On Fractional Model Reference Adaptive Control

This paper extends the conventional Model Reference Adaptive Control systems to fractional ones based on the theory of fractional calculus. A control law and an incommensurate fractional adaptation law are designed for the fractional plant and the fractional reference model. The stability and tracking convergence are analyzed using the frequency distributed fractional integrator model and Lyapunov theory. Moreover, numerical simulations of both linear and nonlinear systems are performed to exhibit the viability and effectiveness of the proposed methodology.


Introduction
Fractional calculus dates back to the end of the 17th century. Over three hundred years, a firm theoretical foundation has been established due primarily to Liouville, Grünwald, Letnikov, Riemann, and Caputo. In the last three decades, many scientific studies have shown the importance of fractional calculus and its applications in mathematics, physics, chemistry, material, engineering, finance, and even social science [1][2][3]. The stability of fractional differential equations and fractional control has gained rapid development very recently [4][5][6].
Inspired by contributions from [25,26], this paper aims at going further by applying an incommensurate fractional adaptation law to fractional plant and fractional reference model. Furthermore, the stability and tracking convergence of the fractional adaptive system are analyzed based on the continuous frequency distributed model of fractional integrator.
The rest of the paper is organized as follows. Section 2 reviews some basic definitions and theorems about fractional calculus. Section 3 designs a control law and a fractional adaptation law for fractional linear MRAC systems along with numerical simulations. Section 4 extends the proposed schemes to fractional nonlinear systems. Finally, Section 5 concludes this paper with some remarks on future study.

Basic Definitions and Preliminaries
Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operator , where and are the bounds of the operation and ∈ R. The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, 2 The Scientific World Journal the Riemann-Liouville definition, and the Caputo definition [1][2][3].
Definition 1. The Grünwald-Letnikov derivative definition of order is described as Definition 2. The Riemann-Liouville derivative definition of order is described as Definition 3. The Caputo definition of fractional derivatives can be written as In the following, we use the Caputo approach to describe fractional systems and the Grünwald-Letnikov approach to perform numerical simulations. To simplify the notation, we denote the fractional derivative of order as instead of 0 in this paper.
Lemma 4 (the continuous frequency distributed model [33]). The fractional integrator − , 0 < < 1 is a linear frequency distributed system, with input V( ) and output ( ). Its frequency distributed state ( , ) verifies the differential equation (for the elementary frequency ) as follows: and the output ( ) of the fractional integrator is the weighted integral (with weight ( )) of all the contributions ( , ) ranging from 0 to ∞ as follows: The relations (4) and (5) define the frequency distributed model of the fractional integrator.

Adaptive Control of Fractional Linear Systems
In this section, we extend the conventional MRAC systems to fractional ones based on the theory of fractional calculus. Firstly, a fractional plant and an incommensurate fractional reference model are described by the fractional differential equations. Then, a control law and an incommensurate fractional adaptation law which are generalized from the conventional ones [35,36] are designed. Finally, the stability and tracking performance of the fractional adaptive system are analyzed based on the continuous frequency distributed model of fractional integrator.

Fractional Adaptive Control Design.
Consider the following fractional differential equation: where 1 is the fractional order lying between (0, 1), is the plant output, is the input, and and are constant plant parameters that are assumed to be unknown. The reference model is specified by a fractional differential equation as follows: where and are constant parameters and ( ) is a bounded external reference signal.
Our objective of the fractional adaptive control design is to construct a control law and a fractional adaptation law to make the fractional plant (6) track the fractional reference model (7) on the basis of system stability.
Let us assume the sign of the parameter to be known and design the control law to be wherê( ) and̂( ) are variable feedback gains to be decided later.
Define the tracking error and the parameter errors̃=̂− * , where * = / and * = ( − )/ are the same as the conventional case. Subtracting (7) from (6) derives the dynamics of tracking error as follows: To adjust the two parameters in the control law (8), an adaptation law can be chosen in the fractional form as follows where 0 < 2 < 1, 0 < 3 < 1.
Note that the control law (8) and the fractional adaptation law (12) are generalized from the conventional MRAC systems [35,36].
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Analysis of Stability and Tracking Convergence.
In the following, we will prove that the fractional plant (6) can be controlled with the control law (8) and the adaptation law (12). The tracking error system (11) and adaptation law (12) constitute the following closed-loop adaptive system: Based on the continuous frequency distributed model of the fractional integrator in Lemma 4, the above adaptive system is exactly equivalent to the infinite dimensional ODEs as follows: with ( ) = (sin( )/ ) − , = 1, 2, 3.
In the above continuous frequency distributed model, Then, Substituting the first equation of (14) into (16) gives Substituting the second equation of (14) into the integral term of (17) yields Similarly, one derives Finally, lets define Then, one derives Owing to Lemma 5, / is negative semidefinite, implying the stability of the fractional adaptive system (13).
This proves that the fractional MRAC problem (6)-(7) can be solved by using the control law (8) and the fractional adaptation law (12).

