## 1 Introduction  
Singular systems, also called descriptor systems or implicit systems, are important in the control field since their general representation. Lots of practical systems can be described as singular state-space models [1; 2; 3; 4; 5; 6]. For instance, DC motor systems, bio-economic models, electrical circuit systems, and network control systems. Before studying the stabilization issues for discrete-time singular systems, regularity and causality need to be discussed. Because of the algebraic equations for the singular models, there may exist more than one solution. It results in the unwanted impulse terms that emerged in state trajectories to destroy singular systems' stability. In the past few decades, the above properties are discussed with matrix decompositions [7; 8]. Based on the matrix decomposition techniques, some sufficient definitions concerning regular, causal, and admissible were presented [9]. Then, numerous related control issues were investigated in [9; 10; 11; 12], e.g., the event-triggered control, sliding mode control, output feedback control, and robust \(H_{\infty}\) control, etc. On the other hand, nonlinearity always exists in practical systems. To discuss the stabilization issue of nonlinear systems, the T-S fuzzy modeling approach was first proposed in [13]. Compared to the Mamdani fuzzy models, the calculations of fuzzy controller design and the systems' structure of the T-S fuzzy models are relatively complex. However, the stability analysis of T-S fuzzy models is more systematic with a mathematical foundation, i.e., it possesses well mathematical basis and structure extension. Thus, the T-S fuzzy models attracted numerous researchers' attention in the past two decades. The T-S fuzzy modeling approach is a conceptually simple and effective method to approximate nonlinear behaviors. Via reasonable fuzzy sets and fuzzy inferences, several linear subsystems are constructed by a series of IF-THEN fuzzy rules. Then, the final output of the T-Ssystems can be obtained with membership functions. Some practical applications of the T-S fuzzy models, such as temperature control [14] and prediction of generated electricity [15], have been investigated to address their advantages. For stabilizing the T-S fuzzy systems, PDC technology [16] was applied to construct fuzzy control schemes. This idea is that for each local subsystem, the controller can be designed with linear control techniques. Some normal systems, such as the ship steering system [17], ship fin stabilizing system [18], and drum boiler system [19], have been discussed. This modeling approach and controller design technique are not only applicable for normal systems but also for singular systems. Several stabilization issues of T-S fuzzy singular systems were researched by PDC-based fuzzy controllers [20; 21; 22]. In the past decade, the state feedback controller was constructed to stabilize the T-S fuzzy singular systems. To overcome the problem that the system's states are not always measurable, some control schemes have been investigated, for example, static output feedback [11], observer-based feedback [23], and dynamic output feedback [24]. However, if only using a traditional state feedback controller to singular systems, the complex model transformations for system matrices are unavoidable. To simplify the derivation process of the stability criteria, the PD feedback scheme was used to discuss control problems in [25; 26; 27; 28; 29; 30; 31; 32]. For instance in [25] the stabilization issues for continuous-time and discrete-time linear switched singular systems were studied via PD control strategy with the free-weighting matrix technique. In [26], the passive \(H_{\infty}\) control for the discrete-time linear switched singular systems was studied by using the Schur lemma. The related PD control problems were studied in [27; 28; 29; 30; 31; 32]. The PD observer issue for discrete-time singular systems was studied in [33]. In [25; 26], the stabilization via PD state feedback fuzzy control scheme was studied.