Integral Cost Function Research Articles

UAV path planning based on improved dung beetle algorithm with multiple strategy integration

This research presents a multi-strategy augmented dung beetle algorithm (CDBO) for UAV path planning in intricate 3D environments. Its main objective is to address the path planning challenge by formulating an integrated cost function model and environment representation that aligns the optimisation task with the navigation prerequisites and safety constraints of the UAV. Firstly, the algorithm initialises the particle population based on the Lévy flight principle to improve the species diversity and adequately search the solution space. Subsequently, an exponentially decreasing inertia weighting strategy is introduced to improve the convergence speed of the algorithm. In addition, the algorithm uses an adaptive [Formula: see text]-distribution with perturbed variation of the updated positions so that the algorithm avoids falling into a local optimum. Finally, to enhance the robustness of the system, the algorithm incorporates an elite retention strategy based on the Pareto principle. In this study, rigorous tests using 12 CEC2022 test functions and Wilcoxon rank sum test are conducted as benchmarks for the performance of the algorithm. The results show that the enhanced dung beetle algorithm outperforms the traditional algorithm, highlighting its efficiency and robustness. In addition, the efficacy of the enhanced dung beetle algorithm was validated in a variety of complex 3D environments, affirming its ability to generate optimal paths. This study further highlights the importance of effective algorithms in UAV path planning, addressing the key issue of optimising flight operations in intricate spatial scenarios.

A Robust Parallel Predictive Torque Control Based on a Proportional Integral Cost Function for PMSM

A Robust Parallel Predictive Torque Control Based on a Proportional Integral Cost Function for PMSM

Observer-based guaranteed-cost bipartite time-varying formation tracking control for multiagent systems subject to communication delays

The paper centers on the guaranteed-cost bipartite time-varying formation tracking control for multiagent systems with nonuniform time-varying communication delays under the framework of a distributed observer. An integral quadratic cost function is presented, founded on formation tracking and observation error. Firstly, distributed state observers, using the relative measurement output between neighbors, are established so that the leader and follower can accurately estimate each agent's unmeasurable state. Subsequently, a bipartite formation tracking protocol is proposed, incorporating estimation information and nonuniform time-varying communication delays. This protocol aims to achieve bipartite time-varying formation tracking for multiagent systems. Next, a new augmented Lyapunov-Krasovskii functional with delay-product and triple integral terms is constructed. By integrating the second-order Bessel-Legendre inequality technique, the delay-dependent reciprocally convex combination inequality method, and the free weight matrix technique, a new guaranteed-cost bipartite time-varying formation tracking criterion is developed, offering less conservatism. Compared to certain existing delay conditions, the stability conditions assure a greater allowable upper bound of delay. Finally, a numerical example is conducted to confirm the validity and superiority of the theoretical findings.

Improving the insulin therapy for diabetic patients using optimal impulsive disturbance rejection: Continuous time approach

The paper proposes a new model-based optimization approach to improve the clinical efficiency of compensatory insulin bolus treatment in diabetic patients, aiming to mitigate the consequences of diabetes. The most important contribution of this paper is a novel methodology for determining the optimal parameters of insulin treatment, namely the size and timing of insulin boluses, to effectively compensate for carbohydrate intake. This concept can be seen as the so-called optimal model-based bolus calculator. The presented theoretical framework deals with the problem of optimal disturbance rejection in impulsive systems by minimizing an integral quadratic cost function. The methodology considers a personalized empirical transfer function model with static gains and time constants as the only parameters assumed to be known, making the bolus calculator more straightforward to implement in clinical practice. Contrary to other techniques, the proposed methodology considers impulsive insulin administration in the form of boluses, which is more feasible than continuous infusion. In contrast to the conventional bolus calculator, the proposed algorithm allows for maximizing therapy performance by optimizing the relative time of insulin bolus administration with respect to carbohydrate intake. Another feature to highlight is that the solution of the optimization problem can be obtained analytically, hence no numerical iterative solvers are required. Additionally, the continuous-time domain approach allows for a much finer adjustments of the insulin administration timing compared to discrete-time models. The proposed approach was validated in an in-silico study, which demonstrated the importance of systematically determined insulin–carbohydrate ratio and the relative delay between disturbance and its compensation. The results showed that the proposed optimal bolus calculator outperforms the traditional suboptimal formula.

