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Volume 5, Issue 4, March 2019, Pages 241 - 244

Time-Varying Lyapunov Function for Mechanical Systems

Authors
Bin Zhang1, *, Yingmin Jia2
1School of Automation, Beijing University of Posts and Telecommunications, Haidian District, Beijing 1008760, China
2The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University (BUAA), Haidian District, Beijing 100191, China
*Corresponding author. Email. zb362301@126.com
Corresponding Author
Bin Zhang
Received 28 October 2018, Accepted 21 November 2018, Available Online 30 March 2019.
DOI
10.2991/jrnal.k.190220.007How to use a DOI?
Keywords
Mechanical systems; time-varying systems; strict Lyapunov function; subsystems
Abstract

In this paper, a general method for constructing time-varying Lyapunov functions is provided for mechanical systems. A new class 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.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
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 and constructions of Lyapunov functions. Converse Lyapunov theory guarantees the existence of smooth Lyapunov functions for some systems. However, construction methods give specific structures of Lyapunov function candidates. In this paper, we deal with the second research topic: construction methods. We aim to provide a generic approach for finding strict Lyapunov functions of nonlinear time-varying mechanical systems described by Euler–Lagrange equation. 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 principle with non-strict Lyapunov functions was provided for power systems 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 kinds of 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.

As we all know, some approaches have been well established to build 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 of mechanical systems. We first consider a mechanical system with time-varying frictions. With the control input, the closed-loop system is obtained. Then, we divide the closed-loop system into two subsystems. 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 shown that the derivative of V(t, x) is bounded. Based on the derivatives of the time-varying Lyapunov function along the two different subsystems, we provide the time-varying Lyapunov function for the whole system. It is shown uniform asymptotical stability can be obtained 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−ρμ(q˙))sat(q˙)+k(t)q = τ
where q(t),q˙(t),q¨(t)∈Rn represent the position, velocity, and acceleration, respectively; M ∈ Rn×n is the positive definite inertia of the system; ρ is a positive constant; μ(·) is a positive definite function related to the Stribeck effect; k(t) denotes a time-varying spring stiffness. For system (1), it is assumed that the following properties hold.

Property 1.

The matrix M is positive definite. Moreover, we define λmax as the maximum eigenvalue of M and λmin as the minimum eigenvalue of M.

Property 2.

The derivative of k(t) is bounded such that [Equation (2)]

|k(t)|≤k^ ∀t≥0
where k^ is a positive constant.

Before presenting the main results, we give some lemmas first.

Lemma 1.

Zhang et al. [10] Let a,b,p,q be positive real numbers. If 1p+1q=1 , then [Equation (3)]

ab≤app+bqq

Lemma 2.

Zhang et al. [10] Let ai > 0, bi > 0, i = 1,2, …, n, p > 1, and q > 1. If 1p+1q=1 , then [Equation (4)]

∑i=1naibi≤(∑i=1naip)1p(∑i=1nbiq)1q

Lemma 3.

Zhang et al. [10] For xi ∈ R, i = 1,2, …, n, and 0 < m < 1, then the following inequality holds [Equation (5)]:

|x1|m+⋅⋅⋅+|xn|m≤(|x1|+⋅⋅⋅+|xn|)m

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

Proof.

Let ai = |xi|m, bi = 1, i = 1, 2, …, n, p=1m , and q=11−m . Then based on Lemma 2, we obtain Equation (7)

|x1|m+⋅⋅⋅+|xn|m≤(∑i=1n(|xi|m)1m)m(∑i=1n111−m)1−m=n1−m(|x1|+⋅⋅⋅+|xn|)m
which completes the proof.

3. SUBSYSTEMS

For the uniform asymptotic stability problem of time-varying mechanical system (1), we propose the following control input [Equation (8)]

τ=−αMq˙+Cq˙−k0q
where α and k0 are control parameters. Let q = x1 and q˙=x2 . Then the closed-loop system (1) and (8) can be written in a compact form as Equation (9)
x˙1=x2x˙2=−(k(t)+k0)M−1x1−αx2−M−1(Fc(t)+Fs(t)e−ρμ(x2))sat(x2)

We define [Equation (10)]

f(t,x)=( x2f˜(t,x))
where function f˜(t,x)=−(k(t)+k0)M−1x1−αx2−M−1(Fc(t)+Fs(t)e−ρμ(x2))sat(x2) . We also define f1(t,x)=(     x2−(k(t)+k0)M−1x1−αx2) and f2(t,x)=(       0−M−1(Fc(t)+Fs(t)e−ρμ(x2))sat(x2)) . Then, we can see that f(t, x) = f1(t, x) + f2(t, x). In this subsection, we consider two subsystems x˙=f1(t, x) and x˙=f2(t, x) , respectively.

Inspired by the results in Malisoff and Mazenc [7], we consider the following Lyapunov function [Equation (11)]

V(t,x)=A((k(t)+k0)x1Tx1+x2TMx2)+x1Tx2
where A > 0 is a positive constant. In this paper, we choose A>max{1λmin,1k1+k0} .

It can be verified directly that [Equation (12)]

V(t,x)=A((k(t)+k0)x1Tx1+x2TMx2)+x1Tx2>−12(x1Tx1+x2Tx2)

The derivative of V(t, x) along subsystem x˙=f1(t, x) shows [Equation (13)]:

V˙(t,x)=∂V∂t+∂V∂xf1(t,x)≤−l1x1Tx1−l2x2Tx2
where we have l1=(k1+k0)1λmax−12α−Ak^ and l2=32α−1. Therefore, we can see that if we choose α>23 and k0>(12α+Ak^)λmax−k1 , subsystem x˙=f1(t, x) is uniformly asymptotically stable.

