NJ2. K. A. Zimenko, A. E. Polyakov, Convergence speed increase of a finite-time stable control system, Scientific and Technical Journal of Information Technologies, Mechanics and Optics, Vol. 17 (6), 2017, p. 1018-1024.

NJ3. K. A. Zimenko, A. E. Polyakov, D. V. Efimov, A. S. Kremlev, Finite-time Stability of System of Series-Connected Integrators, Izvestiya Vysshikh Uchebnykh Zavedeniy. Priborostroenie, 2015, 58 (9), p. 681-686 (in Russian).

NJ4. S. Parsegov, A. Polyakov, P. Shcherbakov, Nonlinear fixed-time control protocol for uniform allocation of agents on a segment, Doklady Mathematics, Vol. 87(1), 2013, p. 133-136. (Translated from Proc. of the Russian Acad. of Sci.)

NJ5. V. Strygin, M. Yefremov, A. Polyakov, S. Prosin, The Algorithm of Angular Stabilization of Spacecraft with Flexible Dynamical Elements under Uncertain Conditions, Proceedings of Voronezh State University, No.1, 2006. (in Russian)

NJ6. A. Polyakov, K.S. Kobychev, Stabilization of mechanical systems via relay delayed control, Chernozemny anthology of scientific research. Series: Appl. Maths and Comp. Sciences, Voronezh, 2006, Vol. 1(2), pp. 15-26. (in Russian)

NJ7. V. Strygin, A. Polyakov, M. Kryachkov On some air-to-fuel ratio control algorithms for automobile engines, Proceedings of Voronezh State University, Series: Mathematics and Physics, No.1, 2005. (in Russian)

NJ8. V. Strygin, L. Fridman, A. Polyakov, Local Stabilization of Relay Systems With Delay, Doklady Mathematics, Vol. 64(1), 2001, p.106-108. (Translated from Proceedings of the Russian Academy of Sciences) (IF:0.289)

## Reviewed national conferences

NC1. M. Kryachkov, A. Polyakov, V. Strygin, Stabilization of uncertain system without quantitative information on system state, Control in Technical Systems, Saint-Petersburg, 2010 (in Russian).

NC2. A. Polyakov, A. Poznyak, Lyapunov Function Design for Finite-Time Convergence Analysis of The Second Order Sliding Mode Controllers, X International Workshop "Stability and Oscillations of Nonlinear Control Systems", 3 - 6 of June, Moscow, 2008 (in Russian).

NC3. S.A. Prosin, A. Polyakov, V.V. Strygin, Modeling, control and realization of an Aircraft Landing, X International Workshop "Stability and Oscillations of Nonlinear Control Systems", 3-6 June 2008, Moscow (in Russian).

NC4. M. Yefremov, A. Polyakov, S. Prosin, V. Strygin, An Universal Algorithm of Semiglobal Stabilization of Mechanical Systems with Uncertainties, Proc. of 9-th All-Russian Congress on Theoretical and Applied Mechanics, 22 - 28 August 2006, N. Novgorod, Russia (in Russian).
## B. Selected articles

The following my articles are attached below

* A. Polyakov, A. Poznyak, Lyapunov Function Design for Finite-Time Convergence Analysis: "Twisting" Controller for Second Order Sliding Mode Realization, Automatica, 2009. 45(2), p. 444-448

* A. Polyakov, Nonlinear feedback design for fixed-time stabilization of Linear Control Systems, IEEE Transactions on Automatic Control, 2012, 57(8), p. 2106-2110

* A. Polyakov, D. Efimov, W. Perruquetti, Finite-time and Fixed-time Stabilization: Implicit Lyapunov Function Method, Automatica, 2015, 51(1), p.332-340

* A. Polyakov, L. Hetel, Relay Control Design for Robust Stabilization in a Finite-Time, IEEE Transactions on Automatic Control, 2017, 62(4), p. 1985-1991
# Lyapunov Function Design for Finite-Time Convergence Analysis: "Twisting" Controller for Second Order Sliding Mode Realization

Andrei Polyakov and Alex Poznyak*

Automatic Control Department, CINVESTAV-IPN, A.P. 14-740, C.P. 07360 Mexico D.F, e-mail: apolyakov(apoznyak)@ctrl.cinvestav.mx,

## Abstract

A generalization of the Zubov's method of a Lyapunov functions design is presented. It is based on the characteristic method application and related to resolving of the first-order partial differential equation of a special type. A successful resolution of this equation guarantees a finite-time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right-hand side. The suggested method is illustrated by its application to the, so-called, "twisting" controller stability analysis. The constructed Lyapunov function as well as its level lines sections is graphically illustrated.

Key words: Lyapunov function; finite-time convergence; characteristic method; high-order sliding mode

# 1 Introduction

## 1.1 On the Lyapunov function approach

In the seminal work published by A.M.Lyapunov in the end of the 19-th century there were presented some very simple (but, philosophically, very profound) theorems (after referred to as the direct Lyapunov's method) for deciding the stability or instability of an equilibrium point of an ODE. The idea of this approach consists in the generalization of the concept of "energy" and its "power" which usefulness lies in the fact that the decision on stability can be made by investigating the differential equation itself (in, fact, its right-hand side only) but not by finding its exact solution. The considered class of ODE contained basically the so-called regular differential equations with right-hand sides continuous on both variables: time and state. Later several authors [5], [18] considered more wide class of differential equations where right-hand side was discontinuous on its variables. Modern approaches usually treat such class of differential equations as differential inclusions (DI). This class of systems includes as a subclass the so-called Variable Structure Systems (VSS') which has attracted a considerable research interest in the last three decades

[17], [8], [9], [13]. Although Lyapunov-type methods have been widely used in practice [3], [16], the need for non-smooth Lyapunov functions in particular has been recognized for non-smooth system models [1]. So, to analyze VSS' with sliding modes, where "kinks" form an essential part of dynamics, a non-smooth Lyapunov-like analysis is required [12].

## 1.2 Motivation

One of the most important features of VSS' systems with sliding modes is a finite-time convergence to a sliding surface (or manifold), or in other words, a finite-time reaching phase $t_{reach}$. The corresponding Lyapunov functions are non-smooth in this case. For example (see [17]), for the first-order sliding mode systems, the simplest representative of which is

$$ \dot{x}(t) = -r \operatorname{sign}[x(t)], \quad r > 0 \quad (1) $$

with

$$ \operatorname{sign}[x] := \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \in [-1, 1] & \text{if } x = 0 \end{cases} \quad (2) $$

evidently, the Lyapunov function is $V(x) = |x|$, and hence, its time-derivative (for $x \neq 0$) taken on the tra-

<footnote>Email address: apolyakov(apoznyak)@ctrl.cinvestav.mx (Andrei Polyakov and Alex Poznyak*).</footnote>
jectories of (1) satisfies

$$ \dot{V}(x(t)) = -r $$

that implies $0 \le V(x(t)) = V(x(0)) - rt$ and defines $t_{reach} = V(x(0))/r$. For the second-order sliding modes (see [8]), such as

$$ \ddot{x}(t) = -r_1 \operatorname{sign}[x(t)] - r_2 \operatorname{sign}[\dot{x}(t)], \quad r_1 > r_2 > 0 \quad (3) $$

the finite-time convergence analysis and the reaching time estimation have been done by a geometrical setting [10]. Such proof seems to be difficult to extend to multidimensional case. Besides, in [12] to analyze the convergence of $x(t)$ governed by (3) there has been suggested the Lyapunov function approach with the Lyapunov function

$$ V(x, \dot{x}) = r_1 |x| + \frac{\dot{x}^2}{2} \quad (4) $$

which guarantees only asymptotic (non finite-time) convergence since in this case it turns out that $\dot{V}(x(t), \dot{x}(t)) \le 0$. The modern approaches to the finite-time convergence analysis of high-order sliding mode systems is based on homogeneity principle [2], [11], [16], [6]. However, it does not allow to estimate the reaching time. Thus, the unified approach to the problems of the finite-time convergence analysis and the reaching time estimation of the high-order sliding mode systems has never been presented before.

So, the motivation of this study is based on the following remarks:

- the geometrical approach for the convergence analysis and reaching time estimation can not be definitely extended to multidimensional case;

- the Lyapunov functions providing finite-time convergence of the high-order sliding systems have never been designed;

- the finite-time converged Lyapunov function permits to estimate the reaching time even in the presence of a bounded deterministic noise.

## 1.3 Structure of the paper

In the next section the main ideas of the Lyapunov function design providing a finite-time convergence are discussed. Then, using the suggested approach, the process of the Lyapunov function designing for the so-called "twisting" controller is given. Finally, it is illustrated by a numerical example.

# 2 The Method of the Lyapunov Function Design with Finite-Time Convergence

## 2.1 System description

Consider an affine control system given by

$$ \begin{cases} \dot{x} = g(x, y) \\ \dot{y} = a(x, y) + b(x, y)u + f(t, x, y) \end{cases} \quad (5) $$

where $x \in \mathbb{R}^k, y \in \mathbb{R}^n$ are components of the state-space vector, $g: \mathbb{R}^k \times \mathbb{R}^n \to \mathbb{R}^k, a: \mathbb{R}^k \times \mathbb{R}^n \to \mathbb{R}^n$ are smooth system vector functions, $u \in \mathbb{R}^m$ is vector of control inputs, $b: \mathbb{R}^k \times \mathbb{R}^n \to \mathbb{R}^{n \times m}$ are control-gain matrix, and $f: \mathbb{R}_+ \times \mathbb{R}^k \times \mathbb{R}^n \to \mathbb{R}^n$ is an uncertain measurable but bounded function (in fact, a deterministic noise), i.e.,

$$ |f_j(t, x, y)| \le C_j, \quad \forall x \in \mathbb{R}^k, \forall y \in \mathbb{R}^n, \forall t \ge 0, j = \overline{1, n} \quad (6) $$

Supposing that a stabilizing control $u$ is already designed as

$$ u = \bar{u}(x, y) \quad (7) $$

(which, in fact, is admitted to be discontinuous) we have to prove the stability (asymptotic or finite-time) of zero solution $(0, 0)$ of the system (5) with the control (7) using the Lyapunov function technique.

## 2.2 The generalization of Zubov's Method

Let $V(x, y)$ be a function differentiable almost everywhere. Then its derivative on the trajectories of (5) is estimated as follows

$$ \begin{aligned} \dot{V} &= \langle \nabla_x V, g \rangle + \langle \nabla_y V, a + b\bar{u} + f \rangle \\ &\le \langle \nabla_x V, g \rangle + \langle \nabla_y V, a + b\bar{u} \rangle + \langle |\nabla_y V|, C \rangle \end{aligned} \quad (8) $$

where $\langle \cdot, \cdot \rangle$ means the scalar product in an Euclidian space, $|\nabla_y V| = (|\frac{\partial V}{\partial y_1}|, |\frac{\partial V}{\partial y_2}|, \dots, |\frac{\partial V}{\partial y_n}|)^T$ and $C = (C_1, C_2, \dots, C_n)^T$ is the vector with non-negative components. Denoting

$$ h_j(x, y, \gamma_j) := a_j(x, y) + b_j(x, y)\bar{u}(x, y) + \gamma_j, \quad j = \overline{1, n} $$

the inequality (8) might be rewritten (in the component-wise form) as

$$ \frac{dV}{dt} \le \sum_{i=1}^{k} \frac{\partial V}{\partial x_i} g_i(x, y) + \sum_{j=1}^{k} \frac{\partial V}{\partial y_j} h_j(x, y, C_j \operatorname{sign}[\frac{\partial V}{\partial y_j}]) $$

Let us try to find the Lyapunov function $V(x, y)$ as an absolute continuous positive definite solution of the following partial differential equation

$$ \sum_{i=1}^{k} g_i(x, y) \frac{\partial V}{\partial x_i} + \sum_{j=1}^{k} h_j(x, y, \gamma_j) \frac{\partial V}{\partial y_j} = -qV^\rho \quad (9) $$
where $\gamma_j = C_j \text{sign}[\frac{\partial V}{\partial y_j}]$ and $q > 0, \rho > 0$ are some positive parameters.

**Remark 1** a) In the case $\rho < 1$ we obtain the Lyapunov function with finite time convergence $t_{reach} \le \frac{1}{q(1-\rho)}[V(x(0), y(0))]^{1-\rho}$.

b) For $\rho \ge 1$ the positive definite solution of (9) implies only an asymptotic convergence

To solve the partial differential equation (9) we need to define precisely the piecewise constant function $\gamma_j$, which depends on sign[$\frac{\partial V}{\partial y_j}$]. Fortunately, since $\gamma_j$ takes only two values $C_j$ and $-C_j$, we may consider (9) supposing that $\gamma_j$ is a one of these two constants. Then, obtaining $V(x, y)$ as a solution of (9), one can try define the parameters $\gamma_j$ in an appropriate way.

The idea of the Lyapunov function design as the solution of the partial differential equation was presented by V.I Zubov [19]. The well-known variable gradient method (see, for example, [7]) is also based on this idea. The presented generalization of the Zubov's method permits the right-hand side of (9) to be depended on the function $V$ too. Only this assumption allows us to find the Lyapunov function with finite-time convergence.

## 2.3 Method of Characteristics

The solution of (9) can be found using the so-called Method of Characteristics [4]. According to this method we can formulate the following lemma.

**Lemma 1** If an absolute continuous positive definite function $V(x, y)$ satisfies the following system of ODE (in the symbolic form)

$$ \frac{dx_1}{g_1(x, y)} = \dots = \frac{dx_k}{g_k(x, y)} = \frac{dy_1}{h_1(x, y, \gamma_1)} = \dots \\ = \frac{dy_n}{h_n(x, y, \gamma_n)} = \frac{dV}{-qV^\rho} \quad (10) $$

for $\|x\|^2 + \|y\|^2 > 0$, then the same function $V(x, y)$ is a solution of (9).

**PROOF.** For $\|x\|^2 + \|y\|^2 > 0$ we have $V(x, y) > 0$, $dx_i = -g_i(x, y) \frac{dV}{qV^\rho}$, $dy_j = -h_j(x, y, \gamma_i) \frac{dV}{qV^\rho}$ and

$$ dV = \sum_{i=1}^{k} \frac{\partial V}{\partial x_i} dx + \sum_{j=1}^{n} \frac{\partial V}{\partial y_j} dy \\ = - \left[ \sum_{i=1}^{k} \frac{\partial V}{\partial x_i} g_i(x, y) + \sum_{j=1}^{n} \frac{\partial V}{\partial y_j} h_j(x, y, \gamma_i) \right] \frac{dV}{qV^\rho} $$

implies (9).

Suppose that the system of ODE (10) can be solved and its first integrals are

$$ \varphi_i(V, x, y, \gamma, q, \rho) = \text{const} := c_i, \quad i = \overline{1, n+k} \quad (11) $$

Since any function of constants is a constant too (in the partial case, zero), the function $V(x, y, \gamma, q, \rho)$ can be found as a solution of nonlinear algebraic equation

$$ \Phi(\varphi_1(V, x, y, \gamma, q, \rho), \dots, \varphi_{n+k}(V, x, y, \gamma, q, \rho)) = 0 \quad (12) $$

where $\Phi(\varphi_1, \dots, \varphi_{n+k})$ is an arbitrary function. An explicit analytical solution of the last equation (if it exists) defines the Lyapunov function candidate. However, the function $\Phi$ and the parameters $\gamma, q, \rho$ should be chosen in such a way that the function $V(x, y)$ should be an absolute continuous and positive definite. Only such functions $V(x, y)$ may be considered as a Lyapunov function candidate.

**Remark 2** Obviously, this approach does not provide a formal algorithm for a Lyapunov function design, but only helps to derive a specific form of a Lyapunov function candidate that reduces the problem of a Lyapunov function design to the problem of the correct parameters definition which guarantee the absolute continuity and positive definiteness of the obtained "energetic" function.

Below we will illustrate the proposed method designing the Lyapunov function with finite time convergence ($0 < \rho < 1$) for the second order sliding system with "twisting" controller.

# 3 Lyapunov Function Design for Twisting Control Algorithm

## 3.1 System description and Problem formulation

Consider the controlled system given by

$$ \begin{cases} \dot{x} = y \\ \dot{y} = f(t, x, y) + u(t) \end{cases} \quad (13) $$

where $x, y \in \mathbb{R}$ are the scalar state variables, $f(t, x, y)$ is a measurable but unknown function treated hereafter as an external noise or perturbation, $u \in \mathbb{R}$ is the so-called "twisting" control (see [8])

$$ u(t) = -r_1 \text{sign}[x(t)] - r_2 \text{sign}[y(t)] \quad (14) $$

where $r_1, r_2 > 0$ are control parameters, and the operator $\text{sign}[\cdot]$ is defined by (2). The solution of (13) is understood in the Filippov's sense (see [5]).

It is supposed that

$$ |f(t, x, y)| \le C, \forall x, y \in \mathbb{R} \text{ and } \forall t \ge 0 \quad (15) $$
with a known constant C.

Now the main problem is to design the Lyapunov function which provides the finite-time convergence of the solutions of (13) to the origin (0, 0) under the control (14) and the assumptions above.

## 3.2 Partial Differential Equation

For an arbitrary absolute continuous function $V(x, y)$ we have

$$ 
\begin{aligned}
\frac{dV}{dt} &= \frac{\partial V}{\partial x}\dot{x} + \frac{\partial V}{\partial y}\dot{y} = y\frac{\partial V}{\partial x} + (u + f(t, x, y))\frac{\partial V}{\partial y} \le \\
& y\frac{\partial V}{\partial x} + (u + C \operatorname{sign}[\frac{\partial V}{\partial y}])\frac{\partial V}{\partial y} = y\frac{\partial V}{\partial x} - \operatorname{sign}[x]\gamma\frac{\partial V}{\partial y}
\end{aligned}
$$
$$
\gamma = r_1 + r_2 \operatorname{sign}[xy] - \gamma_0, \gamma_0 = C \operatorname{sign}[x\frac{\partial V}{\partial y}] \quad (16)
$$

According to the proposed method the equation (9) becomes to be as

$$ y \frac{\partial V}{\partial x} - \operatorname{sign}[x] \gamma \frac{\partial V}{\partial y} = -k V^{1/2} \quad (17) $$

where $k > 0$ is a positive parameter. Supposing that $\gamma_0$ is a constant let us find the solution of (17). The corresponding system (10) of the characteristics ODE is

$$ \frac{dx}{y} = \frac{dy}{-\operatorname{sign}[x]\gamma} = \frac{dV}{-k\sqrt{V}} \quad (18) $$

Its first integrals are as follows

$$ \varphi_1(x, y) = |x| + \frac{y^2}{2\gamma}, \quad \varphi_2(V, y) = \frac{y \operatorname{sign}[x]}{\gamma} - \frac{2\sqrt{V}}{k} \quad (19) $$

Select $\Phi$ as

$$ \Phi(\varphi_1, \varphi_2) = k_0 \sqrt{\varphi_1} + \varphi_2 = 0 \quad (20) $$

where $k_0$ is a real parameter. Substitution (19) into (20) leads to the equation

$$ \frac{2\sqrt{V}}{k} = \frac{y \operatorname{sign}[x]}{\gamma} + k_0 \sqrt{|x| + \frac{y^2}{2\gamma}} \quad (21) $$

Obviously, the right-hand side of (21) makes sense for all $|x| > 0$ if and only if $\gamma > 0$, or equivalently, when $r_1 > r_2 + C$. Since the left-hand side of (21) is always non-negative we have

$$ k_0 > - \frac{\operatorname{sign}[xy] \sqrt{2/\gamma}}{\sqrt{1+2\gamma|x|/y^2}} \quad (22) $$

So, the function $V(x, y)$ can be found from (21) as

$$ V(x, y) = \frac{k^2}{4} \left( \frac{y \operatorname{sign}[x]}{\gamma} + k_0 \sqrt{|x| + \frac{y^2}{2\gamma}} \right)^2 \quad (23) $$

## 3.3 Removing of Discontinuities

The next step consists in the removing of the discontinuities of the function (23) by special selection of the parameters $k$ and $k_0$.

Considering the partial limits of the function $V(x, y)$ when $x$ tends to zero for any fixed $y$ and when $y$ tends to zero for any fixed $x$, one derive

$$ 
\begin{aligned}
& \text{if } x \to 0 \text{ then } V(x, y) \to \frac{k^2}{4} \left( \frac{\operatorname{sign}[xy]}{\gamma} + \frac{k_0}{\sqrt{2\gamma}} \right)^2 y^2 \\
& \text{if } y \to 0 \text{ then } V(x, y) \to \frac{k^2 k_0^2}{4} |x|
\end{aligned}
$$

Hence, to eliminate the discontinuities on the lines $x = 0$ and $y = 0$ it is sufficient to resolve the following system

$$ 
\left\{ 
\begin{array}{l}
k^2 \left( \frac{\operatorname{sign}[xy]}{\gamma} + \frac{k_0}{\sqrt{2\gamma}} \right)^2 = \bar{k}^2 \\
k^2 k_0^2 = 1
\end{array}
\right. 
$$

for some positive value $\bar{k}$. This gives

$$ k = \sqrt{\frac{\gamma}{2} \left| \sqrt{2\gamma}\bar{k} - 1 \right|} > 0, \quad k_0 = \sqrt{\frac{2}{\gamma} \frac{\operatorname{sign}[xy]}{\sqrt{2\gamma}k-1}} \quad (24) $$

Therefore, the function $V(x, y)$ (23) in view of the continuity property, discussed above, should satisfy

$$ V(0, y) = \frac{\bar{k}^2}{4} y^2 \quad \text{and} \quad V(x, 0) = \frac{|x|}{4} \quad (25) $$

Combining (24) with (22) we can formulate the following simple result.

**Lemma 2** If $r_1 > r_2 + C$ and $r_2 > C$ and the parameter $\bar{k}$ in (24) is selected in a such way that the following inequalities hold

$$ \frac{1}{\sqrt{2(r_1+r_2-C)}} < \bar{k} < \frac{1}{\sqrt{2(r_1-r_2+C)}} \quad (26) $$

then $k_0 > 0$ and the condition (22) holds for all $xy \neq 0$.

**PROOF.** 1) Consider the case $xy > 0$. Then we need to check the validity of the following inequality $(\sqrt{2\gamma_+} \bar{k} - 1)^{-1} > -(1 + 2\gamma_+ |x|/y^2)^{-1/2}$ with $\gamma_+ = r_1 + r_1 + C \operatorname{sign}[x \frac{\partial V}{\partial y}]$ which directly results from the inequalities $\bar{k} > (2(r_1 + r_2 - C))^{-0.5} \ge (2\gamma_+)^{-0.5}$.

