Adaptive Fractional Fuzzy Sliding Mode Control for Multivariable Nonlinear Systems

Abstract

This paper presents a robust adaptive fuzzy sliding mode control method for a class of uncertain nonlinear systems. The fractional order calculus is employed in the parameter updating stage. The underlying stability analysis as well as parameter update law design is carried out by Lyapunov based technique. In the simulation, two examples including a comparison with the traditional integer order counterpart are given to show the effectiveness of the proposed method. The main contribution of this paper consists in the control performance is better for the fractional order updating law than that of traditional integer order.

1. Introduction

Fuzzy systems have been applied to many control problems effectively because they do not need accurate mathematical model of the controlled system and they can also cooperate with human expert knowledge. It is well known that fuzzy systems and neural networks can uniformly approximate any nonlinear continuous function over some compact set [1]. Based on the universal approximation theorem [1], some adaptive fuzzy control methods [2–5] have been developed for MIMO nonlinear uncertain systems. The stability analysis of the underlying closed loop system has been carried out by means of Lyapunov approach. To deal with the ubiquitous fuzzy approximation errors and external disturbances, these controllers are usually augmented by a robust compensator, which can be a sliding mode control [2, 3, 6–8] or an 𝐻_∞ control [3]. And many important results have been given. In [9], fuzzy logic systems are used to obtain an adaptive boundary layer. Erbatur et al. [10] utilize the concept of fuzziness for reducing the adverse effects of chattering of the sliding mode control. The parameter’s adaptation law is introduced in [11], where quicker reaching with suppressed oscillations is designed with a comparison with classical sliding mode controller. In [12], a robust control method for uncertain chaotic systems which comprises a nonlinear inversion-based controller with a fuzzy robust compensator is proposed.

Parameter tuning in adaptive control systems is an important part of the overall mechanism alleviating the influence associated with the changes in the parameters and uncertainties of the systems. Many studies can be found in the past two decades, and the domain of adaptation has become a blend of techniques of dynamical systems theory, optimization, soft computing, and heuristics. Now, tuning of parameters based on some set of observations has been facilitated [2, 3, 8, 13–16]. In [16], an in-depth discussion for parameter tuning in continuous as well as discrete time is proposed.

A common feature of all these control methods and the cited research is the fact that the differentiation and integration are performed in integer order. Up to now, with the development of complex engineering applications and natural science, fractional calculus as well as fractional differential equation theory and their applications begin to attract more and more attention from physicists to engineers [17–19]. Particularly for gradient descent rule, which is considered in the integer order in [16], it has been designed in fractional order in [18], where the integer-order integration is replaced with an integration with fractional order of 1.25. On the other hand, sliding mode control (SMC) is an effective technique to robustly control uncertain nonlinear systems [20–22]. The main idea of SMC is to switch the control input to drive the states of the system from the initial states onto some predefined sliding surface. Once the system operates on the sliding surface, it has desired properties such as stability, disturbance rejection capability, and tracking capability. SMC to accommodate fractional order nonlinear systems has not yet attracted much attention, due primarily to the mathematical difficulties in stability analysis. Moreover, there are only limited published results which concern fractional order chaotic systems under SMC.

In the stability analysis of fractional order systems, the Lyapunov function 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑥(𝑡) is often used. The 𝛼th-order of 𝑉(𝑡) can be given as ₀𝐷^𝛼_𝑡𝑉(𝑡)=₀𝐷^𝛼_𝑡𝑥(𝑡)^𝑇𝑥(𝑡)+𝑥^𝑇(𝑡)₀𝐷^𝛼_𝑡𝑥(𝑡)+2Λ, (1) where Λ=∑^∞_𝑘=1(Γ(1+𝛼)/(Γ(1+𝑘)Γ(1−𝑘+𝛼)))₀𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧. In order to obtain the results by using the stability theorems of fractional order systems, in [8, 23], the authors assume that the bounded condition |Λ|≤𝑏|𝑥(𝑡)| holds, where 𝑏 is a positive constant. In this paper, we will prove the condition and establish a fundamental lemma. This lemma is established for stability analysis of fractional order systems, especially for Mittag-Leffler stability [24] analysis of fractional order nonlinear systems.

As a result of the discussion above, the absence of methods designed by fractional differintegration in robust control is visible. The objective of this paper is to fill this gap to the extent that covers the following aspects: (1) better robustness and system uncertainties rejection capabilities than those using conventional integer-order operators; (2) conditions for hitting in finite time; and (3) sliding mode control corporate with fractional order adaptation law. And the aforementioned ideas constitute the major contributions of this paper advancing the subject area to the fractional order adaptation methods.

