## 3 Feedback-Inversion Feedforward Design
### Bound Generation Inequality  
Equation (6) can be written element-wise by  
\[\frac{1}{1+l_{ii0}}=\frac{t_{eij}}{\Big{(}m_{ij}-p_{ij0}m_{ii0}p_{ii0}^{-1} \Big{)}-\Big{(}v_{ij}m_{ij}+\sum_{k\
eq j}^{n}v_{ik}m_{kj}\Big{)}+\Big{(}v_{ ij}t_{eij}+\sum_{k\
eq j}^{n}v_{ik}t_{kj}\Big{)}} \tag{7}\]  
For \(i=j\)  
Figure 3: Equivalent disturbance attenuation structure for the feedback controller design in model matching problem.  
From Equation (7), we have the following equations.  
\[\frac{1}{1+l_{i0}}=\frac{t_{eii}}{\left(m_{ii}-p_{i0}m_{ii}p_{i0}^{-1}\right)- \left(v_{ii}m_{ii}+\sum_{k\
eq i}^{n}v_{ik}m_{ki}\right)+t_{eii}\!\left(v_{ii}+ \sum_{k\
eq i}^{n}v_{ik}\frac{t_{eii}}{t_{eii}}\right)} \tag{8}\]  
Further,  
\[\frac{1}{1+l_{i0}}=\frac{t_{eii}}{t_{eii}\!\left(v_{ii}+\sum_{k\
eq i}^{n}v_{ ik}\frac{t_{eii}}{t_{eii}}\right)\!-\!\left(v_{ii}m_{ii}+\sum_{k\
eq i}^{n}v_{ik}m_{ ki}\right)} \tag{9}\]  
Here, the term \(\left(\frac{t_{eii}}{t_{eii}}\right)\) captures the interaction specifications. Equation (9), in terms of the absolute magnitude, can be expressed as follows:  
\[\left|\frac{1}{1+l_{ii0}}\right|\leq\frac{|t_{eii}|}{\left|v_{ii}m_{ii}+\sum_{k \
eq i}^{n}v_{ik}m_{ki}\right|+\left|t_{eii}\!\left(v_{ii}+\sum_{k\
eq i}^{n}v _{ik}\frac{t_{eii}}{t_{eii}}\right)\right|} \tag{10}\]  
Replacing the \(|t_{eii}|\) term by the specification inequality (2) in (10) gives the following equation  
\[\left|\frac{1}{1+l_{ii0}}\right|\leq\frac{b_{eii}}{\left|v_{ii}m_{ii}+\sum_{k \
eq i}^{n}v_{ik}m_{ki}\right|+b_{eii}\!\left|v_{ii}+\sum_{k\
eq i}^{n}v_{ik} \frac{t_{eii}}{t_{eii}}\right|} \tag{11}\]  
For \(i\
eq j\):  
From Equation (7),  
\[\frac{1}{1+l_{i0}}=\frac{t_{eij}}{\left(m_{ij}-p_{ij0}m_{ij}p_{j0}^{-1}\right) -\left(v_{ij}m_{ij}+\sum_{k\
eq j}^{n}v_{ik}m_{kj}\right)+t_{ejj}\!\left(v_{ ij}+\sum_{k\
eq j}^{n}v_{ik}\frac{t_{eij}}{t_{eij}}\right)} \tag{12}\]  
Equation (12), in terms of absolute magnitude, the above inequality can be written further as (using the triangle inequality),  
\[\left|\frac{1}{1+l_{i0}}\right|\leq\frac{\left|t_{eij}\right|}{\left[\left(m _{ij}-p_{ij0}m_{ij}p_{j0}^{-1}\right)-\left(v_{ij}m_{ij}+\sum_{k\
eq j}^{n}v_ {ik}m_{kj}\right)\right]+\left|t_{ejj}\!\left(v_{ij}+\sum_{k\
eq j}^{n}v_{ik} \frac{t_{eij}}{t_{eij}}\right)\right|} \tag{13}\]  
Inequality (13) can be then written further as follows.  
\[\left|\frac{1}{1+l_{i0}}\right|\leq\frac{\left|t_{eij}\right|}{\left|m_{ij}-p _{ij0}m_{ij}p_{j0}^{-1}\right|+\left|\left(v_{ij}m_{ij}+\sum_{k\
eq i}^{n}v_{ ik}m_{kj}\right)\right|+\left|t_{ejj}\!\left(v_{ij}+\sum_{k\
eq i}^{n}v_{ik} \frac{t_{eij}}{t_{eij}}\right)\right|} \tag{14}\]  
Replacing the \(|t_{eij}|\) and \(|t_{eij}|\) terms in (14) by its specification inequality (2) gives,  
\[\left|\frac{1}{1+l_{ii0}}\right|\leq\frac{b_{eij}}{\left|m_{ij}-p_{ij0}m_{ij} p_{j0}^{-1}\right|+\left|\left(v_{ij}m_{ij}+\sum_{k\
eq i}^{n}v_{ik}m_{kj} \right)\right|+b_{eij}\!\left|\left(v_{ij}+\sum_{k\
eq i}^{n}v_{ik}\frac{t_{ eij}}{t_{eij}}\right)\right|} \tag{15}\]  
The nominal loop transmission function \(l_{ii0}\) is designed to satisfy the Equations (11) and (15), i.e.,  
\[\left|\frac{1}{1+l_{ii0}}\right|\leq min\left(\frac{b_{eii}}{\left|p_{eii} \right|},\frac{b_{eij}}{\left|p_{eij}\right|}\right) \tag{16}\]where, \(\left|p_{eiviv}\right|=\left|p_{ii}m_{ii}+\sum_{k\
eq i}^{n}v_{ik}m_{ki}\right|+b_{ eii}\left|v_{ii}+\sum_{k\
eq i}^{n}v_{ik}\frac{t_{ki}}{t_{eii}}\right|\)  
\[\left|p_{eiiv}\right|=\left|m_{ij}-p_{ij0}m_{jj}p_{jj0}^{-1}\right|+\left|(v_{ ij}m_{ij}+\sum_{k\
eq j}^{n}v_{ik}m_{kj})\right|+b_{ejj}\left|(v_{ij}+\sum_{k\
eq i}^{ n}v_{ik}\frac{t_{ekj}}{t_{ejj}})\right|\]  
Inequality (16) is used to generate the TES bounds on \(l_{ii0}\). Here, \(i^{th}\) equivalent SISO system is termed as channel-\(ii\).  
**Remark 1**.: _Inequality (17) is valid for low frequency range specifications [15]._