# Inherent Stability of Multibody Systems with Variable-Stiffness Springs via Absolute Stability Theory

Authors: Tianyang Hua; Yinlong Hu
Journal: Complexity
 (2021)

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## Abstract

In this paper, the inherent stability problem for multibody systems with variable-stiffness springs (VSSs) is studied. Since multibody systems with VSSs may consume energy during the variation of stiffness, the inherent stability is not always ensured. The motivation of this paper is to present sufficient conditions that ensure the inherent stability of multibody systems with VSSs. The absolute stability theory is adopted, andN-degree-of-freedom (DOF) systems with VSSs are formulated as a Lur’e form. Furthermore, based on the circle criterion, sufficient conditions for the inherent stability of the systems are obtained. In order to verify these conditions, both frequency-domain and time-domain numerical simulations are conducted for several typical low-DOF systems.


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## Body

## 1. Introduction

Vibration control can generally be classified into passive control, active control, and semiactive control. Semiactive control integrates the advantages of both excellent control performance of active control and high reliability of passive control, which can be realized by online adjusting the characteristic parameters (such as stiffness, damping, etc.) of originally passive devices [1]. Since semiactive control devices are obtained based on their passive counterparts, it is expected to inherit the stability of passive control, which means that even though failures and improper control actions of the parameter adjustments may occur, the semiactive control system can remain stable. Such kind of property is known as the inherent stability of semiactive control systems [2]. However, it has been demonstrated that inherent stability of semiactive control systems only conditionally holds for some special semiactive devices [2].Variable-stiffness spring (VSS) is a commonly used semiactive control device because of its effectiveness in modifying the resonance frequencies of vibration control systems [3]. In [4], the VSS-based system is applied in seismic building control, where VSS is aimed at establishing a nonresonant state against earthquake excitation. Up to now, different realizations for VSS have been proposed. For example, a resettable VSS (RVSS) is proposed to enhance the performance of a tuned mass damper system [5]. In [6], a VSS is developed by integrating the magnetorheological fluid into a damper. Based on the force-current analogy between mechanical systems and electrical circuits, a novel electrical VSS (EVSS) is proposed to reduce the suspension deflection of a seat suspension [7]. Moreover, the performance benefits of VSSs in robotic applications have been widely demonstrated [8, 9].A graphical representation on the relationship among vibration control, passivity, and inherent stability is shown in Figure1. Note that the terminologies “passive” (seen in Definition 4) and “active” are mutually exclusive with clear boundary in definition according to whether energy can be internally generated. However, whether semiactive systems are “passive” or “active” involves some abuse of notation [2]. This means that semiactive control can be further classified into “active” semiactive control and “passive” semiactive control. For example, in [2], it is demonstrated that certain semiactive devices including nonnegative variable-damping dampers, resettable-stiffness springs, and resettable-inertance inerters are “passive”; in [10–12], the VSSs are actually “active” due to the less energy required for adjusting the parameters. For “passive” semiactive control systems, inherent stability can be guaranteed due to the passivity formalism [13], as shown in the green lines in Figure 1. For “active” semiactive control systems, inherent stability can not be guaranteed. In [14], it is proved that instabilities can be induced by VSSs. However, for “active” VSS-based semiactive control systems, how to ensure inherent stability has not yet been studied, which is the main concern of this paper, as shown in the red line in Figure 1. Since the stiffness influences the potential energy of a system, a certain amount of energy is generally required to adjust the stiffness. This means that “active” VSSs are more common than the “passive” ones. Therefore, it is of great theoretical and practical significance to investigate the inherent stability problem of VSSs-based systems.Figure 1
The relationship among passivity, stability, and vibration control.The VSS-based systems are typical nonlinear control systems, where the nonlinearities are induced by the variable stiffness within a sector reflecting the bounds of VSSs. For nonlinear mechanical systems, several control strategies have been proposed to ensure stability and performance, such as sliding mode backstepping [15], adaptive finite-time control [16, 17], etc. In [16], the problem of finite-time fault-tolerant consensus protocols for a class of uncertain multiple mechanical systems is investigated, and the proposed scheme can guarantee that the position errors and the velocity errors between any two mechanical systems converge to a small neighbourhood of zero in finite time. In [17], an adaptive fuzzy finite-time control scheme is proposed to guarantee the stability of a class of nonlinear systems with unknown nonlinearities. Note that, for the aforementioned papers, stability of the close-loop system is achieved. However, the inherent stability studied in this paper implies the stability for a group of control laws within certain sectors, instead of a special control law in the aforementioned papers. Therefore, in this paper, the inherent stability problem for the multibody systems with VSSs is studied via the absolute stability theory, and the general multibody system with VSSs is transformed into a Lur’e-type system described as a linear passive system with feedback-connected nonlinearities. The most celebrated methods to analyze the absolute stability of a Lur’e-type systems are circle criterion (CC) and Popov criterion (PC). CC can deal with more diverse nonlinearities including time-varying ones than PC [18]. Since the stiffness of VSSs is time-varying in general, CC is adopted to verify the absolute stability of the multibody systems with VSSs. Furthermore, for several typical low-DOF systems, sufficient conditions imposed on the upper and lower bounds of VSSs are derived and verified via numerical examples in both frequency domain and time domain.The main contributions of this paper are as follows:(1)
The inherent stability problem for the multibody systems with “active” VSSs is investigated, which has not been studied before but with great theoretical and practical significance. Sufficient conditions for the inherent stability of the general multibody systems with “active” VSSs are derived and analyzed via the absolute stability theory.(2)
The inherent stability of several typical low-DOF systems with “active” VSSs is analyzed. Moreover, extensive numerical examples based on low-DOF systems are conducted to show the effectiveness of the results of this paper.The structure of this paper is organized as follows. Definitions and lemmas used in this paper are summarized in Section2. In Section 3, the general multibody systems with VSSs are modeled as Lur’e-type systems, and the absolute stability of these systems is demonstrated. Numerical simulation of the absolute stability of several low-DOF systems is given in Section 4. Conclusions are drawn in Section 5.


