − (n + 4))e2 −

i=1

n

θ˜ ∂πi−1 ˆ˙  ˆ˙ + − (μn − θ) x¯ i φi (Xi ) − θ r1 ∂θˆ i=2

∂πi  1 ˜ d1 + r2 (wn − dˆ˙1 ) x¯ i+1 + ˆ1 r2 ∂ d i=1

n

ki x¯ i2

i=1

Similar to the aforementioned steps and by the definition of u, we can get

Then, substituting (48)–(49) into (47), we obtain n V˙ n ≤ −(λmin (Q) − (n + 4))e2 − ki x¯ i2

(53)

k0 ˜ 2 n−1ˆ 2 2 − (z) + θ + σ¯ + ψ12 d2 ψ12 (z) 2r1 4 n

2 +dˆ 2 P2 (54) ψi2 (z)

+ δn (Xn ), |δn (Xn )| ≤ εn .

θ 2 T a2 1 1 x¯ n ξn ξn + n + x¯ n2 + ε2n 2 2an 2 2 2 1 ˆ nT ξn . x¯ n u = − 2 x¯ n2 θξ 2an

(52)

On the other hand, we can get

Similarly, for any given positive constant εn , the fuzzy logic system Tn ξn (Xn ) is used to approximate the unknown πˆ n (Xn )

−¯xn πˆ n (Xn ) ≤

(51)

i=1

πˆ n (Xn ) = −(¯xn−1 + kn x¯ n + f¯ n ), kn > 0.

n−1

i

i=1

x¯ n ∂πn−1 + + − (ˆx2 + f1 (y)) 2 4¯xn ∂y n−2 9 ∂πn−1 2 r2 ∂πi 2 + x¯ n − x¯ i+1 ψ11 (|y|) 4 ∂y 4 i=1 ∂dˆ 1

πˆ n (Xn ) =

n a2

σn

i=1 2 dˆ 1 ψ11 (|y|)

Tn ξn (Xn )

(50)

i=1

where f¯ n

∂πi  1 ˜ d 2 + r3 (τ − dˆ˙2 ) x¯ i+1 ˆ2 n r3 ∂ d i=1 n−1

+

i=1

(48)

where σ¯ = σn +

(49)

k0 2 θ . 2r1

As each function ψi2 is smooth and ψi2 (0) = 0, there exists a smooth κ∞ -function β such that the following equality holds: n

2 n−1ˆ 2 2 2 ˆ ψi2 (z) d2 ψ12 (z) + d2 P ψ12 (z) + 4 i=1 ≤ β(z2 ). Let c = min

λ

 − (n + 4) , 2ki , k0 ; i = 1, . . . , n > 0. λmax (P)

min (Q)

LIU et al.: ADAPTIVE FUZZY OUTPUT-FEEDBACK CONTROLLER DESIGN FOR NONLINEAR SYSTEMS

Then, (54) can be expressed as

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Therefore, the following inequality holds:

V˙ n ≤ −cVn + σ¯ + β(z ). 2

(55)

Remark 2: References [36]–[39] used the fuzzy system to approximate the unknown functions fi (y) (i = 1, 2, . . . , n). For any nth-order nonlinear system, if Ni fuzzy sets for y are used in [36]–[39], there will be a total of nNi + 1 parameters to be estimated in their adaptive fuzzy control schemes. This means that there will be many parameters needed to be estimated online. However, the algorithm proposed in this paper contains only three adaptive parameters that need to be updated online, no matter how many rules in the fuzzy system are used, and this algorithm can reduce the computation burden significantly. In addition, the fuzzy logic system in this paper is directly used to approximate the unknown control signal rather than unknown function, which makes the control law and the parameter adaptive laws simpler than those in [36]–[39]. As a result, the proposed controller is easy to implement in practical system. C. Analysis of Input-to-State Practically Stability In order to invoke the small-gain theorem in [43], we organize it as follows. First, from Lemma 1, for any 0 < c1 < c, we can see that if c1 Vn − cVn + σ¯ + β(z2 ) ≤ 0

(56)

V˙ n ≤ −c1 Vn .

