Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 686857, 8 pages

http://dx.doi.org/10.1155/2015/686857

## LPV Control with Pole Placement Constraints for Synchronous Buck Converters with Piecewise-Constant Loads

School of Electrical Engineering, University of Ulsan (UOU), Ulsan 680-749, Republic of Korea

Received 5 June 2014; Revised 12 August 2014; Accepted 26 August 2014

Academic Editor: Guido Maione

Copyright © 2015 Hwanyub Joo and Sung Hyun Kim. 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

This paper addresses the output regulation problem of synchronous buck converters with piecewise-constant load fluctuations via linear parameter varying (LPV) control scheme. To this end, an output-error state-space model is first derived in the form of LPV systems so that it can involve a mismatch error that temporally arises from the process of generating a feedforward control. Then, to attenuate the mismatch error in parallel with improving the transient behavior of the converter, this paper proposes an LMI-based stabilization condition capable of achieving both and pole-placement objectives. Finally, the simulation and experimental results are provided to show the validity of our approach.

#### 1. Introduction

Drawing on the development of electronic technology, switching DC-DC converters have been widely and successfully applied to a variety of power conversion systems such as DC power supplies, DC motor drivers, and power generation systems (see [1–4] and the references therein). Recently, with the growing interest in linear matrix inequalities (LMIs) [5], some advanced control techniques have been investigated regarding to output regulation of DC-DC buck (step-down) converters that produce a lower output voltage than the input voltage, especially for T-S fuzzy control [2, 6–10]. Indeed, the asynchronous buck converters operating in a large-signal domain are generally modeled in terms of nonlinear systems. Thus, based on the T-S fuzzy model derived from the averaging method for one-time-scale discontinuous systems (AM-OTS-DS), [9] has successfully designed an integral T-S fuzzy control with respect to the output regulation problem of the asynchronous buck converter. Meanwhile, in the case of synchronous buck converters [11–13], the use of the low-side FET plays an important role in eliminating the voltage drop across the power diode of the nonsynchronous converter, which allows buck converters to be modeled with linear time-varying systems.

In general, the operation of the DC-DC converter is usually affected by the fluctuation of output loads [3, 9, 12, 13]. For this reason, it is of great importance to consider the presence of a wide load range in the problem of regulating the output voltage and current levels of DC-DC converters; that is, it has become a hot topic to maintain high efficiency in a great load fluctuation. Moreover, due to the fact that conventional pulse-width modulation (PWM) buck converters have poor efficiency under light load [14], numerous research efforts have been invested to improve the efficiency of the PWM converters with a wide load range (see [12, 15–17] and the references therein). However, a remarkable point is that most of references cited above have paid considerable attention at the hardware level to cover such problem. Further, [13] used a reduced system model with the limits in theoretically capturing the dynamic behavior of piecewise-constant load fluctuation in the process of implementing the robust periodic eigenvalue assignment algorithm [18]. In other words, limited work has been found in terms of the control theory. Motivated by the concern, this paper proposes a suitable approach in light of the control theory to take the effect of load fluctuations into account.

This paper addresses the output regulation problem of synchronous buck converters with piecewise-constant load fluctuations. To consider the presence of such load fluctuations, we derive an output-error state-space model in the form of linear parameter varying (LPV) systems [19–21], thereby converting the underlying regulation problem into the stabilization problem. Here, it is worth noticing that a mismatch error that temporally arises from the process of generating a feedforward control is clearly incorporated into the LPV model and it is attenuated by the -synthesis technique [22, 23]. However, design provides little control over the transient behavior [24, 25]. Hence, to attenuate the mismatch error in parallel with improving the transient behavior of the converter, this paper proposes an LMI-based stabilization condition capable of achieving both and pole-placement objectives. Finally, the simulation and experimental results are provided to show the validity of our approach.

*Notation.* The notations and mean that is positive semidefinite and positive definite, respectively. In symmetric block matrices, is used as an ellipsis for terms induced by symmetry. For any square matrix , . Lebesgue space consists of square-integrable functions on .

#### 2. Modeling for DC to DC PWM Buck Converter

The equivalent circuit for a class of synchronous DC-DC buck converters and the corresponding closed-loop control system are depicted in Figure 1, where the following notations are used.(i) denotes the static drain to source resistances of the high-side and low-side power MOSFETs, respectively.(ii) and denote the power input and output voltages, respectively, where it is assumed that is time-invariant.(iii) and denote the inductor current and the capacitor voltage, respectively.(iv) and denote the inductance and capacitance selected by the given design specifications including the switching frequency of MOSFETs.(v) and denote the equivalent series resistances of the inductor and capacitor.(vi) and denote the piecewise-constant load resistance subject to and the duty ratio of PWM buck converter.Then, based on averaging method for one-time-scale discontinuous system (AM-OTS-DS) [26], the mathematical model of the buck converter under consideration is described as follows:Further, combining (2) and (3) yieldsby which (1) and (2) can be rewritten as follows:Let be a unique equilibrium point of and assume that the value of is piecewise-constant; that is, for (see Figure 1). Then, by (6), the equilibrium point of , that is, , is given byFurther, by (3), the equilibrium point of is given by . Hence, from (1), the equilibrium point of , that is, , is given by