UAV Trajectory Optimization for Maximum Soaring in Windy Environment

Optimal trajectory planning in a windy environment is a complex problem when airplane performance characteristics are considered. This paper introduces a novel form of Legendre pseudospectral optimization to solve boundary value problems in UAV trajectory planning. The proposed architecture applies Legendre–Gauss–Lobatto and Hamilton–Jacobi–Bellman equations to generate candidate pieces of trajectories with respect to the UAV dynamic constraints. Analytical performance-based solutions are also developed for sample cases to achieve an optimal criterion in trajectory planning. Moreover, the notion of wind soaring is exploited to use the beneficial effects of tailwind velocities in trajectory planning. Integral cost functions are handled by Gauss-type quadrature rules. Simulations demonstrate the efficiency of the pseudospectral method compared with other solvers, in eliminating the difficulties of boundary value problems by employing the boundary points in the interpolation equation. The effectiveness of the proposed approach is demonstrated through dynamic simulations.

Advanced Flowrate Control of Petroleum Products in Transportation: An Optimized Modified Model Reference PID Approach

Efficient flowrate control is paramount for the seamless operation and reliability of petroleum transportation systems, where precise control of fluid movement ensures not only operational efficiency but also safety and cost-effectiveness. The main aim of this paper is to develop a highly effective modified model reference PID controller, tailored to ensure optimal flowrate control of petroleum products throughout their transportation. Initially, the petrol transportation process is analyzed to establish a suitable mathematical model based on vital factors like pipeline diameter, length, and pump attributes. However, using a basic first-order time delay model for petrol transportation systems is limiting due to inaccuracies, variable delay issues, safety oversights, and real-time control complexities. To improve this, the delay portion is approximated as a third-order transfer function to better reflect complex physical conditions. Subsequently, the PID controller is synthesized by modifying its structure to address flowrate control issues. These modifications primarily focus on the controller’s derivative component, involving the addition of a first-order filter and alterations to its structure. To optimize the proposed controller, the genetic, black hole, and zebra optimization techniques are employed, aiming to minimize an integral time absolute error cost function and ensure that the outlet flow of the controlled system closely follows the response of an appropriate reference model. They are chosen for their proficiency in complex optimization to enhance the controller's effectiveness by optimizing parameters within constraints, adapting to system dynamics, and ensuring optimal conditions. Through simulations, it is demonstrated that the proposed controller significantly enhances the stability and efficiency of the control system, while maintaining practical control signals. Moreover, the proposed modifications and intelligent tuning of the PID controller yield remarkable improvements compared to previous related work, resulting in a 36% reduction in rise time, a 63% reduction in settling time, an 80% reduction in overshoot, and a 98% reduction in cost value.

Optimal Control Design and Online Controller-Area-Network Bus Data Analysis for a Light Commercial Hybrid Electric Vehicle

In this article, a hybrid powertrain for the Volkswagen (VW) Crafter is designed using the Model-In-The-Loop (MIL) method. An enhanced Proportional-Integral (PI) control technique based on integral cost functions is developed by carrying out a time-based simulation in MATLAB/Simulink software to realize the optimal fuel economy of the vehicle. Moreover, a comparative study is conducted between the vehicle’s hybrid and pure electric versions to assess the optimal battery energy consumption per unit distance traveled. Communication within our vehicles’ Electronic Control Units (ECUs) is facilitated by a message-based protocol called a Controller Area Network (CAN). Consequently, this paper presents an online CAN Bus data analysis using the Hardware-In-The-Loop (HIL) method. This method uses a standard frame, J1939 CAN protocol, implemented with Net CAN Plus 110 hardware. A graphical user interface is developed on a host Personal Computer (PC) using LabVIEW for decoding the acquired raw CAN data to physical values. The simulation results reveal that the proposed controller is promising and suitable for realizing optimal performance over the HIL method.

Consensus as a Nash Equilibrium of a Stochastic Differential Game

In this paper a consensus has been constructed in a social network which is modeled by a stochastic differential game played by agents of that network. Each agent independently minimizes a cost function which represents their motives. A conditionally expected integral cost function has been considered under an agent’s opinion filtration. The dynamic cost functional is minimized subject to a stochastic differential opinion dynamics. As opinion dynamics represents an agent’s differences of opinion from the others as well as from their previous opinions, random influences and stubbornness make it more volatile. An agent uses their rate of change of opinion at certain time point as a control input. This turns out to be a non-cooperative stochastic differential game which have a feedback Nash equilibrium. A Feynman-type path integral approach has been used to determine an optimal feedback opinion and control. This is a new approach in this literature. Later in this paper an explicit solution of a feedback Nash equilibrium opinion is determined.

