A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system

Abstract

The cheap LQ regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory. In fact, solving the LQ regulator problem is equivalent to keep the output of the related Hamiltonian system identically zero. The solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the original system, and the optimal feedback gain is computed as the friend matrix of the resolving subspace. This study yields a new computational framework for the cheap LQ regulator, relying only on the very basic and simple tools of the geometric approach, namely the algorithms for controlled and conditioned invariant subspaces and invariant zeros.

New results in impulse-free continuous-time cheap LQ optimal control

“…Valuable results on this subject have also been presented in [19], [21], [20], which are based on linear matrix inequalities. A different approach for the cheap LQ problem was introduced in [16], where geometric techniques were used to recast the cheap control problem as a perfect decoupling problem, to the end of characterising in strict geometric terms the subspace of initial conditions for which the optimal control is non-impulsive.…”

The generalised discrete algebraic Riccati equation arising in LQ optimal control problems: Part I

“…Lemma 3.1: Let X be a solution of the discrete Lyapunov equation (14). Then, ker X is A-invariant and is contained in ker Q.…”

“…Proof: Let λ ∈ C be on the unit circle and such that (A + λ I n ) is invertible. We can re-write (14) as…”

On the geometry of the continuous-time generalized algebraic Riccati equation arising in LQ optimal control

“…When R is allowed to be singular, the corresponding LQ problem is called singular, see e.g. [13], [20], [17], [16], [14]. In particular, in [13] and [20] it was proved that an optimal solution of the singular LQ problem exists for all initial conditions if the class of controls is extended to include distributions.…”