Nonlinear Optimal Control Problems Research Articles

A Gauss–Newton-based decomposition algorithm for Nonlinear Mixed-Integer Optimal Control Problems

For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is a convex approximation of the original MINLP and relies on the Gauss–Newton approximation. We analyze the algorithm in terms of approximation properties and establish a first-order consistency result. Then, we investigate the proposed approach considering a numerical case study of Mixed-Integer Optimal Control (MIOC) of a renewable energy system. The investigation shows that the proposed formulation can yield an improved integer solution regarding the objective of the original MINLP compared with the established Combinatorial Integral Approximation (CIA) algorithm.

Application of generalized Lucas wavelet method for solving nonlinear fractal-fractional optimal control problems

Different types of fractional derivatives have recently been noticed by researchers and used in modeling phenomena due to their characteristics. Furthermore, fractional optimal control problems have been the focus of many researchers because they reflect the real nature of different models. Hence, this article considers a class of nonlinear fractal-fractional optimal control problems in the Atangana–Riemann–Liouville sense with the Mittag-Leffler non-singular kernel. In this study, a numerical method based on the generalized Lucas wavelets and the Ritz method is presented to obtain approximate solutions. Then, the generalized Lucas wavelets and an extra pseudo-operational matrix of the Atangana–Riemann–Liouville derivative are introduced. We demonstrate the advantage of the proposed method through three numerical examples.

A novel numerical method for solving optimal control problems using fourth-degree hat functions

This paper focuses on solving a class of nonlinear optimal control problems by constructing novel hat functions based on fourth-order polynomials, namely, fourth-degree hat functions (FDHFs). The FDHFs are used in this method to approximate the state equations and cost function. In fact, the FDHFs enable us to turn the optimal control problem under consideration into a nonlinear optimal control problem with unknown coefficients that is easy to solve by any numerical method. The main advantages of the proposed method are its simplicity, ease of application, low computational expense, and avoidance of numerical integration. Several examples have been discussed to demonstrate the efficacy and applicability of the suggested method.

Constrained nonlinear high-efficiency model predictive technics for test mass capture

This paper presents a nonlinear high-efficiency model predictive control (NHMPC) with constraints designed for the test mass (TM) capture phase of the drag-free satellite about gravitational wave observatory. To avoid collisions between test mass and satellite cavity, TM capture is the essential technology for drag-free satellite. Test mass is located inside an electrostatic suspension and locked by a clamp mechanism initially. The test masses are released with high initial offsets and velocities when the mechanism is retracted. The purpose of this phase is to guarantee the TM to be positioned at the cage center and attitude aligned with the local cage frame. Due to the low actuation authority of electrostatic suspension along with critical initial offsets and velocities, it is a challenging task to design the attitude and translation control schemes with simultaneous consideration of system performance and energy consumption. For the problem given above, the TM capture can be reformulated into a nonlinear quadratic optimal control problem with the state and input constraints. A nonlinear model predictive control (NMPC) structure is also designed to handle the noises by forming a closed loop. This control framework can realize optimality and robustness in a compromise. To improve the speed of solving high-dimensional nonlinear optimal control online for MPC, the indirect Chebyshev pseudospectral method with constraints is employed. The convergence and stability of the control loop are analyzed. Simulation examples show that the feasibility and performance of the designed control loop are verified compared with the existing methods and computation times in the millisecond range can be achieved.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

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.

Optimal Charging Scheduling of Electric Vehicles: The Co-Charging Case

A <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">co-charging</i> parking system is a charging system in which the chargers are automated, i.e., they can automatically connect to or disconnect from vehicles presented in the parking. This system makes it possible to have less chargers than the number of parking places. This paper studies an optimal scheduling problem aiming at charging all vehicles to required battery level while minimizing the total charging cost. We formulate this scheduling problem as a non-linear optimal control problem with mixed integer constraints and mixed discrete and continuous time. The problem is solved using a Branch and Bound Strategy coupled with Pontryagin Maximum Principle (PMP) and Interior Point Methods (IPMs) for optimal control. The presented method is tested on numerous configurations and shows promising results both in term of precision and in term of convergence rate.

