Control For Energy and Sustainability

EPSRC Programme Grant

Project UT-A: Optimal Control

Manager: Richard Vinter

Investigators: Jorge Goncalves, David Limebeer and Richard Vinter

Research Staff: Paola Falugi (Research Associate), Giacomo Perantoni (PhD)

Collaborators: Eric Kerrigan

Start date: 01/10/2009

Linked Projects: PS-B, PS-C, EET-A, EET-C and UT-C

Summary. Optimal Control is concerned with the calculation of open-loop control strategies that optimise a performance index. While widely applied in aerospace (in flight path selection, for example) and in optimisation-based feedback controller design (see Project UT-C), optimal control methods have yet to achieve their full potential in applications to air traffic management, fuel efficient transport and electrical power systems. In these application areas, the underlying dynamic models, the constraints on system variables and the required structure of the controls in some respects fall outside the standard theory.

This project aims to provide the necessary extensions required to broaden its applicability in these fields. The extensions will take account of

1) Hybrid Models (to describe systems experiencing abrupt changes of topology).

2) Unconventional Constraints (to accommodate the 'min-max' type constraints arising in vehicle collision avoidance, for example)

3) The need for fast, efficient computational schemes for the solving optimal control problems arising in optimization based controller design.

4) Feedback Implementation of Optimal Controls (to counter modelling error and disturbances; effective, generic procedures are currently lacking).

This project is linked to the applications projects PS-B, PS-C, EET-A, EET-C and into the optimization routines employed in UT-C.

Current Status. Research on this project has continued to be directed towards development and refinement of software for the computation of optimal controls developed at Imperial College and towards providing a simple user interface for the benefit of research throughout the consortium, fundamental research into optimal control, and applications in the field of transportation.

Software: An open-source MATLAB toolbox ICLOCS has been developed for the formulation and solution of optimal control problems with general control, state and endpoint constraints. The software allows for the presence of design parameters in the dynamic models considered, and also permits free end-times. The software is an implementation of the 'direct method', according to which the optimal control problem is reduced by time discretization to a nonlinear programme (NLP), which is then solved using a open-source package (IPOPT). It offers the user considerable flexibility of the use choice of a discretization scheme, with numerous options for computation enhancement. The software delivers not just an approximation to the optimal control, but sensitivity information and an error analysis for assessment of the quality of the approximation.

The toolbox is a valuable resource that has been used by research groups at Imperial, Cambridge and Oxford on a range of practical problems. These include optimal control problems arising in the implementation of optimization-based controller design methods for lap-time optimization in Formula I, the control of a solar plant, satellite attitude control, fluid mixing, energy harvesting and for other purposes.

Theory of state-constrained optimal control: A feature of many dynamic optimisation problems arising in control of mechanical systems and, more broadly, in the field of transportation, is the presence of path-wise constraints on state variables, which take account of, for example, forbidden regions of a flight envelop or obstacles that must be avoided. While very general optimality conditons are available, such as the Pontryagin Maximum Principle and Dynamic Programming, the investigation of the circumstances under which they supply useful information or just apply in a degenerate form, leading to a breakdown of either analytical or computational attempts at solution, is an undeveloped are of research. Our work has revealed a number of essential features of optimal control problems for which non-degenerate optimality conditions can be derived. But it has also exposed for the first time unexpected pathologies regarding the sensitivity properties of control system with multiple state constraints, invalidating earlier claimed optimality conditions, and has attracted considerable interest.

Other topics: research has carried out in the area of second order optimality conditions. This has centred on the derivation of second order optimality conditions, which confirm the optimality of putative minimizers obtained by either analytic or computational techniques. New second order sufficient conditions have been obtained which, for the first time, allow the minimizers to be non-unique. These have been successfully applied to problems of optimal periodic flight control and other areas, where non-uniqueness naturally arises. Research has also been conducted into game theoretic approaches to the problem of confining the state of a system, with disturbance inputs, to a ‘safe’ region of the state space for as long as possible, resulting in new insights into the structure of the solution to such problems and new numerical methods. New first order conditions have also been derived for optimal control problems with time-delays, which, for the first time, allow the final time to be a choice variable. Finally, a longstanding open research problem has been resolved, concerning the validity of necessary conditions referred to as the Hamiltonian inclusion, which is a generalization of the Hamilton’s system of equations in the Calculus of Variations, when the velocity set fails to be convex.

