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

Abdelghani Djeddi, Abdelaziz Aouiche, Chaima Aouiche and Yazeed Alkhrijah ()
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Abdelghani Djeddi: Department of Electrical Engineering, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Abdelaziz Aouiche: Department of Electronics and Communications, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Chaima Aouiche: Department of Electronics and Communications, Echahid Cheikh Larbi Tebessi University, Tebessa 12002, Algeria
Yazeed Alkhrijah: Department of Electrical Engineering, Engineering College, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia

Mathematics, 2025, vol. 13, issue 11, 1-26

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.

Keywords: fractional-order Takagi–Sugeno systems; fractional-order proportional-integral unknown input observer; partial state augmentation; unmeasurable premise variables (UPVs); Lyapunov stability; linear matrix inequalities (LMIs) (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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