A Primal-Dual Method for Optimal Control and Trajectory Generation in High-Dimensional Systems

Presented is a method for efficient computation of the Hamilton-Jacobi (HJ) equation for time-optimal control problems using the generalized Hopf formula. Typically, numerical methods to solve the HJ equation rely on a discrete grid of the solution space and exhibit exponential scaling with dimension. The generalized Hopf formula avoids the use of grids and numerical gradients by formulating an unconstrained convex optimization problem. The solution at each point is completely independent, and allows a massively parallel implementation if solutions at multiple points are desired. This work presents a primal-dual method for efficient numeric solution and presents how the resulting optimal trajectory can be generated directly from the solution of the Hopf formula, without further optimization. Examples presented have execution times on the order of milliseconds and experiments show computation scales approximately polynomial in dimension with very small high-order coefficients.Comment: Updated references and funding sources. To appear in the proceedings of the 2018 IEEE Conference on Control Technology and Application

A Parallel Decomposition Scheme for Solving Long-Horizon Optimal Control Problems

We present a temporal decomposition scheme for solving long-horizon optimal control problems. In the proposed scheme, the time domain is decomposed into a set of subdomains with partially overlapping regions. Subproblems associated with the subdomains are solved in parallel to obtain local primal-dual trajectories that are assembled to obtain the global trajectories. We provide a sufficient condition that guarantees convergence of the proposed scheme. This condition states that the effect of perturbations on the boundary conditions (i.e., initial state and terminal dual/adjoint variable) should decay asymptotically as one moves away from the boundaries. This condition also reveals that the scheme converges if the size of the overlap is sufficiently large and that the convergence rate improves with the size of the overlap. We prove that linear quadratic problems satisfy the asymptotic decay condition, and we discuss numerical strategies to determine if the condition holds in more general cases. We draw upon a non-convex optimal control problem to illustrate the performance of the proposed scheme

Capacity of UAV-Enabled Multicast Channel: Joint Trajectory Design and Power Allocation

This paper studies an unmanned aerial vehicle (UAV)-enabled multicast channel, in which a UAV serves as a mobile transmitter to deliver common information to a set of $K$ ground users. We aim to characterize the capacity of this channel over a finite UAV communication period, subject to its maximum speed constraint and an average transmit power constraint. To achieve the capacity, the UAV should use a sufficiently long code that spans over its whole communication period. Accordingly, the multicast channel capacity is achieved via maximizing the minimum achievable time-averaged rates of the $K$ users, by jointly optimizing the UAV's trajectory and transmit power allocation over time. However, this problem is non-convex and difficult to be solved optimally. To tackle this problem, we first consider a relaxed problem by ignoring the maximum UAV speed constraint, and obtain its globally optimal solution via the Lagrange dual method. The optimal solution reveals that the UAV should hover above a finite number of ground locations, with the optimal hovering duration and transmit power at each location. Next, based on such a multi-location-hovering solution, we present a successive hover-and-fly trajectory design and obtain the corresponding optimal transmit power allocation for the case with the maximum UAV speed constraint. Numerical results show that our proposed joint UAV trajectory and transmit power optimization significantly improves the achievable rate of the UAV-enabled multicast channel, and also greatly outperforms the conventional multicast channel with a fixed-location transmitter.Comment: To appear in the IEEE International Conference on Communications (ICC), 201

Double smoothing technique for infinite-dimensional optimization problems with applications to optimal control

In this paper, we propose an efficient technique for solving some infinite-dimensional problems over the sets of functions of time. In our problem, besides the convex point-wise constraints on state variables, we have convex coupling constraints with finite-dimensional image. Hence, we can formulate a finite-dimensional dual problem, which can be solved by efficient gradient methods. We show that it is possible to reconstruct an approximate primal solution. In order to accelerate our schemes, we apply double-smoothing technique. As a result, our method has complexity O (1/[epsilon] ln 1/[epsilon]) gradient iterations, where [epsilon] is the desired accuracy of the solution of the primal-dual problem. Our approach covers, in particular, the optimal control problems with trajectory governed by a system of ordinary differential equations. The additional requirement could be that the trajectory crosses in certain moments of time some convex sets.convex optimization, optimal control, fast gradient methods, complexity bounds, smoothing technique

Intermittent predictive control of an inverted pendulum

Intermittent predictive pole-placement control is successfully applied to the constrained-state control of a prestabilised experimental inverted pendulum

Fast Second-order Cone Programming for Safe Mission Planning

This paper considers the problem of safe mission planning of dynamic systems operating under uncertain environments. Much of the prior work on achieving robust and safe control requires solving second-order cone programs (SOCP). Unfortunately, existing general purpose SOCP methods are often infeasible for real-time robotic tasks due to high memory and computational requirements imposed by existing general optimization methods. The key contribution of this paper is a fast and memory-efficient algorithm for SOCP that would enable robust and safe mission planning on-board robots in real-time. Our algorithm does not have any external dependency, can efficiently utilize warm start provided in safe planning settings, and in fact leads to significant speed up over standard optimization packages (like SDPT3) for even standard SOCP problems. For example, for a standard quadrotor problem, our method leads to speedup of 1000x over SDPT3 without any deterioration in the solution quality. Our method is based on two insights: a) SOCPs can be interpreted as optimizing a function over a polytope with infinite sides, b) a linear function can be efficiently optimized over this polytope. We combine the above observations with a novel utilization of Wolfe's algorithm to obtain an efficient optimization method that can be easily implemented on small embedded devices. In addition to the above mentioned algorithm, we also design a two-level sensing method based on Gaussian Process for complex obstacles with non-linear boundaries such as a cylinder