L1 Optimal Control

Collaborators: John Doyle

L1 optimal control is a useful framework because it studies the relationship of the input and output of a system using the infinity signal norm. In many applications, the characteristics of the input signals and the desired behavior of the output signals are specified with maximum and minimum values. These kind of speficiations are naturally captured by the infinity norm of a signal. Apart from these considerations, l1 optimal control can be studied using linear algebra and linear programming which are more accessible than complex analysis which is required by traditional H2 or H optimal control.

One of the aims of this work is a simple and accessible explanation as to why oscillations naturally arise due to tradeoffs in feedback systems, and how these can be aggravated by delays and unstable poles and zeros. We present an entirely time domain model using discrete time dynamics and l1 norm performance. A simple waterbed effect is that imposing zero steady state response to a step naturally create oscillations that double the response to periodic disturbances. We show how this tradeoff is further aggravated not only by unstable poles and zeros, but also delays, in a way clearer than in the frequency domain versions.


  1. Y. P. Leong and J. C. Doyle, “Effects of delays, poles, and zeros on time domain waterbed tradeoffs and oscillations,” IEEE Control Systems Letters, vol. 1, no. 1, pp. 122–127, 2017.