2) When $xy < 0$ we have to prove $(1 - \sqrt{2\gamma_-} \bar{k})^{-1} > (1 + 2\gamma_- |x|/y^2)^{-0.5}$ with $\gamma_- = r_1 - r_1 + C \operatorname{sign}[x \frac{\partial V}{\partial y}]$ that results from $0 < \bar{k} < (2(r_1 - r_2 + C))^{-0.5} \le (2\gamma_-)^{-0.5}$.
3) Since $\sqrt{2\gamma_+}k - 1 > 0$ and $\sqrt{2\gamma_-}k - 1 < 0$ we have $\text{sign}[xy] = \text{sign}[\sqrt{2\gamma}k - 1]$ and $k_0 > 0$ for all $xy \neq 0$.

So, the Lyapunov function (23) becomes

$$V(x, y) = \begin{cases} \frac{k^2}{4} \left( \frac{y \operatorname{sign}[x]}{\gamma} + k_0 \sqrt{|x| + \frac{y^2}{2\gamma}} \right)^2 & \text{if } xy \neq 0 \\ \bar{k}^2 y^2/4 & \text{if } x=0 \\ |x|/4 & \text{if } y=0 \end{cases} \quad (27)$$

with $k, k_0$ are as in (24) and $\bar{k}$ satisfying (26).

### 3.4 Main Theorem

The rigorous consideration of the partial derivative $\frac{\partial V}{\partial y}$ shows that $\text{sign}[\frac{\partial V}{\partial y}]$ switches on the line $L\sqrt{|x|} \text{sign}[x] + y = 0$ where $L$ is a constant depending on the parameters of the system (13)-(15). However, defining $\gamma$ as function of $\text{sign}[L\sqrt{|x|} \text{sign}[x] + y]$ one has to definitely remove new discontinuities of the function $V(x, y)$. So, the more preferable definition of $\gamma$ is

$$\gamma := r_1 + (r_2 - C) \text{sign}[xy] \quad (28)$$

that leads to the following claim.

**Theorem 3** If $r_1 > r_2 + C$, $r_2 > C$ then the function $V(x, y)$ (27) with $\gamma$ given by (28) has the properties:

1) $V(x, y)$ is absolute continuous in all space $\mathbb{R}^2$ and continuously differentiable when $xy \neq 0$;

2) the time derivative of $V(x, y)$ on the trajectories of (13)-(15) satisfies

$$\frac{dV}{dt} \leq -k_{\min} \frac{r_1 - r_2 - C}{r_1 - r_2 + C} \sqrt{V} \quad (29)$$

almost always with $k_{\min} := \min(k)$;

3) the corresponding guaranteed reaching time is

$$t_{\text{reach}} \leq \frac{2(r_1 - r_2 + C)}{k_{\min}(r_1 - r_2 - C)} \sqrt{V(x(0), y(0))} \quad (30)$$

**PROOF.** 1) Obviously that the function $V(x, y)$ may have discontinuities only on the lines $x = 0$ and $y = 0$. Since the parameters $k$ and $k_0$ (24) are selected in such a way that discontinuities have to be removed (see (25)), the function $V(x, y)$ is continuous by the accepted construction.

2) The partial derivatives of the function $V(x, y)$ are

$$\frac{\partial V}{\partial x} = \frac{k^2}{2} \left( \frac{1}{\sqrt{1+2\gamma|x|/y^2}} \frac{k_0}{\sqrt{2\gamma}} + \frac{k_0^2 \text{sign}[x]}{2} \right)$$

$$\frac{\partial V}{\partial y} = \frac{k^2}{2} \left( \frac{y}{\gamma^2} + \frac{k_0 y^2}{2\gamma} + \frac{k_0 \text{sign}[x](y^2 + \gamma|x|)}{\gamma^2 \sqrt{|x| + y^2/(2\gamma)}} \right)$$

Since $\gamma, k$ and $k_0$ switch only on the lines $x = 0$ and $y = 0$, it follows that $\frac{\partial V}{\partial x}$ and $\frac{\partial V}{\partial y}$ are continuous for all $x \neq 0$ and $y \neq 0$. Moreover, the left and right limits on these lines are finite everywhere in any bounded region.

3) Consider then the time derivative of the function $V(x, y)$ on the trajectories of the system (13):

$$\frac{dV}{dt} = k\sqrt{V} \left( -\frac{\tilde{\gamma}}{\gamma} + \sqrt{\frac{\gamma}{2}} \frac{k_0 \text{sign}[xy]}{\sqrt{1+2\gamma|x|/y^2}} \left(1 - \frac{\tilde{\gamma}}{\gamma}\right) \right)$$

where $\tilde{\gamma} = r_1 + r_2 \text{sign}[xy] - f(t, x, y) \text{sign}[x]$, $\gamma = r_1 + r_2 \text{sign}[xy] - C \text{sign}[xy]$ (see (28)). Since, $1 \leq \frac{\tilde{\gamma}}{\gamma}$ for $xy > 0$ and $1 \geq \frac{\tilde{\gamma}}{\gamma} \geq \frac{r_1 - r_2 - C}{r_1 - r_2 + C}$. for $xy < 0$ the inequality

$$-\frac{\tilde{\gamma}}{\gamma} + \sqrt{\frac{\gamma}{2}} \frac{k_0 \text{sign}[xy]}{\sqrt{1+2\gamma|x|/y^2}} \left(1 - \frac{\tilde{\gamma}}{\gamma}\right) \leq -\frac{r_1 - r_2 - C}{r_1 - r_2 + C}. \quad (31)$$

holds. So, we have $\frac{dV}{dt} \leq -k_{\min} \frac{r_1 - r_2 - C}{r_1 - r_2 + C} \sqrt{V}$ for $x, y \neq 0$.

4) To complete the proof it is enough to notice that the system (13)-(15) doesn't have nontrivial solutions with $x(t) \equiv 0$ or $y(t) \equiv 0$, or in other words, the trajectories cross these switching lines avoiding the movements along them (see [12]). Thus, the inequality (29) holds almost always along the trajectory of the system (13)-(15).

**Remark 3** Notice that the Main Theorem conditions $r_1 > r_2 + C, r_2 > C$ coincide with ones in [9].

**Remark 4** It is not difficult to show that using the designed Lyapunov function (27) the local stability (with a finite-time convergence) of (13)-(14) takes place if the condition (15) holds only in a local neighborhood of the origin $D_0$ provided that the system starts from within domain $D(c^{\max})$, where $D(c) = \{(x, y)^T \in \mathbb{R}^2 : V(x, y) \leq c\}$ and $c^{\max} = \max_{\forall c:D(c) \subseteq D_0} c$.

## 4 Numerical Example

Consider the system (13)-(15) with $r_1 = 1.3, r_2 = 0.7, C = 0.4$. The Lyapunov function (27) for $\bar{k} = 0.9/\sqrt{2}$ is

$$V(x, y) = \begin{cases} \frac{(0.9\sqrt{\bar{\gamma}}-1)^2}{8\bar{\gamma}} \left( |y| + \frac{\sqrt{2\bar{\gamma}|x|+y^2}}{0.9\sqrt{\bar{\gamma}}-1} \right)^2 & \text{if } xy \neq 0 \\ 0.10125y^2 & \text{if } x = 0 \\ 0.25|x| & \text{if } y = 0 \end{cases}$$
<figure><img src="image_9.png" /><figcaption>Fig. 1. The Lyapunov function</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 2. The level lines of the Lyapunov function</figcaption></figure>

where $\gamma = 1.3 + 0.3 \operatorname{sign}[xy]$ and it is presented in Fig. 1. Fig. 2 shows the level lines of the Lyapunov function.

## 5 Conclusion

In this paper we present a method of a Lyapunov functions design providing a finite-time convergence of system dynamics to the origin for a class of ODE with discontinuous right-hand side. It is based on Characteristic Method application and related with resolution of the first-order partial differential equation of a special type. Even the suggested approach does not provide a formal algorithm for a Lyapunov function design, it helps to derive a specific form of a Lyapunov function candidate that reduces the problem of a Lyapunov function design to the problem of the correct parameters definition guaranteeing the absolute continuity and positive definiteness of the obtained "energetic" function. The suggested method is successfully illustrated by its application to the "twisting" controller stability analysis.

## References

[1] Bacciotti, A., & Ceragioli F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions, *ESAIM: Control, Optimisation and Calculus of Variations*, 4, 361-376.

[2] Bacciotti, A., & Rosier, L. (2001). Liapunov functions and stability in control theory, Lecture notes in control and information sciences (Vol. 267). New York: Springer.

[3] Edwards, C., & Spergeon, S. (1998). *Sliding mode control: theory and applications*. Taylor&Francis.

[4] El'sgol'ts, L.E. (1961). *Differential Equations*. International Monographs on Advanced Mathematics and Physics. Hindustan Publisher Company, India, Dheli.

[5] Filippov, A.F. (1988). *Differential Equations with Discontinuous Right-hand Sides*. Kluwer Academic Publishers, Dordrecht-Boston-London.

[6] Hong Y., Huang J., Xu Y. (2001). On an output feedback finite-time stabilization problem. *IEEE Trans. Automatic Control*, 46(2), 305-309.

[7] Khalil H.K. (2002). *Nonlinear systems*, third edition, Prentice Hall, Upper Saddle River, New Jersey.

[8] Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. *Int. J. Control*, 58 (6), 1247-1263.

[9] Levant, A., & Fridman, L. (2004). Robustness issues of 2-sliding mode control. *Variable Structure Systems: from Principles to Implementations* (edited by A. Sabanovic, L. Fridman, S. Spurgeon), The IEE, London, 131-156.

[10] Levant, A. (2007). Principles of 2-sliding mode design, *Automatica*, 43 (4), 576-586.

[11] Orlov, Y. (2005). Finite Time Stability and Robust Control Synthesis of Uncertain Switched Systems. *SIAM J. Control Optim*, 43(4), pp. 1253-1271

[12] Orlov, Y. (2006). Extended Invariance Principle and Other Analysis Tools for Variable Structure Systems. *Advances in Variable Structure and Sliding Mode Control* (edited by C. Edwards, E.F. Colet, L. Fridman), Springer, Berlin-Heidelberg-New York, 3-22

[13] Pan, Y., & Furuta, K. (2006). Variable Structure Control Systems using Sectors for Switching Rule Design. *Advances in Variable Structure and Sliding Mode Control* (edited by C. Edwards, E.F. Colet, L. Fridman), Springer, Berlin-Heidelberg-New York, 89-106.

[14] Ryan, E. P. (1998). An integral invariance principle for differential inclusions with applications in adaptive control, *SIAM J. Control Optim.*, 36(3), pp. 960-980.

[15] Sontag, E.D., & Sussman, H.J. (1995) Nonsmooth control-Lyapunov functions. *Proc. 34th IEEE Conf. Decision Control*, USA, New Orleans, 2799-2805.

[16] Shtessel, Y. B., Shkolnikov, I. A. & Levant, A. (2007). Smooth second-order sliding modes: Missile guidance application. *Automatica*, 43(8), pp. 1470-1476.

[17] Utkin, V.I. (1992). *Sliding Modes in Control Optimization*. Springer Verlag, Berlin.

[18] Utkin, V.I., Guldner, J., & Shi, J. (1999). *Sliding Modes in Electromechanical Systems*. Taylor&Francis, London.

[19] Zubov, V.I. (1964). *Methods of A.M. Lyapunov and their applications*. Groningen: P. Noordho: Limited.
# Nonlinear feedback design for fixed-time stabilization of Linear Control Systems

Andrey Polyakov

► To cite this version:

Andrey Polyakov. Nonlinear feedback design for fixed-time stabilization of Linear Control Systems. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2012, 57 (8), pp.2106-2110. <hal-00757561>

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# Nonlinear Feedback Design for Fixed-Time
Stabilization of Linear Control Systems

Andrey Polyakov

**Abstract**

Nonlinear control algorithms of two types are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into selected neighborhood of the origin independently on initial conditions. The control design procedure uses block control principles and finite-time attractivity properties of polynomial feedbacks. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

**Index Terms**

finite-time stability, polynomial feedback, second order sliding mode control

## I. INTRODUCTION

Finite-time stability and stabilization problems have often been a subject of research [13], [4], [19], [21]. The control theory provides many systems that exhibit finite-time convergence to the equilibrium. Frequently such systems appear in observation problems when finite-time convergence of the observed states to the real ones is required [3]. The high order sliding mode control algorithms also provide finite-time convergence to the origin [16], [20], [17], [22]. Typically such controllers have mechanical and electromechanical applications [2], [11], [5].

A. Polyakov is with Non-A INRIA - LNE, Parc Scientifique de la Haute Borne 40, avenue Halley Bat.A, Park Plaza 59650 Villeneuve d'Ascq, France, andrey.polyakov@inria.fr
The paper deals with an extension *global finite-time stability* concept that is related to possible
predefining of guaranteed convergence(settling) time *independently on initial conditions*. The
corresponding property is called in this paper by *fixed-time stability*. Such phenomenon was
discovered in [18],[10],[23],[6], where observers with predetermined finite convergence time
have been developed. The present paper mostly addresses the control design problem for linear
plants providing *fixed-time convergence to the given set*. The developed design procedure requires
only controllability of the system, i.e. rank[$B, AB, ..., A^{n-1}B$]=n. Some controllers are presented
for *fixed-time stabilization of the origin*. These results are restricted by the case rank[$B, AB$]=n.

The controls considered in this paper basically have polynomial forms. Polynomial state
feedback control systems have considerable attention in nonlinear control [8]. This class of
control systems appears in models of a wide range of applications such as chemical processes,
electronic circuits and mechatronics, biological systems etc. This paper studies a special property
of polynomial feedbacks, which is expressed in fixed-time attraction of solutions of the closed-
loop system into any selected neighborhood of the origin.

Usually finite-time stability is closely related with homogeneity property of the system. While
being asymptotically stable and homogeneous of negative degree, the system is shown to ap-
proach the equilibrium point in finite time [16], [20]. The concept of *homogeneity in bi-limit*
introduced in [1] generalizes this property providing that an asymptotically stable system is *fixed-
time stable* if it is homogeneous of negative degree in 0-limit and homogeneous of positive degree
in $\infty$-limit. Unfortunately, homogeneous approach does not allow to adjust or even estimate the
settling time. To overcome this problem the paper introduces a special modification of the so-
called "nested" (terminal) second order sliding mode control algorithm [17] that provides fixed-
time stability of the origin and allows to adjust the global settling time of the closed-loop system.

All control algorithms presented in the paper are robust with respect to system disturbances
and plant parameters variations in the case when the, so-called, *matching condition* [24] holds,
that is, to guarantee successful elimination of system uncertainties or external disturbances they
should act in the same subspace as an admissible control.

The paper is organized as follows. The next section discuses finite-time and fixed-time stability concepts. It also extends Lyapunov Function Method to fixed-time stability case. Section 3 states the main problem. Sections 4 and 5 consider control design algorithms for single-input and multi-input systems. Finally, numerical simulations examples and conclusions are presented.
## II. FINITE-TIME STABILITY AND SOME FURTHER EXTENSIONS

Consider the following system

$$ \dot{x} = g(t, x), \quad x(0) = x_0 \qquad (1) $$

where $x \in \mathbb{R}^n$ is the vector of system states, $g: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ is a nonlinear function. The case of discontinuous right-hand sides of the system (1) is not excluded, so solutions of (1) are understood in the sense of Filippov [12]. Assume that the system (1) has zero equilibrium point.

**Definition 1 ([4], [20]).** The equilibrium point $x = 0$ of the system (1) is said to be globally finite-time stable if it is globally asymptotically stable and any solution $x(t, x_0)$ of (1) reaches the equilibria at some finite time moment, i.e. $x(t, x_0) = 0, \forall t \ge T(x_0)$, where $T: \mathbb{R}^n \to \mathbb{R}_+ \cup \{0\}$ is the so-called settling-time function.

The finite-time stability property may demonstrate homogeneous systems with negative degree [4], [20] or the so-called sliding homogeneous systems [16]. For example, any solution of the system $\dot{x} = -x^{-1/3}, x \in \mathbb{R}$ converges to zero equilibrium point in finite time $T(x_0) := \frac{2}{3}\sqrt[3]{|x_0|^2}$.

**Definition 2.** The equilibrium point $x = 0$ of the system (1) is said to be globally fixed-time stable if it is globally finite-time stable and the settling-time function $T(x_0)$ is bounded by some positive number $T_{\max} > 0$, i.e. $T(x_0) \le T_{\max}$ for $\forall x_0 \in \mathbb{R}^n$.

For example, the system $\dot{x} = -x^{1/3} - x^3, x \in \mathbb{R}$ has the fixed-time stable zero equilibrium. Any solution $x(t, x_0)$ of this system converges to the origin in a finite-time and for any $x_0 \in \mathbb{R}$ the equality $x(t, x_0) = 0$ holds for all $t \ge 2.5$.

In the paper [6] the closed-loop control systems that demonstrate dynamics described in Definition 2 was called by "uniformly finite-time convergent". The term "stability" is used here just because this notion is more traditional and natural for describing the referred properties of the system's equilibrium points, but the term "uniformly finite-time stable" can not be used for this purpose, since it has different natural meaning discussed in [20].

The fixed-time stability can be simply extended to the case of attractive set.

**Definition 3.** The set $M$ is said to be globally finite-time attractive for the system (1) if any solution $x(t, x_0)$ of (1) reaches $M$ in some finite time moment $t = T(x_0)$ and remains there, i.e. $x(t, x_0) \in M, \forall t \ge T(x_0)$, where $T: \mathbb{R}^n \to \mathbb{R}_+ \cup \{0\}$ is the settling-time function.
**Definition 4.** The set M is said to be globally fixed-time attractive for the system (1) if it is globally finite-time attractive and the settling-time function T(x₀) is bounded by some positive number T_max > 0, i.e. T(x₀) ≤ T_max for x₀ ∈ ℝⁿ.

The property of global finite-time attractivity may demonstrate even some linear systems, but the only nonlinear ones may have fixed-time attractive sets.

Denote by D*φ(t) the upper right-hand derivative of a continuous function φ(t), i.e. D*φ(t) := sup lim<sub>h→+0</sub> φ(t+h) - φ(t) / h. The following simple result extends the Lyapunov function method providing the background for the fixed-time stability analysis.

**Lemma 1.** If there exists a continuous radially unbounded function V : ℝⁿ → ℝ⁺ ∪ {0} such that 1) V(x)=0 ⇒ x∈M; 2) any solution x(t) of (1) satisfies the inequality D*V(x(t)) ≤ -(αV<sup>p</sup>(x(t))+βV<sup>q</sup>(x(t)))<sup>k</sup> for some α,β,p,q,k > 0 : pk < 1, qk > 1 then the set M ⊂ ℝⁿ is globally fixed-time attractive for the system (1) and T(x₀) ≤ 1/α<sup>k</sup>(1-pk) + 1/β<sup>k</sup>(qk-1), ∀x₀ ∈ ℝⁿ.

*Proof:* Due to 2) we have D*V(x(t)) ≤ -α^k V^pk(x(t)) if V(x(t)) ≤ 1 and D*V(x(t)) ≤ -β^k V^qk(x(t)) for V(x(t)) > 1. Hence, for any x(t) such that V(x(0)) > 1 the last inequality guarantees V(x(t)) ≤ 1 for t ≥ 1/(β^k(qk-1)) and for any x(t) such that V(x(t₀)) ≤ 1 we derive V(x(t)) = 0 for t ≥ t₀ + 1/(α^k(1-pk)). Therefore, V(x(t)) = 0 for all t ≥ 1/(α^k(1-pk)) + 1/(β^k(qk-1)) and any solution x(t) of (1). Applying the condition 1) we finish the proof of this lemma. ■

Replacement of the condition 1) in Lemma 1 by V(x)=0 ⇔ x=0 helps to check invariance of the set M. So, for M = {0} Lemma 1 allows to analyze the fixed-time stability of the origin.

### III. PROBLEM STATEMENT AND BASIC ASSUMPTIONS

Consider the control system of the form

$$ \dot{x} = Ax + Bu + f(t, x) \quad (2) $$

where x ∈ ℝⁿ is the vector of system states, A ∈ ℝⁿ×ⁿ is the system matrix, B ∈ ℝⁿ×^m is the matrix of control gains, u ∈ ℝ^m is the vector of control inputs, and the function f : ℝ_+ × ℝⁿ → ℝⁿ describes systems uncertainties.

We study the system (2) under the following classical assumptions: **1.** The pair (A, B) is controllable, i.e. rank[B, AB, A²B, ..., A^(n-1)B] = n. **2.** The uncertain function f(t, x) satisfies the so-called matching condition (see, for example, [24]), i.e. f(t, x) = Bγ(t, x), where γ(t, x))
is an unknown but bounded function. The function $\gamma(t, x)$ is assumed to be bounded by some
known function $\gamma_0(t, x) \ge 0$, i.e. $\|\gamma(t, x)\|_{\infty} \le \gamma_0(t, x)$ for $\forall t > 0$ and $\forall x \in \mathbb{R}^n$.

Denote a closed ball of radius $r > 0$ with the center in the origin by $B_r$, i.e. $B_r := \{x \in \mathbb{R}^n : \|x\|_{\infty} \le r\}$, $\|x\|_{\infty} := \max_{i=1,2,...,n} |x_i|$.

This paper address two following problems:

* to design a feedback control $u = u(t, x)$ for the system (2), which provides the fixed-time attractivity property of the given ball $B_r$ for a predefined global settling-time estimate $T_{\max}$;

* to design a feedback control $u = u(t, x)$, which guarantees fixed-time stability of the origin of the closed-loop system (2) for a predefined global settling-time estimate $T_{\max}$.

IV. FIXED-TIME CONTROLLERS FOR SINGLE INPUT SYSTEMS

A. Fixed-time Attractivity

Consider the case $m=1$. Then the linear transformation $y=Gx$, $G=[A^{n-1}b, A^{n-2}b, ..., Ab, b]^{-1}$ brings (may be after reordering) the initial system (2) to the Brunovsky form

$$
\dot{y}_1 = y_2, \quad \dot{y}_2 = y_3, \quad \dots, \quad \dot{y}_n = \sum_{i=1}^{n} a_i y_i + u(y) + \gamma(t, y) \quad (3)
$$

where $a_i \in \mathbb{R}$ and the uncertain term $\gamma$ is described in the previous section, $\|\gamma(t, y)\|_{\infty} < \gamma_0(t, y)$.

