Now, the synthesis of a HOSM control is recalled. The advantage of this methodology [74] is such that the time of convergence \(t_f\) is stated *a priori* and the robustness is ensured during the entire response of the system.

The synthesis of the high-order sliding mode controller is designed in two steps:

(1) A linear continuous finite-time convergent control law is used in order to *induce reference trajectories* for system (4.60), which defines the sliding manifold on which the system evolves as early as \(t = 0\).

(2) A discontinuous control law is designed in order to maintain the system trajectories on the sliding manifold which ensures the establishment a \(r\)th order sliding mode.

Consider an uncertain nonlinear system of the form

$$\begin{aligned} \dot{x}=f(x)+g(x)u \end{aligned}$$

(4.60)

with

\(x\in \mathcal{X} \subset \mathfrak {R}^n\) the state variable and

\(u \in \mathfrak {R}\) the input control. For a sake of clarity, only single input–single output case is considered in the sequel.

Let \(\sigma _{c}(x,t)\) the sliding variable with a relative degree equal to \(r\).

The

\(r\)th order sliding mode control approach allows the finite-time stabilization to zero of the sliding variable

\(\sigma _{c}\) and its

\(r-1\) first time derivatives by defining a suitable discontinuous control function. Then, the output

\(\sigma _{c}\) satisfies equation

$$\begin{aligned} \sigma _{c}^{(r)}= & {} \varphi _{1}(x,t)+\varphi _{2}(x)u \end{aligned}$$

(4.61)

with

\({\displaystyle \varphi _{2}(x) = L_{g}L_{f}^{r-1} \sigma _{c}}\) and

\({\displaystyle \varphi _{1}(x)= L_{f}^{r}\sigma _{c}}\). System (

4.60) has to satisfy:

Then, the

\(r\)th order sliding mode control of (

4.60) with respect to the sliding variable

\(\sigma _c\) is equivalent to the finite-time stabilization of

$$\begin{aligned} \dot{Z}_{c1} = A_{11}Z_{c1} + A_{12}Z_{c2},&\dot{Z}_{c2} = \varphi _{1} + \varphi _{2} u \end{aligned}$$

(4.62)

with

\(Z_{c1}:=[\sigma _{c}~\dot{\sigma }_{c}~ \cdots ~\sigma _{c}^{(r-2)}]^T\) and

\(Z_{c2}=\sigma _{c}^{(r-1)}\).

\({A_{11}}_{(r-1)\times (r-1)}\) and

\({A_{12}}_{(r-1)\times 1} \) are such that

\(Z_{c1}\) dynamics reads as linear Brunovsky form.

**Controller synthesis**

The synthesis of a high-order sliding mode controller for (4.60) consists of two steps.

*Step 1: Switching variable design* Let

\(S\) denote the switching variable defined as

$$\begin{aligned} S= & {} \sigma _{c}^{(r-1)}-{\displaystyle \mathcal F}^{(r-1)}(t)+\lambda _{r-2}\left[ \sigma _{c}^{(r-2)}- {\displaystyle \mathcal F}^{(r-2)}(t)\right] \\&\quad +\, \cdots \, +\lambda _{0}\left[ \sigma _{c}(x,t) - {\displaystyle \mathcal F}(t)\right] ,\nonumber \end{aligned}$$

(4.63)

with

\(\lambda _{r-2},\ldots ,\lambda _{0}\) defined such that

\(P(z) = z^{(r-1)} + \lambda _{r-2}z^{(r-2)}+ \cdots + \lambda _{0}\) is a Hurwitz polynomial in the complex variable

\(z\). The function

\({\displaystyle \mathcal F}(t)\) is a

\(C^r -\) one defined such that

\(S(t=0)=0\) and

\(\sigma _{c}^{(k)}(x(t_f), t_f) -{\displaystyle \mathcal F}^{(k)}(t_f) = 0\) \((0 \le k \le r-1)\). Then, from initial and final conditions the problem consists in finding the function

