Least Norm Problems and Minimum Energy Control

Lecture slides: least-norm-problems-and-minimum-energy-control.pdf Download least-norm-problems-and-minimum-energy-control.pdf

Reading suggestion: This lecture requires some knowledge of linear operators and Hilbert spaces. In an attempt to keep the videos light I have tried to introduce the concepts in a more intuitive manner, with few definitions and not so much rigour. Be warned, there is quite a steep learning curve here, but hopefully a rewarding one if you put in the time! You can find some more info about scalar products Links to an external site. and Hilbert spaces Links to an external site. on Wikipedia.

We now shift focus to optimisation and optimal control. The central object of study is still the linear equation $y=Ax$, but we now try to determine how we should solve it when this equation has many solutions (ie which $x$ should we pick). One choice is to pick the smallest $x$. We show how to do this for a natural (and convenient) notion of size.

   

We now connect this problem to an optimal control problem. In particular when studying controllability we saw that there might be many control inputs that drive a state-space system from an initial condition to the origin. But which input uses the least energy? Since our system dynamics are linear, this is completely analogous to the matrix least squares problem we just solved! We show how to extend the least squares method (our input is now a function rather than a vector) to solve this problem