Introduce the nonlinear coordinate transformation $s = \Phi(y)$ by the following formulas:

$$
s_i = y_i + \varphi_i, \quad i=1,2,...,n, \quad \varphi_1=0, \quad \varphi_{i+1} = \alpha_i (y_i + \varphi_i) + \beta_i (y_i + \varphi_i)^3 + \sum_{k=1}^{i} \frac{\partial \varphi_i}{\partial y_k} y_{k+1},
$$

where $\alpha_i, \beta_i, i = 1, 2, ..., n-1$ are some positive numbers. The presented coordinate transformation is smooth and nonsingular. The inverse transformation $y = \Phi^{-1}(s)$ is defined as follows:

$$
y_i = s_i + \psi_i, \quad i=1,2,...,n, \quad \psi_1=0, \quad \psi_{i+1} = -\alpha_i s_i - \beta_i s_i^3 + \sum_{k=1}^{i} \frac{\partial \psi_i}{\partial s_k} (-\alpha_k s_k - \beta_k s_k^3 + s_{k+1}).
$$

Formal calculations show that the system (3) is equivalent to

$$
\dot{s}_1 = -\alpha_1 s_1 - \beta_1 s_1^3 + s_2, \quad \dot{s}_2 = -\alpha_2 s_2 - \beta_2 s_2^3 + s_3, \quad \dots, \quad \dot{s}_n = \sum_{i=1}^{n} a_i y_i + \sum_{k=1}^{n-1} \frac{\partial \varphi_n}{\partial y_k} y_{k+1} + u + \gamma(t, y) \quad (4)
$$

and the feedback control of the form

$$
u = -\alpha_n s_n - \beta_n s_n^3 - \sum_{i=1}^{n} a_i y_i - \sum_{k=1}^{n-1} \frac{\partial \varphi_n}{\partial y_k} y_{k+1}, \quad \text{where } s = \Phi(y) \quad (5)
$$

provides fixed-time attractivity property of the closed-loop system (4). The next lemma gives a
background for an appropriate adjustment of the control parameters $\alpha_i$ and $\beta_i$.
**Lemma 2.** Let 1) $\varepsilon_i > 0$, $i = 1, 2, ..., n$ and $T_{\max} > 0$ are some positive numbers; 2) $\alpha_1 = \bar{\alpha}_1 + \frac{\varepsilon_2}{\varepsilon_1}$, ..., $\alpha_{n-1} = \bar{\alpha}_{n-1} + \frac{\varepsilon_n}{\varepsilon_{n-1}}$, $\alpha_n := \alpha_n(t, y) = \bar{\alpha}_n + \frac{\gamma_0(t, y)}{\varepsilon_n}$, where $\bar{\alpha}_i$ are nonnegative numbers and $\|\gamma(t, y)\|_{\infty} < \gamma_0(t, y)$; 3) $\beta_i \ge \frac{\bar{\alpha}_i/\varepsilon_i^2}{\exp(2\bar{\alpha}_i \frac{T_{\max}}{n})-1}$, $i = 1, 2, ..., n$. Then any solution $s(t)$ of the system (4)-(5) satisfies the inequalities $|s_i(t)| \le \varepsilon_i$ for $\forall t \ge T_{\max}$.

*Proof:* Denote $V_i(t) = |s_i(t)|$, $i = 1, 2, ..., n$. For all $t > 0$ the inequality $D^*V_n(t) \le -\alpha_n V_n(t) - \beta_n V_n^3(t) + \gamma_0$ holds. Hence, $\dot{V}_n(t) \le -\bar{\alpha}_n V_n(t) - \beta_n V_n^3(t)$ for $t > 0$: $V_n(t) \ge \varepsilon_n$. Studying the last inequality we derive $\frac{\beta_n}{\bar{\alpha}_n} V_n^2(t) + 1 \le (1 - \frac{\beta_n V_n^2(0)}{\beta_n V_n^2(0)+\bar{\alpha}_n}) e^{-2\bar{\alpha}_n t}^{-1}$. Hence, $V_n(t) \le \varepsilon_n$ for all $t \ge T_{\max}/n$. The similar considerations for $i = n-1$ give $V_i(t) \le \varepsilon_i$ for all $t \ge (n-i+1)T_{\max}/n$. The last step ($i=1$) provides $|s_i(t)| < \varepsilon_i$ for all $t > T_{\max}$. ■

**Corollary 1.** Selection of the parameters $\varepsilon_i = \varepsilon > 0$ and $\bar{\alpha}_i = 1$ for $i = 1, 2, ..., n$ under conditions of Lemma 2 implies the inequality $\|s(t)\|_{\infty} \le \varepsilon$ for all $t > T_{\max}$, where $s(t)$ is an arbitrary solution of the system (4)-(5) with control parameters $\alpha_i = 2, i = 1, 2, ..., n-1$; $\alpha_n(t, x) = 1 + \gamma_0(t, x)/\varepsilon$ and $\beta_i = \beta := \varepsilon^{-2} (\exp(\frac{2T_{\max}}{n}) - 1)^{-1}$, $i = 1, 2, ..., n$.

The presented corollary shows the way for designing of the control, which provides the fixed-time attractivity property of the ball $B_{\varepsilon}$ for the system (4). The radius of the ball $\varepsilon$ and the global settling-time $T_{\max}$ can be selected in arbitrary way. Show now that the designed controller will guarantee the same fixed-time attractivity property for the original system (3).

Denote a set of all polynomials of the order $k$ by $\mathbb{P}^k$.

**Corollary 2.** If conditions of Corollary 1 hold, then there exist polynomials $p_i \in \mathbb{P}^i$, $i = 1, 2, ..., n$ with nonnegative coefficients such that $|y_i(t)| \le p_i(q)\varepsilon$, $\forall t > T_{\max}$, where $q = (\exp(\frac{2T_{\max}}{n}) - 1)^{-1}$ and $y(t) = (y_1(t), ..., y_n(t))^T$ is an arbitrary solution of the system (3), (5).

*Proof:* Due to Corollary 1 we have that $|s_i| \le \varepsilon$ for all $t > T_{\max}$. Let $z_i > 0$, $i = 1, 2, ..., n$ and $\tilde{\psi}_1=0$, $\tilde{\psi}_{i+1}=\tilde{\psi}_{i+1}(z_1, z_2, ..., z_{i+1}) := 2z_i + \beta z_i^3 + \sum_{k=1}^i \frac{\partial \tilde{\psi}_i}{\partial z_k}(2z_k + \beta z_k^3 + z_{k+1})$. It is easy to see that $|\psi_i(s_1, \dots, s_i)| \le \tilde{\psi}_i(|s_1|, \dots, |s_i|)$. Show now that $\tilde{\psi}_i(\varepsilon, \dots, \varepsilon) = \varepsilon \sum_{j=0}^{i-1} \mu_{ij}(\beta\varepsilon^2)^j$, where $\mu_{ij}$ are some nonnegative numbers. Obviously, $\tilde{\psi}_1=0$ and $\tilde{\psi}_2=2z_1+\beta z_1^3$ satisfy the required representation. Assume that it fulfils for $\tilde{\psi}_i$ and show the same property for $\tilde{\psi}_{i+1}$. Due to recursive polynomial structure of $\tilde{\psi}_i$ we conclude that for $z_j=\varepsilon, j = 1, 2, ..., i \frac{\partial \tilde{\psi}_i(z_1, \dots, z_i)}{\partial z_k} = \sum_{j=0}^{i-1} \hat{\mu}_{ij}^k(\beta\varepsilon^2)^j$, where $\hat{\mu}_{ij}^k$ are some
nonnegative numbers. Hence, $\tilde{\psi}_{i+1}(\varepsilon, \dots, \varepsilon) = \varepsilon(1+\beta\varepsilon^2)+\varepsilon \sum_{k=1}^{i} \sum_{j=0}^{i-1} \hat{\mu}_{ij}^k (\beta\varepsilon^2)^j (2+\beta\varepsilon^2) = \varepsilon \sum_{j=0}^{i} \mu_{i+1j} (\beta\varepsilon^2)^j$
and the required property is proven. Finally taking into account $\beta\varepsilon^2 = q$ and $|y_i(t)| \le |s_i(t)| + |\psi_i(s_1(t), \dots, s_i(t))| \le p_i(q)\varepsilon$, where $p_i(q) := 1 + \sum_{j=0}^{i-1} \mu_{ij}q^j$, we finish the proof. ■

Remark that the proof of Corollary 2 gives an algorithm of construction of the polynomials $p_i(q) \in \mathbb{P}^i$ introduced in its statement. These polynomials are needed for adjustment of control parameters to guarantee convergence of all solutions of the original system (2) into the required neighborhood of the origin. The explicit form of the corresponding polynomial can be obtained using the recursive formulas for $\tilde{\psi}_i$ calculated in symbolic computation packages such as Mathematica or Symbolic Toolbox of MATLAB. In particular the first three polynomials have the form: $p_1(q) = 1, p_2(q) = 3+q, p_3(q) = 9+12q+3q^2$.

Taking into account $p_i(q) \le p_{i+1}(q)$ we may formulate the theorem.

**Theorem 1.** If $m=1$ and the control $u$ has the form (5) with parameters $\alpha_i=2, i=1,2,\dots,n-1$, $\alpha_n(t,x)=1+\gamma_0(t,x)/\varepsilon$, $\beta_i=q/\varepsilon^2$, $i=1,2,\dots,n$, $\varepsilon=r/(\|G^{-1}\|_{\infty}p_n(q))$, $q = (\exp(\frac{2T_{\max}}{n}) - 1)^{-1}$, where $r>0$ and $T_{\max} > 0$ are arbitrary numbers and the polynomial $p_n \in \mathbb{P}^n$ is introduced in Corollary 2, then the ball $B_r$ is the globally fixed-time attractive set of the closed-loop system (2) with the settling-time function bounded by $T_{\max}$.

Theorem 1 mostly declares the theoretical possibility for designing of a nonlinear control providing fixed-time attractivity property of the closed-loop linear control system. The practical applications may require more precise adjustment of control parameters basing on Lemma 2.

## B. Fixed-time stability

The scheme presented above allows to design the control $u_a(t,x) := u(t,x)$ for the single input case of the system (2), which guarantees fixed-time attractivity property of an arbitrary ball $B_r$ with the center in the origin. The simplest combination of this controller with some control law $u_f(x)$ providing local finite-time stability of the origin for the system (2) gives us a hybrid control algorithm $u_{fx}(t,x) = \begin{cases} u_a(t,x) & \text{for } x \notin B_r \\ u_f(t,x) & \text{for } x \in B_r \end{cases}$, which can afford fixed-time stability of the origin for the system (2). Some design procedures of local finite-time controllers can be found, for example, in [16] or [14]. However, the existed controllers providing local finite-time stability do not have precise algorithms for adjustment of control parameters to predefine the
local settling time estimate (excepting the so-called "twisting" second order ($n = 2$) sliding mode
system, see, for example, [22]). To realize the presented hybrid scheme we need to prevent the
appearance of switching (chattering) regimes [9] around the boundary of the ball $B_r$, which
may slow down or even prevent the convergence process. Therefore the "solid" (non-hybrid)
controller providing fixed-time stability of the origin for the system (2) is more preferable. The
designing such controller is a difficult problem, which is solved in this paper only for linear
plants satisfying the additional assumption $\text{rank}[B, AB] = n$. This condition definitely restricts
the class of controllable systems, but it still covers a lot of real-life control systems. For example,
mechanical systems with maximum numbers of control inputs [5] obviously satisfy this condition.
For $m=1$ the presented assumption implies $n=2$.

Introduce the following denotation $z^{[q]} = |z|^q \text{sign}[z]$, $z, q \in \mathbb{R}$, which presents the involution operation without loss of number's sign. Remark, $z^{[1/k]} = z^{1/k}$ and $z^{[k]} = z^k$ for odd integers $k$.

**Theorem 2.** Let $T_{\max} > 0$ be an arbitrary positive number, $m=1, n=2$ and the control $u(t, x)=u_{fx}(t, Gx)$, $x \in \mathbb{R}^2$ has the form $u_{fx}(t, y)=-a_1y_1-a_2y_2-\frac{\alpha_1+3\beta_1y_1^2+2\gamma_0(t,y)}{2}\text{sign}[s]-(\alpha_2s+\beta_2s^{[3]})[\frac{1}{2}]$, where $s=y_2+(y_2^{[2]}+\alpha_1y_1+\beta_1y_1^{[3]})^{[0.5]}$, $0.5\alpha_1=\alpha_2=0.5\beta_1=\beta_2=64T_{\max}^{-2}$. Then the origin of the closed-loop system (2) is globally fixed-time stable with the settling-time function bounded by $T_{\max}$.

*Proof:* I. Show that $s = 0$ is a sliding surface of the closed-loop system (2). Consider the Lyapunov function candidate $V(y_1, y_2) := |s(y_1, y_2)|$ and calculate its total upper right-hand Dini derivative along the trajectories of the closed-loop system (3) $D^*V(y_1(t), y_2(t)) = \dot{y}_2(t) \text{sign}[s(t)] + (|y_2(t)|\dot{y}_2(t) \text{sign}[s(t)] + \frac{\alpha_1+3\beta_1y_1^2(t)}{2}y_2(t) \text{sign}[s(t)]) / |y_2^{[2]}(t)+\alpha_1y_1(t)+\beta_1y_1^{[3]}(t)|^{\frac{1}{2}}$ for $s \neq 0$. Taking into account the equality $(\alpha_2s + \beta_2s^{[3]})^{[\frac{1}{2}]} \text{sign}[s] = (\alpha_2|s| + \beta_2|s|^{[3]})^{\frac{1}{2}}$ we obtain $\dot{y}_2 \text{sign}[s] = (a_1y_1 + a_2y_2 + u_{fx}(y) + \gamma(t, x)) \text{sign}[s] = -\frac{\alpha_1+3\beta_1y_1^2}{2} - (\alpha_2|s| + \beta_2|s|^{[3]})^{1/2} - (\gamma_0 - \gamma)$ for $s \neq 0$. Hence, we conclude that $D^*V \le -(\alpha_2V + \beta_2V^3)^{1/2}$ and following the proof of Lemma 1 we guarantee $V(t) := |s(t)| = 0$ for all $t \ge T_s := \frac{2}{\sqrt{\alpha_2}} + \frac{2}{\sqrt{\beta_2}} = \frac{T_{\max}}{2}$.

II. The equality $s=0$ implies $2y_2^{[2]}+\alpha_1y_1+\beta_1y_1^3=0$ and the sliding motion equation takes the form $\dot{y}_1=-\left(\frac{\alpha_1}{2}y_1+\frac{\beta_1}{2}y_1^3\right)^{[1/2]}$. Using Lemma 1 with the Lyapunov function $V(y_1)=|y_1|$ we show $y_1=0$ for all $t \ge \frac{T_{\max}}{2}+\frac{2\sqrt{2}}{\sqrt{\alpha_1}}+\frac{2\sqrt{2}}{\sqrt{\beta_1}}=T_{\max}$. Finally we remark that $y_1=0$ and $s=0$ imply $y_2=0$. $\blacksquare$

Remark that for $\beta_1 = 0$ the switching line $s = 0$ coincides with the sliding surface of "nested"
controller [17], which provides finite-time stability of the closed-loop system. The polynomial
terms $\beta_1 y_1^{[3]}$ and $\beta_2 s^{[3]}$ are introduced to guarantee the fixed-time stability property.
# V. FIXED-TIME CONTROLLERS FOR MULTI INPUT SYSTEMS

## A. Block Decomposition

Introduce denotations: rown($W$) and coln($W$) are the number of rows and columns of the matrix $W$, respectively; ker($W$) and range($W$) are the null space and the column space of $W$; null($W$) is the matrix with columns defining the orthonormal basis of ker($W$).

To adapt the fixed-time control design scheme developed for the single-input case we need to decompose the original multi input system (2) to a block from [7]. The required coordinate transformation can be constructed using supporting matrices provided by Algorithm 1.

**Algorithm 1. I. Initialization:** $A_0 = A$, $B_0 = B$, $T_0 = I_n$, $k = 0$. **II. Recursion:** While rank($B_k$) < rown($A_k$) do $A_{k+1} = B_k^\perp A_k (B_k^\perp)^T$, $B_{k+1} = B_k^\perp A_k \tilde{B}_k$, $T_{k+1} = \begin{pmatrix} B_k^\perp \\ \tilde{B}_k \end{pmatrix}$, $k=k+1$, where $B_k^\perp := (\text{null}(B_k^T))^T$, $\tilde{B}_k := (\text{null}(B_k^\perp))^T$.

This algorithm can be easily realized in a computational software system such as MATLAB.

**Lemma 3.** If the pair $(A, B)$ is controllable then 1) Algorithm 1 terminates after finite number of steps $k \le n-1$; 2) the matrices $T_i \in \mathbb{R}^{\text{rown}(B_i) \times \text{rown}(B_i)}$, $i = 1, 2, \dots, k$ are orthogonal; 3) the orthogonal coordinate transformation $y = Gx$

$$ G = \begin{pmatrix} T_k & 0 \\ 0 & I_{w_k} \end{pmatrix} \begin{pmatrix} T_{k-1} & 0 \\ 0 & I_{w_{k-1}} \end{pmatrix} \dots \begin{pmatrix} T_2 & 0 \\ 0 & I_{w_2} \end{pmatrix} T_1, \text{ where } w_i := n - \text{rown}(T_i) \quad (6) $$

reduces the original system (2) to the following block form

$$ \left\{ \begin{aligned} \dot{y}_1 &= A_{11}y_1 + A_{12}y_2 \\ \dot{y}_2 &= A_{21}y_1 + A_{22}y_2 + A_{23}y_3 \\ \dots \\ \dot{y}_k &= A_{k1}y_1 + \dots + A_{kk}y_k + A_{k,k+1}(u + \gamma(t, y)) \end{aligned} \right. \quad (7) $$

where $y = (y_1, \dots, y_k)^T$, $y_i \in \mathbb{R}^{n_i}$, $n_i := \text{rank}(B_{k-i})$, $A_{k,k+1} = \tilde{B}_0 B_0$ and $A_{ij} \in \mathbb{R}^{n_i \times n_j}$ are blocks of the matrix $G A G^T$ such that $\text{rank}(A_{i,i+1}) = n_i$, $i = 1, 2, \dots, k$

*Proof:* **I.** Denote $B_0 = B$, $A_0 = A$ and $r_0=\text{rank}(B_0)$. Controllability of $(A_0, B_0)$ implies $r_0 > 0$. For $r_0 = n$ Algorithm 1 stops after initialization showing that the original system does not require transformation. **II.** Consider the case $r_0 < n$. In this case $\dim(\text{range}(B_0))=r_0$,
$\dim(\ker(B_0^T))=n-r_0$ and $\mathbb{R}^n=\text{range}(B_0)\oplus\ker(B_0^T)$. Columns of the matrix $\text{null}(B_0^T) \in \mathbb{R}^{n \times (n-r_0)}$ by definition are the orthonormal basis for $\ker(B_0^T) \subset \mathbb{R}^n$, so $\ker(B_0^T)=\text{range}(\text{null}(B_0^T))$. Since $\text{range}(B_0)$ is an orthogonal complement to $\ker(B_0^T)$ then $\text{range}(B_0)=\ker(\text{null}(B_0^T)^T)$ and columns of the matrix $(B_0^{\perp})^T=\text{null}(B_0^T)$ together with columns of the matrix $(\tilde{B}_0)^T=\text{null}(\text{null}(B_0^T)^T)$ provide the orthonormal basis in $\mathbb{R}^n$. So, the block matrix $T_1$ is square and orthogonal. Since $B_0^T \text{null}(B_0^T)=0$ then the coordinate transformation $(\tilde{y}_1 \ \tilde{y}_2)^T=T_1x$, $\tilde{y}_1 \in \mathbb{R}^{n-r_0}$, $\tilde{y}_2 \in \mathbb{R}^0$ gives
$$ \dot{\tilde{y}}_1 = \tilde{A}_{11}\tilde{y}_1 + \tilde{A}_{21}\tilde{y}_2, \quad \dot{\tilde{y}}_2 = \tilde{A}_{21}\tilde{y}_1 + \tilde{A}_{22}\tilde{y}_2 + \tilde{B}_0B_0(u + \gamma) $$

where $\tilde{A}_{11}, \tilde{A}_{12}, \tilde{A}_{21}, \tilde{A}_{22}$ are the blocks of the matrix $T_1AT_1^T$. Obviously that $A_{11} = A_1$ and $A_{12} = B_1$, where matrices $A_1$ and $B_1$ are generated by the recursive step of Algorithm 1.

Controllability of the pair $(A_0, B_0)$ implies controllability of $(A_1, B_1)$ [24], so we can repeat the recursive step of Algorithm 1 for $A_1$ and $B_1$. Since for each recursive step we have $\text{rown}(A_{i+1}) = \text{row}(B_{i+1}) = \text{rown}(A_i) - \text{rank}(B_i)$ and $\text{rank}(B_i) > 0$ then Algorithm 1 terminates in finite steps and the coordinate transformation $y = Ax$ reduces the original system (2) to the block form (7). Finally $\text{rank}(A_{i+1})=\text{rank}(\tilde{B}_{k-i}B_{k-i})=\text{rank}(T_{k-i}B_{k-i})=\text{rank}(B_{k-i}) = n_i$. ■

This paper considers the restricted robust problem statement assuming that all uncertainties and disturbances satisfy the matching condition. The control design for more general case can be done using the robustness analysis of block controllability forms presented in [15].

## B. Fixed-time attractivity

Further considerations are presented for the system (2) transformed to the block form (7). Since $\text{rank}(A_{i+1})=\text{row}(A_{i+1})=n_i$ then $A_{i+1}A_{i+1}^T$ is invertible and $A_{i+1}^+ = A_{i+1}^T(A_{i+1}A_{i+1}^T)^{-1}$ is the right inverse matrix of $A_{i+1}$. Denote $z^{[p]}=(z_1^{[p]},...,z_k^{[p]})^T$ for $z=(z_1,...,z_k)^T \in \mathbb{R}^k$.