\({\displaystyle \mathcal F}(t)\) such that

$$\begin{aligned} \begin{array}{llll} \sigma _{c,t=0}={\displaystyle \mathcal F}(0), \sigma _{c,t=t_f}={\displaystyle \mathcal F}(t_f)=0, \dot{\sigma }_{c,t=0}=\dot{{\displaystyle \mathcal F}}(0), \\ \\ \dot{\sigma }_{c,t=t_f}=\dot{{\displaystyle \mathcal F}}(t_f)=0, \ldots , \sigma _{c,t=0}^{(r-1)}= {\displaystyle \mathcal F}^{(r-1)}(0), \\ \\ \sigma _{c,t=t_f}^{(r-1)}= {\displaystyle \mathcal F}^{(r-1)}(t_f)=0. \end{array} \end{aligned}$$

A solution for

\({\displaystyle \mathcal F}(t)\) for (

\(1\le j \le r\)) [77], is given by

$$\begin{aligned} {\displaystyle \mathcal F}(t)=K_{c}Te^{Ft}\sigma _{c}^{(r-j)}(0) \end{aligned}$$

(4.64)

with

\(F\) being a

\(2r \times 2r\)-dimensional stable matrix (strictly negative eigenvalues),

\(T\) being a

\(2r \times 1\)-dimensional vector, and

\(K_{c}\) is a

\(1 \times 2r\)-dimensional gain matrix such system (

4.64) is fulfilled.

Now, we have the following lemma:

Then, the gain matrix

\(K_{c}\) is given by

$$\begin{aligned} K_{c}=\left[ \sigma _{c}^{(r-1)}(0)~0~\sigma _{c}^{(r-2)}(0)~0~\cdots ~\sigma _{c}(0)~0\right] {\displaystyle \mathcal K}^{-1}. \end{aligned}$$

(4.66)

Furthermore, the switching variable

\(S\) is expressed as

$$\begin{aligned} S= & {} \sigma _{c}^{(r-1)}-K_{c}TF^{(r-1)}e^{Ft}\sigma _{c}^{(r-j)}(0)\nonumber \\&\quad +\,\,\lambda _{r-2}\left[ \sigma _{c}^{(r-2)}- K_{c}TF^{(r-2)}e^{Ft}\sigma _{c}^{(r-j)}(0)\right] \\&\quad +\,\cdots \,+\lambda _{0}\left[ \sigma _{c}(x,t)- K_{c}Te^{Ft}\sigma _{c}^{(r-j)}(0)\right] \nonumber . \end{aligned}$$

(4.67)

Equation

\(S=0\) describes the desired dynamics which satisfy the finite time stabilization of

\([\sigma _{c}^{(r-1)} \sigma _{c}^{(r-2)}\,\cdots \,\sigma _{c}]^T\) to zero. The

*switching manifold* on which system (

4.62) is forced to slide, via a discontinuous control

\(v\), is defined as

$$\begin{aligned} \mathcal{S}= & {} \{x|S=0\}. \end{aligned}$$

(4.69)

Given Eq. (

4.66), one gets

\(S(t=0)=0\), at the initial time, the system still evolves on the switching manifold which implies that there is no reaching phase.

*Step 2: Discontinuous control design*

The attention is now focused on the design of the discontinuous control law \(u\) which forces the system trajectories of (4.62) to slide on \(\mathcal{S}\) in order to reach the origin in finite time and so to maintain the system at the origin.

Condition (

4.70) allows to satisfy the attractivity condition

$$\dot{S}S\le -\eta |S|$$

with

\(\eta >0\). By using the same procedure established in Theorem

4.2, there exist gains

\(\alpha _{c}\) such that for the Lyapunov function

$$ W= \dfrac{1}{2}S^{T}S $$

whose time derivative satisfy the inequality

$$\begin{aligned} \dot{W}= \dot{S}S \le -\eta |S|\le -\eta \sqrt{W}. \end{aligned}$$

(4.71)

Integrating the above inequality, we have

$$\begin{aligned} \sqrt{W(t)} \le \sqrt{W(t_{0})}-\dfrac{\eta }{2}t. \end{aligned}$$

(4.72)

Let

\( \sqrt{W(t_{0})}-\dfrac{\eta }{2}t_{f}=0 \), then the time of convergence is given by

$$\begin{aligned} t_{f}= \dfrac{2\sqrt{W(t_{0})}}{\eta }. \end{aligned}$$

(4.73)

Then, the convergence in finite time of the tracking dynamics is guarantee.