Unknown input based observer synthesis for uncertain Takagi–Sugeno fuzzy systems

Abstract

Observer design for a Takagi-Sugeno (T-S) fuzzy system with uncertainties is extremely difficult because the estimation error is incapable of approaching zero asymptotically due to the existence of uncertainty terms. In this study, by regarding the uncertainty as an unknown input, with some particular derivation, the authors successfully synthesize a fuzzy observer which guarantees that the error will converge to zero asymptotically. Based on Lyapunov theory and linear matrix inequality tools, the main theorem is derived for the fuzzy observer synthesis. This study does not limit the size of uncertainties, but the uncertainties have to satisfy a specific matching condition in order to use the unknown input concept. Finally, a numerical example is given to show that the proposed approach is effective in estimating system's states subject to system uncertainties.

Observer synthesis for uncertain Takagi–Sugeno fuzzy systems with multiple output matrices

“…It means that before obtaining matrix from LMIs, it is really difficult to expect the uncertainties satisfy constraint (21). Furthermore, a new approach to design an observer for uncertain T–S fuzzy systems using the unknown inputs method has been studied in the paper [44]. Unfortunately, the proposed method in [44] merely applied for T–S fuzzy systems with the single output matrix , because with the multiple output matrices , the constraints (14) of [44] will be turned into non‐convex constraints and extremely difficult to solve with the existing computational tools.…”

“…Furthermore, a new approach to design an observer for uncertain T–S fuzzy systems using the unknown inputs method has been studied in the paper [44]. Unfortunately, the proposed method in [44] merely applied for T–S fuzzy systems with the single output matrix , because with the multiple output matrices , the constraints (14) of [44] will be turned into non‐convex constraints and extremely difficult to solve with the existing computational tools. In other words, the observer design method in [44] cannot handle the non‐linear systems with non‐linear output equation.…”

“…Additionally, the robust observer‐based control method, which combined both observer and controller to stabilise system and make estimated states approach to the real states, was discussed in [33–43]. Recently, a new approach was proposed in [44], where a mathematical transformation is used to convert the uncertain terms of the T–S fuzzy system into unknown inputs, and then the observer design based on the unknown input method was achieved. Unfortunately, this approach has two drawbacks as follows.…”

“…One is that the output matrix is a single common C but not multiple C ; it means that this approach merely dealt with the non‐linear systems with linear output. The other is that the uncertain parts must satisfy a conservative matching condition with a common in Assumption 1 of [44].…”

“…A special mathematical transformation of [45] is used in this paper to turn the traditional T–S fuzzy model into a new form, so that the matching conditions of the uncertainties can be relaxed and the multiple output matrices can be handled as well. Thus, the main contributions of this paper are three‐fold: (i) introduce new method based on the unknown inputs concept to design observer for uncertain T–S fuzzy system with multiple output matrices ; (ii) reduce the conservatism of method in [44] by using the varying matrices instead of common matrix D ; (ii) synthesise observer for uncertain T–S fuzzy system without upper bounds of uncertainties condition.…”