2. Problem Formulation and Preliminaries

2.1. Problem Formulation

Consider the following MIMO nonlinear dynamic system which can be described by 𝑦^(𝑟1)₁=𝑓₁(𝑥)+^𝑝_𝑗=1𝑔_1𝑗(𝑥)𝑢_𝑗,⋮𝑦^(𝑟𝑝)_𝑝=𝑓_𝑝(𝑥)+^𝑝_𝑗=1𝑔_𝑝𝑗(𝑥)𝑢_𝑗, (2) where 𝑥=[𝑥₁,𝑥̇₁,…,𝑥^(𝑟1₁⁻¹⁾,…,𝑥_𝑝,𝑥̇_𝑝,…,𝑥^(𝑟𝑝_𝑝⁻¹⁾]^𝑇∈𝑅^𝑟, 𝑟=𝑟₁+𝑟₂+⋯+𝑟_𝑝 is the system state vector which is assumed to be available for measurement. 𝑢=[𝑢₁,…,𝑢_𝑝]^𝑇∈𝑅^𝑝 and 𝑦=[𝑥₁,…,𝑥_𝑝]^𝑇∈𝑅^𝑝 are control input and output vector, respectively. 𝑓_𝑖(𝑥), 𝑖=1,…,𝑝, are unknown nonlinear functions and 𝑔_𝑖𝑗(𝑥), 𝑖,𝑗=1,…,𝑝 are unknown constant control gains.

If we denote 𝑦^(𝑟)=𝑥^(𝑟1)₁,…,𝑥^(𝑟𝑝)_𝑝^𝑇,𝐹(𝑥)=𝑓₁(𝑥),…,𝑓_𝑝(𝑥)^𝑇,𝐺(𝑥)=𝑔₁₁(𝑥)⋯𝑔_1𝑝(𝑥)𝑔⋮⋱⋮_𝑝1(𝑥)⋯𝑔_𝑝𝑝(𝑥), (3) then, system (2) can be rewritten as 𝑦^(𝑟)=𝐹(𝑥)+𝐺(𝑥)𝑢. (4)

The objective of this paper is to construct a control input 𝑢 such that the output 𝑦 tracks the specified desired signal 𝑦_𝑑=[𝑦_𝑑1,…,𝑦_𝑑𝑝]∈𝑅^𝑝 with all involved signals in the closed loop system keep bounded. To meet the objective, let us make the following assumptions.

Assumption 1. The desired trajectory signal [𝑦_𝑑1,…,𝑦^(𝑟1)_𝑑1,…,𝑦_𝑑𝑝,…,𝑦^(𝑟𝑝)_𝑑𝑝] is continuous, bounded, and available for measurement.

Assumption 2. The control gain matrix 𝐺(𝑥) is positive definite and satisfies 12𝐺̇⁻¹(𝑥)≤𝛽(𝑥), (5) where 𝛽(𝑥)>0 is an unknown continuous nonlinear function.

Remark 3. Assumption 2 is not restrictive. There are many systems, such as electrical machines and robotic systems which satisfy Assumption 2. And we only assume the existence of the nonlinear function 𝛽(𝑥) and not its knowledge.