## 2. Definitions and Lemmas

In this section, some definitions and lemmas related to the absolute analysis of the multibody systems with VSSs are given.Definition 1 (see [18]).
The definitions of the sector terminology in both the scalar and multivariable cases are as follows:(i)
Consider a scalar functionΨt,y with Ψt,0≡0, t≥0; it is said to belong to the sector K1,K2 if(1)K1y2≤Ψt,yy≤K2y2,∀t≥0,whereK1,K2∈R and K1<K2.(ii)
Consider a specific multivariable functionΨt,y with Ψt,0≡0, t≥0, which can be decoupled to Ψt,y=Ψ1t,y1,Ψ2t,y2,…,Ψpt,ypT; it is said to belong to the sector K1,K2 if(2)Ψt,y−K1yTΨt,y−K2y≤0,whereK1=diagα1,α2,…,αp, K2=diagβ1,β2,…,βp, and βi>αi.Definition 2 (see [19]).
Consider the Lur’e system(3)x˙=Ax+Bu,y=Cx,u=−Ψt,y,where A,B is controllable, A,C is observable, and Ψt,y is a memoryless, possibly time-varying, nonlinearity, which is piecewise continuous in t and locally Lipschitz in y. In addition, Ψt,y satisfies a sector condition per Definition 1. The system is absolutely stable if the origin is globally uniformly asymptotically stable for any nonlinearity in the given sector. It is absolutely stable with a finite domain if the origin is uniformly asymptotically stable.Definition 3 (see [18]).
Denote a diskDK1,K2 for Ψt,y∈K1,K2 where K1 and K2 are constant, as the closed disk in the complex plane centred at −K1+K2/2K1K2+j0 and with radius K2−K1/2K1K2.Definition 4 (see [2]).
A system whose input isus and output is ys, for which it is defined a lower bounded (Lyapunov-like) storage function Vt, is said to be “passive” if it satisfies the following equation:(4)V˙t=ysTus−gt,where gt≥0, ∀t∈0,T.Lemma 1 (see [19]).
The system described in Definition2 is absolutely stable if(i)
Ψ∈K1,∞ and GsI+K1Gs−1 is strictly positive real(ii)
Ψ∈K1,K2, with K=K2−K1=KT>0, and I+K2GsI+K1Gs−1 is strictly positive realLemma 2 (see [19]).
Consider a scalar system of equation (3), where A,B,C,D is a minimal realization of Gs and Ψ∈K1,K2. Then, the system is absolutely stable if one of the following conditions is satisfied, as appropriate:(i)
If0<K1<K2, the Nyquist plot of Gjω does not enter the disk DK1,K2 and encircles it m times in the counterclockwise direction, where m is the number of poles of Gs with positive real parts(ii)
If0=K1<K2, Gs is Hurwitz, and the Nyquist plot of Gjω lies to the right of the vertical line defined by Res=−1/K2(iii)
IfK1<0<K2, Gs is Hurwitz, and the Nyquist plot of Gjω lies in the interior of the disk DK1,K2If the sector condition is satisfied only on an intervala,b, the foregoing conditions ensure that the system is absolutely stable with a finite domain.Lemma 3 (see [20]).
The transfer matrixGs is a strictly positive real matrix if Gs−ε is a positive real matrix for some ε>0.Lemma 4 (see [21]).
LetG be a real m×m matrix-valued rational function of the form(5)Gs=CsI−A−1B+D,if the linear matrix inequalities (LMIs)(6)ATX+XAXB−CBTX−CT−D−DT≤0,X≥0,have a solution X of the LMIs.Lemma 5 (see [20]).
Consider them×m rational matrix Gs=CsI−A−1B+D. Then Gs is a strictly positive real matrix if and only if(i)
All elements ofGs are analytic in Res≥0(ii)
Gjω+GT−jω>0, ∀ω∈R(iii)
limω⟶∞ω2Gjω+GT−jω>0, if detD+DT=0 or limω⟶∞Gjω+GT−jω>0,ifdetD+DT≠0Lemma 6 (see [22]).
Consider them×m rational matrix Gs=CsI−A−1B+D. If all the eigenvalues of A have negative real part, then there is a β>0 such that G is analytic in Res>−β.


