Time-delayed Optimal Control Problems Research Articles

An iterative method to solve the time delay optimal control problem with bounded control variable

This paper presents an iterative approach for solving the optimal control problem (OCP) with time-delayed terms and bounded control variables. The projection function is used to construct the iterative method to approximate the control law. Employing the approximation of control law the approximation of state and the co-state variables are obtained. To the best of our knowledge, this paper is the first attempt to solve bounded and delayed OCPs with a projection function. The convergence theorem for the proposed method is also proven. Several numerical examples are added, showing the developed method's effectiveness to solve the time-delayed OCPs with bounded control variables.

Sequential time scaling transformation technique for time-delay optimal control problem

The time-scaling transformation technique used within the computational framework of control parameterization serves as an effective method for the optimization of control switching time as well as the control value for time-delay optimal control problems. However, the conventional time-scaling transformation stipulates that the switching times for all control components must be identical, which may not be feasible in real-world scenarios. To address this limitation, we introduce the sequential time-scaling transformation technique for solving time-delay optimal control problems. By sequentially applying control parameterization and time-scaling transformation to each of the control variable, the new technique enables the configuration of a varying number of segments for the control, and the switching time for each control component can be adaptively selected. Numerical results demonstrate that our approach not only achieves superior cost performance but also offers enhanced flexibility in control strategy.

Optimization over the Pareto front of nonconvex multi-objective optimal control problems

Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.

An efficient method for a variety of fractional time-delay optimal control problems with fractional performance indices

An efficient method for a variety of fractional time-delay optimal control problems with fractional performance indices

The Ritz numerical method and hybrid functions (block-pulse functions and legendre polynomials) for a class of two-dimensional time-delay optimal control problems

In this paper, we provided a numerical method to solve a class of two dimensional time-delay optimal control problems (2DTDOCPs) with quadratic cost functional using Ritz method and orthogonal Legendre Block-Pulse functions. First, the state and control vectors are approximated as a series of hybrid functions(block-pulse functions and Legendre polynomials) with unknown coefficients. Then, we derive an equation with unknown coefficients by substituting these approximations in the cost functional. A system of algebraic equations is obtained by applying the optimal conditions for this equation. Solving this system and substituting the coefficients into approximating the guessed functions, the state and control functions are obtained. By increasing the number of blocks, as well as the basic functions, we get more accurate solutions. The convergence of proposed method is discussed, and finally, we will present some examples to show the validity and applicability of proposed method, and evaluate its accuracy and efficiency. Moreover, our results are compared to previous results to show the superiority of this technique.

Razumikhin and Krasovskii approaches for safe stabilization

This paper studies the stabilization and safety problems of nonlinear time-delay systems. Following both Razumikhin and Krasovskii approaches, we propose novel control Lyapunov functions/functionals for the stabilization problem and novel control barrier functions/functionals for the safety problem. The proposed control Lyapunov and barrier functions/functionals extend the existing ones from the delay-free case to the time-delay case, and allow for designing the stabilizing and safety controllers in closed-form. Since analytical solutions to time-delay optimal control problems are hard to be achieved, a sliding mode control based approach is developed to merge the proposed control Lyapunov and barrier functions/functionals. Based on the sliding surface functional, a feedback control law is established to investigate the stabilization and safety objectives simultaneously. In particular, the properties of the sliding surface functional are analyzed, and further how to construct the sliding surface functional is discussed. Finally, the proposed approaches are illustrated via two numerical examples from the connected cruise control problem of automotive systems and the synchronization problem of multi-agent systems.

Optimal control for multistage uncertain random dynamic systems with multiple time delays

A complex system exhibiting uncertainty as well as randomness is an uncertain random dynamic system. A novel idea of an uncertain random matrix is introduced as an interaction between uncertainty and randomness. Thereby a time-delayed uncertain random dynamic system is formulated. And then the corresponding time-delayed optimal control problem is first presented in this paper. To tackle a problem like that, a method is proposed to translate it into an equivalent optimization problem without time delay. The relevant recursion equations are developed with the dynamic programming technique to solve the converted optimization problem. Solving the related recursion equations yields the answer to such an optimization problem. The presented approach offers a unified framework for tackling the discrete-time uncertain random optimal control problem involving time delays. In this framework, three algorithms are successfully devised in doing with time-delayed optimal control problems for dynamic systems regarding linear, quadratic, and cubic controls. Meanwhile, the optimal solutions for numerical examples are found by the raised algorithms. Finally, a problem of optimal control for a two-spool turbofan engine is investigated as an application of our results.

