# Robust Stability Analysis Based on LMI for Haptic Interface Systems with Uncertain Delay

Authors: Yanwen Liu; Fanwei Meng; Bowen Guan; Shuhao Zhang
Journal: Complexity
 (2018)

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## Abstract

This paper presents the robust stability analysis based on LMI for force feedback haptic interface systems with uncertain delay. A full-order state observer is introduced to estimate states of the haptic device so as to directly obtain the feedback force, which expands the stability range of the system environment parameters when comparing with that of the system without an observer. By using the delay partitioning idea and the augmented state vector, the Lyapunov-Krasovskii functional is constructed, and then the new stability criterion based on LMI is given in the paper. With this new stability criterion, the observer gains can also be obtained directly. At last, a concrete example is used to prove the effectiveness and less conservativeness of this new approach.


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## Body

## 1. Introduction

In recent years, the haptic interaction technique has been widely used in various fields such as remote surgery, virtual assembly, CAD (computer-aided design), maintenance in space, undersea or nuclear facilities, and telerobot and virtual prototyping [1]. The haptic interaction system is a typical force feedback system, which consists of three parts: a haptic device, a human operator, and a remote or virtual environment. The haptic device and the environment form a closed-loop feedback system, which is known as the haptic interface system [2, 3]. For this system, stability is the basic requirement. Since people with subjective initiative are also included in the system and the instabilities may hurt people, the importance of stability for this kind of system becomes more important [4, 5].Many researchers use passivity as an analysis and design tool for the haptic interface systems [1, 3, 6–8]. For example, Colgate et al. consider the passivity of a haptic system and derive the passivity condition by using the frequency domain lifting technique [3]. Colgate et al. give the same passivity condition by considering the whole haptic interface system as a pure discrete time system. Woo and Lee derive a passivity condition for haptic interface systems including the impedance of the operator [9]. By using the real frequency characteristics of the system, Liu et al. give new passivity and active boundaries [5]. Bianchini et al. present a passivity condition in terms of LMI [10].Although these passive conditions are almost accepted by many researchers, the passive design is only a very conservative design approach, and the range of stability is generally much larger than that of the passivity [11, 12]. Thus, for a stable human-computer interaction, the condition for passivity is only a sufficient but not necessary condition, and seeking for a less conservative stability condition becomes a crucial issue to researchers [13–16].It needs to be emphasized that for most of the control systems, especially for computer control systems, there are usually various time delays due to program calculation, data processing, and communication [15]. Similarly, there are also such delays in the haptic interface systems, which as a major source of instability may not be ignored. Therefore, to study the stability of the system, we must consider the influence of time delays [17]. In order to obtain a more conservative delay-dependent stability criterion for the system with time delays, many methods have been proposed based on the Lyapunov-Krasovskii functional [18–21]. By far, researchers have made significant progress in this field. For example, the descriptor system approach [22], the free-weighting matrix technique [23, 24], and the delay partitioning approach [25–27].In this paper, the haptic interface system with a virtual environment for virtual wall implementation is studied. This system is in fact a typical sampled-data control system with an uncertain time-varying delay; thus, most of the research on such system is based on the discrete time control theory. For instance, based on LMIs, Dang et al. give the stability conditions of the haptic interface systems implemented by the virtual wall [2, 13, 15, 17]. In literatures [13, 15], observers are also used to directly obtain the states of the haptic device, but the state estimate errors are not included in the Lyapunov-Krasovskii functional and the observer gains cannot be directly obtained by solving the stability criterion in terms of LMIs (linear matrix inequality). Allowing this, a full-order state observer is used in this paper to estimate the states of the haptic device so as to calculate the force feedback [1, 3, 12]. The observer gains can be obtained by solving the stability criteria in terms of LMI. An augmented vector composed of states and state estimate errors is defined and used in the Lyapunov-Krasovskii functional candidate. Moreover, the utilization of the delay partitioning idea may further improve the conservatism of the stability condition given in this paper.Other parts of the paper are arranged as follows. Section2 gives the structure diagram and the model of the haptic interface system. In Section 3, the discretized state space model for the plant is constructed, and the full-order state observer is presented. Section 4 gives the mathematical preliminaries used in the paper. By using the delay partitioning idea and the augmented state vector, a novel Lyapunov-Krasovskii functional is constructed in Section 5, and then the new stability criterion based on LMI is given in this section. In Section 6, the effectiveness and less conservativeness of the proposed approach are testified through a simulation example. The conclusions are given in Section 7.


