Abstract

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, when no constraints are violated, while they 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 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 four structure exploiting algorithms that were used in applications:

(a) a convex time transformation for time optimal robot arm control [4]

(b) online active set strategy for an optimal pre-filter for machine tools [3]

(c) nonlinear real-time iterations for model predictive control of power generating kite systems [2]

(d) a duality and Fourier based approach to optimal clipping in hearing aids.

The talk will present joint work with J. Swevers, M. Moonen, J. De Schutter, T. Van Waterschoot, L. Vanden Broeck, D. Verscheure, B. Houska, H.J. Ferreau, and B. Defraene.

References

[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.

[2] A. Ilzhoefer, B. Houska, and M. Diehl. Nonlinear MPC of kites under varying wind conditions for a new class of large scale wind power generators. International Journal of Robust and Nonlinear Control, 17(17):15901599, 2007.

[3] L. Van den Broeck, M. Diehl, and J. Swevers. Embedded optimization for input shaping. IEEE Transactions on Control System Technology, 2009. In press.

[4] D. Verscheure, B. Demeulenaere, J. Swevers, J. De Schutter, and M. Diehl. Time-optimal path tracking for robots: a convex optimization approach. IEEE Transactions on Automatic Control, 2009. Accepted for publication.