Method to Solve the Nonlinear Infinite Horizon Optimal Control Problem With Application to the Track Control of a Mobile Robot

Abstract

We present a numerical method to solve the infinite time horizon optimal control problem for low dimensional nonlinear systems. Starting from the linear-quadratic approximation close to the origin, the extremal field is efficiently calculated by solving the Euler-Lagrange equations backward in time. The resulting controller is given numerically on an interpolation grid. We use the method to obtain the optimal track controller for a mobile robot. The result is a globally asymptotically stable nonlinear controller, obtained without any specific insight into the system dynamics.

Optimal tracking of a mobile robot with path determined by Random Particle Optimization and genetic algorithms

“…This section considers a linear model for the robot and presents results of the computed command vector, optimal control and resulting trajectories (Eqs. [10][11][12].…”

“…The system follows a command vector (10) from which the optimal control is found as (11) Finally, the optimal trajectory from the discretized state transition equation is thus readily found from (12) III.…”

“…Earlier methods for optimal control include sliding mode control and the artificial potential approach [9,11]. The Linear Quadratic regulator formulation [1][2][3] has been used for linear systems with quadratic performance indicators to obtain optimal control strategies and have also been used for nonlinear analysis [10] which is difficult conceptually and numerically. Path-tracking controller consisting of feedforward as well as feed-back components have been used [6] with good results.…”

Optimal control of a class of nonlinear systems using Euler-Lagrange back integration method

“…Bu çözümde tasarlanan denetleyicide, sistem parametreleri bilinmeden, değer fonksiyonunun optimal denetleyicinin değer fonksiyonuna yakınsamasını sağlanmıştır. Doğrusallaştırılmış halleri denetlenebilir olan doğrusal olmayan sistemler için optimal dinamik denklemleri zamanda geriye doğru çözülerek optimal denetleyici tasarlanabilir [9][10][11][12]. Bu yöntemde zamanda geriye doğru çözümün başlangıç noktası, doğrusallaştırılmış sistemin denge noktası yakınlarındaki bir hiper elipsoit yüzeyidir.…”

“…Daha sonra, bu optimal yollar optimal geri beslemede kullanılmaktadır. 10.17341/gazimmfd.875563 Anahtar Kelimeler: Doğrusal olmayan optimal denetim, HJB denklemi, doğrusal girişli doğrusal olmayan sistemler, genişlik öncelikli arama, Euler-Lagrange geri integrasyon yöntemi, üçgenleme Optimal control of a class of nonlinear systems using Euler-Lagrange back integration method H I G H L I G H T S  Optimal control of nonlinear, linearizable input affine systems  Euler-Lagrange back integration method  Tessellation of optimal paths in the state-space…”