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1. Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems

   https://doi.org/10.23919/ACC45564.2020.9147847  

   Caluya, Kenneth F.; Halder, Abhishek (July 2020, Proceedings of the 2020 American Control Conference)

   null (Ed.)

   In this paper, we study the feedback synthesis problem for steering the joint state density or ensemble subject to multi-input state feedback linearizable dynamics. This problem is of interest to many practical applications including that of dynamically shaping a robotic swarm. Our results here show that it is possible to exploit the structural nonlinearities to derive the feedback controllers steering the joint density from a prescribed shape to another while minimizing the expected control effort to do so. The developments herein build on our previous work, and extend the theory of the Schrödinger bridge problem subject to feedback linearizable dynamics. 

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2. Reflected Schrödinger Bridge: Density Control with Path Constraints

   https://doi.org/10.23919/ACC50511.2021.9482813  

   Caluya, Kenneth F.; Halder, Abhishek (January 2021, 2021 American Control Conference (ACC))

   How to steer a given joint state probability density function to another over finite horizon subject to a controlled stochastic dynamics with hard state (sample path) constraints? In applications, state constraints may encode safety requirements such as obstacle avoidance. In this paper, we perform the feedback synthesis for minimum control effort density steering (a.k.a. Schrödinger bridge) problem subject to state constraints. We extend the theory of Schrödinger bridges to account the reflecting boundary conditions for the sample paths, and provide a computational framework building on our previous work on proximal recursions, to solve the same. 

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3. Implicit Trajectory Planning for Feedback Linearizable Systems: A Time-varying Optimization Approach

   https://doi.org/10.23919/ACC45564.2020.9147997  

   Zheng, Tianqi; Simpson-Porco, John; Mallada, Enrique (July 2020, American Control Conference)

   We develop an optimization-based framework for joint real-time trajectory planning and feedback control of feedback-linearizable systems. To achieve this goal, we define a target trajectory as the optimal solution of a time-varying optimization problem. In general, however, such trajectory may not be feasible due to , e.g., nonholonomic constraints. To solve this problem, we design a control law that generates feasible trajectories that asymptotically converge to the target trajectory. More precisely, for systems that are (dynamic) full-state linearizable, the proposed control law implicitly transforms the nonlinear system into an optimization algorithm of sufficiently high order. We prove global exponential convergence to the target trajectory for both the optimization algorithm and the original system. We illustrate the effectiveness of our proposed method on multi-target or multi-agent tracking problems with constraints. 

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4. Wasserstein Proximal Algorithms for the Schr\"{o}dinger Bridge Problem: Density Control with Nonlinear Drift

   https://doi.org/10.1109/TAC.2021.3060704  

   Caluya, Kenneth; Halder, Abhishek (January 2021, IEEE Transactions on Automatic Control)

   null (Ed.)

   We study the Schr{\"o}dinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite time horizon, subject to a controlled stochastic differential evolution of the state vector. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the ``Schr\"{o}dinger factors". We show that if the drift is a gradient vector field, or is of mixed conservative-dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. We employ a proximal algorithm developed in our prior work to solve these IVPs and compute the Schr\"{o}dinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples. 

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5. Exploiting the experts: Learning to control unknown SISO feedback linearizable systems from expert demonstrations

   https://doi.org/10.1109/CDC45484.2021.9683193  

   Sultangazin, Alimzhan; Fraile, Lucas; Tabuada, Paulo (December 2021, 2021 60th IEEE Conference on Decision and Control (CDC))

   It was shown, in recent work by the authors, that it is possible to learn an asymptotically stabilizing controller from a small number of demonstrations performed by an expert on a feedback linearizable system. These results rely on knowledge of the plant dynamics to assemble the learned controller from the demonstrations. In this paper we show how to leverage recent results on data-driven control to dispense with the need to use the plant model. By bringing these two methodologies — learning from demonstrations and data-driven control — together, this paper provides a technique that enables the control of unknown nonlinear feedback linearizable systems solely based on a small number of expert demonstrations. 

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