Nonconvex Optimal Control Problems Research Articles

High-Order Control Lyapunov–Barrier Functions for Real-Time Optimal Control of Constrained Non-Affine Systems

This paper presents a synthesis of higher-order control Lyapunov functions (HOCLFs) and higher-order control barrier functions (HOCBFs) capable of controlling nonlinear dynamic systems while maintaining safety. Building on previous Lyapunov and barrier formulations, we first investigate the feasibility of the Lyapunov and barrier function approach in controlling a non-affine dynamic system under certain convexity conditions. Then we propose an HOCLF form that ensures convergence of non-convex dynamics with convex control inputs to target states. We combine the HOCLF with the HOCBF to ensure forward invariance of admissible sets and guarantee safety. This online non-convex optimal control problem is then formulated as a convex Quadratic Program (QP) that can be efficiently solved on board for real-time applications. Lastly, we determine the HOCLBF coefficients using a heuristic approach where the parameters are tuned and automatically decided to ensure the feasibility of the QPs, an inherent major limitation of high-order CBFs. The efficacy of the suggested algorithm is demonstrated on the real-time six-degree-of-freedom powered descent optimal control problem, where simulation results were run efficiently on a standard laptop.

Optimal switching of vaccination for an infectious disease model

We study a switched SIHR (Susceptible Infected Hospitalized Recovered) model for infectious diseases. The aim is to find the most effective switching signal to manage the disease's effects as effectively as maintaining a continuous vaccination program. A nonlinear, non-convex optimal control problem of the switched system is converted into a linear and convex optimal control problem by applying the theory of moments and semi-definite programming. Finally, some numerical simulations are performed to compare continuous vaccination programs and optimal switching of vaccination control. For more information see https://ejde.math.txstate.edu/Volumes/2024/59/abstr.html

On the convergence of gradient projection methods for non-convex optimal control problems with affine system

On the convergence of gradient projection methods for non-convex optimal control problems with affine system

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

Convergence of solutions to nonlinear nonconvex optimal control problems

In this paper, we consider nonlinear nonconvex optimal control problems and study convergence conditions of their solutions. To be more precise, we first introduce a generalized boundedness condition and discuss its relations with some typical existing ones in the literature. Next, combining this condition with the Gronwall Lemma, we investigate the boundedness property of solutions to state equations and the compactness of feasible sets of the reference problems. Then, based on these obtained results, convergence conditions in the sense of Painlevé-Kuratowski for such problems are formulated. Finally, at the end of the paper, applications to two practical situations, problems of fuel-optimal frictionless horizontal motion of a mass point and glucose models, are also presented.

Fast Homotopy for Spacecraft Rendezvous Trajectory Optimization with Discrete Logic

This paper presents a computationally efficient optimization algorithm for solving nonconvex optimal control problems that involve discrete logic constraints. Traditional solution methods require binary variables and mixed-integer programming (MIP), which is prohibitively slow and computationally expensive. This paper proposes a faster and computationally cheaper algorithm that can produce locally optimal solutions in seconds. This is achieved by blending sequential convex programming and numerical continuation into a single iterative solution process. The algorithm approximates discrete logic constraints with smooth functions and uses a homotopy parameter to control the accuracy of this approximation. The homotopy parameter is updated such that, by the time the algorithm converges, the smooth approximations enforce the exact discrete logic. The effectiveness of this approach is numerically demonstrated for a realistic rendezvous scenario inspired by the Apollo Transposition and Docking maneuver. In less than 15 s of cumulative solver time, the algorithm finds a fuel-minimizing trajectory that obeys the following discrete logic constraints: thruster minimum impulse-bit, range-triggered approach cone, and range- triggered plume impingement. The optimized trajectory uses significantly less fuel than reported NASA design targets.

