# A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems
## 4 Observability
### Partial Observability
\begin{array}{cccccccc}20& 20&20&20&20&20&20\\ 20&20&20&20&20&20&20\\ 20&20&20&20&20&20&20\\ 20&20&20&20&20&20&20\\ 20&20&20&20&20&20&20\\ 20&20&20&20&5&5&5\\ 0&0&0&0&15&15&15\\ 0&-20&20&20&-15&-20&20&20\\ -20&0&0&-20&-20&-15&-15&-20\\ -20&-20&20&0&-20&-20&20&15\\ -20&-20&-20&-20&-20&-20&-20\\ -20&-20&-20&-20&-20&-20&-20\\ 5&5&5&20&20&20&20\\ 0&0&0&0&0&0&0\\ -20&-20&-20&-20&-20&-20&-20\\ -20&-20&-20&-20&-20&-20\\ -20&-20&20&0&-20&20&15\\ 20&-20&-20&-20&-20&-20&-15\\ -20&-20&-20&-20&-20&-20\\ -20&-20&-20&-20&-20&-20\\ -20&-20&20&0&-20&20&15\\ -20&-20&-20&-20&-20&-20\\ -20&-20&-20&-20&-20&-20\\ -20&-20&20&0&20&20&15\\ 20&-20&-20&-20&-20&-20&-15\\ -20&-20&-20&-20&-20&-20\\ \end{array} \right)_{\lambda}, \tag{61}\]  
in which there are neither linear subspace nor cone parts but only a polytope generated by the eight vertices given by the column vectors of the last matrix. This means that we have infinitely many solutions to satisfy the indicated demand, that is, to distribute the indicated water amounts from the tanks to the consumption points, and then, not all the flows of the different links can be determined uniquely. Thus, the problem of observability, that is, determining which variables can be observed uniquely, arises. We note that while the uniqueness property is basic in mathematics, it is not in engineering, where one looks for multiple solutions that permit solving the problem caused for some unexpected failures. In other words, if some pipes in the network fail, making some solutions unfeasible, we want other possibilities to solve our problem. For example, in the water supply example, taking a look to the solution (61), one finds that there are no linear subspace nor cone parts and that for rows 1, 2, 3, 4, 11, 13, 15, 16, and 19, all columns have the same values. Thus, no matter what values of the \(\lambda\) linear convex combinations we choose, the final result will be the same for the \(x\) unknowns with the corresponding indices. On the contrary, the remaining rows contain different values, leading to infinitely many solutions. Consequently, we have full observability for nine unknowns. The remaining 10 unknowns are not fully observable, but we have very interesting information about them, because we can know immediately their lower and upper bounds, which are the minima and maxima values in their corresponding rows. For example, \(5\leq x_{5}\leq 20\), \(0\leq x_{6}\leq 15\) or \(-20\leq x_{14}\leq 0\). Another important conclusion that can be derived from the information in (61) is about what pipes can and what cannot be removed for making the satisfaction of the water demand possible. A simple look at the rows of the matrix in (61) shows that the link flows \(x_{1},x_{2},x_{3},x_{4},x_{11},\)\(x_{15},x_{16},\) and \(x_{19}\) are fixed and reach the maximum capacity of 20 units, where the positive values refer to flows in the direction indicated by the arrows in Figure 2, and the negative values refer to the contrary direction. These links cannot be removed; otherwise the required flows will no longer be warranted. In addition, the link \(x_{13}\) is null and can be removed. In addition, the links with a zero in their corresponding rows, or with the zero inside the segment limited by their minima and maxima, can be removed. Since all the columns are particular solutions to the problem, removing the zero links in a column is possible without failing to obtain a solution. For example, we can remove links \(6,8,13,\) and \(14\) because columns 3 and 4 contain zeros in those rows. In addition, any linear convex combination of those two columns provides a solution without those links, that is, we have infinitely many solutions. To find all solutions with these four links removed we can add to the system of inequalities (60) the equations \(x_{6}=x_{8}=x_{13}=x_{14}=0\), and then, we obtain the new polytope of all solutions:  
\[\left(