Suboptimal control of linear systems with delays in state and input by orthonormal basis

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.…”

“…Hence we select ξ = 20. By choosing k = 2, from (29) and (30) we see n 0 = 0 and n d = 2 and n min1 = −2. According to Remark 1, we set n e0 = 2, so n min1 = 0 and n 1 = 5.…”

Optimal control of linear multi‐delay systems based on a multi‐interval decomposition scheme

“…We briefly review some recent papers that are relevant to the method developed in the current work. In , the linear Legendre multi‐wavelets (LLMW) as an orthonormal basis for L 2 [0,1] have been used to numerically solve optimal control problems with delays in state and control variables. The implementation of LLMW produces a piecewise linear function to control input.…”

“…This is a major limitation of this numerical scheme. Although the procedure developed in is relatively simple, it does not provide a satisfactory approximation to more general optimal control problems. Additionally, the operational matrix of delay associated to the LLMW is not sparse.…”

“…All of the aforementioned properties are demonstrated through the paper. It should be also pointed out that the present work is an extension of the delayed optimal control problem considered in , , and . In , a hybrid of block‐pulse functions and Legendre polynomials has been introduced for solving optimal control problems with a single delay.…”