Computing fuzzy trajectories for nonlinear dynamic systems

Abstract

One approach for representing uncertainty is the use of fuzzy sets or fuzzy numbers. A new approach is described for the solution of nonlinear dynamic systems with parameters and/or initial states that are uncertain and represented by fuzzy sets or fuzzy numbers. Unlike current methods, which address this problem through the use of sampling techniques and do not account rigorously for the effect of the uncertain quantities, the new approach is not based on sampling and provides mathematically and computationally rigorous results. This is achieved through the use of explicit analytic representations (Taylor models) of state variable bounds in terms of the uncertain quantities. Examples are given that demonstrate the use of this new approach and its computational performance.

Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design

“…Removing the difference of the two matrices in (17) with the help of the Schur complement formula [36,37] leads to Equation ( 13) which completes the proof of Theorem 1.…”

“…Recently, corresponding set representations were developed which allow to express either a notion of variables certainly belonging to a solution set or certainly not belonging to the set (denoted as a relative distance measure interval arithmetic [15]); alternatively, the notion of thick intervals, thick boxes, thick functions, and thick ellipsoids can be seen as an approach to handle this kind of uncertainty on set boundaries [16]. Moreover, combinations of fuzzy and interval methods as those presented in [17] represent other currently investigated techniques that aim at expressing uncertainty on state boundaries.…”

A robust feeding strategy to maintain set‐point glucose in mammalian fed‐batch cultures when input parameters have a large error

“…Similar trajectories as we used them can be generalized in non-linear systems where upper and lower probabilities are represented as fuzzy numbers. 14 When the error ranges are well known, ordinary differential equations (ODEs) can be set up to calculate trajectories which are based on the maximum lumped uncertainty like in guidance systems. 15 Biotechnological applications increasingly consider the propagation of input uncertainty to ensure good modeling practice, 16 to achieve high quality of the developed models even though sensor input is not ideal, 17,18 and to meet end-product specifications by ensuring that the trajectories in which product quality was accepted were never breached.…”

“…This resulted in uncertainty trajectories which would evolve over time as the simulation progressed. Similar trajectories as we used them can be generalized in non‐linear systems where upper and lower probabilities are represented as fuzzy numbers . When the error ranges are well known, ordinary differential equations (ODEs) can be set up to calculate trajectories which are based on the maximum lumped uncertainty like in guidance systems .…”

A New Method to Solve Numeric Solution of Nonlinear Dynamic System

“…There is a fixed rule in the dynamic system to describe how the point changes over time in the space, and a continuous dynamic system is often represented as a set of differential equations [8]…”