By analogy to single input case introduce the nonlinear coordinate transformation $s = \Phi(y)$, $s = (s_1, ..., s_k)^T$, $s_i \in \mathbb{R}^{n_i}$, $y = (y_1, ..., y_k)^T$, $y_i \in \mathbb{R}^{n_i}$ by the recurrent formulas: $s_i=y_i+\varphi_i$, $i=1, 2, ..., k$, $\varphi_1=0$, $\varphi_{i+1}=A_{i+1}^+(\alpha_i(y_i+\varphi_i)+\beta_i(y_i+\varphi_i)^{[3]}+\sum_{j=1}^i A_{ij}y_j+\sum_{r=1}^i \frac{\partial \varphi_i}{\partial y_r}\sum_{j=1}^{r+1} A_{rj}y_j)$, where $\alpha_i > 0$, $\beta_i > 0$ and $i = 1, 2, ..., k-1$. The presented coordinate transformation is smooth and nonsingular. The inverse transformation $y = \Phi^{-1}(s)$ is defined as follows: $y_i=s_i+\psi_i$, $i=1, 2, ..., k$, $\psi_1=0$, $\psi_{i+1}=A_{i+1}^+(\sum_{k=1}^i \frac{\partial \psi_i}{\partial s_k}(A_{i+1}s_{k+1}-\alpha_ks_k-\beta_ks_k^{[3]})-\alpha_i s_i-\beta_i s_i^{[3]}-\sum_{j=1}^i A_{ij}(s_j+\psi_j))$.
Applying the transformation $s = \Phi(y)$ to the system (7) we obtain

$$ egin{cases} 
\dot{s}_1 = -\alpha_1 s_1 - \beta_1 s_1^3 + A_{12} s_2 \\ 
\dot{s}_2 = -\alpha_2 s_2 - \beta_2 s_2^3 + A_{23} s_3 \\ 
\dots \\ 
\dot{s}_k = \sum_{i=1}^{k} A_{ki} y_i + \sum_{i=1}^{k-1} \frac{\partial \varphi_k}{\partial y_i} \sum_{j=1}^{i+1} A_{ij} y_j + A_{k+1}(u + \gamma(t, y)) 
\end{cases} 
\quad (8) $$

So, the feedback control providing fixed-time attractivity property to the system (8) has the form

$$ u = -A_{k+1}^{+} \left( \alpha_k s_k + \beta_k s_k^3 + \sum_{i=1}^{k} A_{ki} y_i + \sum_{i=1}^{n-1} \frac{\partial \varphi_k}{\partial y_i} \sum_{j=1}^{i+1} A_{ij} y_j \right), \quad s = \Phi(Gy), \alpha_k, \beta_k > 0 \quad (9) $$

Let $\varepsilon$ and $T_{max}$ be positive numbers, $\alpha_i=1+\|A_{i+1}\|_{\infty}$, $i=1,2,...,k-1$, $\alpha_k=1+\frac{\|A_{k+1}\|_{\infty}\gamma_0(t,y)}{\varepsilon}$ and $\beta_i=\beta:=\frac{q}{\varepsilon^2}$, $i=1,2,...,k$, $q=(\exp(\frac{2T_{max}}{n}) - 1)^{-1}$ then repeating the proof of Lemma 2 we can show that any solution $s(t)$ of the system (8)-(9) satisfies the inequality $\|s(t)\|_{\infty} \le \varepsilon, \forall t \ge T_{max}$.

By analogy with single input case let us show now that there exist polynomials $p_i \in \mathbb{P}^i$, $i = 1, 2, ..., k$ with nonnegative coefficients such that $\|y_i(t)\|_{\infty} \le p_i(q)\varepsilon$ for $\forall t > T_{max}$, where $y(t) = (y_1(t), ..., y_k(t))^T$ is an arbitrary solution of the system (7), (9).

Denote the vector-modulus and the matrix-modulus operation by $\|\cdot\|$, i.e. $\|v\| = (|v_1|, ..., |v_r|)^T$ for the vector $v = (v_1, ..., v_r)^T$ and $\|W\| := \{|w_{ij}|\}$ for the matrix $W = \{w_{ij}\}$. The inequalities $\|v_1\| \le \|v_2\|$ and $\|W_1\| \le \|W_2\|$ are understood in a component-wise sense.

Let $\tilde{\psi}_1=0$, $\tilde{\psi}_{i+1}=\|A_{i+1}^+\| \left( \alpha_i z_i + \beta z_i^3 + \sum_{j=1}^i \|A_{ij}\|(z_j+\tilde{\psi}_j) + \sum_{k=1}^i \frac{\partial \tilde{\psi}_k}{\partial z_k} (\alpha_k z_k+\beta z_k^3+\|A_{i+1}\|z_{k+1}) \right)$, where $z_i \in \mathbb{R}_+^{n_i}$, $i=1,2,...,k$. Following the proof of Corollary 2 we can show that $\|\psi_i(s_1,...,s_i)\| \le \tilde{\psi}_i(\|s_1\|, ..., \|s_i\|)$, $\tilde{\psi}_i(\varepsilon\bar{e}_{n_1}, ..., \varepsilon\bar{e}_{n_i}) = \varepsilon \sum_{j=0}^i (\beta\varepsilon^2)^j M_{ij}\bar{e}_{n_j}$, where $\bar{e}_r=(1,...,1)^T \in \mathbb{R}_+^r$ and $M_{ij} \in \mathbb{R}^{n_i \times n_j}$ are matrices with nonnegative elements. Hence it directly follows that $\|y_i\|_{\infty} \le \|s_i\|_{\infty}+\|\psi_i(s_1,...,s_i)\|_{\infty} \le \varepsilon+\|\tilde{\psi}_i(\varepsilon\bar{e}_{n_1},...,\varepsilon\bar{e}_{n_i})\|_{\infty}=\varepsilon+\varepsilon \sum_{j=0}^i (\beta\varepsilon^2)^j \|M_{ij}\|_{\infty}:=p_i(q)\varepsilon$.

Similarly to single input case the polynomials $p_i(q)$ can be calculated using some symbolic computation package. For instance, $p_1(q)=1$ and $p_2(q)=1+\|A_{12}^+\|_{\infty}(1+\|A_{12}\|_{\infty}+q+\|A_{11}\|_{\infty})$. Therefore, we have just proven the main theorem for the multi input case.

**Theorem 3.** If the control $u$ has the form (9) with parameters $\alpha_i = 1+\|A_{i+1}\|_{\infty}$, $i = 1, 2, ..., k-1$, $\alpha_k = 1 + \frac{\gamma_0 \|A_{k+1}\|_{\infty} \|G\|_{1} p_k(q)}{r}$, $\beta_i = \frac{q \|G\|_{1}^2 p_k^2(q)}{r^2}$, $i = 1, 2, ..., k$, $q=(\exp(\frac{2T_{max}}{n}) - 1)^{-1}$, where $r > 0$ and $T_{max} > 0$ are arbitrary positive numbers, then the ball $B_r$ is the globally fixed-time attractive set of the closed-loop system (2) with the settling-time function bounded by $T_{max}$.
### C. Fixed-time stability

In this subsection we assume rank[$B, AB$]=n. Then Algorithm 1 terminates after the first recursive step, so the transformation $y = Gx$ brings the original system to the form

$$ egin{cases} \dot{y}_1 = A_{11}y_1 + A_{12}y_2 \\ \dot{y}_2 = A_{21}y_1 + A_{22}y_2 + A_{23}(u + \gamma) \end{cases} \tag{10} $$

where $y_1 \in \mathbb{R}^{n_1}, y_2 \in \mathbb{R}^{n_2}, n_2 = \text{rank}(B), n_1 = n - n_2, \text{rank}(A_{12}) = n_1 \le \text{rank}(A_{23}) = n_2$. Denote $A_{12}^{\perp} = \text{null}(A_{12})^{\perp}$.

**Theorem 4.** Let the controller $u(t, y), y = (y_1^T, y_2^T)^T \in \mathbb{R}^n$ has the form:

$$ u = A_{23}^{\perp}(u_{eq}(y) + u_d(t, y) + u_p(y)), \tag{11} $$

where $u_{eq} = -A_{12}^{\perp}((A_{11}^2 + A_{12}A_{21})y_1 + (A_{11}A_{12} + A_{12}A_{22})y_2) - (A_{12}^{\perp})^+ A_{12}^{\perp}(A_{21}y_1 + A_{22}y_2)$, $u_d = -\frac{\alpha_1+3\beta_1\|y_1\|_{\infty}^2+2\|A_{12}A_{23}\|_{\infty}\gamma_0(t,y)}{2}A_{12}^{\perp}\text{sign}[s_1] - \gamma_0(t,y)\|A_{12}^{\perp}A_{23}\|_{\infty}(A_{12}^{\perp})^{\perp}\text{sign}[s_2]$, $u_p = -A_{12}^{\perp}(\alpha_2 s_1 + \beta_2 s_1^{[3]})^{[\frac{1}{2}]} - (A_{12}^{\perp})^{\perp}(\alpha_3 s_2 + \beta_3 s_2^{[3]})^{[\frac{1}{2}]}$, $s_1 = A_{11}y_1 + A_{12}y_2 + ((A_{11}y_1 + A_{12}y_2)^{[2]} + \alpha_1 y_1 + \beta_1 y_1^{[3]})^{[\frac{1}{2}]}$, $s_2 = A_{12}^{\perp}y_2$

with $0.5\alpha_1=\alpha_2=4\alpha_3=0.5\beta_1=\beta_2=4\beta_3=64T_{\max}^{-2}$, $T_{\max} > 0$. Then the origin of the closed-loop system (10) is globally fixed-time stable with the settling-time function bounded by $T_{\max}$.

*Proof:* First of all remark that $A_{12}(A_{12}^{\perp})^{\perp} = 0$ and $A_{12}^{\perp}A_{12}^{\perp} = 0$. Denote $z = A_{11}y_1 + A_{12}y_2$. In this case the system (10) can be rewritten in the form

$$ \dot{y}_1 = z, \quad \dot{z} = -A_{12}u_d - A_{12}u_q + A_{12}A_{23}\gamma, \quad \dot{s}_2 = -A_{12}^{\perp}u_d - A_{12}^{\perp}u_p + A_{12}^{\perp}A_{23}\gamma $$

and $s_1 = z + (z^{[2]} + \alpha_1 y_1 + \beta_1 y_1^{[3]})^{[\frac{1}{2}]}$. Following the proof of Theorem 2 we similarly show that $y_1(t) = 0$ and $z(t) = 0$ for all $t \ge T_{\max}$. Hence $A_{12}y_2(t) = 0$ for all $t \ge T_{\max}$. Taking into account formulas for $u_d$ and $u_p$ we obtain $\dot{s}_2 = -\gamma_0(t, y)\|A_{12}^{\perp}A_{23}\|_{\infty}\text{sign}[s_2] - (\alpha_3 s_2 + \beta_3 s_2^{[3]})^{[\frac{1}{2}]} + A_{12}^{\perp}A_{23}\gamma$. Hence it can be easily shown that $s_2(t) = A_{12}^{\perp}y_2(t) = 0$ for all $t \ge T_{\max}$. ■

Theorem 4 extends the "nested" second order sliding mode control and the fixed-time controller presented in Theorem 2 to the multi input case.

Remark that the control (11) has the form closed to classical sliding mode controllers [24]. The equivalent control term $u_{eq}$ compensates the inner dynamic of the system, the switching term $u_d$ ensures the sliding mode existence and the new additional term $u_p$ has the form required for fixed-time stability.

For $n_1=n_2$ we have $\text{ker}(A_{12})=\emptyset$, so the terms with $A_{12}^{\perp}$ and $(A_{12}^{\perp})^{\perp}$ presented in (11) disappear.
VI. NUMERICAL EXAMPLES

Consider as a benchmark example the system (2) with

$$
A = \begin{pmatrix} 1 & -3 & 2 \\ -2 & 0 & 3 \\ 0 & -1 & 4 \end{pmatrix}, B = \begin{pmatrix} 2 & 0 \\ -1 & 1 \\ 0 & -3 \end{pmatrix}, f = \begin{pmatrix} 2 \\ 0 \\ -3 \end{pmatrix}, \sin(t), G = \begin{pmatrix} 0.4286 & 0.8571 & 0.2857 \\ -0.8571 & 0.4857 & -0.1714 \\ -0.2857 & -0.1714 & 0.9429 \end{pmatrix}.
$$

The transformation $y = Gx$ brings the system to the block from (10) with $y_1 \in \mathbb{R}$, $y_2 \in \mathbb{R}^2$,
$A_{11} = -0.5918$, $A_{23} = \begin{pmatrix} -2.2 & 1.0 \\ -0.4 & -3.0 \end{pmatrix}$, $A_{21} = \begin{pmatrix} 1.2980 \\ 0.7184 \end{pmatrix}$, $A_{22} = \begin{pmatrix} 3.0612 & -0.8367 \\ -0.5510 & 2.5306 \end{pmatrix}$,
$A_{12} = \begin{pmatrix} -0.4449 & 4.9469 \end{pmatrix}$. For this case we have $\gamma(t,x)=(\sin(t),\sin(t))^T$ and $\gamma_0(t,x)=1$.

A. Example of fixed-time attractivity

Let $T_{max}=6$, $r=1$. Following Theorem 3 we select the controller $u(t, x)$ in the form (9) with the control parameters $\alpha_1=6.3918$, $\alpha_2 = 13.333$, $\beta_1=\beta_2=0.0327$. Fig.1(a) and Fig. 1(b) show evolutions of the system states for $x(0)=(1,0,0)$ and $x(0)=(10,0,0)$. Fig. 1(c) presents the plot for the control magnitude. Numerical experiments show that the predetermined settling time

<figure><img src="image_11.png" /><figcaption>Fig. 1. Simulation results for fixed-time attraction.</figcaption></figure>

estimate $T_{max}=6$ is too conservative. Obtained convergence times to the ball $B_1$ are less than 1.

B. Example of fixed-time stability

According to Theorem 4 the controller of the form (11) with parameters $\alpha_1=\beta_1=1$, $\alpha_2=\beta_2=0.5$, $\alpha_3 = \beta_3 = 0.25$ provides fixed-time stability of the considered system guaranteeing the global settling-time estimate $T_{max}=8$. Simulations results approve declared property (see Fig. 2).
<figure><img src="image_12.png" /><figcaption>Fig. 2. Simulation results for fixed-time stabilization.</figcaption></figure>

In contrast to polynomial feedback numerical simulations for the fixed-time stabilizing con-
troller demonstrates very small conservatism in settling-time estimate showing that for $x(0) =$
$(10^7, 0, 0)$ the settling-time is equal approximately to 7.2.

VII. CONCLUSIONS

In this paper the new control problems are considered. They consist in designing of the feedback control for linear uncertain plant that guarantees convergence of system trajectories to the origin or into selected neighborhood of the origin in predetermined finite time *independently on initial conditions*. The described property is called in this paper by *fixed-time stability* and attractivity, respectively. Controllers that provides fixed-time attractivity to the closed-loop systems are designed in the form of polynomial feedbacks under the classical assumption on controllability of the pair (A, B). Fixed-time stabilizing controllers presented for more restrictive case rank[*B*, *AB*] = *n* are modification of the second order sliding mode control algorithm. It is proven that all controllers are robust with respect to matched uncertainties and disturbances.

Numerical experiments approve the declared fixed-time properties for the closed-loop system
with the proposed control algorithms. However, they show that for polynomial feedbacks the
real convergence time is less than the predetermined. This fact probably points out ineffective
usage of the control resource. So, the problem of optimal selection of the control parameters in
order to minimize the control energy and magnitude appears. But this is another story.

VIII. ACKNOWLEDGEMENTS

The author wishes to thank professor Boris T. Polyak for helpful discussions and remarks.
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# Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Approach

Andrey Polyakov<sup>a,b</sup>, Denis Efimov<sup>a,b</sup>, Wilfrid Perruquetti<sup>b,a</sup>

## Abstract

Theorems on Implicit Lyapunov Functions (ILF) for finite-time and fixed-time stability analysis of nonlinear systems are presented. Based on these results, new nonlinear control laws are designed for robust stabilization of a chain of integrators. High order sliding mode (HOSM) algorithms are obtained as particular cases. Some aspects of digital implementations of the presented algorithms are studied, it is shown that they possess a chattering reduction ability. Theoretical results are supported by numerical simulations.

**Key words:** Lyapunov Methods, Robust Control, Sliding Mode Control

## 1 Introduction and related works

Many practical applications require severe time response constraints (for security reasons, or simply to improve productivity). That is why, finite-time stability and stabilization problems have been intensively studied, see [2], [3], [4], [5], [6]. Time constraint may also appear in observation problems when a finite-time convergence of the state estimate to the real values is required: [7], [8], [9], [10], [11].

Let us stress that finite-time stability property is frequently associated with HOSM controls, since these algorithms should ensure finite-time convergence to a sliding manifold [12], [13], [14], [15]. Typically, the associated controllers have mechanical and electromechanical applications [16], [17], [18], [19].

The theoretical background of HOSM control systems is very well developed [12], [20], [21]. However, applications of the existing HOSM control algorithms are complicated, since there are a few constructive algorithms for tuning the HOSM control parameters. Most of them are restricted to the second order sliding mode systems [21], [15], [22], [23].

Fixed-time stability, that demands boundedness of the settling-time function for a globally finite-time stable system, was studied in [23], [22], [24]. This property was originally discovered in the context of homogeneous systems [25]. Fixed-time stability looks promising if a controller (observer) has to be designed in order to provide some required control (observation) precision in a given time and independently of initial conditions.

The main tool for analysis of finite-time and fixed-time stability is the Lyapunov function method (see, for example, [4], [5], [24]), which is lacking for constructive design in the nonlinear case. This paper deals with ILF method [26], which relies on Lyapunov functions defined, implicitly, as solutions to an algebraic equation. Stability analysis does not require solving this equation, since the implicit function theorem [27] helps to check all stability conditions directly from the implicit formulation. The similar approach was presented in [28] for control design and called the controllability function method (see, also [29]).

This paper addresses the problem of a control design for the robust finite-time and fixed-time stabilization of a chain of integrators. The ILF method is used to design the control laws together with Lyapunov functions for closed-loop systems. This method allows us to analyze robustness of the closed-loop system and to design a high order sliding mode control algorithm, which rejects bounded matched exogenous disturbances. Finite-time and fixed-time stability conditions were obtained in the form of Linear Matrix Inequalities (LMI). They provide simple constructive schemes for

<footnote><sup>a</sup>Non-A INRIA - LNE, Parc Scientifique de la Haute Borne 40, avenue Halley Bat.A, Park Plaza 59650 Villeneuve d'Ascq, France.</footnote>

<footnote><sup>b</sup>LAGIS UMR CNRS 8219, Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France.</footnote>

<footnote>* The preliminary version of this paper was presented at IFAC Symposium on Nonlinear Control Systems 2013 [1]. This work is supported by ANR Grant CHASLIM (ANR-11-BS03-0007)

Email addresses: andrey.polyakov@inria.fr (Andrey Polyakov), denis.efimov@inria.fr (Denis Efimov), wilfrid.perruetti@inria.fr (Wilfrid Perruquetti).</footnote>
tuning the control parameters in order to predefine the required convergence time and/or to guarantee stability and robustness with respect to disturbances of a given magnitude.

Through the paper the following notation will be used:

* $\mathbb{R}$ is the set of real numbers, $\mathbb{R}_+ = \{x \in \mathbb{R} : x > 0\}$;

* $\frac{dV}{dt}|_{(.)}$ is the time derivative of a function $V$ along the solution of a differential equation numbered as (.);

* $\| \cdot \|$ is the Euclidian norm in $\mathbb{R}^n$;

* $\mathrm{diag}\{\lambda_i\}_{i=1}^n$ is the diagonal matrix with the elements $\lambda_i$;

* a continuous function $\sigma: \mathbb{R}_+ \to \mathbb{R}_+$ belongs to the class $\mathcal{K}$ if it is monotone increasing and $\sigma(h) \to 0^+$ as $h \to 0^+$;

* for a symmetric matrix $P = P^T$ the minimal and maximal eigenvalues are denoted by $\lambda_{\min}(P)$ and $\lambda_{\max}(P)$,

* $\mathrm{int}(\Omega)$ is the interior of the set $\Omega \subseteq \mathbb{R}^n$.

## 2 Problem statement

The paper deals with finite-time and fixed-time stabilization problems for the disturbed chain of integrators. Note that a control design scheme developed for such systems usually admits extension to feedback linearizable nonlinear multi-input multi-output systems [30]. The problem statement presented below is also typical for high-order sliding mode control design [12].

Consider a linear single input system of the following form

$$ \dot{x}(t) = Ax(t) + bu(t) + d(t, x(t)), \quad t > 0, \quad (1) $$

$$ A = \begin{pmatrix} 0 & 1 & 0 & \dots & 0 \\ \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{pmatrix} \quad \text{and} \quad b = \begin{pmatrix} 0 \\ \dots \\ 0 \\ 1 \end{pmatrix}, $$

where $x \in \mathbb{R}^n$ is the state vector, $u \in \mathbb{R}$ is the control input, and the function $d: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ describes the system uncertainties and disturbances. The whole state vector $x$ is assumed to be measured. Let the function $d$ be measurable locally bounded uniformly in time, i.e. $\sup_{t \in \mathbb{R}_+, x \in \mathbb{R}^n, \|x\| < \delta} \|d(t, x)\| < \infty$ for any $\delta > 0$. Both the control function $u$ and the function $d$ are admitted to be discontinuous with respect to $x$. For example, the function $d$ may describe unknown dry friction of a mechanical model. The analysis of such systems requires a special mathematical framework. In this paper we use Filippov theory [31].

The goal of the paper is to develop control laws such that the origin of the closed-loop system (1) will be globally asymptotically stable and all its trajectories will reach the origin in a finite time or in the fixed time $T_{\max} \in \mathbb{R}_+$.

In addition, the control algorithms to be developed should have effective schemes for tuning the control parameters and assigning of the settling time.

The control design is based of ILF approach to finite-time and fixed-time stability analysis, which is developed in the next section.

## 3 Preliminaries

### 3.1 Finite-time and fixed-time stability

Consider the system defined by

$$ \dot{x}(t) = f(t, x(t)), \quad t \in \mathbb{R}_+, \quad x(0) = x_0, \quad (2) $$

where $x \in \mathbb{R}^n$ is the state vector, $f: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ is a nonlinear vector field locally bounded uniformly in time. If $f$ is a locally measurable function that is discontinuous with respect to the state variable $x$ then a solution of the Cauchy problem (2) is understood in the sense of Filippov [31], namely, as an absolutely continuous function satisfying the differential inclusion

$$ \dot{x}(t) \in K[f](t, x(t)) $$

for almost all $t \in [0, t^*)$, where $t^* \in \mathbb{R}_+$ or $t^* = +\infty$.

The set-valued function $K[f] : \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}^n$ is defined for any fixed $(t, x) \in \mathbb{R}_+ \times \mathbb{R}^n$ as follows

$$ K[f](t, x) = \bigcap_{\varepsilon>0} \bigcap_{N:m(N)=0} \co f(t, B(x, \varepsilon)\setminus N), $$

where $\co(M)$ defines the convex closure of the set $M \subset \mathbb{R}^n$, $B(x, \varepsilon)$ is the ball with the center at $x \in \mathbb{R}^n$ and the radius $\varepsilon \in \mathbb{R}_+$, the equality $m(N) = 0$ means that the Lebesgue measure of the set $N \subset \mathbb{R}^n$ is zero.

Let the origin be an equilibrium of (2), i.e. $0 \in K[f](t, 0)$. The system (2) may have non-unique solutions and may admit both weak and strong stability (see, for example, [31]). This paper deals only with the strong stability properties, which ask for stable behavior of all solutions of the system (2). The next definition of uniform finite-time stability is just a compact representation of Definition 2.5 from [13].

**Definition 1** The origin of system (2) is said to be globally uniformly finite-time stable if it is globally uniformly asymptotically stable (see, for example, [13]) and there exists a locally bounded function $T: \mathbb{R}^n \to \mathbb{R}_+ \cup \{0\}$, such that $x(t, x_0) = 0$ for all $t \ge T(x_0)$, where $x(\cdot, x_0)$ is an arbitrary solution of the Cauchy problem (2). The function $T$ is called the settling-time function.

