Time Control Problem Research Articles

Comparative study of PI-controller and neurocontroller performances in optimal by settling time control problems

When developing control systems, an important issue arises of choosing an operator, which forms the control function. The standard approach is to use a PI- or PID-controller, a more advanced approach involves ANNs training for this purpose. A comparative analysis of the PI- and neurocontroller performances makes it possible to establish the disadvantages and advantages of each of the compared controllers, which is an important scientific and applied problem. The purpose of the work was to conduct a comparative analysis of the performance of the PI-controller and the neurocontroller based on a set of evaluation indicators for plants of the second and third orders. Such a comparison was carried out by using an approach to the synthesis of both controllers, which involved the minimization of a complex objective function. The latter is obtained as a result of reducing the problem of optimal control with constraints to the problem of unconstrained optimization. The analysis showed that according to the settling time indicator (optimization criterion), the neurocontroller has an advantage of 6.1...96.2% for the modelled plants. At the same time, according to other indicators of the control quality, the PI-controller has an advantage. In addition, the synthesis of a neurocontroller in terms of finding the minimum of the objective function is a more difficult problem. For its solution, a bigger number of iterations of the VCT-PSO optimization algorithm is required. It is rationally to set more than 1000 iterations and swarm population in the range 30…50 particles. A comparative analysis by the settling time of the neurocontroller and PI-controller, which is tuned according to engineering methods, showed significant reserves for improving this indicator. Thus, if the requirements for settling time minimization are quite strict, then it is advisable to use a neurocontroller. The obtained results will make it possible to develop recommendations for the rational choice of the control operator when solving practical problems of the control systems synthesis

Cooperative attack strategy during the terminal guidance phase for loitering munitions based on impact angle and impact time constraints

In order to solve the cooperative attack problem during the terminal guidance phase for loitering munition swarm, a cooperative attack strategy for loitering munitions with impact angle constraints is proposed. By using the small deviation method, an offset proportional guidance law with impact angle constraints is derived in order to solve the problem of directional strike in the longitudinal plane. Combining with the present guidance scheme, a method for estimating the remaining time of proportional control and angle control is derived. In the lateral plane, a time control bias term with time-varying coefficients and no singularity in the division of zero is designed to solve the time control problem. In addition, taking the remaining time as the coordination variable, a two-layer cooperative architecture is used to solve the problem of coordinated directional strike of multiple loitering munitions. The simulation results show that the cooperative guidance strategy has a good performance of impact angle control and cooperative attack for low-speed loitering munitions.

Interior of the Integral of a Set-Valued Mapping and Problems with a Linear Control System

The dependence of the radius of a ball centered at zero inscribed in the values of the integral of a set-valued mapping on the upper integration limit is studied. For some types of integrals, exact asymptotics of the radius with respect to the upper limit are found when the upper limit tends to zero. Examples of finding this radius are considered. The results obtained are used to derive new sufficient conditions for the uniformly continuous dependence of the minimum time and solution-point in the linear minimum time control problem on the initial data. We also consider applications in some algorithms with a reachability set of a linear control system

Linear Convergence of a Policy Gradient Method for Some Finite Horizon Continuous Time Control Problems

Linear Convergence of a Policy Gradient Method for Some Finite Horizon Continuous Time Control Problems

Adaptive Prescribed Finite Time Control for Strict-Feedback Systems

In this paper we address the prescribed finite time control problem for a class of strict-feedback systems with unknown parameters. A backstepping based adaptive prescribed-time control algorithm is proposed for second-order strict-feedback systems, where the global stability of system is ensured, and a dynamic surface control (DSC) based adaptive prescribed-time algorithm is designed for the system in high-order case. For the DSC based method, a novel first-order filter is constructed to guarantee the boundedness of “virtual error”, which avoids the so-called “Differential Explosion” problem, however, the control result is only semi-global. Both the developed algorithms are capable of ensuring the regulation in prescribed time with a unique converging feature in that the convergence time is independent of any initial conditions and other design parameters that can be pre-assigned freely by the designer according to the control requirements. The key to achieve the objective in prescribed finite time is the introduction of a descending power time-varying feedback into the controller design. Both the theory analysis and simulation confirm the effectiveness of the proposed methods.

Modeling and Optimal Control Analysis on Armed Banditry and Internal Security in Zamfara State

In Nigeria, Banditry which includes kidnapping, is currently a national threat that led to the loss of lives, livelihoods, crippled economic activities, broken social cohesion and above all led to a massive displacement of people especially in Zamfara State. We present a deterministic mathematical model for controlling the spread of armed banditry using job creation, ()1ut and efforts to make armed banditry unprofitable, ()2ut. We formulate a fixed time control problem subject to the model dynamics with aim to obtain the suitable optimal combination of the two control measures that will minimize the cost of the control efforts as well as the incidence of the menace. The simulations was carried out using the estimate initial conditions and parameter values obtained from the existing literature and hypothetical values. The Pontryagin’s maximum principle was employed to characterize the optimality system and solve the system numerically. The results obtained on simulations suggests that the application of any of the controls is effective in decreasing the population profile of the informers and the bandits in a finite time. Furthermore when both controls are combined that is efforts to make armed banditry unprofitable ()2utshould be at upper bound for a longer duration when compared with the job creation control, ()1ut.

