Embedded Optimization - Convex and Beyond

Moritz Diehl
Katholieke Universiteit Leuven

Many branches of engineering employ linear mappings between some input and output sequences, most prominently in control engineering and in signal processing. Examples are PID or other linear controllers, the Kalman Filter, as well as the many filters used in sound processing e.g. in loudspeakers or hearing aids. These linear maps are usually only useful for one special set of conditions and need to be adapted whenever the conditions change.

A completely different approach is the following: we generate a map between inputs and outputs via embedded optimization, i.e. the outputs are generated as the solution of parametric optimization problems that are solved again and again, each time for different values of the input parameters. This approach directly generates a nonlinear map between inputs and outputs, and allows us to easily incorporate constraints and user defined objectives. It can be shown that this approach is able to generate any continuous input-output map even if we require the optimization problems to be convex in both inputs and outputs, which is the most favourable case [1].

The structure of the embedded optimization problems needs to be exploited to the maximum, as many applications require sampling times in the order of milli or even microseconds. We present three
structure exploiting algorithms, two convex and one non-convex one, that were used in applications:

(a) optimal clipping in hearing aids

(b) online active set strategy for fast mechatronic point-to-point motions without residual vibrations

(c) nonlinear real-time iterations for model predictive control of tethered airplanes for wind power generation

[1] M. Baes, M. Diehl, and I. Necoara. Every continuous nonlinear control system can be obtained by parametric convex programming. IEEE Transactions on Automatic Control, 53(8):19631967, September 2008.

Prof. Dr. Moritz Diehl Electrical Engineering Department (ESAT)and Optimization in Engineering Center (OPTEC)K.U.Leuven Belgium. The talk will present joint work with J. Swevers, M. Moonen, J. De Schutter, T. Van Waterschoot, L. Vanden Broeck, B. Houska, H.J. Ferreau, Kurt Geebelen, and B. Defraene.

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