A parsimonious algorithm for the solution of continuous-time constrained LQR problems with guaranteed convergence

This paper presents an efficient computational method for solving the input-constrained, continuous time, infinite horizon, linear quadratic regulator problem to within a user specified tolerance. The infinite dimensional input trajectory is approximated with a piecewise linear function on a finite time discretization to ensure input constraint satisfaction. This approximate problem is then a standard finite dimensional quadratic program and is solved by conventional methods, generating an upper bound for the optimal value function. The finite time discretization is then refined by subdividing the intervals estimated to cause the largest decrease in the cost function. Convergence of the solution of this discretized problem towards the optimal continuous-time solution, as the discretization is refined, is proved. Exploiting the strict convexity of the original infinite dimensional problem, the gradient of the cost function with respect to the continuous-time input can be computed to generate a lower bound for the optimal cost. For computational efficiency, a lower bound for the solution of the discretized control at a very fine discretization can be used instead. The algorithm terminates when the difference between the upper and lower bounds meets a user supplied tolerance.