(57)

V˙ 0

≤ −c2 α0 (z) + c2 α0 (z) − α0 (z) + γ(Vn ) + d0 ≤ −c2 α0 (z) (64)

as long as the following inequality holds: γ(Vn ) d0 + α0 (z) ≥ 1 − c2 1 − c2 or equivalently V0 ≥ α1 ◦ α−1 0

(65)

γ(V ) d0  n . + 1 − c2 1 − c2

(66)

Consequently, based on (64)–(66), we know that if

2γ(V ) 

2d  n 0 −1 , α V0 ≥ max α1 ◦ α−1 ◦ α 1 0 0 1 − c2 1 − c2

(67)

then (64) holds. Finally, to check the conditions of the small-gain theorem [43, Th. 1], we choose

2γ(s)  2 2β(α−1 1 (s )) . (68) , χ2 (s) = α1 ◦ α−1 χ1 (s) = 0 c − c1 1 − c2 Based on (62) and (67)–(68), the following inequality holds: s (69) χ1 (s) ◦ χ2 (s) = < s. 2 Therefore, from Lemma 1, we obtain that the closed-loop system is ISpS, and all the signals in the closed-loop system are semi-globally uniformly ultimately bounded.

then IV. Simulation Examples

And c1 Vn − cVn + σ¯ + β(z2 ) ≤ 0 is equivalent to

In this section, two examples will be used to test the effectiveness of the proposed scheme. Example 1 [36]. Consider the following second-order nonlinear system: z˙ = −z + 0.125y2

Vn ≥

β(z2 ) σ¯ + . c − c1 c − c1

(58)

From (3), we can get

y = x1 .

z ≤

α−1 1 (V0 (z)).

Based on (56)–(59), we know that if

2β(α−1 (V (z))2 ) 2σ¯  0 1 , Vn ≥ max c − c1 c − c1 2 ¯ β(α−1 (V (z)) ) σ 0 1 ≥ + c − c1 c − c1

(59)

(60)

then (57) holds. Second, for any 0 < c2 < 1 and s > 0, choose κ∞ -function γ that satisfies (61) γ0 (|y|) ≤ γ(Vn ) 

 1 − c2 c − c1 γ(s) < α0 ◦ α−1 α1 β−1 s . (62) 1 2 4 Substituting (61) into (3), we obtain ∂V0 q(z, y) ∂z

x˙ 1 = x2 + y2 + 0.5z2 x˙ 2 = u + y2 sin(y) + z2

≤ −α0 (z) + γ0 (|y|) + d0 ≤ −α0 (z) + γ(Vn ) + d0 .

(63)

(70) 2

Choose ψi1 (|y|) = 0(i = 1, 2), ψ12 (z) = ψ22 (z) = z , then |1 (z, y)| = 0.5z2 ≤ 0 × ψ11 (|y|) + 0.5ψ12 (z), |2 (z, y)| = z2 ≤ 0 ×ψ21 (|y|)+ψ22 (z), that is, Assumption 1 holds. Take V0 (z) = z2 , then α1 (z) = 0.5z2 ≤ V0 (z) ≤ 1.5z2 = α2 (z), ∂V0 q(z, x) ≤ − 15 z2 + 81 y2 , that is, Assumption 2 is satisfied. ∂z 8 For the fuzzy control, seven fuzzy sets are characterized by the following membership functions:  −(x + 1.5)2   −(x + 1)2  , μFi2 = exp μFi1 = exp 2 2  −(x + 0.5)2   −(x)2  μFi3 = exp , μFi4 = exp 2 2  −(x − 0.5)2   −(x − 1)2  μFi5 = exp , μFi6 = exp 2 2  −(x − 1.5)2  μFi7 = exp . 2 The virtual control signal (16), the control law (17), and the adaptation laws (18)–(20) are used.

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Fig. 1.

Trajectories of y in Case 1 and Case 2 for Example 1.

Fig. 2.

Trajectories of x2 in Case 1 and Case 2 for Example 1.

Fig. 4.

Trajectories of θˆ in Case 1 and Case 2 for Example 1.

Fig. 5.

Trajectories of u in Case 1 and Case 2 for Example 1. TABLE I

Performance Comparisons Between Case 1 and Case 2 With the Output, Observer Errors, and Control Indexes for Example 1

Fig. 3.

Trajectories of z in Case 1 and Case 2 for Example 1.

