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

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 L2 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.