# On Fractional Model Reference Adaptive Control

Authors: Bao Shi; Jian Yuan; Chao Dong
Journal: The Scientific World Journal
 (2014)

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## Abstract

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.


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## Body

## 1. 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 [1–3]. The stability of fractional differential equations and fractional control has gained rapid development very recently [4–6].Several pioneering attempts to develop fractional order control methodologies have been made, such as TID control [7], CRONE control [8], fractional PID control [9], and fractional lead-lag compensator [10]. Basic ideas and technical formulations of the above four fractional control schemes with comparative comments have been addressed in [11]. By applying fractional calculus to advanced nonlinear control theory, several fractional nonlinear control schemes have been proposed very recently, such as fractional sliding mode control [12–24], fractional adaptive control [25–29], and fractional optimal control [30–32]. Particularly, in [25], the authors have presented two ideas to extend the conventional Model Reference Adaptive Control (MRAC) by using fractional order parameter adjustment rule and fractional reference model. In [26], a fractional model reference adaptive control algorithm for SISO plants has been proposed in frequency domain, which guarantees the stability and ability to reject disturbances.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. Section2 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.


## 2. Basic Definitions and Preliminaries

Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operatoraDtα, where a and t are the bounds of the operation and a∈ℝ. The three most frequently used definitions for the general fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [1–3].Definition 1.
The Grünwald-Letnikov derivative definition of orderα is described as
(1)aDtαf(t)=lim⁡h→0⁡1hα∑j=0∞‍(-1)j(αj)f(t-jh).Definition 2.
The Riemann-Liouville derivative definition of orderα is described as
(2)aDtαf(t)=1Γ(n-α)dndtn∫at‍f(τ)dτ(t-τ)α-n+1.Definition 3.
The Caputo definition of fractional derivatives can be written as(3)aDtαf(t)=1Γ(n-α)∫at‍f(n)(τ)dτ(t-τ)α-n+1,n-1<α<n.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 orderα as Dα instead of 0Dtα in this paper.Lemma 4 (the continuous frequency distributed model [33]).
The fractional integratorD-α, 0<α<1 is a linear frequency distributed system, with input v(t) and output x(t). Its frequency distributed state z(ω,t) verifies the differential equation (for the elementary frequency ω) as follows:
(4)∂z(ω,t)∂t=-ωz(ω,t)+v(t),
and the output x(t) of the fractional integrator is the weighted integral (with weight μ(ω)) of all the contributions z(ω,t) ranging from 0 to ∞ as follows:
(5)x(t)=∫0∞‍μ(ω)z(ω,t)dω,
with μ(ω)=(sin(απ)/π)ω-α.
The relations (4) and (5) define the frequency distributed model of the fractional integrator.Lemma 5.
The quadratic formW=W1+W2 is positive semidefinite if ai≥0, where W1=∑i=1mW1i and W2=∑i=1maiW2i with W1i=∫0∞‍μi(ω)ωzi2dω and W2i=xi2, i=1,2,…,m [34].


## 3. 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.


### 3.1. Fractional Adaptive Control Design

Consider the following fractional differential equation:(6)Dα1y=-apy+bpu,
where α1 is the fractional order lying between (0,1), y is the plant output, u is the input, and ap and bp are constant plant parameters that are assumed to be unknown.The reference model is specified by a fractional differential equation as follows:(7)Dα1ym=-amym+bmr(t),
where am and bm are constant parameters and r(t) 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 parameterbp to be known and design the control law to be
(8)u=a^r(t)r+a^y(t)y,
where a^r(t) and a^y(t) are variable feedback gains to be decided later.Define the tracking error(9)e=y-ym,
and the parameter errors
(10)a~r=a^r-ar*,a~y=a^y-ay*,
where ar*=bm/bp and ay*=(ap-am)/bp are the same as the conventional case.Subtracting (7) from (6) derives the dynamics of tracking error as follows:
(11)Dα1e=-ame+bp(a~rr+a~yy).To adjust the two parameters in the control law (8), an adaptation law can be chosen in the fractional form as follows
(12)Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey,
where 0<α2<1, 0<α3<1.Note that the control law (8) and the fractional adaptation law (12) are generalized from the conventional MRAC systems [35, 36].


