## Overview

*Control Systems**(optimal control**of non-linear systems, non-linear feedback design, computation of optimal controls, distributed parameter control systems, hybrid control systems, differential games)**Filtering and Estimation**Calculus of Variations**Non-Linear Analysis*

**OPTIMAL CONTROL**

Optimal Control concerns control strategies for dynamic systems that optimise a criterion of best performance.

*Optimal Orbit Transfer*

This is an example of a problem where optimization is a design objective. The object is to find a control strategy to transfer a vehicle to a circular orbit of maximal radius (under constraints on the amount of fuel comsumed and on the rate of fuel consumption).

Optimal Control theory predicts bang-bang thrust, with continuously varying thrust angle. Many much more complex problems arising in mission planning have been investigated (‘gravity assist’ in outer planets exploration, minimise atmospheric heating, etc.)

*Optimal Control of a Growth/Consumption Model. *

Dynamic optimization problems are encountered also in the field of economics. One example is the `growth versus consumption' problem of neoclassical macro-economics, based on the Ramsey model of economic growth. The question here is, what balance should be struck between investment and consumption to maximize overall consumption on social programmes over a fixed period time? The optimal exhibits a first phase of maximum investment during which economic output builds up to some critical value, followed by a second phase of intermediate investment over which economic output is maintained and, finally, a third phase over which there is no investment because the remaining time is too small for the benefits of investment to show through.

*Optimal Control in Anti-Cancer Treatment.* Chemotherapy is a treatment aimed at destroying cancer cells by means of a cocktail of drugs, administered either at specific times or continuously. It is typically part of a complex overarching treatment plan, in which chemotherapy is following up by procedures, surgical or drug-based, for inhibiting renewed tumor growth. Traditionally, chemotherapy treatments have been based on the the maximum tolerated dose paradigm. But a side effect of chemotherapy, a `two-edged sword', is damage to normal cells. Modern day treatments aim to improve outcomes by balancing destruction of cancer cells and suppression of side effects. Empirical design of treatment plans based on clinical trials is time consuming and extremely expensive. Mathematical models of the underlying phamaco-dynamic processes involved have an important role, because they can be used to simulate on the computer the effects of different treatment strategies, simply and at low cost. Optimal Control is the appropriate tool for designing optimal treatment strategies based on these models.

**References:**

*General:*

R. B. VINTER, Optimal Control Birkhäuser, Boston, 2000.

A substantially extended and updated version of this publication, taking account of the latest advances in the theories of necessary conditions and dynamic programming, but also included an extended overview chapter which serves as an introduction to the field, is now complete.

P. Bettiol and R. B. Vinter, Dynamic Optimization, Springer Monographs in Mathematics, Springer, Berlin, (expected publication date, Dec, 2023).

*Selected Journal Articles (Optimal Control):*

The following papers concern properties of optimal controls. Many of them relate to optimal control problems with path-wise state constraints. Topics include first order necessary conditions, generalizing the Pontryagin Maximum Principle or the Euler Lagrange condition of the classical Calculus of Variations, Sensitivity Relations, ‘distance estimates’ concerning the proximity of a given state trajectories to the subset of state trajectories satisfying a state constraint and applications regarding multiplier non-degeneracy, minimizer regularity, etc., conditions under which relaxation procedures do not introduce an ‘infimum gap’, optimality conditions expressed in terms of the Hamiltonian generalizing the classical condition `the Hamiltonian is constant along optimal trajectories for autonomous optimal control problems’. Research is also reported on new insights into longstanding questions in non-smooth optimal concerning conditions for validity of the Hamiltonian inclusion. Several papers provide a systematic exploration of properties of optimal trajectories for problems in which the time dependence of the dynamic constraint is discontinuous, the control is impulsive, there are multiple player (Games Theory) or the state space is infinite dimensional.

Bettiol and R. Vinter, Improved first order necessary conditions for dynamic optimization problems with free end-times, under review.

Colombo, F. Rampazzo and R. B. Vinter, Discontinuous Solutions to the Hamilton Jacobi Equation under Second Order Interiority Hypotheses, under review.

Fusco, M. Motta and R. B. Vinter, Optimal impulsive control for time delay systems, under review.

M. Marchini and R. B. Vinter, The Maximum Principle for Lumped-Distributed Control Systems, under review.

Bernis, P. Bettiol and R. B. Vinter, Solutions to the Hamilton-Jacobi equation for state constrained Bolza problems with discontinuous time dependence, J. Diff. Eqns., 341, 2022, pp.589-619.