Let us define the tracking error as 𝑒_𝑖=𝑦_𝑑𝑖−𝑦_𝑖,𝑖=1,…,𝑝, (6) and the sliding mode surface as 𝑠_𝑖=𝑑𝑑𝑡+𝜆_𝑖^𝑟𝑖−1𝑒_𝑖(𝑡),𝑖=1,…,𝑝. (7) Equation (7) can be rewritten as 𝑠_𝑖=𝑒^(𝑟𝑖_𝑖⁻¹⁾+𝑟_𝑖−1𝜆_𝑖𝑒^(𝑟𝑖_𝑖⁻²⁾+𝑟+⋯_𝑖−1𝜆^𝑟𝑖_𝑖⁻²𝑒_𝑖̇+𝜆^𝑟𝑖_𝑖⁻¹𝑒_𝑖. (8) Notice that if we select 𝜆_𝑖>0, then the roots of the polynomial 𝐻_𝑖(𝑠)=𝑠^𝑟1−1+(𝑟_𝑖−1)𝜆_𝑖𝑠^𝑟𝑖−2+⋯+(𝑟_𝑖−1)𝜆^𝑟𝑖_𝑖⁻²𝑠̇+𝜆^𝑟𝑖_𝑖⁻¹=0 are all in the left-half complex plane. In other words, the objective of this paper becomes the design of controller to force the filtered tracking error 𝑠_𝑖(𝑡)→0. Differentiating 𝑠_𝑖 with respect to time we obtain 𝑠_𝑖̇=𝑣_𝑖−𝑓_𝑖(𝑥)−^𝑝_𝑗=1𝑔_𝑖𝑗(𝑥)𝑢_𝑗, (9) with 𝑣_𝑖=𝑦^(𝑟𝑖)_𝑑1+𝛾_{1,𝑟𝑖−1}𝑒^(𝑟𝑖₁⁻¹⁾+⋯+𝛾_1,1𝑒_𝑖̇, 𝛾_𝑖,𝑗=((𝑟_𝑖−1)!/((𝑟_𝑖−𝑗)!(𝑗−1)!))𝜆^𝑟𝑖_𝑖^−𝑗, 𝑖=1,…,𝑝 and 𝑗=1,…,𝑟_𝑖−1. Denote 𝑠=[𝑠₁,…,𝑠_𝑝]^𝑇 and 𝑣=[𝑣₁,…,𝑣_𝑝]^𝑇; then (9) can be rewritten in the following compact form: 𝑠̇=𝑣−𝐹(𝑥)−𝐺(𝑥)𝑢. (10) Thereafter, (10) will be used in the construct of the controller as well as the stability analysis.

2.2. Description of the Fuzzy Logic System

The fuzzy logic system that employs singleton fuzzification, sum-product inference, and center-of-sets defuzzification, as shown in Figure 1, can be modeled by 𝛼(𝑥)=∑^𝑁_𝑗=1𝜃_𝑗∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖𝑥_𝑖∑^𝑁_𝑗=1∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖𝑥_𝑖, (11) where 𝛼(𝑥) is the output of the fuzzy system, 𝑥 is the input vector, 𝜇_𝐹^𝑗_𝑖(𝑥_𝑖) is 𝑥_𝑖’s membership of the 𝑗th rule, and 𝜃_𝑗 is the centroid of the 𝑗th consequent set. Equation (6) can be rewritten as following equation: 𝛼(𝑥)=𝜃^𝑇𝜓(𝑥), (12) with =[𝜃₁,…,𝜃_𝑁]^𝑇, 𝜓(𝑥)=[𝑝₁(𝑥),𝑝₂(𝑥),…,𝑝_𝑁(𝑥)]^𝑇, and the fuzzy basis function can be written as 𝑝_𝑗(𝑥)=∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)/∑^𝑁_𝑗=1[∏^𝑛_𝑖=1𝜇_𝐹^𝑗_𝑖(𝑥_𝑖)]. Suppose there are 𝑁 rules of the fuzzy system used to approximate the unknown function 𝛼(𝑥).

Figure 1 

Structure of a fuzzy inference system.

Rule i. if 𝑥₁ is 𝐹^𝑖₁ and ⋯ and 𝑥_𝑛 is 𝐹^𝑖_𝑛 then 𝛼(𝑥) is 𝐵^𝑖,𝑖=1,2,…,𝑁.

3. Adaptive Fractional Fuzzy Controller Design

3.1. Ideal Controller

Suppose that the functions 𝐺(𝑥) and 𝐹(𝑥) are known in advance. From (10) we know 𝐺₁(𝑥)𝑠̇=𝐺₁(𝑥)𝑣−𝐺₁(𝑥)𝐹(𝑥)−𝑢, (13) where 𝐺₁(𝑥)=𝐺⁻¹(𝑥).

Then we can construct the following ideal controller 𝑢^∗: 𝑢=𝑢^∗=𝐺₁(𝑥)𝑣−𝐺₁(𝑥)𝐹(𝑥)+𝐾+𝛽(𝑥)𝐼𝑠, (14) where 𝐾=diag[𝑘₁,…,𝑘_𝑝] with 𝑘_𝑖>0, 𝑖=1,2,…,𝑝.

Theorem 4. Consider system (2). If Assumptions 1 and 2 are satisfied. The control input (14) can guarantee that all signals in the closed loop system will remain bounded and the tracking errors and their derivatives converge to origin asymptotically; that is, 𝑒_𝑖^(𝑗)→0, 𝑡→∞, 𝑖=1,…,𝑝, and 𝑗=1,…,𝑟_𝑗−1.

