# Robust LQR Control for PWM Converters with Parameter-Dependent Lyapunov Functions
## 1 Introduction  
Control systems for power converters typically must satisfy several specifications and requirements, while dealing with uncertainty or operating point dependence at the same time. Since worst-case models may not exist or be different for each specification, the conventional industry standard approaches, such as the ones based on voltage-mode [1, 2] and current-mode [2, 3, 4] controllers, rely on expert knowledge, simulation and iteration in order to find an appropriate controller. As an alternative to this manual iteration, the automatic synthesis of controllers for switched-mode power converters has been one active topic of research in the last decade. These approaches are of interest because they can take into account the requirements together with the uncertainty or the nonlinearities of the converter to provide robust stability and performance, and they can do all that by imposing conditions beforehand. Methods based on linear matrix inequalities (LMIs) have been some of the most successful approaches to the synthesis of robust controllers for power converters. The first attempts [5, 6, 7] demonstrated how uncertainty could be modeled and how the transient and frequency domain specifications could be taken into account. More recently, the efforts have been focused on approaches that do not require full state feedback [8], that improve the robustness [9] or the performance properties [10]. Although these papers employ averaged models of the converters, other approaches have also tackled the problem from a hybrid system perspective [11, 12]. One of the open problems in the topic is the fact that the results may be conservative. The synthesized controller may not offer the best possible performance, when compared with conventionally tuned controllers, such as current-mode controllers. One possible solution to this conservativeness was shown in [13], where excellent robustness and tight regulation were achieved simultaneously, at the expense of control complexity. One of the causes behind the conservativeness of LMI methods in [7] is the fact that the stability of the system is ensured no matter how large the derivative of the uncertain parameters may be. Specifically, when the uncertainty is characterized by being norm bounded, time varying and evolving in a set of polyhedral vertices, one difficulty remains: how to find an adequate mathematical representation for it, as well as for its rate of variation [14]. Nonetheless, several ways for representing both the derivative of the time-dependent parameter and the parameter itself have been proposed in the literature [14; 15; 16; 17]. Different approaches to control these uncertain systems have been reported, such as state feedback gain-scheduling control [18], output feedback [15], linear quadratic Gaussian (LQG) or linear quadratic regulator (LQR) controllers [19; 20; 21] and gain-scheduled linear quadratic regulators (LQRs) [22; 23]. In this paper, we propose a new method to synthesize robust LQR controllers for pulse-width modulated (PWM) converters, with the objective to improve the LQR synthesis that was proposed in [7]. The method is based on the results introduced in [14; 15], such that the proposed approach can consider the time derivative of the uncertain parameters. As a consequence, the new LQR formulation can obtain less conservative results. This reduced conservativeness can be seen as a new degree of freedom. With this method, practicing engineers can synthesize controllers for larger sets of uncertainty (i.e., with improved robustness) or controllers that provide tighter regulation (i.e., improved performance) when compared with the previous method. The approach has been verified with the synthesis of a controller for a boost converter, such that a direct comparison with [7] has been carried out. Note that the proposed method could also be used in other switched-mode power converters, such as the buck converter (which was also treated in [7]). This paper is organized as follows.