Asymptotic stability of the time-independent (autonomous) system always implies its uniform asymptotic stability (see, for example, [32]). For finite-time stable systems this is not true in general case (see, for example, [24]), since Definition 1 additionally asks a uniformity of the settling time with respect to initial conditions.

The origin of system $\dot{x}(t) = -|x(t)|^{0.5} \mathrm{sign}[x(t)]$, $x \in \mathbb{R}$ is globally uniformly finite-time stable, since its settling-time function $T$ is locally bounded: $T(x_0) = 2\sqrt{|x_0|}$.
**Definition 2 ([23])** The origin of system (2) is said to be globally fixed-time stable if it is globally uniformly finite-time stable and the settling-time function T is globally bounded, i.e. $\exists T_{\max} \in \mathbb{R}_+$ such that $T(x_0) \le T_{\max}$, $\forall x_0 \in \mathbb{R}^n$.

The presented definition just asks more: strong uniformity of finite-time stability with respect to initial condition. The origin of system $\dot{x}(t) = -(|x|^{0.5}(t) + |x|^{1.5}(t)) \text{sign}(x(t))$, $x \in \mathbb{R}$, is globally fixed-time stable, since its settling time function $T(x_0) = 2 \arctan(\sqrt{|x_0|})$ is bounded by $\pi \approx 3.14$.

The uniformity of finite-time and fixed-time stability with respect to system disturbances can also be analyzed. For instance, finite-time stability, which is uniform (in some sense) with respect to both initial conditions and system disturbances, was called equiuniform finite-time stability [13].

## 3.2 Implicit Lyapunov Function Method

This subsection introduces some stability theorems further used for control design. They refine the known results about global uniform asymptotic, finite-time and fixed-time stability of differential inclusions to the case of implicit definition of Lyapunov function. The next theorem extends Theorem 2 from [26].

**Theorem 3** If there exists a continuous function

$$Q: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R} \\ (V, x) \to Q(V, x)$$

satisfying the conditions

C1) $Q$ is continuously differentiable outside the origin;

C2) for any $x \in \mathbb{R}^n \setminus \{0\}$ there exists $V \in \mathbb{R}_+$ such that $Q(V, x) = 0$;

C3) let $\Omega = \{(V, x) \in \mathbb{R}_+ \times \mathbb{R}^n : Q(V, x)=0\}$ and $\lim_{\substack{x \to 0 \\ (V, x) \in \Omega}} V=0^+$, $\lim_{\substack{V \to 0^+ \\ (V, x) \in \Omega}} \|x\|=0$, $\lim_{\substack{V = +\infty \\ (V, x) \in \Omega}} V = +\infty$;

C4) $\frac{\partial Q(V, x)}{\partial V} < 0$ for all $V \in \mathbb{R}_+$ and $x \in \mathbb{R}^n \setminus \{0\}$;

C5) $\sup_{t \in \mathbb{R}_+, y \in K[f](t,x)} \frac{\partial Q(V, x)}{\partial x} y < 0$ for all $(V, x) \in \Omega$;

then the origin of (2) is globally uniformly asymptotically stable.

**Proof.** The conditions C1), C2), C4) and the implicit function theorem [27] imply that the equation $Q(V, x) = 0$ implicitly defines a unique function $V: \mathbb{R}^n \setminus \{0\} \to \mathbb{R}_+$ such that $Q(V(x), x) = 0$ for all $x \in \mathbb{R}^n \setminus \{0\}$. The function $V$ is continuously differentiable outside the origin and

$$\frac{\partial V}{\partial x} = - \left[ \frac{\partial Q(V, x)}{\partial V} \right]^{-1} \frac{\partial Q(V, x)}{\partial x} \quad \text{for } Q(V, x) = 0, x \neq 0.$$

Due to the condition C3) the function $V$ can be continuously prolonged at the origin by setting $V(0) = 0$. In addition, it is radially unbounded and positive definite. Denote $Z(V, x) = \left[ \frac{\partial Q}{\partial V} \right]^{-1} \sup_{t \in \mathbb{R}_+, y \in K[f](t,x)} \frac{\partial Q(V, x)}{\partial x} y$ and $W(x) = Z(V(x), x)$. The conditions C4 and C5 imply $W(x) > 0$ for

$x \neq 0$. Let $x(t, x_0)$ be a solution of the system (2) with initial condition $x(0, x_0) = x_0$ then the function $V(x(t, x_0))$ is differentiable for almost all $t$ such that $x(t, x_0) \neq 0$ and $\frac{d}{dt}V(x(t, x_0)) \le -W(x(t, x_0))$. Finally, we finish the proof using Theorem 4.1 from [33] and, for example, Lemmas 4, 6 from [24]. ■

Evidently, the conditions of Theorem 3 mainly repeat (in the implicit form) the requirements of the classical theorem on global asymptotic stability (see, for example, [33]). Indeed, Condition C1) asks for smoothness of the Lyapunov function. Condition C2) and the first two limits from Condition C3) provide its positive definiteness. The last limit from Condition C3) implies radial unboundedness of the Lyapunov function. Condition C5) guarantees the negative definiteness of the total derivative of the Lyapunov function calculated along trajectories of the system (2). The only specific condition is C4), which is imposed by implicit function theorem (see, for example, [27]). This condition is required in order to guarantee that the Lyapunov function is (uniquely) well-defined by the equation $Q(V, x) = 0$.

**Theorem 4** If there exists a continuous function $Q: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}$ that satisfies the conditions C1)-C4) of Theorem 3 and the condition

C5bis) there exist $c > 0$ and $0 < \mu \le 1$ such that $\sup_{t \in \mathbb{R}_+, y \in K[f](t,x)} \frac{\partial Q(V, x)}{\partial x} y \le cV^{1-\mu} \frac{\partial Q(V, x)}{\partial V}$ for $(V, x) \in \Omega$,

then the origin of the system (2) is globally uniformly finite-time stable and $T(x_0) \le \frac{V^\mu}{c\mu}$, where $Q(V_0, x_0) = 0$.

**Proof.** Theorem 3 implies global uniform asymptotic stability of the origin of (2). The uniform finite-time stability of the origin follows from the differential inequality

$$\frac{dV(x(t, x_0))}{dt} \le -cV^{1-\mu}(x(t, x_0)),$$

which, due to the condition C5bis), holds for almost all $t$ such that $x(t, x_0) \neq 0$. For a detailed analysis of the obtained differential inequality see, for example, [4], [5], [24]. ■

<figure><img src="image_13.png" /></figure>

**Theorem 5** Let there exist two functions $Q_1$ and $Q_2$ satisfying the conditions C1)-C4) of Theorem 3 and the conditions C6) $Q_1(1, x) = Q_2(1, x)$ for all $x \in \mathbb{R}^n \setminus \{0\}$;

C7) there exist $c_1>0$ and $0 < \mu < 1$ such that the inequality

$$\sup_{t \in \mathbb{R}_+, y \in K[f](t,x)} \frac{\partial Q_1(V, x)}{\partial x} y \le c_1 V^{1-\mu} \frac{\partial Q_1(V, x)}{\partial V},$$

holds for all $V \in (0, 1]$ and $x \in \mathbb{R}^n \setminus \{0\}$ satisfying the equation $Q_1(V, x) = 0$;

C8) there exist $c_2>0$ and $\nu>0$ such that the inequality

$$\sup_{t \in \mathbb{R}_+, y \in K[f](t,x)} \frac{\partial Q_2(V, x)}{\partial x} y \le c_2 V^{1+\nu} \frac{\partial Q_2(V, x)}{\partial V},$$

holds for all $V \ge 1$ and $x \in \mathbb{R}^n \setminus \{0\}$ satisfying the equation $Q_2(V, x) = 0$, then the system (2) is globally fixed-time stable with the settling-time estimate $T(x_0) \le \frac{1}{c_1\mu} + \frac{1}{c_2\nu}$.

**Proof.** Let the two functions $V_1$ and $V_2$ be defined by the equations $Q_1(V, x) = 0$ and $Q_2(V, x) = 0$ (see, the proof of Theorem 3). Consider the sets $\Sigma_1 = \{x \in \mathbb{R}^n : V_1(x) > 1\}$,
$\Sigma_2 = \{x \in \mathbb{R}^n : V_2(x) > 1\}$ and prove that $\Sigma_1 = \Sigma_2$. Suppose the contrary, i.e. $\exists z \in \mathbb{R}^n$ such that $z \in \Sigma_1$ and $z \notin \Sigma_2$. On the one hand, $Q_1(V_1, z) = 0$ implies $V_1 > 1$ and $Q_1(1, z) > Q_1(V_1, z) = 0$ due to Condition C4). On the other hand, $Q_2(V_2, z) = 0$ implies $V_2 \le 1$ and $Q_2(1, z) \le Q_2(V_2, z) = 0$. The contradiction follows from Condition C6).

Therefore, due to C6) and C4) the function $V: \mathbb{R}^n \to \mathbb{R}$ defined by the equality

$$V(x) = \begin{cases} V_1(x) & \text{for } V_1(x) < 1, \\ V_2(x) & \text{for } V_2(x) > 1, \\ 1 & \text{for } V_1(x) = V_2(x) = 1, \end{cases}$$

is positive definite, continuous in $\mathbb{R}^n$ and continuously differentiable for $x \notin \{0\} \cup \{x \in \mathbb{R}^n : V(x) = 1\}$. The function $V$ is Lipschitz continuous outside the origin and has the following Clarke's gradient [34]:

$\nabla_C V(x) = \xi \nabla V_1(x) + (1 - \xi) \nabla V_2(x), \quad x \in \mathbb{R}^n,$
where $\xi = 1$ for $0 < V_1(x) < 1$, $\xi = 0$ for $V_2(x) > 1$, $\xi = [0, 1]$ for $V_1(x) = V_2(x) = 1$ and $\nabla V_i$ is the gradient of the function $V_i$, $i = 1, 2$. Hence, due to conditions C7) and C8), the inequality

$$\frac{dV(x(t,x_0))}{dt} \le \begin{cases} -c_1 V^{1-\mu}(x(t,x_0)) & \text{for } V(x(t,x_0)) < 1, \\ -c_2 V^{1+\nu}(x(t,x_0)) & \text{for } V(x(t,x_0)) > 1, \\ -\min\{c_1, c_2\} & \text{for } V(x(t,x_0)) = 1, \end{cases}$$
holds for almost all $t$ such that $x(t, x_0) \ne 0$, where $x(t, x_0)$ is a solution of the system (2) with the initial condition $x(0) = x_0$. This implies the fixed-time stability of the origin of the system (2) with the estimate of settling-time function given above. Please see [23] or [24] for more details. ■

# 4 Control design using Implicit Lyapunov Function Method

## 4.1 Finite-Time Stabilization

Introduce the implicit Lyapunov function candidate

$$Q(V, x) := x^T D_r(V^{-1}) P D_r(V^{-1}) x - 1, \quad (3)$$

where $V \in \mathbb{R}_+$, $x \in \mathbb{R}^n$, $P \in \mathbb{R}^{n \times n}$, $P > 0$, $D_r(\lambda) := \text{diag}\{\lambda^{r_i}\}$ is the matrix with $r = (r_1, ..., r_n)^T \in \mathbb{R}_+$ and $r_i = 1 + (n - i)\mu$, $0 < \mu \le 1$.

Denote $H_\mu := \text{diag}\{-r_i\}_{i=1}^n$.

The function $Q$ is an implicit analog of the quadratic Lyapunov function. Indeed, for $\mu = 0$ the equality $Q(V, x) = 0$ gives $V(x) = \sqrt{x^T P x}$. For $\mu = 1$ it coincides with the implicit Lyapunov function considered in [26].

**Theorem 6 (Finite-time Stabilization)** Let 1) $\mu \in (0, 1]$, $\alpha, \beta, \gamma \in \mathbb{R}_+$ such that $\alpha > \beta$ and the system of matrix

inequalities

$$\left\{ \begin{array}{l} AX + XA^T + by + y^T b^T + \alpha X + \beta I_n \le 0, \\ -\gamma X \le XH_\mu + H_\mu X < 0, \quad X > 0 \end{array} \right. \quad (4)$$

is feasible for some $X = X^T \in \mathbb{R}^{n \times n}$ and $y \in \mathbb{R}^{1 \times n}$;

## 2) the control u has the form

$$u(V, x) = V^{1-\mu} k D_r(V^{-1}) x, \quad (5)$$

where $k := yX^{-1}$, $V \in \mathbb{R}_+$ is such that $Q(V, x) = 0$ and $Q$ is defined by (3) with $P = X^{-1}$;

## 3) the function $d(t, x)$ satisfy the inequality

$$\beta^2 V^{-2\mu} \ge \sup_{t \in \mathbb{R}_+} d^T(t, x) D_r^2(V^{-1}) d(t, x), \quad (6)$$

for all $V \in \mathbb{R}_+$, $x \in \mathbb{R}^n$ such that $Q(V, x) = 0$;

then the closed-loop system (1) is globally uniformly finite-time stable and $T(x) \le \frac{\gamma V^\mu}{\mu(\alpha - \beta)}$, where $Q(V, x) = 0$.

**Proof.** It can be shown that the following inequalities

$$\frac{\lambda_{\min}(P)\|x\|^2}{\max\{V^{2+2(n-1)\mu}, V^2\}} \le Q(V, x) + 1 \le \frac{\lambda_{\max}(P)\|x\|^2}{\min\{V^{2+2(n-1)\mu}, V^2\}}$$

hold for all $V \in \mathbb{R}_+$ and $x \in \mathbb{R}^n$. The function (3) satisfies conditions C1)-C3) of Theorem 3. The condition C4) of Theorem 3 also holds, since $0 > \frac{\partial Q}{\partial V} = V^{-1}x^T D_r(V^{-1}) (H_\mu P + P H_\mu) D_r(V^{-1}) x \ge -\gamma V^{-1}$.

Taking into account that $D_r(V^{-1}) A D_r^{-1}(V^{-1}) = V^{-\mu} A$ and $D_r(V^{-1}) b u(x) = V^{-\mu} b k D_r(V^{-1}) x$ we obtain

$$\frac{\partial Q}{\partial x}(Ax + bu(x) + d(t, x)) =$$

$$\begin{pmatrix} z_1 \\ z_2 \end{pmatrix}^T W \begin{pmatrix} z_1 \\ z_2 \end{pmatrix} + \beta^{-1} V^\mu z_2^T z_2 - \alpha V^{-\mu} z_1^T P z_1,$$

where $z_1 = D_r(V^{-1})x$, $z_2 = D_r(V^{-1})d$, the matrix

$$W := \begin{pmatrix} \frac{1}{V^\mu} (P(A+bk) + (A+bk)^T P + \alpha P) & P \\ P & -\frac{V^\mu}{\beta} I_n \end{pmatrix}$$

is negative semidefinite due to (4) and the Schur complement. Since $z_1^T P z_1 = 1$ and $z_2^T z_2 \le \beta^2 V^{-2\mu}$ by (6) then the condition C5bis) of Theorem 4 holds for $c = (\alpha - \beta)/\gamma$.

Let us make remarks about the presented control scheme:

* The practical implementation of the control (5) requires to find the solution $V(x)$ of the equation $Q(V, x) = 0$, which can be solved numerically and on-line using the current value of the state vector. A simple numerical scheme that can be utilized for this purpose is presented in Section 5.

* If $n = 2$ and $\mu = 1$ the function $V(x)$ can be found analytically. Indeed, the equation $Q(V, x) = 0$ becomes $V^4 - p_{22}x_2^2V^2 - 2p_{12}x_1x_2V - p_{11}x_1^2 = 0$, where $\{p_{ij}\}$ are elements of the matrix $P > 0$ and
$(x_1, x_2)^T = x \in \mathbb{R}^2$. The roots of the obtained quartic (with respect to $V$) equation can be found using, for example, Ferrari formulas. Due to Theorem 6 the equation has a unique positive root for any $(x_1, x_2)^T \in \mathbb{R}^2$.

Denote $z(x_1, x_2) = \frac{2p_{22}x_2^2 - C_1(x_1, x_2) - C_2(x_1, x_2)}{3}$, where

$C_i(x_1, x_2) = \sqrt[3]{(-1)^i \sqrt{\Delta_1^2(x_1, x_2) - 4\Delta_0^3(x_1, x_2) + \Delta_1(x_1, x_2)}}$, $\Delta_0(x_1, x_2) = p_{22}^2 x_2^4 - 12p_{11} x_1^2$ and $\Delta_1(x_1, x_2) = p_{22}^3 x_2^6 - 180p_{12}^2 x_1^2 x_2^2$. The roots of the quartic equation coincide with roots of two quadratic equations [35]:

$$V^2 + (-1)^i \sqrt{z(x_1, x_2)V + r_i(x_1, x_2)} = 0, \quad i = 1, 2,$$
where $r_i(x_1, x_2) = \frac{p_{22}x_2^2 + z(x_1, x_2)}{2} + (-1)^i \frac{p_{12}x_1x_2}{\sqrt{z(x_1, x_2)}}$.

The explicit representations of roots are rather cumbersome. They are omitted in order to safe the space.

* The implicit restriction (6) to the system disturbances and uncertainties takes an explicit form when $\mu = 1$ and the matching condition [14] holds, i.e. $d_i(t, x) \equiv 0$ for $i = 1, ..., n-1$ and $d = (d_1, ..., d_n)^T$. In this case the condition (6) becomes $|d_n(t, x)| \le \beta$ and the term $\beta I_n$ in (4) can be replaced with $\beta E$, where the matrix $E = \{e_{ij}\} \in \mathbb{R}^{n \times n}$ has only one nonzero element: $e_{n,n} = 1$. So, the finite-time control (5) designed for $\mu = 1$ rejects the bounded disturbances realized **HOSM** algorithm. The HOSM version of (5) has a discontinuity only at zero similarly to the quasi-continuous HOSM algorithm [20].

* For $\mu = 1$ the control (5) is globally bounded. Indeed, the equality $x^T D_r(V^{-1}) P D_r(V^{-1}) x = 1$ implies $\|D_r(V^{-1})x\|^2 \le \frac{1}{\lambda_{\min}(P)}$ and

$$\|u(x)\| \le \|k\| \cdot \|D_r(V^{-1})x\| \le \frac{\|k\|}{\sqrt{\lambda_{\min}(P)}}.$$

Hence the condition $\|u(x)\| \le u_0$, $u_0 \in \mathbb{R}_+$ can be formulated as additional LMI:

$$\begin{pmatrix} X & y^T \\ y & u_0^2 \end{pmatrix} \ge 0. \qquad (7)$$

Indeed, the inequality $u^2 = x^T D_r(V^{-1}) k^T k D_r(V^{-1}) x \le u_0^2 = u_0^2 x^T D_r(V^{-1}) P D_r(V^{-1}) x$ holds for all $V \in \mathbb{R}_+$ and $x \in \mathbb{R}^n$ such that $Q(V, x) = 0$. Hence $k^T k \le u_0^2 P$ and Schur complement gives (7) for $X = P^{-1}$, $y = kX$.

* The advantage of the control design scheme presented in Theorem 6 is related to simplicity of tuning the control parameters, which is based on LMI technique. The parameters $\gamma$ and $\alpha$ are introduced in (4) in order to tune the settling time of the closed-loop system.

**Corollary 7** Let the conditions of Theorem 6 hold and the control law is modified as $u(x) = u(V_0, x)$, where $V_0 \in \mathbb{R}_+$ is an arbitrary fixed number, then the ellipsoid

$$\Pi(V_0) = \{x \in \mathbb{R}^n : x^T (D_r(V_0) X D_r(V_0))^{-1} x \le 1\} \quad (8)$$

is strictly positively invariant set of the closed-loop system (1), i.e. $x(t_0) \in \Pi(V_0) \Rightarrow x(t) \in \text{int} \Pi(V_0)$, $t > t_0$, where

$x(\cdot)$ is any trajectory of the closed-loop system (1) and $t_0 \ge 0$ is an arbitrary instant of time.

**Proof.** Rewrite the matrix inequality (4) in the form: $PA + A^T P + Pbk + k^T b^T P + \alpha P + \beta P^2 \le 0$. Hence we derive $D_r(V_0^{-1})(PA + A^T P + Pbk + k^T b^T P + \alpha P + \beta P^2)D_r(V_0^{-1}) \le 0$. Denoting $P_0 = D_r(V_0^{-1})P D_r(V_0^{-1}) > 0$ and taking into account $D_r^{-1}(V_0^{-1})A D_r(V_0^{-1}) = V_0^\mu A$, $D_r^{-1}(V_0^{-1})b = V_0 b$ we obtain the following matrix inequality:

$$P_0 A + A^T P_0 + P_0 b k_0 + k_0^T b^T P_0 + \frac{\alpha P_0 + \beta P_0 D_r^2(V_0) P_0}{V_0^\mu} \le 0,$$

where $k_0 = V_0^{-1-\mu} k D_r(V_0^{-1})$, $\alpha > \beta > 0$ and $P_0 > 0$. This means that the matrix $A + bk_0$ is Hurwitz, i.e. $u(x) = u(V_0, x) = k_0 x$ is a stabilizing linear feedback control for the system (1) with the Lyapunov function $\tilde{V}(x) = x^T P_0 x$. Note that using Schur Complement the obtained matrix inequality can be transformed into the form $W_0 \le 0$, where

$$W_0 = \begin{pmatrix} P_0(A+bk_0)+(A+bk_0)^T P_0 + \frac{\alpha}{V_0^\mu} P_0 & P_0 \\ P_0 & -\frac{V_0^\mu D_r^2(V_0^{-1})}{\beta} \end{pmatrix}.$$

Taking into account the inequality (6) for $V = V_0$ (or equivalently $\tilde{V} = 1$) we derive

$$\begin{aligned} \left. \frac{d}{dt} \tilde{V} \right|_{(1)} &= x^T (P_0 A + A^T P_0 + P_0 b k_0 + k_0^T b^T P_0) x + 2d^T P_0 x \\ &= \begin{pmatrix} x \\ d \end{pmatrix}^T W_0 \begin{pmatrix} x \\ d \end{pmatrix} + \frac{V_0^\mu}{\beta} d^T D_r^2(V_0^{-1}) d - \frac{\alpha \tilde{V}}{V_0^\mu} \\ &\le \frac{\beta - \alpha}{V_0^\mu} < 0 \quad \text{for } \tilde{V}(x) = 1. \end{aligned}$$

The obtained inequality implies that the ellipsoid (8) is strictly positively invariant. ■

The next corollary proves stability of the sampled-time realization for the ILF-based control algorithm (5).

