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1. Method for solving chance constrained optimal control problems using biased kernel density estimators

   https://doi.org/10.1002/oca.2675  

   Keil, Rachel_E ; T Miller, Alexander ; Kumar, Mrinal ; Rao, Anil_V ( October 2020 , Optimal Control Applications and Methods)

   A method is developed to numerically solve chance constrained optimal control problems. The chance constraints are reformulated as nonlinear constraints that retain the probability properties of the original constraint. The reformulation transforms the chance constrained optimal control problem into a deterministic optimal control problem that can be solved numerically. The new method developed in this paper approximates the chance constraints using Markov Chain Monte Carlo sampling and kernel density estimators whose kernels have integral functions that bound the indicator function. The nonlinear constraints resulting from the application of kernel density estimators are designed with bounds that do not violate the bounds of the original chance constraint. The method is tested on a nontrivial chance constrained modification of a soft lunar landing optimal control problem and the results are compared with results obtained using a conservative deterministic formulation of the optimal control problem. Additionally, the method is tested on a complex chance constrained unmanned aerial vehicle problem. The results show that this new method can be used to reliably solve chance constrained optimal control problems.

    

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2. Structure Detection Method for Solving State-Variable Inequality Path Constrained Optimal Control Problems

   Byczkowski, Cale A. ; Rao, Anil V. ( January 2023 , American Astronautical Society)

   A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

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3. A Convex Optimization Approach to Chance-Constrained Linear Stochastic Drift Counteraction Optimal Control

   https://doi.org/10.1109/CDC45484.2021.9683303  

   Tang, Sunbochen ; Li, Nan ; Kolmanovsky, Ilya ; Zidek, Robert ( December 2021 , Proceedings of 2021 60th IEEE Conference on Decision and Control (CDC))

   In this paper, we propose a convex optimization approach to chance-constrained drift counteraction optimal control (DCOC) problems for linear systems with additive stochastic disturbances. Chance-constrained DCOC aims to compute an optimal control law to maximize the time duration before the probability of violating a prescribed set of constraints can no longer be maintained to be below a specified risk level. While conventional approaches to this problem involve solving a mixed-integer programming problem, we show that an optimal solution to the problem can also be found by solving a convex second-order cone programming problem without integer variables. We illustrate the application of chance-constrained DCOC to an automotive adaptive cruise control example. 

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4. Data-Driven Chance Constrained Control using Kernel Distribution Embeddings

   Thorpe, Adam ; and Oishi, Meeko ( May 2021 , Proceedings of The 4th Annual Learning for Dynamics and Control Conference)

   We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finite subset of stochastic open-loop control trajectories, we relax the original problem to a linear program over the control parameters that can be efficiently solved using standard convex optimization techniques. We demonstrate our proposed approach in simulation on a system with nonlinear non-Markovian dynamics navigating in a cluttered environment. 

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5. Efficient Scenario Generation for Heavy-Tailed Chance Constrained Optimization

   https://doi.org/10.1287/stsy.2021.0021  

   Blanchet, Jose ; Zhang, Fan ; Zwart, Bert ( March 2024 , Stochastic Systems)

   We consider a generic class of chance-constrained optimization problems with heavy-tailed (i.e., power-law type) risk factors. As the most popular generic method for solving chance constrained optimization, the scenario approach generates sampled optimization problem as a precise approximation with provable reliability, but the computational complexity becomes intractable when the risk tolerance parameter is small. To reduce the complexity, we sample the risk factors from a conditional distribution given that the risk factors are in an analytically tractable event that encompasses all the plausible events of constraints violation. Our approximation is proven to have optimal value within a constant factor to the optimal value of the original chance constraint problem with high probability, uniformly in the risk tolerance parameter. To the best of our knowledge, our result is the first uniform performance guarantee of this type. We additionally demonstrate the efficiency of our algorithm in the context of solvency in portfolio optimization and insurance networks.

   Funding: The research of B. Zwart is supported by the NWO (Dutch Research Council) [Grant 639.033.413]. The research of J. Blanchet is supported by the Air Force Office of Scientific Research [Award FA9550-20-1-0397], the National Science Foundation [Grants 1820942, 1838576, 1915967, and 2118199], Defense Advanced Research Projects Agency [Award N660011824028], and China Merchants Bank.

    

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