Original Optimal Control Problem Research Articles

Sequential Optimal Trajectory Planning Scheme for Robotic Manipulators along Specified Path Based on Direct Collocation Method

Robotic manipulators play a pivotal role in modern intelligent manufacturing and unmanned production systems, often tasked with executing specific paths accurately. However, the input of the robotic manipulators is trajectory which is a path with time information. The resulting core technology is trajectory planning methods which are broadly classified into two categories: maximum velocity curve (MVC) methods and multiphase direct collocation (MPDC) methods. This paper concentrates on addressing challenges associated with the latter methods. In MPDC methods, the solving efficiency and accuracy are greatly influenced by the number of discretization nodes. When dealing with systems with complex dynamics, such as robotic manipulators, striking a balance between solving time and path discretization errors becomes crucial. We use a mesh refinement (MR) algorithm to find a suitable number of nodes under the premise of ensuring the path discretization error. So, the actual device can effectively implement the planned solutions. Nonetheless, the conventional application of the MR algorithm requires solving the original problem in each iteration; these processes are extremely time-consuming and may fail to solve when dealing with a complex dynamic system. As a result, we propose a sequential optimal trajectory planning scheme to solve the problem efficiently by dividing the original optimal control (OC) problem into two stages: path planning (PP) and trajectory planning (TP). In the PP stage, we employ a DC method based on arc length and an MR algorithm to identify key nodes along the specified path. This aims to minimize the approximation error introduced during discretization. In the TP stage, the identified key nodes serve as boundary conditions for an MPDC method based on time. This facilitates the generation of an optimal trajectory that maximizes motion performance, considering constant velocity in Cartesian space and dynamic constraints while keeping the path discretization error. Simulation and experiment are conducted with a six-axis robotic manipulator, ROCR6, and show significant potential for a wide range of applications in robotics.

Convex distributed robust model predictive control for collision and obstacle avoidance

AbstractThe article proposes a convex distributed robust model predictive control algorithm for collision avoidance in multi‐agent systems with additive disturbances, which utilizes the concept of tube model predictive control to address the disturbances. To tackle the coupled collision avoidance constraints, each agent integrates assumed nominal position and input trajectories from its neighbors, rather than relying on actual ones. Compatibility constraints are then designed using the normal vector of the constructed separating hyperplanes for collision avoidance and the residual collision avoidance margin of the optimal solutions at the last time step to restrict deviations between assumed and actual trajectories and ensure consistency among the agents. The nonconvex collision and obstacle avoidance constraints are convexified using the concept of safe sets, transforming them into time‐varying closed polyhedral constraints while accounting for the impact of disturbances. Particularly, the residual collision avoidance margin is incorporated into the construction of safe sets for the satisfaction of coupled collision avoidance constraints. Consequently, the original collision avoidance optimal control problems can be efficiently and simultaneously solved for all agents using standard quadratic programming techniques. Under the design of local terminal constraint sets for each agent, the proposed algorithm ensures a rigorous analysis of robust constraint satisfaction for collision avoidance constraints, as well as an assessment of recursive feasibility and closed‐loop stability. The effectiveness of the algorithm is demonstrated through an illustrative example, and simulation results validate its ability to successfully achieve collision and obstacle avoidance even in the presence of disturbances.

To Charge In-Flight or Not: A Comparison of Two Parallel Hybrid Electric Aircraft Configurations via Optimal Control

This study examines two configurations for parallel hybrid electric aircraft, one with a mechanical connection between the engines and the electric motors and the other without. A previously proposed power allocation algorithm in the context of aircraft energy management is applied to these two configurations for a 19-seat conceptual hybrid electric aircraft. The original optimal control problem is then transformed into a finite-dimensional optimization, and the second-order sufficient conditions for the global optimality of the obtained solution are validated. This is followed by a sensitivity analysis of the fuel consumption to the initial aircraft weight and flight duration. Simulation and theoretical results clarify the limited benefits of charging the battery in-flight for this class of hybrid electric aircraft for the purpose of reducing fuel consumption and CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> emissions.