**Corollary 8** Let $\{t_i\}_{i=0}^\infty$ be a strictly increasing sequence of arbitrary time instants, $0 = t_0 < t_1 < t_2 < ...$ such that $\lim_{i \to \infty} t_i = +\infty$. Let all conditions of Theorem 6 hold and the control $u$ is applied as follows: $u(t) = u(V_i, x(t))$ for $t \in [t_i, t_{i+1})$, where $V_i > 0 : Q(V_i, x(t_i)) = 0$. Then the origin of the system (1) is globally asymptotically stable.

**Proof.** Let $V(x)$ be a positive definite function implicitly defined by the equation $Q(V, x) = 0$ and $x(t)$ be a trajectory of the closed-loop system (1) with the sampled control application described above. Let us prove that the sequence $\{V(x(t_i))\}_{i=1}^\infty$ is monotone decreasing and tends to 0. This, obviously, implies convergence of $x(t)$ to the origin.

Consider the time interval $[t_i, t_{i+1})$ and the function $\tilde{V}_i(x) = x^T P_i x$, where $P_i = D_r(V^{-1}(x(t_i))) P D_r(V^{-1}(x(t_i))) > 0$
0. The switching control $u(x) = u_i(x)$ on this interval takes the form $u_i(x) = u(V(x(t_i)), x) = k_i x$, where $k_i := V^{1-\mu}(x(t_i))k D_r(V^{-1}(x(t_i)))$. Repeating the proof of Corollary 7 we show $\tilde{V}_i(x(t_i)) > \tilde{V}_i(x(t))$ for all $t \in (t_i, t_{i+1})$. Since $V(x)$ is such that $Q(V(x), x) = 0$ then for all $t \in (t_i, t_{i+1})$

$$ 
\begin{aligned}
Q(V(x(t_i)), x(t)) &= x^T(t)D_r(V^{-1}(x(t_i)))PD_r(V^{-1}(x(t_i)))x(t) - 1 \\
&= \tilde{V}_i(x(t)) - 1 < \tilde{V}_i(x(t_i)) - 1 = Q(V(t_i), x(t_i)) = \\
&= 0 = Q(V(x(t)), x(t)).
\end{aligned}
$$

For any given $x \in \mathbb{R}^n \setminus \{0\}$ the function $Q(V, x)$ is monotone decreasing for all $V \in \mathbb{R}^+$ (see Condition C4) of Theorem 3 and the proof of Theorem 6). Then the obtained inequality implies $V(x(t)) < V(x(t_i))$, $\forall t \in (t_i, t_{i+1}]$. Moreover $V(x(t)) \le V(x(0))$ for all $t > 0$, i.e. the origin of the closed-loop system (1) is Lyapunov stable. The proven properties of closed-loop system also imply that $\|x(t)\|$ and $\|d(t, x(t))\|$ are bounded by some constants for all $t > 0$.

Since the function $V(x)$ is positive definite then the monotone decreasing sequence $\{V(x(t_i))\}_{i=1}^\infty$ converge to some limit. Let us show now that this limit is zero. Suppose the contrary, i.e. $\lim_{i \to \infty} V(x(t_i)) = V_* > 0$ or equivalently
$$ \forall \varepsilon > 0 \quad \exists N = N(\varepsilon) : 0 \le V(x(t_i)) - V_* < \varepsilon, \forall i \ge N. $$
Let us represent the sampled control law $u$ in the form

$$ 
\begin{aligned}
u(V(x(t_i)), x) &= u(V_*, x) + \Delta_i x, \\
\Delta_i &= V^{1-\mu}(x(t_i))kD_r(V^{-1}(x(t_i))) - V_*^{1-\mu}kD_r(V_*^{-1}).
\end{aligned}
$$

Since the control function $u = u(V, x)$ is continuous in $\mathbb{R} \times \mathbb{R}^n$ and then there exists $\gamma \in K$ (possibly depended on $\mu, k, V_*, n$) such that $\|\Delta_i\| \le \gamma(\varepsilon)$ for all $i \ge N(\varepsilon)$. This means that for $t > t_{N(\varepsilon)}$ the closed-loop system (1) can be presented in the form

$$ \dot{x} = (A + bk_*)x + b\Delta_i x + d(t, x) \quad (9) $$

where $k_* = V_*^{1-\mu}kD_r(V_*^{-1})$. Consider the Lyapunov function candidate $\tilde{V}(x) = x^T P_* x$, where $P_* = D_r(V_*^{-1})PD_r(V_*^{-1})$. Under the assumptions made above we have $\tilde{V}(x(t)) \ge 1$ for all $t > 0$ and repeating the proof of Corollary 7 we derive

$$ 
\begin{aligned}
\left. \frac{d\tilde{V}(x(t))}{dt} \right|_{(1)} &\le \frac{V_*^\mu d^T D_r^2(V_*^{-1})d}{\beta} - \frac{\alpha \tilde{V}(x(t))}{V_*^\mu} + 2x^T(t) P_* b \Delta_i x(t) \\
&\le \frac{-\alpha + \beta}{V_*^\mu} + 2x^T(t) P_* b \Delta_i x(t) \\
&\quad + \frac{V_*^\mu (d^T D_r^2(V_*^{-1})d - \beta^2 V_*^{-2\mu})}{\beta}.
\end{aligned}
$$

Since $\|x(t)\|$ and $\|d(t, x(t))\|$ are bounded by some constants for all $t > 0$ then for sufficiently small $\varepsilon > 0$ we have
$$ \left. \frac{d\tilde{V}(x(t))}{dt} \right|_{(1)} \le \frac{-\alpha + \beta}{2V_*^\mu} < 0 \text{ for all } t > t_{N(\varepsilon)}. $$

the condition $V(x(t)) \ge V^*$ for all $t > 0$. Consequently, $\lim_{i \to \infty} V(x(t_i)) = 0$. Hence, we derive the asymptotic stability of the closed-loop system (1). ■

The proven corollary shows that the sampled-time realization of the developed "implicit" control scheme preserves the asymptotic stability to the origin of the closed-loop system (1) independently on the sampling period. Between two switching instants the unperturbed system is linear, so analysis of its discrete-time version can be studied using discretization schemes of linear systems (see, e.g. [36]).

## 4.2 Fixed-time Stabilization

Consider now two functions:

$$ Q_1(V, x) = x^T D_{r_1} (V^{-1}) P D_{r_1} (V^{-1}) x - 1, $$

$$ Q_2(V, x) = x^T D_{r_2} (V^{-1}) P D_{r_2} (V^{-1}) x - 1, $$

where $P \in \mathbb{R}^{n \times n}$, $P = P^T > 0$, $r_1 = (1 + (n-1)\mu, 1 + (n-2)\mu, ..., 1 + \mu, 1)^T \in \mathbb{R}^n$, $r_2 = (1, 1+\nu, ..., 1+(n-2)\nu, 1+(n-1)\nu)^T \in \mathbb{R}^n$, $0 < \mu \le 1$ and $\nu \in \mathbb{R}_+$. Denote $H_\nu = \text{diag}\{-(r_2)_i\}_{i=1}^n$.

**Theorem 9 (Fixed-time stabilization) If 1) the system of matrix inequalities**

$$ 
\left\{
\begin{array}{l}
AX + XA^T + by + y^T b^T + \alpha X + \beta I_n \le 0, \\
-\gamma_1 X \le X H_\mu + H_\mu X < 0, \\
-\gamma_2 X \le X H_\nu + H_\nu X < 0, \\
X > 0,
\end{array}
\right.
\quad (10)
$$

is feasible for some $X = X^T \in \mathbb{R}^{n \times n}$, $y \in \mathbb{R}^{1 \times n}$ and numbers $\mu \in (0, 1]$, $\nu, \alpha, \beta, \gamma_1, \gamma_2 \in \mathbb{R}_+$ such that $\alpha > \beta$;

2) the control $u$ has the form

$$ u(V, x) = \begin{cases} V^{1-\mu} k D_{r_1} (V^{-1}) x & \text{for } x^T P x < 1, \\ V^{1+n\nu} k D_{r_2} (V^{-1}) x & \text{for } x^T P x \ge 1, \end{cases} \quad (11) $$

where $k = yX^{-1}$, $P = X^{-1}$ and $V$ defined by

$$ V = V(x) \text{ such that } \begin{cases} Q_1(V, x) = 0 & \text{for } x^T P x < 1, \\ Q_2(V, x) = 0 & \text{for } x^T P x \ge 1. \end{cases} $$

3) the disturbance function $d$ satisfy

$$ \beta^2 V^{-2\mu} \ge \sup_{t \in \mathbb{R}_+} d^T(t, x) D_{r_1}^2 (V^{-1}) d(t, x) \text{ if } x^T P x < 1, $$

$$ \beta^2 V^{2\nu} \ge \sup_{t \in \mathbb{R}_+} d^T(t, x) D_{r_2}^2 (V^{-1}) d(t, x) \text{ if } x^T P x \ge 1; $$

then the closed-loop system (1) is fixed-time stable with the settling-time estimate: $T(x) \le (\gamma_1/\mu + \gamma_2/\nu)(\alpha - \beta)^{-1}$.

**Proof.** The functions $Q_1(V, x)$ and $Q_2(V, x)$ satisfy the conditions C1-C4) of Theorem 3. Since $Q_1(1, x) = Q_2(1, x)$, then the condition C6) of Theorem 5 also holds.
Repeating the proof of Theorem 6 it can be shown that
$$ \frac{\partial Q_1}{\partial x}(Ax+bu(x)+d(t,x)) \le \frac{\alpha-\beta}{\gamma_1} V^{1-\mu} \frac{\partial Q_1}{\partial V} \quad \text{for } x^T P x \le 1, $$
$$ \frac{\partial Q_2}{\partial x}(Ax+bu(x)+d(t,x)) \le \frac{\alpha-\beta}{\gamma_2} V^{1+\nu} \frac{\partial Q_2}{\partial V} \quad \text{for } x^T P x \ge 1. $$
Applying Theorem 5 we finish the proof. ■

If $\mu = 1$, $|d_n(t, x)| \le \beta$ and the matching condition holds then Theorem 9 provides a HOSM control with fixed-time reaching phase.

Let us call the controls (5) and (11) by the finite-time ILF control and the fixed-time ILF control, respectively. The ILF control with $\mu = 1$ we call the HOSM ILF control.

It is worth to stress that Corollaries 7 and 8 stay true for the case of fixed-time control (11) application. We do not prove the finite-time and fixed-time stability properties for the sampled-time realization of the ILF controls. However, the motion of the sampled system will be close to the original one if the sampling period is sufficiently small.

## 5 Aspects of practical implementation

Corollaries 7 and 8 give some remarks on possible implementation of the developed control scheme. A detailed study of sampled-time and discrete-time versions of the presented control algorithms goes beyond the scope of this paper providing the subject for a future research. In this section we provide just some general ideas to be used for ILF control implementation.

The control scheme (5) can be realized in digital control devices, which allow us to solve the equation $Q(V, x) = 0$ numerically and on-line for any given $x \in \mathbb{R}^n \setminus \{0\}$. Rather simple numerical procedures can be utilized for this purpose. The function $Q(V, x)$ satisfy the properties (C1)-(C4) of Theorem 3. So, for each fixed $\bar{x} \in \mathbb{R}^n \setminus \{0\}$ the function $\bar{Q}(V) = Q(V, \bar{x})$ is monotone decreasing and has the unique zero on the interval $(0, +\infty)$. In this case we may use, for example, the bisection method in order to solve the scalar equation $\bar{Q}(V) = 0$.

Let the control $u(V, x)$ be given by (5) and the parameter $V$ may change its value at some time instants $t_0 = 0, t_i > 0, i = 1, 2, ...$. Namely, let the control signal $u(t)$ be defined as $u(t) = u(V_i, x(t))$ for $t \in [t_i, t_{i+1})$ and $V_i \in \mathbb{R}_+$. Recall that $u(V_i, x)$ is a linear stabilizing feedback for any $V_i \in \mathbb{R}_+$ (see, the proof of Corollary 7). Denote $x_i = x(t_i)$. If $V_i = V(x_i)$ then stability of the sampling control scheme follows from Corollary 8.

The simplest algorithm for finding the switching control parameter $V_i$ is given below.

### Algorithm 10 (Implementation of the ILF control)

INITIALIZATION: $a = V_{\min}$; $V_0 > V_{\min}$; $b = V_0$; STEP :

If $x_i^T D_r(b^{-1}) P D_r(b^{-1}) x_i > 1$ then $a = b$; $b = 2b$;
else if $x_i^T D_r(a^{-1}) P D_r(a^{-1}) x_i < 1$ then
  $b = a$; $a = \max\{\frac{a}{2}, V_{\min}\}$;
else $c = \frac{a+b}{2}$;
If $x_i^T D_r(c^{-1}) P D_r(c^{-1}) x_i < 1$ then $b = c$;
else $a = \max\{V_{\min}, c\}$;
endif;
endif;
$V_i = b$,

where $V_{\min} \in \mathbb{R}_+$ is a minimal admissible value of $V_i$.

INITIALIZATION defines a linear feedback $u(V_0, x)$ on the first sampling interval $[t_0, t_1)$ by means of selection of the value $V_0 \in \mathbb{R}_+$. STEP of the algorithm is applied at each sampling time instances $t_i$ in order to define the value $V_i$ (and the linear feedback $u(V_i, x)$) for the next interval $[t_i, t_{i+1})$. Let $x_i \in \mathbb{R}^n$ is a current state vector. If at the sampling instant $t_i$ STEP of the algorithm is repeated many times for the same $x_i$ (for example, there exists a loop containing STEP) then Algorithm 10 realizes: 1) a localization of the unique positive root of the equation $Q(V, x_i) = 0$, i.e. $V(x_i) \in [a, b]$; 2) improvement of the obtained localization by means of the bisection method, i.e. $|b - a| \to 0$. Such an application of Algorithm 10 allows us to calculate $V(x_i)$ with a high precision, however, it requests a high computational capability of a control device.

Therefore, it is more reasonable to realize STEP of Algorithm 10 just once or a few (2 or 3) times at each sampling instant. Some additional considerations are needed in order to show that the algorithm will work properly in this case.

Denote $\Pi(V_i) = \{x \in \mathbb{R}^n : x^T D_r(V_i^{-1}) P D_r(V_i^{-1}) x \le 1\}$. If the set of initial conditions is known then the value $V_0 \in \mathbb{R}_+$ can be always selected such that $x_0 \in \Pi(V_0)$. Otherwise, applying Algorithm 10 for sufficiently small sampling period there always exists a finite number $i^* \in \{0, 1, 2, ...\}$ such that $x_{i^*} \in \Pi(V_{i^*})$. Corollary 7 implies that the ellipsoid $\Pi(V_i)$ is strictly positively invariant for the closed-loop system (1) with the feedback law $u(x) = u(V_i, x)$ of the form (5). Monotonicity condition $\frac{\partial Q(V, x)}{\partial V} < 0$ implies that $\Pi(V') \subset \Pi(V'')$ for $V' < V$". In order to guarantee stability of the sampled-time realization of the developed control, on each STEP of Algorithm 10 the upper estimate (i.e. $V_i = b$) is selected in order to design the linear feedback $u(V_i, x)$ for the next interval $[t_i, t_{i+1})$. Such selection ensures $V_{i+1} \in (0, V_i]$ if $x_i \in \Pi(V_i)$ and the sequence $\{V_i\}_{i=i^*}^{\infty}$ generated by Algorithm 10 is non-increasing. The bisection procedure will operate until $x(t) \notin \Pi(V_{\min})$. So, it can be shown that $V_i \to V_{\min}$ as $i \to \infty$.

The parameter $V_{\min}$ defines the lower possible value of $V$ and the "minimal" attractive invariant set $\Pi(V_{\min})$ for the closed-loop system. This parameter cannot be selected arbitrary small due to finite numerical precision of digital devices. Fixed-time control application can be realized with small changes of Algorithm 10.
**Remark 11** An additional advantage of the developed control scheme is related to the possible reduction of the chattering effect for HOSM ILF control application. Indeed, the HOSM control of the form (5) with $\mu = 1$ has the unique discontinuity point $x = 0$. According Algorithm 10, near discontinuity point the feedback law is defined as follows: $u(x) = kD(V_{\min})x, \forall x \in \Pi(V_{\min})$, where $V_{\min} \in \mathbb{R}_+$, i.e. we always have a linear continuous control inside the ellipsoid $\Pi(V_{\min})$. Such a modification of the control law obviously follows the classical idea of the chattering reduction developed for the first order sliding mode algorithms [14]. Namely, for practical realization of the sliding mode control, the discontinuous feedback law can be replaced with a high-gain linear feedback if the system state is close to the switching manifold [14].

# 6 Numerical examples

## 6.1 HOSM ILF control

Consider the system (1) for $n = 4$, $d_1(t, x) = d_2(t, x) = d_3(t, x) = 0$ and $|d_4(t, x)| \le \beta = 0.2$. We take $d_4(t, x) = 0.1 \sin(t) + 0.1 \cos(x_4)$ and $x(0) = (2, 0, 0, 0)^T$.

The numerical solution of ODE for the closed-loop system has been obtained using the explicit Euler method with a fixed step size $h \in \mathbb{R}_+$. In order to show the effectiveness of the developed control scheme with respect to the chattering reduction, we select quite a large step size $h = 0.1$.

The Fig. 1(a) shows the simulation results for the closed-loop system with the HOSM ILF control (5) ($\mu = 1$) that is restricted by $|u(x)| \le 1$ (see, (7)) and applied by the scheme presented in Algorithm 10 for $V_{\min} = 0.1$ and $h = 0.1$.

The parameters of the HOSM ILF control (5) were selected by means of solving the LMI system (4), (7) for $\mu = 1$ and $\alpha = 0.5, \beta = 0.2, \gamma = 10, u_0 = 1$:

$$P = \begin{pmatrix} 0.0058 & 0.0318 & 0.0644 & 0.0517 \\ 0.0318 & 0.1983 & 0.4351 & 0.3798 \\ 0.0644 & 0.4351 & 1.0798 & 1.0393 \\ 0.0517 & 0.3798 & 1.0393 & 1.3857 \end{pmatrix}, \\ k = (-0.0330, -0.2522, -0.7823, -1.1386).$$

In order to compare the obtained results with some existing HOSM controller, the simulations also have been done for the fourth order sliding mode (4-SM) nested controller [12] of the form

$$u = -\operatorname{sign}\left[x_4 + 0.5\left(|x_1|^3 + |x_2|^4 + |x_3|^6\right)^{\frac{1}{12}} \operatorname{sign}(\phi)\right], \\ \phi = x_3 + \frac{1}{4}\left(|x_1|^3 + |x_2|^4\right)^{\frac{1}{6}} \operatorname{sign}\left[x_2 + 0.15|x_1|^{\frac{3}{4}} \operatorname{sign}[x_1]\right].$$

The effective procedures for parameters adjustment of high

order sliding mode control algorithms are not developed yet for $n \ge 3$. So, the parameters of this controller have been selected manually. The simulations of the 4-SM nested control algorithm were initially done for the small sampling period: $h = 10^{-3}$. They showed good performances of this control: fast convergence rate and rejection of matched bounded disturbances. The chattering effect appears in the system, when the sampling period becomes larger. The simulation results with $h = 0.1$ for the 4-SM nested control are depicted on the Fig. 1(b). The Fig. 1(c) shows the control inputs generated by two considered algorithms.

Note that the comparison of the sliding mode control laws for the sufficiently large sampling period $h = 0.1$ is motivated by practical reasons. For instance, the HOSM control is frequently demonstrated on the control problems for mobile robotics (see, for instance, [12]). However, the computational restrictions of autonomous mobile systems does not allow us to spend to much computational power for the generation of control actions. In practice, the reasonable sampling period can be $0.05 - 0.1$ second [37].

## 6.2 Fixed-time ILF control

Define the fixed-time control $u$ in the form (11) with $n = 3$, $\mu = 0.5, \nu = 0.1$, where the matrix

$$P = \begin{pmatrix} 5.0575 & 4.5868 & 1.4558 \\ 4.5868 & 7.4884 & 2.2003 \\ 1.4558 & 2.2003 & 1.7959 \end{pmatrix}$$

and the vector $k = (-2.9319, -5.6235, -2.6998)$ are obtained from the LMI (10) with $\beta = 0, \alpha = 1, \gamma_1 = 4.5, \gamma_2 = 4.5$. In this case the settling time estimate provided by Theorem 9 gives $T(x_0) \le 54$.

The numerical simulation has been done using the Euler method with a fixed step size $h = 0.01$. The control has been applied using Algorithm 10 with $V_{\min} = 0.001$. The simulation results are presented on Fig. 2 for $x_0 = (1, 0, 0)^T$ and $x_0 = (9, 0, 0)^T$. They show a "week" dependence of the convergence time on the initial conditions. Definitely, it requires a rather high magnitude of the control.

# 7 Conclusion

The paper develops the Implicit Lyapunov Function method for finite-time and fixed-time stability analysis. Using this theoretical framework new algorithms of finite-time and fixed-time stabilization for a chain of integrators and high order sliding mode control design are developed. The obtained control schemes have the following advantages:

* The control design algorithms are constructive. The schemes for tuning the control parameters have LMI representations.

* For $\mu = 1$ the algorithms provide high order sliding mode control rejecting matched bounded disturbances.
* The digital implementation of the developed control scheme admits the effective chattering reduction by means of tuning the parameter $V_{\min}$.

* The fixed-time ILF controllers allow us to prescribe the convergence time independently of the initial condition.

### and disadvantages:

* The algorithms are applicable only in digital controllers.

* The practical realization of the developed control schemes asks for additional computational power of the digital control device, which is required for on-line computation of the ILF value at the current state.

The developed Implicit Lyapunov Function method for finite-time and fixed-time stability analysis is promising to tackle many other problems such as fast observation and estimation, development of fast adaptation algorithms or fast consensus protocols, etc.

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<figure><img src="image_14.png" /><figcaption>(a) The HOSM ILF control.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) The 4-SM nested control [12].</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(c) Comparison of control inputs.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(a) $x_0 = (1, 0, 0)^T$</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(b) $x_0 = (9, 0, 0)^T$</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(c) Control inputs.</figcaption></figure>

<caption>Fig. 2. The simulation results for the fixed-time ILF control.</caption>
# Relay Control Design for Robust Stabilization in a Finite-Time

Andrey Polyakov and Laurentiu Hetel

**Abstract**— The problem of robust finite-time stabilization of perturbed multi-input linear system by means of generalized relay feedback is considered. A new control design procedure, which combines convex embedding technique with Implicit Lyapunov Function (ILF) method, is developed. The sufficient conditions for both local and global finite-time stabilization are provided. The issues of practical implementation of the obtained implicit relay feedback are discussed. Our theoretical result is supported by numerical simulation for a Buck converter.

## I. INTRODUCTION

Theory of relay automatic control systems has a long outstanding history. Relay feedbacks appeared in the early technological developments of the 19th century. However, the systematic theoretical study of relay control methods was provided in 1950s [1], [2]. The modern frequency domain approach to analysis and design of the relay systems can be found in [3].