This research on underlying theory has resulted in five journal papers including six papers in SIAM J. Control and Optimization and two full papers in IEEE Transactions Optimal Control, and presentations at five international meetings including three IEEE Decision and Control conferences, and a paper in the Journal of Differential Equations.

Applications: The main applications aspects of this project centre on the work of the PhD student at Oxford University which, following the completion a modelling phase of the project, now address path-following and minimum lap time control strategies for road vehicles, including optimization based strategies. The Oxford research activity in applied optimal control is being carried out with Ricardo, aimed at merging advances in optimal control theory, and modelling and controller hardware in the context of automotive power-trains.

Vehicle Modelling: The limit performance of road vehicles is dictated by the tyres. The tyres impose limits on the acceleration and braking performance in straight running, while restricting the top speed in cornering. As a result, high-fidelity tyre models are a prerequisite to a meaningful vehicular optimal control study. Current high-fidelity tyre models make use of so called ‘magic formulae’ that are complicated, and under some circumstances non-smooth as well as requiring a large number of parameters. Another setback is that these models have little or no basis in physics. In an attempt to address all of these shortcomings this project has focused thus far on the development of a new type of motorcycle model based on the LuGre friction model. This model sets out to capture the essence of complicated tyre friction phenomena with a physics-based model of modest complexity.

Optimal Control of Vehicles: Extensive Optimal control studies have been undertaken, aimed at optimizing vehicle operation, in various senses. Special attention has been given to achieving an appropriate level of model complexity, which adequately captures vehicle behaviour but for which the computation of optimal controls is a tractable problem. Control objectives considered have included the minimisation of the lap time around a circuit, the completion of manoeuvres for the satisfaction of safety standards, the avoidance of moving obstacles and trajectory tracking. The numerical optimal control framework adopted permits the inclusion of the vehicle parameters as optimisation variables.

This research on vehicle modelling and optimal control has been presented at numerous conferences, and provides material for several papers, which have either appeared or have been submitted for journal publication.


[BFV14] P. Bettiol, H. Frankowska, R.B. Vinter, Improved sensitivity relations in state constrained optimal control, Applied Mathematics and Optimization (online version), July 2014

[GV14] C. Gavriel and R. B. Vinter, Second Order Sufficient Conditions for Optimal Control Problems with Non-unique Minimizers: An Abstract Framework, Applied Mathematics & Optimization, April 2014 (electronic version)., April 2014

[PV14] M. Palladino and R. B. Vinter, Minimizers That Are Not Also Relaxed Minimizers, SIAM J. Control and Optim. 52, 4, 2014, pp. 2164-2179., 2014

[V14] R. Vinter, The Hamiltonian Inclusion for Non-Convex Velocity Sets, SIAM J. Control and Optim. , 52, 2, 2014, pp. 1237-125, 2014

[LPR14] D. J. N. Limebeer, G. Perantoni and A. V. Rao, Optimal control of Formula One car energy recovery systems, International Journal of Control, Volume 87, Issue 10, 2014, 2014

[PL13a] G Perantoni and D J N Limebeer, Time-optimal control of rolling bodies, International Journal of Control, Special Issue: Systems, Modelling and Feedback Control: A Special Issue Dedicated to Sir Alistair G.J. MacFarlane, Volume 86, 11, 2013., 2013

[PL13] G. Perantoni and D. J. N. Limebeer, Optimal Control of a Two-Mass Skate Bicycle Without Steering, IEEE International Conference on Industrial Technology (ICIT 2013), Cape Town, South Africa, 2013, 2013

[BFMV13] A. Bocchia, P. Falugi, H. Maurer and R. B. Vinter, Free time Optimal Control Problems with Time Delays, Proc. 52nd Conference on Decision and Control, Florence, 2013, 2013

[BFV13] P. Bettiol, H. Frankowska and R. B. Vinter, Sensitivity interpretations of the co-state trajectory for opimal control problems with state constraints, Proc. 52nd Conference on Decision and Control, Florence, 2013, 2013

[BBV13a] P. Bettiol, A. Bocchia and R B Vinter, Stratified Necessary Conditions for Differential Inclusions with State Constraints, SIAM Journal on Control and Optimization, 51, 5, pp3903-3917, 2013

[BV13a] P. Bettiol and R. B. Vinter, Estimates on Trajectories in a Closed Set with Corners for (t,x) Dependent Data, Mathematical Control and Related Fields, 3 ,3 , 2013, pp.245-267., 2013

[PLA12] G. Perantoni, D. J. N. Limebeer and M. R. Arthington, A Direct Method for Optimal Control and Optimal Design of Two-Wheeled Vehicles, presented at The 11th International Symposium on Advanced Vehicle Control, Seoul, South Korea, 9-12 September 2012., 9-12 September 2012.