Proof. Substituting the ideal control input (14) to the tracking error dynamics (13) gives 𝐺₁(𝑥)𝑠̇=−𝛽(𝑥)𝐼𝑠−𝐾𝑠. (15)
Multiplying 𝑠^𝑡 to both sides of (15) we have 𝑠^𝑇𝐺₁(𝑥)𝑠̇=−𝑠^𝑇𝛽(𝑥)𝐼𝑠−𝑠^𝑇𝐾𝑠. (16)
Let us define the following Lyapunov function: 1𝑉=2𝑠^𝑇𝐺₁(𝑥)𝑠. (17)
Its derivative with respect to time can be given by 𝑉̇=𝑠^𝑇𝐺₁(𝑥)𝑠̇+12𝑠^𝑇𝐺̇₁(𝑥)𝑠. (18)
By using Assumption 2 and (16), we can obtain 𝑉̇=𝑠^𝑇𝐺₁(𝑥)𝑠̇+12𝑠^𝑇𝐺̇₁(𝑥)𝑠≤−𝑠^𝑇(𝑥)𝐼1𝐾+𝛽𝑠+2𝐺̇₁(𝑥)‖𝑠‖²≤−𝑠^𝑇𝐾𝑠. (19)
From (19) we can conclude that 𝑠_𝑖→0 as 𝑡→∞. Therefore, the tracking errors and their derivatives converge to origin. And this ends the proof of Theorem 4.

3.2. Fractional order Adaption Law of Parameters of Fuzzy Systems

Let 0<𝛼<1. The Riemann-Liouville (R-L) definition of the 𝛼th-order fractional integration can be given as ₀𝐷_𝑡^−𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏, (20) and the 𝛼th-order fractional derivative can be given as ₀𝐷^𝛼_𝑡𝑓(𝑡)=𝑓^(𝛼)(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝜏)^−𝛼𝑓(𝜏)𝑑𝜏, (21) where Γ(⋅) represents the Gamma function Γ(𝛼)=∫^∞₀𝑒^−𝑡𝑡^𝛼−1𝑑𝑡. From the above definition, we can get the following properties of the fractional calculus [8, 17]: ₀𝐷^𝛼_𝑡𝑡^𝛽=Γ𝛽+1Γ𝑡𝛽+1−𝛼^𝛽−𝛼₀,𝐷_𝑡^−𝛼₀𝐷^𝛼_𝑡𝑓(𝑡)=𝑓(𝑡)−𝑓^𝛼−1(0)𝑡^𝛼−1Γ(𝛼)₀𝐷_𝑡^−𝛼₀𝐷^𝛼_𝑡𝑓(𝑡)=𝑓(𝑡). (22)

In the rest of this section, we assume that the target output of the fuzzy system is known such that the approximation error is available for parameter updating process. Let 𝑧 and 𝑧_𝑑 be the output of the fuzzy system (12) and the target output, respectively. Then we have 𝑧=𝜃^𝑇𝜓(𝑥). (23)

Let the approximation error of the fuzzy system be 𝑧=𝑧−𝑧_𝑑. (24)

Now we are ready to give the following results.

Theorem 5. Suppose the following boundedness conditions hold: ^∞𝑁_𝑘=1𝑖=1𝛼𝑘⁰𝐷_𝑡^𝛼−𝑘𝜃_𝑖0𝐷^𝑘_𝑡𝜓_𝑖≤𝑎₁,_∞𝑘=1Γ(1+𝛼)Γ(1+𝑘)Γ(1−𝑘+𝛼)⁰𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧≤𝑎₂|𝑧|,₀𝐷^𝛼_𝑡𝑧_𝑑≤𝑎₃, (25)(26)(27) where 𝑎_𝑖>0, 𝑖=1,2,3 are constants. If the adaption law is chosen as ₀𝐷^𝛼_𝑡𝜓𝜃=−𝑏𝜓²sign(𝑧), (28) where 𝜃=[𝜃₁,…,𝜃_𝑝]^𝑇 and 𝜓=[𝜓₁,…,𝜓_𝑝]^𝑇.
Then the approximation error will converge to zero within some finite time satisfying 𝑏−𝑎₁Γ()𝑡1+𝛼^𝛼ℎ≤⁰𝐷_𝑡^𝛼−1𝑧(0)+₀𝐷_𝑡^𝛼−1𝑧_𝑑(0)Γ(𝛼)𝑡_ℎ^𝛼−1+𝑧_𝑑𝑡_ℎ (29) if 𝑏>𝑎₁+𝑎₂+𝑎₃ is satisfied.

