Simple Adaptive Asymptotic Tracking Scheme for Parametric Strict-Feedback Nonlinear Systems with Additive Disturbance

Abstract

We consider the same class of systems in a previous paper, that is, a class of parametric strict-feedback nonlinear systems with additive disturbance. By using the backstepping technique, adaptive control algorithm is developed. Unlike the existing control schemes for systems with additive disturbance, no knowledge is assumed on two times continuous differentiability of the disturbance. Also, the developed backstepping controller does not require the uncertain parameters to be restricted within a known interval. Furthermore, overestimation of the system parameters is removed by employing the tuning functions method. It is shown that the proposed controller guarantees not only the stability in the sense of Lyapunov definition, but also the asymptotic tracking.

1. Introduction

Disturbance can be encountered in many physical systems. The control problem of such systems has received much attention in the past decades. For example, in [1], a class of single-input-single-out linear systems with unmodelled dynamics and input and output disturbances were considered. A dynamic normalizing signal and a smooth switching-σ function were introduced to design adaptive backstepping controller, and the mean-squared tracking error bound was established. In [2], adaptive control of the perturbed nonlinear output feedback systems was investigated. The Lyapunov functions with flat zone were employed, and the flat-zone modifications were incorporated into the intermediate control functions. The output error was proved to converge to arbitrary small interval around zero. However, in most of control literature, asymptotic regulation or tracking could not be achieved, and the system stability was established only in the sense that all the closed-loop signals were bounded (see, e.g., [1–25]).

Recently, Cai et al. proposed a new class of adaptive controllers to deal with the control problem of nonlinear systems with disturbance [26–28]. Two special cases were firstly reported in the control conferences (see [26, 27]). In [26], the tracking problem for nonlinear parametric strict-feedback systems of relative degree two in the presence of matched additive disturbance were studied. In [27], the regulation problem for 𝑛th-order nonlinear systems in the strict-feedback form was addressed. Then, in a more general case, the tracking control of strict-feedback 𝑛th-order nonlinear systems with parametric uncertainties and additive disturbance was considered. The main contributions of these works were that continuous adaptive controllers were designed and asymptotic tracking performance was ensured. The basic control idea lied in the combination of robust control technique in [29] and sufficiently smooth projection operator [30]. While the mathematical manipulations in [26–28] are very elegant, the proposed control schemes are very complex. To be more important, in order to design the asymptotic tracking controller, several restrictive conditions are imposed to the controlled system, which will be stated as follows.

First, it is reasonable to require that the external disturbance is bounded. However, in [26–28], the disturbance was assumed to be two times continuous differentiable. Moreover, the first-order and second-order derivatives of the disturbance were required to be bounded by some known constants. These assumptions on the disturbance may be very limited in reality and in turn the practicality of the presented control schemes would be greatly reduced. Then, the system parameters must be constrained within some known interval. The purpose of making such assumption was to employ the smooth projection operator of [30]. On the one hand, the smooth projection algorithm can ensure the boundedness of the parameter estimates. On the other hand, the smoothness of such algorithm is critical for backstepping since the recursive design procedure requires multiple differentiations of the adaptive law. However, the lower and upper bounds of unknown system parameters may be not available to the designer in the control practice. Furthermore, in [26, 28], the control gain 𝛽 was selected to satisfy the following condition: 𝛽>‖ℎ(𝑡)‖_𝐿∞+‖𝑑(̇𝑡)‖_𝐿∞+1/𝑐_𝑛(‖ℎ̇(𝑡)‖_𝐿∞+‖𝑑(̈𝑡)‖_𝐿∞). It is easy to determine ‖𝑑(̇𝑡)‖_𝐿∞ and ‖𝑑(̈𝑡)‖_𝐿∞, which is due to the system assumptions. However, the explicit expressions for ‖ℎ(𝑡)‖_𝐿∞ and ‖ℎ̇(𝑡)‖_𝐿∞ were not provided in [26, 28]. Actually, it is very difficult to specify the value of the design parameter 𝛽. As a result, the choice of 𝛽 in [26, 28] is only in the theoretical sense and it is not applicable to the practice. Finally, except for the last step of the recursive design, the unknown system parameters were repeatedly estimated in other steps of backstepping design. It can be computed that the total dynamic order of the parameter estimates is (𝑛−1)𝑝, where 𝑛 is the dimension of the controlled nonlinear system and 𝑝 is the dimension of the unknown system parameter. That is, the overestimation problem arose. It is well known that many parameter update laws will increase system complexity and implementation difficulty. So it is not desirable to design adaptive controller with too many parameter estimates in reality.