Maio and R. B. Vinter, Optimal Control of a Growth/Consumption Model, Optimal Control, Applications and Methods, 2021.

D. Q. Mayne and R. B. Vinter, First-Order Necessary Conditions in Optimal Control, JOTA, 189, 2021, pp. 716-743.

R. B. Vinter, Optimal Control Problems with Time Delays: Constancy of the Hamiltonian, SIAM J. Control and Optim., 57, 2019, pp. 2574-2602.

M. Motta, F. Rampazzo and R. B. Vinter, Normality and Gap Phenomena in Optimal Unbounded Control., ESAIM Control Optim. Calc. Var., 24 , 2018, pp. 1645-73.

R. B. Vinter, State constrained optimal control problems with time delays, J. Math. Analysis and Applications, 457, 2018, pp. 1696-1712.

P. Bettiol and R. B. Vinter, The Hamilton Jacobi Equation For Optimal Control Problems with Discontinuous Time Dependence, SIAM J. Control and Optim., 55, 2017, pp. 1199-1225.

A. Boccia and R. B. Vinter, Optimal control problems with time delays, SIAM J. Control Optim., 55, 2016, pp. 2905-2935.

R. B. Vinter, Multifunctions of Bounded Variation, Journal of Differential Equations, 260, 4, 2016, pp. 3350-3379.

Palladino and R. B. Vinter, Regularity of the Hamiltonian Along Optimal Trajectories, SIAM J. Control Optim., 53, 2, 2015, pp. 1892-1919. *(SIAM J. Control and Optim, Best Paper award 2014-15) *

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

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

TARGET TRACKING

*Aims:*

The aim of this research is to develop and assess new, high precision algorithms for difficult tracking problems involving single and multiple targets, applicable in situations where traditional tracking algorithms perform badly or fail altogether. The algorithms are Bayesian; they are based on probabilistic modeling and the recursive construction of approximations to the evolving condition distribution of target motion, given the observations. The problems considered include such features as ill conditioned bearings only measurements, target models with unknown parameters and tracking in high clutter environments. Research efforts have centred on developing and assessing a new algorithm, called the shifted Rayleigh filter, for bearings-only tracking of a single target. It takes its name form the fact that certain coefficients appearing in the algorithm can be interpreted as moments of a shifted Rayleigh distribution. Attention has also been given to developing similar algorithms for range-only tracking problems.

*The Shifted Rayleigh Filter*

In common with other moment matching algorithms, the shifted Rayleigh filter makes use of a normal approximation to the prior distribution of target motion. It is unusual, however, in incorporating an exact calculation of the updated distribution, to take account of a new measurement. Thus the only approximation introduced by the algorithm is to replace a conditional distribution by a matched normal distribution, at a single point in each iteration. The isolation of the approximation in this way is important because it simplifies the analysis of tracker performance and permits the construction of error bounds.

Paper [3], in which full details of the underlying analysis appear, supplies a theoretical justification of the shifted Rayleigh algorithm. The paper also confirms that the algorithm is competitive with other moment matching algorithms and particle filters in a ‘benign’ scenario, which has been the basis of earlier comparative studies.

The conference papers provide an assessment of the shifted Rayleigh filter, applied to more challenging bearings only tracking problems where, according to earlier simulation studies reported in the literature, standard moment matching algorithms, such as the extended Kalman filter, fail to provide useful estimates. Paper [7] reports on a comparative study of a particle filter and the shifted Rayleigh filter, where the purpose is to estimate the position of a moving target from noisy, bearings only measurements taken by six drifting sonobuoys, whose positions are estimated from bearings only measurements taken by a stationary monitoring sensor. Simulation studies reveal that the shifted Rayleigh filter performs favourably compared with the particle filter, while reducing the computational burden by an order of magnitude. Paper [8] concerns the application of the shifted Rayleigh filter a high clutter variant on the preceding tracking problem. Here, the filter provides excellent estimates, even in scenarios in which the clutter probability is 67% and standard deviations on the bearings only measurements are in excess of 16 degrees. Paper [5] assesses the performance of the shifted Rayleigh filter for a challenging scenario in which the extended Kalman filter fails altogether (The target passes under the sensor platform).