## 3. Absolute Stability Analysis ofN-DOF Multibody Systems with VSSs

Consider aN-DOF multibody system with a dynamic equation as follows:(7)Mpq¨t+Cpq˙t+Kpqt=Ft,where qt∈RN×1 is the generalized coordinate vector of each DOF and Mp, Cp, and Kp∈RN×N are the mass, damping, and stiffness matrices of the N-DOF multibody system, respectively. Ft∈RN×1 is the force applied by VSSs and is imposed on the N-DOF multibody system. Note that equation (7) represents a general description of multibody systems, which can be used to model vehicles, bridges, buildings, etc. The main problem of this paper is as follows.Problem 1.
Consider aN-DOF system with nk VSSs where the stiffness can be adjusted online within certain ranges. Determine the ranges for the stiffness of VSSs where the inherent stability can always be guaranteed.
Since it has been proved in [14] that instability can be induced by “active” VSSs, Problem 1 is intended to propose sufficient conditions for the “active” VSSs to guarantee the inherent stability on condition that the stiffness of VSSs is adjusted within certain ranges. In practice, the stiffness of VSSs is always bounded due to the physical realizations. For example, for variable-stiffness leaf springs [23], the stiffness is limited by the geometry; for a variable stiffness mechanism [24], the stiffness is limited by the rotation amount of the springs, etc. Therefore, apart from the theoretical interest of studying Problem 1, the ranges obtained from Problem 1 can provide practical guidance for the selection and manufacture of VSSs.
Assume that there arenk VSSs to be applied to the multibody system, where nk≤nn+1/2 (since VSS has two mechanical terminals, there are at most nn+1/2 VSSs that can be applied in a N-DOF system). The locations for connecting VSSs are numbered as 0,1,2,…,n, where location 0 is the inertial frame (mechanical ground) and locations 1,2,…,n correspond to the N-DOFs, respectively. For a VSS kl, l=1,…,nk, its two terminals are connected to the locations il and jl, where il and jl are some integers between 0,…,n and jl>il.Proposition 1.
Ifil≠0, eil,jl is a column vector with the ilth entry as 1, the jlth entry as −1, and others as 0; otherwise, if il=0, then eil,jl is a column vector with the jlth entry as 1 and others as 0.Proof.
Ifil≠0, the relevant parts of the dynamic equation can be obtained as(8)milq¨il+⋯=klsqjl−qil,mjlq¨jl+⋯=−klsqjl−qil,where the abridged parts are not affected by the ilth VSS. Therefore, eil,jl=0,0,…,−1,0,…,1,0,…,0T, where line il is −1 and line jl is 1 in this vector.
Ifil=0, the relevant parts of the dynamic equation can be obtained as(9)mjlq¨jl+⋯=−klsqjl,where the abridged parts are not affected by the ilth VSS. Therefore, eil,jl=0,0,…,1,0,…,0T, where line jl is 1 in this vector.
According to Proposition1, if il≠0, eil,jl is a column vector with the il entry as 1, jl entry as −1, and others as 0; otherwise, if il=0, then eil,jl is a column vector with the jl entry as 1 and others as 0.
Then, the dynamic equation of the multibody system withnk VSSs can be represented as(10)Mpq¨t+Cpq˙t+Kpqt+∑l=1nkklteil,jleil,jlTqt=0,where klt is the stiffness of the VSS l and eil,jl∈RN×1 is the position vector for VSS l.
DenoteHk∈RN×nk with(11)Hk=ei1,j1,ei2,j2,…,eink,jnk,and then, equation (10) can be rewritten as(12)Mpq¨t+Cpq˙t+Kpqt+HkKstHkTqt=0,where Kst=diagk1st,…,knkst.
Therefore,(13)q¨t=−Mp−1Cpq˙t−Mp−1Kpqt−Mp−1HkKstHkTqt.
Denoting(14)x=qtT,q˙tTT,one can obtain the state-space representation of the N-DOF system with VSSs as(15)x˙=Ax+Bu,y=Cx,u=−Ψt,y,where(16)A=0N×NIN×N−Mp−1Kp−Mp−1Cp,B=0N×nkMp−1Hk,C=HkT0nk×N,Ψt,y=Ksty.Remark 1.
Note that, in [2], the inherently stability of “passive” VSS-based semiactive systems is studied, while in this paper, multibody systems with VSSs are “active.” In the following, the passivity of these systems is analyzed via energy consumption.
Note thatHk is a matrix containing pairs (or singleton) of connection points such that the vector of relative displacements across the devices is qc=HkTq. Consider a storage function that is equal to the potential energy Ep stored in VSSs:(17)Vt=∫0qcqcTKstdqc.
The derivative ofVt equals(18)V˙t=∂∫0qcqcTKstdqc∂qc⋅q˙c+∂∫0qcqcTKstdqc∂Kst⋅K˙st=q˙cTKstqc+12qcTK˙stqc.
According to Definition4, for the VSSs-based systems, assume that the output is ys=q˙c, and the external power input us=Kstqc. Substituting equation (18) into equation (4), the dissipated power can be solved as follows:(19)gt=−12qcTK˙stqc.
In [2], to ensure the passivity, the resettable-stiffness spring is proposed. However, the restriction is that the adjustment actions of the spring only take place at the instant that qc=0. In this paper, “active” multibody systems with VSSs are studied. Compared with the restriction to ensure the passivity, it should only satisfy that K˙st>0, ∃t∈0,T, which is common in practice. To the best of our knowledge, this is for the first time in the spring-related research field.Remark 2.
Note that, in Definition1, the bounds of the sector are better to be constants or diagonal matrices for the absolute stability analysis, and the matrix Kst in equation (15) is always diagonal no matter what the position and number of VSSs are. Therefore, a Lur’e-type form (15) can represent the general multibody systems with VSSs.
The transfer function of equation (15) is(20)Gs=CsI−A−1B+D.
Let(21)Hs=I+K¯sGsI+K¯sGs−1,where K¯s and K¯s are the upper and lower bound of the matrix Ks.
From Lemma4, the transfer function Hs−εN is positive real, if there exists a symmetric, positive-definite matrix XN, which satisfies(22)ANTXN+XNANXNBN−CNBNTXN−CNT−DN−DNT≤0,where AN,BN,CN,DN is converted by the transfer function Hs−εN.
From Lemma3, the transfer function Hs is strictly positive real if there exist some εN, which satisfies that Hs−εN is positive real.
According to Lemma1, if the transfer function Hs is strictly positive real, the N-DOF multibody system with VSSs is absolutely stable.


## 4. Numerical Simulation

In this section, numerical simulations of the absolute stability of several low-DOF systems are conducted. In Section4.1, the absolute stability problem of semiactive control systems with only one VSS is studied. Section 4.2 derives the sufficient conditions for semiactive control systems with several VSSs.


