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Colloquium on Optimisation and Control University of Sheffield Monday April 24 th 2006

Colloquium on Optimisation and Control University of Sheffield Monday April 24 th 2006 Sensitivity Analysis and Optimal Control Richard Vinter Imperial College. Sensitivity Analysis Sensitivity analysis: the effects of parameter changes on the solution of an optimisation problem:

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Colloquium on Optimisation and Control University of Sheffield Monday April 24 th 2006

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  1. Colloquium on Optimisation and Control University of Sheffield Monday April 24th 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 m vector parameter Minimize over and s.t. Data: Value function: (describes how minimum cost changes with ) (no constraints case)

  4. Links With Lagrange Multipliers Special case: over Minimize 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’

  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 , for , so By ‘minimality’: (since ) Hence , where (Caution: analysis not valid unless V is differentiable.)

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

  7. Pontryagin Maximum Principle Take a minimizer Define ‘Hamiltonian’: Then, for some ‘co-state arc’ (adjoint equation) (max. of Hamiltonian cond.) (transversality cond.) where (maximised Hamiltonian) C is the normal cone at x ‘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) For some choice of co-state p(.) +1 -1 (Valid for non-differentiable value functions)

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

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

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

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

  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 Bunt phase Stealth phase • Six degree of freedom model of skid-to-turn missile • (two controls: normal acceleration demains • Select cost function to achieve motion, within constraints.

  15. 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

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