Optimal Problem Research Articles

Optimal control problems for a parabolic system inspired by a cancer therapy

In this paper, we consider optimal control problems for a parabolic system modeling a therapy, based on oncolytic viruses, for the glioma brain cancer. Using several techniques typical of functional analysis, we prove the global in time well posedness of the control model, the existence of optimal controls for specific objective functionals, which are natural for cancer therapies, and we derive necessary conditions for optimality.

Event-triggered design for optimal output consensus of high-order multi-agent systems

In this paper, we study the optimal output consensus problem for networked linear multi-agent systems. Existing distributed algorithms usually rely on continuous communication among neighbouring agents or continuous update of each agent's local controller. To save communication and computation resources, we apply event-triggered technique to this optimal output consensus problem. We first constructively develop an event-triggered algorithm with a set of applicable parameters relying on event-triggered communication and control and show its effectiveness in solving the problem by rigorous proofs. Then we extend it to the case where the triggering conditions only have to be checked in some sampling time instants. Two simulation examples are given to illustrate the efficacy of our designs.

On the regularity of optimal potentials in control problems governed by elliptic equations

Abstract In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV \mathrm{BV} one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV \mathrm{BV} regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases.

Optimal Gliding Trajectories for Descent on Mars

In this paper, optimal gliding trajectories are analyzed, formulating an optimal control problem to be solved computationally via nonlinear programming. A multi-objective terminal cost function that minimizes the velocity, altitude, horizontal flight path angle, and the error of a desired terminal point is formulated. The results are validated via Monte Carlo simulations, analyzing the influence of the variations in the lift-to-drag ratios in the atmospheric entry. The results show feasible solutions for minimizing the distance to the desired final point, which have significant practical implications for the controlled gliding entry of a spaceplane on Mars. In this sense, the results also show that the longitudes of the landing points do not change too much for almost all trajectories, while the latitude of these points is in the interval of about 4 degrees for the majority of the trajectories. This suggests that it is possible to implement an optimal control approach with reasonable accuracy for the controlled gliding entry of a spaceplane on Mars, even in the presence of some uncertainties regarding the aerodynamic performance of the spacecraft, such as the lift-to-drag ratio.

RADIAL BASIS NEURAL NETWORK FOR THE SOLUTION OF OPTIMAL CONTROL PROBLEMS VIA SIMULINK

This study uses radial basis neural network to solve optimum control problems via simulink. Using Pontryaginí's principle, the optimal control problem's optimum system is constructed. MATLAB is used to simulate the optimality system and generate the simulink architecture for the trial value. A radial basis neural network is then used to train the system and produce the optimal solution. This approach's effectiveness is evaluated using a few control problems, and it is shown to be effective because of the reliable, accurate, and consistent results that are produced. The performance of this strategy is superior to that of other approaches.

Two-stage coordinated scheduling of hydrogen-integrated multi-energy virtual power plant in joint capacity, energy, and ancillary service markets

With the emergence of distributed resources (DRs), developing an effective, profitable, and green management method is of great importance. Virtual power plant (VPP) can aggregate various DRs, optimize their schedules, and participate in electricity markets uniformly to earn profit. The optimal scheduling problem in spot market has been widely studied, however, few studies have extended the services of VPP to the long-term scale. This paper proposes a coordinated scheduling method for a multi-energy virtual power plant (MEVPP), considering the integration of hydrogen facilities. A holistic market framework, including capacity market, energy market (EM), and ancillary service market (ASM) is developed jointly for the coordination among the annual capacity schedule, daily energy schedule, and daily reserve schedule of MEVPP. A hybrid hydrogen storage system including daily hydrogen storage and seasonal hydrogen storage is modeled to achieve both intraday and cross-seasonal peak shaving. Stochastic programming is used to tackle the uncertainties and risk management is considered by conditional value at risk. With the help of numerical analysis in case studies, it can be concluded that the proposed coordinated scheduling method can improve the economy of MEVPP. The profit from joint market decreases the total cost by about 52 %. The capacity adequacy and flexibility of MEVPP can be improved by the coordination of capacity and reserve services. The hybrid hydrogen storage system can realize the efficient integration of hydrogen and increase hydrogen production and consumption by about 9 %.