### 4.1. Absolute Stability Analysis of the System with One VSS

Theorem 1.
Consider a 1-DOF system with one VSS, seen in Figure2; the dimensionless dynamic equation is(23)x¨1+2ξ1ω1x˙1+ω12+ω1s2tx1=0,where ξ1=c1/2k1m1, ω1=k1/m1, ω1st=k1st/m1, and 0≤ω¯1s≤ω1st≤ω¯1s (m1, c1, k1, k1st, and x1 represent mass, damping, stiffness, variable stiffness, and displacement of the 1-DOF system, respectively). If it satisfies equation (24), the 1-DOF system with one VSS is absolutely stable.(24)a>0,ora≤0,4ω12+ω¯1s2ω12+ω¯1s2−a2>0,where a=4ξ12−2ω12−ω¯1s2+ω¯1s2.Figure 2
A 1-DOF model with a VSS.Proof.
Denotex1=x1,x˙1T; equation (23) can be converted to(25)x˙1=A1x1+B1u1,y1=C1x1,u1=−Ψ1t,y1,where(26)A1=01−ω12−2ξ1ω1,B1=01,C1=10,Ψ1t,y1=ω1s2y1.
The transfer function of equation (25) is(27)G1s=C1sI−A1−1B1=1s2+2ξ1ω1s+ω12.
Let(28)H1s=1+ω¯1s2G1s1+ω¯1s2G1s−1=1+ω¯1s2−ω¯1s2s2+2ξ1ω1s+ω12+ω¯1s2.
The transfer functionH1s can be converted to state-space realization Aα,Bα,Cα,Dα, where(29)Aα=01−ω12−ω¯1s2−2ξ1ω1,Bα=0ω¯1s2−ω¯1s2,Cα=10,Dα=1.
It is clear that all the eigenvalues ofAα have negative real part. Then, according to Lemma 6, all elements of H1s are analytic in Res≥0, which satisfies list 1 of Lemma 5.
According to list 1 of Lemma5, it is obvious that all elements of H1s are analytic in Res≥0. According to list 2 of Lemma 5,(30)H1jω+H1T−jω=1+ω¯1s2−ω¯1s2−ω2+ω12+ω¯1s2−ω2+ω12+ω¯1s22+2ξ1ω1ω2.
In order to satisfy thatH1jω+H1T−jω>0, ∀ω∈R, it can be obtained that(31)fω>0,∀ω∈R,where fω=ω4+2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2ω2+ω12+ω¯1s2ω12+ω¯1s2.
According to the position of the symmetry axis, the following inequality can be obtained:(32)2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2≥0,f0>0,ω¯1s<ω¯1s,or(33)2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2<0,fminω>0;ω¯1s<ω¯1s.
By simplifying the inequalities equations (32) and (33), the following sufficient condition can be obtained: (34)a>0,ora≤0,4ω12+ω¯1s2ω12+ω¯1s2−a2>0,where a=4ω12+ω¯1s2ω12+ω¯1s2.
Therefore, in order to satisfy thatH1jω+H1T−jω>0, ∀ω∈R, equation (34) should be satisfied.
According to list 3 of Lemma5, note that D=1 in H1s, and then(35)limω⟶∞H1jω+H1T−jω=2>0.
Therefore, list 3 of Lemma5 is obviously satisfied in all conditions.
In summary, if equation (34) is satisfied, H1s is a strictly positive real matrix.
According to Lemma1, note that Ψ∈ω¯1s2,ω¯1s2, and(36)H1s=1+ω¯1s2G1s1+ω¯1s2G1s−1,is strictly positive real. Therefore, the 1-DOF system with one VSS is absolutely stable.Example 1.
Consider a 1-DOF system, seen in Figure2, and choose ω1=1, ξ1=2, ω¯1s=1. In the following, the range of ω¯1s is calculated by equation (32) or (33). From equation (32), the range of ω¯1s is calculated as 1,13, and from equation (33), the range is 13,5.321. Therefore, ω¯1s can be set from 1 to 5.321 to ensure the absolute stability of the 1-DOF system. Figure 3 shows the Nyquist plot and disk plot of the 1-DOF system; the system is absolutely stable, according to Lemma 2.
Apart from verification in the frequency domain, time-domain simulation is also considered as follows. Two kinds of nonlinearities are considered. For the first one, the selectedω1s is given in equation (37), which is in the range ω1s∈1,5.321. The other selected ω¯1s is given in equation (38), which is out of the range ω1s∈1,5.321.(37)ω1s=sint+2,(38)ω1s=7,x1tx˙1t>0,14,x1tx˙1t≤0.
Time-domain simulation is shown in Figure4. Figure 4(a) shows the phase trajectory of state x1 when the selected ω¯1s is in the stable range. The phase trajectory from a specific initial condition x10=1,x˙10=0, which converge to the origin, indicates the stability of the system. Figure 4(b) shows the phase trajectory of state x1 when the selected ω¯1s is out of the stable range. The phase trajectory from a specific initial condition x10=1,x˙10=0, which is divergent, indicates the instability of the system.Figure 3
The Nyquist diagram withω¯1s∈1,5.321.Figure 4
The phase portrait of a 1-DOF systems with different nonlinearities in the feedback path. (a) The phase trajectory of statex1 with the selected ω¯1s in the stable range. (b) The phase trajectory of state x1 with the selected ω¯1s out of the stable range.
(a)(b)Remark 3.
Note that Lemmas3 and 4 can also verify whether H1s is strictly positive real, apart from Lemma 5. If there exist positive ε1 and symmetric, positive-definite matrix X1 satisfying equation (39), the 1-DOF system with one VSS is absolutely stable.(39)A11TX1+X1A11X1B11−C11B11TX1−C11T−D11−D11T≤0,where A11,B11,C11,D11 is converted by H1s−ε1.
In the following, numerical simulation is given to compare the conservatism. The values ofω¯1s, ω1, and ξ1 are the same as Example 1. Equation (33) is feasible when ω¯1s varies from 1 to 5.3. Figure 5 shows the Nyquist plot and disk plot of the 1-DOF system; the system is absolutely stable, according to Lemma 2.
Note that the range ofω1s is smaller than that in Example 1; the LMI condition in this remark is more conservative than that in Theorem 1. Therefore, it is better to analyze the absolute stability of a 1-DOF system with variable stiffness spring by equation (24) in Theorem 1.Figure 5
The Nyquist diagram withω¯1s∈1,5.3.Example 2.
A comparison between the method proposed in this paper and the traditional stability analysis method with a quadratic Lyapunov function for the 1-DOF system shown in Figure2 is conducted. The values of ω1 and ξ1 are the same as those in Example 1, and a VSS with a periodic stiffness, i.e.,(40)ω1s2=kα+kβsint,which can be found in robot manipulators such as in [25], is adopted.
Letkβ=0.5 and kα≥0.5. For the traditional stability analysis method with a quadratic Lyapunov function, one can select a quadratic Lyapunov function as(41)Vt=12x1TQx1,where Q=11/21/21. It can be checked that if 1/2≤kα≤2+21/2, then V>0 and V˙≤0, indicating that the 1-DOF system is stable.
By using the method proposed in this paper, from equation (24), one obtains that the 1-DOF system with a VSS is always stable whenever ω1s varies in the interval 0,26. This means that the stability condition obtained by the traditional quadratic Lyapunov function method only holds for specific cases such as the periodic variable stiffness in equation (40), while the condition obtained by the method proposed in this paper is applicable for any type of VSS’s variable stiffness within a sector. To further show this fact, the following variable stiffnesses are considered:(42)ω1s2=3+0.5sint,(43)ω1s2=10+0.5sint.
Figure6(a) shows the sector, seen as equation (44), induced by a VSS. The red solid lines represent the bound of the sector 0,24. The black dashed line and the blue dotted line represent the periodic variable stiffness, (42) and (43), respectively.(44)Ψ1t,y1=ω1s2y1,y1∈−1,1.
It is clear that the case with equation (42) is stable by using the quadratic Lyapunov function in equation (41). However, for the case with equation (43), the stability cannot be verified by using the quadratic Lyapunov function in equation (41), while since it is within the sector as shown in Figure 6(a), the stability of the case with equation (43) can be checked by using the method proposed in this paper, where Figure 6(b) shows that both cases are stable. This example demonstrates that the method proposed in this paper is applicable for a group of nonlinearities within a certain sector, instead of a specific nonlinearity.Figure 6
A comparison between the method proposed in this paper and the traditional quadratic Lyapunov function method. (a) The sector region induced by a VSS. (b) The phase portrait of a 1-DOF system with a VSS.
(a)(b)


