Control of electric motors and drives via convex optimization
Keywords: Mechanical Engineering.

Electric motors and drives have been in common use for more than a century. Traditionally, alternating-current electric motors are driven by applying symmetric, multiphase sinusoidal voltage waveforms to their terminals. However, due to the wide availability of switching power converters as well as microcontrollers for controlling them, it is now possible to drive motors using specialized, nonsinusoidal waveforms, allowing us to design the drive voltage waveform along with the motor. This is one of our main points in this work: techniques and technologies now exist for controlling unconventional electric motors. We first consider optimal voltage and current waveform design for electric motors, then considers the design of feedback controllers for switched-mode power converters that could be used to implement these waveforms. First we give energy-optimal current waveforms for a permanent magnet synchronous motor that result in a desired average torque. Our formulation generalizes previous work by including a general back-EMF wave shape, voltage and current limits, an arbitrary phase winding connection, a simple eddy current loss model, and a trade-off between power loss and torque ripple. Determining the optimal current waveforms requires solving a convex optimization problem. We show how to use the alternating direction method of multipliers to find the optimal current in milliseconds or hundreds of microseconds, depending on the processor used, which allows the possibility of generating optimal waveforms in real time. We next present a method for finding current waveforms for induction motors that minimize resistive loss while achieving a desired average torque output. Our method is not based on reference-frame theory for electric machines, and therefore directly handles induction motors with asymmetric winding patterns, nonsinusoidally distributed windings, and a general winding connection. We do not explicitly handle torque ripple or voltage constraints. Our method is based on converting the torque control problem to a nonconvex linear-quadratic control problem, which can be solved by using a (tight) semidefinite programming relaxation. We then address the problem of finding current waveforms for a switched reluctance motor that minimize a user-defined combination of torque ripple and RMS current. The motor model we use is fairly general, and includes magnetic saturation, voltage and current limits, and highly coupled magnetics (and therefore, unconventional geometries and winding patterns). We solve this problem by approximating it as a mixed-integer convex program, which we solve globally using branch and bound. We demonstrate our approach on an experimentally verified model of a fully pitched switched reluctance motor, for which we find the globally optimal waveforms, even for high rotor speeds. After considering the motor design problem, we consider control of switched-mode power electronic converters. First we consider the theoretical problem of controlling general switched-affine dynamical systems i.e., the problem of selecting a sequence of affine dynamics from a finite set in order to minimize a sum of convex functions of the system state. We develop a new reduction of this problem to a mixed-integer convex program (MICP), based on perspective functions. Relaxing the integer constraints of this MICP results in a convex optimization problem, whose optimal value is a lower bound on the original problem value. We show that this bound is at least as tight as similar bounds obtained from two other well-known MICP reductions (via conversion to a mixed logical dynamical system, and by generalized disjunctive programming), and our numerical study indicates it is often substantially tighter. Finally, we consider the problem of controlling switched-mode power converters using model predictive control. Model predictive control requires solving optimization problems in real time, limiting its application to systems with small numbers of switches and a short horizon. We propose a technique for using off-line computation to approximate the model predictive controller. This is done by dividing the planning horizon into two segments, and using a quadratic function to approximate the optimal cost over the second segment. The approximate model predictive algorithm minimizes the true cost over the first segment, and the approximate cost over the second segment, allowing the user to adjust the computational requirements by changing the length of the first segment. We conclude with two simulated examples.