An Experience-Correction Method of Fractional-Order Systems Subject to Intermittent Sampled-Data Transmission: A Forgetting Curve Perspective

Abstract

In this article, we establish an experience-correction (E-C) framework to deal with sampled-data fractional-order systems in the manner of general integer-order discrete-time model at sampling instants by coping with the memory effect. The designed E-C controller can be partitioned into local input and history input, respectively. Generated by current sampling, the local input is in a sense analogous to the input of integer-order system. In particular, the history input can correct all previous experience via incorporating earlier local inputs. Emphasize that the weights of earlier local inputs, norm of E-C controller together with corresponding sampling error are proved to descend to origin no slower than the Pareto function, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{O}(t^{-\alpha})$</tex-math> </inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula> represents the derivative’s order. Similar to the well-known forgetting curve, we witness a rapid decline in the proportion of earlier past in E-C controller, which is with respect to one of the memory failures: transience, the deterioration of memory with the passage of time. The closer the order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula> is to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1$</tex-math> </inline-formula> , the faster the forgetting is. The above analysis accounts for a phenomenon not previously understood: well-performed simulation results could still be expected in existing research who ignored memory effect and only considered local input. This is owing to the fractional orders are usually hypothesized to be larger than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.5$</tex-math> </inline-formula> and thus the weight of earlier past decays sharply over time. Furthermore, both absolute and proportional decline of earlier past’s weight will slow down when time goes by, which is consistent with memory consolidation and Jost’s law in psychology. Overall, the E-C method keeps an elegant formulation which theoretically describes the evolution of hereditary effect. Meanwhile, it is also suitable for first-order systems. Thereafter, we address related applications in fractional-order multiagent systems (MASs) subject to intermittent sampled-data communication. Numerical simulations are provided finally to substantiate the effectiveness of theoretical results.