A Simple and Efficient Algorithm for Nonlinear Model Predictive Control

We present PANOC, a new algorithm for solving optimal control problems arising in nonlinear model predictive control (NMPC). A usual approach to this type of problems is sequential quadratic programming (SQP), which requires the solution of a quadratic program at every iteration and, consequently, inner iterative procedures. As a result, when the problem is ill-conditioned or the prediction horizon is large, each outer iteration becomes computationally very expensive. We propose a line-search algorithm that combines forward-backward iterations (FB) and Newton-type steps over the recently introduced forward-backward envelope (FBE), a continuous, real-valued, exact merit function for the original problem. The curvature information of Newton-type methods enables asymptotic superlinear rates under mild assumptions at the limit point, and the proposed algorithm is based on very simple operations: access to first-order information of the cost and dynamics and low-cost direct linear algebra. No inner iterative procedure nor Hessian evaluation is required, making our approach computationally simpler than SQP methods. The low-memory requirements and simple implementation make our method particularly suited for embedded NMPC applications

Direct Collocation for Numerical Optimal Control of Second-Order ODE

Mechanical systems are usually modeled by second-order Ordinary Differential Equations (ODE) which take the form $\ddot{q} = f(t, q, \dot{q})$. While simulation methods tailored to these equations have been studied, using them in direct optimal control methods is rare. Indeed, the standard approach is to perform a state augmentation, adding the velocities to the state. The main drawback of this approach is that the number of decision variables is doubled, which could harm the performance of the resulting optimization problem. In this paper, we present an approach tailored to second-order ODE. We compare it with the standard one, both on theoretical aspects and in a numerical example. Notably, we show that the tailored formulation is likely to improve the performance of a direct collocation method, for solving optimal control problems with second-order ODE of the more restrictive form $\ddot{q} = f(t, q)$.Comment: Submitted to IEEE European Control Conference 2023 (ECC23). Contains 7 pages including 4 figure

A radial basis function method for solving optimal control problems.

This work presents two direct methods based on the radial basis function (RBF) interpolation and arbitrary discretization for solving continuous-time optimal control problems: RBF Collocation Method and RBF-Galerkin Method. Both methods take advantage of choosing any global RBF as the interpolant function and any arbitrary points (meshless or on a mesh) as the discretization points. The first approach is called the RBF collocation method, in which states and controls are parameterized using a global RBF, and constraints are satisfied at arbitrary discrete nodes (collocation points) to convert the continuous-time optimal control problem to a nonlinear programming (NLP) problem. The resulted NLP is quite sparse and can be efficiently solved by well-developed sparse solvers. The second proposed method is a hybrid approach combining RBF interpolation with Galerkin error projection for solving optimal control problems. The proposed solution, called the RBF-Galerkin method, applies a Galerkin projection to the residuals of the optimal control problem that make them orthogonal to every member of the RBF basis functions. Also, RBF-Galerkin costate mapping theorem will be developed describing an exact equivalency between the Karush–Kuhn–Tucker (KKT) conditions of the NLP problem resulted from the RBF-Galerkin method and discretized form of the first-order necessary conditions of the optimal control problem, if a set of conditions holds. Several examples are provided to verify the feasibility and viability of the RBF method and the RBF-Galerkin approach as means of finding accurate solutions to general optimal control problems. Then, the RBF-Galerkin method is applied to a very important drug dosing application: anemia management in chronic kidney disease. A multiple receding horizon control (MRHC) approach based on the RBF-Galerkin method is developed for individualized dosing of an anemia drug for hemodialysis patients. Simulation results are compared with a population-oriented clinical protocol as well as an individual-based control method for anemia management to investigate the efficacy of the proposed method

A direct memetic approach to the solution of multi-objective optimal control problems

This paper proposes a memetic direct transcription algorithm to solve Multi-Objective Optimal Control Problems (MOOCP). The MOOCP is first transcribed into a Non-linear Programming Problem (NLP) with Direct Finite Elements in Time (DFET) and then solved with a particular formulation of the Multi Agent Collaborative Search (MACS) framework. Multi Agent Collaborative Search is a memetic algorithm in which a population of agents combines local search heuristics, exploring the neighbourhood of each agent, with social actions exchanging information among agents. A collection of all Pareto optimal solutions is maintained in an archive that evolves towards the Pareto set. In the approach proposed in this paper, individualistic actions run a local search, from random points within the neighbourhood of each agent, solving a normalised Pascoletti-Serafini scalarisation of the multi-objective NLP problem. Social actions, instead, solve a bi-level problem in which the lower level handles only the constraint equations while the upper level handles only the objective functions. The proposed approach is tested on the multi-objective extensions of two well-known optimal control problems: the Goddard Rocket problem, and the maximum energy orbit rise problem

