Asymptotic Stability for Some Types of Nonlinear Fractional Order Differential-Algebraic Control Systems

The aim of this paper is to study the asymptotically stable solution of nonlinear single and multi fractional differential-algebraic control systems, involving feedback control inputs, by an effective approach that depends on necessary and sufficient conditions.


Introduction
The nonlinear fractional order differential-algebraic control systems appear in a variety of theories and applications. The theory of fractional descriptor ordinary and fractional partial differential equations, with different types of derivatives, have recently been addressed by several researchers for different problems. It is well known that descriptor systems or differential-algebraic systems are the major research fields of the control theory. During the past two decades, differential-algebraic systems attracted much attention due to the comprehensive applications in economics singular systems, not only those containing differential or difference equations as normal systems but also algebraic equations. Thus, their description is considered as being more general. Their class of systems has been widely studied, not only because of theoretical interest but also because of its extensive applications in areas such as robotics and power systems. The necessary and sufficient conditions for the solvability, positivity, and asymptotic stability and stabilization of the fractional descriptor linear systems were established [1-7, 8, 9]. Earlier works [10,11] studied the partial eigenvalue assignment for stabilization of descriptor fractional discrete-time linear systems or by derivative state feedback. In other investigations [12,10], the stabilization problem of singular fractional-order systems with

ISSN: 0067-2904
Hasan Iraqi Journal of Science, 2021, Vol. 62, No. 2, pp: 623-638 624 fractional commensurate fractional order, via static output feedback, was studied . The stability problem of descriptor second-order systems was also considered [13]. Lyapunov equations for stability of second-order systems were established by using Lyapunov method. The robust admissibility problem in singular fractional-order continuous time systems was also studied with a static output feedback controller that is designed for the uncertain closed-loop system to be admissible [14]. Other articles studied the robust stability and stabilization of uncertain fractional-order differential-algebraic nonlinear systems [15,13].
Our intersect in this paper is to study the asymptotic stability of nonlinear fractional order differential-algebraic control systems, involving feedback input controls. Also, we aim to study single-fractional (15)(16) and multi-fractional (21-22) order differentialalgebraic control equations .
The following definitions and results are needed later on. Definition (1.1), [16] Let be a function such that .

. Nonlinear Fractional Order Differential-Algebraic Control Systems
The following two types of nonlinear fractional order differential-algebraic control systems are presented.

Single-Fractional Order Differential -Algebraic Control Equations
Consider the nonlinear single fractional order differential -algebraic control system:  (7) and (8) become , Now, we consider the following related linear feedback control system are constants. For the nonlinear multi-fractional order differential -algebraic control system in equations (10-11) with equations (12-13), we obtain: Thus,

), [19]
Let v(t) be a nonnegative function that is locally integrable on [0,T), let a(t) be a nonnegative,nondecreasing continuous function that is defined on [0,T), and let a(t) <M . Suppose that z(t) is nonnegative and locally integrable on Suppose that the following nonlinear fractional order differential-algebraic control system (17)(18) with feedback control (14) satisfies the following conditions: where By taking the Laplace inverse transformation to (19), we obtain The following single-fractional order differential-algebraic equations (19)(20) with Caputo derivative are asymptotically stable, as follows

Multi -Fractional Order Differential -Algebraic Control Equation
Consider the following nonlinear multi-fractional order differentialalgebraic control system: where [ ( ( We have that

Conclusions
We studied the asymptotic stability for the proposed multi-fractional differential-algebraic control systems, involving multi control inputs, which needed to be transformed to single-fractional differential systems, using sufficient and necessary conditions.