Integrated learning self-triggered control for model-free continuous-time systems with convergence guarantees

This paper presents an integrated self-triggered control strategy with convergence guarantees for model-free continuous-time systems using reinforcement learning. To consider the control cost and triggering consumption in the self-triggered scheme simultaneously, an integrated cost function is proposed. With this integrated cost function, the trade-off between the triggering occupation and control performance could be adjusted according to different requirements. Then, the actor-critic framework of reinforcement learning is employed to learn the control inputs and triggering intervals by minimizing the corresponding integrated cost function. Considering the divergent characteristics between the control inputs and triggering intervals, two different actors are utilized to learn the triggering strategy and control policy, respectively. Also, the convergence of the developed model-free self-triggered control learning algorithm is proved to ensure the limited learning duration of both the control policy and triggering strategy. The proposed framework can be used to design self-triggered controllers for a wide range of engineering systems with unknow dynamics, including control of aircraft, robots, chemical processes, and other automated systems. Finally, the effectiveness and superiorities of the proposed method are verified by an illustrative example.

Optimal design of fuzzy-PID controller for automatic generation control of multi-source interconnected power system

This paper suggests a fuzzy logic controller (FLC) structure from seven membership functions (MFs) and its input–output relationship rules to design a secondary controller to reduce load frequency control (LFC) issues. The FLC is coupled to a proportional–integral–derivative (PID) controller as the proposed FPID controller, which is tuned by an optimized water cycle algorithm (WCA). The proposed WCA: FPID scheme was implemented with two models from the literature under the integral time absolute error cost function. Initially, a two-area non-reheat unit was implemented, and the gains of PID and FPID controllers were adjusted to verify the suitability of WCA in solving LFC issues. Then, in order to identify the robustness of the closed-loop system, sensitivity analysis is carried out. Additionally, a two-area non-reheat unit was tested under the governor dead band nonlinearity. To guarantee the suitability of the proposed FPID controller, a model with a mixture of power plants, such as reheat, hydro, and gas unit in each area was carried out with and without the HVDC link, which can increase practical issues with LFC. The proposed controller's robustness was studied for all models under numerous scenarios, step load perturbations (SLP), and different objective functions. Simulation results proved that the proposed FPID controller provided superior performance compared to recently reported techniques in terms of peaks and settling time.

Adaptive actor-critic learning-based robust appointed-time attitude tracking control for uncertain rigid spacecrafts with performance and input constraints

This paper addresses the prescribed performance attitude tracking control problem for a rigid spacecraft with input constraint and bounded disturbance on the special orthogonal group (SO(3)). By introducing a proper pseudo-Morse function as the attitude error function, the tracking problem on matrix manifold SO(3) is reformulated as an equivalent stabilization problem in the associated Lie algebra (so(3)). To on-line compensate the system uncertainties, an actor-critic neural network (NN) architecture is incorporated into the controller design process. The critic NN intends to reconstruct the long-term integral cost function and evaluate the prescribed performance attitude tracking quality. Based on the output of the critic NN, the actor NN is utilized to reconstruct the unknown system uncertainties and generate the auxiliary compensation control. By introducing the actor-critic architecture into the logarithm barrier Lyapunov function method, a model-free attitude tracking controller is proposed. Furthermore the input saturation limit is considered to support the normal operation of the rigid spacecraft. Besides, an adaptive robustifying term is designed to enhance the disturbance attenuation performance. By incorporating the Lyapunov stability analysis, we prove that the designed adaptive robust controller can ensure that all signals are uniformly ultimately bounded (UUB), and prescribed performance constraints are not destroyed. Finally, numerical simulations are included to verify the performance of the proposed adaptive robust tracking control strategy with intelligent actor-critic learning mechanism.

Universal Singular Optimal Control: Affine Systems

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

Optimality for Control Problem with PDEs of Second-Order as Constraints

This paper deals with a class of second-order partial differential equation (in short, PDE) constrained optimal control problems. More specifically, by using appropriate variational techniques, we state necessary conditions of optimality associated with this class of optimization problems, defined by controlled curvilinear integral cost functionals involving partial derivatives of second-order. The importance of the considered problem is provided by its applications in mechanics and physics. Compared with other research works, here we develop a new mathematics context that extends the results obtained so far, both through the use of controlled curvilinear integrals and also by considering partial derivatives of second-order. In addition, to emphasize the usefulness of the main results, an illustrative example is provided.