In this paper, we develop a simple adaptive backstepping control scheme for the same class of nonlinear systems as in [26–28], where the uncertain parameters and the matched external disturbance are considered simultaneously. The main contributions of the paper are as follows. (i) In our design, the external disturbance is only required to be bounded. Its upper bound is not necessary to be known and is estimated by designing adaptive law. (ii) The system parameters are not restricted to be within a known interval, and they are only assumed to be unknown constants. The projection algorithm or the smooth projection algorithm is not utilized in our design. Common adaptive laws without any modifications are constructed. (iii) The design parameters are not related to the external disturbance, the reference signal, the known system functions, and the initial conditions of the controlled system, and they can be freely chosen. Thus, the control performance of the closed-loop system can be greatly improved by choosing appropriate design constants. (iv) The tuning functions method [31] is employed to remove overestimation. Except for the estimate of the upper bound of the disturbance, the number of the parameter estimates is equal to that of the unknown system parameters. Besides showing the closed-loop stability in the Lyapunov sense, the tracking error is proven to converge to zero asymptotically.

This paper is organized as follows. After introducing the class of considered systems in Section 2, we present the main results of this paper in Section 3. In Section 4, an example with simulation to illustrate the effectiveness of the proposed design method is given. Finally, this paper is concluded by Section 5.

2. Problem Statement

We consider the same class of SISO nonlinear systems as in [26–28], which are given as 𝑥̇_𝑖=𝑥_𝑖+1+𝜑^𝑇_𝑖𝑥₁,…,𝑥_𝑖𝑥̇𝜃,𝑖=1,2,…,𝑛−1,_𝑛=𝑢+𝜑^𝑇_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛𝜃+𝑑(𝑡), (1) where 𝑥₁,𝑥₂,…,𝑥_𝑛, are system states, 𝑢 is control input, 𝜑_𝑖∈𝑅^𝑝, 𝑖=1,2,…,𝑛, are known and smooth nonlinear functions, 𝜃∈𝑅^𝑝 is unknown constant vector, and 𝑑(𝑡) is bounded additive disturbance. The objective is to enable the output 𝑦=𝑥₁ to follow a desired trajectory 𝑦_𝑟, while all the closed-loop signals remain bounded.

Assumption 1. The reference signal 𝑦_𝑟 and its first 𝑛 derivatives are known and bounded.

To prove the asymptotic convergence of the tracking error, the following lemma is used.

Lemma 2 (Barbalat’s lemma [32]). Let 𝑓∶[0,∞)↦𝑅. If 𝑓,𝑓̇∈𝐿_∞ and 𝑓∈𝐿₂, then 𝑓(𝑡)→0 as 𝑡→∞.

Remark 3. When 𝑑(𝑡)=0, the system (1) is the standard parametric strict-feedback nonlinear system. The disturbance 𝑑(𝑡) appears in the same equation as the control input 𝑢. That is, the matched disturbance is considered. Assumptions (A2) and (A3) in [26–28] are removed in this paper. We will design the asymptotic adaptive controller under the relaxed conditions.

3. Main Results

In our design, the first (𝑛−1) steps are the same as those in common adaptive backstepping control with tuning functions. To avoid repetition, the design steps are summarized as follows (see [31, Page 158, Table 4.1, (4.189)–(4.192)]).

