## 1 Introduction  
Descriptor systems (also known as singular systems) play an important role in modern control theory, allowing us to model and analyze constrained dynamical systems [1, 2, 3]. These restrictions can be naturally imposed on a system (as a consequence of physical laws, e.g., the law of conservation of energy) or determined by the engineer (e.g., the constrained area of work).  
In the second half of the 20 th century, many papers and monographs on descriptor systems were written, laying the foundations for this theory [4, 5, 6, 7, 8, 9, 10, 11]. An overview of the state of the art in the field of descriptor systems theory can be found in [1, 2, 3, 7]. The stability of such systems was examined in [2, 3, 7, 12, 13]. Static and dynamic feedback control in descriptor systems was also investigated for state feedback [2, 7, 14] and the output feedback [2, 7, 15, 16, 17]. Descriptor systems theory can be used in many areas, such as electrical and mechanical engineering, robotics, fluid mechanics, chemical engineering, economics, and demography, see e.g., [2, 4, 8, 18, 19, 20].  
In recent years, the analysis and synthesis problems of dynamical systems described by fractional-order differential (or difference) equations have attracted a lot of attention [3, 20, 21, 22, 23].  
The notion of the practical stability of positive fractional discrete-time systems was introduced in [24] and conditions for practical stability were provided in [24, 25]. The stability of discrete-time linear systems with delays was investigated in [26, 27].  
Superstable systems are a subclass of asymptotically stable systems, in which dynamics are more restricted. Such systems provide some practically important properties, e.g., superstability (as opposed to stability) remains under the presence of time-varying and nonlinear perturbations, which allows researchers to solve problems relating to the synthesis of robust systems easily. Moreover, superstable systems ensure the elimination of peaks or sharp increases in the state vector trajectory [28, 29, 30].  
In this article, the superstabilizing state-feedback control problem in DDFL systems is studied. The main advantage of the presented approach is that it can be applied to the analysis of descriptor systems properties which are determined by matrix entries, such as positivity and superstability. This study is an extension of the results presented in [31].  
The organization of the paper is as follows. In Section 2 the considered state-space model is introduced. Section 3 is devoted to the application of the Drazin inverse to the analysis of DDFL systems. In Section 4 an equivalent model of this class of dynamical systems is presented. Methods for investigating stability and superstability are given in Section 5. In Sections 6 and 7 descriptor systems with static and dynamic state-feedback are studied and procedures for the computation of the gain matrices such that the closed-loop system is superstable are given. A numerical example showing the efficacy of the discussed approach is presented in Section 8. In Section 9 some concluding remarks and open problems are provided.  
The following system of symbols will be used in the paper: \(\mathbb{R}\) for the set of real numbers, \(\mathbb{R}^{n\times m}\) for the set of \(n\times m\) real matrices, \(\mathbb{C}\) for the set of complex numbers, \(\mathbb{Z}_{+}\) for the set of nonnegative integers, and \(\mathbb{I}_{n}\) for the \(n\times n\) identity matrix.