Minimizing control energy in a class of bounded‐control linear‐quadratic regulator problems

Abstract

SUMMARYMinimal‐control‐energy strategies are substantiated and illustrated for linear‐quadratic problems with penalized endpoints and no state‐trajectory cost, when bounds in control values are imposed. The optimal solution for a given process with restricted controls, starting at a known initial state, is shown to coincide with the saturated solution to the unrestricted problem that has the same coefficients but starts at a generally different initial state. This result reduces the searching span for the solution: from the infinite‐dimensional set of admissible control trajectories to the finite‐dimensional Euclidean space of initial conditions. An efficient real‐time scheme is proposed here to approximate (eventually to find) the optimal control strategy, based on the detection of the appropriate initial state while avoiding as much as possible the generation and evaluation of state and control trajectories. Numerical (including model predictive control) simulations are provided, compared, and checked against the analytical solution to ‘the cheapest stop of a train’ problem in its pure‐upper‐bounded brake, flexible‐endpoint setting. Copyright © 2013 John Wiley & Sons, Ltd.

A Cost Reduction Procedure for Control-Restricted Nonlinear Systems

“…This case-study is a modification of the 'cheapest stop of a train' problem already considered in [1,7,10]. Here, a friction term is included in the model as proposed elsewhere [14,15] to obtain the following nonlinear system:…”

Whither Discrete Time Model Predictive Control?

“…[5]- [7] and references therein). On the other hand, methods specifically tailored to constrained linear systems are less common, but some interesting results and methods can be found in [8]- [14]. We remark that one distinguishing feature of our method is that it computes a solution to P∞(x) that is accurate to a user defined tolerance.…”

Online suboptimal control of linearized models

“…The generation of a suboptimal control strategy by approximating the unknown parameters (Ŝ,x 0 ) has already been published in the minimal-control-energy situation (Costanza et al, 2014), and the extension to the general LQR problem was announced in Costanza and Rivadeneira (2013), although there the updating of the unknown parameters still required the simulation of state trajectories. In this paper, explicit algebraic formulas are given for parameters updating, avoiding ODE integrations, and thus reducing the computer time in the process of decreasing the total cost.…”

“…The decisive theoretical finding may be phrased as follows: the optimal solution to a given restricted LQR problem can be generated by saturating the solution of another unrestricted LQR problem, with the same dynamics and cost objective as the original one, but starting at a different initial condition and subject to a quadratic final penalization with a different matrix coefficient. Offline and online schemes were developed to detect this new initial condition and final penalization matrix (Costanza, Rivadeneira, & González, 2014). An online algorithm in this direction is the main contribution of this article from the practical point of view.…”