Step  i. 𝑖=1,2,…,𝑛−1, 𝑧_𝑖=𝑥_𝑖−𝑦_𝑟^(𝑖−1)−𝛼_𝑖−1,𝛼_𝑖𝑥_𝑖,𝑦_𝑟^(𝑖−1),𝜃=−𝑧_𝑖−1−𝑐_𝑖𝑧_𝑖−𝜔^𝑇_𝑖𝜃+^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗⋅𝑥_𝑗+1+𝜕𝛼_𝑖−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)+𝜕𝛼_𝑖−1𝜕𝜃Γ𝜏_𝑖+^𝑖−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ𝜔_𝑖,𝜏_𝑖𝑥_𝑖,𝑦_𝑟^(𝑖−1),𝜃=𝜏_𝑖−1+𝜔_𝑖𝑧_𝑖,𝜔_𝑖𝑥_𝑖,𝑦_𝑟^(𝑖−2),𝜃=𝜑_𝑖−^𝑖−1_𝑗=1𝜕𝛼_𝑖−1𝜕𝑥_𝑗𝜑_𝑗,𝑥_𝑖=𝑥₁,𝑥₂,…,𝑥_𝑖^𝑇,𝑦_𝑟^(𝑖)=𝑦_𝑟,𝑦̇_𝑟,…,𝑦_𝑟^(𝑖)𝑇, (2)(3)(4)(5)(6)(7) where 𝑦_𝑟⁽⁰⁾=𝑦_𝑟, 𝑧_𝑖 is error variable, 𝑧₀=0, 𝛼_𝑖 is virtual control law, 𝛼₀=0, 𝜏_𝑖 is tuning function, 𝜏₀=0, 𝜃 is the estimate of 𝜃 with the estimation error 𝜃=𝜃−𝜃, 𝑐_𝑖>0 is design constant, and Γ is adaptive gain. In the above steps, the Lyapunov functions are, respectively, defined as 𝑉₁=12𝑧²₁+12𝜃^𝑇Γ⁻¹𝑉𝜃,_𝑖=𝑉_𝑖−1+12𝑧²_𝑖,𝑖=2,…,𝑛−1, (8) and the derivative of 𝑉_𝑛−1 is governed by 𝑉̇_𝑛−1=−^𝑛−1_𝑗=1𝑐_𝑗𝑧²_𝑗+𝑧_𝑛−1𝑧_𝑛−𝜃^𝑇Γ⁻¹𝜃̇−𝜏_𝑛−1+^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ𝜏_𝑛−1−̇𝜃. (9) The detailed deductions are similar to the corresponding parts in Section 4.2 of [31] (see [31, Page 140–146, (4.87)–(4.131)]). In this section, only the last step of the design, that is, Step n, is elaborated in detail.