Optimal Trajectory Planning for Connected and Automated Vehicles in Lane-Free Traffic With Vehicle Nudging

The paper presents a movement strategy for Connected and Automated Vehicles (CAVs) in a lane-free traffic environment with vehicle nudging by use of an optimal control approach. State-dependent constraints on control inputs are considered to ensure that the vehicle moves within the road boundaries and to prevent collisions. An objective function, comprising various weighted sub-objectives, is designed, whose minimization leads to vehicle advancement at the desired speed, when possible, while avoiding obstacles. A nonlinear optimal control problem (OCP) is formulated for the minimization of the objective function subject to constraints for each vehicle. A computationally efficient Feasible Direction Algorithm (FDA) is called, on event-triggered basis, to compute in real-time the numerical solution for finite time-horizons within a Model Predictive Control (MPC) framework. The approach is applied to each vehicle on the road, while running simulations on a lane-free ring-road, for a wide range of vehicle densities and different types of vehicles. From the simulations, which create myriads of driving episodes for each involved vehicle, it is observed that the proposed approach is highly efficient in delivering safe, comfortable and efficient vehicle trajectories, as well as high traffic flow outcomes. The approach is under investigation for further use in various lane-free road infrastructures for CAV traffic.

Safety-Critical and Flexible Cooperative On-Ramp Merging Control of Connected and Automated Vehicles in Mixed Traffic

Cooperative on-ramp merging control for connected and automated vehicles (CAVs) can effectively improve traffic throughput and vehicle fuel efficiency at highway on-ramp merging bottlenecks. However, in the mixed traffic scenario where CAVs and human-driven vehicles (HDVs) coexist, the uncertain maneuvers of human drivers pose a major challenge to merging control in terms of safety and flexibility. To this end, this paper proposes a hierarchical cooperative on-ramp merging control strategy for CAVs to optimize flexible trajectories with safety guarantees in mixed traffic. First, the on-ramp merging control problem for CAVs is considered in the case of a three-vehicle coordination, resulting in an optimal control problem (OCP) coordinating on-ramp and main-lane CAVs for efficient operation while satisfying multiple safety-critical constraints. Second, a two-level hierarchical control architecture is developed to solve the OCP with mixed state-control constraints. The upper-level planner solves an unconstrained OCP with Pontryagin’s Minimum Principle to calculate an expected merging position, which is embedded in the variable time headway of safe merging constraints in the lower-level controller. Then, the controller converts the nonlinear OCP with safety-critical constraints to a quadratic programing (QP) problem by exploiting Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). By solving the QP efficiently, the time and energy efficient trajectory for each CAV is obtained. In addition, a receding horizon control framework is employed, which enables CAVs to determine flexible merging opportunity and tackle the disturbances caused by HDVs. Finally, comprehensive simulation results show that the proposed cooperative on-ramp merging strategy has potential in enabling merging flexibility, improving traffic efficiency and energy economy in real time.

Trajectory Optimization of Chance-Constrained Nonlinear Stochastic Systems for Motion Planning Under Uncertainty

In this article, we present generalized polynomial chaos-based sequential convex programming (gPC-SCP) to compute a suboptimal solution for a continuous-time chance-constrained stochastic nonlinear optimal control (SNOC) problem. The approach enables motion planning for robotic systems under uncertainty. The gPC-SCP method involves two steps. The first step is to derive a surrogate problem of deterministic nonlinear optimal control (DNOC) with convex constraints by using gPC expansion and the distributionally robust convex subset of the chance constraints. The second step is to solve the DNOC problem using sequential convex programming for trajectory generation and control. We prove that in the unconstrained case, the optimal value of the DNOC converges to that of SNOC asymptotically and that any feasible solution of the constrained DNOC is a feasible solution of the chance-constrained SNOC. We also present the predictor–corrector extension (gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> ) for real-time motion trajectory generation in the presence of stochastic uncertainty. In the gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> method, we first predict the uncertainty using the gPC method and then optimize the motion plan to accommodate the uncertainty. We empirically demonstrate the efficacy of the gPC-SCP and the gPC-SCP <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^\text{PC}$</tex-math></inline-formula> methods for the following two test cases: first, collision checking under uncertainty in actuation and physical parameters and second, collision checking with stochastic obstacle model for 3DOF and 6DOF robotic systems. We validate the effectiveness of the gPC-SCP method on the 3DOF robotic spacecraft testbed.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

Fitness Dependent Optimizer Based Computational Technique for Solving Optimal Control Problems of Nonlinear Dynamical Systems

