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In this study, the Lyapunov technique is used to analyze the observer-based control problem for polynomial fuzzy fractional order (PFFO) models. The case of polynomial matrices with unmeasurable states is considered to increase the applicability of the PFFO models in the design problem. In this regard, we offer two design procedures. First, the design conditions are presented in a one-step procedure. In this design, the non-convex conditions are transformed in a set of sum of squares (SOS) by introducing a new symbolic variable except the state vector β and its estimated β ^ . The obtained SOS conditions are presented involve three independent symbolic variables which increase both computing complexity and conservatism. To get around this shortcoming, a second design method is developed. By using the suggested method, SOS conditions requiring just two separate symbolic variables may be obtained. The architecture is shown in two stages; however, the observer and controller gains are computed in a single stage in order to further minimize conservatism. In order to demonstrate the utility of the suggested theoretical analysis, a simulated example is then provided. [ABSTRACT FROM AUTHOR]

This research addresses the problem of globally stabilizing a distinct category of fractional-order power systems (F-OP) by employing an observer-based methodology. To address the inherent nonlinearity in these systems, we leverage a Takagi–Sugeno (TS) fuzzy model. The practical stability of the proposed system is systematically established through the application of a sum-of-squares (SOS) approach. To demonstrate the robustness and effectiveness of our approach, we conduct simulations of the power system using SOSTOOLS v3.00. Our study contributes to advancing the understanding of F-OP and provides a practical framework for their global stabilization. [ABSTRACT FROM AUTHOR]

The delay-dependent state, Fault Estimation (FE) and Fault Tolerant Control problems for polynomial fuzzy systems with time delay are investigated in this paper. Firstly, a polynomial adaptive fuzzy observer is used to estimate system states and actuator faults. A polynomial Lyapunov–Krasovskii functional including double integral term is introduced in order to obtain delay-dependent sufficient conditions in terms of Sum Of Squares (SOS) which can be solved via SOSTOOLS and a Semi-Definite Program solver. Then, by exploiting the FE information, a polynomial fuzzy fault tolerant controller is designed to guarantee the stability of the system and to compensate the impact of actuator faults. Finally, several examples are given to illustrate the use of the present delay-dependent result. [ABSTRACT FROM AUTHOR]

In this paper, a robust fault estimation for polynomial systems with time delay via a polynomial adaptive observer is presented and described. In fact, in contrast to previous research works on actuator faults, the learning rate utilized to govern the convergence speed of states and faults errors is not predetermined but is derived using the sum of squares optimization problem. Furthermore, the H ∞ optimization approach was created to reduce the impact of disturbances on state estimation error and fault estimation error. Finally, an illustrative practical example is given to validate the effectiveness of the suggested methodology. [ABSTRACT FROM AUTHOR]

This paper investigates both fault estimation and fault-tolerant control problems for polynomial systems with state delay in presence of faults. Before studying the control problem, a polynomial observer is proposed to jointly estimate state and fault vectors. Based on the obtained informations, a polynomial controller is designed to compensate fault effects and to stabilize the closed-loop systems even in presence of state delay. The existence of both polynomial observer and controller is proved via satisfying sufficient conditions based on a sum of square (SOS) approach. Polynomial observer and controller gains are computed using SOSTOOLS. Finally, a numerical example is given to demonstrate that the SOS approach provides a significant improvement in reaction to fault occurrence for time delay systems. Furthermore, a tunnel diode circuit and a mass–spring–damper system are used to illustrate the applicability of the design method. [ABSTRACT FROM AUTHOR]

This paper presents a new Line-integral polynomial Lyapunov functional approach for observer-based control for a class of polynomial fuzzy systems with time delay and unmeasured state variables. To guarantee the global asymptotic stability and estimation error convergence, a design method is proposed. In this work we consider a Line-integral polynomial Lyapunov function in which the Lyapunov matrices are polynomial matrices depending not only of the estimated states but also of the estimated delayed states and we use the dual system to reduce the computational efforts. The design conditions are established in terms of Sum Of Squares (SOS) which can be numerically and symbolically solved via the recent developed SOSTOOLS and a Semi-Definite Program solver. Finally, numerical examples are proposed to show the validity and applicability of the proposed results. [ABSTRACT FROM AUTHOR]

This article is concerned with the design of polynomial fuzzy controller for a class of polynomial fuzzy model subject to both state delay and actuator saturation. Based on polytopic model of the input saturation, two design methods are proposed. In the first method, a specific Lyapunov Krasovskii function is proposed to transform the nonconvex sum of squares (SOS) conditions into convex SOS ones. The second method overcomes the restriction of the first method by bounding the state derivative. The obtained results are formulated in terms of SOS matrices which can be symbolically and numerically solved via the SOSTOOLS and the SeDuMi. Moreover, an attractive region of initial states that ensures asymptotic stability of polynomial fuzzy model is determined. Two numerical examples are given to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]