A Generalized Framework of Linear Multivariable Control

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

$\begin{array}{l}\hfill A_2=\left[\begin{array}{ll}0& 1\\ 0& 0\end{array}\right],A_1=\left[\begin{array}{ll}1& 0\\ 0& 0\end{array}\right],A_0=\left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right],\end{array}$

$\begin{array}{l}\hfill B_{}\end{array}$