## 5 Stability and Superstability Analysis
### Superstability Analysis  
The value of the free response of an asymptotically stable system decreases to zero over time, but it may considerably increase in the initial part of the trajectory. In superstable systems, which are a subclass of asymptotically stable systems, state variables are limited by the value of the norm of the state vector, which decreases monotonically to zero over time [28; 29; 30]. Furthermore, the problems of static output stabilization, the simultaneous stabilization of more than one system, robust stabilization under matrix uncertainty, etc., are solved easily for superstable systems [29]. In this paper the following vector and matrix norms will be used:  
1. the infinity-norm of a vector \(x_{i}=[x_{i,k}]\in\mathbb{R}^{n}\) \[\|x_{i}\|=\max_{1\leq k\leq n}|x_{i,k}|,\] (39)
2. the infinity-norm of a matrix \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) \[\|A\|=\max_{1\leq i\leq n}\left(\sum_{j=1}^{n}|a_{ij}|\right).\] (40)  
**Definition 5** ([29]). : _A matrix \(A=[a_{ij}]\in\mathbb{R}^{n\times n}\) of the discrete-time linear system_  
\[x_{i+1}=Ax_{i} \tag{41}\]  
_is called superstable if_  
\[\sigma(A)=\sigma=1-\|A\|>0 \tag{42}\]  
_or equivalently_  
\[\|A\|<1, \tag{43}\]  
_where the quantity \(\sigma\) is called the superstability degree of the matrix \(A\)._  
A superstable matrix is also a stable one, but the reverse implication is not true (a stable matrix may not be a superstable one). **Theorem 6** ([29]). : _For the superstable discrete-time linear system (41) the following holds:_  
\[\|x_{i}\|\leq\sigma^{i}\|x_{0}\|,\;\;i\in\mathbb{Z}_{+}. \tag{44}\]Now let us consider the DDFL system (1). From (25) and (26) for \(u_{i}=0\), \(i\in\mathbb{Z}_{+}\) we have  
\[x_{i}=x_{1,i}=\bar{E}\bar{E}^{D}x_{i},\quad i\in\mathbb{Z}_{+} \tag{45}\]  
since \(x_{2,i}=0\). Taking into account (45) and \((\mathbb{I}_{n}-\bar{E}\bar{E}^{D})\bar{E}\bar{E}^{D}=0\), Equation (25) for \(u_{i}=0\), \(i\in\mathbb{Z}_{+}\) can also be written as  
\[x_{i+1}=Fx_{i}+\sum_{j=1}^{L}c_{j}\bar{E}E^{D}x_{i-j}, \tag{46}\]  
where  
\[\bar{F}=\bar{A}_{1\alpha}+\bar{G}(\mathbb{I}_{n}-\bar{E}\bar{E}^{D}) \tag{47}\]  
and the matrix \(\bar{G}\in\mathbb{R}^{n\times n}\) is arbitrary. In descriptor systems the matrix \(\bar{A}_{1\alpha}\) acts as a pseudo-state matrix. From the solution of Equation (23) it follows that \(\bar{A}_{1\alpha}\) may contain insignificant entries that are further reduced through multiplication by \(x_{0}\in\mathbb{I}\bar{E}^{D}\). To eliminate such entries from the matrix \(\bar{A}_{1\alpha}\) we can use the term \(\bar{G}(\mathbb{I}_{n}-\bar{E}\bar{E}^{D})\), which does not change the solution to the state equation [3; 31]. Taking into consideration (16) and (45) the solution to Equation (46) for \(v=x_{0}\) can be expressed by  
\[x_{i}=\Phi_{i}^{(\bar{F})}\bar{E}\bar{E}^{D}x_{0}, \tag{48}\]  
where  
\[\|A\|=\max_{1\leq i\leq n}\left(\sum_{j=1}^{n}|a_{ij}|\right).\]0  
Let us introduce the definition of practical superstability for DDFL systems, analogous to the definition of practical stability given in Section 5.1. Definition 6: The DDFL system (1) is called practically superstable for a given length \(L\) of practical implementation if the DDFL system (46) is superstable. If the DDFL system (46) is superstable for \(L\to\infty\), then the DDFL system (1) is called superstable (independent of \(L\)). Theorem 6.1: The DDFL system (46) is superstable if there exists an arbitrary matrix \(\bar{G}\in\mathbb{R}^{n\times n}\) such that  
\[\|A\|=\max_{1\leq i\leq n}\left(\sum_{j=1}^{n}|a_{ij}|\right).\]1  
or  
\[\|A\|=\max_{1\leq i\leq n}\left(\sum_{j=1}^{n}|a_{ij}|\right).\]2  
where the matrix \(\bar{F}\) is defined by (47) and  
\[\|A\|=\max_{1\leq i\leq n}\left(\sum_{j=1}^{n}|a_{ij}|\right).\]3  
Proof: From Theorem 6 it follows that for a superstable system we have \(\|x_{i+1}\|<\|x_{i}\|\). Let us assume that the DDFL system (46) is superstable.