Step n. Define 𝑧_𝑛=𝑥_𝑛−𝑦_𝑟^(𝑛−1)−𝛼_𝑛−1. (10) From (1) and (10), we obtain 𝑧̇_𝑛=𝑢+𝜑^𝑇_𝑛𝜃+𝑑(𝑡)−𝑦_𝑟^(𝑛)−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗⋅𝜑^𝑇_𝑗𝜃+𝑥_𝑗+1−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)−𝜕𝛼_𝑛−1𝜕𝜃̇𝜃=𝑢+𝜔^𝑇_𝑛𝜃−𝑦_𝑟^(𝑛)−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝑥_𝑗+1−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)−𝜕𝛼_𝑛−1𝜕𝜃̇𝜃+𝑑(𝑡), (11) where 𝜔_𝑛=𝜑_𝑛−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝜑_𝑗. (12) We define the Lyapunov function as 𝑉_𝑛=𝑉_𝑛−1+12𝑧²_𝑛+12𝛾⁻¹𝑑²=12^𝑛_𝑗=1𝑧²_𝑗+12𝜃^𝑇Γ⁻¹1𝜃+2𝛾⁻¹𝑑², (13) where 𝑑=𝑑−𝑑,𝑑=sup_𝑡≥0|𝑑(𝑡)|, (14)𝑑 denotes the upper bound of 𝑑(𝑡), 𝑑 is the estimate of 𝑑, 𝑑 is parameter estimation error, and 𝛾 is adaptive gain. With (9) and (11), the derivative of 𝑉_𝑛 is given by 𝑉̇_𝑛=−^𝑛−1_𝑗=1𝑐_𝑗𝑧²_𝑗−𝜃^𝑇Γ⁻¹𝜃̇−𝜏_𝑛−1+^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃⋅Γ𝜏_𝑛−1−̇𝜃+𝑧_𝑛𝑧_𝑛−1+𝑢+𝜔^𝑇_𝑛𝜃−𝑦_𝑟^(𝑛)−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝑥_𝑗+1−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)−𝜕𝛼_𝑛−1𝜕𝜃̇𝜃+𝑧_𝑛𝑑(𝑡)−𝛾⁻¹𝑑̇𝑑. (15) By introducing 𝜏_𝑛=𝜏_𝑛−1+𝜔_𝑛𝑧_𝑛 (16) and noting 𝜃=𝜃+𝜃 (17) (15) can be rewritten as 𝑉̇_𝑛=−^𝑛−1_𝑗=1𝑐_𝑗𝑧²_𝑗−𝜃^𝑇Γ⁻¹𝜃̇−𝜏_𝑛+^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃⋅Γ𝜏_𝑛−1−̇𝜃+𝑧_𝑛𝑧_𝑛−1+𝑢+𝜔^𝑇_𝑛𝜃−𝑦_𝑟^(𝑛)−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝑥_𝑗+1−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)−𝜕𝛼_𝑛−1𝜕𝜃̇𝜃+𝑧_𝑛𝑑(𝑡)−𝛾⁻¹𝑑̇𝑑. (18) Due to Γ𝜏_𝑛−1−̇𝜃=Γ𝜏_𝑛−1−Γ𝜏_𝑛+Γ𝜏_𝑛−̇𝜃=−Γ𝜔_𝑛𝑧_𝑛+Γ𝜏_𝑛−̇𝑧𝜃,_𝑛𝑑(𝑡)≤𝑧_𝑛𝑑, (19) we have 𝑉̇_𝑛≤−^𝑛−1_𝑗=1𝑐_𝑗𝑧²_𝑗−𝜃^𝑇Γ⁻¹𝜃̇−𝜏_𝑛+^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃⋅Γ𝜏_𝑛−̇𝜃+𝑧_𝑛𝑧_𝑛−1+𝑢+𝜔^𝑇_𝑛𝜃−𝑦_𝑟^(𝑛)−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝑥_𝑗+1−^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)−^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ𝜔_𝑛−𝜕𝛼_𝑛−1𝜕𝜃̇𝜃+𝑧_𝑛𝑑−𝛾⁻¹𝑑̇𝑑. (20) Then, adaptive control law is designed as 𝑢=−𝑧_𝑛−1−𝑐_𝑛𝑧_𝑛−𝜔^𝑇_𝑛𝜃+𝑦_𝑟^(𝑛)+^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑥_𝑗𝑥_𝑗+1+^𝑛−1_𝑗=1𝜕𝛼_𝑛−1𝜕𝑦_𝑟^(𝑗−1)𝑦_𝑟^(𝑗)+^𝑛−1_𝑗=2𝑧_𝑗𝜕𝛼_𝑗−1𝜕𝜃Γ𝜔_𝑛+𝜕𝛼_𝑛−1𝜕𝜃Γ𝜏_𝑛−𝑧_𝑛𝑑²𝑧_𝑛(𝑡),̇𝑑+𝛿𝜃=Γ𝜏_𝑛,̇𝑑=𝛾𝑧_𝑛, (21)(22)(23) where 𝑐_𝑛 is positive design constant, 𝑑(0)≥0, and 𝛿(𝑡) is a time-varying function and satisfies 𝛿(𝑡)>0,^𝑡₀𝛿(𝜏)𝑑𝜏≤𝛿, (24) with positive constant 𝛿.

