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Open AccessArticle

Observer-Based Exponential Stabilization for Time Delay Takagi–Sugeno–Lipschitz Models

by Omar Kahouli, Hamdi Gassara, Lilia El Amraoui and Mohamed Ayari

Mathematics 2025, 13(19), 3170; https://doi.org/10.3390/math13193170 - 3 Oct 2025

Cited by 1 | Viewed by 914

This paper addresses the problem of observer-based control (OBC) for nonlinear systems with time delay (TD). A novel hybrid modeling framework for nonlinear TD systems is first introduced by synergistically combining TD Takagi–Sugeno (TDTS) fuzzy and Lipschitz approaches. The proposed methodology broadens the range of representable systems by enabling Lipschitz nonlinearities to fulfill dual functions: they may describe essential dynamic behaviors of the system or represent aggregated uncertainties, depending on the specific application. The proposed TDTS–Lipschitz (TDTSL) model class features measurable premise variables while accommodating Lipschitz nonlinearities that may depend on unmeasurable system states. Then, through the construction of an appropriate Lyapunov–Krasovskii (L-K) functional, we derive sufficient conditions to ensure exponential stability of the augmented closed-loop model. Subsequently, through a decoupling methodology, these stability conditions are reformulated as a set of linear matrix inequalities (LMIs). Finally, the proposed OBC design is validated through application to a continuous stirred tank reactor (CSTR) with lumped uncertainties. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis: Theory, Methods and Applications)

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25 pages, 1198 KB

Open AccessArticle

State Estimation Based State Augmentation and Fractional Order Proportional Integral Unknown Input Observers

by Abdelghani Djeddi, Abdelaziz Aouiche, Chaima Aouiche and Yazeed Alkhrijah

Mathematics 2025, 13(11), 1786; https://doi.org/10.3390/math13111786 - 27 May 2025

Viewed by 1046

Abstract

This paper presents a new method for the simultaneous estimation of system states and unknown inputs in fractional-order Takagi–Sugeno (FO-TS) systems with unmeasurable premise variables (UPVs), by introducing a fractional-order proportional-integral unknown input observer (FO-PIUIO) based on partial state augmentation. This approach permits the estimation of both states and unknown inputs, which are essential for system monitoring and control. Partial state augmentation allows the integration of unknown inputs into a partially augmented model, ensuring accurate estimates of both states and unknown inputs. The state estimation error is formulated as a perturbed system. The convergence conditions for the state estimation errors between the system and the observer are derived using the second Lyapunov method and the

L_{2}

approach. Compared to traditional integer-order unknown input observers or fuzzy observers with measurable premise variables, in our method, fractional-order dynamics are combined with partial state augmentation uniquely for the persistent estimation of states along with unknown inputs in unmeasurable premise variable systems. Such a combination allows for robust estimation even under uncertainties in systems and long memory phenomena and is a significant step forward from traditional methods. Finally, a numerical example is provided to illustrate the performance of the proposed observer. Full article

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7 pages, 594 KB

Open AccessProceeding Paper

Observer Design for Takagi–Sugeno Fuzzy Systems with Unmeasurable Premise Variables Based on Differential Mean Value Theorem

by Wail Hamdi, Mohamed Yacine Hammoudi and Anouar Boukhlouf

Eng. Proc. 2023, 58(1), 28; https://doi.org/10.3390/ecsa-10-16008 - 15 Nov 2023

Cited by 8 | Viewed by 1651

Abstract

In this work, we present the design of an observer for Takagi–Sugeno fuzzy systems with unmeasurable premise variables. Moving away from Lipschitz-based and

L_{2}

attenuation-based methods—which fall short in eliminating the mismatching terms in the estimation error dynamics—we leverage the differential mean [...] Read more.

