Closed-Loop Expression of Fixed-end-point LQ Optimal Control for Digital Systems

Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

“…We remark that the problem considered in this paper is pretty general, since we allow each one of the initial and terminal states to be either fixed or quadratically weighted in the performance index, thus encompassing the standard LQ problem, in which the initial state is assigned and the terminal state is quadratically weighted, and the so-called fixed end-point LQ problem, in which the extreme states are both sharply assigned. We refer to [7], [14] and references therein for a discussion on the presence of constraints in the initial and terminal states. See also [2] for a feedback compensation scheme that guarantees optimality only for a given set of initial states.…”

A Feedback Compensation Scheme for the Linear Quadratic Regulator with Zero End-Point

“…Many studies have been devoted in recent literature to the fixed end-point linear quadratic regulator, both in the continuous and in the discrete-time domain: see to this purpose [1], [3], [7], [8], [9], [10] and references therein. For example in [9] and [7], closed-form expressions of the input function are presented for the solution of the zero end-point LQ, whereas in [10] and [3] the continuous-time point-to-point LQ is considered.…”

“…For example in [9] and [7], closed-form expressions of the input function are presented for the solution of the zero end-point LQ, whereas in [10] and [3] the continuous-time point-to-point LQ is considered. The former presents the expression of the optimal control law in finite terms by means of the controllability gramians, the latter exploits a closedform parametrization of the solutions of the Hamiltonian differential equation.…”

“…From(7), it follows that the overall set of optimal parameters p o and q o is parametrized in the null-space of matrix M as…”