## 4 Modified Functional Observer with Unknown Input
### Stability Analysis in the Frequency Domain
For this case, the effect of the disturbances can be alleviated, since feedback into the system of the equivalent unknown inputs accurately estimated by the functional observer can effectively counteract actual unknown inputs. According to Equations (50), (51), and (53), if \(\mathrm{G_{dd}}(s)\approx I_{m}\) can be appropriately designed and the functional observer gives a good estimate of the EID, such that \(\bar{a}_{f}(t)\to u_{f}(t)\), then \(\bar{d}_{e}(t)\to d_{e}(t)\) when \(t\to\infty\) (see Equation (48)). Specifically, for any positive scalars \(\varepsilon_{1}<<1\), \(\varepsilon_{2}<<1\), \(\varepsilon_{3}<<1\) and assuming that the bounded \(\mathrm{G_{ufo}}(s)\
eq 0\) has been properly determined, if  
\[1-\big{|}\mathrm{G_{dd}^{ij}}(s)\big{|}\leq\varepsilon_{1};\ (i=1,\ldots,m) \tag{54}\]  
and  
\[\sum_{j=1}^{m}\big{|}\mathrm{G_{dd}^{ij}}(s)\big{|}\leq\varepsilon_{2};\ (i;j=1,\ldots,m;\ j\
eq i), \tag{55}\]  
then from Equations (51) and (53), we can obtain  
\[\|\mathrm{G_{yd}^{ij}}(s)\|_{\infty}\leq\varepsilon_{3};\ (i=1,\ldots,p;j=1, \ldots,m), \tag{56}\]  
such that \(y_{c}(t)\to 0\) when \(t\to\infty\). In other words, let \(\Delta\mathrm{y_{c}}(s)\) be defined as the error between the disturbed and undisturbed outputs, then \(y_{c}(t)\to 0\) in Equation (53) implies that \(\Delta\mathrm{y_{c}}\big{(}s)=\!\!\big{[}C(sI_{n}-A)^{-1}B+D\big{]}\!\big{(} \mathrm{d_{e}}(s)-\mathrm{d_{e}}(s)\big{)}\), i.e., \(\mathrm{d_{e}}(s)\approx\mathrm{d_{e}}(s)\). According to Equations (37a) and (37b), it is true that \(\mathrm{\widetilde{d}_{e}}(s)\approx\mathrm{d_{e}}(s)\) in steady state. As previously mentioned, the design of \(\mathrm{G_{dd}}(s)\) should honor Equations (54) and (56), i.e., \(\mathrm{G_{dd}}(s)\approx I_{m}\), to effectively estimate the EID and cancel the unknown disturbance. Nevertheless, it is well-known that \(\big{\|}\mathrm{G_{yd}}(s)\big{\|}_{\infty}\) in Equation (56) cannot be ideally minimized throughout the frequency range \(\omega_{c}\in[0,\ \infty)\). Therefore, the natural frequency of the unknown low-frequency disturbances to be suppressed is assumed to be below \(\omega_{c}\), which is true for a wide class of servo systems. Thus, it is suggested to design \(\mathrm{G_{dd}}(s)\) as a low-pass-filter-type dynamic system, such that \(\max\limits_{\omega\in\Omega}\!\big{\|}\mathrm{G_{yd}}(j\omega)\big{\|}_{ \infty}<1\), where \(\Omega\in[0,\ \omega_{c})\) instead, to filter the higher frequency noise [24]. For simplicity, \(\mathrm{G_{dd}}(s)\) is directly constructed as the diagonal matrix  
\[\mathrm{G_{dd}}(j\omega)=\mathrm{diag}\big{[}\mathrm{G_{dd}^{11}}(j\omega_{11}),\mathrm{G_{dd}^{22}}(j\omega_{22}),\ldots,\mathrm{G_{dd}^{mm}}(j\omega_{mm}) \big{]}, \tag{57}\]  
with the \(1^{st}\)-order filter (with least phase lag and simple structure)  
\[\mathrm{G_{dd}}(j\omega)=\frac{1}{\frac{1}{\omega_{i}}s+1}=\frac{1}{\tau_{ii}s+ 1}=\frac{\omega_{cii}}{s+\omega_{cii}}, \tag{58}\]where \(\omega_{ii}\in\Omega_{ii}\), with \(\Omega_{ii}\subseteq\Omega,\ i=1,2,\cdots,m,\ \omega_{cii}\) as the cut-off frequency of the \(i^{th}\) diagonal element, and \(\tau_{ii}\) as the corresponding time constant. Its state-space representation \((A_{f},B_{f},C_{f})\) can be obtained from \(G_{\text{dd}}(s)\) by an appropriate transformation. More details about selecting the low-pass filter from Equations (37a) and (37b) can be found in [24]. As in [4], the proposed approach preserves some independence in the design of the state observer, even though the dynamics of such an observer are associated with the EID estimator.