A novel high accurate numerical approach for the time-delay optimal control problems with delay on both state and control variables

<abstract><p>In this study, we intend to present a numerical method with highly accurate to solve the time-delay optimal control problems with delay on both the state and control variables. These problems can be seen in many sciences such as medicine, biology, chemistry, engineering, etc. Most of the methods used to work out time delay optimal control problems have high complexity and cost of computing. We extend a direct Legendre-Gauss-Lobatto spectral collocation method for numerically solving the issues mentioned above, which have some difficulties with other methods. The simple structure, convergence, and high accuracy of our approach are the advantages that distinguish it from different processes. At first, by replacing the delay functions of state and control variables in the dynamical method, we propose an equivalent system. Then discretizing the problem at the collocation points, we achieve a nonlinear programming problem. We can solve this discrete problem to obtain the approximate solutions for the main problem. Moreover, we prove the gained approximate solutions convergent to the exact optimal solutions when the number of collocation points increases. Finally, we show the capability and the superiority of the presented method by solving some numeral examples and comparing the results with those of others.</p></abstract>

Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

Application of periodic Bernoulli polynomials in approximating the solution of linear time-delay optimal control problems

In this paper, a numerical method based on application of periodic Bernoulli polynomials for solving optimal control of linear time-delay systems with quadratic performance index is presented. Some important properties of Bernoulli polynomials are reviewed and a new operational matrix of delay based on these functions is introduced. Also an error bound for approximating a function with periodic Bernoulli polynomials is presented. Using operational matrices of differentiation, delay and the integration of the cross product of two Bernoulli polynomials, the linear time delay optimal control problem with quadratic objective function is converted to a nonlinear optimization problem. Finally, numerous illustrative examples are studied to show the efficiency and accuracy of the proposed method compared with some other methods.

A new gradient computational formula for optimal control problems with time-delay

<p style='text-indent:20px;'>In this paper, we consider a class of time-delay optimal control problem (TDOCP) with canonical equality and inequality constraints. By applying control parameterization method together with time-scaling transformation, a TDOCP can be readily solved by gradient-based optimization methods. The partial derivative of the cost as well as the constraint functions with respect to the decision variables are obtained by variational approach, which is inefficient when the discretization for the control function is relatively dense. For general optimal control problem without time-delay, co-state approach is an effective way to compute the gradients, however, when time-delay is involved in the dynamic system, the co-state system is not known. In this paper, we derive the co-state system for TDOCP to compute the gradients of the cost and constraints. Numerical results show that the computational efficiency is much higher when compared with the traditional variational approach.</p>

Solving infinite-horizon optimalcontrol problems of the time-delayedsystems by a feed forward neural network model

ABSTRACT A numerical method using neural network for solving infinite-horizon time-delayed optimal control problems is studied. The problem is first transformed, using a Páde approximation, to one without a time-delayed argument. By a suitable change of variable, the obtained non-delay infinite-horizon optimal control problem is converted to a finite-horizon nonlinear optimal control problem. We try to approximate the solution of Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. We then minimize the error function where weights and biases associated with all neurons are unknown. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method.

A symplectic pseudospectral method for constrained time-delayed optimal control problems and its application to biological control problems

ABSTRACT Time-delays are often involved in modelling biological processes, which arise more difficulties for the control task. In fact, once the control goal is given, a constrained optimal control problem can be formulated to describe the biological control mission. However, it’s almost impossible to obtain analytical solutions for such complicated problems, especially for those with high nonlinearity and complex constraints. In this paper, a symplectic pseudospectral method for solving nonlinear optimal control problems of constrained stated-delayed system is developed. The original nonlinear problem is transformed into a series of linear-quadratic optimal control problems by the successive convexification technique at the first implementation. By applying the local Legendre-Gauss-Lobatto pseudospectral method and the parametric variational principle, each of the transformed linear-quadratic problems is converted into a standard linear complementarity problem which can be solved easily by the Lemke’s method. The cost functional, system dynamics and constraints can be generally nonlinear functions of state, control and time variables. and three kinds of inequality constraints, i.e. pure-state, pure-control and mixed state-control, can be treated under a uniform framework. The developed symplectic pseudospectral method is finally validated by an optimal harvesting problem and an optimal vaccination problem.