### 3.2. 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:
(13)Dα1e=-ame+bp(a~rr+a~yy),Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey.Based on the continuous frequency distributed model of the fractional integrator in Lemma4, the above adaptive system is exactly equivalent to the infinite dimensional ODEs as follows:
(14)∂z1(ω,t)∂t=-ωz1(ω,t)-ame+bp(a~rr+a~yy),e(t)=∫0∞‍μ1(ω)z1(ω,t)dω,∂z2(ω,t)∂t=-ωz2(ω,t)--sign⁡(br)γer,a~r(t)=∫0∞‍μ2(ω)z2(ω,t)dω,∂z3(ω,t)∂t=-ωz3(ω,t)--sign⁡(by)γey,a~y(t)=∫0∞‍μ3(ω)z3(ω,t)dω,
with μi(ω)=(sin(αiπ)/π)ω-αi, i=1,2,3.In the above continuous frequency distributed model,z1(ω,t), z2(ω,t), and z3(ω,t) are the true state variables, while e(t), a~r(t), and a~y(t) are the pseudo state variables.Let us define two types of Lyapunov functions as follows:(i)
vi(w,t): the monochromatic Lyapunov functions corresponding to the elementary frequency;(ii)
Vi(t): the Lyapunov functions summing all the monochromatic vi(w,t) with the weighting functions μi(ω), i=1,2,3.Namely,(15)v1(w,t)=12z12,v2(w,t)=|bp|2γz22,v3(w,t)=|bp|2γz32,Vi(t)=∫0∞‍μi(ω)vi(ω,t)dω,i=1,2,3.
Then,
(16)dV1dt=∫0∞‍μ1(ω)∂v1(ω,t)∂tdω.Substituting the first equation of (14) into (16) gives
(17)dV1dt=∫0∞‍μ1(ω)z1[(a~rr+a~yy)-ωz1(ω,t)-ame+bp(a~rr+a~yy)]dω=-∫0∞‍μ1(ω)ωz12dω+∫0∞‍μ1(ω)z1[-ame+bp(a~rr+a~yy)]dω=-∫0∞‍μ1(ω)ωz12dω+[-ame+bp(a~rr+a~yy)]×∫0∞‍μ1(ω)z1dω.Substituting the second equation of (14) into the integral term of (17) yields
(18)dV1dt=-∫0∞‍μ1(ω)ωz12dω-ame2+ebp(a~rr+a~yy).
Similarly, one derives
(19)dV2dt=-|bp|γ∫0∞‍μ2(ω)ωz22dω-ebpa~rr,dV3dt=-|bp|γ∫0∞‍μ3(ω)ωz32dω-ebpa~yy.
Finally, lets define
(20)V(t)=V1(t)+V2(t)+V3(t).
Then, one derives
(21)dVdt=-∫0∞‍μ1(ω)ωz12dω-|bp|γ∫0∞‍μ2(ω)ωz22dω-|bp|γ∫0∞‍μ3(ω)ωz32dω-ame2.Owing to Lemma5, dV/dt 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).


### 3.3. Numerical Simulations

Consider the control of the fractional plant with known fractional orderα1=0.9 and unknown parameters ap and bp. The sign of bp is assumed to be positive. The fractional reference model is chosen to be
(22)D0.9ym=-4ym+4sin(3t),
that is, α1=0.9, am=4, bm=4, r(t)=sin(3t).The adaptation gain is chosen to beγ=1, while the fractional orders of the adaptation law are chosen as α2=0.4, α3=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 y(t)=y(0+)=10, ym(t)=ym(0+)=10, a^r(t)=a^r(0+)=1, a^y(t)=a^y(0+)=1, for -∞≤t≤0.The numerical simulations of the behavior of the fractional linear adaptive system are illustrated in Figure1. For interpretations of the references to the color in the upper left figure, the reader is referred to the web version of this paper.Fractional adaptive control of the fractional linear system (6). (a) Tracking performance (red line represents the state of the fractional plant, while the blue line represents the state of the fractional reference model); (b) control parameter a^r(t); (c) control parameter a^y(t); (d) control input u(t); (e) estimation of parameter ap(t); (f) estimation of parameter ay(t).
(a)
(b)
(c)
(d)
(e)
(f)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.,α1=1 in system (13)) and the fractional adaptation law (12), the closed-loop adaptive system is described by
(23)dedt=-ame+bp(a~rr+a~yy),Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey.By converting the last two FDEs into infinite-dimensional ODEs as (14) and introducing Lyapunov function as
(24)V(t)=12e2+|bp|2γ∫0∞‍μ2(ω)z22dω+|bp|2γ∫0∞‍μ3(ω)z32dω,
one derives
(25)dVdt(t)=-|bp|γ∫0∞‍μ2(ω)ωz22dω-|bp|γ∫0∞‍μ3(ω)ωz32dω-ame2.By Lemma5, dV/dt is negative semidefinite, which implies the stability of the fractional adaptive system (23). This proves that the integer order plant (α1=1 in system (6)) can be controlled with the control law (8) and the fractional adaptation law (12).Remark 7.
From the analysis in Remark6, it is evident that the values of the fractional orders α1, α2, and α3 can be extended to the following values: 0<α1≤1, 0<α2≤1, and 0<α3≤1.