Non-Overlapping Domain Decomposition for 1D Optimal Control Problems Governed by Time-Fractional Diffusion Equations on Coupled Domains: Optimality System and Virtual Controls

We consider a non-overlapping domain decomposition method for optimal control problems of the tracking type governed by time-fractional diffusion equations in one space dimension, where the fractional time derivative is considered in the Caputo sense. We concentrate on a transmission problem defined on two adjacent intervals, where at the interface we introduce an iterative non-overlapping domain decomposition in the spirit of P.L. Lions for the corresponding first-order optimality system, such that the optimality system corresponding to the optimal control problem on the entire domain is iteratively decomposed into two systems on the respective sub-domains; this approach can be framed as first optimize, then decompose. We show that the iteration involving the states and adjoint states converges in the appropriate spaces. Moreover, we show that the decomposed systems on the sub-domain can in turn be interpreted as optimality systems of so-called virtual control problems on the sub-domains. Using this property, we are able to solve the original optimal control problem by an iterative solution of optimal control problems on the sub-domains. This approach can be framed as first decompose, then optimize. We provide a mathematical analysis of the problems as well as a numerical finite difference discretization using the L1-method with respect to the Caputo derivative, along with two examples in order to verify the method.

Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach

We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example.

Eco-speed optimization model for active rear-end collision avoidance of connected and automated vehicles on freeways

This paper proposed an eco-speed optimization model for active rear-end collision avoidance of connected and automated vehicles on freeways in the scene of dangerous sources downstream, and the optimal speed curve was obtained to guide connected and automated vehicles to travel or stop safely and smoothly to avoid rear-end collision accidents. To implement the eco-speed optimization model, a traffic state prediction model based on confidence interval was proposed for forecasting the state of the front low-speed traffic flow. To improve the computational efficiency, the original optimal control problem was decomposed into several sub-problems based on the hyperbolic tangent function and only the leading vehicle was optimized. To eliminate the deviation caused by traffic variation during the period of solving the eco-speed optimization model, a prediction-based optimization strategy was adopted. To verify the performances of the eco-speed optimization model, the control strategy of in-vehicle driving alert systems was taken as the study case for comparison. The results show that the approximate model can basically ensure the computational accuracy and significantly improve the computational efficiency, that the proposed traffic state prediction model can ensure traffic safety, and that the speed control strategy obtained in this paper can make the vehicle decelerate in advance to avoid rear-end collision, smooth the vehicle’s speed curve, save the fuel consumption by 24.08%, restrain traffic oscillation and improve the safety of the traffic flow.

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.

The variational optimality condition in the problem of minimizing the finite state norm by a composite system of hyperbolic and ordinary differential equations

An optimal control problem for a system of linear first-order hyperbolic equations is studied. The boundary conditions are determined from controlled systems of ordinary differential equations. A nonclassical exact formulae for the increment of a linear performance index (a finite state norm) is suggested. Based on this result, a variational optimality condition is proved. The original optimal control problems for a hyperbolic system is reduced to the problem for systems of ordinary differential equations.

Optimal bilinear control of the logarithmic Schrödinger equation

We investigate the optimal bilinear control of the 3D logarithmic nonlinear Schrödinger equation arises from quantum physics. To overcome the difficulty caused by the absence of the Lipschitz continuity at the origin due to the logarithmic nonlinearity, we use an approximate scheme. By studying the approximate systems and passing to a limit, we finally get the minimizer of the original optimal control problem.

A symplectic method for trajectory planning of general tractor-trailer systems

Trajectory planning is a key issue in the motion control of tractor-trailer system. In this paper, a unified differential-algebraic equation approach to establish the kinematic models of general tractor-trailer systems is developed. By analyzing practical physical limitations and transportation requirements, the trajectory planning of tractor-trailer system is formulated as a nonlinear constrained optimal control problem. Aiming at efficient numerical solution, a novel symplectic method is proposed. In the proposed method, a symplectic discretization is first implemented to transform the necessary conditions for the original optimal control problem into a set of non-smooth equations. Then, a composite strategy that integrates preconditioned iteration techniques is employed for robust solution. The proposed symplectic method is validated by three examples with different system configurations. Simulation results demonstrate that our method outperforms pseudospectral methods in computational efficiency and robustness to initial guesses. Specifically, a multi-stage strategy can be adopted to achieve robust solution in the presence of obstacles, making the system successfully travel across narrow corridors. An animation of the simulation results is available at www.bilibili.com/video/BV1EN4y1P7Ar.