Meta-Reinforcement Learning for Spacecraft Proximity Operations Guidance and Control in Cislunar Space

Innovative lightweight and model-free guidance algorithms are essential to achieve full autonomy of spacecraft and address future space exploration challenges. In the near future, this type of technology will be essential for cislunar space proximity operations, such as NASA’s Artemis program and its Lunar Gateway project. In this scenario, a meta-reinforcement learning algorithm is employed to address the real-time optimal guidance problem of a spacecraft in the cislunar environment. Non-Keplerian orbits pose complex dynamics, where classic control theory may be less adaptable and more computationally expensive compared to machine learning approaches. Meta-RL stands out for its ability to learn how to learn through experience, training on various tasks to enhance efficiency, and effectiveness in tackling new ones. By modeling a stochastic optimal control problem within the circular restricted three-body problem framework as a Markov decision process, an LSTM-based network agent is trained using proximal policy optimization, considering operational constraints and stochastic effects for safety and robustness evaluation. Additionally, an MLP-based agent and an optimal control pseudo-spectral solution are assessed for comparison. The resulting tool autonomously guides spacecraft in cislunar proximity operations, approximating the optimal control solution with a versatile algorithmic framework. This ensures both robustness and computational efficiency.

Neural Network-Based Robust Guaranteed Cost Control for Image-Based Visual Servoing of Quadrotor.

In this article, a neural network (NN)-based robust guaranteed cost control design is proposed for image-based visual servoing (IBVS) control of quadrotors. According to the dynamics of three subsystems (yaw, height, and lateral subsystems) derived from the quadrotor IBVS dynamic model, the main control design is to solve the robust control problem for the time-varying lateral subsystem with angle constraints and uncertain disturbances. Considering the system dynamics, a two-loop structure is conducted. The outer loop uses the linear quadratic regulator to solve the Riccati equation for the lateral image feature system, and the inner loop adopts the optimal robust guaranteed cost control to solve the lateral velocity system. For the lateral velocity system, the optimal robust control problem is transformed to solve the modified Hamilton-Jacobi-Bellman equation of the corresponding optimal control problem utilizing adaptive dynamic programming. The implementation is accomplished with the time-varying NN and the designed estimated weight update law. In addition, the stability and effectiveness are proved by the theoretic proof and simulations.

A Two Phases Multiobjective Trajectory Optimization Scheme for Multi-UGVs in the Sight of the First Aid Scenario.

Timely delivery of first aid supplies is significant to saving lives when an accident happens. Among the promising solutions provided for such scenarios, the application of unmanned vehicles has attracted ever more attention. However, such scenarios are often very complex, while the existing studies have not fully addressed the trajectory optimization problem of multiple unmanned ground vehicles (multi-UGVs) against the scenario. This study focuses on multi-UGVs trajectory optimization in the sight of first aid supply delivery tasks in mass accidents. A two-stage completely decoupling fuzzy multiobjective optimization strategy is designed. On the first stage, with the proposed timescale involved tridimensional tunneled collision-free trajectory (TITTCT) algorithm, collision-free coarse tunnels are build within a tridimensional coordinate system, respectively, for the UGVs as the corresponding configuration space for a further multiobjective optimization. On the second stage, a fuzzy multiobjective transcription method is designed to solve the decoupled optimal control problem (OCP) within the configuration space with the consideration of priority constrains. Following the two-stage design, the computational time is significantly reduced when achieving an optimal solution of the multi-UGV trajectory planning, which is crucial in a first aid task. In addition, other objectives are optimized with the aspiration level reflected. Simulation studies and experiments have been curried out to testify the effectiveness and the improved computational performance of the proposed design.

Multiobjective optimization of collaborative robotic task sequence assignment problems under collision-free constraints

This paper proposes a multiobjective optimization approach to address the challenge of collaborative manufacturing with multiple robot arms. Given the necessity for multiple robot arms to work together, the potential for collisions between robotic motions is a significant concern, and the automated task sequence assignment for robots becomes increasingly complex. Previous research has either simplified the collision-free conditions in a limited working area, or employed a master-slave approach to obtain only a local solution. Consequently, we propose a unified global optimization approach for simultaneously addressing various collaborative manufacturing issues, including robotic task sequence assignment (RTSA), multiple inverse kinematics (IK) selection, joint-space collision-free operations and multiple manufacturing objectives. As the optimal collaborative RTSA problem is a combinatorial optimization problem with non-deterministic polynomial-time hard (NP-hard) complexity, this paper presents a hybrid nondominated sorting genetic algorithm III (NSGA-III) method that integrates a Hamming-distance-based method and a greedy strategy within NSGA-III to improve population diversity and solution quality. To validate the efficacy of the proposed approach, simulation experiments were conducted on cooperative manufacturing scenarios, with two objectives: task completion time and task load balancing. The experimental results demonstrate that the proposed approach is effective in obtaining collision-free Pareto solutions. Furthermore, the proposed hybrid NSGA-III method obtains superior solutions compared to the original method in the studied problem, as measured by two performance indices.