Optimal control of the treatment and the vaccination in an epidemic switched system using polynomial approach

We study the problem of optimal switching on a treatment-vaccination model with four switches. These switches provide a more realistic approach when dealing with the control of an epidemic. We have used a relatively new approach to investigate optimal switching. The main idea behind this approach is to convert the switched system, a nonlinear, nonconvex optimal control problem, into an equivalent optimal control problem with a linear and convex structure. This allows us to obtain an equivalent convex formulation better suited to being solved by high-performance numerical computing. Toward the end, we perform some numerical simulations and draw the conclusion that the presence of treatment is essential, whereas vaccination control can be used at intervals with progressively larger gaps between the intervals.

Trajectory planning for satellite cluster reconfigurations with sequential convex programming method

This paper presents trajectory planning algorithms for large-scale satellite clusters reconfiguration based on sequential convex programming, and these algorithms consider fuel consumption and collision avoidance. Firstly, the trajectory planning problem is formulated as a nonconvex optimal control problem with nonlinear dynamics and nonconvex path constraints. Secondly, the original nonlinear continuous optimal control problem is transformed into a discrete convex optimization subproblem through linearization and discretization. Collision avoidance strategy between discrete points and obstacle avoidance constraints are considered in the convex subproblem to ensure that the trajectories are collision-free. Thirdly, coupled and decoupled sequential convex programming methods are proposed for rapidly generating collision-free and fuel-efficient trajectories. Finally, comparative numerical simulations are presented to demonstrate that as the number of satellites increases, 94 to 99 percent performance improvement over the pseudo-spectral method is achieved in terms of computational efficiency.

Spherically Constrained Relative Motion Trajectories in Low Earth Orbit

This paper considers a satellite inspection mission where the deputy satellite is required to remain a fixed distance away from the chief satellite. This constraint on the deputy’s state is modeled as the surface of a sphere. Finding minimum energy solutions to travel from one point to another point on the sphere requires solving a nonconvex optimal control problem. Various novel suboptimal algorithms are proposed that can guarantee a feasible solution. These algorithms are benchmarked by optimal values obtained via a global continuation method, and their computational performance is analyzed in a statistical setting.

Invariant ideals and their applications to the turnpike theory

AbstractIn this paper, the turnpike property is established for a nonconvex optimal control problem in discrete time. The functional is defined by the notion of the ideal convergence and can be considered as an analogue of the terminal functional defined over infinite-time horizon. The turnpike property states that every optimal solution converges to some unique optimal stationary point in the sense of ideal convergence if the ideal is invariant under translations. This kind of convergence generalizes, for example, statistical convergence and convergence with respect to logarithmic density zero sets.

Relaxation in nonconvex optimal control problems described by evolution Riemann-Liouville fractional differential inclusions

<p style='text-indent:20px;'>In this paper, we are concerned with the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a Riemann-Liouville fractional differential control system with mixed nonconvex feedback constraints on the control. At First, the existence results for Riemann-Liouville fractional semilinear differential control systems are discussed by using Schauder's fixed point theorem. Under some reasonable assumptions, we prove that the relaxation problem has an optimal solution, and that for each optimal solution there has a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously. Finally, we give an example to illustrate our main results.</p>

Real-Time Sequential Conic Optimization for Multi-Phase Rocket Landing Guidance

We introduce a multi-phase rocket landing guidance framework that can handle nonlinear dynamics and does not mandate any additional mixed-integer or nonconvex constraints to handle discrete temporal events/switching. To achieve this, we first introduce sequential conic optimization (seco), a new paradigm for solving nonconvex optimal control problems that is entirely devoid of matrix factorizations and inversions. This framework combines sequential convex programming (SCP) and first-order conic optimization and can solve unified multi-phase trajectory optimization problems in real-time. The novel features of this framework are: (1) time-interval dilation, which enables multi-phase trajectory optimization with free-transition-time; (2) single-crossing compound state-triggered constraints, which are entirely convex if the trigger and constraint conditions are convex; (3) virtual state, which is a new approach to handling artificial infeasibility in SCP methods that preserves the shapes of the constraint sets; and, (4) the use of the proportional-integral projected gradient method (pipg), a high-performance first-order conic optimization solver, in tandem with the penalized trust region (ptr) SCP algorithm. We demonstrate the efficacy and real-time capability of seco by solving a relevant multi-phase rocket landing guidance problem with nonlinear dynamics and convex constraints only, and observe that our solver is 2.7 times faster than a state-of-the-art convex optimization solver.

