## 3 Property of the EID Estimator  
Consider a continuous-time controllable and observable system with stable zeros and unknown disturbances, described by  
\[\dot{x}_{c}(t)=Ax_{c}(t)+Bu_{c}(t)+G_{i}\overline{d}_{i}(t), \tag{7a}\] \[y_{c}(t)=Cx_{c}(t)+Du_{c}(t)+G_{o}\overline{d}_{o}(t), \tag{7b}\]  
where \(x_{c}(t)\in\Re^{n}\), \(u_{c}(t)\in\Re^{m}\), and \(y_{c}(t)\in\Re^{p}\), respectively, represent the state, input, and output vectors; \(A\in\Re^{n\times n}\), \(B\in\Re^{n\times n}\), \(C\in\Re^{p\times n}\), and \(D\in\Re^{p\times n}\) are known matrices; \(\overline{d}_{i}(t)\in\Re^{l_{i}}\) and \(\overline{d}_{o}(t)\in\Re^{l_{o}}\) are the mismatched input and output disturbance vectors, respectively; \(G_{i}\in\Re^{n\times l_{i}}\) and \(G_{o}\in\Re^{p\times l_{o}}\) are matrices. Here, \(G_{i}\), \(G_{o}\), \(\overline{d}_{i}(t)\) and \(\overline{d}_{o}(t)\) are assumed unknown. Let matrices \(B\) and \(C\) be full column and full row ranks, respectively, and \(\text{rank}(CB)=p\leq m\leq n\). There is no constraint on the number of unknown inputs in \(\overline{d}_{i}(t)\), i.e., there may be more unknown inputs than control inputs and measured outputs, and also the disturbances may be state-dependent. Section 9.5 in [19] presents a discussion on the effects of state-dependent uncertainties appearing in the process model as either additive or multiplicative disturbances (which can be included in a sensitivity analysis of combined estimation and control) and the associated algorithms. Additionally, other robust control approaches concerning state-dependent disturbances can be found in [11,20]. It is well-known that it is difficult or even impossible to estimate the components of \(\overline{d}_{i}(t)\) and \(\overline{d}_{o}(t)\) in a precise manner. It is also known that in most disturbed servo systems, the unknown inputs are estimated for disturbance rejection purposes to maintain the desired performance. An alternative is to assume the existence of a net EID, \(d_{e}(t)\in R^{m}\), of the unknown disturbances \(\overline{d}_{i}(t)\) and \(\overline{d}_{o}(t)\) entering the plant through the control input channel \(B\) and the direct feedthrough term \(D\), meaning Equations (7a) and (7b) can be expressed as  
\[\dot{x}_{e}(t) =Ax_{e}(t)+B[u_{c}(t)+d_{e}(t)], \tag{8a}\] \[y_{e}(t) =Cx_{e}(t)+D[u_{c}(t)+d_{e}(t)], \tag{8b}\]  
where \(x_{e}(t)\in\Re^{n}\) and \(y_{e}(t)\in\Re^{p}\) are the equivalent state and output vectors, respectively, with \(x_{e}(0)=x_{c}(0)\). Notice that \(x_{e}(t)\
eq x_{c}(t)\) for \(t>0\), because in general \(Bd_{e}(t)\
eq G_{i}\overline{d}_{i}(t)\), even for \(d_{o}(t)=0\). Similar to the case of a strictly proper system (i.e., no \(Du_{c}(t)\) term) [14], the definition of net EID of an input-output feedthrough system with both mismatched input and output disturbances is analogous to the definition given in [4]. **Definition 1**. : _Let the control inputs of Equations (7a) and (7b), (8a) and (8b) be \(u_{c}(t)=0\); let the output of Equations (7a) and (7b) with mismatched disturbances \(G_{i}\overline{d}_{i}(t)\) and \(G_{o}\overline{d}_{o}(t)\) be \(y_{c}(t)\); and let the output of Equations (8a) and (8b) with matched disturbance \(d_{e}(t)\) be \(y_{e}(t)\). The disturbance \(d_{e}(t)\) is known as the net equivalent input disturbance (EID) of the mismatched disturbances \(G_{i}\overline{d}_{i}(t)\) and \(G_{o}\overline{d}_{o}(t)\) if \(y_{e}(t)=y_{c}(t)\), with \(x_{e}(0)=x_{c}(0)\) for \(t\geq 0\)._  
It is important to remark that \(y_{e}(t)=y_{c}(t)\) does not imply that \(x_{e}(t)=x_{c}(t)\), because given \(y_{c}(t)\) and the output matrix \(C\in\Re^{p\times n}\) with \(p<n\), \(y_{c}(t)\) will be equal to \(C_{c}x_{c}(t)\) for an infinite number of \(x_{c}(t)\).