840 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 51, NO. 4, AUGUST 2004 Selection of Optimal Closed-Loop Controllers for DC–DC Voltage Regulators Based on Nominal and Tolerance Design Margherita Di Lorenzo del Casale, Member, IEEE, Nicola Femia, Member, IEEE, Patrizia Lamberti, and Virginia Mainardi Abstract—This paper discusses a tolerance design approach for the feedback compensation networks of dc–dc switching regulators, identifying the most reliable solutions among different feasible alternatives that fulfil closed-loop design constraints. A voltage-mode-regulated dc–dc buck converter is considered as a case study. Given the performance and stability constraints, as tol- erance ranges for crossover frequency and phase margin, feasible design solutions are sought by means of Monte Carlo and Interval Arithmetic computations. The search space is a set of available commercial values of – parameters and related tolerances. Best design is identified by a weighted fitness function, exploring the set of solutions provided by different design approaches. The results presented in the paper highlight that the tolerance design approach allows one to find compensation networks that fit the given performance/robustness priorities better than those ones found by means of the classical nominal design approach. Index Terms—DC–DC converters, feedback control, optimiza- tion, robustness, tolerance design. I. INTRODUCTION O PTIMIZATION of switching converters is a topic of great interest. The two most frequently adopted approaches are based either on deterministic or on stochastic methods. In the first case, Lagrangian multipliers are often employed [1], [2]. In [3], the Sequential Quadratic Programming Method is applied to the joined design of static converters and control circuitry. Gradient methods are also used, as in [4], to the op- timal design of a power-factor corrector (PFC) boost converter. Deterministic methods are usually rather onerous to be coded, and they often require derivatives in explicit form. Strong nonlinearity of objective and constraint functions may also cause convergence failures. They are not reliable in multimodal problems, which involve suboptima. Stochastic methods are much easier to use: they do not need derivatives in explicit analytical form and they are robust with respect to nonlinearity and nonmonotonicity of objective and constraint functions. However, they require a careful tuning of the numerical algo- rithm’s parameters, which may strongly influence the results Manuscript received December 12, 2002; revised October 30, 2003. Abstract published on the Internet May 20, 2004. M. Di Lorenzo del Casale is with the Dipartimento di Ingegneria Elettrica, Università di Palermo, I-90128 Palermo, Italy (e-mail: dilorenzo@unipa.it). N. Femia, P. Lamberti, and V. Mainardi are with the Dipartimento di Ingeg- neria dell’Informazione, ed Ingegneria Elettrica, Università di Salerno, I-84084 Fisciano, Italy (e-mail: femia@unisa.it; plamberti@unisa.it; vmainardi@ unisa.it). Digital Object Identifier 10.1109/TIE.2004.831737 of computations. Monte Carlo, simulated annealing, and tabu search techniques are mostly used [5]. Only a few cases of tolerance design of switching converters by stochastic methods are considered in the literature [6]. Among stochastic methods, evolutionary algorithms have been increasingly used in the re- cent past to optimize the design of switching voltage regulators [7], fuzzy and proportional–integral–derivative (PID) digital controllers [8], and to minimize electromagnetic emissions of switching converters [9]. Tolerances of parameters are only rarely accounted for in converters optimization [6]. Neverthe- less, present and future switching converters are expected to operate with increasing switching frequency, wider ranges of variation of load and source, and higher static and dynamic performances; thus, tolerances, uncertainties, and sensitivity must be adequately accounted for in robust design. Design of offline PC power supplies and microprocessor voltage regulation modules (VRMs), interface converters in hybrid distributed power systems using renewable sources, and motor drives, are just some of the most challenging fields by the point of view of tolerance-conditioned design problems. As an example, the maximum allowed output voltage deviation off the reference setting for a VRM supplying an Intel Pentium 4 microprocessor is 50 mV, for typical operation voltages from 1.55 to 1.75 V [10]; such tight tolerance range includes initial dc output voltage set-point error, component aging effects, output ripple and noise, full ambient temperature range and warmup, static operation, and dynamic output load changes from minimum-to-maximum or maximum-to-minimum loads. Therefore, the emerging requirement is to peak robustness of performance, which means much more than the simply peaking performance itself. The main goal of this work is to propose a novel tolerance de- sign approach for analog compensation networks of switching voltage regulators. This paper focuses on the design of a basic voltage-mode feedback compensator for a nonisolated dc–dc buck converter; however, the same approach can be easily ap- plied more generally. The tolerance design approach is charac- terized by the following issues. • Commercial values and related tolerances of components are used from the beginning of the design. • Monte Carlo (MC) technique is adopted to explore the space of design solutions. • Interval Arithmetic (IA) is used to manage computations including tolerances of components and to verify the fea- sibility of design solutions. 0278-0046/04$20.00 © 2004 IEEE