## 1 Introduction
As an alternative to separately estimating input or output disturbances, as in most of the existing UIO design approaches presented recently, in this paper it is proposed to estimate the net EID of the system. Based on the proof of the existence of EID for strictly proper systems [14], an EID is developed in Section 3 of this paper as an extension to the case of proper continuous-time systems. Given a controllable and observable proper system with all stable zeros and unknown mismatched independent input disturbances \(d_{i}(t)=G_{i}\overline{d}_{i}(t)\) and output disturbances \(d_{o}(t)=G_{o}\overline{d}_{o}(t)\), an EID \(d_{e}(t)\) exists that enters the plant through the control input channel \(B\) and the input-output direct feedthrough term \(D\), such that the two systems--one with \(\{d_{i}(t)\) and \(d_{o}(t)\}\) and the other with the matched disturbances \(\{Bd_{e}(t)\) and \(Dd_{e}(t)\}\)--share the same outputs but differ in their states [15; 16]. The simultaneous estimation of the system state and the EID of a strictly proper regular system subject to mismatch-dependent input and output disturbances was discussed in [11]. Based on the advanced functional observer originally presented in [10] for strictly proper regular systems, a modified functional observer for regular systems with an input-output direct feedthrough term is developed in Section 4 to estimate the net EID under the effects of disturbances. The EID method in conjunction with the modified functional observer can deal with proper systems with mismatch and independent input and output disturbances, i.e., \(d_{i}(t)=G_{i}\overline{d}_{i}(t)\) and \(d_{o}(t)=G_{o}\overline{d}_{o}(t)\), where \(G_{i}\in\Re^{n\times d_{i}}\) and \(G_{o}\in\Re^{p\times d_{o}}\) are unknown and there is no additional constraint for \(p\geq l=\max(l_{i},\ l_{o})\). The main contributions of this paper can be briefly discussed as follows. Consider a practical singular analog system in which the input and output are measurable. From a theoretical point of view, no existing MATLAB toolbox can be used to simulate the states of the singular analog system to construct the outputs, even if a mathematical model of the singular analog system \((E_{r},A_{r},\ B_{r},\ C_{r})\) is available, due to the singular matrix \(E_{r}\) in the term \(E_{r}\dot{x}(t)\). In addition, an approach for directly solving the tracker-based generalized algebraic Riccati equation for the singular analog system \((E_{r},A_{r},\ B_{r},\ C_{r})\) has not yet been developed in the literature. Using the proposed equivalent reduced-order proper regular model, these issues can be indirectly overcome, and also the controller can be designed. Now, since no mathematical models exist for unknown singular analog systems, the off-line observer-Kalman-filter identification (OKID) method [17] can be applied to construct the aforementioned equivalent reduced-order proper regular model for the singular system. As a result, in this paper, such a method is utilized to construct the newly developed modified functional observer (using the system output to estimate the integrated control input and estimated EID) and the EID estimator (to estimate the EID), such that the desired control input can consequently be determined. The developed equivalent system can also be utilized to directly realize the observer, and the control objective is then to estimate and feed the EID back through the control input channels to counteract the negative effects induced by the input and output disturbances, without any of the aforementioned constraints on the disturbances. This paper also provides more details on improving the disturbance rejection performance based on a net EID estimation approach for a system with \(l_{i}\) unknown input disturbances and \(l_{o}\) output disturbances imposed simultaneously, where \(l_{i},l_{o}>p\), with \(p\) representing the number of outputs. The rest of this paper is organized as follows.