Design of penalty functions for optimal control of linear dynamical systems under state and input constraints

Abstract

International audienceThis paper addresses the problem of solving a constrained optimal control for a general single-input single output linear time varying system by means of an unconstrained method. The exposed methodology uses a penalty function approach, commonly considered in finite dimensional optimization problem, and extended here it to the considered infinite dimensional (functional optimization) case. The main novelty is that both the bounds on the control variable and on a freely chosen output variable are considered and studied theoretically. It is shown that a relatively simple and constructive choice of penalty functions allows to completely alleviate the usual difficulties of handling such constraints in optimal control. An illustrative example is provided to show the potential of the method

Investigating the ability of various buildings in handling load shiftings

“…In (11), the expression serves to keep the output away from the constraint (see [15]). The power 2.1 guarantees the wellbehavness of the method (see [8]). Now, to compute the optimal control, an interior method is used.…”

“…For this problem, the constraints are (5) and (6) and the considered criterion is the energy consumed over the whole week. To solve the state constraint optimal control problem a maximum principle-based interior method is used [5], [6], [7], [8], [14], which has been adapted for the energy consumption problem. The criterion is given by the following:…”

“…with the dynamics and the state constraint y ∈ [y − , y + ] seen above, and where T = 7 days. The first step, in this algorithm is to operate the following change in variable 2 on the control variable u: u φ(ν) = u + e kν 1 + e kν (8) This change in variable is such that ν is an unconstrained variable and the optimization problem is now:…”

“…This article follows this approach and studies the load shifting opportunities on five thermal models ranging from poorly to well insulated buildings. The contribution consists in a dynamic optimization under constraints method in continuous time allowing to accurately compute optimal trajectories (when they exist) [5], [6], [7], [8]. By acting on the duration of load shifting, one can figure out the maximum duration of a complete heating load shifting while maintaining an acceptable level of comfort and shifting the consumption.…”

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

“…This point is critical because interiority is a requirement to avoid ill-posedness and computational failure of implemented algorithms. The problem of interiority in infinite-dimensional optimization has been addressed in [30] for input-constrained optimal control, and in [31][32][33], respectively, for linear systems, single input single output nonlinear systems, and multi-variable nonlinear systems with cubic input constraints. These contributions provide penalty functions guaranteeing the interiority of the solutions.…”