Remark 4. It is noted that the first (𝑛−1) steps of the control design are the same as those in the standard backstepping procedure [31]. The difference between the tuning functions design of [31] and our scheme lies in the final step of the recursive design. In (21), the additional term −𝑧_𝑛𝑑²/(|𝑧_𝑛|𝑑+𝛿(𝑡)) is incorporated into the actual control law in order to suppress the influence of the disturbance on the closed-loop stability. Condition (24) implies that 𝛿(𝑡) is a positive time-varying integral function, which will play an important role in the subsequent stability analysis.

Substituting (21)–(23) into (20) yields 𝑉̇_𝑛≤−^𝑛_𝑗=1𝑐_𝑗𝑧²_𝑗−𝑧²_𝑛𝑑²𝑧_𝑛(𝑡)+𝑧𝑑+𝛿^𝑛𝑑=−^𝑛_𝑗=1𝑐_𝑗𝑧²_𝑗+𝑧_𝑛𝑑𝛿(𝑡)𝑧_𝑛(𝑡).𝑑+𝛿 (25) In view of positive initial condition in (23) (see below (23)), we have 𝑑(𝑡)≥0, 𝑡≥0. By using the inequality 𝑎𝑏/(𝑎+𝑏)≤𝑏, for all 𝑎≥0, 𝑏>0, (25) becomes 𝑉̇_𝑛≤−^𝑛_𝑗=1𝑐_𝑗𝑧²_𝑗+𝛿(𝑡). (26) Integrating (26) on [0,𝑡] and using (24), we get 𝑉_𝑛(𝑡)+^𝑛_𝑗=1^𝑡₀𝑐_𝑗𝑧²_𝑗(𝜏)𝑑𝜏≤𝑉_𝑛(0)+^𝑡₀𝛿(𝜏)𝑑𝜏≤𝑉_𝑛(0)+𝛿. (27) From (13) and (27), it follows that 𝑉_𝑛 and thus 𝑧_𝑗, 𝑗=1,…,𝑛,𝜃,𝑑 are bounded. From (2) and (10), we have 𝑥_𝑖=𝑧_𝑖+𝑦_𝑟^(𝑖−1)+𝛼_𝑖−1,𝑖=1,…,𝑛. (28) According to Assumption 1 and the boundedness of 𝑧₁, 𝑥₁ is bounded. Since 𝛼₁ is a smooth function of 𝑥₁, 𝑦_𝑟, and 𝜃, we conclude that 𝛼₁ is bounded. Let 𝑖=2 in (28). We obtain the boundedness of 𝑥₂. By expecting the expressions in (3) and (4) with 𝑖=2, it is shown that 𝛼₂ is bounded. Let 𝑖=3 in (28). Then, we have the boundedness of 𝑥₃. Continuing in this way, we can prove that 𝛼₃,𝑥₄,…,𝛼_𝑛−1,𝑥_𝑛, are bounded. Noting −𝑧_𝑛𝑑²𝑧_𝑛(𝑡)≤𝑧𝑑+𝛿^𝑛𝑑𝑧_𝑛𝑑+𝛿(𝑡)𝑑≤𝑑 (29) and using the boundedness of 𝑥_𝑛, we prove that 𝑢 in (21) is bounded.

Furthermore, differentiating (2) with 𝑖=1 gives 𝑧̇₁=𝑥̇₁−𝑦̇_𝑟=𝑥₂+𝜑^𝑇₁𝜃−𝑦̇_𝑟. (30) Using the boundedness of all the closed-loop signals and Assumption 1, we know that 𝑧̇₁∈𝐿_∞. Noting (27), we get ^𝑛_𝑗=1^𝑡₀𝑐_𝑗𝑧²_𝑗(𝜏)𝑑𝜏≤𝑉_𝑛(0)+𝛿, (31) which particularly implies that ^𝑡₀𝑧²₁(𝜏)1𝑑𝜏≤𝑐₁𝑉_𝑛(0)+𝛿. (32) That is, 𝑧₁∈𝐿₂. Thus, by Lemma 2, we obtain that lim_𝑡→∞𝑧₁(𝑡)=0.

