Neighbouring-extremal control for singular dynamic optimisation problems. Part II: multiple-input systems

Dynamic optimisation provides a unified framework for improving process operations while taking operational constraints into account. In the presence of uncertainty, measurements can be incorporated into the optimisation framework for tracking the optimum. For non-singular control problems, neighbouring-extremal (NE) control can be used to force the first-order variation of the necessary conditions of optimality (NCO) to zero along interior arcs. An extension of NE control to singular control problems has been proposed in the companion paper for single-input problems. In this article, a generalisation to multiple-input systems is presented. In order for these controllers to be tractable from a real-time optimisation perspective, an approximate NE feedback law is proposed, whose application guarantees, under mild assumptions, that the first-order variation of the NCO converges to zero exponentially. The performance of multi-input NE control is illustrated by the case study of a steered car.

[1]  Dominique Bonvin,et al.  A Two-Time-Scale Control Scheme for Fast Unconstrained Systems , 2006 .

[2]  Dominique Bonvin,et al.  Use of measurements for enforcing the necessary conditions of optimality in the presence of constraints and uncertainty , 2005 .

[3]  A. Isidori.  Nonlinear Control Systems , 1985 .

[4]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[5]  J. Rawlings,et al.  Feedback control of chemical processes using on-line optimization techniques , 1990 .

[6]  L. Petzold,et al.  Numerical methods and software for sensitivity analysis of differential-algebraic systems , 1986 .

[7]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[8]  Hans Josef Pesch,et al.  The accessory minimum problem and its importance for the numerical computation of closed-loop controls , 1990, 29th IEEE Conference on Decision and Control.

[9]  A. Y. Lee.  Neighboring extremals of dynamic optimization problems with path equality constraints , 1988 .

[10]  A. Bryson,et al.  Optimization and Control of Nonlinear Systems Using the Second Variation , 1963 .

[11]  J. V. Breakwell,et al.  On the conjugate point condition for the control problem , 1965 .

[12]  Christof Büskens,et al.  Sensitivity Analysis and Real-Time Control of Parametric Optimal Control Problems Using Nonlinear Programming Methods , 2001 .

[13]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[14]  H. Gardner Moyer,et al.  3 Singular Extremals , 1967 .

[15]  B.D.O. Anderson,et al.  Singular optimal control problems , 1975, Proceedings of the IEEE.

[16]  H. Maurer.  On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly , 1975, Optimization Techniques.

[17]  W. Rugh.  Linear System Theory , 1992 .

[18]  Dominique Bonvin,et al.  A Two-time-scale Control Scheme for Fast Systems , 2005 .

[19]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[20]  Wolfgang Marquardt,et al.  Productivity optimization of an industrial semi-batch polymerization reactor under safety constraints , 2000 .

[21]  Hans Josef Pesch,et al.  Solution differentiability for parametric nonlinear control problems with control-state constraints , 1995 .

[22]  H. J. Pesch.  Real‐time computation of feedback controls for constrained optimal control problems. part 1: Neighbouring extremals , 1989 .

[23]  Dominique Bonvin,et al.  Real-Time Optimization of Batch Processes by Tracking the Necessary Conditions of Optimality , 2007 .

[24]  H. Kelley.  Guidance theory and extremal fields , 1962 .

[25]  H. J. Pesch.  Real-time computation of feedback controls for constrained optimal control problems. Part 2: A correction method based on multiple shooting , 1989 .

[26]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[27]  H. Kelley.  An optimal guidance approximation theory , 1964 .

[28]  R. Lewis,et al.  Definitions of Order and Junction Conditions in Singular Optimal Control Problems , 1980 .

[29]  Dominique Bonvin,et al.  Neighbouring-extremal control for singular dynamic optimisation problems. Part I: single-input systems , 2009, Int. J. Control.

[30]  H. J. Pesch,et al.  New general guidance method in constrained optimal control, part 1: Numerical method , 1990 .

[31]  Dominique Bonvin,et al.  Dynamic optimization of batch processes: II. Role of measurements in handling uncertainty , 2003, Comput. Chem. Eng..

[32]  Kok Lay Teo,et al.  A Unified Computational Approach to Optimal Control Problems , 1991 .

[33]  E. Kreindler,et al.  On the concepts of controllability and observability of linear systems , 1964 .

[34]  Bala Srinivasan,et al.  Optimal operation of batch processes via the tracking of active constraints. , 2003, ISA transactions.

[35]  H. J. Pesch,et al.  Real-Time Solutions of Bang-Bang and Singular Optimal Control Problems , 2001 .

[36]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[37]  H. Robbins.  A generalized legendre-clebsch condition for the singular cases of optimal control , 1967 .