This paper presents a pragmatic approach established on the hybridization of nature-inspired optimization algorithms and Bernstein Polynomials (BPs), achieving the optimum numeric solution for Nonlinear Optimal Control Problems (NOCPs) of dynamical systems. The approximated solution for NOCPs is obtained by the linear combination of BPs with unknown parameters. The unknown parameters are evaluated by the conversion of NOCP to an error minimization problem and the formulation of an objective function. The Fitness Dependent Optimizer (FDO) and Genetic Algorithm (GA) are used to solve the objective function, and subsequently the optimal values of unknown parameters and the optimum solution of NOCP are attained. The efficacy of the proposed technique is assessed on three real-world NOCPs, including Van der Pol (VDP) oscillator problem, Chemical Reactor Problem (CRP), and Continuous Stirred-Tank Chemical Reactor Problem (CSTCRP). The final results and statistical outcomes suggest that the proposed technique generates a better solution and surpasses the recently represented methods in the literature, which eventually verifies the efficiency and credibility of the recommended approach.

Novel closed-loop controllers for fractional nonlinear quadratic systems

&lt;abstract&gt;&lt;p&gt;A novel closed-loop optimal controller for fractional nonlinear quadratic optimal control problems is introduced. By using a new idea, the optimality conditions for the fractional nonlinear problems are derived. The linearized Riccati fractional order differential equation is derived and a new solution method is given for the first time, which can be applied to integer order nonlinear optimal control problems. The proposed closed-loop controller is applied to illustrative examples. Novel unprecedented processes of designing a variable linear controller and of finding the optimal performance index for integer order nonlinear systems are presented.&lt;/p&gt;&lt;/abstract&gt;

Automatic Scenario Generation for Robust Optimal Control Problems*

Existing methods for nonlinear robust control often use scenario-based approaches to formulate the control problem as nonlinear optimization problems. Increasing the number of scenarios improves robustness, while increasing the size of the optimization problems. Mitigating the size of the problem by reducing the number of scenarios requires knowledge about how the uncertainty affects the system. This paper draws from local reduction methods used in semi-infinite optimization to solve robust optimal control problems with parametric uncertainty. We show that nonlinear robust optimal control problems are equivalent to semi-infinite optimization problems and can be solved by local reduction. By iteratively adding interim globally worst-case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. In particular, we show that local reduction methods find worst case scenarios that are not on the boundary of the uncertainty set. The proposed approach is illustrated with a case study with both parametric and additive time-varying uncertainty. The number of scenarios obtained from local reduction is 101, smaller than in the case when all 214+3x192 boundary scenarios are considered. A validation with randomly drawn scenarios shows that our proposed approach reduces the number of scenarios and ensures robustness even if local solvers are used.

A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems

&lt;abstract&gt;&lt;p&gt;This paper describes a study of the barycentric interpolation collocation method for the optimal control problem governed by a nonlinear convection-diffusion equation. Using Lagrangian multipliers, we obtain the continuous optimality system which is composed of state equations, adjoint equations and optimality conditions. Then, barycentric interpolation collocation methods are applied to discretize the optimality system and the nonlinear term is treated by Newton's iteration. Furthermore, the corresponding consistency analyses of discrete schemes are demonstrated. Finally, the validity of the proposed schemes is demonstrated through several numerical experiments. Compared with the classical finite difference method, collocation schemes can yield the higher-order accurate solutions with fewer nodes.&lt;/p&gt;&lt;/abstract&gt;

Distributionally Robust Differential Dynamic Programming With Wasserstein Distance

Differential dynamic programming (DDP) is a popular technique for solving nonlinear optimal control problems with locally quadratic approximations. However, existing DDP methods are not designed for stochastic systems with unknown disturbance distributions. To address this limitation, we propose a novel DDP method that approximately solves the Wasserstein distributionally robust control (WDRC) problem, where the true disturbance distribution is unknown but a disturbance sample dataset is given. Our approach aims to develop a practical and computationally efficient DDP solution. To achieve this, we use the Kantrovich duality principle to decompose the value function in a novel way and derive closed-form expressions of the distributionally robust control and worst-case distribution policies to be used in each iteration of our DDP algorithm. This characterization makes our method tractable and scalable without the need for numerically solving any minimax optimization problems. The superior out-of-sample performance and scalability of our algorithm are demonstrated through kinematic car navigation and coupled oscillator problems.