Reduced-Order Observer-Based Preassigned Finite-Time Control of Nonlinear Systems and Its Applications

In this article, a preassigned finite time control problem of nonlinear systems in strict-feedback form is investigated. From the perspective of arbitrary settling time, an appropriate preassigned finite-time performance function (PFPF) is constructed, and the preassigned finite-time stability (PAFS) is established, where the settling (convergence) time is not only completely unconcerned with initial conditions and design parameters but also more flexible. Furthermore, the backstepping technique and reduced-order observer are used to obtain the preassigned finite-time control scheme. The stability criteria of PAFS are developed to guarantee that the output can quickly converge to an arbitrarily small zone in preassigned time, and all signals of the closed-loop control system are PAFS. In the end, simulation examples verify the effectiveness of the presented method.

Singular Optimal Linear Quadratic Regulator of Switched Systems

In this paper, we will investigate the singular linear quadratic regulator (LQR) problem of switched linear system in finite time horizon. The proposed method is transforming into a switched LQR problem by adopting linear transformation. Next, we adopt an embedding transformation method to convert the switched LQR problem to a traditional optimal control problem, so the bang-bang-type solution of the embedded optimal control problem in finite time is the optimal solution to the switched LQR problem. The switching sequence of modes and the switching instants can be calculated by solving a closed-form optimal switching condition. The optimal state feedback control law is determined simultaneously. Then, by solving a sequence of Riccati equation, we find some conditions that ensure switched LQR problem can be convert to the singular LQR problem. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed method.

Three-dimensional impact-time-constrained proportional navigation guidance using range-varying gain

This paper presents a novel feedback guidance law to address the impact time control problem of homing missiles in three-dimensional (3D) space. First, a range-varying gain is incorporated into the conventional proportional navigation guidance (PNG) law, which leads to globally precise time-to-go solution and achievable impact time region. Next, an impact time control guidance (ITCG) law is developed by augmenting the proposed varying-gain PNG law with a feedback biased term. Unlike most existing studies, the proposed ITCG law does not involve any linearized approximations or numerical iterative schemes. Furthermore, it can achieve the desired impact time and high-precision interception for a wide launch envelop with arbitrary initial leading angles. Finally, extensive numerical simulations, including comparison studies and Monte Carlo assessments, are carried out to validate the theoretical findings.

Premium control with reinforcement learning

AbstractWe consider a premium control problem in discrete time, formulated in terms of a Markov decision process. In a simplified setting, the optimal premium rule can be derived with dynamic programming methods. However, these classical methods are not feasible in a more realistic setting due to the dimension of the state space and lack of explicit expressions for transition probabilities. We explore reinforcement learning techniques, using function approximation, to solve the premium control problem for realistic stochastic models. We illustrate the appropriateness of the approximate optimal premium rule compared with the true optimal premium rule in a simplified setting and further demonstrate that the approximate optimal premium rule outperforms benchmark rules in more realistic settings where classical approaches fail.

Lyapunov-Based Impact Time Control Guidance Law with Performance Prediction

This paper proposes an impact time control guidance law based on exact nonlinear kinematics equations. To address the impact time control problem of providing enhanced intercept accuracy, we formulated an error variable whose regulation ensures the fulfillment of the required tasks without time-to-go estimation. Based on the Lyapunov stability theory, a desired line-of-sight rate profile that satisfies the convergence of the error variable was constructed, from which the guidance command was designed using the optimal tracking formulation. The simple structure of the proposed guidance law enables the prediction of interceptor behavior during homing, thereby allowing the interceptor to maneuver along feasible trajectories. In addition, although the structure of the proposed guidance law is simple and similar to that of proportional navigation, it is theoretically guaranteed to execute the required mission precisely at the end of homing. Numerical simulations demonstrated that the proposed guidance law achieved effective target interception under various terminal constraint settings.

Invariant ideals and their applications to the turnpike theory

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

Hamiltonian-Driven Adaptive Dynamic Programming With Approximation Errors

In this article, we consider an iterative adaptive dynamic programming (ADP) algorithm within the Hamiltonian-driven framework to solve the Hamilton-Jacobi-Bellman (HJB) equation for the infinite-horizon optimal control problem in continuous time for nonlinear systems. First, a novel function, "min-Hamiltonian," is defined to capture the fundamental properties of the classical Hamiltonian. It is shown that both the HJB equation and the policy iteration (PI) algorithm can be formulated in terms of the min-Hamiltonian within the Hamiltonian-driven framework. Moreover, we develop an iterative ADP algorithm that takes into consideration the approximation errors during the policy evaluation step. We then derive a sufficient condition on the iterative value gradient to guarantee closed-loop stability of the equilibrium point as well as convergence to the optimal value. A model-free extension based on an off-policy reinforcement learning (RL) technique is also provided. Finally, numerical results illustrate the efficacy of the proposed framework.