Proof. Noting that 𝑠=0 corresponds to the fact that the states are on the sliding manifold. While 𝑧=0 represents that the control signal eventually results in 𝑠=0. As a result, the dynamical conclusions of 𝑧=0 are different from 𝑠=0.
Let us define Ξ=^∞𝑁_𝑘=1𝑖=1𝛼𝑘⁰𝐷_𝑡^𝛼−𝑘𝜃_𝑖0𝐷^𝑘_𝑡𝜓_𝑖. (30) Consider the following Lypunov function candidate: 𝑉₁=𝑧². (31)
According to the Leibniz rule of the fractional differentiation, we have ₀𝐷^𝛼_𝑡𝑉₁=₀𝐷^𝛼_𝑡𝑧𝑧+Λ, (32) where Λ=^∞_𝑘=1Γ(1+𝛼)Γ(1+𝑘)Γ(1−𝑘+𝛼)⁰𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧. (33)
Then we have ₀𝐷^𝛼_𝑡𝑧𝑧=₀𝐷^𝛼_𝑡𝑧−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=₀𝐷^𝛼_𝑡^𝑚_𝑖=1𝜃_𝑖𝜓_𝑖−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=^∞𝑁_𝑘=0𝑖=1𝛼𝑘⁰𝐷_𝑡^𝛼−𝑘𝜃_𝑖0𝐷^𝑘_𝑡𝜓_𝑖−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=^𝑁_𝑖=1𝜓_𝑖0𝐷^𝛼_𝑡𝜃_𝑖+^∞𝑁_𝑘=1𝑖=1𝛼𝑘⁰𝐷_𝑡^𝛼−𝑘𝜃_𝑖0𝐷^𝑘_𝑡𝜓_𝑖−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=𝑧𝜓^𝑇₀𝐷^𝛼_𝑡𝜃+Ξ−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=−𝑏𝑧𝜓^𝑇𝜓𝜓²sign(𝑧)+Ξ−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧=−𝑏sign(𝑧)𝑧+Ξ−₀𝐷^𝛼_𝑡𝑧_𝑑𝑧≤−𝑏|𝑧|+|Ξ𝑧|+₀𝐷^𝛼_𝑡𝑧_𝑑𝑧|≤|−𝑏+𝑎₁+𝑎₃|𝑧|. (34)
Substituting (34) into (32) and using (26) yields ₀𝐷^𝛼_𝑡𝑉₁≤𝑏−𝑎₁−𝑎₂−𝑎₃|𝑧|<0. (35)
Since ₀𝐷^𝛼_𝑡𝑉₁<0, as the same discuss in [24], we know that the phase space are attracted by 𝑧=0.
Now, let us prove that first hitting to the switching surface happens in some finite time 𝑡_ℎ. From (34) and the fractional order adaption law (28) we have ₀𝐷^𝛼_𝑡𝑧=−𝑏sign(𝑧)+Ξ−₀𝐷^𝛼_𝑡𝑧_𝑑. (36)
Applying the fractional integration operator ₀𝐷_𝑡^−𝛼_ℎ on both sides of (36) we can obtain 𝑧𝑡_ℎ−₀𝐷_𝑡^𝛼−1_ℎ𝑧(0)𝑡_ℎ^𝛼−1Γ(𝛼)=−𝑏sign(𝑧(0))Γ()𝑡𝛼+1^𝛼ℎ+₀𝐷_𝑡^−𝛼_ℎΞ−₀𝐷^𝛼_𝑡ℎ𝑧_𝑑. (37)
Multiplying sign(𝑧(0)) to both sides of (37) we have −₀𝐷_𝑡^𝛼−1_ℎ𝑧(0)(𝑧(0𝑡sign))^ℎ𝛼−1Γ(𝛼)=−𝑏Γ()𝑡𝛼+1^𝛼ℎ+sign(𝑧(0))₀𝐷_𝑡^−𝛼_ℎΞ−sign(𝑧(0))₀𝐷_𝑡^−𝛼_ℎ0𝐷^𝛼_𝑡ℎ𝑧_𝑑. (38) Noting that sign(𝑧(0))₀𝐷_𝑡^−𝛼_ℎΞ≤₀𝐷_𝑡^−𝛼_ℎ|Ξ|≤₀𝐷_𝑡^−𝛼_ℎ𝑎₁=𝑎₁𝑡^𝛼_ℎΓ(),(𝑧(0𝛼+1sign))₀𝐷_𝑡^−𝛼_ℎ0𝐷^𝛼_𝑡ℎ𝑧_𝑑=sign(𝑧(0))𝑧_𝑑𝑡_ℎ−₀𝐷_𝑡^𝛼−1_ℎ𝑧_𝑑(0)𝑡_ℎ^𝛼−1Γ(𝛼), (39) then we have −₀𝐷_𝑡^𝛼−1_ℎ𝑧(0)(𝑧(0𝑡sign))^ℎ𝛼−1Γ(𝛼)≤−𝑏Γ()𝑡𝛼+1^𝛼ℎ+𝑎₁𝑡_ℎ^𝛼−1Γ()(𝑧(0𝑧𝛼+1−sign))_𝑑𝑡_ℎ+sign(𝑧(0))₀𝐷_𝑡^𝛼−1_ℎ𝑧_𝑑(0)𝑡_ℎ^𝛼−1Γ(𝛼). (40)
After some straightforward manipulators, we can obtain 𝑏−𝑎₁Γ()𝑡1+𝛼^𝛼ℎ≤⁰𝐷_𝑡^𝛼−1𝑧(0)+₀𝐷_𝑡^𝛼−1𝑧_𝑑(0)Γ(𝛼)𝑡_ℎ^𝛼−1+𝑧_𝑑𝑡_ℎ. (41)
And this ends the proof of Theorem 5.