When the sliding mode control methodology [4] was invented, it suggested to utilize a proper fast relay switching strategy in order to maintain the motion of the control system on a prescribed surface in the state space. Indeed, the classical example of the sliding mode system has the form of relay feedback: $\dot{x}(t) = -\operatorname{sign}[x(t)]$, $t > 0$, $x(0) = x_0 \in \mathbb{R}$, where the sign function is defined as follows: $\operatorname{sign}[\rho] = 1$ if $\rho > 0$ and $\operatorname{sign}[\rho] = -1$ if $\rho < 0$. Any trajectory of this system reaches the state $x = 0$ in a finite time and remains thereafter. In fact, finite-time stability frequently accompanies the relay and sliding mode feedback systems [4], [5], [6], [7], [8]. The main application domains of sliding mode approach are electrical and electro-mechanical systems [4], [7]. Historically, being invented in 1960s the sliding mode is one of the first robust control approaches. Indeed, the sliding mode control is nonsensitive with respect to the so-called matched disturbances [9], [4].

Nowadays, the relay control systems can be found in different application domains. For instance, relay actuators (switches) are used in power converters [10] for control of output voltage but the relay (pulsed) jets are utilized in fluid dynamics for control of separated (turbulent) flows [11], [12].

The modern theoretical framework of hybrid dynamical systems [13], [14], [15] includes relay feedbacks as a particular case of switched affine systems [16], [17], [18], [19]. To the best of our knowledge, the finite-time stabilization problems have never been studied in the context of switched affine systems.

The present paper addresses the *robust finite-time stabilization of linear multi-input system using the so-called generalized relay control*. The generalized relay assumes a fixed set of possible values for the whole vector of control inputs [20]. This means, in particular, that the inputs cannot be switched independently. Such type of relay is motivated by practical demands of power electronics [21], where, for example, some switch cannot be turned on while an another one has the same state. Such restrictions in general case do not allow us to apply directly conventional schemes of relay sliding mode control design [4], [6]. In order to tackle this challenge, the technical note uses the convex embedding procedure. Recently, the ideas of convex embedding have been applied in order to design a locally exponentially stabilizing relay switching law based on the existence of a stabilizing linear feedback [20]. Here based on Implicit Lyapunov Function (ILF) method [22], [23], [24], [25] the convex embedding procedure allows us

* to design a finite-time stabilizing feedback for the case of generalized relays;

* to formulate stability conditions and restrictions to control parameters in terms of linear matrix inequalities;

* to obtain an estimate of both matched and mismatched uncertainties and disturbances, which can be rejected by the generalized relay feedback;

* to derive sufficient conditions for global finite-time stabilization and to estimate an attraction domain in the local case;

* to provide a settling-time estimate of the closed-loop system.

One more challenge tackled in the paper is the development of a feedback control rejecting state-dependent discontinuous disturbances. This allows us, for example, to take into account an unknown dry friction in mechanical or electro-mechanical models.

A preliminary version of this paper appeared in [26]. The key differences with respect to [26] are as follows: in this technical note we design a *robust* relay feedback control for *perturbed* multi-input system, present the proofs to all claims (Appendix), provide the global finite-time stabilization conditions (Proposition 7) and study a possible application of the proposed relay control to buck converter (Section VI).

The paper is organized as follows. The section II discusses problem statement and basic assumptions. After that preliminaries are considered. Next, the main result is presented. Finally, numerical simulation example and concluding remarks are given. All proofs are presented in Appendix.

*Notation:* $\mathbb{R}$ is the set of real numbers; $\mathbb{R}_+ = \{x \in \mathbb{R} : x > 0\}$; $\|x\|$ denotes the Euclidian norm of the vector

<footnote>This research is supported by ANR ROCC-SYS (ANR-14-CE27-0008). A. Polyakov (andrey.polyakov@inria.fr) is with Non-A team-project, Inria Lille-Nord Europe, 40, avenue Halley, 59650 Villeneuve d'Ascq, France. L. Hetel (laurentiu.hetel@ec-lille.fr) and A. Polyakov are with CRISTAL UMR CNRS 9189, 59650 Villeneuve d'Ascq, France.</footnote>
$x \in \mathbb{R}^n$; Range($B$) is the column space of the matrix $B \in \mathbb{R}^{n \times m}$; diag{$\lambda_1, ..., \lambda_n$} is a diagonal matrix with elements $\lambda_i$; the order relation $P > 0$($< 0, \ge 0, \le 0$) for $P \in \mathbb{R}^{n \times n}$ means that $P$ is symmetric and positive (negative) definite (semidefinite); if $P > 0$ then the matrix $P^{1/2} := B$ is such that $B^2 = P$; $\lambda_{\max}(P)$ and $\lambda_{\min}(P)$ denote maximum and minimum eigenvalues of the symmetric matrix $P \in \mathbb{R}^{n \times n}$; co($U$) is the convex closure of the set $U \subset \mathbb{R}^n$; int{$U$} denotes the interior of the set $U$; $\mathcal{B}(r) = \{u \in \mathbb{R}^m : \|u\| \le r\}$ is the ball of the radius $\varepsilon$ in $\mathbb{R}^m$.

## II. PROBLEM STATEMENT

Let us consider a model of a control system described by the ordinary differential equation (ODE):

$$ \dot{x}(t) = Ax(t) + Bu(t) + d(t, x(t), u(t)), \quad t \in \mathbb{R}_+, \quad (1) $$

where $x(t) \in \mathbb{R}^n$ is the state vector, $u(t) \in \mathbb{R}^m$ is the vector of control inputs, $A \in \mathbb{R}^{n \times n}$ is the system matrix, $B \in \mathbb{R}^{n \times m}$ is the matrix of control gains and the locally measurable function $d: \mathbb{R}^{n+m+1} \to \mathbb{R}^n$ describes the exogenous disturbances and parametric uncertainties.

It is assumed that the matrices $A$ and $B$ are known, $\text{rank}(B) = m \le n$ and the pair $(A, B)$ is controllable; the whole state vector $x$ can be measured and utilized for control purposes. The control input $u$ is assumed to be a generalized relay, i.e. it can take values from a given discrete set:

$$ u(t) \in \mathcal{U} := \{v_1, v_2, ..., v_N\}, \quad v_i \in \mathbb{R}^m, \quad t \in \mathbb{R}_+, \quad (2) $$

where $N$ is a natural number. In addition, the assumption

$$ 0 \in \int \{co(\mathcal{U})\} \subset \mathbb{R}^m \quad (3) $$

is possessed in order to guarantee the existence of the locally stabilizing relay control (see, [20] for the details). As we will see further, this configuration includes as a particular case the classical sliding control generated by sign functions. This control configuration may also be related to the simplex method in [27], [28] and to the stabilization of switched affine systems [17], [18]. Filippov theory of differential equations with discontinuous right-hand sides [29] is utilized below in order to take into account the discontinuity of the control law and possible discontinuity of the disturbances.

The control aims are

* to stabilize the origin of the system (1) in a finite time,

* to describe a class of uncertainties $d$, which can be rejected by relay feedback control,

* to specify a set of admissible initial conditions (i.e. the domain of finite-time attraction).

Following the ideas of [20] the relay stabilizing control law can be designed in two steps. Initially, some (possibly continuous) stabilizing feedback should be selected. For this purpose the method of the Implicit Lyapunov Functions (ILF) is utilized [22], [24], [25], [30]. Next, a proper convex embedding procedure [20] is applied in order to construct the relay switching law in the form

$$ u(t) \in u_r(t, x(t)) = \underset{v \in \mathcal{U}}{\operatorname{argmin}} \Gamma^T(t, x(t))v, \quad (4) $$

where $\Gamma: \mathbb{R}^{n+1} \to \mathbb{R}^m$ is a continuous (outside the origin) nonlinear function to be defined. The inclusion in (4) indicates that argmin is not unique in general case. In particular, if $m = 1$ and $\mathcal{U} = \{-1, 1\}$ then $u_r(t, x) = -\text{sign}[\Gamma(t, x)]$ similarly to the sliding mode control [4], where

$$ \overline{\text{sign}}[\rho] = \begin{cases} 1 & \text{if } \rho > 0, \\ -1 & \text{if } \rho < 0, \\ \{-1, 1\} & \text{if } \rho = 0. \end{cases} $$

Note that in order to define the control input according to the formula (4) we just need to select the minimum of $\Gamma^T(t, x(t))v$ over finite set values $v \in \mathcal{U}$. This operation does not need applying any finite or infinite dimensional optimization procedure. Just $N$ scalar products $\Gamma^T(t, x(t))v_i$, $i = 1, ..., N$ must be calculated at each instant of time.

## III. PRELIMINARIES

### A. Finite-Time Stability

Let us consider the differential inclusion [29] of the form

$$ \dot{x}(t) \in F(t, x), \quad t \in \mathbb{R}_+, \quad (5) $$

where $x \in \mathbb{R}^n$ is the state vector, $F: \mathbb{R}^{n+1} \Rightarrow \mathbb{R}^n$ is a multivalued map that is convex-valued, compact-valued and upper semi-continuous. It is well known [29] that an absolute continuous function $x: (a, b) \subset \mathbb{R}^n$ is called Caratheodory solution of (5) if it satisfies the differential inclusion (5) almost everywhere on the time interval $(a, b)$.

Let the origin be an equilibrium point of the system (5), i.e. $0 \in F(t, 0)$. Only strong uniform stability properties of the system (5) are studied in this paper, so the corresponding words "strong uniform" will be omitted below for shortness and simplicity of the presentation.

**Definition 1 ([31], [32], [5]):** The origin of system (5) is said to be **finite-time stable** if it is asymptotically stable and finite-time attractive, i.e. for any initial condition $x(0) = x_0 \in \mathcal{M} \setminus \{0\}$ there exists $T(x_0) \in \mathbb{R}_+$ such that $x(t) = 0$ for all $t \ge T(x_0)$, where $\mathcal{M}$ is a neighborhood of the origin and $T$ is called the **settling-time function** of the system (5). If $\mathcal{M} = \mathbb{R}^n$ then the origin is globally finite-time stable.

According to [29] the discontinuous system (1), (4) can also be treated as differential inclusion (5) properly constructed by means of the so-called Filippov regularization procedure. Below we deal only with Filippov solutions of the control system (1), (4).

### B. Implicit Lyapunov Function Method

The next theorem is utilized below in order to design the feedback law.

**Theorem 2:** [25] If there exists a continuous function $Q: \mathbb{R}_+ \times \mathbb{R}^n \to \mathbb{R}$ that satisfies the conditions

C1) $Q$ is continuously differentiable in $\mathbb{R}_+ \times \mathbb{R}^n \setminus \{0\}$;

C2) for any $x \in \mathbb{R}^n \setminus \{0\}$ there exist $V \in \mathbb{R}_+$ such that $Q(V, x) = 0$;

C3) let $\Omega = \{(V, x) \in \mathbb{R}_+ \times \mathbb{R}^n : Q(V, x) = 0\}$ and

$$ \lim_{(V,x) \in \Omega} V = 0^+, \quad \lim_{(V,x) \in \Omega} \|x\| = 0, \quad \lim_{(V,x) \in \Omega} \|x\| \to \infty $$
C4) the inequality $\frac{\partial Q(V,x)}{\partial V} < 0$ holds for all $V \in \mathbb{R}_+$ and $x \in \mathbb{R}^n \setminus \{0\}$;

C5) there exist $c \in \mathbb{R}_+$ and $\mu \in (0,1]$ such that $\sup_{t \in \mathbb{R}_+, y \in F(t,x)} \frac{\partial Q(V,x)}{\partial x} y \le cV^{1-\mu} \frac{\partial Q(V,x)}{\partial V}$, $(V,x) \in \Omega$;

then the origin of system (5) is globally finite time stable with the following settling time estimate: $T(x_0) \le \frac{V_0^\mu}{c\mu}$, where $V_0 \in \mathbb{R}_+$: $Q(V_0, x_0) = 0$.

Theorem 2 provides the sufficient conditions of finite-time stability for implicit definition of Lyapunov function. The conditions C1)-C4) guarantee existence and uniqueness of a continuously differentiable (outside the origin) positive definite radially unbounded function $V: \mathbb{R}^n \to \mathbb{R}_+$, which is implicitly defined by the equation $Q(V,x)=0$. The implicit function theorem [33] gives $\frac{\partial V}{\partial x} = -\left[\frac{\partial Q}{\partial V}\right]^{-1} \frac{\partial Q}{\partial x}$. Due to C4) and C5) the estimate $\dot{V} \le \sup_{t \in \mathbb{R}_+, y \in F(t,x)} \frac{\partial V}{\partial x} y \le -cV^{1-\mu}$ implies the finite-time stability of the origin of (5) if $\mu \in (0,1]$. We refer the reader to [31], [23], [32], [5], [6], [25] for more details about finite-time stability and implicit Lyapunov function method.

**Corollary 3:** If the conditions C1)-C4) of Theorem 2 are fulfilled then the set

$$ \varepsilon(V_0) = \{z \in \mathbb{R}^n : Q(V_0, z) \le 0\} \quad (6) $$

is the $V_0$-level set $\{s \in \mathbb{R}^n : V(s) \le V_0\}$ of the positive definite function $V: \mathbb{R}^n \to \mathbb{R}_+$ implicitly defined by the equation $Q(V, s) = 0$.

*Proof:* Indeed, if $\tilde{s} \in \mathbb{R}^n$ is such that $V(\tilde{s}) = \alpha$, where $\alpha \in \mathbb{R}_+$ then $Q(\alpha, \tilde{s}) = 0$, i.e. $\tilde{s} \in \varepsilon(\alpha)$. The condition C4) implies that $Q(V', \tilde{s}) < 0$ (i.e $\tilde{s} \in \varepsilon(V')$) for any $V' > \alpha$ and $Q(V'', \tilde{s}) > 0$ (i.e. $\tilde{s} \notin \varepsilon(V')$) for any $V'' < \alpha$. ■

Corollary 3 allows us to adapt Theorem 2 to local finite-time stability analysis possessing the condition C5) locally, i.e. $0 < V < \bar{V}$ and $x \in \varepsilon(\bar{V})$ for some given $\bar{V} \in \mathbb{R}_+$. The level set $\varepsilon(\bar{V})$ specifies the finite-time attraction domain $\mathcal{M}$ in this case (see, Definition 1).

# IV. IMPLICIT RELAY FEEDBACK LAW

## A. Block Decomposition

Let us initially decompose the original multi-input system (1) to a block form [34]. In order to make the paper self-contained the block decomposition procedure studied in [35], [30] is very briefly discussed in Appendix. It constructs the non-singular coordinate transformation

$$ s = \Theta x \quad (7) $$

reducing the original system (1) to the block form

$$ \dot{s}(t) = \tilde{A}s(t) + \tilde{B}u(t) + \tilde{d}(t, s(t), u(t)), \quad (8) $$

where $\tilde{d}(t, s, u) = \Theta d(t, \Theta^{-1}s, u) + K_{lin}s$ and $K_{lin} \in \mathbb{R}^{n \times n}$,

$$ \tilde{A} = \begin{pmatrix} 0 & A_{12} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & \dots & A_{k-1,k} \\ 0 & \dots & \dots & 0 \end{pmatrix}, \quad \tilde{B} = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ A_{kk+1} \end{pmatrix} \in \mathbb{R}^{n \times n_k}, \quad (9) $$

the matrices $A_{i i+1} \in \mathbb{R}^{n_i \times n_{i+1}}$ are of full row rank, $j = 1, 2, \dots, k$, $n_1 + \dots + n_k = n$, $n_k = m$ and $A_{k k+1} \in \mathbb{R}^{m \times m}$ is a nonsingular matrix, where $k$ is the number of blocks.

The function $\tilde{d}$ may be discontinuous. Following Filippov regularization procedure [29] we construct the compact and convex-valued upper semi-continuous map $\Delta: \mathbb{R}^{n+1} \to \mathbb{R}^n$:

$$ \Delta(t, s) = \bigcap_{\delta > 0} \bigcap_{\mu(N)=0} \mathrm{co} \left( \tilde{d}(t, s + B(\delta) \setminus N, \mathrm{co}(U)) \right), \quad (10) $$

which collects all possible perturbations of the system (1).

## B. Relay Feedback Design

Let us introduce the ILF function [30] of the form

$$ Q(V, s) = s^T D_r(V^{-1}) P D_r(V^{-1}) s - 1, \quad (11) $$

where $s = (s_1, \dots, s_k)^T$, $s_i \in \mathbb{R}^{n_i}$, $V \in \mathbb{R}_+$, $D_r(\lambda)$ is the so-called homogeneous dilation matrix [30]

$$ D_r(\lambda) = \begin{pmatrix} \lambda^{r_1} I_{n_1} & 0 & \dots & 0 \\ 0 & \lambda^{r_2} I_{n_2} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & 0 & \lambda^{r_k} I_{n_k} \end{pmatrix}, \quad (12) $$

where $\lambda \in \mathbb{R}_+$ is a non-negative scaling parameter, $r_i = 1 + (k-i)\mu$, $i = 1, 2, \dots, k$ are scaling weights with $0 < \mu \le 1$, and $P \in \mathbb{R}^{n \times n}$, $P > 0$. Denote $H_\mu := \mathrm{diag}\{r_i I_{n_i}\}_{i=1}^k$.

**Theorem 4:** Let $\mu \in (0,1)$, $\alpha > 0$, $X \in \mathbb{R}^{n \times n}$ and $Y \in \mathbb{R}^{n_k \times n_k}$ satisfy the system of matrix inequalities:

$$ \begin{gathered} \tilde{A}X + X\tilde{A}^T + \tilde{B}Y + Y^T\tilde{B}^T + \alpha(H_\mu X + XH_\mu) + R \le 0, \quad (13) \\ XH_\mu + H_\mu X > 0, \quad X > 0, \end{gathered} $$

then there exists a function $V: \mathbb{R}^n \to \mathbb{R}_+$ implicitly defined by the equation $Q(V, s)=0$ with $Q$ is given by (11) for $P=\bar{X}^{-1}$. Let $\beta \in (0, \alpha)$, $V_{\max} \in \mathbb{R}_+ \cup \{+\infty\}$ and the inequality

$$ \delta^T D_r(V^{-1}) R^{-1} D_r(V^{-1}) \delta \le \beta V^{-2\mu} s^T D_r(V^{-1}) (H_\mu P + P H_\mu) D_r(V^{-1}) s, \quad (14) $$

holds for $\delta \in \Delta(t, s)$, $V \in (0, V_{\max})$, $s \in \mathbb{R}^n: Q(V, s) = 0$ and for almost all $t \in \mathbb{R}_+$, then the control (4) with

$$ \Gamma^T(t, x) = x^T \Theta^T D_r \left( \frac{1}{V(\Theta x)} \right) P D_r \left( \frac{1}{V(\Theta x)} \right) \tilde{B}, \quad (15) $$

locally stabilizes the origin of the system (1) in a finite time and the settling-time function $T$ can be estimated as follows

$$ T(x_0) \le \frac{V_0^\mu}{(\alpha - \beta)\mu}, \quad \forall x_0 \in \mathbb{R}^n: \Theta x_0 \in \varepsilon(\bar{V}), \quad (16) $$

where $V_0 \in \mathbb{R}_+$: $Q(V_0, \Theta x_0) = 0$ and $\varepsilon(\bar{V})$ is the finite-time attraction domain (6) with a positive $\bar{V} \in \mathbb{R}_+$: $\bar{V} \le V_{\max}$.

All proofs are given in Appendix, where the estimate of the finite-time attraction domain is also provided.

The system of matrix inequalities (13) can be easily solved using LMI toolbox of MATLAB or, for example, SeDuMi solver. The solution of (13) also can be constructed analytically using the proof (see, [30]) of the next proposition.

**Proposition 5 ([30]):** If the pair $(A, B)$ is controllable then the system of matrix inequalities (13) is feasible for any $\mu \in \mathbb{R}_+$.
Obviously, if $\tilde{d} \equiv 0$ then $\Delta \equiv \{0\}$ and (14) holds for any $R$. The matrix $R$ is introduced in (13) is order to characterize disturbance functions $d$, which do not affect the finite-time stability property of the closed-loop system. The formula (14) gives the implicit restrictions to the system disturbances. In order to provide the explicit ones let us introduce the matrices $E_i$, $i = 1, 2, \dots, k$ by the following formula

$$E_i = \begin{pmatrix} 0_{n_1} & \dots & 0_{n_1 \times n_i} & \dots & 0_{n_1 \times n_k} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ 0_{n_i \times n_1} & \dots & I_{n_i} & \dots & 0_{n_i \times n_k} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ 0_{n_k \times n_1} & \dots & 0_{n_k \times n_i} & \dots & 0_{n_k} \end{pmatrix}. \quad (17)$$

Note that if $E_i \tilde{d} \neq 0$ just for some $i = i_1, i_2, \dots, i_p$ then Theorem 4 stays true even when the term $R$ in the LMI (13) is replaced with $(E_{i_1}+E_{i_2}+\dots+E_{i_p})R(E_{i_1}+E_{i_2}+\dots+E_{i_p})$.

**Proposition 6:** Let $X \in \mathbb{R}^{n \times n}$ be a solution of the LMI system (13) with $R = \text{diag}\{\beta_1 I_{n_1}, \beta_2 I_{n_2}, \dots, \beta_k I_{n_k}\}$, $\beta_i \in \mathbb{R}_+: \beta = \beta_1 + \dots + \beta_k < \alpha$ and $P = X^{-1}$. If

$$\delta^T E_i \delta \le \beta_i^2 \gamma \begin{cases} (\lambda_{\min}(P)s^T s)^{1+(k-i-1)\mu} & \text{if } s^T P s \le 1, \\ (\lambda_{\min}(P)s^T s)^{\frac{1+(k-i-1)\mu}{1+(k-i-1)\mu}} & \text{if } s^T P s > 1, \end{cases} \quad (18)$$

for $\delta \in \Delta(t, s)$, $\gamma = \lambda_{\min}(P^{-\frac{1}{2}}(PH_\mu + H_\mu P)P^{-\frac{1}{2}})$ and $i = 1, 2, \dots, k$ then the inequality (14) holds for any $V \in \mathbb{R}_+: Q(V, s) = 0$.

In the view of Corollary 3 the inequality $V \le 1$ is equivalent to $s^T P s \le 1$, so the restriction (18) can be used locally for $V_{\max} = 1$ (see, (14)).

The class of the disturbances which can be rejected by the proposed control law is essentially depended on the parameter $\mu$. If $\mu = 1$ then, obviously, condition (18) implies $\delta^T E_k \delta \le \beta_k^2 \gamma$ = const for any $s \in \mathbb{R}^n$, i.e. the constructed relay feedback rejects the so-called bounded matched disturbances [4]. The mismatched disturbances $E_i \delta$, $i = 1, 2, \dots, k-1$ should vanish as $s \to 0$ and admit polynomial estimates close to the origin (see (18)).

