Mohammed S. Al-Numay

Digital tracking control for PWM systems with unacceptable zeros

Perfect output tracking, which requires inversion of the input-output dynamics, is not always a practical control objective. Difficulties are encountered for systems with zeros which are unstable, or stable but lightly damped. When the zeros are unacceptable in the above sense, perfect output tracking would require the control to be either unbounded, or bounded but highly oscillatory. In this paper, an approximate output tracking control design method is introduced for pulsewidth modulated (PWM) systems with unacceptable zeros. The design method is applied to the nonlinear sampled-data model of the PWM system, and is based on output redefinition. Although the new approach leads to approximate rather than perfect output tracking, it guarantees bounded and nonoscillatory responses.

A new discrete-time simulation method for switched systems using averaging

Abstract A new simulation method for switched systems which provides the discrete-time response of the one-cycle-average value of any signal in the system is reported. It is compared to the more traditional simulation methods with respect to accuracy and speed through a numerical simulation example of boost converter. This method is shown to be both fast and to give exact average values at switching instants.

Modeling, Dynamics, Bifurcation Behavior and Stability Analysis of a DC–DC Boost Converter in Photovoltaic Systems

A study of a DC–DC boost converter fed by a photovoltaic (PV) generator and supplying a constant voltage load is presented. The input port of the converter is controlled using fixed frequency pulse width modulation (PWM) based on the loss-free resistor (LFR) concept whose parameter is selected with the aim to force the PV generator to work at its maximum power point. Under this control strategy, it is shown that the system can exhibit complex nonlinear behaviors for certain ranges of parameter values. First, using the nonlinear models of the converter and the PV source, the dynamics of the system are explored in terms of some of its parameters such as the proportional gain of the controller and the output DC bus voltage. To present a comprehensive approach to the overall system behavior under parameter changes, a series of bifurcation diagrams are computed from the circuit-level switched model and from a simplified model both implemented in PSIM© software showing a remarkable agreement. These diagrams show that the first instability that takes place in the system period-1 orbit when a primary parameter is varied is a smooth period-doubling bifurcation and that the nonlinearity of the PV generator is irrelevant for predicting this phenomenon. Different bifurcation scenarios can take place for the resulting period-2 subharmonic regime depending on a secondary bifurcation parameter. The boundary between the desired period-1 orbit and subharmonic oscillation resulting from period-doubling in the parameter space is obtained by calculating the eigenvalues of the monodromy matrix of the simplified model. The results from this model have been validated with time-domain numerical simulation using the circuit-level switched model and also experimentally from a laboratory prototype. This study can help in selecting the parameter values of the circuit in order to delimit the region of period-1 operation of the converter which is of practical interest in PV systems.

Piecewise-linear methods for digital control of input-switched PWM systems

Pulse width modulation (PWM) is a widely used strategy for operating switch controlled power electronic circuits and systems. One method of designing digital controllers for PWM systems is based on inversion of nonlinear discrete-time maps which define the input-output relation (for output control) or the input-state relation (for state control). Unfortunately, any direct attempt to implement this type of control would generally require either a great deal of on-line computation or a very large memory space for data storage. In this paper, piecewise-linear methods are used to simplify the discrete-time map inversion process without incurring any significant degradation in accuracy. The result is that substantially improved control performance is now achievable with a better balance between computation and storage requirements.

Prediction of subharmonic oscillation in switching regulators: from a slope to a ripple standpoint

ABSTRACT New exact critical conditions for predicting subharmonic instability in switching regulators are approximated by simple design-oriented expressions valid under practical conditions. These simplified expressions contain the ripple and slope information of the feedback control signal. Depending on the converter topology, the controller used and values of parasitic parameters, either the slope or the ripple can be dominant in predicting instability. A discussion on the validity of this interpretation is illustrated through six different examples of switching regulators using the concept of the spectral radius and the relative degree of the system loop. Using this approach, the boundary between the desired stable region and the subharmonic instability can be easily obtained. The theoretical results are validated by means of numerical simulations.

Discrete-Time Model for PWM Converters in Discontinuous Conduction Mode

A new discrete-time model for pulse-width modulated (PWM) converters operating in the discontinuous conduction mode (DCM) which leads to the exact discrete-time mathematical representation of the averaged values of the output signal is proposed. This model can also provide the averaged values of other internal signals with little increase in simulation time. The use of piecewise linear (PL) iteration method dramatically reduces the simulation time, while introducing a little simulation approximation. It is compared to other existing models with respect to accuracy and simulation speed through a numerical example of boost converter. This method gives the exact one-cycle-average (OCA) values of signals at switching instants if PL iteration is not used and, by far, more accurate than other methods if PL iteration is not used. Numerical simulations demonstrate the superiority of the proposed method in terms of accuracy and speed.

One-cycle-averaging for digital PWM control system design

Pulse-width modulation (PWM) is a widely used strategy for operating switch controlled systems. A model for PWM systems is reported which provides the discrete-time response of the one-cycle-average value of any signal in the system. The model typically has unstable zero dynamics, as well as a nonlinear dependence on the duty ratio input. Using scheduled output redefinition, an approximate design model is derived which has stable zero dynamics. A controller based on discrete-time input-output linearization is then developed using this approximate design model. Although this approach leads to approximate rather than perfect output tracking, it guarantees internal stability.

Approximate output tracking control for PWM systems with unacceptable zeros

Perfect output tracking, which requires inversion of the input-output dynamics, is not always a practical control objective. Difficulties are encountered for systems with zeros which are unstable, or stable but lightly damped. When the zeros are unacceptable in the above sense, perfect output tracking would require the control to be either unbounded, or bounded but highly oscillatory. In this paper, an approximate output tracking control design method is introduced for pulse-width modulated (PWM) systems with unacceptable zeros. The design method is applied to the nonlinear sampled-data model of the PWM system, and is based on output redefinition.

Boundaries of Subharmonic Oscillations Associated With Filtering Effects of Controllers and Current Sensors in Switched Converters Under CMC

Subharmonic oscillation is widely studied in switching converters under current mode control (CMC). Its boundary of occurrence in the design parameter space is well known in the case of an infinite-bandwidth current sensor. However, in a practical implementation, either a limited-bandwidth current sensor or an additional filter is used. This could have an effect on the system dynamics particularly for relatively high switching frequencies. This paper discusses the effect of the additional dynamics due to the finite-bandwidth current sensor or the additional filter on the stability and subharmonic oscillation boundaries in switching converters under CMC. It is shown that a simplified model, taking into account the dynamics of the inductor current and the sensor/filter, is enough to obtain accurate results concerning the prediction of the occurrence of subharmonic instabilities in switched converters. Design-oriented equations describing the occurrence of subharmonic oscillations are used to show some problems related to the additional dynamics on the ramp compensator design in a single-switch dc-dc converter working in continuous conduction mode. Some proposed solutions are also discussed. Numerical simulations and experimental measurements corroborate the theoretical predictions.

Bifurcation behavior in a two-loop DC-DC quadratic boost converter

The dynamic behavior and stability analysis of a quadratic boost converter for high conversion ratio applications is addressed. After studying the stability of the system by using the monodromy matrix, a closed form stability condition is used for predicting the boundary of subharmonic oscillation in the system in terms of the duty cycle and the slope of the ramp modulator. The derived theoretical conditions are validated by numerical simulations using a system-level switched model obtaining a good matching between the results. This work provides a convenient means of stability boundary determination in the parameter space hence facilitating the design of quadratic boost converters.