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Register a new userA Neural Network Approach for High-Dimensional Optimal Control
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We propose a neural network approach for solving high-dimensional optimal control problems arising in real-time applications. Our approach yields controls in a feedback form and can therefore handle uncertainties such as perturbations to the systems state. We accomplish this by fusing the Pontryagin Maximum Principle (PMP) and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizing the value function with a neural network. We train our neural network model using the objective function of the control problem and penalty terms that enforce the HJB equations. Therefore, our training algorithm does not involve data generated by another algorithm. By training on a distribution of initial states, we ensure the controls optimality on a large portion of the state-space. Our grid-free approach scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate the effectiveness of our approach on several multi-agent collision-avoidance problems in up to 150 dimensions. Furthermore, we empirically observe that the number of parameters in our approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.
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We propose a neural network approach for solving high-dimensional optimal control problems. In particular, we focus on multi-agent control problems with obstacle and collision avoidance. These problems immediately become high-dimensional, even for moderate phase-space dimensions per agent. Our approach fuses the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman (HJB) approaches and parameterizes the value function with a neural network. Our approach yields controls in a feedback form for quick calculation and robustness to moderate disturbances to the system. We train our model using the objective function and optimality conditions of the control problem. Therefore, our training algorithm neither involves a data generation phase nor solutions from another algorithm. Our model uses empirically effective HJB penalizers for efficient training. By training on a distribution of initial states, we ensure the controls optimality is achieved on a large portion of the state-space. Our approach is grid-free and scales efficiently to dimensions where grids become impractical or infeasible. We demonstrate our approachs effectiveness on a 150-dimensional multi-agent problem with obstacles.
Control of complex systems involves both system identification and controller design. Deep neural networks have proven to be successful in many identification tasks, however, from model-based control perspective, these networks are difficult to work with because they are typically nonlinear and nonconvex. Therefore many systems are still identified and controlled based on simple linear models despite their poor representation capability. In this paper we bridge the gap between model accuracy and control tractability faced by neural networks, by explicitly constructing networks that are convex with respect to their inputs. We show that these input convex networks can be trained to obtain accurate models of complex physical systems. In particular, we design input convex recurrent neural networks to capture temporal behavior of dynamical systems. Then optimal controllers can be achieved via solving a convex model predictive control problem. Experiment results demonstrate the good potential of the proposed input convex neural network based approach in a variety of control applications. In particular we show that in the MuJoCo locomotion tasks, we could achieve over 10% higher performance using 5* less time compared with state-of-the-art model-based reinforcement learning method; and in the building HVAC control example, our method achieved up to 20% energy reduction compared with classic linear models.
Mean field optimal control problems are a class of optimization problems that arise from optimal control when applied to the many body setting. In the noisy case one has a set of controllable stochastic processes and a cost function that is a functional of their trajectories. The goal of the optimization is to minimize this cost over the control variables. Here we consider the case in which we have $N$ stochastic processes, or agents, with the associated control variables, which interact in a disordered way so that the resulting cost function is random. The goal is to find the average minimal cost for $Nto infty$, when a typical realization of the quenched random interactions is considered. We introduce a simple model and show how to perform a dimensional reduction from the infinite dimensional case to a set of one dimensional stochastic partial differential equations of the Hamilton-Jacobi-Bellman and Fokker-Planck type. The statistical properties of the corresponding stochastic terms must be computed self-consistently, as we show explicitly.
Recent research has shown that supervised learning can be an effective tool for designing optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of these neural network (NN) controllers is still not well understood. In this paper we use numerical simulations to demonstrate that typical test accuracy metrics do not effectively capture the ability of an NN controller to stabilize a system. In particular, some NNs with high test accuracy can fail to stabilize the dynamics. To address this we propose two NN architectures which locally approximate a linear quadratic regulator (LQR). Numerical simulations confirm our intuition that the proposed architectures reliably produce stabilizing feedback controllers without sacrificing performance. In addition, we introduce a preliminary theoretical result describing some stability properties of such NN-controlled systems.
This paper is concerned with the distributed linear quadratic optimal control problem. In particular, we consider a suboptimal version of the distributed optimal control problem for undirected multi-agent networks. Given a multi-agent system with identical agent dynamics and an associated global quadratic cost functional, our objective is to design suboptimal distributed control laws that guarantee the controlled network to reach consensus and the associated cost to be smaller than an a priori given upper bound. We first analyze the suboptimality for a given linear system and then apply the results to linear multiagent systems. Two design methods are then provided to compute such suboptimal distributed controllers, involving the solution of a single Riccati inequality of dimension equal to the dimension of the agent dynamics, and the smallest nonzero and the largest eigenvalue of the graph Laplacian. Furthermore, we relax the requirement of exact knowledge of the smallest nonzero and largest eigenvalue of the graph Laplacian by using only lower and upper bounds on these eigenvalues. Finally, a simulation example is provided to illustrate our design method.
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