### 4.2. Absolute Stability Analysis of the Systems with VSSs

Theorem 2.
Consider a 2-DOF system with VSSs, seen in Figure7; the dimensionless dynamic equation is(45)x¨1+2ξ1ω1+2ξ2ω2α1x˙1+ω12+ω22α1x1+ω01s2t+ω12s2tx1−2ξ2ω2α1x˙2−ω22α1x2−ω12s2tx2=0,α1x¨2+2ξ2ω2α1x˙2+ω22α1x2+ω12s2t+ω02s2tx2−2ξ2ω2α1x˙1−ω22α1x1−ω12s2tx1=0,where ξ1=c1/2k1m1, ω1=k1/m1, ξ2=c2/2k2m2, ω2=k2/m2, α1=m2/m1, ω01st=k01st/m1, ω12st=k12st/m1, ω02st=k02st/m1, and ω01sω12s,orω02s varies from ω¯01sω¯12s,orω¯02s to ω¯01sω¯12s,orω¯02s. m1m2, c1c2, k1k2, k01stk12st,k02st, and x1x2 represent mass, damping, stiffness, variable stiffness, and displacement of the 2-DOF system, respectively. If there exist positive ε2 and symmetric, positive-definite matrix X2 satisfying equation (46), the 2-DOF system with VSSs is absolutely stable.(46)A21TX2+X2A21X2B21−C21B21TX2−C21T−D21−D21T≤0,whereA21,B21,C21,D21 is converted by H2s−ε2.Figure 7
A 2-DOF model with VSSs.Proof.
Denotex1=x1,x2,x˙1,x˙2T; equation (45) can be converted to(47)x˙2=A2x2+B2u2,y2=C2x2,u2=−Ψ2t,y2,where(48)A2=00100001−ω22α1−ω12ω22α1−2ξ2ω2α1−2ξ1ω12ξ2ω2α1ω22−ω222ξ2ω2−2ξ2ω2,B2=000000−1100−1/α1−1/α1,C2=1000−11000100,Ψ2t,y2=diagω01s2,ω12s2,ω02s2y2.
The transfer function of equation (47) is(49)G2s=C1sI−A2−1B2=−s2+2ω2ξ2s+ω22dens2den−2ω2ξ2s+ω22den∗−1+α1s2+2ω1ξ1s+ω12α1⋅den−s2+2ω1ξ1s+ω12α1⋅den∗∗−s2+2ω1ξ1+2α1ω2ξ2s+ω12+α1ω22α1⋅den,where ∗ denotes the symmetrical element, and(50)den=s4+2ω1ξ1+2ω2ξ2+2ω2ξ2α1s3+ω12+ω22+ω22α1+4ω1ω2ξ1ξ2s2+2ω12ω2ξ2+2ω1ω22ξ1s+ω12ω22.
Then(51)H2s=I+K22G2sI+K21G2s−1,where K21=diagω¯01s2,ω¯12s2,ω¯02s2 and K22=diagω¯01s2,ω¯12s2,ω¯02s2. Equation (51) can be converted to the state space A21,B21,C21,D21.
From Lemma3, the transfer function H2s is strictly positive real if there exist some ε2, which satisfies that H2s−ε2 is positive real.
From Lemma4, the transfer function H2s−ε2 is positive real if there exists a symmetric, positive-definite matrix X2, which satisfies(52)A21TX2+X2A21X2B21−C21B21TX2−C21T−D21−D21T≤0.Example 3.
Chooseω1=2, ξ1=6, ω2=3, ξ2=7, α1=3, ω¯01s=0, ω¯01s=3, ω¯12s=0, ω¯12s=2, ω¯02s=0, ω¯02s=1, and ε2=0.1; the symmetric and positive-definite matrix X2 is calculated by the standard numerical simulation toolbox.(53)X2=14.9948−9.84011.11480.4961−121.998422.6820−0.4832−1.3486∗93.1342−25.2808−8.2322−43.7687−14.3113−4.82−7.2607∗∗34.710210.171522.1161.49924.4151.9909∗∗∗3.18786.1410.74260.9826−0.142∗∗∗∗1176.4−186.752510.92522.7945∗∗∗∗∗38.3025−11.2653−6.8299∗∗∗∗∗∗106.252546.7287∗∗∗∗∗∗∗24.697.
According to Theorem2, if there exist some ε2>0 and a symmetric and positive-definite matrix X2, the 2-DOF system with VSSs is absolutely stable.
Apart from verification in the frequency domain, time-domain simulation is also considered as follows. Two kinds of nonlinearities are considered. For the first one, the selectedω01s, ω12s, and ω02s are given in equation (54), which are in the ranges ω01s∈0,3, ω12s∈0,2, and ω02s∈0,1. The other selected ω01s, ω12s, and ω02s are given in equation (55), which are out of the ranges ω01s∈0,3, ω12s∈0,2, and ω02s∈0,1.
In the following, a time-domain simulation is conducted, similar to that in Example1. The values of ω01s, ω12s, and ω02s are, respectively, selected as equations (54) and (55):(54)ω01s=sint+1.5,ω12s=0.5sint+0.6,ω02s=0.1sint+1.2,(55)ω01s=sint+4,ω12s=sint+4,ω02s=sint+2.
Numerical simulation is shown in Figure8, where similar results as Example 1 can be found. The system can be stable when the selected values are in the stable ranges, as shown in Figure 8(a). However, when the selected values are out of the stable ranges, the system will be unstable, as shown in Figure 8(b).Figure 8
The phase portrait of a 2-DOF system with different nonlinearities in the feedback path. (a) The phase trajectory with the selected values in the stable ranges. (b) The phase trajectory with the selected values out of the stable ranges.
(a)(b)