In this work, we present the design of an observer for Takagi–Sugeno fuzzy systems with unmeasurable premise variables. Moving away from Lipschitz-based and

L_{2}

attenuation-based methods—which fall short in eliminating the mismatching terms in the estimation error dynamics—we leverage the differential mean value theorem. This approach not only removes these terms but also streamlines the factorization of the estimation error dynamics, making it directly proportional to the estimation error. To ensure the asymptotic convergence of the estimation error, we apply the second Lyapunov theorem, which provides sufficient stability conditions described as linear matrix inequalities. A numerical example applied on a three-tank hydraulic system is presented to demonstrate the observer’s effectiveness. Full article

(This article belongs to the Proceedings of The 10th International Electronic Conference on Sensors and Applications)

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20 pages, 6534 KB

Open AccessEditor’s ChoiceArticle

Observer-Based Controller Using Line Integral Lyapunov Fuzzy Function for TS Fuzzy Systems: Application to Induction Motors

by Rabiaa Houili, Mohamed Yacine Hammoudi, Mohamed Benbouzid and Abdennacer Titaouine

Machines 2023, 11(3), 374; https://doi.org/10.3390/machines11030374 - 10 Mar 2023

Cited by 6 | Viewed by 2803

Abstract

This paper deals with the stabilization problem of a nonlinear system described by a Takagi–Sugeno fuzzy (TSF) model with unmeasurable premise variables via a robust controller. Applying the sector nonlinearity techniques, the nonlinear system is represented by a decoupled fuzzy model. Then, we design a robust observer-based controller for the obtained fuzzy system by utilizing the differential mean value approach. The observer and controller gains are obtained by the separation principle, in which the problem is solved in the sum of linear matrix inequalities (LMIs). The paper presents two main contributions: A state feedback controller is designed using differential mean value (DMVT) which ensures robust stabilization of the nonlinear system. Additionally, the Luenberger observer is extended to the Takagi–Sugeno fuzzy models. The second contribution is to reduce conservatism in the obtained conditions, a non-quadratic Lyapunov function (known as the line integral Lyapunov fuzzy candidate (LILF)) is employed. Two examples are provided to further illustrate the efficiency and robustness of the proposed approach; specifically, the Takagi–Sugeno fuzzy descriptor of an induction motor is derived and a robust observer-based controller applied to the original nonlinear system. Full article

(This article belongs to the Special Issue 10th Anniversary of Machines—Feature Papers in Electrical Machines and Drives)

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16 pages, 1255 KB

Open AccessArticle

Fractional Order Unknown Inputs Fuzzy Observer for Takagi–Sugeno Systems with Unmeasurable Premise Variables

by Abdelghani Djeddi, Djalel Dib, Ahmad Taher Azar and Salem Abdelmalek

Mathematics 2019, 7(10), 984; https://doi.org/10.3390/math7100984 - 16 Oct 2019

Cited by 37 | Viewed by 4380

Abstract

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO. Full article

(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)

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21 pages, 319 KB

Open AccessArticle

A New Observer Design for Fuzzy Bilinear Systems with Unknown Inputs

by Jun Yoneyama

Designs 2017, 1(2), 10; https://doi.org/10.3390/designs1020010 - 21 Nov 2017

Cited by 6 | Viewed by 3886

Abstract

An observer design for a class of nonlinear systems with unknown inputs is considered. Takagi–Sugeno fuzzy bilinear systems represent a wide class of nonlinear systems, and these systems with unknown inputs are an ideal model for many physical systems. For such systems, a design method for obtaining an observer that estimates the state of the system is proposed. A parallel distributed observer (PDO), which is constructed with local linear observers and the appropriate grade of the membership functions, is a conventional observer for Takagi–Sugeno fuzzy bilinear systems. However, it is known that its design conditions have conservativeness. In this paper, to reduce the conservatism in the design conditions, non-PDO with new design conditions is proposed. Our design conditions are derived from a multiple Lyapunov function, which depends on the membership function with time-delay in the premise variables. This method eventually reduces the conservatism and enables us to construct an observer for a wide class of nonlinear systems. When the premise variables are the state variables that are not measurable, Takagi–Sugeno fuzzy bilinear systems can represent a wider class of nonlinear systems. Hence, an observer design for fuzzy bilinear systems with unmeasurable premise variables is also proposed. Finally, numerical examples are given to illustrate our design methods. Full article

(This article belongs to the Special Issue Optimization, Health Monitoring and Control Methods for Modern Complex Systems)

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