Linear programming based optimality conditions and approximate solution of a deterministic periodic optimal control problem in discrete time

This paper is focused on a discrete time periodic optimal control problem. We establish linear programming based optimality conditions and present a method for the construction of near optimal controls. A numerical example is included to demonstrate the results.

Appointed Fixed Time Observer-Based Sliding Mode Control for a Quadrotor UAV Under External Disturbances

In this article, a study of the appointed fixed time control problem is presented for the quadrotor unmanned aerial vehicle attitude control system subject to external disturbances. Based on the appointed fixed time sliding mode variable and the adaptive technique, a novel sliding mode observer is first established to estimate the external disturbances within the appointed settling time. Subsequently, an appointed fixed time controller is proposed to track the desired attitude regardless of the initial states. To circumvent the singularity problem of the above controller, a novel improved adaptive sliding mode control law is designed by employing a nonlinear function and a modified sliding mode observer. Moreover, the chattering phenomenon is attenuated by replacing the sign function with hyperbolic tangent function. The appointed practical fixed time stability of the closed-loop attitude control system is proved and analyzed rigorously utilizing the Lyapunov theory. Finally, the effectiveness and fine performance of the proposed schemes are illustrated by numerical simulation results.

Equivalence of three kinds of optimal control problems for linear heat equations with memory

This paper studies an equivalence theorem for three different kinds of optimal control problems, which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.

Minimal time control problem of a linear heat equation with memory

This paper studies a minimal time control problem for a linear heat equation with memory. The purpose of such a problem is to find a control (among certain control constraint set), which steers the solution of the heat equation with memory from a given initial state to a given target as soon as possible. In this paper, we study the existence of optimal control to this problem, show the bang–bang property of the optimal control and build up a necessary and sufficient condition of the optimal control.

A Macroeconomic SIR Model for COVID-19

The COVID-19 pandemic and subsequent lockdowns highlight the close and delicate relationship between a country’s public health and economic health. Models that combine macroeconomic factors with traditional epidemic dynamics to calculate the impacts of a disease outbreak are therefore extremely useful for policymakers seeking to evaluate the best course of action in such a crisis. We developed a macroeconomic SIR model that considers herd immunity, behavior-dependent transmission rates, remote workers, and the indirect externalities of lockdowns. It is formulated as an exit time control problem where a social planner is able to prescribe separate levels of the lockdown low-risk and high-risk portions of the adult population. The model predicts that by considering the possibility of reaching herd immunity, high-risk individuals are able to leave lockdown sooner than in models where herd immunity is not considered. Additionally, a behavior-dependent transmission rate (which represents increased personal caution in response to increased infection levels) can lower both output loss and total mortality. Overall, the model-determined optimal lockdown strategy, combined with individual actions to slow virus transmission, is able to reduce total mortality to one-third of the model-predicted no-lockdown level of mortality.

Optimal Guidance and Control of a Low-Altitude Skid-to-Turn Vehicle

The problem of optimal guidance of a low-altitude skid-to-turn vehicle is considered. In particular, the skid-to-turn vehicle is both guided and controlled using a recently developed computational optimal guidance and control method that employs Gaussian quadrature collocation. First, this recently developed method is used to re-solve the optimal control problem on a shrinking horizon at constant guidance cycles. The performance of this computational guidance and control method is then analyzed and compared against a classical LQ follower. It is found that, although the computational efficiencies of both methods are comparable, the computational method for optimal guidance and control is significantly more accurate than that of the LQ follower. The results of this study show that the recently developed computational method for optimal guidance and control has the potential to improve performance over traditional methods when solving an optimal control problem in real time.

Fixed time stability of a class of chaotic systems with disturbances by using sliding mode control

The main goal of this article is to consider the fixed time control problem of perturbed chaotic systems by virtue of sliding mode control. For this aim, this article presents a novel fixed time stability theorem at first by the Lyapunov tools. Then combining the obtained stability theorem and sliding mode technique, a new sliding mode surface is constructed and some novel controllers are designed appropriately to stabilize the discussed chaotic system. The proposed controllers have two main advantages: (1) The control criteria is robust against the effects of perturbations. (2) The convergence time, which is only dependent on the control parameters regardless of the initial conditions, is bounded by a fixed constant. Finally two typical systems are taken as the numerical examples to verify the validity of the control strategy.