**Proposition 7:** Theorem 4 stays true for $\mu = 1$ if the matrix inequality (13) is supplied with the additional condition

$$\begin{pmatrix} X & Y^T \\ Y & u_0^2 I_m \end{pmatrix} \ge 0, \quad (19)$$

where $0 < u_0 \le u_{\max} = \sup_{r \in \mathbb{R}_+: B(r) \subseteq \text{co}\{U\}} r$. Moreover, the closed-loop system (1), (4), (15) is globally finite-time stable if $V_{\max} = +\infty$.

This proposition provides sufficient condition for global robust finite-time stability of the closed-loop relay system (1), (4), (15). In particular, it is globally finite-time stable if (13) and (19) holds and $\tilde{d}$ satisfies (18) for $\mu = 1$.

## V. PRACTICAL IMPLEMENTATION

In order to realize the control algorithm (4), (15) in practice we need to know $V$. In some cases the function $V$ can be calculated analytically [36]. The function $V$ can also be approximated numerically on a grid constructed in the finite-time attraction domain $\varepsilon(\bar{V})$. Finally, the relay control

<figure><img src="image_20.png" /><figcaption>Fig. 1. Scheme of Buck converter</figcaption></figure>

law (4) can be applied by means of on-line estimation of $V$. The following corollary may be utilized for this purpose.

**Corollary 8:** If 1) the conditions of Theorem 4 hold; 2) $\{t_i\}_{i=0}^{+\infty}$ is an arbitrary sequence of time instances such that $0 = t_0 < t_1 < t_2 < \dots$ and $\lim_{i \to +\infty} t_i = +\infty$; 3) the relay control $u_r$ has the form (4) with the sampled computation of the switching function $\Gamma(t, x) = \tilde{\Gamma}_i(\Theta x)$ for $t \in [t_i, t_{i+1})$,

$$\tilde{\Gamma}_i^T(s) = s^T D_r (V_i^{-1}) P D_r (V_i^{-1}) \tilde{B}, \quad Q(V_i, s(t_i)) = 0.$$

Then the system (1), (4) is globally asymptotically stable.

The corollary shows that asymptotic stability of the closed loop system is maintained even if the switching function $\Gamma$ can be updated only at some isolated instants of time. The estimation of the switching parameter $V_i$ can be obtained using the simple bisection method. We refer the reader to [25], [30] for more details about practical implementation of the control algorithms based on implicit Lyapunov function.

## VI. EXAMPLE: CONTROL OF BUCK CONVERTER

Let us consider the model of Buck converter (Fig. 1) studied in [10]: $x = (v_C \ i_L)^T$, $x \in \mathbb{R}^2$, $U = \{0, v_{in}\}$,

$$A = \begin{pmatrix} -1/(R_o C_0) & 1/C_o \\ -1/L & -R_c/L \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 1/L \end{pmatrix},$$

where $i_L$ - inductor current, $v_C$ - capacitor voltage, $v_{in} = 100$ V, $R = 2$ $\Omega$, $L = 500$ $\mu H$, $C_o = 470\mu F$ and $R_o = 50$ $\Omega$. In the active mode, the switches $S_1$ and $S_2$ operate as follows: if $S_1$ is opened then $S_2$ is closed and vise versa.

Let us apply the coordinate transformation:

$$s = \Theta(x - x^*), \quad \Theta = \begin{pmatrix} 1 & 0 \\ -1/R_o & 1 \end{pmatrix}, \quad x^* = \begin{pmatrix} v_{out} \\ v_{out}/R_o \end{pmatrix},$$

where $v_{out} \in (0, v_{in}/(1 + R_c/R_o))$ is the required output voltage. We derive the following relay control system

$$\dot{s}(t) = \begin{pmatrix} 0 & 1/C_o \\ 0 & 0 \end{pmatrix} s + B\hat{u} + \tilde{d}(s),$$

where $\hat{u} \in \{v_1, v_2\}$,

$$\begin{aligned} v_1 &= -v_{out}(1 + R_c/R_o), \\ v_2 &= v_{in} - v_{out}(1 + R_c/R_o) \end{aligned}$$

and $\tilde{d}(s) = -(1/L \quad R_c/L - 1/(R_o C_o))s$. The obtained system satisfies the assumption (3) and all conditions of Theorem 4. The system of matrix inequalities (13), (19) gives

$$P = \begin{pmatrix} 0.1768 & 0.1172 \\ 0.1172 & 0.1059 \end{pmatrix}, \quad K = \begin{pmatrix} -0.0960 & -0.0707 \end{pmatrix}$$
<figure><img src="image_21.png" /><figcaption>Fig. 2. Evolution of the system state</figcaption></figure>

for the matrix $R = 10^5 I_2 \in \mathbb{R}^{2 \times 2}$, for parameters $\mu = \alpha = 1$ and $u_0 = \min(|v_1|, v_2)$, $v_{out} = 60$. The switching surface $\Gamma$ is defined by (15) and the dilation matrix $D_r$ has the weights $r_1 = 2$ and $r_2 = 1$.

The Fig. 2 shows the simulation results of the control application for $x(0) = (0, 0)^T$ and the sampling period $10^{-5}$. During the simulations the value of the implicit Lyapunov function has been calculated on-line using bisection method.

It was expectable that relay finite-time stabilizing algorithm should be a sort of (high order) sliding mode control. This fact is indirectly confirmed by simulations, since the chattering phenomenon appear. Fortunately, the oscillations are observed only for the current $i_L$. They are not destructive, since $i_K$ is the fast variable of the system.

## VII. CONCLUSION

The paper presents relay feedback control algorithm for stabilization of linear multi-input system provided non-asymptotic transitions. The control design procedure combines the ILF method and convex embedding technique. This approach allows us to develop a simple procedure for implicit switching surface design using LMIs. The algorithm of practical implementation of the obtained implicit relay feedback is presented and tested on the numerical example. Its sampled-time analysis is considered as the subject for future research.

## VIII. APPENDIX

### A. Transformation to Block Form

Let us denote by $\text{rown}(W)$ the number of rows of a matrix $W$ and by $\text{null}(W)$ the matrix that has the columns defining an orthonormal basis of the null space of a matrix $W$.

Let the orthogonal matrices $T_i$ be defined by the following simple algorithm:

**Initialization:** $A_0 = A$, $B_0 = B$, $T_0 = I_n$, $k = 0$.

**Loop:** While $\text{rank}(B_k) < \text{rown}(A_k)$ do

$$ 
\begin{aligned}
A_{k+1} &= B_k^{\perp} A_k (B_k^{\perp})^T, & B_{k+1} &= B_k^{\perp} A_k \hat{B}_k, \\
T_{k+1} &= \begin{pmatrix} B_k^{\perp} \\ \hat{B}_k \end{pmatrix}, & k &= k+1,
\end{aligned}
$$

$$ \text{where } B_k^{\perp} = (\text{null}(B_k^T))^T, \quad \hat{B}_k = (\text{null}(B_k^{\perp}))^T. $$

It was proven (see, e.g. [35]) that

$$ GAG^T = \begin{pmatrix} A_{11} & A_{12} & 0 & \dots & 0 \\ A_{21} & A_{22} & A_{23} & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ A_{k-1 1} & A_{k-1 2} & \dots & A_{k-1 k-1} & A_{k-1 k} \\ A_{k1} & A_{k2} & \dots & A_{kk-1} & A_{kk} \end{pmatrix}, \quad (20) $$

$$ \tilde{B} = GB = \begin{pmatrix} 0 & 0 & \dots & 0 & A_{k(k+1)}^T \end{pmatrix}^T, $$

$$ G = \begin{pmatrix} T_k & 0 \\ 0 & I_{w_k} \end{pmatrix} \begin{pmatrix} T_{k-1} & 0 \\ 0 & I_{w_{k-1}} \end{pmatrix} \dots \begin{pmatrix} T_2 & 0 \\ 0 & I_{w_2} \end{pmatrix} T_1, \quad (21) $$

$$ \text{where } w_i := n - \text{rown}(T_i), \quad A_{k+1} = \hat{B}_0 B_0, \quad A_{ij} \in \mathbb{R}^{n_i \times n_j}, \\ n_i := \text{rank}(B_{k-i}), \quad i, j = 1, 2, \dots, k \text{ and } \text{rank}(A_{i \ i+1}) = n_i. $$

Recall that the $B$ has full column rank ($\text{rank}(B) = m$). Consequently, $A_{k+1}$ is square and nonsingular. Since $\text{rank}(A_{i \ i+1}) = n_i = \text{rown}(A_{i \ i+1})$ then $A_{i \ i+1} A_{i \ i+1}^T$ is invertible and $A_{i \ i+1}^T = A_{i \ i+1}^T (A_{i \ i+1} A_{i \ i+1}^T)^{-1}$ is the right inverse matrix of $A_{i \ i+1}$. Introduce the linear coordinate transformation $s = \Phi y$, $s = (s_1, \dots, s_k)^T$, $s_i \in \mathbb{R}^{n_i}$, $y = (y_1, \dots, y_k)^T$, $y_i \in \mathbb{R}^{n_i}$ by the formulas:

$$ 
\begin{aligned}
s_i &= y_i + \varphi_i, & i &= 1, 2, \dots, k, & \varphi_1 &= 0, \\
\varphi_{i+1} &= A_{i \ i+1}^+ \left( \sum_{j=1}^{i} A_{ij} y_j + \sum_{r=1}^{i} \frac{\partial \varphi_i}{\partial y_r} \sum_{j=1}^{r+1} A_{rj} y_j \right). & & &
\end{aligned}
\quad (22) $$

The presented coordinate transformation is linear and nonsingular. The inverse transformation $y = \Phi^{-1}s$ is defined as follows: $y_i = s_i + \psi_i$, $i = 1, 2, \dots, k$, $\psi_1 = 0$, $\psi_{i+1} = A_{i \ i+1}^+ (\sum_{k=1}^i \frac{\partial \psi_i}{\partial s_k} A_{i \ i+1} s_{k+1} - \sum_{j=1}^i A_{ij} (s_j + \psi_j))$.

Applying the transformation $s = \Theta x$ with $\Theta = \Phi G$ to (1)

$$ \text{we derive } \dot{s} = \begin{pmatrix} 0 & A_{12} & \dots & 0 \\ \dots & \dots & \dots & \dots \\ 0 & \dots & \dots & A_{k-1 k} \\ \tilde{A}_{k1} & \dots & \dots & \tilde{A}_{kk} \end{pmatrix} s + \tilde{B}u + \Theta d, \text{ that is} $$

$$ \text{equivalent to (8) with } K_{\text{lin}} = \tilde{B} A_{k \ k+1}^{-1} (\tilde{A}_{k1} \dots \tilde{A}_{kk}). $$

### B. Proof of Theorem 4

The proof is divided into several steps. First, based on implicit Lyapunov function method we design a continuous feedback ensuring global finite-time stabilization. Next, we provide an estimation of the domain in which the continuous feedback may be re-configured as a relay using convex embedding procedure. We continue with the Filippov regularization of the closed-loop relay system. At last, we show the local finite-time stability of the closed-loop system when Filippov solutions are being considered. The construction of the switching surface $\Gamma$ is the most specific part of proof, since, usually [20], it is depended on the Lyapunov function, which does not have an explicit representation in our case.

1) The continuous feedback law providing finite-time stabilization: In the proof we use a result presented in [30]. Let $\mu \in (0, 1)$, $\alpha > 0$, $X \in \mathbb{R}^{n \times n}$ and $Y \in \mathbb{R}^{n_k \times n}$ satisfy the system of matrix inequalities (13). Let $\Theta \in \mathbb{R}^{n \times n}$ be given by (7), $K := YX^{-1}$ and the positive definite function
$V: \mathbb{R}^n \to \mathbb{R}$ satisfying $Q(V(s), s) = 0$ with $Q$ defined by (11) and $P := X^{-1}$. It is known [30] that the function $V$ is smooth outside the origin and the continuous feedback law

$$u_{con}(x(t)) \stackrel{\text{def}}{=} \tilde{u}(\Theta x(t)), \quad (23)$$

$$\tilde{u}(s) = V^{1-\mu}(s)KD_r(V^{-1}(s))s, \quad (24)$$

ensures the inequality

$$\frac{\partial V}{\partial s} (\tilde{A}s + \tilde{B}u_{con}(s) + \delta) \le -(\alpha - \beta)V^{1-\mu}(s), \quad (25)$$

for any $\delta$ satisfying (14). Hence, for $V_{\max} = \infty$ the origin of the closed-loop system (1) is globally finite-time stable and the settling-time function is bounded as follows

$$T(x_0) = \frac{V_0^{\mu}}{\mu(\alpha - \beta)}, \quad (26)$$

where $V_0 \in \mathbb{R}_+$: $Q(V_0, \Phi G x_0) = 0$. The function $V$ is the Lyapunov function of the closed-loop system [30], which is defined implicitly by means of equation $Q(V, s) = 0$. If $V_{\max} < +\infty$ then the given feedback law guarantees local finite-time stabilization for $x_0 \in \varepsilon(V_{\max})$, where $\varepsilon(V_{\max})$ is the level set of the Lyapunov function $V$ (see, Corollary 3).

2) Estimation of attraction domain: As follows we show how the continuous feedback (23), (24) can be re-configured as a relay feedback using convex optimization arguments. The representation of the continuous feedback (23), (24) as a convex combination of the relay vectors $v_i, i = 1, \dots, N$,

$$u_{con}(s) = \sum_{i=1}^{N} \alpha_i(s)v_i, \quad \alpha_i(s) \ge 0, \quad \sum_{i=1}^{N} \alpha_i(s) = 1 \quad (27)$$

can be applied locally for this purpose. In contrast to the feedback law (23), (24) the relay control (4) is always bounded. Therefore, the representation (27) holds locally in the domain defined as follows

$$C_U = \{z \in \mathbb{R}^n : u_{con}(z) \in \text{co}(U)\}. \quad (28)$$

Since $u_{con}$ is the continuous function [30] and $u_{con}(0) = 0$, then $C_U$ is a nonempty compact set. Moreover, due to (3) the inclusion $0 \in \text{int}\{C_U\}$ holds. Let $\bar{V} \in \mathbb{R}_+$ be defined as

$$\bar{V} = \min \left\{ V_{\max}, \sup_{V \in \mathbb{R}_+ : \varepsilon(V) \subset C_U} V \right\},$$

where $\bar{V} > 0$ due to $0 \in \text{int}\{C_U\}$. Below we show that $\varepsilon(\bar{V})$ is the finite-time attraction domain of the system (1),(4),(15).

3) Filippov regularization of the closed-loop relay system: Since the closed-loop system (1), (4), (15) has a discontinuous right-hand side, classical solutions may not exist. In order to describe the system behavior, we considered solution $x(t)$ in the sense of Filippov and construct the differential inclusion

$$\dot{x}(t) \in F(t, x(t)),$$

$$F(t, x) = \left\{ Ax + Bco \left( \underset{v \in U}{\operatorname{argmin}} \Gamma^T(t, x)v \right) + \Theta^{-1} \Delta(t, \Theta x) \right\}.$$

Obviously, the obtained inclusion contains all Filippov solutions of the closed-loop system (1), (4), (15). For its finite-time stability analysis Theorem 2 can be utilized.

4) Finite-time stability of the closed-loop relay system: Let us show that the function the implicit Lyapunov function (11) satisfying (25) is also Lyapunov function for the system (1) closed by the relay feedback (4), (15). It is sufficient to show that

$$\frac{\partial V}{\partial s}(\tilde{A}s + \delta) + \sup_{z \in \text{co}(\underset{v \in U}{\operatorname{argmin}} \tilde{\Gamma}^T(s)v)} \frac{\partial V}{\partial s} \tilde{B}z \le -(1 - \beta)V^{1-\mu}(s)$$

for all $\delta \in \Delta(t, s)$ and all $s \in \mathbb{R}^n \setminus \{0\} : s \in \varepsilon(\bar{V})$, where

$$\tilde{\Gamma}(s) = \left( s^T D_r(V^{-1}(s)) P D_r(V^{-1}(s)) \tilde{B} \right)^T. \quad (29)$$

Obviously, $\Gamma(t, x) = \tilde{\Gamma}(\Theta x)$ (see, the formula (15)). Recall that the implicit function theorem [33] gives $\frac{\partial V}{\partial s} = -[\frac{\partial Q}{\partial V}]^{-1} \frac{\partial Q}{\partial x}$. So, we derive the following representation:

$$\frac{\partial V}{\partial s} = 2V(s) \frac{s^T D_r(V^{-1}(s)) P D_r(V^{-1}(s))}{s^T D_r(V^{-1}(s))(H_\mu P + P H_\mu) D_r(V^{-1}(s))s},$$

where $H_\mu P + P H_\mu > 0$ (see, (13)). Hence, we conclude that the relation

$$\underset{v \in U}{\operatorname{argmin}} \tilde{\Gamma}^T(s)v = \underset{v \in U}{\operatorname{argmin}} \frac{\partial V}{\partial s} \tilde{B}v$$

holds for any $s \in \mathbb{R}^n \setminus \{0\}$. Moreover, the inequality $\frac{\partial V}{\partial s} \tilde{B}z \le \frac{\partial V}{\partial s} \tilde{B}v_i$, holds for any $v_i \in U$ and any $z \in \underset{v \in U}{\operatorname{argmin}} \tilde{\Gamma}^T(s)v$. Using the standard convexity arguments, it can be easily shown that this inequality also holds for any $z \in \text{co}(\underset{v \in U}{\operatorname{argmin}} \tilde{\Gamma}^T(s)v)$. Therefore, we derive

$$\frac{\partial V}{\partial s} \tilde{B}z = \sum_{i=1}^{N} \tilde{\alpha}_i \frac{\partial V}{\partial s} \tilde{B}z \le \sum_{i=1}^{N} \tilde{\alpha}_i \frac{\partial V}{\partial s} \tilde{B}v_i$$

$$\forall \tilde{\alpha}_i \ge 0 : \sum_{i=1}^{N} \tilde{\alpha}_i = 1 \text{ and } \forall z \in \text{co} \left( \underset{v \in U}{\operatorname{argmin}} \tilde{\Gamma}^T(s)v \right).$$

Finally, taking into account the representation (27) of the control law (23), (24) and the equality (25) we derive with $\tilde{\alpha}_i = \alpha_i(s), i = 1, \dots, N$, the representation $\frac{\partial V}{\partial s}(\tilde{A}s + \delta) + \sum_{i=1}^{N} \alpha_i(s) \frac{\partial V}{\partial s} \tilde{B}v_i \le -(\alpha - \beta)V^{1-\mu}$, $\sum_{i=1}^{N} \alpha_i(s) = 1, \alpha_i(s) \ge 0$ for any $\delta \in \Delta(t, s)$ and any $s \in \varepsilon(\bar{V})$. The obtained representation completes the proof.

### C. Proof of Corollary 8

In [30] it was proven that for any fixed $V_i \in \mathbb{R}_+$ the control (23), (24) with $V = V_i$ is the linear stabilizing feedback for the system (8) and the corresponding quadratic Lyapunov function $V_i$ is defined as follows $V_i(s) = s^T P_i s$, where $P_i := D_r(V_i^{-1}) P D_r(V_i^{-1}) > 0$. Hence, the ellipsoid $\varepsilon(V_i)$ defined in Corollary 3 is strictly positively invariant set.

Corollary 9 ([30]): If 1) the conditions (13) of Theorem 4 hold; 2) $\{t_i\}_{i=0}^{+\infty}$ is an arbitrary sequence of time instances such that $0 = t_0 < t_1 < t_2 < \dots$ and $\lim_{i \to +\infty} t_i = +\infty$; 3) the control has the form (23) with $\tilde{u}(s) = \tilde{u}_{V_i}(s)$ on each time interval $[t_i, t_{i+1})$, where $\tilde{u}_V(s)$ is defined by (24) and $V_i \in \mathbb{R}_+: Q(V_i, s(t_i)) = 0$. Then the closed-loop system (1) is globally asymptotically stable.
The convex embedding technique for design relay feedback law using existing linear stabilizing feedback is studied in [20]. Applying this technique to the switching linear feedback described in this corollary (see the condition 3)) we finish the proof of Corollary 8. Note that the ellipsoids $\epsilon(V_i)$ are also strictly positively invariant for the relay feedback.

### D. Proof of Proposition 6

For proof of this proposition we use the ideas presented in [30]. Obviously, $\gamma I_n \le P^{1/2} H_\mu P^{-1/2} + P^{-1/2} H_\mu P^{1/2}$ or equivalently $\gamma P \le PH_\mu + H_\mu P$. On the one hand, we have $\gamma = \gamma s^T D_r(V^{-1}) P D_r(V^{-1}) s \le s^T D_r(V^{-1}) (PH_\mu + H_\mu P) D_r(V^{-1}) s$, where $(V, s) \in \mathbb{R}_+ \times \mathbb{R}^n$ such that $Q(V, s) = 0$. On the other hand,

$$1 = s^T D_r(V^{-1}) P D_r(V^{-1}) s \ge \begin{cases} \lambda_{\min}(P) V^{-2} s^T s & \text{for } s^T P s \le 1, \\ \lambda_{\min}(P) V^{-2-2(k-1)\mu} s^T s & \text{for } s^T P s > 1. \end{cases}$$

Hence, we derive $\delta^T D_r(V^{-1}) R^{-1} D_r(V^{-1}) \delta = \sum_{i=1}^k \beta_i^{-1} V^{-2-2(k-i)\mu} \delta^T E_i \delta \le \frac{\gamma}{V^{2\mu}} (\beta_1 + \dots + \beta_k) \le \beta V^{-2\mu} s^T D_r(V^{-1}) (PH_\mu + H_\mu P) D_r(V^{-1}) s$, i.e. the inequality (14) holds.

### E. Proof of Proposition 7

If $\mu = 1$ then the control function $\tilde{u}_v$ is continuous outside the origin and bounded for all $x \in \mathbb{R}^n$. Indeed, since $s^T D_r(V^{-1}) P D_r(V^{-1}) s = 1 \to \|D_r(V^{-1}) s\|^2 \le \frac{1}{\lambda_{\min}(P)}$ and $\tilde{u}^2(s) \le s D_r(V^{-1}) K^T K D_r(V^{-1}) s \le \frac{\lambda_{\max}(K^T K)}{\lambda_{\min}(P)}$.

Hence, in order to restrict globally the control magnitude by $\|\tilde{u}(s)\| \le u_0$ the inequality $K^T K \le u_0^2 P$ should be added to (13). Therefore, the inequality (19) guarantees that $C_U$ is nonempty and all steps of the proof of Theorem 4 can be repeated for $\mu = 1$. Moreover, $\{s \in \mathbb{R}^n : \|u_{con}(s)\| \le u_{max}\} \subset C_U$, so $\bar{V} = \min\{V_{max}, +\infty\} = V_{max}$.

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