Similarly, the derivative of V(t, x) along subsystem x˙=f2(t, x) shows [Equation (14)]:

∂V∂xf2(t, x)≤2n1λminFc(t)+Fs(t)V12

In the following, we define time-varying parameter w(t)=n1λmin‖Fc(t)+Fs(t)‖ .

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 the derivative of V(t, x) along system x˙=f(t, x) , which is shown as following Equation (15)

V˙(t,x)|f(t,x)=∂V∂t+∂V∂x(f1(t,x)+f2(t,x))

Therefore, motivated by the results in Malisoff and Mazenc [7], we provide the following time-varying Lyapunov function [Equation (16) ]

V#(t,x)=∫0V1(1+s)0.5ds+ϕ∫t0t(1T∫ττ+Tw(v)dv−w(τ))dτ ln(1+V)
where T, ϕ are positive constants. It can be verified directly that if k0 and α are chosen large enough, then there exist T and ϕ such that [Equation (17)]
V˙#(t,x)|f(t,x)=∂V∂t+∂V∂x(f1(t,x)+f2(t,x))<0

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 α and k0 large enough, then the closed loop system is uniformly asymptotically stable. Moreover, (16) is a strict Lyapunov function.

Proof.

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

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

Mq¨+Cq˙+(Fc(t)+Fs(t)e−ρμ(q˙))sat(q˙)+k(t)q+ H(q,q˙)=τ
where H(q,q˙) is the unmodeled dynamics. For system (18), we assume that [Equation (19)]
limλ→0H(t,λq,λq˙)λ=0

Without detailed proof, we provide the following theorem for the robustness case.

Theorem 2.

Consider system (18) with unmodeled dynamics. We assume that (19) is satisfied. If the conditions in Theorem 1 hold, then we have that the closed-loop system is uniformly asymptotically stable.

5. NUMERICAL EXAMPLE

We give a numerical example to demonstrate the effectiveness of the proposed 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.

Figure 1

The state trajectories of system.

6. CONCLUSION

The stability problems have been considered in this paper for time-varying mechanical systems. A class of strict Lyapunov functions has been established, in which time-varying terms have been 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

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

Authors Introduction

Dr. Bin Zhang

He received his PhD degrees in control theory and applications from Beihang University, in 2016. He is with the School of Automation, Beijing University of Posts and Telecommunication. His research interests include multi-agent systems.

Dr. Yingmin Jia

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 and applications from Beihang University, Beijing, China, in 1990 and 1993, respectively. In 1993, he joined the Seventh Research Division, Beihang University, where he is currently a Professor of automatic control.

REFERENCES

[1]HK Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, 2002.
[2]A Bacciotti and L Rosier, Liapunov Functions and Stability in Control Theory, Springer Verlag, New York, 2005.
[3]D Han, A El-Guindy, and M Althoff, Estimating the domain of attraction based on the invariance principle, in 2016 IEEE 55th Conference on Decision and Control (CDC), IEEE (Las Vegas, NV, USA, 2016), pp. 5569-5576. https://doi.org/10.1109/cdc.2016.7799125
[4]NG Bretas and LFC Alberto, Lyapunov function for power systems with transfer conductances: extension of the invariance principle, in IEEE Transactions on Power Systems, IEEE (New York, NY, USA, 2003), pp. 769-777. https://doi.org/10.1109/tpwrs.2003.811207
[5]Y Hong, Y Xu, and J Huang, Finite-time control for robot manipulators, Syst. Control Lett., Vol. 46, 2002, pp. 243-253. https://doi.org/10.1016/s0167-6911%2802%2900130-5
[6]B Zhang and Y Jia, On weak-invariance principles for nonlinear switched systems, in IEEE Transactions on Automatic Control, IEEE (New York, NY, USA, 2014), pp. 1600-1605. https://doi.org/10.1109/tac.2013.2292730
[7]M Malisoff and F Mazenc, Constrictions of Strict Lyapunov Functions, Springer Verlag, New York, 2009.
[8]M Malisoff and F Mazenc, Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems, Automatica, Vol. 41, 2005, pp. 1973-1978. https://doi.org/10.1016/j.automatica.2005.05.015
[9]P Giesl and S Hafstein, Review on computational methods for Lyapunov functions, Discrete Continuous Dyn. Syst. Ser. B, Vol. 20, 2015, pp. 2291-2331. https://doi.org/10.3934/dcdsb.2015.20.2291
[10]B Zhang, Y Jia, J Du, and J Zhang, Finite-time synchronous control for multiple manipulators with sensor saturations and a constant reference, IEEE Transactions on Control Systems Technology, IEEE, 2014, pp. 1159-1165. https://doi.org/10.1109/tcst.2013.2271447
<Previous Article In Issue
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Journal
Journal of Robotics, Networking and Artificial Life
Volume-Issue
5 - 4
Pages
241 - 244
Publication Date
2019/03/30
ISSN (Online)
2352-6386
ISSN (Print)
2405-9021
DOI
10.2991/jrnal.k.190220.007How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

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TY  - JOUR
AU  - Bin Zhang
AU  - Yingmin Jia
PY  - 2019
DA  - 2019/03/30
TI  - Time-Varying Lyapunov Function for Mechanical Systems
JO  - Journal of Robotics, Networking and Artificial Life
SP  - 241
EP  - 244
VL  - 5
IS  - 4
SN  - 2352-6386
UR  - https://doi.org/10.2991/jrnal.k.190220.007
DO  - 10.2991/jrnal.k.190220.007
ID  - Zhang2019
ER  -