A Nonlinear-System Approach to Analysis and
Design of Power-Electronic Converters With
Saturation and Bilinear Terms
Tingshu Hu, Senior Member, IEEE
Abstract—Power-electronic converters are intrinsically nonlinear. This paper proposes a Lyapunov approach to analysis and design of a class of nonlinear systems arising from power-electronic converters. The system has a bilinear term as the product of the state and the input—the duty cycle, which is subject to strict constraint (or saturation). The nonlinearities and the input saturation are considered in this paper by using piecewise-quadratic Lyapunov functions and by describing the system with a piecewiselinear differential inclusion. The problems considered include controller design for robust stability, and estimation of stability region and tracking domain. These analysis and design problems are converted into numerically efﬁcient optimization algorithms involving linear-matrix inequalities (LMIs). A buck–boost dc–dc converter is used to demonstrate the proposed methods. The optimization results show that a simple state-feedback law can be constructed to achieve practically global stabilization and tracking, which is theoretically conﬁrmed by the Lyapunov approach. An experimental buck–boost converter is constructed to verify the tracking of a square reference varying almost between the upper and the lower limit.
Index Terms—Bilinear system, linear-matrix inequality (LMI), power-electronic converters, saturation, stability, tracking.
IGH-EFFICIENCY power-electronic converters use
pulsewidth modulators (PWMs) to adjust the duty cycle of switched-mode semiconductor devices. The duty cycle is used as the input to control the output voltage, current, or power. An important task in the converter design is to construct a feedback law that adjusts the duty cycle based on available measurement so that the output voltage, current, or power, follows a given reference. The feedback control system is subject to input saturation due to the hard limit on the duty cycle, which is constrained within [0, 1], or a smaller range caused by the limitation of the switching devices and nonidealities of the circuit elements. Furthermore, the differential equation of the model obtained by averaging
Manuscript received January 14, 2010; revised April 13, 2010; accepted June 15, 2010. Date of current version February 4, 2011. This work was supported in part by the National Science Foundation (NSF) under Grant ECS-0621651 and Grant ECCS-0925269. Recommended for publication by Associate Editor B. Choi.
The author is with the Department of Electrical and Computer Engineering, University of Massachusetts, Lowell, MA 01854 USA (e-mail: firstname.lastname@example.org).
Color versions of one or more of the ﬁgures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identiﬁer 10.1109/TPEL.2010.2054115
over one switching cycle (the state-space averaging model developed by Middlebrook ) has a bilinear term, which is the product of the duty cycle and the state. In traditional design, the saturation has usually been neglected and the bilinear term has been discarded as high-order terms to obtain a linearized model under the small-signal assumption (e.g., see [23, p. 351], or [7, p. 221]). Although switched-mode power conversion is
a well-established technology and numerous high-performance
power-electronic converters have been constructed by neglecting these nonlinearities, there are still continuing research efforts devoted toward better understanding the theory behind the actual nonlinear system (see , , , –, , , , , , and  for a sample of recent literature.)
One basic problem of a power-electronic circuit is stability. When the linearized model is used for control design, the circuit will be locally stable when the variations...