Paper Info
Title
Inversion in indirect optimal control: constrained and unconstrained cases
Abstract
This paper focuses on using non linear inversion in optimal control problems. This technique allows us to rewrite the stationarity conditions derived from the calculus of variations under a higher order form with a reduced number of variables. After a brief tutorial overview of the multi- input multi-output cases for which the cost functions have a positive definite Hessian with respect to control variables, we address the case of linear systems with a control affine cost to be minimized under input constraints. This is the main contribution of this paper. We study the switching function between singular and regular arcs and explain how higher order stationarity conditions can be obtained. An example from the literature (energy optimal trajectory for a car) is addressed. I. INTRODUCTION Lately, inversion has been used in direct methods of numerical optimal control (i.e. collocation). In such meth- ods, coefficients are used to approximate both states and inputs (12) with basis functions. The numerical impact of the relative degree (as defined in (13)) of the output chosen to cast the optimal control problem into a nonlinear programming problem was emphasized in (18), (14). Choos- ing outputs with maximum relative degrees is the key to efficient variable elimination that lowers the number of re- quired coefficients (see for example (17), (7)). In differential equations, in constraints, and in cost functions, unnecessary variables are substituted with successive derivatives of the chosen outputs. When combined to a NLP solver (such as NPSOL (10) for instance), this can induce drastic speed-ups in numerical solving (5), (15), (20), (16), (1). In (9), we focused on indirect methods (i.e. methods using adjoint variables) for numerically solving optimal control problems. In this framework, we explained how to use the ge- ometric structure of the dynamics. In the single-input single- output (SISO) case (with an n-dimensional state), assuming the cost is quadratic in the control variables, we emphasized that r, the relative degree of the primal system, plays a role in the adjoint (dual) dynamics. The two-point boundary value problem (TPBVP) obtained from the calculus of variations can be rewritten by eliminating many variables. In fact, only n r variables are required. In the case of full feedback linearizability, the primal and adjoint dynamics take the form of a 2n-degree differential equation in a single variable: the linearizing output. More generally, we addressed the F. Chaplais is with the Centre Automatique et Systemes, ´ general case of multi-inputs multi-outputs (MIMO) systems. Noting m the number of inputs, and r the total relative degree, we obtained a similar reduction of variables results. Numerous adjoint variables could be easily recovered once the optimal solution was known, providing direct insight into neighboring extremals and post-optimal analysis. Further, a strong positive impact (in terms of accuracy and CPU usage) of dealing with the obtained higher order representation of the TPBVP was underlined.
Year
DOI
Venue
2007
10.1109/CDC.2007.4434074
New Orleans, LA
Keywords
Field
DocType
MIMO systems,inverse problems,optimal control,control affine cost,indirect optimal control,linear systems,multi-input multi-output cases,nonlinear inversion
Affine transformation,Mathematical optimization,Optimal control,Nonlinear system,Linear-quadratic-Gaussian control,Linear system,Computer science,Control theory,Positive-definite matrix,Hessian matrix,Control variable
Conference
ISSN
ISBN
Citations 
0191-2216 E-ISBN : 978-1-4244-1498-7
978-1-4244-1498-7
0
PageRank 
References 
Authors
0.34
3
2
Name
Order
Citations
PageRank
Chaplais, F.100.34
Petit, N.2143.56