Indirect model reference adaptive control for a class of fractional order systems

doi:10.1016/j.cnsns.2016.03.016

Communications in Nonlinear Science and Numerical Simulation

Volume 39, October 2016, Pages 458-471

https://doi.org/10.1016/j.cnsns.2016.03.016 Get rights and content

Highlights

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A constrained gradient estimation method is proposed.
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An indirect model reference adaptive controller is designed and fractional order update laws are utilized to update the parameters.
•
The asymptotical convergence of tracking error is proven rigorously without using certainty equivalence principle.

Abstract

This article focuses on the indirect model reference adaptive control problem for fractional order systems. A constrained gradient estimation method was established firstly, since parameter estimation is part and parcel of the whole control problem. Then a novel adaptive control law is designed, from which the two problems, i.e., parameter estimation and reference tracking, can be unified perfectly. On these basis, an effective control scheme is established. The stability of the resulting closed-loop system is analyzed rigorously via indirect Lyapunov method and frequency distributed model. Finally, a careful simulation study is reported to illustrate the effectiveness of the proposed scheme.

Introduction

The idea of fractional calculus has been known since the development of the regular integer order calculus, with the initial works being associated with Leibniz and $L^{'} H \hat{o} spital$ [1], [2]. In spite of being such an old topic, the developments in this field were rather slow especially compared with the integer order calculus. Yet, in recent decades, fractional order calculus has been widely used to describe a complex system more concisely and precisely like viscoelastic structures [3], heat conduction [4] and so on. It is also used to design a controller showing better performances than that of the traditional integer order case, like the PI^λD^μ controllers [5], the sliding mode controllers [6], the fractional order adaptive controllers [7], [8], [9], and many other controllers.

The traditional controllers are almost designed based on the precise system models, like the state feedback control, the pole placement control and so on. Yet, parameter variations, uncertainties and disturbances always exist in real plant, for which the designed controller can not show desirable performances all the time or even cannot guarantee the controlled system is stable. The model reference adaptive control (MRAC) is an efficient control strategy to guarantee the system works at a satisfying state all the time as the controller parameters are adjusted with the changing of the controlled system [10]. Furthermore, fractional order MRAC has been concerned by many scholars including the fractional order direct MRAC (FO-DMRAC) and the fractional order indirect MRAC (FO-IMRAC) recently. For the direct case, the authors in [11] present the fractional order MIT adjustment rules and the fractional order reference models by applying the fractional calculus to the MRAC scheme. Ref. [8] investigates the MRAC with fractional order parameter adjustment law, and compares it with the integer order case. However, the above methods are only for the integer order plants. Recently, authors in Ref. [12] study the control problem of fractional order systems by using fractional order MRAC and presents the rigorous stability analysis via the indirect Lyapunov method [13]. Ref. [14] discusses the problem of the DMRAC for fractional order systems with the system order 1 < α < 2. For the indirect case, there is only one paper [15] talking about this problem to the largest of the authors’ knowledge. It uses the discrete-time approximation method to transform the fractional order system to an integer order system and then uses the traditional IMRAC to control the system. But the authors in Ref. [15] do not solve the key problem of the IMRAC for fractional order systems to some extent.

As is known to all, IMRAC is an important part of MRAC and it can solves some problem that DMRAC cannot solve like MRAC for the systems with unstable zeroes [10]. But there are only few researches about it because IMRAC is much more complex than the DMRAC. On one hand, the IMRAC firstly estimates the system parameters and then the controller parameters are calculated from the estimated system parameters. Thus, we must find an estimation method suitable for the fractional order systems. In Ref. [16], a gradient parameter estimation method is proposed and fractional order integration is used to update the parameters. But it only considers the unconstrained parameter estimation, which sometimes cannot satisfy our requirements as there always exists some constraints on the system parameters. On the other hand, certainty equivalence principle (CEP) is often used in IMRAC [17] with two most important foundations, which are maximal plant uncertainty parameterization and stable controller parameter adaptation, yet whether CEP is suitable for the fractional order case has not been proven rigorously. Thus CEP may not be used in the FO-IMRAC simply as the integer order case, which makes the FO-IMRAC much more difficult to deal with.

