Continuous time constrained linear quadratic regulator - convex duality approach

Abstract

A continuous time infinite horizon linear quadratic regulator with input constraints is studied. On the theoretical side, optimality conditions, both in the open loop and feedback form, are shown together with smoothness of the value function and local Lipschitz continuity of the optimal feedback. Arguments are self-contained, use basic ideas of convex conjugacy, and in particular, use a dual optimal control problem. A method of calculating the optimal and stabilizing feedback without relying on discrete optimization is outlined.

Mean field game based control of dispersed energy storage devices with constrained inputs

“…In [Goebel et Subbotin, 2005], a control problem dual to the constrained LQR problem is studied, which has no constraint but non-quadratic cost. The value function is then described as a convex conjugate of the dual value function, and an optimal feedback solution is constructed based on the gradient of the optimal value function, which can be numerically propagated backwards from the solution to an appropriate Riccati equation.…”

Linear control of time-domain constrained systems

“…In this case the bounds in (12) become time-varying. Univariate positive polynomial constraints (meaning polynomials in only one variable), such as (13) with λ ∈ D 0 = [0, 1] ⊆ R, can be transformed into LMI conditions, see [19] for the details. Once we transformed the design problem into a polynomial optimization problem as formulated in Problem 2.1, there are appropriate tools available for solving the problem.…”

“…Besides these predictive control methods, that are typically suited for a discrete-time context, there are only a few methods available in the literature that can directly synthesize controllers incorporating time-domain constraints in the continuous-time setting. For instance in the case of input constraints, [13,17] consider the linear quadratic regulator problem with positivity constraints on the input, while various control problems with amplitude and rate constraints on the input signal are solved in [33]. The latter line of work has also been extended to stabilization and output regulation problems with amplitude and rate constraints on certain output variables, see, e.g., [32].…”

Finite-Dimensional Characterization of Optimal Control Laws Over an Infinite Horizon for Nonlinear Systems

“…The second approach relies on expressing the dynamics of a system in terms of a linear and a nonlinear component and presents a sequence which provides an approximation of the flow of the Hamiltonian system on the stable manifold. A different approach, still utilising the relationship between optimal control problems and their associated Hamiltonian systems, is taken in [19]. Therein, solutions for the LQR problem with input constraints are solved via a numerical procedure which exploits the property that the optimal solution is associated with a specific trajectory of the associated Hamiltonian system.…”