Mathematical Problems in Engineering

Volume 2016, Article ID 6935081, 7 pages

http://dx.doi.org/10.1155/2016/6935081

## Fractional-Order Terminal Sliding-Mode Control for Buck DC/DC Converter

^{1}State Key Laboratory Base of Eco-Hydraulic Engineering in Arid Area, Xi’an University of Technology, Xi’an 710048, China^{2}Institute of Water Resources and Hydro-Electric Engineering, Xi’an University of Technology, Xi’an 710048, China^{3}College of Electronics and Information, Xi’an Polytechnic University, Xi’an 710048, China^{4}School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 11 March 2016; Revised 14 June 2016; Accepted 3 July 2016

Academic Editor: Anna Pandolfi

Copyright © 2016 Ningning Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, the combination of fractional calculus (FC) and sliding-mode control (SMC) has been gaining more and more interests due to fusion characteristics of SMC and FC. This paper presents the fractional-order terminal sliding-mode control (FTSMC) which has a new fractional-order sliding surface and assures the finite time convergence of the output voltage error to the equilibrium point during the load changes. TSMC is a special case of FTSMC. Through mathematical analysis, the system can reach the sliding-mode surface in finite time. The theoretical considerations have been verified by numerical simulations. And a Buck DC/DC converter application is presented and compared to illustrate the effectiveness of the proposed method. It is shown that the novel fractional terminal sliding-mode control exhibits considerable improvement in terms of a faster output voltage response during load changes.

#### 1. Introduction

The concept of fractional calculus as an extension of ordinary calculus can stretch back to 1695. In the letter to L’Hospital, Leibniz proposed the possibility of generalizing the operation of half-order derivative [1]. Though fractional calculus has a long history, only in recent years have the applications of fractional calculus to physics and engineering become an important aspect of modern technology. In comparison with the classical elementary calculus, the main advantage of fractional calculus is that it can provide an elegant description for the memory and hereditary properties of various real objects. The list of the applied fields of fractional calculus has been ever growing and includes the electrode-electrolyte polarization, viscoelastic fluids, chaotic systems, and power converters [2–6]. Most striking of all, the fractional differentials and integrals are applied to the control theory, when the controller or the controlled system is described by fractional calculus [7–11]. In 1996, Oustaloup et al. developed the first fractional-order controller which is the so-called CRONE [12]. Then some fractional-order control strategies are proposed one after another, such as fractional-order PID controllers [13], fractional-order sliding-mode controllers [14–16], fractional-order optimal controllers [17], and fractional-order adaptive controllers [18, 19].

DC/DC power converters, which work in switch mode, are applied in a wide variety of applications, including DC motor drives, active filters, computers, power supplies, and medical instrumentation. The Buck converter is one of the simplest but most useful power converters that can convert a DC input to a DC output at a lower voltage. There are some other basic DC/DC converters, namely, the Boost, Buck-Boost, Cuk, Zeta, and Sepic. Each of these converters consists of the passive power switch, the active power switch, and the storage elements. The main objective of most closed-loop feedback controlled DC/DC converters is to ensure that the converter operates with fast dynamical response, small steady-state output error, and low overshoot, while maintaining high efficiency and low noise emission in terms of rejection of input voltage changes, parameter uncertainties, and load variations [20]. The designing of high performance control strategy is always a challenge, because DC/DC converters are inherently time-varying variable-structure nonlinear systems. Nonlinear control strategies [21–23] are better candidates in DC/DC converters than other linear ones. Among the nonlinear control strategies, the sliding-mode control (SMC) has been well known due to its fast dynamic response, robustness to disturbances, guaranteed stability, parameter variations, and simplicity in implementation. Moreover, compared with other nonlinear control strategies, the SMC method is easy to implement and has a high degree of flexibility in the designing process.