Optimal Control for Kinematic Bicycle Model With Continuous-Time Safety Guarantees: A Sequential Second-Order Cone Programming Approach

The optimal control problem for the kinematic bicycle model is considered where the trajectories are required to satisfy the safety constraints in the continuous-time sense. Based on the differential flatness property of the model, necessary and sufficient conditions in the flat space are provided to guarantee safety in the state space. The optimal control problem is relaxed to the problem of solving three second-order cone programs (SOCPs) sequentially, which find the safe path, the trajectory duration, and the speed profile, respectively. Solutions of the three SOCPs together provide a sub-optimal solution to the original optimal control problem. Simulation examples and comparisons with state-of-the-art optimal control solvers are presented to demonstrate the effectiveness of the proposed approach.

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.

Optimal control of nonlinear systems with integer‐valued control inputs and stochastic constraints

AbstractPractical industrial process is usually a dynamic process including uncertainty. Stochastic constraints can be used for industrial process modeling, when system sate and/or control input constraints cannot be strictly satisfied. Thus, optimal control of nonlinear systems with stochastic constraints can be available to address practical industrial process problems with integer‐valued control inputs. In general, obtaining an analytical solution of this optimal control problem is very difficult due to the discrete nature of the control inputs and the complexity of stochastic constraints. To obtain a numerical solution, this problem is formulated as a finite dimensional static constrained optimization problem. First, the integer‐valued control input is relaxed into a continuous‐valued control input by imposing a penalty term on the objective function. Convergence results show that the relaxed solution obtained is an integer solution as long as the penalty parameter is sufficiently large. In addition, it should be noted that no any constraint is introduced in the novel relaxation technique, which can effectively avoid introducing any extra extreme point for the original optimal control problem. Next, a novel smooth approximation function is used to construct a subset of feasible region for this optimal control problem. It is proved that the smooth approximation can converge uniformly to the stochastic constraints as the adjusting parameter reduces. Following that, a numerical computation method is proposed for solving the original optimal control problem. Finally, in order to illustrate the effectiveness of the proposed method, an electric vehicle energy management problem is extended by considering some stochastic constraints. The numerical results show that the proposed method is less conservative compared with the existing typical approaches and can obtain a stable and robust performance when considering the initial condition small perturbations.

A general numerical scheme for the optimal control of fractional Birkhoffian systems

This paper gives a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d’Alembert principle which includes the control as an external force term. Following the strategy of variational integrators, the fractional Pfaff–Birkhoff–d’Alembert principle is directly discretized to develop the equivalent discrete fractional forced Birkhoff equations that served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by existing algorithms. An illustrative example is given to show the efficiency and simplicity of the proposed method.

Rapid optimization of continuous trajectory for multi-target exploration propelled by electric sails

As an innovative spacecraft propellantless propulsion technology, the electric sail can generate electric fields that interact with the solar wind, generating continuous propulsive thrust. However, unlike conventional electric thrusters, the thrust vector of the electric sail is constrained, and the attitude cannot be adjusted quickly. Consequently, the position and velocity of each flight trajectory at the connecting node and the direction of propulsion acceleration must be continuous. In this study, the continuous trajectory optimization of the electric sail–based probe in multi-target exploration is investigated. An integrative Bezier shaping approach (IBSA) is proposed to realize the rapid optimization of a multi-target exploration trajectory with continuous acceleration. The relationship between boundary constraints, intermediate constraints, and Bezier basis function coefficients is derived, transforming the original continuous optimal control problem of multiple stages into a problem of nonlinear programming that naturally satisfies the boundary and intermediate constraints. Numerical simulation results demonstrate that the IBSA can obtain a better interplanetary trajectory with a shorter calculation time than that obtained with the traditional Bezier shaping approach using piecewise optimization. Furthermore, the simulation results are successfully applied to the initial value estimation of the Gauss pseudospectral method (GPM). Compared with the GPM, the proposed shaping approach differs from the objective function by approximately 2%–6%, but requires approximately 1.5% of the computational time. During the preliminary mission design stage, the IBSA can be used for rapid feasibility assessments of different electric sail–based probe mission profiles.

Robust Low-Thrust Trajectory Optimization Using Convex Programming and a Homotopic Approach

A robust algorithm to solve the low-thrust fuel-optimal trajectory optimization problem for interplanetary spacecraft is developed in this article. The original nonlinear optimal control problem is convexified and transformed into a parameter optimization problem using an arbitrary-order Gauss–Lobatto discretization scheme with nonlinear control interpolation. A homotopic approach that considers the energy-to-fuel smoothing path is combined with an adaptive second-order trust-region mechanism to increase performance. The overall robustness is assessed in several fuel-optimal transfers with poor initial guesses. The results show a superior performance in terms of convergence and computational time compared to standard convex programming approaches in the literature.

