Journal of Robotics, Networking and Artificial Life
Vol. 5(4); March (2019), pp. 241–244
DOI: https://doi.org/10.2991/jrnal.k.190220.007; ISSN 2405-9021 print; 2352-6386 online
https://www.atlantis-press.com/journals/jrnal

Research Article

Time-Varying Lyapunov Function for Mechanical Systems
Bin Zhang1,*, Yingmin Jia2
School of Automation, Beijing University of Posts and Telecommunications, Haidian District, Beijing 1008760, China
The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering,Beihang University (BUAA), Haidian District, Beijing 100191, China12ARTICLE INFOABSTRACT

Article History

In this paper, a general method for constructing time-varying Lyapunov functions is provided for mechanical systems. A newclass of time-varying Lyapunov functions is provided and sufficient conditions of uniform asymptotical stability are established.
Different from the existing results on this subject, we remove the periodic restrictions and persistency of excitation restrictions
in our work. Moreover, the results are extended into the robustness cases with unmodeled dynamics. Based on the Lyapunov
methods, it is shown that uniform asymptotical stability can be obtained by using the control input provided in our work.

Received 28 October 2018
Accepted 21 November 2018
Keywords
Mechanical systems
time-varying systems
strict Lyapunov functionsubsystems© 2019 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
In this paper, we investigate the stability problem of time-varying
mechanical systems. It is well known that Lyapunov functions
provide fundamental tools for stability analysis and controller
design. In terms of Lyapunov functions, two research directions
are widely investigated, that is, converse Lyapunov theory andconstructions of Lyapunov functions. Converse Lyapunov theory
guarantees the existence of smooth Lyapunov functions for somesystems. However, construction methods give specific structuresof Lyapunov function candidates. In this paper, we deal with thesecond research topic: construction methods. We aim to provide ageneric approach for finding strict Lyapunov functions of nonlinear time-varying mechanical systems described by Euler–Lagrangeequation. Several approaches for construction of strict Lyapunov
functions have been proposed under various frameworks and control objectives [1,2].
In Han et al. [3], a new method for estimation the domain of
attraction was shown by using non-strict Lyapunov functions.
In Bretas and Alberto [4], an extension of invariance principlewith non-strict Lyapunov functions was provided for powersystems with transmission losses. In Hong et al. [5], strict
Lyapunov functions were established for finite-time control
problem of robot manipulators. In Zhang and Jia [6], two kindsof weak-invariance principles for nonlinear switched systems
were developed and accurate convergent regions were obtained.
The readers can refer to Malisoff and Mazenc [7], Mazenc [8],and Giesl and Hafstein [9] for more detailed discussions on this
topic and a list of related references.
Corresponding author. Email. zb362301@126.com*As we all know, some approaches have been well established tobuild explicit strict Lyapunov functions for time-invariant systems,
such as variable gradient method, Krasovskii method and so on.
But construction of strict Lyapunov functions for time-varying
systems is a challenging problem and there are no general methods.
Motivated by the above considerations, we provide a general
method for constructing time-varying Lyapunov functions ofmechanical systems. We first consider a mechanical system with
time-varying frictions. With the control input, the closed-loopsystem is obtained. Then, we divide the closed-loop system into twosubsystems. For subsystem one, it is shown that the origin is uniformly asymptotically stable and a class of time-varying Lyapunov
function V(t, x) is established. For the subsystem two, it is shownthat the derivative of V(t, x) is bounded. Based on the derivativesof the time-varying Lyapunov function along the two different subsystems, we provide the time-varying Lyapunov function for thewhole system. It is shown uniform asymptotical stability can beobtained with the control input provided in this paper.