Numerical Simulations.
Consider the control of the fractional plant with known fractional order 1 = 0.9 and unknown parameters and . The sign of is assumed to be positive. The fractional reference model is chosen to be 0.9 that is, 1 = 0.9, = 4, = 4, ( ) = sin(3 ). The adaptation gain is chosen to be = 1, while the fractional orders of the adaptation law are chosen as 2 = 0.4, As for the initialization issue, we refer to the method proposed by Lorenzo and Hartley in [37], where it is addressed that the initial conditions for fractional differential equations with order between 0 and 1 are a constant function of time. Therefore, the initial conditions of the fractional plant, the fractional model, and the fractional adaptation law are chosen, respectively, as ( ) = (0 + ) = 10, ( ) = (0 + ) = 10,̂( ) =̂(0 + ) = 1,̂( ) =̂(0 + ) = 1, for −∞ ≤ ≤ 0.
The numerical simulations of the behavior of the fractional linear adaptive system are illustrated in Figure 1. For interpretations of the references to the color in the upper left figure, the reader is referred to the web version of this paper.

Remark 6.
In [25], the authors have designed a commensurate fractional adaptation law for the integer order SISO systems. Benefits from the use of fractional calculus are also illustrated mainly via numerical simulations. However, detailed theoretical analysis is left out in their work. In the following, we give the theoretical analysis of the fractional control for the integer order plant.
With the first order error dynamics (i.e., 1 = 1 in system (13)) and the fractional adaptation law (12), the closed-loop adaptive system is described by The Scientific World Journal By converting the last two FDEs into infinite-dimensional ODEs as (14) and introducing Lyapunov function as one derives By Lemma 5, / is negative semidefinite, which implies the stability of the fractional adaptive system (23). This proves that the integer order plant ( 1 = 1 in system (6)) can be controlled with the control law (8) and the fractional adaptation law (12).

Extension to Fractional Nonlinear Systems
In this section, we extend the fractional control method previously proposed to fractional nonlinear systems. The fractional nonlinear plant is described by the fractional differential equation as follows: where is a known nonlinear function. The fractional reference model is chosen as (7). Instead of using control law (8) and adaptation law (12), we now use the following control law: The Scientific World Journal and the adaptation law Similarly, one can easily analyze the stability and tracking convergence of the above fractional nonlinear adaptive system based on the continuous frequency distributed model of fractional integrator.
The following example demonstrates the behavior of the fractional nonlinear adaptive system.
Consider the fractional nonlinear plant with known fractional order 1 = 0.9 and unknown parameters , , and . The sign of is assumed to be positive. ( ) is chosen to be 2 . The fractional reference model is chosen to be the same as (22).
The initial conditions of the fractional plant, the fractional model, and the fractional adaptation law are chosen, respectively, as ( ) = (0 + ) = 10, ( ) = (0 + ) = 10, The adaptation gain is chosen to be = 1, while the fractional orders of the adaptation law are chosen as 2 = 0.9, 3 = 0.6, 4 = 0.6. Figures 2 and 3 illustrate the numerical simulations of the behavior of the fractional nonlinear adaptive system. For interpretations of the references to the color in the upper left figure of Figure 2, the reader is referred to the web version of this paper.

Concluding Remarks
Based on the theory of fractional calculus, this paper has extended the conventional MRAC systems to fractional ones by designing a control law and a fractional adaptation law The Scientific World Journal 7 for the fractional plant and fractional reference model. The stability and tracking convergence have been analyzed using the frequency distributed fractional integrator model and Lyapunov theory. Moreover, numerical simulations of both linear and nonlinear systems have been performed to exhibit the viability and effectiveness of the proposed methodology.
As for the future perspectives, our research efforts will be focused on the following.
(i) How the fractional orders of the adaptation law affect the performance of the control system.
(ii) The optimal design of the fractional orders of the adaptation law.
(iii) Superiority of fractional MRAC systems compared to the conventional ones.