A novel approach for the numerical investigation of optimal control problems containing multiple delays

Abstract

SummaryThis paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block-pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points.

Legendre Wavelets with Scaling in Time-delay Systems

“…, r can be obtained by (12). Remark 2 It should be noted that the presented expansions of the integrals of delayed terms which were introduced in [6] and have been used in [4,7], differ from the expansions presented in similar works for those integrals, that is, t 0 E µ (t )x(t − h µ )dt and t 0 F ν (t )u(t − h ν )dt , for example, some of the literature are [29][30][31][32] and those listed in [6]. When we use our expansions, by callingP T ξẼ µξ θ µξ andP T ξF νξ ζ νξ , we can see (43) and (44) provide exact results and there is no need to define a matrix for integrating the desired wavelet vector from 0 to delay(s), that is, the constant matrix Z have been defined in the literature.…”

Hybrid Functions Direct Approach and State Feedback Optimal Solutions for a Class of Nonlinear Polynomial Time Delay Systems

“…Statement of the Problem. Consider the system defined by (28), (29), and (33) with an initial condition (0) = 0 . Our objective is firstly to find the optimal open loop control * ( ), which minimizes the performance index:…”

“…It is worth noting that parametrization relies basically on orthogonal functions or wavelets [20][21][22]; however that tool have been used to solve various other problems of dynamic systems like identification (see [8]), tracking control (see [23]), observer based control (see [24]), or minimum time control (see [25]). The main characteristic of this pseudo-spectral technique is that it allows transforming complex dynamic optimization problems to solving a set of algebraic equations in the least square sense in the linear systems case [26,27] or permits formulating an equivalent nonlinear static programming problem for problems related to nonlinear systems [13,28,29].…”

“…In the nonlinear time delay optimal control problems context, an approach using hybrid functions which consist of block pulse functions and orthonormal Taylor series (see [15,29]) had been proposed, where authors propose to solve the necessary and sufficient condition equations for stationary emanating from the Hamiltonian based on state and control coefficients over the basis. Similarly, [28] propose a direct approach based on a hybrid of block pulse functions and Lagrange interpolating polynomials in order to convert the original optimal problem containing multiple delay into a mathematical programming one, where the resulting optimization problem is solved numerically by the Lagrange multipliers method. Reference [27] proposed similar approach based on hybrid functions of block pulse and Bernouilli polynomials, while [30] uses biorthogonal cubic Hermite spline multiwavelets in addition to block pulse functions to constitute the hybrid basis.…”

Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

“…The pilot models for aircraft control design purposes are another problem with time delay models that are used in human operator models, frequently. We can find some applications of the optimization control problems in the work of Bohannan 18 and Suárez et al 19 Furthermore, there are several numerical methods to solve delay optimal control problems such as method based on biorthogonal cubic Hermite spline multiwavelets, 20 linear programming, 21 hybrid-Lagrange interpolating polynomials, 22,23 hybrid-Bernoulli polynomials, 24 a discretization method, 25 method in the work of Basin, 26 Legendre multiwavelets, 27 multi-interval decomposition scheme, 28 generalized Bessel functions, 29 method based on iterative dynamic programming, 30 recursive shooting method, 31 iterative method, 32 and discretization method. 33 On the other hand, Fibonacci polynomials have been defined using the following general formula 34 :…”