2. PROBLEM STATEMENT
Consider the following mechanical system with time-varying
friction [Equation (1)]
Mq + Cq + (Fc (t ) + Fs (t )e - rm (q ) )sat(q ) + k(t )q = t(1)where q(t ), q (t ), q(t ) ∈ Rn represent the position, velocity, andacceleration, respectively; M ∈ Rn×n is the positive definite inertia of the system; r is a positive constant; m(·) is a positive definite
function related to the Stribeck effect; k(t) denotes a time-varying

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spring stiffness. For system (1), it is assumed that the following
properties hold.
Property 1. The matrix M is positive definite. Moreover, we define
lmax as the maximum eigenvalue of M and lmin as the minimum
eigenvalue of M.

x 1 = x2x 2 = −(k(t ) + k0 )M −1x1 − a x2
− M −1 (Fc (t ) + Fs (t )e − rm ( x2 ) )sat(x2 )We define [Equation (10)]Property 2. The derivative of k(t) is bounded such that [Equation (2)]| k(t ) |£ k "t ³ 0(9)

 x f (t , x ) =   2  f (t , x )

(2)

(10)

where function f (t , x ) = −(k(t ) + k0 )M −1x1 − a x2 − M −1 (Fc (t ) + Fs (t )e − rm ( x2 ) )sawhere k is a positive constant.

x2
− rm ( x2 )−1−1
Before presenting thef mainsomelemmas first.
(t , x ) =results,−(k(t )we+ kgive)sat(x2 ). We also define f1 (t , x ) = −10 )M x1 − a x 2 − M (Fc (t ) + Fs (t )e−(k(t)+k)
M−xax
012
Lemma 1. Zhang et al. [10] Let a,b,p,q be positive real numbers. If0and f 2 (t , x ) = 
1 1 , then [Equation (3)]− rm ( x2 )−1
 . Then, we can see−
M(F(t)+F(t)e)
xsat()+ =1
cs2 
p qthat f(t, x) = f1(t, x) + f2(t, x). In this subsection, we consider two suba p bq
systems x = f1 (t , x ) and x = f 2 (t , x ), respectively.
ab ≤+(3)p q
Inspired by the results in Malisoff and Mazenc [7], we consider thefollowing Lyapunov function [Equation (11)]Lemma 2. Zhang et al. [10] Let ai > 0, bi > 0, i = 1,2, …, n, p > 1, and1 1
q > 1. If + = 1, then [Equation (4)]V (t , x ) = A((k(t ) + k0 )x1T x1 + x2T Mx2 ) + x1T x2
(11)
p q
11

æ n
ö p æ n öqaibi £ ç å aip ÷ ç å biq ÷åi =1
è i =1 ø è i =1 øn(4)Lemma 3. Zhang et al. [10] For xi ∈ R, i = 1,2, …, n, and 0 < m < 1,then the following inequality holds [Equation (5)]:mmm| x1 | + ⋅⋅⋅ + | xn | ≤ (| x1 | + ⋅⋅⋅ + | xn |)(5)

Lemma 4. For xi ∈ R, i = 1,2, …, n, and 0 < m < 1, then the following inequality holds [Equation (6)]:| x1 |m + ⋅⋅⋅ + | xn |m ≤ n1− m (| x1 | + ⋅⋅⋅ + | xn |)m

(6)

11.
Proof. Let ai = |xi|m, bi = 1, i = 1, 2, …, n, p = , and q =1− mm
Then based on Lemma 2, we obtain Equation (7)| x1 |m + ⋅⋅⋅ + | xn |mm1 n  n 1 ≤  ∑ (| xi |m )m   ∑ 11− m  i =1  i =1

= n1− m (| x1 | + ⋅⋅⋅ + | xn |)mwhich completes the proof.

1− mwhere A > 0 is a positive constant. In this paper, we choose 11 .
A > max , lmin k1 + k0 
It can be verified directly that [Equation (12)]V (t , x ) = A((k(t ) + k0 )x1T x1 + x2T Mx2 ) + x1T x2
1> - x1T x1 + x2T x2
2()

(12)

The derivative of V(t, x) along subsystem x = f1 (t , x ) shows[Equation (13)]:∂V ∂VV (t , x ) =+f1 (t , x ) ≤ −l1x1T x1 − l2 x2T x2∂t∂x(13)

11
 and l = 3 a − 1.
− a − Ak
2
2lmax 22
æ1ö l - k
Therefore, we can see that if we choose a > and k0 > ç a + Ak
÷ max 1
3
è2
ø
(7)
1
æ ö l - k , subsystem x = f (t , x ) is uniformly asymptotically stable.
k0 > ç a + Ak
1
÷ max 1
è2øSimilarly, the derivative of V(t, x) along subsystem x = f 2 (t , x )shows [Equation (14)]:
where we have l1 = (k1 + k0 )1

