Globally Feedback Linearizable Time-Invariant Systems: Optimal Solution for Mayer’s Problem

Abstract

This paper discusses the optimal solution of Mayer’s problem for globally feedback linearizable time-invariant systems subject to general nonlinear path and actuator constraints. This class of problems includes the minimum time problem, important for engineering applications. Globally feedback linearizable nonlinear systems are diffeomorphic to linear systems that consist of blocks of integrators. Using this alternate form, it is proved that the optimal solution always lies on a constraint arc. As a result of this optimal structure of the solution, efficient numerical procedures can be developed. For a single input system, this result allows to characterize and build the optimal solution. The associated multi-point boundary value problem is then solved using direct solution techniques. [S0022-0434(00)02002-5]

State and feedback linearizations of single-input control systems

“…This can be used to improve the dynamical behavior of chaotic systems as it can be seen for the Lorenz control system in [26]. Feedback linearization techniques have also been applied to optimal control problems (e.g.…”

“…As mentioned by the authors, such problems frequently arise in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. Mayer's problem has also been considered in [1] (see also [26]) and an optimal solution for globally feedback linearizable time-invariant systems, subject to path and actuator constraints, obtained. Recall that Mayer's problem consists of determining u(t) and x(t) with t ∈ [t 0 , t f ] that minimize a functional cost J = Φ(x(t f ), t f ) subject to the dynamicsẋ = f (x) + g(x)u and inequality constraintss(x, u) ≤ 0,c(x) ≤ 0 when initial states are given and terminal states satisfy Ψ(x(t 0 ), x(t f )) = 0.…”

Feedback Linearizable Feedforward Systems: A Special Class

“…It appears in the proof that the coordinates changes are globally defined though the inverse of the feedback is only locally guaranteed. Using this equivalent linear form 3, it is proved in [14] that Mayer's optimal solution for globally feedback linearizable systems always lies on a constraint arc which allows to characterize and build such optimal solution for single-input systems. For multi-input systems, efficient numerical procedures (like pseudospectral method [4]) can be developed.…”

“…In [4], the authors used pseudospectral method to solve optimal control subject to feedback linearizable dynamics. Mayer's problem has been considered in [1] (see also [14]) and an optimal solution for globally feedback linearizable time-invariant systems obtained. Recall that Mayer's problem consists of finding u(t) and x(t) with t 2 [t 0 ; t f ] that minimize a functional cost J = 8(x(t f ); t f ) subject to _ x = f(x) + g(x)u and inequality constraintss(x; u) 0,c(x) 0 with initial and terminal states satisfying 9(x(t 0 ); x(t f )) = 0.…”

“…Feedback linearization techniques have been used to improve the dynamical behavior of chaotic systems (see Lorenz system in [11], [14]) and have been applied to optimal control problems with a recent regain of interest. In [4], the authors used pseudospectral method to solve optimal control subject to feedback linearizable dynamics.…”