\(\delta\) function:
Consider a sampled switched system with bounded perturbation
of the form \(\{\frac{dy(t)}{dt}=f_u(y(t),w(t))\}_{u\in U}\)
satisfying the equation:
\begin{multline}\label{eq:H1}
\langle f_u(y_1,w_1)-f_u(y_2,w_2), y_1-y_2\rangle
\\
\leq
\lambda_u\|y_1-y_2\|^2 + \gamma_u \|y_1-y_2\|\|w_1-w_2\|
\end{multline}
where for all \(u\in U\),
there exist constants \(\lambda_u\in\mathbb{R}\)
and \(\gamma_u\in\mathbb{R}_{\geq 0}\) such that,
for all \(y_1,y_2\in {\cal S}\) and \(w_1,w_2\in {\cal W}\).
Recall that \(\lambda_u\) has to be computed in the absence of perturbation
(\(w=0\)). The additional constant \(\gamma_u\) is used for taking into account
the perturbation \(w\).
Given \(\lambda_u\), the constant \(\gamma_u\) can be computed itself
using a nonlinear optimization solver. Instead of computing them globally
for \(S\), it is advantageous to compute \(\lambda_u\) and \(\gamma_u\) locally depending on the subregion of \(S\) occupied by the system state during a considered interval of time.
Consider a point \(y_0\in {\cal S}\) of \(\varepsilon\)-representative \(z_0\in {\cal X}\).
We have, for all \(u\in U\), \(t\in[0,\tau]\) and \(w(t)\) with \(\|w(t)\|\leq \omega\):
\begin{equation}
\|Y^{u}_{y_0, w}(t)-\tilde{Y}^u_{z_0}(t)\|\leq \delta^u_{\varepsilon,{\cal W}}(t)
\end{equation}
with:
- if \(\lambda_{u} <0\),
\begin{multline}
\delta^u_{\varepsilon,{\cal W}}(t) =
\left( \frac{C_{u}^2}{-\lambda_{u}^4} \left( - \lambda_{u}^2 t^2 - 2 \lambda_{u} t + 2 e^{\lambda_{u} t} - 2 \right) \right. \\
+ \left. \frac{1}{\lambda_{u}^2} \left( \frac{C_{u} \gamma_{u} |{\cal W}|}{-\lambda_{u}} \left( - \lambda_{u} t + e^{\lambda_{u} t} -1 \right) \right. \right. \\ + \left. \left. \lambda_{u} \left( \frac{\gamma_{u}^2 (|{\cal W} |/2)^2}{-\lambda_{u}} (e^{\lambda_{u} t } - 1) + \lambda_{u} \varepsilon^2 e^{\lambda_{u} t} \right) \right) \right)^{1/2}
\end{multline}
- if \(\lambda_{u} >0\),
\begin{multline}
\delta^u_{\varepsilon,{\cal W}}(t) = \frac{1}{(3\lambda_{u})^{3/2}} \left( \frac{C_u^2}{\lambda_{u}} \left( - 9\lambda_{u}^2 t^2 - 6\lambda_{u} t + 2 e^{3\lambda_{u} t} - 2 \right) \right. \\
+ \left. 3\lambda_{u} \left( \frac{C_u \gamma_{u} |{\cal W}|}{\lambda_{u}} \left( - 3\lambda_{u} t + e^{3\lambda_{u} t} -1 \right) \right. \right. \\
+ \left. \left. 3\lambda_{u} \left( \frac{\gamma_{u}^2 (|{\cal W} |/2)^2}{\lambda_{u}} ( e^{3\lambda_{u} t } - 1) + 3\lambda_{u} \varepsilon^2 e^{3\lambda_{u} t} \right) \right) \right)^{1/2}
\end{multline}
- if \(\lambda_{u} = 0\),
\begin{multline}
\delta^u_{\varepsilon,{\cal W}}(t)=
\left( {C_{u}^2} \left( - t^2 - 2 t + 2 e^{ t} - 2 \right) \right. \\
+ \left. \left( {C_{u} \gamma_{u} |{\cal W}|} \left( - t + e^{ t} -1 \right) \right. \right. \\ + \left. \left. \left({\gamma_{u}^2 (|{\cal W} |/2)^2} ( e^{ t } - 1) + \varepsilon^2 e^{ t} \right) \right) \right)^{1/2}
\end{multline}
where \(|{\cal W}|\) is the diameter of \({\cal W}\) (maximum distance between 2 points of \({\cal W}\)).
NB: In the examples, the tubes \({\cal B}(t)\) are represented by the green curves which correspond to the borders of the tube.