Control via interconnection and damping assignment of linear time-invariant systems: a tutorial

Interconnection and damping assignment is a controller design methodology that regulates the behaviour of dynamical systems assigning a desired port-Hamiltonian structure to the closed-loop. A key step for the application of the method is the solution of the so-called matching equation that, in the case of nonlinear systems, is a partial differential equation. It has recently been shown that for linear systems the problem boils down to the solution of a linear matrix inequality that, moreover, is feasible if and only if the system is stabilisable - making the method universally applicable. It has also been shown that if we narrow the class of assignable structures e. g. to mechanical instead of the larger port-Hamiltonian - the problem is still translated to a linear matrix inequality, but now stabilisability is not sufficient to ensure its feasibility. It is additionally required that the uncontrolled modes are simple and lie on the j omega axis, which is consistent with the considered scenario of mechanical systems without friction. The purpose of this article is to present these important results in a tutorial, self-contained form - invoking only basic linear algebra methods.