Model-based experimental design for nonlinear dynamical systems with unknown state delay and continuous state inequalities

This paper studied the model-based design of experiment (MBDoE) for nonlinear dynamical systems with unknown state delay and continuous state inequalities (SDCSI). The design problem is formulated as a time-delay optimal control problem of a dynamic system governed by augmented sensitivity-state equations (ASSE). Consider the co-existence of control delay and multiple state delays in the ASSE, each continuous state inequality constraint is approximated by an integral constraint using a constraint-handling technique based on local smoothing approximation and constraint transcription, which guaranteed the satisfaction of state inequality constraints during the experiment. In addition, an efficient numerical procedure is derived to determine the gradients of the objective function and constraints, which involved integrating an auxiliary impulsive time-delay system backward in time. The procedure is combined with standard gradient-based optimization methods to solve the MBDoE-SDCSI problem. The performance of the proposed scheme is illustrated through the experimental design of a fed-batch biomass fermentation process.

Linear optimal control of time delay systems via Hermite wavelet

To solve the time delay optimal control problem with quadratic performance index, a direct numerical method based on Hermite wavelet has been proposed in the present study. The idea is to convert the time delay optimal control problem into a quadratic programming problem. To do so, various time functions in the system are expanded as their truncated series and the properties of the operational matrices of integration, delay and product of two Hermite wavelet vectors are used as well. These matrices are utilized to reduce the solution of optimal control with time delay system, to the solution of a quadratic programming with linear constraints. Finally, three examples of time varying and time invariant coefficients are given to compare the results with some of the existing methods.

Time-scaling transformation for optimal control problem with time-varying delay

This paper focuses on the solution to nonlinear time-delay optimal control problems with time-varying delay subject to canonical equality and inequality constraints. Traditional control parameterization in conjunction with time-scaling transformation could optimize control parameters and switching times at the same time when the time-delays in the dynamic system are taken as constants. The purpose of this paper is to extend this method to solve the dynamic systems with time-varying delay. We introduce a hybrid time-scaling transformation that converts the given time-delay system into an equivalent system defined on a new time horizon with fixed switching times. Meanwhile, we obtain the value of time-delay state utilizing the relationship between the new time scale and the original one. After computing the gradients of the cost and constraints with respect to the control heights and its durations, we could solve the equivalent optimal control problem using gradient based optimization method.

Solving optimal control problems of the time-delayed systems by a neural network framework

ABSTRACTA numerical method using neural networks for solving time-delayed optimal control problems is studied. The problem is first transformed into one without a time-delayed argument, using a Páde approximation. We try to approximate the solution of the Hamiltonian conditions based on the Pontryagin minimum principle (PMP). For this purpose, we introduce an error function that contains all PMP conditions. We then minimise the error function where weights and biases associated with all neurons are unknown. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method.

Optimal Control Problems with Time Delays: Constancy of the Hamiltonian

This paper concerns necessary conditions of optimality for optimal control problems with time delays in the state variable. It is well known that, when there are no time delays and the dynamics are autonomous, the standard necessary conditions in the form of a maximum principle can be supplemented by an extra condition, namely “constancy of the Hamiltonian" along optimal trajectories (and associated costate trajectories). This property, possibly supplemented by other invariance principles, has been used to investigate properties of optimal trajectories, such as solution regularity, without the need to solve the underlying extremal equations. In classical mechanics, for example, the constancy of the Hamiltonian condition can be used to derive a conservation of energy principle from Hamilton's principle of least action. While the maximum principle has been generalized to cover time delays, the validity of constancy of the Hamiltonian-type conditions has not been previously investigated. We provide the first “extra" optimality condition of this nature for autonomous, time delay optimal control problems. The new “constancy of the Hamiltonian" condition involves a correction term, without which the condition is not valid. We illustrate the significance of this condition by applications to minimizer regularity and conservation laws in nonclassical Hamiltonian mechanics.

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

ABSTRACTThis work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.

Optimal control of linear time-varying systems with state and input delays by Chebyshev wavelets

This paper presents a novel method for finding the optimal control, state and cost of linear time-delay systems with quadratic performance indices. The basic idea here is to convert a time-delay optimal control problem into a quadratic programming one which can be easily solved using MATLABr. To implement this idea we choose a state and control parameterization method by using Chebyshev wavelets. The inverse time operational matrix of Chebyshev wavelets is introduced and applied for parameterizing state and control terms containing inverse time. The method is also applicable to linear quadratic time-delay systems with combined constraints. Illustrative examples demonstrate the validity and applicability of the approach which new expansions for initial vector functions of state and control variables are defined.