Remark 6. The assumptions (25)–(27) in Theorem 5 are rather stringent. We know that the control system will be stable if these conditions hold; yet, imposing such conditions makes the proposed design valid only in a local region. In the simulation, we present examples to show that the aforementioned region is large enough to achieve a highly satisfactory performance.

Remark 7. The boundedness condition (26) can also be seen in [23, pp. 6927].

In this paper, we will prove the boundedness conditions and establish a fundamental lemma. This lemma is established for stability analysis of fractional order systems, especially for Mittag-Leffler stability [24] analysis of fractional order nonlinear systems. As an example, we will prove the boundedness condition (26).

Lemma 8. Let Λ=^∞_𝑘=1Γ(1+𝛼)Γ(1+𝑘)Γ(1−𝑘+𝛼)⁰𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧. (42) Then there exists some positive constant 𝑎₂>0 such that ^∞_𝑘=1Γ(1+𝛼)Γ(1+𝑘)Γ(1−𝑘+𝛼)⁰𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧≤𝑎₂|𝑧|. (43)

Proof. Since ₀𝐷^𝑘_𝑡𝑧₀ exists, it is obvious that ₀𝐷^𝑘_𝑡𝑧₀ are bounded. As a result, there exists 𝑀 such that ₀𝐷^𝑘_𝑡𝑧₀≤𝑀. (44)
On the other hand, because 0<𝛼<1, we have ₀𝐷_𝑡^𝛼−𝑘𝑧≤𝐾‖𝑧‖. (45)
It is known that the Gamma function is nonzero everywhere along the real line, and there is in fact no complex number 𝑧 for which Γ(𝑧)=0. As a result the reciprocal Gamma function 1/Γ(𝑧) is an entire function. There exists a lower bound 𝐿 such that 0<𝐿≤|Γ(1−𝛼+𝑘)| for 𝑘=1,2,3,….
Since Γ(𝑘)/(Γ(𝑘+1))=1/𝑘 and the infinite series ∑^∞_𝑘=11/(Γ(𝑘+1)) is convergence, there exists an upper bound 𝐻>0 such that 0<^∞_𝑘=11Γ(𝑘+1)<𝐻. (46)
From above discussion, we can obtain the following inequality: ^∞_𝑘=1Γ(1+𝛼)Γ(1+𝑘)Γ(1−𝑘+𝛼)⁰𝐷^𝑘_𝑡𝑧₀𝐷_𝑡^𝛼−𝑘𝑧≤𝑎₂|𝑧| (47) in which 𝑎₂=Γ(1+𝛼)𝑀𝐾𝐻𝐿. (48)

3.3. Controller Design

Since the nonlinear functions 𝐹(𝑥) and 𝐺(𝑥) are uncertain, the ideal controller in Section 3.1 cannot be used directly. Firstly, let us rearrange the ideal controller (14) as 𝑢^∗=𝐺₁(𝑥)𝑣−𝐺₁(𝑥)𝐹(𝑥)+(𝑥)𝐼𝑠(𝑥)𝐾+𝛽=𝛾+𝐾𝑠, (49) where 𝛾(𝑥)=𝐺₁(𝑥)𝑣−𝐺₁(𝑥)𝐹(𝑥)+𝛽(𝑥)𝐼𝑠=[𝛾₁(𝑥),…,𝛾_𝑝(𝑥)]^𝑇 represents the system uncertainty. Then from the discussions in Section 3.2, the unknown function 𝛾_𝑗(𝑥) can be approximated by the fuzzy logic system as 𝛾_𝑗(𝑥)=𝜃^𝑇_𝑗𝜓(𝑥),𝑗=1,2,…,𝑝. (50)