According to the parameters design guideline described in the previous section, and to check the effects of the main design parameters ai (i = 1, 2) and k0 on the control performance, the design parameters are chosen as the following two cases. Case 1: a1 = 0.8, a2 = 0.6, k0 = 5, l1 = 144, l2 = 24, r1 = 8.5, r2 = 0.8, r3 = 0.9, and Q = [8, 0; 0, 8]. Case 2: a1 = 0.2, a2 = 0.4, k0 = 15, and the other design parameters are chosen similarly as Case 1. For the above two cases, the initial conditions are all chosen as [x1 (0), x2 (0), z(0), xˆ 1 (0), xˆ 2 (0)] = [0.1, 0.5, 0, 0, 0] ˆ and [θ(0), dˆ 1 (0), dˆ 2 (0)] = [0, 0, 1]. The simulation results are shown in Figs. 1–5. To compare with the control performances between Case 1 and Case 2, define the indexes of the output and control as the sum of the  squared output and control actions, M M 2 2 i.e., [y(k)] and k=1 k=1 [u(k)] , define the indexes of the of the squared errors, i.e., M observer errors 2as thesum M [x (k) − x ˆ (k)] and [x ˆ 2 (k)]2 , where M is 1 1 k=1 k=1 2 (k) − x the number of sampling data. The output, observer errors, and control indexes are calculated from 0 to 20 s with a sampling period of 0.01 s. The comparison results are presented in Table I. From Figs. 1, 2, 5, and Table I, we can see that the smaller the design parameters ai , and the larger k0 , the faster

the convergence rates of the output. However, control energy becomes larger when ai are smaller and k0 is larger. Therefore, in practice, carefully choices of the design parameters should be made according to the actual demands. By comparing the simulation results with the ones in [36], we can see that the control scheme in this paper not only guarantees that the closed-loop system is bounded and the system output converges to a small neighborhood of the origin, but also requires a smaller control gain. In addition, the proposed scheme contains only three adaptive parameters, while the method in [36] had 11 adaptive parameters. Therefore, the computation burden of the proposed method is reduced greatly. Example 2. To further test the effectiveness of the proposed adaptive scheme, we consider the following third-order nonlinear system with unmodeled dynamics and dynamical disturbances: z˙ = −3z + 0.2y2 x˙ 1 = x2 + 0.5y2 + y ln(10 + y2 ) + 0.8yz2 y2 x˙ 2 = x3 + + 0.5y sin z3 1 + y2 x˙ 3 = u + e−y y2 sin(y) − 0.6y2 cos z y = x1 .

LIU et al.: ADAPTIVE FUZZY OUTPUT-FEEDBACK CONTROLLER DESIGN FOR NONLINEAR SYSTEMS

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Fig. 6.

Trajectories of y in Case 1 and Case 2 for Example 2.

Fig. 9.

Fig. 7.

Trajectories of x2 in Case 1 and Case 2 for Example 2.

Fig. 10.

Adaptation parameter θˆ in Case 1 and Case 2 for Example 2.

Fig. 8.

Trajectories of x3 in Case 1 and Case 2 for Example 2.

Fig. 11.

Control input signal u in Case 1 and Case 2 for Example 2.

Choose ψ11 (|y|) = ψ31 (|y|) = y2 , ψ21 (|y|) = y4 , ψ12 (z) = ψ22 (z) = z4 , ψ32 (z) = 0, then |1 (z, y)| = 0.8|y|z2 ≤ 0.4 × ψ11 (|y|) + 0.4ψ12 (z), |2 (z, y)| = 0.5|y sin z3 | ≤ 1 × ψ21 (|y|) + 38 ψ22 (z), |3 (z, y)| = 0.6y2 | cos z| ≤ 0.6 × 8 ψ31 (|y|) + 0 × ψ32 (z), that is, Assumption 1 holds. Take V0 (z) = z2 , then α1 (z) = 0.5z2 ≤ V0 (z) ≤ 1.5z2 = 0 q(z, x) ≤ −5.8z2 + 0.2y4 , that is, Assumption 2 is α2 (z), ∂V ∂z satisfied. Take the fuzzy membership functions defined in Example 1. Then, choose the virtual control functions π1 , π2 , the control u and the parameter adaptive laws of dˆ 1 , dˆ 2 , θˆ as follows, respectively: π1 (X1 ) π2 (X2 ) u θˆ˙

θˆ x¯ 1 ξ T (X1 )ξ1 (X1 ) 2a12 1 θˆ = − 2 x¯ 2 ξ2T (X2 )ξ2 (X2 ) 2a2 θˆ = − 2 x¯ 3 ξ3T (X3 )ξ3 (X3 ) 2a3 3 r1 2 T = r1 P2 + x¯ ξ (Xi )ξi (Xi ) − k0 θˆ 2 i i 2a i i=1

=−

Trajectories of z in Case 1 and Case 2 for Example 2.

