Brief paperA new framework for solving fractional optimal control problems using fractional pseudospectral methods☆
Introduction
Fractional optimal control problems (FOCPs) can be regarded as a generalization of classical integer optimal control problems (IOCPs) in the sense that the dynamics are described by fractional differential equations (Agrawal, 2004). There are various definitions of fractional derivatives and the two most important types are the Riemann–Liouville derivatives and the Caputo derivatives. It is noteworthy here that in distinct contrast with the integer derivatives (which are locally defined in the epsilon neighborhood of a chosen point), the fractional derivatives are nonlocal in nature as they are globally defined by a definite fractional integral over a domain. Moreover, the fractional derivatives involve singular kernel/weight functions, and the solutions of fractional differential equations are usually singular near the boundaries of the domain (Chen, Shen, & Wang, 2016). More background information on the fractional calculus can be found in Oldham and Spanier (2006) and Sabatier, Agrawal, and Tenreiro Machado (2007).
Because of the complexity of most applications, FOCPs/IOCPs are often solved numerically. In recent years, a class of numerical methods called pseudospectral methods (Elnagar, Kazemi, & Razzaghi, 1995; Fahroo & Ross, 2001; Benson, Huntington, Thorvaldsen, & Rao, 2006; Huntington, 2007; Garg et al., 2010, Garg et al., 2011; Francolin, Benson, Hager, & Rao, 2015) has become increasingly popular in the numerical solution of IOCPs. The basic principle of pseudospectral methods is to approximate the state using a set of basis functions and discretize the dynamic constraints using collocation at a specified set of points. As a result, a continuous optimal control problem is transcribed to a finite-dimensional nonlinear programming problem (NLP) which is then solved using well-known optimization software such as SNOPT (Gill, Murray, & Saunders, 2005) and IPOPT (Biegler & Zavala, 2009). The basis functions are typically Lagrange interpolating polynomials and the collocation points are usually chosen based on Gaussian-type quadrature rules. Basically there are two primary implementation forms for pseudospectral methods: differential and integral. Although differential and integral pseudospectral methods are quite different, recent work (Tang, Liu, & Hu, 2016) has shown that they are equivalent for collocation at the Jacobi–Gauss (JG) and flipped Jacobi–Gauss–Radau (FJGR) points. Inspired by the aforementioned global property of the fractional derivatives and the fact of IOCPs being special cases of FOCPs, the first author has recently proposed the notion of fractional pseudospectral integration matrices (FPIMs) and developed integral fractional pseudospectral methods for solving FOCPs (Tang, Liu, & Wang, 2015). However, to the best of our knowledge, differential fractional pseudospectral methods for solving FOCPs have not yet received attention. Moreover, a relevant question that comes along is: does the equivalence between classical pseudospectral methods (Tang et al., 2016) still hold for fractional pseudospectral methods?
The aim of this paper is to develop new fractional pseudospectral methods and to prove the equivalence between them via a suitable Birkhoff interpolation. The present work is strikingly different from our previous work (Tang et al., 2015, Tang et al., 2016) in the sense of pseudospectral scheme and Birkhoff interpolation, and establishes a new unified framework for solving fractional optimal control problems using fractional pseudospectral methods. Specifically, the main contributions of this work are as follows:
- (1)
We propose the notion of fractional pseudospectral differentiation matrices (FPDMs) and develop differential fractional pseudospectral methods for solving FOCPs. Moreover, we propose the notion of -FPIMs by employing the basis of weighted Lagrange interpolating functions (Weideman & Reddy, 2000).
- (2)
We take a distinctive route to prove the equivalence between the proposed fractional pseudospectral methods from the perspective of Caputo fractional Birkhoff interpolation.
- (3)
We provide exact, efficient, and stable approaches to compute FPDMs/-FPIMs even at millions of Jacobi-type points.
- (4)
We extend the framework of Garg et al. (2010) to fractional pseudospectral methods with collocation at the Jacobi-type points, and that of Tang et al. (2015) to containing differential fractional pseudospectral methods.
The rest of this paper is organized as follows. In Section 2, some preliminaries are presented for subsequent developments. In Section 3, the definitions and computation of FPDMs are presented. This is followed by the definitions and computation of -FPIMs in Section 4. The detailed implementation of differential fractional pseudospectral methods is provided in Section 5. In Section 6, the equivalence mentioned above is proved by using the Caputo fractional Birkhoff interpolation. In Section 7, some comments on fractional pseudospectral methods are made. Numerical results on two benchmark FOCPs are shown in Section 8. Finally, Section 9 is for some concluding remarks.
Section snippets
Some preliminaries
In this section, we present the definitions of the Riemann–Liouville fractional integrals and the Caputo fractional derivatives. Definition 1 The left and right Riemann–Liouville fractional integrals of real order of a function , are defined, respectively, as where is the Gamma function. It is noteworthy here that for , the fractional integrals coincide with the usual iterated integralsKilbas, Srivastava, & Trujillo, 2006
Definitions and computation of FPDMs
In this section, the definitions and computation of FPDMs are presented.