Advanced PSO Algorithms Development with Combined lbest and gbest Neighborhood Topologies

Abstract This paper introduces an innovative approach integrating global best (gbest) and local best (lbest) PSO communication topologies. The algorithm initiates with lbest and seamlessly transitions to gbest, with the switching rate controlled by the parameter “a”. Rational values of “a” is determined through numerical experiments. A comparative methodology employing two estimation criteria is used to showcase the improved performance of the modified PSO-based algorithms. Furthermore, the efficacy of this approach is demonstrated in addressing two optimal control problems within dynamical systems. Results highlight the modified algorithms’ superiority in terms of the total number of successful runs and statistical indicators. Consequently, these advanced algorithms prove effective for applications such as artificial neural network training, controller gains determination, and similar problem domains.

ADP-based online compensation hierarchical sliding-mode control for partially unknown switched nonlinear systems with actuator failures

This article investigates an adaptive dynamic programming-based online compensation hierarchical sliding-mode control problem for a class of partially unknown switched nonlinear systems with actuator failures and uncertain perturbations under an identifier-critic neural networks architecture. Firstly, by introducing a cost function related to hierarchical sliding-mode surfaces for the nominal system, the original control problem is equivalently converted into an optimal control problem. To obtain this optimal control policy, the Hamilton–Jacobi–Bellman equation is solved through an adaptive dynamic programming method. Compared with conventional adaptive dynamic programming methods, the identifier-critic network architecture not only overcomes the limitation on the unknown internal dynamic but also eliminates the approximation error arising from the actor network. The weights in the critic network are tuned via the gradient descent approach and the experience replay technology, such that the persistence of excitation condition can be relaxed. Then, a compensation term containing hierarchical sliding-mode surfaces is used to offset uncertain actuator failures without the fault detection and isolation unit. Based on the Lyapunov stability theory, all states of the closed-loop nonlinear system are stable in the sense of uniformly ultimately boundedness. Finally, numerical and practical examples are given to demonstrate the effectiveness of our presented online compensation control strategy.

Virtually coupled train set control subject to space-time separation: A distributed economic MPC approach with emergency braking configuration

The emerging virtual coupling technology aims to operate multiple train units in a Virtually Coupled Train Set (VCTS) at a minimal but safe distance. To guarantee collision avoidance, the safety distance should be calculated using the state-of-the-art space-time separation principle that separates the Emergency Braking (EB) trajectories of two successive units during the whole EB process. In this case, the minimal safety distance is usually numerically calculated without an analytic formulation. Thus, the constrained VCTS control problem is hard to address with space-time separation, which is still a gap in the existing literature. To solve this problem, we propose a Distributed Economic Model Predictive Control (DEMPC) approach with computation efficiency and theoretical guarantee. Specifically, to alleviate the computation burden, we transform implicit safety constraints into explicitly linear ones, such that the optimal control problem in DEMPC is a quadratic programming problem that can be solved efficiently. For theoretical analysis, sufficient conditions are derived to guarantee the recursive feasibility and stability of DEMPC, employing compatibility constraints, tube techniques and terminal ingredient tuning. Moreover, we extend our approach with globally optimal and distributed online EB configuration methods to shorten the minimal distance among VCTS. Finally, experimental results demonstrate the performance and advantages of the proposed approaches.

Energy Maximization Absorption of Wave Energy Converter Based on Fourier Pseudo-Spectral Method and Adaptive Dynamic Programming

In this paper, a novel noncausal control framework to address the energy maximization problem of wave energy converters (WECs) with physical constraints is proposed. The energy maximization problem of WECs is a constrained optimal control problem. The proposed control framework converts this problem into a reference trajectory-tracking problem through the Fourier pseudo-spectral method (FPSM) and utilizes the online tracking adaptive dynamic programming (OTADP) algorithm to realize real-time trajectory tracking for practical use in the ocean environment. Using the wave prediction technique, the optimal trajectory is generated online through a receding horizon (RH) implementation. A critic neural network (NN) is applied to approximate the optimal cost value function and calculate the error-tracking control by solving the associated Hamilton-Jacobi-Bellman (HJB) equation. The proposed WEC control framework improves computational efficiency and makes it feasible for the online control of the WEC in practice. Simulation results show the effects of the receding-horizon implementation of FPSM with different window lengths and window functions, while verifying the performances of tracking control and energy absorption of WECs in two different sea conditions.