[PL12] G. Perantoni and D. J. N. Limebeer, A Dynamic Motorcycle Tyre Model Based on LuGre Friction, presented at The 11th International Symposium on Advanced Vehicle Control, Seoul, South Korea, 9-12 September 2012., September 2012

[FKV12] P. Falugi, P. A. Kountouriotis and R. B. Vinter, Controllers that Confine a System to a Safe Region in the State Space, IEEE Trans. Automat. Contr. 57, 2012, pp. 2778-278., 2012

[GLV11] C. Gavriel, S. Lopez and R. B. Vinter, Regularity of Minimizers for Higher Order Variational Problems in One Independent Variable, Annual Reviews in Control, 35, 2011, pp. 172-177., 2011

[BFV11] P.Bettiol, H.Frankowska and R.B.Vinter, L infinity estimates on trajectories confined to a closed subset, Journal of Differential Equations, 252, 2012, pp. 1912-1933., 2012

[V11] R. B. Vinter, Regularity of Minimizers for Second Order Variational Problems in One Independent Variable, Discrete and Continuous Dynamical Systems (DCDS-A), vol. 29, no. 2, 2011, pp. 547-557., 2011

[GV11] C. Gavriel and R. B. Vinter, Regularity of Minimizers for Second Order Variational Problems in One Independent Variable, Discrete and Continuous Dynamical Systems (DCDS-A), vol. 29, no. 2, 2011, pp. 547-557. , 2011

[BV11] P.Bettiol and R.B.Vinter, Trajectories Satisfying a State Constraint: Improved Estimates and New Non-Degeneracy Conditions, IEEE Transactions on Automatic Control, Vol 56, no 5, pp 1090-1096, 2011

[BBV11] P.Bettiol, A Bressan and R.B.Vinter, Estimates for Trajectories Confined to a Cone in R^n, SIAM Journal on Control and Optimisation, Vol 49, no 1, pp 21-42, 2011

[WPK10] E. J. V. Wyk, P. Falugi, and E. C. Kerrigan, Imperial College London Optimal Control Software (ICLOCS) (Solves nonlinear optimal control problems with constraints.),, 2010

[GV10] C. Gavriel and R. B. Vinter, Second order sufficient conditions for optimal control problems with non-unique minimizers, Proc. 2010 American Control Conference 2010, Baltimore Maryland, 2010, 2010

[BV10a] P. Bettiol and R. B. Vinter, Existence of Feasible Approximating Trajectories Satisfying Multiple State Constraints, 49th IEEE Conference on Decision and Control, Atlanta Georgia, December 15-17 2010,

[BV10] P.Bettiol and R.B.Vinter, Sensitivity Interpretations of the Co-State Variable for Optimal Control Problems with State Constraints, SIAM Journal on Control and Optimisation 48 (5), pp.3297-3317, 2010

[BBV10] P.Bettiol, A.Bressan and R.B.Vinter, On trajectories satisfying a state constraint: W(1,1) Estimates and Counter-Examples, SIAM Journal on Control and Optimisation, vol 48, no 7, pp 4664-4679, 2010

[CGMMPZ10] S. Cordiner, S. Galeani, F. Mecocci, V. Mulone, G. Perantoni and L. Zaccarian, Dynamic input allocation of torque references for a parallel HEV, Proceedings of the 49th IEEE Conference on Decision and Control, CDC 2010, December 15-17, 2010, Atlanta, Georgia, USA, pp. 4920-4925., December 2010

[HL10] A. J. Hazell and D. J. N. Limebeer, A Framework for Discrete-Time H2 Preview Control, ASME Journal of Dynamic Systems, Measurement and Control, 132,3, May, 2010

[BV09] P.Bettiol and R.B.Vinter, Existence of feasible approximating trajectories for differential inclusions with obstacles as state constraints, 48th IEEE Conference on Decision and Control, Shanghai, P.R. China, Dec. 2009