Let us denote 𝛾(𝑥)=𝜃𝜓(𝑥)=𝜃₁𝜓(𝑥),…,𝜃_𝑝𝜓(𝑥)^𝑇, (51) then the controller can be designed as 𝑢=𝜃^𝑇𝜓(𝑥)+𝐾𝑠+𝐾₁sign(𝑠), (52) where 𝐾₁sign(𝑠) is a robust term used to cancel the approximation error of the fuzzy systems.

Remark 9. Noting that there are 𝑝 fuzzy logic systems are employed to approximate the unknown function 𝛾(𝑥). And for every fuzzy system, the parameters are updated by the adaption law (28); that is, there are 𝑝 fractional adaption laws which are used in this paper.

From above discussions, now we are ready to give the following theorem.

Theorem 10. Consider system (2). Suppose that Assumptions 1 and 2 are satisfied. If the control input is defined by (52) with fraction adaption laws (28), then all signals in the closed loop system will remain bounded and the tracking errors and their derivatives converge to origin asymptotically; that is, 𝑒_𝑖^(𝑗)→0, 𝑡→∞, 𝑖=1,…,𝑝, 𝑗=1,…,𝑟_𝑗−1.

From Theorems 4 and 5, we can easily get Theorem 5. Here we omit the proof of Theorem 10.

4. Simulation Results

Two nonlinear systems are utilized to show the effectiveness of the proposed hybrid control scheme.

4.1. Example  1

Firstly, let us use the following MIMO system: 𝑥̇₁₁=𝑥₁₂,𝑥̇₁₂=𝑥₂₁−0.3sin𝑥₁₁𝑥₁₂+𝑥²₁₂+4+cos𝑥₁₁𝑢₁+1+sin²𝑥₂₁𝑢₂,𝑥̇₂₁=𝑥₂₂,𝑥̇₂₂=𝑥³₂₂+𝑒^𝑥11−1+𝑥²₁₂+1+sin²𝑥₂₁𝑢₁+1+sin²𝑥₂₂𝑢₂,𝑦₁=𝑥₁₁,𝑦₂=𝑥₂₁. (53)

System (53) can be written as the following compact form: 𝑥̈=𝐹(𝑥)+𝐺(𝑥)𝑢, (54) where 𝐹(𝑥)=^{𝑥21−0.3sin(𝑥11𝑥12)+𝑥212}_{𝑥322+𝑒𝑥11−1+𝑥212}, 𝐺(𝑥)=^{4+cos(𝑥11)1+sin2(𝑥21)}_{1+sin2(𝑥21)1+sin2(𝑥22)}.

The fuzzy systems have [𝑥₁₁,𝑥₁₂,𝑥₂₁,𝑥₂₂]^𝑇 as input. For each variable of these fuzzy systems, we define five Gaussian membership functions distributed on the interval [−10,10]. The Gaussian membership functions are shown in Figure 2. So there are 5⁴=625 rules that are used in the simulation. The initial value of 𝜃_𝑗, 𝑗=1,2,3,4, is chosen as 0.

Figure 2 

Membership functions in fuzzy system.

The design parameters are chosen as 𝜆=3, 𝑏=10, 𝐾=diag[5,5], and 𝐾₁=𝐾. The initial value of the system and the desired signal are chosen as 𝑥(0)=[1,−1,2,−2]^𝑇 and 𝑦_𝑑=[0,0]^𝑇, respectively.

Figure 3 shows the simulation results of the proposed method. Figures 3(a) and 3(b) indicate that the tracking errors are bounded and converge to zero rapidly. Figure 3(c) shows the boundedness of control inputs. Figure 3(d) gives the time response of the sliding surface. From these results, we can conclude that the sliding surface and the tracking errors converge in the vicinity of the origin. To show the change and boundedness of the fuzzy system parameters, time responses of ‖𝜃₁‖ and ‖𝜃₂‖ are depicted in Figure 4.

(a)

(b)

(c)

(d)

Figure 3 

Simulation results for Example  1: (a) tracking errors: 𝑒1 (solid line) and 𝑒1̇ (dashed line). (b) Tracking errors: 𝑒2 (solid line) and 𝑒2̇ (dashed line). (c) Control inputs: 𝑢1 (solid line) and 𝑢2 (dashed line). (d) Sliding surface: 𝑠1 (solid line) and 𝑠2 (dashed line).

