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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]

Fractional systems have been widely utilized in various fields, such as mathematics, physics and finance, providing a versatile framework for precise measurements and calculations involving partial quantities. This paper aims to develop a novel polynomial controller for a power system (PS) with fractional-order (FO) dynamics. It begins by studying the practical stability of a general class of tempered fractional-order (TFO) nonlinear systems, with broad applicability and potential for expanding its applications. Afterward, a polynomial controller is designed to guarantee the practical stability of the PS, encompassing the standard constant controller as a specific instance. The design conditions for this controller are resolved using the sum of squares (SOS) approach, a powerful technique for guaranteeing stability and control design. To showcase the practical value of the analytical findings, simulations of the PS are conducted utilizing SOSTOOLS. [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]

Summary: This paper deals with some synthesis problems for a class of positive polynomial systems with time delay, for which the state vector takes nonnegative values whenever the initial conditions are nonnegative. First, the synthesis of state feedback controllers is solved using a Lyapunov‐Krasovskii approach, including the requirement of positiveness. In a second part, observers are included. In both cases, the design conditions are presented in terms of sum of squares, which can be numerically and symbolically solved via SOSTOOLS and a semidefinite program solver. Finally, four numerical examples are given to illustrate the proposed approach. [ABSTRACT FROM AUTHOR]