1. Colloquium on Optimisation and Control University of Sheffield Monday April 24 th 2006 Sensitivity Analysis and Optimal Control Richard Vinter Imperial College

2. Sensitivity Analysis Sensitivity analysis: the effects of parameter changes on the solution of an optimisation problem: Practical Relevance: Resource economics (economic viability of optimal resource extraction in changing environment) Design (buildings to withstand earthquakes,..) Theoretical Relevance: Intimate links with theory of constrained optimization (Lagrange multipliers, etc.) Intermediate step in mini-max optimisation ‘parametric’ approaches to MPC

3. The Value Function Minimizeover s.t. and Data: m vector parameter Value function: (describes how minimum cost changes with ) (no constraints case)

4. Links With Lagrange Multipliers Minimize over s.t. (m vector parameter is value of equality constraint function ) Lagrange multiplier rule: Fix. Suppose is a minimiser for. Then for some m vector ‘Lagrange multiplier’ Special case:

5. Value function: Fact: The Lagrange multiplier has interpretation: ( is the gradient of the value function associated with perturbations of the constraint ) Show this: For any,, sofor By ‘minimality’: (since ) Hence, where (Caution: analysis not valid unless V is differentiable.)

6. Consider now the optimal control problem: Minimize s.t. Most significant value function is associated with perturbation of initial data: (data: ) sets and Minimize s.t. t x Domain of andand and

7. Pontryagin Maximum Principle Take a minimizer Define ‘Hamiltonian’: Then, for some ‘co-state arc’ where (adjoint equation) (max. of Hamiltonian cond.) (transversality cond.) (maximised Hamiltonian) is the normal cone at x C ‘normal vector’ at x

8. Sensitivity Relations in Optimal Control Gradients of value function w.r.t. ‘initial data’ are related to co-state variable What if V is not differentiable? Interpret sensitivity relation in terms of set valued ‘generalized gradients’: (definition for ‘Lipshitz functions’, these are ‘almost everywhere’ differentiable) +1 (Valid for non-differentiable value functions) For some choice of co-state p(.)

9. Generalizations Minimize s.t. Dynamics and cost depend on par. Obtain sensitivity relations (gradients of V’ ) by ‘state augmentation’. with extra state equationIntroduce expressible in terms of co-state arcs for state augmented problem, nominal value )( and

10. Application to ‘robust’ selection of feedback controls Classical tracker design: Step 1: Determine nominal trajectory using optimal control Step 2: Design f/b to track the nominal trajectory (widely used in space vehicle design) Can fail to address adequately conflicts between performance and robustness Alternatively, Integrate design steps 1 and 2 Append ‘sensitivity term’ in the optimal control cost to reduce effect of model inaccuracies This is ‘robust optimal control’

11. Robust Optimal Control 1) Model dynamics: 3) Model variables requiring de-sensitisation: 4) Feedback control law: Example: magnitude of deviation from desired terminal location : Objective: find sub-optimal control which reduces sensitivity of to deviation of from. 2) Model cost:

12. Sensitivity Relations For control, let be state trajectory for. The `sensitivity function’ has gradient: where the arc p (.) solves (Write

13. Optimal Control Problem with Sensitivity Term Minimize s.t. and Pink blocks indicate extra terms to reduce sensitivity is sensitivity tuning parameter 0 values sensitiveinsensitive

14. Trajectory Optimization for Air-to-Surface Missiles with Imaging Radars Researchers: Farooq, Limebeer and Vinter. Sponsors: MBDA, EPSRC ‘Terminal guidance strategies for air-to-surface missile using DBS radar seeker’. Specifications include: Stealthy terrain phase, followed by climb and dive phase (‘bunt’ trjectory) Sharpening radars impose azimuthal plane constraints on trajectory Stealth phase Bunt phase Six degree of freedom model of skid-to-turn missile (two controls: normal acceleration demains Select cost function to achieve motion, within constraints.

16. References: R B VINTER, Mini-Max Optimal Control, SIAM J. Control and Optim., 2004 V Papakos and R B Vinter, A Structured Robust Control Technique, CDC 2004 A Farooq and D J N Limebeer, Trajectory Optimization for Air-to-Surface Missiles with Imaging Radars, AIAA J., to appear