Convergence of discrete-time approximations of constrained linear-quadratic optimal control problems

Continuous-time linear constrained optimal control problems are in practice often solved using discretization techniques, e.g. in model predictive control (MPC). This requires the discretization of the (linear time-invariant) dynamics and the cost functional leading to discrete-time optimization problems. Although the question of convergence of the sequence of optimal controls, obtained by solving the discretized problems, to the true optimal continuous-time control signal when the discretization parameter (the sampling interval) approaches zero has been addressed in the literature, we provide some new results under less restrictive assumptions for a class of constrained continuous-time linear quadratic (LQ) problems with mixed state-control constraints by exploiting results from mathematical programming extensively. As a byproduct of our analysis, a regularity result regarding the costate trajectory is also presented.

[1]  G. Thompson,et al.  Optimal Control Theory: Applications to Management Science and Economics , 2000 .

[2]  Elijah Polak,et al.  On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems , 1993, Math. Program..

[3]  E. Polak,et al.  A Dual Method for Optimal Control Problems with Initial and Final Boundary Constraints , 1973 .

[4]  Richard B. Vinter,et al.  Optimal Control , 2000 .

[5]  D. Wishart.  Introduction to the Mathematical Theory of Control Processes. Volume 1—Linear Equations and Quadratic Criteria , 1969 .

[6]  J. Daniel.  On the convergence of a numerical method for optimal control problems , 1969 .

[7]  Suresh P. Sethi,et al.  A Survey of the Maximum Principles for Optimal Control Problems with State Constraints , 1995, SIAM Rev..

[8]  J. Cullum.  Discrete Approximations to Continuous Optimal Control Problems , 1969 .

[9]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[10]  James A. Primbs,et al.  Feasibility and stability of constrained finite receding horizon control , 2000, Autom..

[11]  Jong-Shi Pang,et al.  Differential variational inequalities , 2008, Math. Program..

[12]  J. Daniel.  On the approximate minimization of functionals , 1969 .

[13]  William W. Hager,et al.  Second-Order Runge-Kutta Approximations in Control Constrained Optimal Control , 2000, SIAM J. Numer. Anal..

[14]  Richard B. Vinter,et al.  Regularity properties of optimal controls for problems with time-varying state and control constraints , 2006 .

[15]  Maxim V. Subbotin,et al.  Continuous Time Linear Quadratic Regulator With Control Constraints via Convex Duality , 2007, IEEE Transactions on Automatic Control.

[16]  Richard Bellman,et al.  Introduction to the mathematical theory of control processes , 1967 .

[17]  B. Brogliato.  Some results on optimal control with unilateral state constraints , 2009 .

[18]  van Stef Stef Eijndhoven,et al.  Linear quadratic regulator problem with positive controls , 1997, 1997 European Control Conference (ECC).

[19]  W. Hager,et al.  Dual Approximations in Optimal Control , 1984 .

[20]  W. Hager.  Lipschitz Continuity for Constrained Processes , 1979 .

[21]  Lars Grüne,et al.  On the Infinite Horizon Performance of Receding Horizon Controllers , 2008, IEEE Transactions on Automatic Control.

[22]  Rafal Goebel,et al.  Convex Optimal Control Problems with Smooth Hamiltonians , 2005, SIAM J. Control. Optim..

[23]  Anton A. Stoorvogel,et al.  Linear quadratic regulator problem with positive controls , 1998 .

[24]  Stephen E. Wright,et al.  Consistency of Primal-Dual Approximations for Convex Optimal Control Problems , 1995 .

[25]  J. Cullum.  An explicit procedure for discretizing continuous, optimal control problems , 1971 .

[26]  William W. Hager.  The Ritz–Trefftz Method for State and Control Constrained Optimal Control Problems , 1975 .

[27]  William W. Hager,et al.  The Euler approximation in state constrained optimal control , 2001, Math. Comput..