## 4.1. Absolute Stability Analysis of the System with One VSS

Theorem 1.
Consider a 1-DOF system with one VSS, seen in Figure2; the dimensionless dynamic equation is(23)x¨1+2ξ1ω1x˙1+ω12+ω1s2tx1=0,where ξ1=c1/2k1m1, ω1=k1/m1, ω1st=k1st/m1, and 0≤ω¯1s≤ω1st≤ω¯1s (m1, c1, k1, k1st, and x1 represent mass, damping, stiffness, variable stiffness, and displacement of the 1-DOF system, respectively). If it satisfies equation (24), the 1-DOF system with one VSS is absolutely stable.(24)a>0,ora≤0,4ω12+ω¯1s2ω12+ω¯1s2−a2>0,where a=4ξ12−2ω12−ω¯1s2+ω¯1s2.Figure 2
A 1-DOF model with a VSS.Proof.
Denotex1=x1,x˙1T; equation (23) can be converted to(25)x˙1=A1x1+B1u1,y1=C1x1,u1=−Ψ1t,y1,where(26)A1=01−ω12−2ξ1ω1,B1=01,C1=10,Ψ1t,y1=ω1s2y1.
The transfer function of equation (25) is(27)G1s=C1sI−A1−1B1=1s2+2ξ1ω1s+ω12.
Let(28)H1s=1+ω¯1s2G1s1+ω¯1s2G1s−1=1+ω¯1s2−ω¯1s2s2+2ξ1ω1s+ω12+ω¯1s2.
The transfer functionH1s can be converted to state-space realization Aα,Bα,Cα,Dα, where(29)Aα=01−ω12−ω¯1s2−2ξ1ω1,Bα=0ω¯1s2−ω¯1s2,Cα=10,Dα=1.
It is clear that all the eigenvalues ofAα have negative real part. Then, according to Lemma 6, all elements of H1s are analytic in Res≥0, which satisfies list 1 of Lemma 5.
According to list 1 of Lemma5, it is obvious that all elements of H1s are analytic in Res≥0. According to list 2 of Lemma 5,(30)H1jω+H1T−jω=1+ω¯1s2−ω¯1s2−ω2+ω12+ω¯1s2−ω2+ω12+ω¯1s22+2ξ1ω1ω2.
In order to satisfy thatH1jω+H1T−jω>0, ∀ω∈R, it can be obtained that(31)fω>0,∀ω∈R,where fω=ω4+2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2ω2+ω12+ω¯1s2ω12+ω¯1s2.
According to the position of the symmetry axis, the following inequality can be obtained:(32)2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2≥0,f0>0,ω¯1s<ω¯1s,or(33)2ξ1ω12−2ω12+ω¯1s2−ω¯1s2−ω¯1s2<0,fminω>0;ω¯1s<ω¯1s.
By simplifying the inequalities equations (32) and (33), the following sufficient condition can be obtained: (34)a>0,ora≤0,4ω12+ω¯1s2ω12+ω¯1s2−a2>0,where a=4ω12+ω¯1s2ω12+ω¯1s2.
Therefore, in order to satisfy thatH1jω+H1T−jω>0, ∀ω∈R, equation (34) should be satisfied.
According to list 3 of Lemma5, note that D=1 in H1s, and then(35)limω⟶∞H1jω+H1T−jω=2>0.
Therefore, list 3 of Lemma5 is obviously satisfied in all conditions.
In summary, if equation (34) is satisfied, H1s is a strictly positive real matrix.
According to Lemma1, note that Ψ∈ω¯1s2,ω¯1s2, and(36)H1s=1+ω¯1s2G1s1+ω¯1s2G1s−1,is strictly positive real. Therefore, the 1-DOF system with one VSS is absolutely stable.Example 1.
Consider a 1-DOF system, seen in Figure2, and choose ω1=1, ξ1=2, ω¯1s=1. In the following, the range of ω¯1s is calculated by equation (32) or (33). From equation (32), the range of ω¯1s is calculated as 1,13, and from equation (33), the range is 13,5.321. Therefore, ω¯1s can be set from 1 to 5.321 to ensure the absolute stability of the 1-DOF system. Figure 3 shows the Nyquist plot and disk plot of the 1-DOF system; the system is absolutely stable, according to Lemma 2.
Apart from verification in the frequency domain, time-domain simulation is also considered as follows. Two kinds of nonlinearities are considered. For the first one, the selectedω1s is given in equation (37), which is in the range ω1s∈1,5.321. The other selected ω¯1s is given in equation (38), which is out of the range ω1s∈1,5.321.(37)ω1s=sint+2,(38)ω1s=7,x1tx˙1t>0,14,x1tx˙1t≤0.
Time-domain simulation is shown in Figure4. Figure 4(a) shows the phase trajectory of state x1 when the selected ω¯1s is in the stable range. The phase trajectory from a specific initial condition x10=1,x˙10=0, which converge to the origin, indicates the stability of the system. Figure 4(b) shows the phase trajectory of state x1 when the selected ω¯1s is out of the stable range. The phase trajectory from a specific initial condition x10=1,x˙10=0, which is divergent, indicates the instability of the system.Figure 3
