Average Cost Minimization Problems Nathalie T. Khalil Université de Bretagne Occidentale, France Conference, Control of state constrained dynamical systems Università degli Studi di Padova September 29, 2017 nathalie.khalil@univ-brest.fr
Outline Average cost problem Motivating problem 1 Our problem on average cost Link with previous works 2 Motivation Novelty and necessary 3 Link with previous works conditions 4 Novelty and necessary conditions for optimality Proof Conclusion and 5 Conclusion and perspectives Perspective Joint work with Piernicola Bettiol Nathalie T. Khalil Average Cost Minimization Problems 2/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem Link with previous works Novelty and necessary conditions Proof Conclusion and Perspective Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Novelty and necessary conditions Proof Conclusion and Perspective Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof Conclusion and Perspective Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof ◆ u ( t ) control. Evolution of the system affected Conclusion and Perspective Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof ◆ u ( t ) control. Evolution of the system affected Conclusion and Perspective Control System with unknown parameters ˙ x ( t ) = f ( t , x ( t , ω ) , u ( t ) , ω ) x ( 0 , ω ) = x 0 Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof ◆ u ( t ) control. Evolution of the system affected Conclusion and Perspective Control System with unknown parameters ˙ x ( t ) = f ( t , x ( t , ω ) , u ( t ) , ω ) x ( 0 , ω ) = x 0 ◆ ω ∈ Ω unknown set. Uncertainties are taken into account Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof ◆ u ( t ) control. Evolution of the system affected Conclusion and Perspective Control System with unknown parameters ˙ x ( t ) = f ( t , x ( t , ω ) , u ( t ) , ω ) x ( 0 , ω ) = x 0 ◆ ω ∈ Ω unknown set. Uncertainties are taken into account + Performance Criterion ? Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem Cauchy Problem Average cost problem ˙ x ( t ) = f ( t , x ( t )) x ( 0 ) = x 0 Motivating problem ◆ Predicting but no altering the evolution Link with previous works Control System Novelty and necessary ˙ x ( t ) = f ( t , x ( t ) , u ( t )) x ( 0 ) = x 0 conditions Proof ◆ u ( t ) control. Evolution of the system affected Conclusion and Perspective Control System with unknown parameters ˙ x ( t ) = f ( t , x ( t , ω ) , u ( t ) , ω ) x ( 0 , ω ) = x 0 ◆ ω ∈ Ω unknown set. Uncertainties are taken into account + Performance Criterion ? Optimal Control Nathalie T. Khalil Average Cost Minimization Problems 3/17
An Optimal Control Problem For a given µ (probability measure on Ω ) and g ( x , ω ) Average cost problem � Motivating minimize g ( x ( T , ω ); ω ) d µ ( ω ) average cost problem Ω Link with over u : [ 0 , T ] → R m and W 1 , 1 arcs { x ( ., ω ) } previous works Novelty and such that u ( t ) ∈ U ( t ) a.e. t ∈ [ 0 , T ] necessary conditions and, for each ω ∈ Ω , Proof ˙ x ( t , ω ) = f ( t , x ( t , ω ) , u ( t ) , ω ) a.e. t ∈ [ 0 , T ] , Conclusion and Perspective x ( 0 , ω ) = x 0 and x ( T , ω ) ∈ C ( ω ) . Ω (set of unknown parameters) is a complete separable metric space Nathalie T. Khalil Average Cost Minimization Problems 4/17
An Optimal Control Problem For a given µ (probability measure on Ω ) and g ( x , ω ) Average cost problem � Motivating minimize g ( x ( T , ω ); ω ) d µ ( ω ) average cost problem Ω Link with over u : [ 0 , T ] → R m and W 1 , 1 arcs { x ( ., ω ) } previous works Novelty and such that u ( t ) ∈ U ( t ) a.e. t ∈ [ 0 , T ] necessary conditions and, for each ω ∈ Ω , Proof ˙ x ( t , ω ) = f ( t , x ( t , ω ) , u ( t ) , ω ) a.e. t ∈ [ 0 , T ] , Conclusion and Perspective x ( 0 , ω ) = x 0 and x ( T , ω ) ∈ C ( ω ) . Ω (set of unknown parameters) is a complete separable metric space Goal: characterize the optimal control independently of the unknown parameter action Nathalie T. Khalil Average Cost Minimization Problems 4/17
Example from aerospace engineering: Spacecraft 1 Average cost problem q = 1 Motivating ˙ Dynamics 2 Q ( r ) q problem Link with r = I − 1 ( − r × I · r − r × m c ( δ ) − A ( δ ) u ) ˙ previous works Novelty and ˙ δ = u necessary conditions Proof q ∈ R 4 (attitude), r ∈ R 3 (body rate), δ N c − vector of gimbals angles Conclusion and (associate with the onboard control moment gyros CMG), I inertia Perspective matrix, Q ( r ) a given matrix, m c ( δ ) angular momentum of CMG, A ( δ ) is a 3 × N c matrix associated with the control u ∈ U . Goal: minimize the time between two collects of images 1Ross, I. M., Karpenko M., and Proulx J. R. "A Lebesgue-Stieltjes Framework For Optimal Control and Allocation." American Control Conference (ACC) 2015. Nathalie T. Khalil Average Cost Minimization Problems 5/17
✧ Example from aerospace engineering: Spacecraft 2 Average cost if δ ( 0 ) = mean value of δ 0 Uncontrollable system! problem Motivating problem Link with previous works Novelty and necessary conditions Proof Conclusion and Perspective 2Ross, I. M., Karpenko M., and Proulx J. R. "A Lebesgue-Stieltjes Framework For Optimal Control and Allocation." American Control Conference (ACC) 2015. Nathalie T. Khalil Average Cost Minimization Problems 6/17
✧ Example from aerospace engineering: Spacecraft 2 Average cost if δ ( 0 ) = ω ∈ Ω (‘uncertainty’ set) not necessarily compact problem Motivating problem Link with previous works Novelty and necessary conditions Proof Conclusion and Perspective 2Ross, I. M., Karpenko M., and Proulx J. R. "A Lebesgue-Stieltjes Framework For Optimal Control and Allocation." American Control Conference (ACC) 2015. Nathalie T. Khalil Average Cost Minimization Problems 6/17
Example from aerospace engineering: Spacecraft 2 Average cost if δ ( 0 ) = ω ∈ Ω (‘uncertainty’ set) not necessarily compact problem Motivating problem Link with previous works Novelty and necessary Optimal control problem with average cost conditions Proof Satisfactory results ✧ Conclusion and Perspective 2Ross, I. M., Karpenko M., and Proulx J. R. "A Lebesgue-Stieltjes Framework For Optimal Control and Allocation." American Control Conference (ACC) 2015. Nathalie T. Khalil Average Cost Minimization Problems 6/17
Some literature on average control... Zuazua, Average control, Automatica 50 (12), 2014 Average cost problem Motivating Agrachev, Baryshnikov, and Sarychev, Ensemble problem controllability by Lie algebraic methods, ESAIM: Link with previous works Control, Optimisation and Calculus of Variations 22 Novelty and necessary (4), 2016 conditions Proof Caillau, Cerf, Sassi, Trélat, and Zidani, Solving Conclusion and Perspective chance-constrained optimal control problems in aerospace engineering via Kernel Density Estimation, preprint , 2016 Nathalie T. Khalil Average Cost Minimization Problems 7/17
Recommend
More recommend