Definitions and computation of -FPIMs
In this section, the definitions and computation of -FPIMs are presented.
Differential/integral fractional pseudospectral methods
In this section, the differential/integral scaled FOCP of Tang et al. (2015, Eqs. (8)–(11)/Eqs. (12)–(15)) is discretized using differential/integral fractional pseudospectral methods via FPDMs/-FPIMs. We omit the implementation of integral fractional pseudospectral methods since they can be derived directly from Tang et al. (2015, Section 5) by replacing FPIMs with -FPIMs.
Equivalence between fractional pseudospectral methods
In this section, we prove the equivalence between the above fractional pseudospectral methods from the perspective of Caputo fractional Birkhoff interpolation by following Jiao, Wang, and Huang (2016, Section 4.1) where the Jacobi–Gauss–Lobatto (JGL) points are considered.
Some comments on fractional pseudospectral methods
In this section, we make some comments regarding the scopes and features of fractional pseudospectral methods. Remark 17 Similar to FPIMs (see Tang et al., 2015, Remark 14), it is clear from Theorems 5, 6, 11, and 12 that both of FPDMs and -FPIMs can be computed efficiently and stably for millions of Jacobi-type points in an complexity.
Remark 18 It is easy to see that classical pseudospectral methods (Benson et al., 2006; Garg et al., 2010; Francolin et al., 2015) are special cases of the proposed
Examples
In this section, the proposed methods are applied to two benchmark FOCPs taken from the open literature. All computations were performed on a 3.6 GHz Intel Core i7 desktop with 16 GB of 1600 MHz DDR3 RAM running Windows Version 10 and MATLAB Version R2015b.
Conclusions
This paper provided differential and integral fractional pseudospectral methods with equivalence for solving FOCPs, and proved the equivalence from the distinctive perspective of Caputo fractional Birkhoff interpolation. Moreover, this paper provided exact, efficient, and stable approaches for computing the associated fractional pseudospectral differentiation/integration matrices. The performance of the proposed methods was demonstrated on two benchmark FOCPs including a fractional bang–bang
Acknowledgments
The authors would like to express their gratitude to the associate editor and the anonymous reviewers for their constructive comments, which shaped the paper into its final form.
The work of X. Tang was partially supported by the Fundamental Research Funds for the Central Universities (Grant No. 3102016ZY003). The work of Y. Shi was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC RGPIN-2016-05386), and the National Natural Science Foundation of China
Xiaojun Tang received the B.S., M.S., and Ph.D. degrees from Northwestern Polytechnical University, China, in 2002, 2005, and 2010, respectively. He is currently an Associate Professor with the School of Aeronautics, Northwestern Polytechnical University, China. He was a visiting scholar with the Applied Control and Information Processing Lab (ACIPL), University of Victoria, Canada, from September 2015 to September 2016. His research interests include optimal control, state estimation, and
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Xiaojun Tang received the B.S., M.S., and Ph.D. degrees from Northwestern Polytechnical University, China, in 2002, 2005, and 2010, respectively. He is currently an Associate Professor with the School of Aeronautics, Northwestern Polytechnical University, China. He was a visiting scholar with the Applied Control and Information Processing Lab (ACIPL), University of Victoria, Canada, from September 2015 to September 2016. His research interests include optimal control, state estimation, and pseudospectral methods.
Yang Shi received the Ph.D. degree in electrical and computer engineering from the University of Alberta, Edmonton, AB, Canada, in 2005. Now he is a Professor in the Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia, Canada. His current research interests include networked and distributed control systems, model predictive control, system identification, mechatronics, autonomous vehicles, cyber-physical systems, and energy system applications.
Dr. Shi received the University of Saskatchewan Student Union Teaching Excellence Award in 2007, the Faculty of Engineering Teaching Excellence Award at the University of Victoria in 2012. He received the 2017 IEEEE TFS Outstanding Paper award, the Craigdarroch Silver Medal for Excellence in Research of the University of Victoria in 2015, and the JSPS Invitation Fellowship (short-term) in 2013. He serves as Associate Editor for IEEE/ASME Trans. Mechatronics, IEEE Trans. Industrial Electronics, IEEE Trans. Control Systems Technology, IEEE Trans. Cybernetics, ASME Journal of Dynamic Systems, Measurement, and Control. He is currently a Fellow of IEEE, ASME and CSME, and a registered Professional Engineer in British Columbia, Canada.
Li-Lian Wang received his Ph.D. degree in Computational Mathematics from Shanghai University, China in 2000. He is currently an Associate Professor with the Division of Mathematical Sciences in the School of Physical and Mathematical Sciences of Nanyang Technological University, Singapore. His research interests include numerical methods for partial differential equations, computational electromagnetics and image processing based on variational techniques and partial differential equations.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen.
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