The Nyquist diagram withω¯1s∈1,5.321.Figure 4
The phase portrait of a 1-DOF systems with different nonlinearities in the feedback path. (a) The phase trajectory of statex1 with the selected ω¯1s in the stable range. (b) The phase trajectory of state x1 with the selected ω¯1s out of the stable range.
(a)(b)Remark 3.
Note that Lemmas3 and 4 can also verify whether H1s is strictly positive real, apart from Lemma 5. If there exist positive ε1 and symmetric, positive-definite matrix X1 satisfying equation (39), the 1-DOF system with one VSS is absolutely stable.(39)A11TX1+X1A11X1B11−C11B11TX1−C11T−D11−D11T≤0,where A11,B11,C11,D11 is converted by H1s−ε1.
In the following, numerical simulation is given to compare the conservatism. The values ofω¯1s, ω1, and ξ1 are the same as Example 1. Equation (33) is feasible when ω¯1s varies from 1 to 5.3. Figure 5 shows the Nyquist plot and disk plot of the 1-DOF system; the system is absolutely stable, according to Lemma 2.
Note that the range ofω1s is smaller than that in Example 1; the LMI condition in this remark is more conservative than that in Theorem 1. Therefore, it is better to analyze the absolute stability of a 1-DOF system with variable stiffness spring by equation (24) in Theorem 1.Figure 5
The Nyquist diagram withω¯1s∈1,5.3.Example 2.
A comparison between the method proposed in this paper and the traditional stability analysis method with a quadratic Lyapunov function for the 1-DOF system shown in Figure2 is conducted. The values of ω1 and ξ1 are the same as those in Example 1, and a VSS with a periodic stiffness, i.e.,(40)ω1s2=kα+kβsint,which can be found in robot manipulators such as in [25], is adopted.
Letkβ=0.5 and kα≥0.5. For the traditional stability analysis method with a quadratic Lyapunov function, one can select a quadratic Lyapunov function as(41)Vt=12x1TQx1,where Q=11/21/21. It can be checked that if 1/2≤kα≤2+21/2, then V>0 and V˙≤0, indicating that the 1-DOF system is stable.
By using the method proposed in this paper, from equation (24), one obtains that the 1-DOF system with a VSS is always stable whenever ω1s varies in the interval 0,26. This means that the stability condition obtained by the traditional quadratic Lyapunov function method only holds for specific cases such as the periodic variable stiffness in equation (40), while the condition obtained by the method proposed in this paper is applicable for any type of VSS’s variable stiffness within a sector. To further show this fact, the following variable stiffnesses are considered:(42)ω1s2=3+0.5sint,(43)ω1s2=10+0.5sint.
Figure6(a) shows the sector, seen as equation (44), induced by a VSS. The red solid lines represent the bound of the sector 0,24. The black dashed line and the blue dotted line represent the periodic variable stiffness, (42) and (43), respectively.(44)Ψ1t,y1=ω1s2y1,y1∈−1,1.
It is clear that the case with equation (42) is stable by using the quadratic Lyapunov function in equation (41). However, for the case with equation (43), the stability cannot be verified by using the quadratic Lyapunov function in equation (41), while since it is within the sector as shown in Figure 6(a), the stability of the case with equation (43) can be checked by using the method proposed in this paper, where Figure 6(b) shows that both cases are stable. This example demonstrates that the method proposed in this paper is applicable for a group of nonlinearities within a certain sector, instead of a specific nonlinearity.Figure 6
A comparison between the method proposed in this paper and the traditional quadratic Lyapunov function method. (a) The sector region induced by a VSS. (b) The phase portrait of a 1-DOF system with a VSS.
(a)(b)