Motivated by aforementioned reasons, a constrained gradient estimation method (CGEM) for fractional order systems will be studied in this article. Then for a class of commensurate fractional order systems, an indirect model reference adaptive controller is designed to control the system, which guarantees the output of the controlled system tracks the desired output asymptotically. Moreover, we consider a specific filter and a special cost function when using the CGEM, with which the closed-loop stability can be achieved by indirect Lyapunov method rather than using the CEP.

The remainder of this article is organized as follows. Section 2 introduces some preliminary results on fractional order calculus. CGEM is presented in Section 3 and the convergence of the parameters is analyzed via indirect Lyapunov method. Moreover, FO-IMRAC for a class of fractional order systems is proposed based on the CGEM. In Section 4, numerical simulations are provided. Finally, conclusions are given in Section 5.

Section snippets

Preliminaries

Fractional order calculus is the generalization of the classical integer order calculus. There are several definitions for fractional order derivative [18], among which the most commonly used definitions are the Grünwald–Letnikov, Riemann–Liouville, and Caputo derivative definition.

The Riemann–Liouville derivative definition of the order α is described as $\begin{matrix} _{c} D_{t}^{α} f (t) = \frac{d^{n}}{d t^{n}} [\frac{1}{Γ (n - α)} \int_{c}^{t} \frac{f (τ)}{{(t - τ)}^{α - n + 1}} d τ], \end{matrix}$ where $n - 1 < α < n, n \in N,$ $Γ (α) = \int_{0}^{\infty} x^{α - 1} e^{- x} d x$ is the Gamma function.

The Caputo derivative definition has the

Problem formulation and transformation

In this article, we consider a class of commensurate fractional order systems formulated as $\begin{matrix} D^{α} y (t) = - a_{p} y (t) + b_{p} u (t) - c_{p} f (y), \end{matrix}$ where 0 < α < 1, $y (t) \in R$ and $u (t) \in R$ are the system order, system output and control input, respectively. f(·) is a known function. a_p, c_p are unknown constants. b_p is also an unknown constant but its signum i.e. sgn(b_p) is known and its lower bound is known i.e. |b_p| ≥ ρ > 0.

Assume that the related reference model is given by $\begin{matrix} D^{α} y_{m} (t) = - a_{m} y_{m} (t) + b_{m} r (t), \end{matrix}$ where r(t) and y_m(t) are

Illustrative examples

In this section, three examples will be presented to demonstrate the effectiveness of the proposed approaches. Those numerical examples are implemented via the integer-order approximation algorithm in frequency domain and one can refer to Ref. [21] for more details about the algorithm.

Example 1

In this example, we will demonstrate the effectiveness of the proposed CGEM and its robustness to noise. Consider following fractional order system: $\begin{matrix} D^{α} y (t) + a_{0} y (t) = b_{0} u (t), \end{matrix}$ where $α = 0.7, a_{0} = 1, b_{0} = 3,$ $y (t) = 0, t \leq 0$ .

Take a

Conclusions

In the article, CGEM with fractional order update laws has been studied, which can guarantee the dynamics of the estimated parameters more steady than the integer order case. With the proposed estimation method, FO-IMRAC for a class of fractional order systems is presented. Moreover, by selecting suitable filter and cost function, the convergence of the tracking error is achieved via indirect Lyapunov method rather than the commonly used certainty equivalence principle. Finally, three examples

Conflict of interests

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

Acknowledgment

The work described in this paper was fully supported by the National Natural Science Foundation of China (No. 61573332) and the Fundamental Research Funds for the Central Universities (No. WK2100100028).

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View full text

Indirect model reference adaptive control for a class of fractional order systems

Highlights

Abstract

Introduction

Section snippets

Preliminaries

Problem formulation and transformation

Illustrative examples

Conclusions

Conflict of interests

Acknowledgment

ISA Trans

Appl Math Comput

Automatica

Automatica

ISA Trans

Signal Process

Commun Nonlinear Sci Numer Simul

Automatica

Comput Math Appl

Variable-order fractional operators for adaptive order and parameter estimation

IEEE Trans Autom Control