Short communication: A collocation-type method for linear quadratic optimal control problems

Abstract

This communication presents a spectral method for solving time‐varying linear quadratic optimal control problems. Legendre–Gauss–Lobatto nodes are used to construct the mth‐degree polynomial approximation of the state and control variables. The derivative x·(t) of the state vector x(t) is approximaed by the analytic derivative of the corresponding interpolating polynomial. The performance index approximation is based on Gauss–Lobatto integration. The optimal control problem is then transformed into a linear programming problem. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.

Galerkin Optimal Control

“…Due to the properties of the orthogonal family of collocation points (such as those found via the Legendre or Chebyshev polynomial basis) approximations converge at spectral rates [6]. The most widely used Legendre PS method [1,5,6,9,10] is based on the LGL points [36]. However, the Legendre PS method may be based upon LGR or LG nodes as well [36,37].…”

“…The computational strategy of the GOCM-S is to find the feasible solution x N (t) ∈ X and u N (t) ∈ U for the following cases: 9) subject to the Galerkin constraints…”

“…Two recent highlights are the successful use of the Legendre PS method for the first ever zero-propellant attitude maneuver of the International Space Station [4] and the first ever minimum-time rotational maneuver performed in orbit by a NASA space telescope called TRACE [8]. In the Legendre PS method [1,2,5,6,9,10], the problem is discretized at the Legendre-Gauss-Lobatto (LGL) points, Legendre-Gauss-Radau (LGR) points or Legendre-Gauss (LG) points. The states are approximated with globally interpolating Lagrange polynomials and the cost function is typically approximated using Gaussian quadrature rule.…”

Pseudospectral Feedback Control for Three-Axis Magnetic Attitude Stabilization in Elliptic Orbits

“…The basic idea of this method is to seek polynomial approximations for the state, costate, and control functions in terms of their values at the Legendre-Gauss-Lobatto (LGL) points. Thus it is apparent that LQR problems can be easily transformed into quadratic programming (QP) problems (a quadratic cost function subject to linear algebraic constraints) [19]. He and Unbehauen [20] compared pseudospectral techniques to Riccati methods in solving LQR problems and showed that there is a huge reduction in the number of equations to be solved, and the required computer memory storage.…”

“…He and Unbehauen [20] compared pseudospectral techniques to Riccati methods in solving LQR problems and showed that there is a huge reduction in the number of equations to be solved, and the required computer memory storage. Although [19][20][21][22] solved their QP numerically, the analytical solutions were derived using preudospectral methods as shown in [23]. Compared with other analytical control laws that use step-by-step replacements for the states [24], this approach is quite easy to derive and implement.…”