# A State of the Art Review of Systems of Linear Inequalities and Related Observability Problems
## 4 Observability
### Partial Observability  
If the system of linear inequalities neither has a unique solution, nor is incompatible, we can have partial information about the variables. In other words, the system has multiple solutions, and then, not all unknowns can be determined uniquely. However, it is possible that some subset of variables has a unique solution. In that case, we say that those variables are fully observable. For this to be possible, the corresponding rows of the linear subspace and cone parts must be null and those in the polytope must have identical values or the polytope be non-existent or reduced to the zero vector. In addition, we can obtain very easily lower and upper bounds for the remaining variables. In fact, the minimum and maximum values of each row provide these lower and upper bounds, respectively. This, though not full information, can be very valuable in practical applications, especially in cases where exact values of the unknowns are not needed. Another piece of important information that can be derived from the general solution, given in the indicated form is about boundedness or unboundedness. If the row of the linear subspace part of one unknown is not zero, that unknown is unbounded. If the row of the cone part of one unknown contains a positive value, that unknown is unbounded from above, and if it is negative it is unbounded from below. Example of the Water Supply Problem  
This example has two intentions: first, it aims at motivating real applications of systems of linear inequalities and, second, it is used to illustrate the theoretical observability problem using a practical problem. Consider the water supply network in Figure 2, which consists of 19 pipes and 13 nodes, of which nodes 1, 2, 3, and 4, are water tanks, where water is added, and nodes 5 to 13 are consumption nodes, where water is removed from the network. In addition, as explained before, we consider an extra virtual unknown, \(x_{20}\), we pass the independent terms to the left-hand side of the Equation (56), we add the equality \(x_{20}=1\) and, for convenience, the redundant inequality \(-x_{20}\leq 0\), as the first inequality. With this, we obtain the system:  
\[ \begin{array}{rcl}-x_{20}&<&0\\ (\mathbf{A}_{1}|-\mathbf{q})\mathbf{x}^{*}&=&\mathbf{0}_{13}\\ (\mathbf{0}_{19}^{t}-|1)\mathbf{x}^{*}&<&0\\ (\mathbf{I}_{19}|-\mathbf{c})\mathbf{x}^{*}&\leq&\mathbf{0}_{19}\\ (-\mathbf{I}_{19}|-\mathbf{c})\mathbf{x}^{*}&\leq&\mathbf{0}_{19}\\ x_{20}&=&1,\end{array}  \tag{60}\]  
where \(\mathbf{A}_{1}\) and \(\mathbf{q}\) are the matrices in (57), \(\mathbf{x}^{*}\) is the column matrix including the links \(x_{1}\) to \(x_{13}\) and the virtual link \(x_{14}\), \(\mathbf{0}_{k}\) is a zero column vector of dimension \(k\), and \(\mathbf{c}\) is a column vector of dimension 19 with values equal to the link capacities, which are assumed to all be equal to 20 units. The solution of this system can be obtained by obtaining the dual of the cone generated by the row vectors of the system, where the coefficients of the first 13 equations correspond to the linear subspace generators and the coefficients of the remaining 39 equations are the acute cone generators. Obtaining the dual of this cone using the gamma algorithm (see [18] or [6]), and following the process indicated above, we obtain the solution:  
\[\left( \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{5}\\ x_{6}\\ x_{7}\\ x_{8}\\ x_{10}\\ x_{11}\\ x_{12}\\ x_{13}\\ x_{14}\\ x_{15}\\ x_{16}\\ x_{17}\\ x_{18}\\ x_{19}\end{array} \right)=\mathbf{H}_{\lambda}=\left(