## 4.2. Absolute Stability Analysis of the Systems with VSSs

Theorem 2.
Consider a 2-DOF system with VSSs, seen in Figure7; the dimensionless dynamic equation is(45)x¨1+2ξ1ω1+2ξ2ω2α1x˙1+ω12+ω22α1x1+ω01s2t+ω12s2tx1−2ξ2ω2α1x˙2−ω22α1x2−ω12s2tx2=0,α1x¨2+2ξ2ω2α1x˙2+ω22α1x2+ω12s2t+ω02s2tx2−2ξ2ω2α1x˙1−ω22α1x1−ω12s2tx1=0,where ξ1=c1/2k1m1, ω1=k1/m1, ξ2=c2/2k2m2, ω2=k2/m2, α1=m2/m1, ω01st=k01st/m1, ω12st=k12st/m1, ω02st=k02st/m1, and ω01sω12s,orω02s varies from ω¯01sω¯12s,orω¯02s to ω¯01sω¯12s,orω¯02s. m1m2, c1c2, k1k2, k01stk12st,k02st, and x1x2 represent mass, damping, stiffness, variable stiffness, and displacement of the 2-DOF system, respectively. If there exist positive ε2 and symmetric, positive-definite matrix X2 satisfying equation (46), the 2-DOF system with VSSs is absolutely stable.(46)A21TX2+X2A21X2B21−C21B21TX2−C21T−D21−D21T≤0,whereA21,B21,C21,D21 is converted by H2s−ε2.Figure 7
A 2-DOF model with VSSs.Proof.
Denotex1=x1,x2,x˙1,x˙2T; equation (45) can be converted to(47)x˙2=A2x2+B2u2,y2=C2x2,u2=−Ψ2t,y2,where(48)A2=00100001−ω22α1−ω12ω22α1−2ξ2ω2α1−2ξ1ω12ξ2ω2α1ω22−ω222ξ2ω2−2ξ2ω2,B2=000000−1100−1/α1−1/α1,C2=1000−11000100,Ψ2t,y2=diagω01s2,ω12s2,ω02s2y2.
The transfer function of equation (47) is(49)G2s=C1sI−A2−1B2=−s2+2ω2ξ2s+ω22dens2den−2ω2ξ2s+ω22den∗−1+α1s2+2ω1ξ1s+ω12α1⋅den−s2+2ω1ξ1s+ω12α1⋅den∗∗−s2+2ω1ξ1+2α1ω2ξ2s+ω12+α1ω22α1⋅den,where ∗ denotes the symmetrical element, and(50)den=s4+2ω1ξ1+2ω2ξ2+2ω2ξ2α1s3+ω12+ω22+ω22α1+4ω1ω2ξ1ξ2s2+2ω12ω2ξ2+2ω1ω22ξ1s+ω12ω22.
Then(51)H2s=I+K22G2sI+K21G2s−1,where K21=diagω¯01s2,ω¯12s2,ω¯02s2 and K22=diagω¯01s2,ω¯12s2,ω¯02s2. Equation (51) can be converted to the state space A21,B21,C21,D21.
From Lemma3, the transfer function H2s is strictly positive real if there exist some ε2, which satisfies that H2s−ε2 is positive real.
From Lemma4, the transfer function H2s−ε2 is positive real if there exists a symmetric, positive-definite matrix X2, which satisfies(52)A21TX2+X2A21X2B21−C21B21TX2−C21T−D21−D21T≤0.Example 3.
Chooseω1=2, ξ1=6, ω2=3, ξ2=7, α1=3, ω¯01s=0, ω¯01s=3, ω¯12s=0, ω¯12s=2, ω¯02s=0, ω¯02s=1, and ε2=0.1; the symmetric and positive-definite matrix X2 is calculated by the standard numerical simulation toolbox.(53)X2=14.9948−9.84011.11480.4961−121.998422.6820−0.4832−1.3486∗93.1342−25.2808−8.2322−43.7687−14.3113−4.82−7.2607∗∗34.710210.171522.1161.49924.4151.9909∗∗∗3.18786.1410.74260.9826−0.142∗∗∗∗1176.4−186.752510.92522.7945∗∗∗∗∗38.3025−11.2653−6.8299∗∗∗∗∗∗106.252546.7287∗∗∗∗∗∗∗24.697.
According to Theorem2, if there exist some ε2>0 and a symmetric and positive-definite matrix X2, the 2-DOF system with VSSs is absolutely stable.
Apart from verification in the frequency domain, time-domain simulation is also considered as follows. Two kinds of nonlinearities are considered. For the first one, the selectedω01s, ω12s, and ω02s are given in equation (54), which are in the ranges ω01s∈0,3, ω12s∈0,2, and ω02s∈0,1. The other selected ω01s, ω12s, and ω02s are given in equation (55), which are out of the ranges ω01s∈0,3, ω12s∈0,2, and ω02s∈0,1.
In the following, a time-domain simulation is conducted, similar to that in Example1. The values of ω01s, ω12s, and ω02s are, respectively, selected as equations (54) and (55):(54)ω01s=sint+1.5,ω12s=0.5sint+0.6,ω02s=0.1sint+1.2,(55)ω01s=sint+4,ω12s=sint+4,ω02s=sint+2.
Numerical simulation is shown in Figure8, where similar results as Example 1 can be found. The system can be stable when the selected values are in the stable ranges, as shown in Figure 8(a). However, when the selected values are out of the stable ranges, the system will be unstable, as shown in Figure 8(b).Figure 8
The phase portrait of a 2-DOF system with different nonlinearities in the feedback path. (a) The phase trajectory with the selected values in the stable ranges. (b) The phase trajectory with the selected values out of the stable ranges.
(a)(b)


## 5. Conclusions

In this paper, the inherent stability problem for multibody systems with VSSs was investigated. Based on the feature that the inherent stability implied the stability for a group of control laws within certain sectors, absolute stability theory was adopted. The generalN-DOF multibody systems with VSSs were formulated in a Lur’e form, and the circle criterion method was employed to obtain sufficient conditions for the inherent stability of the systems. Numerical examples based typical low-DOF systems were conducted to show the effectiveness of the proposed conditions in this paper. Furthermore, the results of this paper can contribute to adjusting the bounds of the stiffness of VSSs to ensure the inherent stability and can also be regarded as a guidance for designing VSSs-based mechanical control systems.



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