Dual-Loop Optimal Control of a Robot Manipulator and Its Application in Warehouse Automation

The goal of this work is to develop the next generation of coordinated optimal planning and control schemes for real-world robotic applications, with cost-effective intelligent robots that can safely and robustly perform the tasks at hand. This article presents a dual-loop optimal hierarchical control scheme for robotic manipulators consisting of outer and inner loops. The outer kinematic control loop in the operational space provides a joint velocity reference signal to the inner one. The kinematic control is formulated as a closed-loop optimal control approach to trajectory generation where the desired target (possibly time-varying) is obtained by acting upon the feedback from the actual state of the robot. The kinematic control is obtained using a closed-form analytic solution of the Hamilton–Jacobi–Bellman (HJB) equation. The proposed methodology defines the task in terms of the integral cost function, which results in a global optimal solution. The inner dynamic control loop consists of a neural network (NN)-based adaptive critic (AC) optimal tracking control scheme. The online NN approximator-based dynamic controller learns the infinite-horizon cost function related to inner loop error dynamics in continuous time and calculates the corresponding optimal control input to minimize the cost function forward in time. The stability and the performance of the proposed control scheme are shown theoretically via the Lyapunov approach and also verified experimentally using a 7-DOF Barrett WAM robot manipulator. The warehousing applications of the proposed dual-loop control scheme have been demonstrated experimentally in an exact warehouse setting. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article was motivated by our previous experiences in the Amazon Robotics Challenge (ARC) competitions as a participant. Although the major lessons from those events revealed many important issues toward warehouse automation both from robotic vision and grasping aspects, the application aspects of control engineering to solution for these real-world problems could significantly benefit the today’s highly automated society. The intention of this article is to address this situation by designing a cost-effective robot manipulator control scheme by optimizing robot manipulation trajectories in the outer loop along with the actuator input in the inner loop. Therefore, we propose a dual-loop optimal control scheme for robotic manipulations under a cost-optimal framework with guaranteed stability proof via the Lyapunov approach. This results in smoother trajectories with cost-effective control actuator inputs than the state-of-the-art methods. We believe that this work can be beneficial for academic and industrial research to design more advanced and optimized solutions in this domain.

Robust saddle‐point criterion in second‐order partial differential equation and partial differential inequation constrained control problems

AbstractIn this article, for a given class of multi‐dimensional scalar variational control problems (named ) with mixed constraints implying second‐order partial differential equations and inequations, we introduce an auxiliary (modified) class of variational control problems (named ), which is much easier to study, and provide some characterization results of and by using the notions of normal weak robust optimal solution and robust saddle‐point associated with a Lagrange functional corresponding to . For this aim, we consider scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi‐time controlled second‐order Lagrangian.

On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

On a Class of Second-Order PDE&amp;PDI Constrained Robust Modified Optimization Problems

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution.

Second-Order PDE Constrained Controlled Optimization Problems with Application in Mechanics

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.

Actor-critic-based predefined-time control for spacecraft attitude formation system with guaranteeing prescribed performance on SO(3)

In this paper, an attitude formation control problem of multi-spacecraft systems is investigated on the special orthogonal group (SO(3)) under the undirected tree communication topology. A novel appointed-time performance function is developed to characterize the user-assigned transient and steady state performance. To online compensate the system uncertainties, an actor-critic neural network architecture is utilized in the control design process. The critic neural network intends to approximate the long-term integral cost function, which can evaluate the consensus performance of the formation system. Based on the exported reinforcement signal, the actor neural network is introduced to generate the feedforward compensation term to cope with the system uncertainties. By utilizing the barrier Lyapunov function, a model-free decentralized control scheme is constructed with an adaptive robustifying term to enhance the disturbance rejection ability, in which the control input for each spacecraft is produced by using only the local relative information from its neighbors. Moreover, an auxiliary system is devised to cope with the input saturation constraint. By using the Lyapunov stability theory, the stability and convergence of the closed-loop system are analyzed. Finally numerical simulations are conducted to illustrate the feasibility and effectiveness of the proposed adaptive actor-critic learning-based control scheme.

Harmonic-Based Optimal Motion Planning in Constrained Workspaces Using Reinforcement Learning

In this work, we propose a novel reinforcement learning algorithm to solve the optimal motion planning problem. Particular emphasis is given on the rigorous mathematical proof of safety, convergence as well as optimality w.r.t. to an integral quadratic cost function, while reinforcement learning is adopted to enable the cost function's approximation. Both offline and online solutions are proposed, and an implementation of the offline method is compared to a state-of-the-art RRT* approach. This novel approach inherits the strong traits from both artificial potential fields, i.e., reactivity, as well as sampling-based methods, i.e., optimality, and opens up new paths to the age-old problem of motion planning, by merging modern tools and philosophies from various corners of the field.