## 1 Introduction  
Singular analog system models naturally arise in describing large-scale (complex) systems, such as interconnected power systems. In general, a large class of interconnected state-variable subsystems is described as a singular system, even though the entitled state-space representation may not be available. Practical singular systems usually comprise dynamic and non-dynamic subsystems, which are mathematically represented by a combination of differential and algebraic equations. The complex features of singular systems (also known as descriptor systems) often make it difficult to find numerical and analytical solutions for such systems, especially for control tasks. The numerically stable and computationally fast matrix sign function-based method has been used to solve the regulator-based generalized algebraic Riccati equation for optimal control of a linear continuous-time singular system [1]. A technique was developed in [2] to decompose the singular system into a reduced-order regular subsystem and a non-dynamic subsystem. Then, an optimal tracker was developed in [3] based on the equivalent model of the linear singular system. Nevertheless, an approach for solving the tracker-based generalized algebraic Riccati equation for the singular system has not yet been developed in the literature. It was proven in [4] that the tracker-based generalized algebraic Riccati equation cannot be directly solved for a singular system, even with an impulsive model-free and strictly proper transfer function. To overcome this issue, in [4] the singular system is converted into an equivalent regular model with a direct transmission term from input to output. Based on this equivalent proper model, the approach for solving the regulator-based generalized algebraic Riccati equation can then be extended to solve the tracker-based generalized algebraic Riccati equation. In other words, the optimal control methodology for finding the linear quadratic regulator via the matrix sign function [1] can be directly extended to find the optimal tracker for singular systems. The unknown input problem (UIO) involves the estimation of the state of a dynamic system subject to unknown input excitations [5]. Many published works focus on the issue of simultaneously estimating the system state and the unknown input vectors in the presence of unknown disturbances. Specifically, UIOs are involved in practical engineering systems, such as diagnosis and fault detection [6; 7], secure communication [8; 9], or systems where the measurement of the inputs is either practically impossible or too expensive [10; 11], while an equivalent input observer is often required to estimate the exerting force or torque of a robotic system or the cutting force of a machine tool [11]. Other engineering systems, such as industrial biological processes, wastewater treatment processes, and fuel cell stack systems, can be found in [11] and references therein. For a strictly proper system, different approaches are given in [5; 12; 13] to simultaneously estimate the state and unknown input disturbances. However, they are restricted to particular assumptions. For example, it is assumed in [5] that the number of independent signals \(l\) in the unknown input disturbance \(d(t)\) must be no greater than the output dimension and that the matrix of unknown inputs \(G_{n\times l}\) (given as \(d(t)_{n\times 1}=G_{n\times l}\overline{d}(t)_{l\times 1}\)) and some pre-specified rank conditions must be known. Notice that it is more suitable to estimate the net equivalent input disturbance (EID) of the unknown mismatched input disturbance than to estimate the disturbance itself, because the control input is required to suppress the negative effects of the mismatched input disturbance. Regardless, whenever the output number \(p<l\) or the input number \(m<l\), it is impossible to separately estimate the unknown mismatched input disturbance.