### 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 *n*th 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 *n*th 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.