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

Abstract

The aim of this paper is to determine the optimal open loop solution and a nonlinear delay-dependent state feedback suboptimal control for a class of nonlinear polynomial time delay systems. The proposed method uses a hybrid of block pulse functions and Legendre polynomials as an orthogonal base for system’s states and input expansion. Hence, the complex dynamic optimization problem is then reduced, with the help of operational properties of the hybrid basis and Kronecker tensor product lemmas, to a nonlinear programming problem that could be solved with available NLP solvers. A practical nonlinear feedback controller gains are deduced with respect to a least square formalism based on the optimal open loop control results. Simulation results show efficiency of the proposed numerical optimal approach.

A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

“…andẋ(0) = x (1) (0). (43) has the effect of changing the matrix b 1 . Therefore, b 1 in (42) is changed to…”

“…We use an approximation scheme in which a nonlinear delay problem is replaced with a sequence of linear delay problems, for an example see [42]. It should be noted that there are some remarkable schemes, like that proposed in [43], that can be implemented by the fmincon function provided by the optimization toolbox in MATLAB. The results of the scheme by using quadprog are reported in Table 9.…”

A General Framework for Optimal Control of Fractional Nonlinear Delay Systems by Wavelets

“…In this example, we aim to minimize the criterion [37] ∫ 10 0 {x 2 1 (t) + x 2 2 (t) + u 2 (t)} dt subject to…”

“…To overcome drawbacks of classical (conventional) Legendre wavelets, Ref. [37] was used the hybrid functions as a combinations of block pulse and Legendre functions. But we can use Legendre wavelets with scaling [16] by which those drawbacks are eliminated and they have advantages over these hybrid functions, for example Γ lw = I.…”

A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

“…The controller design relies on a tailored direct spectral method (Jaddu, 2002) where shifted Legendre polynomials (SLPs) (Iben Warrad et al, 2015; Perng, 1986) with some important properties are exploited to transform the complex nonlinear optimal problem into a nonlinear programming (NLP) one with multiple equality constraints. Indeed, the important operational properties of the used orthogonal basis, such as the Kronecker product (Bouafoura and Benhadj Braiek, 2019; Iben Warrad et al, 2018) and the integration operational matrices (Hwang and Chen, 1985; Hwang and Shih, 1986; Datta and Mohan, 1995), reduce the differential equations with nonlinearities into easier-to-process algebraic equations (Bichiou et al, 2018) feeding the NLP solver. Closed-loop system with proposed suboptimal control stability is investigated and sufficient conditions are given.…”