## 3.1. Fractional Adaptive Control Design

Consider the following fractional differential equation:(6)Dα1y=-apy+bpu,
where α1 is the fractional order lying between (0,1), y is the plant output, u is the input, and ap and bp are constant plant parameters that are assumed to be unknown.The reference model is specified by a fractional differential equation as follows:(7)Dα1ym=-amym+bmr(t),
where am and bm are constant parameters and r(t) 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 parameterbp to be known and design the control law to be
(8)u=a^r(t)r+a^y(t)y,
where a^r(t) and a^y(t) are variable feedback gains to be decided later.Define the tracking error(9)e=y-ym,
and the parameter errors
(10)a~r=a^r-ar*,a~y=a^y-ay*,
where ar*=bm/bp and ay*=(ap-am)/bp are the same as the conventional case.Subtracting (7) from (6) derives the dynamics of tracking error as follows:
(11)Dα1e=-ame+bp(a~rr+a~yy).To adjust the two parameters in the control law (8), an adaptation law can be chosen in the fractional form as follows
(12)Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey,
where 0<α2<1, 0<α3<1.Note that the control law (8) and the fractional adaptation law (12) are generalized from the conventional MRAC systems [35, 36].


## 3.2. 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:
(13)Dα1e=-ame+bp(a~rr+a~yy),Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey.Based on the continuous frequency distributed model of the fractional integrator in Lemma4, the above adaptive system is exactly equivalent to the infinite dimensional ODEs as follows:
(14)∂z1(ω,t)∂t=-ωz1(ω,t)-ame+bp(a~rr+a~yy),e(t)=∫0∞‍μ1(ω)z1(ω,t)dω,∂z2(ω,t)∂t=-ωz2(ω,t)--sign⁡(br)γer,a~r(t)=∫0∞‍μ2(ω)z2(ω,t)dω,∂z3(ω,t)∂t=-ωz3(ω,t)--sign⁡(by)γey,a~y(t)=∫0∞‍μ3(ω)z3(ω,t)dω,
with μi(ω)=(sin(αiπ)/π)ω-αi, i=1,2,3.In the above continuous frequency distributed model,z1(ω,t), z2(ω,t), and z3(ω,t) are the true state variables, while e(t), a~r(t), and a~y(t) are the pseudo state variables.Let us define two types of Lyapunov functions as follows:(i)
vi(w,t): the monochromatic Lyapunov functions corresponding to the elementary frequency;(ii)
Vi(t): the Lyapunov functions summing all the monochromatic vi(w,t) with the weighting functions μi(ω), i=1,2,3.Namely,(15)v1(w,t)=12z12,v2(w,t)=|bp|2γz22,v3(w,t)=|bp|2γz32,Vi(t)=∫0∞‍μi(ω)vi(ω,t)dω,i=1,2,3.
Then,
(16)dV1dt=∫0∞‍μ1(ω)∂v1(ω,t)∂tdω.Substituting the first equation of (14) into (16) gives
(17)dV1dt=∫0∞‍μ1(ω)z1[(a~rr+a~yy)-ωz1(ω,t)-ame+bp(a~rr+a~yy)]dω=-∫0∞‍μ1(ω)ωz12dω+∫0∞‍μ1(ω)z1[-ame+bp(a~rr+a~yy)]dω=-∫0∞‍μ1(ω)ωz12dω+[-ame+bp(a~rr+a~yy)]×∫0∞‍μ1(ω)z1dω.Substituting the second equation of (14) into the integral term of (17) yields
(18)dV1dt=-∫0∞‍μ1(ω)ωz12dω-ame2+ebp(a~rr+a~yy).
Similarly, one derives
(19)dV2dt=-|bp|γ∫0∞‍μ2(ω)ωz22dω-ebpa~rr,dV3dt=-|bp|γ∫0∞‍μ3(ω)ωz32dω-ebpa~yy.
Finally, lets define
(20)V(t)=V1(t)+V2(t)+V3(t).
Then, one derives
(21)dVdt=-∫0∞‍μ1(ω)ωz12dω-|bp|γ∫0∞‍μ2(ω)ωz22dω-|bp|γ∫0∞‍μ3(ω)ωz32dω-ame2.Owing to Lemma5, dV/dt 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).


