A PfC , qs : f ' EEE in ! f Hoo control of boost converters

The tasks of the controller in a PWM power converter are to stabilize the system and to guarantee an almost constant output voltage in spite of the perturbations in the input voltage and in the output load. We propose to design the controller using Hoo control theory, via the solution of two algebraic Riccati equations. The almost optimal HOO controller is of the same order as the converter and has a relatively low DC gain ( as opposed to the classical approach of maximizing the gain at low frequencies). In a case study, for a typical low power boost converter, we obtain significantly improved frequency responses of the closed loop system, as compared to classical voltage mode, current mode or feed forward control. Pulse width modulation (PWM) DC-DC converters are nonlinear and time variant systems. After linearization of the average model (see, e.g. Sum [5]) we obtain a linear time invariant model that describes approximately the behavior of the system for frequencies up to half of the s\vitching frequency. Boost and buck-boost converters (in continuous conduction mode) have a zero in the right halfplane (RHP) in their transfer function from the control input (the duty cycle) to the output voltage. The tasks of the controller are to stabilize the system and to minimize the fluctuations of the output voltage, in spite of the perturbations in the input voltage and the output load. The classical design method, based on Bode and Nyquist plots, is to impose a high loop gain in a frequency band as wide as possible, without distroying stability, see for example the Unitrode manual [4] or Sum [5]. The existence of the RHP zero limits the achievable loop bandwidth and thus the frequency response of the controlled system. We show that the objectives of the controller match very well ,vith the standard problem dealt with in Hoo control theory (see, e.g., Francis [3] or Doyle et al [2]). The suboptimal solution of the standard problem can be found via the description of the system in state space and the solution of t'vo algebraic Riccati equations, for ,vhich reliable programs are available in MATLAB (in the Robust Control Toolbox). The nearly optimal HCX: controller is of the same order as the converter and has a relatively lo,v DC gain (in contrast ,vith the classical idea of using high gain at lo,v frequencies), and so it is easy to implement. As a design example we took the t:-pical lo'v power boost converter shown in Fig. 1, which transforms a nominal12V input voltage into 24V at its output, with a nominal output current of O.5A. The switching frequency was 240KHz. The role of the current source in the output is to simulate changes in the load ( changes in the output current ). The series resistance of the inductor, of the switch, of the diode and of the capacitor are all shown. The controller imposes the duty cycle D, which is the ratio between the time Ton when the switch is closed and the period T. After linearization of the average model (see Sum [5]), ,ve can translate the problem of designing a controller to the standard problem of Hoo control, as illustrated in Fig. 2. In this diagram, the plant p is the converter, C is the controller to be designed and W is a weight function. The vector w contains the perturbations, the output z is the weighted error signal, the vector y contains the measurement outputs of the plarit and the control input u is the duty cycle. In Hoo control the aim is to find a controller C wich minimizes the Hoo norm of the transfer function T zw from w to z. The Hoo norm is the maximal gain, so that in fact 719 0-7803-2482-X/95 $4.00 @ 19951EEE