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EXPONENTIAL DECAY IN THE SENSITIVITY ANALYSIS OF nonlinear dynamic programming

SIAM JOURNAL ON OPTIMIZATION 2020年第2期30卷 1527-1554页

作者： Na, Sen Anitescu, Mihai Univ Chicago Dept Stat Chicago IL 60637 USA Argonne Natl Lab Div Math & Comp Sci 9700 S Cass Ave Argonne IL 60439 USA

In this paper, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time k, x(k)*, and u(k)*, with respect to the reference signal at time i, d(i), will have exponential decay in terms of vertical bar k - i vertical bar with a decay rate rho independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al. can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.

关键词： sensitivity analysis nonlinear dynamic programming optimization

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A Fast Temporal Decomposition Procedure for Long- Horizon nonlinear dynamic programming

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MATHEMATICS OF OPERATIONS RESEARCH 2023年第2期49卷 653-1302, C2页

作者： Na, Sen Anitescu, Mihai Kolar, Mladen Univ Calif Berkeley Int Comp Sci Inst Berkeley CA 94720 USA Univ Calif Berkeley Dept Stat Berkeley CA 94720 USA Argonne Natl Lab Math & Comp Sci Div Lemont IL 60439 USA Univ Chicago Booth Sch Business Chicago IL 60637 USA

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method.

关键词： nonlinear dynamic programming temporal decomposition sequential quadratic programming augmented Lagrangian

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dynamic stability of two pitching foils using dynamic programming

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MODERN PHYSICS LETTERS B 2020年第3期34卷

作者： Wang, Fan Li, Jia-Jun Chao, Li-Ming Northwestern Polytech Univ Sch Management Xian 710072 Peoples R China Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710072 Peoples R China

In this paper, a theoretical study is conducted to investigate the propulsive performance of two flapping foils with the inline arrangement. At first, we consider the propulsive performance of two flapping foils when they are relative static to each other, that is, the horizontal gap between the foils is fixed. Then, we provide a model to consider the dynamic stability of the two-foil system when the relative movement between them has occurred, that is, the horizontal gap between the two foils can be changed depending on the propulsive force acting on the foil. This work provides helpful suggestions for managing the trajectories and energy cost of multiple bio-inspired robotics.

关键词： nonlinear dynamic programming mathematical modeling energy management

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Optimal energy-management of two cylinders under flow-induced motion using dynamic programming

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MODERN PHYSICS LETTERS B 2019年第34期33卷

作者： Wang, Fan Li, Jia-Jun Northwestern Polytech Univ Sch Management Xian 710072 Shaanxi Peoples R China

In this paper, we performed a work combining numerical and theoretical studies to investigate the optimal energy-management of two cylinders under flow-induced motion using dynamic programming. First, we considered the energy generation of two freely oscillated cylinders with different configurations at Reynolds number of 200. Then, we constructed a theoretical model to maximize the energy-harvesting capacity of two cylinders using dynamic programming. This work can provide helpful suggestions for designing the energy-harvesting apparatus.

关键词： nonlinear dynamic programming energy management flow-induced motion

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