## 8.6 Illustrative example

This example is to display the fact that the initial conditions of β_{NH}(t) and u(t) must satisfy some conditions in order that the system is impulsive free and the fact that the proposed approach is efficient in analyzing the impulsive property of the system.

Consider the following LNHMDE:

$\begin{array}{l}\hfill \left[\begin{array}{ll}\rho +1& {\rho}^{2}\\ 0& 1\end{array}\right]\left[\begin{array}{l}{\beta}_{1(NH)}(t)\\ {\beta}_{2(NH)}(t)\end{array}\right]=\left[\begin{array}{lll}{\rho}^{2}+1& \rho & 1\\ 0& \rho +1& 1\end{array}\right]\left[\begin{array}{l}{u}_{1}(t)\\ {u}_{2}(t)\\ {u}_{3}(t)\end{array}\right],t\ge 0.\end{array}$

(8.32)

with the initial values

$\begin{array}{l}\hfill {\beta}_{NH}(0):=\left[\begin{array}{l}{\beta}_{1(NH)}(0)\\ {\beta}_{2(NH)}(0)\end{array}\right],{{\beta}_{NH}}^{(1)}(0):=\left[\begin{array}{l}{\beta}_{1(NH)}^{(1)}(0)\\ {\beta}_{2(NH)}^{(1)}(0)\end{array}\right]\end{array}$

and

$\begin{array}{l}\hfill u\left(0\right)=\left[\begin{array}{l}{u}_{1}\left(0\right)\\ {u}_{2}\left(0\right)\\ {u}_{3}\left(0\right)\end{array}\right],{u}^{\left(1\right)}\left(0\right)=\left[\begin{array}{l}{u}_{1}^{\left(1\right)}\left(0\right)\\ {u}_{2}^{\left(1\right)}\left(0\right)\\ {u}_{3}^{\left(1\right)}\left(0\right)\end{array}\right]\end{array}$

we have

$\begin{array}{l}\hfill {A}_{2}=\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right],{A}_{1}=\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right],{A}_{0}=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right],\end{array}$

$\begin{array}{l}\hfill {B}_{}\end{array}$

Get *A Generalized Framework of Linear Multivariable Control* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.