The sliding-mode control is an effective robust control strategy with the feature of switching the control law to force the state trajectories of the system from the initial states onto some predefined sliding surface which exhibit desired dynamics. When in the sliding mode, the closed-loop response becomes totally insensitive to both internal parameter uncertainties and external disturbances. Compared with the SMC, the terminal sliding-mode control (TSMC) has some superior properties, such as the state of the system which converges in finite time and the higher steady-state tracking accuracy. Near the equilibrium point the rate of the convergence is being speeded up, so the TSMC is more suitable for high precision control. In recent years, the combination of fractional-order control (FOC) and sliding-mode control (SMC) has been gaining more and more interests from the systems control community. An introductory work on the application in fractional-order sliding mode is reported in [24]; the double integrator is introduced as the plant under control. A fuzzy fractional-order sliding-mode controller is adopted to control nonlinear systems in [25]. In [26], fractional-order sliding-mode control strategies for power electronic Buck converters are presented, where pulse-width modulation (PWM) sets the basis for the regulation of switched mode converters. In [27], authors collect different methods to apply FOC in SMC through the use of fractional-order surfaces which are fractional-order PID or fractional-order PI and propose a direct Boolean control strategy in order to avoid using PWM.

In this paper, we focus on the introduction of the FOC in TSMC for switching systems. The fractional-order terminal sliding-mode control (FTSMC) method has been adopted for controlling the Buck converter. The idea of such method is used to design a novel nonlinear sliding surface function which is a fusion of characteristics of TSMC and FOC. Based on the Lyapunov stability theory, a fractional-order sliding-mode control law is derived to assure finite time convergence of the output voltage error to the equilibrium point. Through mathematical analysis, we obtain the finite time of FTSMC. It is shown that TSMC is a special case of FTSMC. The theoretical considerations have been verified by numerical simulations and a Buck converter application is given to show the superiority of the proposed strategy. The fractional terminal sliding-mode control exhibits considerable improvement in terms of a faster output voltage response during load changes. FTSMC can overcome the influence which disturbances bring about to control system and guarantee that DC/DC converter keeps good dynamic and steady performances. The output can follow the given well, and the disturbances almost do not affect the output.

The rest of this paper is organized as follows. In Section 2, some basic concept and definitions of the fractional calculus are introduced. Then the basic principle of the Buck converter in CCM is introduced. In Section 3, the integer-order TSMC is briefly reviewed. In Section 4, FTSMC is described for the Buck converter. In Section 5, numerical simulation results are presented to verify the theoretical considerations and an application to Buck converter is given to show the superiority of the proposed strategy. Finally, some concluding remarks of this paper are drawn in Section 6.

#### 2. Preliminaries

In this section, firstly, some basic concept and definitions of fractional calculus are introduced. Afterwards, a brief introduction to the basic principle of the Buck converter in CCM is presented.

##### 2.1. The Basis of Fractional Calculus

In fractional calculus, the operator is the differintegral operator which takes both integrals and derivatives in one single expression. It denotes generalization of integrals and derivatives to arbitrary order and can be defined as follows:where can be an arbitrary real number, which denotes the order of the operation, and and are the lower and upper limits.

*Definition 1. *In the fractional calculus, the Gamma function proposed by Euler is defined by the integralwhich satisfies , and converges in the right half of the complex plane.

Several reputed definitions for fractional derivatives are put forward including Riemann-Liouville definition, Grunwald-Letnikov definition, Caputo definition, Weyl definition, and Marchaud definition. Among them, being the best-known one, Riemann-Liouville definition is precisely studied.

*Definition 2 (see [7]). *Riemann-Liouville definition of the th-order fractional derivative operator is given bywhere .

*Definition 3 (see [7]). *Riemann-Liouville definition of the th-order fractional derivative operator is defined byAccording to this definition, the derivative of a time function with , , is evaluated as

*Definition 4 (see [7]). *If the fractional derivative ) and a function are integrable, then

##### 2.2. Brief Introduction of the Buck Converter in CCM

The Buck converter, sometimes called a step-down power stage, is a popular nonisolated power stage topology. A simplified schematic of the Buck converter is shown in Figure 1(a).