Constrained LQR Using Online Decomposition Techniques

This paper presents an algorithm to solve the infinite horizon constrained linear quadratic regulator (CLQR) problem using operator splitting methods. First, the CLQR problem is reformulated as a (finite-time) model predictive control (MPC) problem without terminal constraints. Second, the MPC problem is decomposed into smaller subproblems of fixed dimension independent of the horizon length. Third, using the fast alternating minimization algorithm to solve the subproblems, the horizon length is estimated online, by adding or removing subproblems based on a periodic check on the state of the last subproblem to determine whether it belongs to a given control invariant set. We show that the estimated horizon length is bounded and that the control sequence computed using the proposed algorithm is an optimal solution of the CLQR problem. Compared to state-of-the-art algorithms proposed to solve the CLQR problem, our design solves at each iteration only unconstrained least-squares problems and simple gradient calculations. Furthermore, our technique allows the horizon length to decrease online (a useful feature if the initial guess on the horizon is too conservative). Numerical results on a planar system show the potential of our algorithm.Comment: This technical report is an extended version of the paper titled "Constrained LQR Using Online Decomposition Techniques" submitted to the 2016 Conference on Decision and Contro

Generalized spiked harmonic oscillator

A variational and perturbative treatment is provided for a family of generalized spiked harmonic oscillator Hamiltonians H = -(d/dx)^2 + B x^2 + A/x^2 + lambda/x^alpha, where B > 0, A >= 0, and alpha and lambda denote two real positive parameters. The method makes use of the function space spanned by the solutions |n> of Schroedinger's equation for the potential V(x)= B x^2 + A/x^2. Compact closed-form expressions are obtained for the matrix elements , and a first-order perturbation series is derived for the wave function. The results are given in terms of generalized hypergeometric functions. It is proved that the series for the wave function is absolutely convergent for alpha <= 2.Comment: 14 page

Matrix elements for a generalized spiked harmonic oscillator

Closed-form expressions for the singular-potential integrals are obtained with respect to the Gol'dman and Krivchenkov eigenfunctions for the singular potential V(x) = B x^2 + A/x^2, B > 0, A >= 0. These formulas are generalizations of those found earlier by use of the odd solutions of the Schroedinger equation with the harmonic oscillator potential [Aguilera-Navarro et al, J. Math. Phys. 31, 99 (1990)].Comment: 12 pages in plain tex with 1 ps figur