∂V
1f 2 (t , x ) ≤ 2 n
Fc (t ) + Fs (t )V 2lmin∂x

3. SUBSYSTEMS(14)1F
In the following, we define time-varying parameter w(t ) = nl
For the uniform asymptotic stability problem of time-varyingmin
1w(t ) = n
F (t ) + Fs (t ) .
mechanical system (1), we propose the following control inputlmin c[Equation (8)]t = −a Mq + Cq − k0q(8)where a and k0 are control parameters. Let q = x1 and q = x2. Thenthe closed-loop system (1) and (8) can be written in a compactform as Equation (9)4. MAIN RESULTS
From (13) and (14), we can see that V(t, x) is not a Lyapunov function for system (1), because it is not easy to verify the negativity of

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the derivative of V(t, x) along system x = f (t , x ), which is shown as
following Equation (15)¶V ¶V
V (t , x ) f (t , x ) =( f1 (t , x ) + f 2 (t , x ))+¶t ¶x(15)Therefore, motivated by the results in Malisoff and Mazenc [7], we
provide the following time-varying Lyapunov function [Equation (16)]1ds(1 + s)0.5
t æ 1 t +Tö+f ò ç ò w(v )dv - w(t ) ÷ dt ln(1 + V )t0 T tèø

V # (t , x ) = òV0(16)where T, f are positive constants. It can be verified directly that ifk0 and a are chosen large enough, then there exist T and f such that
[Equation (17)]¶V ¶VV # (t , x ) | f (t , x ) =+( f1 (t , x ) + f2 (t , x )) < 0¶t ¶x(17)

Therefore, we conclude with the following theorem.
Theorem 1. Consider time-varying system (1). If the control input
is chosen as (8) with control parameters a and k0 large enough,then the closed loop system is uniformly asymptotically stable.
Moreover, (16) is a strict Lyapunov function.

Figure 1 | The state trajectories of system.

functions has been established, in which time-varying terms havebeen included. By using the new Lyapunov functions, concise
criterion for uniform asymptotical stability have been provided.
A numerical example has been presented for illustration.

ACKNOWLEDGMENT

Proof. This result can be verified by taking the derivative of V#(t, x)
given in (16) directly. The detailed process is omitted here.

This work is supported by the National Natural Science Foundationof China (Grant No. 61603050).

Then, we can extend the result to the robustness case. We take thefollowing systems with unmodeled dynamics [Equation (18)]

REFERENCES(Mq + Cq + Fc (t ) + Fs (t ) e- rm ( q )) sat (q ) + k (t ) q + H (q, q ) = t (18)where H (q, q ) is the unmodeled dynamics. For system (18), we
assume that [Equation (19)]liml ®0

H (t , lq, lq )l=0(19)Without detailed proof, we provide the following theorem for therobustness case.
Theorem 2. Consider system (18) with unmodeled dynamics. We
assume that (19) is satisfied. If the conditions in Theorem 1 hold, thenwe have that the closed-loop system is uniformly asymptotically stable.

5. NUMERICAL EXAMPLE
We give a numerical example to demonstrate the effectiveness of theproposed theoretical results under different initial values. Two different initial values are chosen as x(0) = (0.5, 0.3) and x(0) = (–0.1, 0.05),
respectively. By using the control input (8), we can see that the state
trajectories converge to the origin asymptotically (Fig. 1). The simulation clearly illustrate the exactness of our results.

6. CONCLUSION
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Authors Introduction
Dr. Bin Zhang

Dr. Yingmin Jia
He received his PhD degrees incontrol theory and applications fromBeihang University, in 2016. Heis with the School of Automation,Beijing University of Posts andTelecommunication. His researchinterests include multi-agent systems.

He received the B.S. degree in control
theory from Shandong University,Jinan, China, in 1982, and the M.S.
and PhD degrees in control theory andapplications from Beihang University,Beijing, China, in 1990 and 1993,
respectively. In 1993, he joined theSeventh Research Division, Beihang
University, where he is currently aProfessor of automatic control.