The above facts prove the following results.

Theorem 5. Suppose that the proposed design procedure is applied to system (1). Then the boundedness of all signals in the resulting closed-loop system is guaranteed and asymptotic tracking is achieved.

4. Simulation

In this section, we illustrate the aforementioned methodologies to the same example system as that in [26], which is described as 𝑥̇₁=𝑥₂+𝜑^𝑇₁𝑥₁𝜃𝑥̇₂=𝑢+𝜑^𝑇₂(𝑥)𝜃+𝑑(𝑡), (33) where 𝜃=[2,1]^𝑇 is assumed to be unknown constant, 𝜑₁, 𝜑₂, and 𝑑(𝑡) are, respectively, set to be 𝜑₁=−𝑥²₁14𝑥³₁,𝜑₂=−𝑥²₁14𝑥³₂,𝑑(𝑡)=0.1sin(𝑡). (34) The desired trajectory is 𝑦_𝑟=0.1sin(𝑡)(1−𝑒^−0.5𝑡2). The design procedure in Section 3 is as follows.

Step 1. Consider 𝑧₁=𝑥₁−𝑦_𝑟,𝜔₁=𝜑₁,𝛼₁=−𝑐₁𝑧₁−𝜔^𝑇₁𝜃,𝜏₁=𝜔₁𝑧₁. (35)

Step 2. Then 𝑧₂=𝑥₂−𝑦̇_𝑟−𝛼₁,𝜔₂=𝜑₂−𝜕𝛼₁𝜕𝑥₁𝜑₁,𝜏₂=𝜏₁+𝑧₂𝜔₂,𝑢=−𝑧₁−𝑐₂𝑧₂−𝜔^𝑇₂𝜃+𝑦_𝑟⁽²⁾+𝜕𝛼₁𝜕𝑥₁𝑥₂+𝜕𝛼₁𝜕𝑦_𝑟𝑦̇_𝑟+𝜕𝛼₁𝜕𝜃Γ𝜏₂−𝑧₂𝑑²𝑧₂(𝑡),̇𝑑+𝛿𝜃=Γ𝜏₂,̇𝑑=𝛾𝑧₂. (36) The simulation is carried out with the following choices: 𝑐₁=8,𝑐₂{},(𝑡)=1,Γ=diag1,1𝛾=0.5,𝛿=0.01𝑒^−0.01𝑡. (37) The initial conditions are set to be 𝑥₁(0)=1,𝑥₂(0)𝜃(0)=[]=0,1,−1^𝑇,𝑑(0)=2. (38) The system responses are shown in Figures 1, 2, 3, and 4. From the simulation results, we observe that asymptotic tracking is achieved in the presence of external disturbance, and plant states, parameter estimate, and control effort are all bounded.

(a)

(b)

Figure 1 

Plant states 𝑥1, 𝑥2.

Figure 2 

Tracking error 𝑧1.

Figure 3 

Parameter estimates 𝜃=[𝜃1,𝜃2]𝑇, 𝑑.

Figure 4 

Control input 𝑢.

5. Conclusions

In this paper, we present an adaptive backstepping control design for a class of nonlinear systems with additive disturbance. Unlike some existing control schemes, the proposed backstepping control does not require the knowledge on the differentiability of the disturbance and the known interval of the model parameters. In our design, a positive time-varying integral function is involved and can ensure perfect tracking. The overestimation problem is solved by introducing the tuning functions. Simulation results are provided to illustrate the effectiveness of our scheme.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61104007, 61203320, 61273123, and 61374090, China Postdoctoral Science Foundation funded project under Grant 2012M511465, Shandong Postdoctoral Science Foundation funded project under Grant 201203031, the project supported by the Young Core Instructor, and Domestic Visitor Foundation from the Education Department of Shandong Province, China.