Robust Fuzzy Adaptive Control with MRAC Configuration for a Class of Fractional Order Uncertain Linear Systems

Bachir Bourouba a,1, Samir Ladaci b,c,2,∗, Rachid Illoul d,3 a Department of Electrical Engineering, Setif-1 University BP: El Bez, Setif 19000, Algeria b Department of E.E.A., National Polytechnic School of Constantine, BP: 75A RP, Ali Mendjeli, Constantine 25000, Algeria c SP-Lab Laboratory, University of Mentouri Constantine 1, Department of Electronics, 25000 Constantine, Algeria d Department of Automatics, National Polytechnic School of Algiers, (ENP) BP 162, Elharrach, Algiers, 16200, Algeria 1 bourouba_b@yahoo.fr; 2 samir.ladaci@enp-constantine.dz; 3 rachid.illoul@g.enp.edu.dz ∗ Corresponding Author


Introduction
Fractional calculus is a 300 years old mathematical concept, but no significant impact was achieved in science and engineering until recent years. Recently, considerable attention has been paid to fractional-order systems whose models are described by fractional-order differential equations and especially, to fractional-order control design since it provides more robustness to model uncertainties and better response in comparison with classical controllers. Many works and applications are found in fractional calculus literature [1] [2] with many applications such as viscoelastic materials modeling [3], health monitoring [4], modeling and control of robotic systems [5], renewable energy systems [6], chaotic systems [7] and others [8]. One has to mention particularly, that fractional order controllers have been extensively used in many applications to achieve robust performance of the systems [9][10] [11]. ISSN: 2775ISSN: -2658 International Journal of Robotics and Control Systems Vol. 1, No. 3, 2021, pp. 326-337 327 certain linear systems is adaptive control. One can refer to the papers [12][13] [14] and the books [15][16] [17] and references therein. Design of direct Model Reference Adaptive Control (MRAC) was the subject for many research efforts for linear plants with structured parametric uncertainties. In this control approach, the principle objective is to obtain a system that follows closely the behavior of a system called reference model [18] [19].
Fractional order adaptive control is one of the most important emerging research topics in the present century [20][21] [22]. Among the proposed fractional adaptive control structures, MRAC-based approaches have attracted many researchers and designers because of its simplicity and efficiency [23]. However, when confronted to model uncertainties and disturbances, other control techniques have to be considered [24][25] [26].
An efficient solution is to combine fuzzy systems and estimators to the controller. Fractional order control has been integrated with fuzzy logic in many successful work in order to improve the control system behavior and robustness as in Fractional order PID control [27], fractional adaptive sliding mode control [28] [29] and fractional adaptive control [30] of uncertain systems.
Based on the results reported in literature, we propose a novel method to control a class of uncertain linear fractional-order systems using the MRAC configuration. The controller is developed to ensure perfect tracking of the reference model behavior despite model uncertainties and additive disturbances. Numerical simulations show the efficiency of the proposed control scheme.
This paper is organized as follows: Mathematical preliminaries definitions are presented in Section 2. Description of the fuzzy logic system is given in Section 3 while the proposed fractional order model reference adaptive control design and stability proof are developed in Section 4. A numerical simulation example illustrates the efficiency of the method in Section 5 and finally concluding remarks are given in Section 6.

Fractional order operators
Fractional order calculus opens a new perspective on integrals and derivatives notion and can be considered as a generalization of integer-order calculus. Let us define the fractional order operator a D γ t , where a denotes the lower limit and t is the upper limit respectively of the operator, and γ ∈ ℜ is the order of integration or differentiation. There are several definitions for fractional order derivatives. The three most commonly used definitions are the Grünwald-Letnikov, Riemann-Liouville, and Caputo derivative definition [31] [32]. if f (t) is a continuous time function, for the order n where n ∈ [0, 1].

Definition 2.1
The Riemann-Liouville's, fractional order derivative of f (t) of order γ is defined as [33],

Definition 2.2
The Caputo fractional order derivative of the function f (t) of order γ is defined as, γ > 0 is defined as, Where h is the sampling period and ω (γ) j are the coefficients of the polynomial: The coefficients can be expressed as: The Grünwald-Letnikov definition of the fractional-order derivative is used in this paper due to its well-understood physical interpretation and implementation easiness.

Remark 1
For the sake of simplicity, we will also note D α t f (t) as f (α) (t).

Some properties of the fractional order derivative
Two general properties of the fractional-order derivative will be used are recalled below: The additive index law: where α and β are real numbers.
property 2.5 The Caputo fractional order derivative is a linear operator: where a and b are real constants.

Stability of fractional order systems
We recall here some important results on the stability analysis of fractional order systems [34]. Theorem 2.6 Let x = 0 be an equilibrium point for the non autonomous fractional order system given by, Then the origin of system (7) is Lyapunov stable.
is decreasing, then the origin of system (7) is Lyapunov uniformly stable.

Lemma 2.7
Assume h(t) ∈ ℜ be a continuous and derivable function. Then, for any time instant t ≥ t 0 , the following inequality holds [34],

Description of the Fuzzy Logic System
Fuzzy Control is the field of control theory based on Fuzzy Set Theory introduced by Zadeh in 1965 [35]. A Fuzzy Logic System (FLS) consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine using a set of fuzzy rules and the defuzzifier. Usually, a FLS is modeled by the knowledge base of the FLS which is constituted of a collection of fuzzy If-then rules of the following form [28][36]: where x = (x 1 , ..., x n ) T and y are the fuzzy logic system input and output, respectively. Fuzzy sets F j i and φ j , associated with the fuzzy functions µ F j i (x i ) and µ φ j ( y) respectively. N is the number of rules. Through singleton function, center average defuzzification, and product inference, the fuzzy logic system can be expressed as [37]: and,f Where where ε > 0 is an arbitrary constant.