2 2 y2 2 ψi1 (|y|) + r2 ψ11 (|y|) + P2 4 4 i=1 3

dˆ˙1

=

dˆ˙2

2 2 y2 2 = + P2 ψi2 (z) + r3 ψ12 (z) 4 4 i=1 3

where x¯ 1 = y, x¯ 2 = xˆ 2 − π1 , x¯ 3 = xˆ 3 − π2 . Similarly, to check the effects of the main design parameters ai (i = 1, 2, 3) and k0 on the control performance, the design parameters are chosen as the following two cases. Case 1: a1 = 0.8, a2 = 0.6, a3 = 0.8, k0 = 5, l1 = 56, l2 = 5, l3 = 5, r1 = 8.6, r2 = 0.8, r3 = 0.9, Q = [8, 0, 0; 0, 8, 0; 0, 0, 8]. Case 2: a1 = 0.2, a2 = 0.4, a3 = 0.6, k0 = 15, and the other design parameters are chosen similarly as Case 1. For the above two cases, the initial conditions are all chosen as [x1 (0), x2 (0), x3 (0), z(0), xˆ 1 (0), xˆ 2 (0), xˆ 3 (0)] = ˆ [−0.1, −0.3, 0.5, 0.2, 0.5, −0.1, 0.1] and [θ(0), dˆ 1 (0), dˆ 2 (0)] = [0, 0, 1]. The simulation results are shown in Figs. 6–11. By employing the output, observer errors, and control indexes defined in Example 1, we can obtain the comparison results of the control performances between Case 1 and Case 2, which are summarized in Table II. From Figs. 6–8, 11, and

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TABLE II Performance Comparisons Between Case 1 and Case 2 With the Output, Observer Errors, and Control Indexes for Example 2

Table II, we can see that it is necessary to choose the appropriate design parameters. If ai are smaller and k0 is larger, then the output converges to the origin faster; meanwhile, the control energy becomes larger. From the simulation results of these two examples, we can see that the proposed control scheme guarantees that these closed-loop systems are bounded and the system outputs converge to a small neighborhood of the origin. What is more, there are only three adaptive parameters to be estimated for the nth-order system.

V. Conclusion In this paper, we have considered a class of nonlinear systems with unmodeled dynamics and unstructured uncertainties. The dynamic disturbances of the system depend on the states and the measured output of the system, and our restrictive conditions are more general than those in the previous articles. The fuzzy logic system is directly used to approximate the unknown control signal rather than the unknown function. By using the backstepping technique and the small-gain approach, a stable adaptive fuzzy output-feedback control scheme has been proposed. The proposed fuzzy adaptive controller guarantees that the closed-loop system is bounded, and it contains only three adaptive parameters that need to be updated online. Therefore, this algorithm is convenient to implement in the practical systems. In the future, by integrating data-driven methods in [45] and [46], we will apply the proposed control scheme to the area of control and monitoring within datadriven framework to achieve more industrial-oriented results.

Acknowledgment The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper.

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Zhi Liu received the B.S. degree from the Huazhong University of Science and Technology, Wuhan, China, in 1997, the M.S. degree from Hunan University, Changsha, China, in 2000, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2004, all in electrical engineering. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou, China. His current research interests include fuzzy logic systems, neural networks, robotics, and robust control.

Fang Wang received the B.S. degree from Qufu Normal University, Qufu, China, in 1997, and the M.S. degree from Shandong Normal University, Jinan, China, in 2004. She is currently pursuing the Ph.D. degree with the Department of Automation, Guangdong University of Technology, Guangzhou, China. Since 2005, she has been with the College of Science, Shandong University of Science and Technology, Qingdao, China. Her current research interests include fuzzy control, neural network control, backstepping control, and adaptive control.

Yun Zhang received the B.S. and M.S. degrees in automatic engineering from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree in automatic engineering from the South China University of Science and Technology, Guangzhou, China, in 1998. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou. His current research interests include intelligent control systems, network systems, and signal processing.

Xin Chen received the B.S. degree from the Changsha Railway Institute, Hunan, China, in 1982, the M.S. degree from the Harbin Institute of Technology, Heilongjiang, China, in 1988, and the Ph.D. degree from the Huazhong University of Science and Technology, Wuhan, China, in 1995, all in mechanical engineering. He is currently a Professor with the Department of Mechatronics Engineering, Guangdong University of Technology, Guangzhou, China. His current research interests include mechatronics and robotics.

C. L. Philip Chen received the M.S. degree from the University of Michigan, Ann Arbor, MI, USA, in 1985, and the Ph.D. degree from Purdue University, West Lafayette, IN, USA, in 1988. He is currently the Dean and Chair Professor with the Faculty of Science and Technology, University of Macau, Macau, China. He is also a Professor and the Chair with the Department of Electrical and Computer Engineering and an Associate Dean for Research and Graduate Studies with the College of Engineering, University of Texas, San Antonio, TX, USA.