## 3.3. Numerical Simulations

Consider the control of the fractional plant with known fractional orderα1=0.9 and unknown parameters ap and bp. The sign of bp is assumed to be positive. The fractional reference model is chosen to be
(22)D0.9ym=-4ym+4sin(3t),
that is, α1=0.9, am=4, bm=4, r(t)=sin(3t).The adaptation gain is chosen to beγ=1, while the fractional orders of the adaptation law are chosen as α2=0.4, α3=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 y(t)=y(0+)=10, ym(t)=ym(0+)=10, a^r(t)=a^r(0+)=1, a^y(t)=a^y(0+)=1, for -∞≤t≤0.The numerical simulations of the behavior of the fractional linear adaptive system are illustrated in Figure1. For interpretations of the references to the color in the upper left figure, the reader is referred to the web version of this paper.Fractional adaptive control of the fractional linear system (6). (a) Tracking performance (red line represents the state of the fractional plant, while the blue line represents the state of the fractional reference model); (b) control parameter a^r(t); (c) control parameter a^y(t); (d) control input u(t); (e) estimation of parameter ap(t); (f) estimation of parameter ay(t).
(a)
(b)
(c)
(d)
(e)
(f)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.,α1=1 in system (13)) and the fractional adaptation law (12), the closed-loop adaptive system is described by
(23)dedt=-ame+bp(a~rr+a~yy),Dα2a^r=-sign⁡(bp)γer,Dα3a^y=-sign⁡(bp)γey.By converting the last two FDEs into infinite-dimensional ODEs as (14) and introducing Lyapunov function as
(24)V(t)=12e2+|bp|2γ∫0∞‍μ2(ω)z22dω+|bp|2γ∫0∞‍μ3(ω)z32dω,
one derives
(25)dVdt(t)=-|bp|γ∫0∞‍μ2(ω)ωz22dω-|bp|γ∫0∞‍μ3(ω)ωz32dω-ame2.By Lemma5, dV/dt is negative semidefinite, which implies the stability of the fractional adaptive system (23). This proves that the integer order plant (α1=1 in system (6)) can be controlled with the control law (8) and the fractional adaptation law (12).Remark 7.
From the analysis in Remark6, it is evident that the values of the fractional orders α1, α2, and α3 can be extended to the following values: 0<α1≤1, 0<α2≤1, and 0<α3≤1.


## 4. 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:(26)Dα1y=-apy-cpf(y)+bpu,
where f 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:
(27)u=a^r(t)r+a^y(t)y+a^f(t)f(y),
and the adaptation law
(28)Dα2a~r=-sign⁡(bp)γer,Dα3a~y=-sign⁡(bp)γey,Dα4a~f=-sign⁡(bp)γef(y),
with a~f=a^f-cp/bp.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 orderα1=0.9 and unknown parameters ap, bp, and cp. The sign of bp is assumed to be positive. f(y) is chosen to be y2. 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, asy(t)=y(0+)=10, ym(t)=ym(0+)=10, a^r(t)=a^r(0+)=1, a^y(t)=a^y(0+)=1, a^f(t)=a^f(0+)=1, for -∞≤t≤0.The adaptation gain is chosen to beγ=1, while the fractional orders of the adaptation law are chosen as α2=0.9, α3=0.6, α4=0.6.Figures2 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.Fractional adaptive control of the fractional nonlinear system (26). (a) Tracking performance (red line represents the state of the fractional nonlinear plant, while, blue line represents the state of the fractional reference model); (b) control parameter a^r(t); (c) control parameter a^y(t); (d) control parameter a^f(t).
(a)
(b)
(c)
(d)Fractional adaptive control of the fractional nonlinear system (26). (a) Estimation of parameter ap(t); (b) parameter bp(t); (c) estimation of parameter cp(t); (d) control input u(t).
(a)
(b)
(c)
(d)


## 5. 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 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.(iv)
Design of fractional MRAC for general integer order or fractional order linear system, namely,Dαx=Ax+Bu, 0<α≤1.



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*Source: 521625-2014-01-16.xml*