A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

Abstract

a b s t r a c tThis paper presents a sparse collocation method for solving the time-dependent Hamilton-Jacobi-Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite timehorizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the timedependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation.

Semi‐feedback optimal control design for nonlinear problems

“…in Equation 15, the state equation will be given as in Equation 20, and then, by considering the initial and final values, state and co-state differential equations can be solved together to find the reference trajectory and optimal control open-loop command.…”

“…In order to calculate the eigenvalues, one should linearize the nonlinear problem when Equation 23 is substituted into Equation 15. Then, the eigenvalues can be found easily using the characteristic equation.…”

Quadrotor System Identification Using the Multivariate Multiplex B-Spline

“…The multiplex spline is a generalization of the tensor-product simplex spline presented by Govindarajan et al 6 In the latter framework the B-coefficients c κ are represented by a univariate simplex spline. In the current discussion c κ is instead described using a multivariate spline.…”

“…6,7 In this framework the B-coefficients of a multivariate simplex spline are defined using a univariate spline function of a different variable. This framework was used by Govindarajan et alto estimate autopilot safety margins, 7 and by Sun et alin an aircraft system identification framework.…”

Optimal Control Framework for Estimating Autopilot Safety Margins

“…A common strategy used to overcome these computational challenges is to use Adaptive Dynamic Programming (ADP) techniques [23] to find approximations of the value function with general function approximators. In this paper, we have used the method from Govindarajan et al [24], that uses multivariate simplex splines [25,26] to find an approximation of V i (t, x).…”