where $x=[x1,x2,...,x8]T$ is the vector of state space variables, namely $x1=\alpha $, $x2=\alpha \u02d9$, $x3=\xi $, $x4=\xi \u02d9$, $x5=w1$, $x6=w2$, $x7=w3$, and $x8=w4$. The augmented variables $wi$ are defined as [22]
Display Formula

(20)$w1=\u222b0\tau e\u2212\epsilon 1(\tau \u2212\sigma )\alpha (\sigma )d\sigma \u2009\u2009,\u2009w2=\u222b0\tau e\u2212\epsilon 2(\tau \u2212\sigma )\alpha (\sigma )d\sigma w3=\u222b0\tau e\u2212\epsilon 1(\tau \u2212\sigma )\xi (\sigma )d\sigma \u2009,\u2009\u2009w4=\u222b0\tau e\u2212\epsilon 2(\tau \u2212\sigma )\xi (\sigma )d\sigma $