There are many tracking problems for which moment matching algorithms are not suitable, notably those when the distributions of interest are multi-modal. But moment matching algorithms offer such substantial computational savings over particle filters, that it is important to explore the range of applicability of such algorithms. Perhaps the most significant as pect of this research is to point to new classes of nonlinear filtering problems for which moment matching algorithms, appropriately applied , are the best available choice.

*Collaborator: J M C Clark*

1. J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, ''A Gaussian mixture filter for range-only trac king,'', Trans. Aut. Control, , IEEE TAC, Vol. 56, No. 5, 2011, pp. 1090-1096.

2. J. M. C. Clark, P. A. Kountouriotis and R. B. Vinter, ''A new Gaussian mixture algorithm for GMTI tracking un der a minimum detectable velocity c onstraint'', Trans. Aut. Control,54 ,12, pp. 2745-2756 , 2009

3. J. M. C. Clark, R. B. Vinter and M. Yaqoob, ‘The Shifted Rayleigh Filter: A New Algorithm for Bearings Only Tracking'', IEEE Trans on Aerospace and Electr o nic Systems'', IEEE Trans. Aero. and Electronic Systems, 43,4, pp. 13 73-1384, 2007

4. J M C Clark, S Robiatti and R B Vinter, ''The Shifted-Rayleigh Filter Mixture Algorithm for Bearings-Only Tracking of Manoeuvring Targets'', IEEE Trans on Signal Processing, 55, 7, pp. 3207-3218, 2007

5. Rajiv Arulampulam, Martin Clark and Richard Vinter, ''Performance of the Shifted Rayleigh Filter in Single-sensor Bearings-only Tracking'', Proc. Fusion 2007, Quebe c, 200 7

6. R. B. Vinter and J. M. C. Clark, ''A New Class of Moment M at ching Filters for Nonlinear Tracking and Estimation Problems'', IEEE Nonlinear Statistical Signal Processing Workshop, Cambridge, (2006)

7. J M C Clark, S Maskell, R B Vinter and M Yaqoob, ''Comparative Study Of the Shifted Rayleigh Filter and a Particle Filter'', 2005 IEEE Aerospac e Conference, Big Sky Montana (Special Session on Monte Carlo Methods).

8. J M C Clark, R B Vinter and M Yaqoob, '' The Shifted Rayleigh Filter for Bearings On ly Tracking’, Proc. Fusion 2005, Philadelphia, 2005

DIFFERENTIAL GAMES AND ROBUST CONTROLLER DESIGN

*Aims*

The goal of this research is to develop new, practical approaches to the design of feedback controllers of nonl inear systems (such as flow control systems and dynamic telecommunication s links), based on the theory of differenti al games. The approach takes account of ‘ worst case’ disturbances and path-wise constraints (representing, fo r example, actuator sa turation or the necessity to avoid ‘dangerous’ regions of the operational profile in an aeronautics or process control context).

*What are Differential Games?*

Differential Games concern the balance of ‘optimal’ strategies applied by two opposing players, who have conflicting notions of ‘best’ performance of the dynamical system they are both trying to control. The field has its origins in pursuit-evasion games in a military context, but now has a much more important role in Robust Controller Design.

*Relevance of Differential Games to Robust Controller Design*

Robust Control concerns the design of control systems whose performance is not degraded by modelling inaccuracies or the presence of disturbances. It is linke d to Differential Games, because disturbances and model changes can be interpreted as ‘strategies’ o f an antagonistic playe r. The Differential Games approach provides controllers that deal with disturbances on a worst case basis.

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, 11 (2012) pp. 2778-278.

J. M. C. Clark and R. B. Vinter, ‘Stochastic Exit Time Problems Arising in Process Control’, Stochastics, 84, 5-6 (2012), pp. 667-681

J. M. C. CLARK, M. R. JAMES and R. B. VINTER, R 20;The Interpretation of Discontinuous State Feedback Control Laws as Non-Anticipative Control Strategie s in Differential Games”, IEEE Transactions Automatic Control, IEEE Transactions Automatic Control 49, 8 ( 2004), pp. 1360-1365.

A. Festa and R. B. Vinter, Decomposition of Differential Games with Multiple Targets, J. Optim. Theory and Applic.,169, 2016, pp. 848-875.

P. Bettiol, M. Quincampoix and R. B. Vinter, Existence and Characterization of the Values of Two Player Differential Games with State Constraints, Applied Mathematics and Optimization, 80, 2019, pp. 765-799.