Appendix
In the method based on the time-state control form, a chained system is transformed into two linear systems: the state-control part, which is an nth order linear system, and the time-control part, which is a first-order linear system. We generally provided the conditions for asymptotically converging the state of the nth order linear system under arbitrary switching of the time axis [18]. The two-wheeled car version of the proof is given below.
Under the assumption that |θ| < π/2, we use the nonlinear coordinate transformation
$$\varvec{z} = (y,\tan \theta )^{T}$$
(A1)
and the input transformation
$$\mu = \frac{{v_{2} }}{{v_{1} \cos^{3} \theta }}$$
(A2)
for Eq. (1). The state x increases with respect to time t when v
1 > 0; therefore, the state x can be regarded as an alternative time scale. Since the derivative of y with respect to x is
$$\begin{aligned} \frac{dy}{dx} & = \frac{dy}{dt}\frac{dt}{dx} \\ & = \tan \theta , \\ \end{aligned}$$
(A3)
the state equation of z with the time scale x is obtained as
$$\frac{d}{dx}{\varvec{z}} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right]\,\mu .$$
(A4)
The state x is controlled by the input v
1.
$$\frac{dx}{dt} = v_{1} \cos \theta$$
(A5)
Eq. (A4) is the controllable canonical form. A state feedback controller for stabilizing the system given by Eq. (A4) can be easily designed using the linear control theory. When z → 0 as x → ∞ by using the stabilizing feedback controller, θ also converges to 0 because of the assumption.
When v
1 < 0, x decreases with respect to the actual time t. Therefore, we have to introduce another time scale x′ = − x which increases with respect to t for analyzing the stability of the system. The state equation of z with x′ is obtained as
$$\frac{d}{dx'}{\varvec{z}} = \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \right]\,\mu .$$
(A6)
Now, a new time scale τ is defined as
$$\tau = \int_{0}^{\,t} {\left| {\,\frac{dx}{dt}} \right|\,dt} .$$
(A7)
The time scale τ is the distance traveled along the x-axis. The two systems (A4) and (A6) are described as a switched system by using the time scale τ [17].
$$\frac{d}{d\tau }{\varvec{z}} = \left\{ {\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} 0 & {\,\,1\,\,} \\ 0 & {\,\,0\,} \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ {\,\,1\,\,} \\ \end{array} } \right]\,\mu \,\,\,\,\,\,\,\,\,(v_{1} > 0)} \\ {\left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} +\left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \right]\,\mu \,\,\,\,\,\,\,\,\,(v_{1} < 0)} \\ \end{array} } \right.$$
(A8)
The input function μ in the controller (2) can be rewritten as
$$\mu = \left\{ \begin{aligned} - \,\,\left[ {\,\begin{array}{*{20}c} {k_{1} } & {\,\,\,\,\alpha k_{2} } \\ \end{array} } \right]\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} > 0) \hfill \\ - \,\,\left[ {\begin{array}{*{20}c} {\,k_{1} } & { - \alpha k_{2} } \\ \end{array} } \right]\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} < 0) \hfill \\ \end{aligned} \right..$$
(A9)
By substituting (A9) into (A8), the closed-loop system becomes
$$\frac{d}{d\tau }{\varvec{z}} = \left\{ \begin{aligned} &\tilde{A}\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} > 0) \hfill \\ &E_{2} \tilde{A}\,E_{2}\, {\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} < 0) \hfill \\ \end{aligned} \right.,$$
(A10)
where
$$\tilde{A} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - k_{1} } & { - \alpha k_{2} } \\ \end{array} } \right]\,\,,\,\,\,\,E_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right].$$
(A11)
Now, consider a Lyapunov candidate function as
$$V = {\varvec{z}}^{T} P\,{\varvec{z}},\quad P = \left[ {\begin{array}{*{20}c} {k_{1} k_{2} }& 0 \\ 0 &{k_{2} } \\ \end{array} } \right].$$
(A12)
If k
1 and k
2 > 0, V > 0 (z ≠ 0). The derivative of V with respect to τ for both v
1 > 0 and v
1 < 0 is calculated as follows:
$$\begin{aligned} \frac{d}{d\tau }V & = \left\{ {\begin{array}{*{20}c} {{\varvec{z}}^{T} \left( {\tilde{A}^{T} P + P\tilde{A}} \right)\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} > 0)} \\ {{\varvec{z}}^{T} \left( {E_{2} \tilde{A}^{T} E_{2} P + PE_{2} \tilde{A}E_{2} } \right)\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} < 0)} \\ \end{array} } \right. \\ & = {\varvec{z}}^{T} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & { - 2\alpha k_{2}^{2} } \\ \end{array} } \right]\,{\varvec{z}} \\ & = - \alpha \,{\varvec{z}}^{T} {\varvec{c}}^{T} {\varvec{c}}\,{\varvec{z}}\,\,, \\ \end{aligned}$$
(A13)
where
$${\varvec{c}} = [\begin{array}{*{20}c} 0 & {\sqrt 2 } \\ \end{array} k_{2} ].$$
(A14)
From the condition α > 0, we have dV/dt ≤ 0. Furthermore, both \(({\varvec{c}},\tilde{A})\) and \(({\varvec{c}},\,\,E_{2} \tilde{A}E_{2} )\) are observable pairs; therefore, it is guaranteed that z → 0 as τ → ∞ under the arbitrary switching of α and the sign of v
1 by Lyapunov’s stability theory.