Figure 4 

Time response of the fuzzy system parameters: ‖𝜃1‖ (solid line) and ‖𝜃2‖ (dashed line).

4.2. Example  2

In this Section, a 3D saturated multiscroll chaotic system will be used. The comparison between our control method and the control method proposed in [12] will be made.

A 3D saturated multiscroll chaotic system can be described by [12, 25] 𝑥̇₁=𝑥₂−𝑑₂𝑠𝑥₂;𝑘₂,ℎ₂,𝑝₂,𝑞₂𝑏+𝑢₁=𝑓₁(𝑥)+𝑢₁,𝑥̇₂=𝑥₃−𝑑₃𝑠𝑥₃;𝑘₃,ℎ₃,𝑝₃,𝑞₃𝑐+𝑢₂=𝑓₂(𝑥)+𝑢₂,𝑥̇₃=−𝑎𝑥₁−𝑏𝑥₂−𝑐𝑥₃+³_𝑖=1𝑑_𝑖𝑠𝑥_𝑖;𝑘_𝑖,ℎ_𝑖,𝑝_𝑖,𝑞_𝑖+𝑢₃=𝑓₃(𝑥)+𝑢₃, (4.2) where the saturated function series 𝑠(𝑥;𝑘,ℎ,𝑝,𝑞) is defined as 𝑠=𝑥;𝑘,ℎ,𝑝,𝑞2𝑞+1if𝑘()𝑥>𝑞ℎ+1𝑥−𝑖ℎ+2𝑖𝑘if||()𝑘𝑥−𝑖ℎ≤1,−𝑝≤𝑖≤𝑞2𝑖+1if()−𝑘𝑖ℎ+1<𝑥<𝑖+1ℎ−1,−𝑝≤𝑖<𝑞2𝑝+1if𝑥<−𝑝ℎ−1. (55) When 𝑎=𝑑₁=0.7, 𝑏=𝑐=𝑑₂=𝑑₃=0.8, 𝑘₁=100, ℎ₁=200, 𝑘₂=𝑘₃=40, ℎ₂=ℎ₃=80, and 𝑝_𝑖=𝑞_𝑖=2, the unforced system (34) has a 3D 6×6×6-grid scroll chaotic attractor, as shown in Figure 5.

(a)

(b)

Figure 5 

3D multiscroll attractor: (a) 𝑥−𝑦 plane and (b) 𝑥−𝑧 plane.

The initial values are chosen as 𝑥₁(0)=5, 𝑥₂(0)=−5, 𝑥₃(0)=−4, 𝜃_𝑖(0)=0, and 𝜀_𝑖̂‌(0)=0, 𝑖=1,2,3. The desired trajectories are 𝑥_𝑑1=𝑥_𝑑2=𝑥_𝑑3=sin𝑡.

The design parameters are chosen as 𝜆=3, 𝑏=10, 𝐾=diag[0.5,0.5,0.5], and 𝐾₁=𝐾. Note that 𝑘_𝑖 are chosen the same as in [12]. The discontinuous function sign(⋅) has been replaced by smooth function  arctan(20⋅)  to cancel the chattering phenomenon.

Figure 6 shows the simulation results of the proposed scheme. From the simulation results, we can see that the tracking significantly decreased by using the proposed method. Compared with the control scheme in [12], the proposed controller can achieve a better performance in the presence of disturbances and system uncertainties.

(a)

(b)

(c)

(d)

Figure 6 

Comparison between our control method and the control method proposed in [12]. (a), (b), (c) Tracking of 𝑥1, 𝑥2, and 𝑥3: control method proposed in [12] (dotted line), desired curve (dashed line), and our control method (solid line). (d) The control inputs: 𝑢1 (dotted line), 𝑢2 (dashed line), and 𝑢3 (solid line).

5. Conclusions

This paper proposes a fractional order integration method for updating the parameters of fuzzy systems. It is shown that the proposed controller is applicable to MIMO nonlinear systems. According to the results in this paper, the fractional order updating law outperforms the updating mechanisms exploiting integer-order operators. To demonstrate the effectiveness of fractional order operators in the fuzzy system parameters updating, this paper investigates a wide range of applications from the domain of adaptive control. Specifically, the adaptive fuzzy sliding mode control method is focused on in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 61001086) and Fundamental Research Funds for the Central Universities (Grant no. ZYGX2011X004).