Direct Policy Optimization using Deterministic Sampling and Collocation

We present an approach for approximately solving discrete-time stochastic optimal-control problems by combining direct trajectory optimization, deterministic sampling, and policy optimization. Our feedback motion-planning algorithm uses a quasi-Newton method to simultaneously optimize a reference trajectory, a set of deterministically chosen sample trajectories, and a parameterized policy. We demonstrate that this approach exactly recovers LQR policies in the case of linear dynamics, quadratic objective, and Gaussian disturbances. We also demonstrate the algorithm on several nonlinear, underactuated robotic systems to highlight its performance and ability to handle control limits, safely avoid obstacles, and generate robust plans in the presence of unmodeled dynamics.Comment: revisions for RA-L 202

Applications of the homotopy analysis method to optimal control problems

Traditionally, trajectory optimization for aerospace applications has been performed using either direct or indirect methods. Indirect methods produce highly accurate solutions but suer from a small convergence region, requiring initial guesses close to the optimal solution. In past two decades, a new series of analytical approximation methods have been used for solving systems of dierential equations and boundary value problems. The Homotopy Analysis Method (HAM) is one such method which has been used to solve typical boundary value problems in nance, science, and engineering. In this investigation, a methodology is created to solve indirect trajectory optimization problems using the Homotopy Analysis Method. Use of the auxiliary convergence control parameter to widen the convergence region and increase the rate of convergence have been demonstrated on multiple optimal control problems. The guaranteed convergence and the ease of selecting the initial guess for trajectory optimization problems makes the method of high signicance. It has been demonstrated that initial guesses for the optimal control problem can be generated using a simple approach based on only the initial boundary conditions. The approach has been demonstrated on the Zermelo\u27s problem and two cases of a 2D ascent problem. It has been established that for free nal-time boundary value problems, nding the convergence region is much harder as compared to xed nal-time cases. To validate the approach, results are compared with those obtained using the MATLAB\u27s bvp4c function. A number of new challenges are discovered and listed during the process

Multiresolution strategies for the numerical solution of optimal control problems

Optimal control problems are often characterized by discontinuities or switchings in the control variables. One way of accurately capturing the irregularities in the solution is to use a high resolution (dense) uniform grid. This requires a large amount of computational resources both in terms of CPU time and memory. Hence, in order to accurately capture any irregularities in the solution using a few computational resources, one can refine the mesh locally in the region close to an irregularity instead of refining the mesh uniformly over the whole domain. Therefore, a novel multiresolution scheme for data compression has been designed which is shown to outperform similar data compression schemes. Specifically, we have shown that the proposed approach results in fewer grid points in the grid compared to a common multiresolution data compression scheme. The validity of the proposed mesh refinement algorithm has been verified by solving several challenging initial-boundary value problems for evolution equations in 1D. The examples have demonstrated the stability and robustness of the proposed algorithm. Next, a direct multiresolution-based approach for solving trajectory optimization problems is developed. The original optimal control problem is transcribed into a nonlinear programming (NLP) problem that is solved using standard NLP codes. The novelty of the proposed approach hinges on the automatic calculation of a suitable, nonuniform grid over which the NLP problem is solved, which tends to increase numerical efficiency and robustness. Control and/or state constraints are handled with ease, and without any additional computational complexity. The proposed algorithm is based on a simple and intuitive method to balance several conflicting objectives, such as accuracy of the solution, convergence, and speed of the computations. The benefits of the proposed algorithm over uniform grid implementations are demonstrated with the help of several nontrivial examples. Furthermore, two sequential multiresolution trajectory optimization algorithms for solving problems with moving targets and/or dynamically changing environments have been developed.Ph.D.Committee Chair: Tsiotras, Panagiotis; Committee Member: Calise, Anthony J.; Committee Member: Egerstedt, Magnus; Committee Member: Prasad, J. V. R.; Committee Member: Russell, Ryan P.; Committee Member: Zhou, Hao-Mi