Robustness of optimal quantum annealing protocols

Abstract Noise in quantum computing devices poses a key challenge in their realization. In this paper, we study the robustness of optimal quantum annealing protocols against coherent control errors, which are multiplicative Hamlitonian errors causing detrimental effects on current quantum devices. We show that the norm of the Hamiltonian quantifies the robustness against these errors, motivating the introduction of an additional regularization term in the cost function. We analyze the optimality conditions of the resulting robust quantum optimal control problem based on Pontryagin’s maximum principle, showing that robust protocols admit larger smooth annealing sections. This suggests that quantum annealing admits improved robustness in comparison to bang-bang solutions such as the quantum approximate optimization algorithm. Finally, we perform numerical simulations to verify our analytical results and demonstrate the improved robustness of the proposed approach.

Data-driven optimal cooperative tracking control for heterogeneous multi-agent systems

This paper presents a novel hierarchical control scheme for solving the data-driven optimal cooperative tracking control problem of heterogeneous multi-agent systems. Considering that followers cannot communicate with the leader, a prescribed-time fully distributed observer is devised to estimate the leader’s state for each follower. Then, the data-driven decentralized controller is designed to ensure that the follower’s output can track the leader’s one. Compared with the existing results, the advantages of the designed distributed observer are that the prescribed convergence time is completely predetermined by the designer, and the design of the observer gain is independent of the global topology information. Besides, the advantages of the designed decentralized controller are that neither the follower’s system model nor a known initial stabilizing control policy is required. Finally, simulation results exemplify the advantage of the proposed method.

Synthesis of a Regulator for a Linear-Quadratic Optimal Control Problem

Synthesis of a Regulator for a Linear-Quadratic Optimal Control Problem

Isoperimetric Constraint Inference for Discrete-Time Nonlinear Systems Based on Inverse Optimal Control.

In this article, the problem of inferring unknown isoperimetric constraints is considered given optimal state and control trajectories that solve the optimal control problem with isoperimetric constraints. By exploiting Pontryagin's principle, the recovery equations for unknown isoperimetric constraints are established. Under verifiable dimensionality condition and matrix rank condition, the proposed method is guaranteed to infer the unknown isoperimetric constraints exactly. Furthermore, the proposed method is extended to multiple trajectory setting. Finally, the effectiveness of the proposed method is illustrated by two simulation examples with various settings.

Improved artificial potential field method based on robot local path information

The artificial potential field (APF) is an important method for robot path planning. However, some information in APF is not fully utilized in practical applications. In this paper, an improved artificial potential field (IAPF) method is presented, in which the local path information is defined and used. And the calculation formulas for various forces in IAPF are given, which include repulsive force (R-force) of obstacle on the robot, the attractive force (A-force) of target on the robot, and the resultant force of R-force and A-force. Then, based on the local path information, a method for solving the robot falling into local optimality problem is proposed and used into IAPF. Finally, IAPF is respectively simulated and discussed in general scenario, complex scenario, and scenarios with the same and different size of circular obstacles. The results show that IAPF has higher efficiency than traditional artificial potential field (TAPF) method and can overcome the local optimality problem. At the same time, IAPF is compared with dynamic window method in the scenarios with the same and different size of circular obstacles. The results show that IAPF is more efficient than the dynamic window approach (DWA) for robot path planning.

Solving optimal predictor-feedback control using approximate dynamic programming

This paper is concerned with approximately solving the optimal predictor-feedback control problem of multiplicative-noise systems with input delay in infinite horizon. The optimal predictor-feedback control, provided by the analytical method, is determined by Riccati–ZXL equations and is hard to obtain in the case of unknown system dynamics. We aim to propose a policy iteration (PI) algorithm for solving the optimal solution by approximate dynamic programming. For convergence analysis of the algorithm, we first develop a necessary and sufficient stabilizing condition, in the form of several new Lyapunov-type equations, which parameterizes all predictor-feedback controllers and can be seen as an important addition to Lyapunov stability theory. We then propose an iterative scheme for the Riccati–ZXL equations computations, along with convergence analysis, based on the condition. Inspired by this scheme, a data-driven online PI algorithm, convergence implied in that of the iterative scheme, is proposed for the optimal predictor-feedback control problem without full system dynamics. Finally, a numerical example is used to evaluate the proposed PI algorithm.