Discrete Mixed-Integer Shooting (DMIS): Algorithm and Application to Plug-In Hybrid Electric Vehicle Energy Management Accounting for Fuel Cranking and Actual Powertrain Efficiency Maps

This article presents an algorithm for mixed-integer nonlinear optimal control problems (OCPs) formulated directly in discrete time. In our current application, we address the energy management of a plug-in hybrid electric vehicle (PHEV) seeking to achieve minimum trip fuel consumption under a terminal state-of-charge (SOC) equality constraint while avoiding powertrain efficiency map approximations in order to maximize its fuel economy (FE). An additional engine cranking state with a nonsmooth state and control transition cost is then incorporated to prevent undesired engine on/off behaviors. We pursue a single shooting-based numerical algorithm, motivated by the success of the computationally efficient Pontryagin’s Minimum Principle (PMP)-based single shooting in achieving similar FE results as dynamic programming (DP) for the power-split optimization problem without the additional engine cranking state shown in prior work. The Discrete Mixed-Integer Shooting (DMIS) algorithm, with costate backward-in-time propagation, is presented here, avoiding unstable shooting iterations that are typically encountered in PMP-based single shooting. When applying DMIS to solve the power-split problem with cranking fuel consideration, we show through simulation that it leads to better FE results than the PMP-based single shooting without the cranking state, thus demonstrating the effectiveness of the DMIS algorithm. Finally, experimental dynamometer test results with the model predictive powertrain controller solved by DMIS for minimum trip fuel consumption are presented and discussed, demonstrating its real-time execution and the cranking fuel reduction capability.

A New Approach for Solving a New Class of Nonlinear Optimal Control Problems Generated by Atangana-Baleanu- Caputo Variable Order Fractional Derivative and Fractional Volterra-Fredholm Integro-Differential Equations

In the sequel, the numerical solution of a new class of nonlinear optimal control problems (OCPs) generated by Atangana-Baleanu-Caputo (ABC) variable order (V-O) fractional derivative (FD) and fractional Volterra-Fredholm integro-differential equations (FVFIDEs) is found by Bezier curve method (BCM). The main idea behind this work is the use of the BCM. In this technique, the solution is found in the form of a rapid convergent series. Using this method, it is possible to obtain BCM solution of the general form of multipoint boundary value problems. To shown the efficiency of the developed method, numerical results are stated as the main results in this study.

Neural ODEs as Feedback Policies for Nonlinear Optimal Control

Neural ordinary differential equations (Neural ODEs) define continuous time dynamical systems with neural networks. The interest in their application for modelling has sparked recently, spanning hybrid system identification problems and time series analysis. In this work we propose the use of a neural control policy capable of satisfying state and control constraints to solve nonlinear optimal control problems. The control policy optimization is posed as a Neural ODE problem to efficiently exploit the availability of a dynamical system model. We showcase the efficacy of this type of deterministic neural policies in the controlled Van der Pol system. This approach represents a practical approximation to the intractable closed-loop solution of nonlinear control problems.

Nonlinear Optimal Control for a Gas Compressor Actuated by a Five-Phase Induction Motor

Abstract The article proposes a nonlinear optimal control method for the dynamic model of a gas centrifugal compressor being actuated by a five-phase induction motor (5-phase IM). To achieve high torque and high power in the functioning of gas compressors, 5-phase IM appear to be advantageous in comparison to three-phase synchronous or asynchronous electric machines. The dynamic model of the integrated compression system, which comprises the gas compressor and the 5-phase IM, is first written in a nonlinear and multivariable state-space form. It is proven that the electrically driven gas-compression system is differentially flat. Next, this system is approximately linearised around a temporary operating point that is recomputed at each sampling interval. The linearisation is based on first-order Taylor series expansion and uses the computation of the Jacobian matrices of the state-space model of the integrated system. For the linearised state-space description of the compressor and 5-phase IM, a stabilising optimal (H-infinity) feedback controller is designed. This controller achieves a solution to the nonlinear optimal control problem of the compressor and 5-phase IM system under model uncertainty and external perturbations. The feedback gains of the controller are computed by solving an algebraic Riccati equation at each iteration of the control method. Lyapunov analysis is used to demonstrate global stability for the control loop. Additionally, the H-infinity Kalman filter is used as a robust state estimator, which allows for implementing sensorless control for the gas compression system.