Fault-Tolerant Optimal Spacecraft Attitude Maneuver: An Incremental Model Approach

This paper addresses the fault-tolerant optimal attitude control problem for a rigid spacecraft with external disturbance, model uncertainties, and actuator faults. Firstly, the incremental nonlinear control technology is used to simplify the attitude control system into an incremental nominal model with a synthetic uncertainty/fault term that is estimated by a nonsingular terminal sliding mode disturbance observer. Secondly, considering a quadratic performance index, the original fault-tolerant optimal control problem is transformed into a guaranteed performance optimal problem of the nominal model. Benefiting from the simplicity of the incremental model, such a modified optimal control problem is solved online by the modified policy iteration adaptive dynamic programming without any initial stable policy assumption. Finally, the attitude angles of the closed-loop system are proved theoretically to be uniformly ultimately bounded at the desired state by using the Lyapunov methods. Simulation results are given to verify the effectiveness of the proposed fault-tolerant optimal attitude control strategy.

PySCP: A Multiple-Phase Optimal Control Software Using Sequential Convex Programming

Optimal control problems are common in aerospace engineering. A Python software program called PySCP is described for solving multiple-phase optimal control problems using sequential convex programming methods. By constructing a series of approximated second-order cone programming subproblems, PySCP approaches to the solution of the original optimal control problem in an iterative way. The key components of the software are described in detail, including convexification, discretization, and the adaptive trust region method. The convexification of the first-order differential dynamic equation is implemented using successive linearization. Six discretization methods, including zero-order hold, first-order hold, Runge-Kutta, and three hp pseudospectral collocation methods, are implemented so that different types of optimal control problems can be tackled efficiently. Adaptive trust region method is employed, and robust convergence is achieved. Both free-final-time problem and fixed-final-time problem can be solved by the software. The application of the software is demonstrated on three optimal control problems with varying complexity. PySCP provides researchers a useful toolkit to solve a wide variety of optimal control problems using sequential convex programming.

Optimization of start-up operation for centrifugal chiller

In this study, we derived an operating profile that minimizes startup time using a rigorous physical model that takes into account the characteristics of the compressors and heat exchangers that make up a centrifugal chiller, as well as the phase change and flow of the refrigerant. As an optimization method, we used the direct method of transforming the original infinite optimal control problem into a finite-dimensional nonlinear programming problem. We were able to derive an operation profile that reduces the startup time while satisfying the constraints compared to conventional operations and confirmed that the model and optimization method are effective for optimal design of equipment and control logic.

Optimal feedback control for a class of fed-batch fermentation processes using switched dynamical system approach

&lt;abstract&gt;&lt;p&gt;This paper considers an optimal feedback control problem for a class of fed-batch fermentation processes. Our main contributions are as follows. Firstly, a dynamic optimization problem for fed-batch fermentation processes is modeled as an optimal control problem of switched dynamical systems, and a general state-feedback controller is designed for this dynamic optimization problem. Unlike the existing switched dynamical system optimal control problem, the state-dependent switching method is applied to design the switching rule, and the structure of this state-feedback controller is not restricted to a particular form. Then, this problem is transformed into a mixed-integer optimal control problem by introducing a discrete-valued function. Furthermore, each of these discrete variables is represented by using a set of 0-1 variables. By using a quadratic constraint, these 0-1 variables are relaxed such that they are continuous on the closed interval $ [0, 1] $. Accordingly, the original mixed-integer optimal control problem is transformed intoa nonlinear parameter optimization problem. Unlike the existing works, the constraint introduced for these 0-1 variables are at most quadratic. Thus, it does not increase the number of locally optimal solutions of the original problem. Next, an improved gradient-based algorithm is developed based on a novel search approach, and a large number of numerical experiments show that this novel search approach can effectively improve the convergence speed of this algorithm, when an iteration is trapped to a curved narrow valley bottom of the objective function. Finally, numerical results illustrate the effectiveness of this method developed by this paper.&lt;/p&gt;&lt;/abstract&gt;