Path Planning for Concentric Tube Robots: A Toolchain with Application to Stereotactic Neurosurgery

We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear and highly nonconvex optimal control problem is solved by introducing a homotopy on the obstacle positions where at one extreme of the parameter the obstacles are removed from the operating space, and at the other extreme they are located at their intended positions. We present a detailed example (with more than a thousand obstacles) from stereotactic neurosurgery with real-world data obtained from labelled MRI scans.

Fully Distributed Model Predictive Control of Connected Automated Vehicles in Intersections: Theory and Vehicle Experiments

We propose a fully distributed control system architecture, amenable to in-vehicle implementation, that aims to safely coordinate connected and automated vehicles (CAVs) at road intersections. For control purposes, we build upon a fully distributed model predictive control approach, in which the agents solve a nonconvex optimal control problem (OCP) locally and synchronously, and exchange their optimized trajectories via vehicle-to-vehicle (V2V) communication. To accommodate a fast solution of the nonconvex OCPs, we apply the penalty convex-concave procedure which solves a convexified version of the original OCP. For experimental evaluation, we complement the predictive controller with a localization layer, being in charge of self-localization, and an estimator, which determines joint collision points with other agents. Experimental tests reveal the efficacy of the proposed control system architecture.

Convergence-Guaranteed Trajectory Planning for a Class of Nonlinear Systems With Nonconvex State Constraints

In this article, we study the problem of trajectory planning for a class of nonlinear systems with convex state/control constraints and nonconvex state constraints with concave constraint functions. This corresponds to a challenging nonconvex optimal control problem. We present how to convexify the nonlinear dynamics without any approximation via a combination of variable redefinition and relaxation. We then prove that the relaxation is exact by designing an appropriate objective function. This exact relaxation result enables us to further convexify the nonconvex state constraints simply by linearization. As a result, an algorithm is designed to iteratively solve the obtained convex optimization problems until convergence to get a solution of the original problem. A unique feature of the proposed approach is that the algorithm is proved to converge and it does not rely on any trust-region constraint. High performance of the algorithm is demonstrated by its application to trajectory planning of UAVs and autonomous cars with obstacle avoidance requirements.

Efficiency-aware nonlinear model-predictive control with real-time iteration scheme for wave energy converters

ABSTRACT Several solutions have been proposed in the literature to maximise the harvested ocean energy, but only a few consider the overall efficiency of the power take-off system. The fundamental problem of incorporating the power take-off system efficiency is that it leads to a nonlinear and non-convex optimal control problem. The main disadvantage of the available solutions is that none solve the optimal control problem in real-time. This paper presents a nonlinear model predictive control (NMPC) approach based on the real-time iteration (RTI) scheme to incorporate the power take-off system's efficiency when solving the optimal control problem at each time step in a control law aimed at maximising the energy extracted. The second contribution of this paper is the derivation of a condensing algorithm for ‘output-only’ cost functions required to improve computational efficiency. Finally, the RTI-NMPC approach is tested using a scaled model of the Wavestar design, demonstrating the benefit of this new control formulation.

Penetration trajectory optimization for the hypersonic gliding vehicle encountering two interceptors

The penetration trajectory optimization problem for the hypersonic gliding vehicle (HGV) encountering two interceptors is investigated. The HGV penetration trajectory optimization problem considering the terminal target area is formulated as a nonconvex optimal control problem. The nonconvex optimal control problem is transformed into a second-order cone programming (SOCP) problem, which can be solved by state-of-the-art interior-point methods. In addition, a penetration strategy that only requires the initial line-of-sight (LOS) angle information of the interceptors is proposed. The convergent trajectory obtained by the proposed method allows the HGV to evade two interceptors and reach the target area successfully. Furthermore, a successive SOCP method with a variable trust region is presented, which is critical to balance the trade-off between time consumption and optimality. Finally, the effectiveness and performance of the proposed method are verified by numerical simulations.