Fractional Order Model Reference Adaptive Control Design
The MRAC control structure is very efficient for plants with parameters that are unknown or varying slowly in the time, with better performance and robustness than fixed controllers. As illustrated in the block diagram of Fig. 1, it is structured in four main parts; the plant, the controller, the reference model and the adjustment mechanism. In order to introduce the fractional order model reference adaptive control problem, we first consider the class of fractional-order uncertain linear system described by: where x(t) ∈ ℜ n is a state vector and u(t) ∈ ℜ is the system input, A ∈ ℜ n × ℜ n and B ∈ ℜ n have unknown constant parameters. δx(t) is regarded as an uncertain dependent nonlinear state perturbation.
In order to introduce the proposed control input, the following assumptions are supposed true: Assumption 4.1 There exists a vector φ ∈ ℜ n , a real function ω(t) and scalar η ∈ ℜ such that:

Assumption 4.2
The states of the system x(t) are measurable.

Assumption 4.3 δx(t) is unknown but bounded.
According to Assumption 4.1, the system given by (12) can be written as, The fractional order reference model was chosen in order to generate the desired trajectory x m (t) which the plant output has to follow. The reference model is: The reference model is evidently stable and the input signal u r (t) ∈ ℜ is a bounded piecewise continuous function. A m ∈ ℜ n × ℜ n and B m ∈ ℜ n are known, x m (t) ∈ ℜ n is the reference model state vector available at each time t. The design objective is to make the tracking error e(t) = x(t) − x m (t) converge to 0 and based on the universal approximation theorem [38], the above fuzzy logic system is capable of uniformly approximating any well-defined nonlinear function over a compact set U c to any degree of accuracy.
Firstly, the ideal controller is defined as, Where the vectors K x and K r are constant but unknown ideal gains and u f (t) is approximation of smooth function ω(t).
Comparing the ideal closed-loop transfer function of the controlled plant to the desired reference model, the model-matching conditions are, In the adaptive control problem, the parameters of A and B are unknown and so, the control (15) cannot be implemented. Therefore, an adaptive controller which has the same structure as in (15) is used as follows, Where K x (t) and K r (t) are the estimates of K x and K r at time t, and we search an adaptive law to generate K x (t) and K r (t) on-line with, In order to develop the adaptive laws for K x (t), K r (t) and θ (t), we need an error equation in terms of tracking error e(t) = x(t) − x m (t) and the parameters errors Based on the certainty equivalence principle and the proposed adaptive control input (18), the closed-loop fractional order error dynamic system can be described by, Let us define the matrix P = P T ∈ ℜ n × ℜ n solution of the following Lyapunov equation: for any chosen matrix Q ∈ ℜ n × ℜ n such that Q = Q T . Then we present the main result of this paper,  (12) in the presence of the uncertainties subject to assumptions 4.1-4.3. The robust adaptive controller defined by (18)- (20) with adaptation laws given by where the positive-definite matrices Γ x = Γ T x ∈ ℜ n × ℜ n and Γ θ = Γ T θ ∈ ℜ p × ℜ p and the scalar γ r ∈ ℜ + are design parameters. Ensures that all the closed-loop signals are bounded, and the tracking errors converge to zero.

Proof of Theorem 4.4:
In order to analyse the closed-loop stability, let us consider the following Lyapunov candidate function, where Γ x and Γ θ are the adaptation rate matrices for K x and K θ respectively. γ is as adaptive scalar gain that affects the speed of the parameter adaptation.
Taking the fractional order derivative of (25) with respect to time and using Lemma 2.7, one has By substituting (22) into (26) we obtain V (α) ≤ e T PA m + A T m P e + ...
Then, in order to make V (α) ≤ 0 we choose the adaptive laws for K x (t), K r (t) and θ (t) as specified in (24), namely Using Barbalat's lemma [15][39], we conclude that the system is Lyapunov stable with all variables bounded and converge to zero: e(t) → 0 exponentially as t → ∞ for all initial conditions of the variables x(0) ∈ ℜ n . This completes the proof.

Simulation Results and Discussion
In this section we will give a numerical example to illustrate the efficiency of the proposed fuzzy robust adaptive scheme. Consider the plant represented by a state-space model of the form (12) with, with the disturbance signal modeled as, and consider the reference model represented with state-space equation (14) with, For the proposed fuzzy adaptive controller design, we chose the following membership functions for error and error derivative, The simulation test parameter settings are as follows: With the initial conditions, Simulation results are shown in Figs. 2-6, from which we can conclude that the stability of the system are guaranteed. Fig. 5 shows that the error signal converges rapidly towards zero. Fig. 6 shows also that all the varying gains are bounded during the control action.
It is also noticeable that error signal e(t) converges relatively slowly to zero, approximately in 30s (response time). This fact can be explained by the learning phase for adaptive control in presence of uncertainties and disturbances. However, the system is able to follow the varying reference perfectly after this limited time.

Conclusion
In this paper, we propose a new robust fractional adaptive control design for a class of fractional order uncertain linear systems. We introduce a fractional order adaptive fuzzy controller using the MRAC configuration in order to deal with external disturbances and model uncertainties while forcing the output of the controlled plant to track the output of a given reference model system, while maintaining the overall closed loop stability and signals convergence. The stability analysis is performed using Lyapunov stability theorem. Simulation results illustrate the efficiency of the proposed fuzzy adaptive controller to deal with uncertain fractional order linear systems. Further research will concentrate on application of this control scheme to real processes and compare its performance to other available control techniques in the same operating conditions. Extension to fractional order linear systems with delays is also a challenge.