Control: H2 and Hinf (H infinity) norm of Impulse Response matrix as Yalmip Objective

[Stefan Rickli]

In order to synthesize a discrete time controller according to this paper (I need a way to give Yalmip an objective function that is the norm of some impulse response matrices.

So I currently have three-dimensional SDP variables (one 2-D matrix for each time step, where time is the 3rd dimension) and I want to compute the H_2 and H_infinity norm of those systems.

Normally if I had a state space representation, P = ss(A,B,C,D), of these systems, I could just call Matlab's

norm(P,2)

or

norm(P,inf) = hinfnorm(P)

But of course Yalmip's SDP variables and these operators don't work together automatically...

I'm no expert on norms of linear systems, so does anyone know how to deal with this?

At the moment I just calculate the sum of all squared impulse response matrix elements, which should give me the squared H_2 norm, if I'm not mistaken (referring to (4.118) ).

(These excerpts were taken from Skogestad: Multivariable Feedback Control - Analysis and Design)

[Johan Löfberg]

In a convex optimization framework, the norm of a linear system is defined by the minimum value of a semidefinite program. For instance, (6.52) in http://web.stanford.edu/~boyd/lmibook/ (where you minimize a variable representing gamma^2 to make it linear)

Of course, if an impusle response models has a simple solution as you indicate, you should use that and not complicate matters with LMIs