Decentralized Multi-UAV Path Planning Based on Two-Layer Coordinative Framework for Formation Rendezvous

Unmanned aerial vehicle (UAV) formation rendezvous path planning problem is one of the important research topics in multiple UAV (multi-UAV) coordinated path planning. Aiming at solving low computational efficiency and poor scalability of the traditional multi-UAV path planning method, the decentralized multi-UAV path planning method suitable for obstacle environments is proposed. Firstly, the UAV rendezvous path planning problem with constraints such as the kinematics of UAVs and collision-free constraints is modeled as a non-convex optimal-control problem. To minimize formation rendezvous time and energy consumption, a two-layer coordinative framework is developed to solve this problem. In the coordination layer, relying only on the information of neighboring UAVs, each UAV in the decentralized communication graph negotiates the desired flight time using a consensus protocol to achieve coordination among UAVs. In the planning layer, the initial non-convex formation rendezvous path planning problem is decoupled into several sub-problems, which can be solved in parallel by path planners distributed on each UAV using sequential convex programming. Finally, numerical simulations are carried out to verify the effectiveness and scalability of the proposed method. The results show that this decentralized multi-UAV path planning method can handle the minimum-time rendezvous path planning problem and optimize the energy consumption in flight, and the computing time does not increase significantly with the enlargement of the UAV swarm. This decentralized framework scales well with the number of UAVs and can be applied for future urban flight and supplies delivery tasks.

Space splitting convexification: a local solution method for nonconvex optimal control problems

Many optimal control tasks for engineering problems require the solution of nonconvex optimisation problems, which are rather hard to solve. This article presents a novel iterative optimisation procedure for the fast solution of such problems using successive convexification. The approach considers two types of nonconvexities. Firstly, equality constraints with possibly multiple univariate nonlinearities, which can arise for nonlinear system dynamics. Secondly, nonconvex sets comprised of convex subsets, which may occur for semi-active actuators. By introducing additional decision variables and constraints, the decision variable space is decomposed into affine segments yielding a convex subproblem which is solved efficiently in an iterative manner. Under certain conditions, the algorithm converges to a local optimum of a scalable, piecewise linear approximation of the original problem. Furthermore, the algorithm tolerates infeasible initial guesses. Using a single-mass oscillator application, the procedure is compared with a nonlinear programming algorithm, and the sensitivity regarding initial guesses is analysed.

Joint Mobility, Communication and Computation Optimization for UAVs in Air-Ground Cooperative Networks

Unmanned aerial vehicles (UAVs) play a significant role in various 5G or Beyond-5G (B5G)-enabled Internet-of-Things (IoT) applications. However, the UAV performance in an air-ground cooperative network is significantly affected by its mobility and air-to-ground (A2G) communication and computation behaviors. In this paper, a UAV-oriented computation offloading system is investigated, where the UAV desires to complete its onboard computation demands with the assistance of a ground edge-computing infrastructure, i.e., a road-side unit (RSU). The objective is to maximize the energy efficiency of the UAV. Specifically, a non-convex constrained optimal control problem is formulated to optimize the overall energy efficiency of UAV by jointly considering the coupled effects of UAV's longitudinal mobility, A2G communication, and computation dynamics. To address the coupled complexity and non-convexity of the original problem, a primal decomposition approach is developed to transform the problem into a convex subproblem and a primary problem, and then a closed-form optimal transmission power control is derived by solving the subproblem, which is dependent on mobility information. By embedding the closed-form optimal power control into the primary problem, a gradient projection-based iterative algorithm is proposed to obtain a joint optimal solution for both the longitudinal acceleration control and the power control, the feasibility and convergence of which is also theoretically proven. Extensive simulations have been conducted to validate the effectiveness of the proposed method in terms of constraint satisfaction and convergence speed, and comparative results also demonstrate that it can outperform other benchmark methods in terms of global energy efficiency.