## 2. Model of the System

Figure1 shows the structure diagram of the haptic system, which is a one degree of freedom system and with an environment for stiff virtual wall implementation. In Figure 1, the haptic device is equivalent to the continuous plant, and the virtual wall environment is equivalent to the discrete controller. td denotes the uncertain delay. For the convenience of the following analysis, we use the discrete-time form delay by defining d=td/T and assuming that dk satisfies 1≤dm≤dk≤dM, where dm and dM are both constant positive scalars. H is a ZOH (zero order hold), f denotes the force signal exerted by a human operator, q is the displacement output of the haptic device, u is the feedback signal from the environment, which is also a force signal, and T is the sampling period.Figure 1
Structure diagram of the haptic interface system.Since a human operator is also a part of the haptic interface system, its dynamic model is also considered in some studies. Simulation and experimental results in the literature [7, 11, 12, 14, 16] show that the human operator may actually make the system more stable, and this conclusion can be further proved by Figure 2 in this paper. Therefore, the dynamic model of the human operator can be ignored in the stability analysis of the haptic interface system.Figure 2
Stability regions in theB–K parametric plane (without delay).If the human operator model is considered and it is assumed that the mass of the haptic device ism, the damping coefficient of the haptic device is b, the mass of the human operator is mh, the damping coefficient of the human operator is bh, the stiffness coefficient of the human operator is kh, then the dynamic model of the human operator and haptic device can be described as
(1)m+mhq¨+b+bhq̇+khq=u+f.In view of the above analysis, the human operator model will be ignored in the following parts of the paper. Thus, the dynamic model of the haptic device (plant) is described asx1=q
(2)mq¨+bq̇=u+f.The state vectorx=x1x2T is chosen as and x2=q̇=v (v is the velocity of the haptic device); the output is chosen as y=x1=q; then the state space expression for the haptic device is
(3)ẋ=Ax+B1u+B2f,y=Cx,where
(4)A=010−bm,B1=B2=01m,C=10.For the stiff virtual wall environment, it is composed of a virtual spring and a damper in mechanical parallel. Defining the stiffness of the spring asKe and the damping coefficient of the damper as Be and assuming the position of the virtual wall is equal to zero, the virtual force is then given by
(5)Fe=Keq+Beq̇,contact;0,noncontact;where q in (5) is the displacement of the haptic device.


## 3. Realization of the State Observer

In order to obtain the discrete state observer, we should first derive the discretized model of the haptic device. By combining ZOH, the sampler, and the delaydk with the haptic device, we can get the following discretized model of the haptic device:
(6)xk+1=Adxk+B1duk−dk+B2dfk,yk=Cdxk,where
(7)Ad=eAT,B1d=B2d=∫0TeAtB1dt,Cd=C.Since the discrete-time signalsq∗ and v∗ are needed to compute the feedback force Fe∗, a full-order state observer is used here to estimate the discrete-time states (q∗ and v∗) of the haptic device.According to the system model in expression (6), the full-order state observer is of the following form:
(8)x^k+1=Adx^k+B1duk−dk+Ldyk−y^k,y^k=Cdx^k.Defining the state estimation error asek=xk−x^k, we can get the following dynamic equation:
(9)ek+1=Ad−LdCdek+B2dfk.Considering the following control law:(10)uk=−Kdx^k=−KeBex^k.And defining the augmented vector as(11)αk=xkek.The following augmented closed-loop system can be obtained:(12)αk+1=A~αk+B~1uk−dk+B~2fk,yk=C~αk,where
(13)A~=Ad00Ad−LdCd,B~1=−B1dKdB1dKd00,B~2=B2dB2d,C~=Cd0.From expression (12), we can get the discrete-time state space (14), which will be used for the next asymptotic stability analysis:
(14)xk+1ek+1=Ad00Ad−LdCdxkek+−B1dKdB1dKd00xk−dkek−dk.


## 4. Mathematical Preliminaries

The following lemmas are prepared for the next derivations.Lemma 1 [28].
For the given matricesX and Y with appropriate dimensions, one has
(15)XTY+YTX≤εXTX+ε−1YTY,∀ε>0.Lemma 2 [13].
For any symmetric positive definite matrixR∈ℝn×n, integers q≥p>0, and vector function x:p,q→ℝn, the following inequality holds
(16)−q−p+1∑i=pqxTiRxi≤∑i=pqxiT−R∑i=pqxi.Lemma 3 (Schur complement) [29].
Given the constant matricesM, N, and Q with appropriate dimensions, where MT=M and QT=Q>0, then M+NTQ−1N<0 if and only if
(17)MNTN−Q<0,
or equivalently(18)−QNNTM<0.


## 5. Main Results

A novel Lyapunov-Krasovskii functional is constructed for the discrete-time haptic interface system (14), and then a new stability criterion based on LMI will be given in this section. The main result is as follows.Theorem 1.
For the given positive integersτ and dM, the discrete-time system (14) is asymptotically stable if the LMI in inequality (19) holds. Where the variables should satisfy ε>0 and P1,P2,Q1,Q2,Q3,Q4,Q5,Q6,R1,R2,R3,R4,Z1,andZ2 are positive matrices with appropriate dimensions,
(19)Ω11′000Z10000000AdTP1Ad−ITZ1Ad−ITZ2000∗Ω22′0000000000000AdTPCdTVCdTV∗∗−R3−2Z200000Z20Z20−KdTB1dTP1−KdTB1dTZ1−KdTB1dTZ2000∗∗∗−R400000000KdTB1dTP1KdTB1dTZ1KdTB1dTZ2000∗∗∗∗Q3−Q1−Z10000000000000∗∗∗∗∗Q4−Q2000000000000∗∗∗∗∗∗Q5−Q300000000000∗∗∗∗∗∗∗Q6−Q40000000000∗∗∗∗∗∗∗∗−Q5−Z2000000000∗∗∗∗∗∗∗∗∗−Q600000000∗∗∗∗∗∗∗∗∗∗−R1−Z20000000∗∗∗∗∗∗∗∗∗∗∗−R2000000∗∗∗∗∗∗∗∗∗∗∗∗−P100000∗∗∗∗∗∗∗∗∗∗∗∗∗−1τ2Z10000∗∗∗∗∗∗∗∗∗∗∗∗∗∗−1ρ2Z2000∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−P200∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−P20∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗−ε−1I<0,where ρ=dM−dm=dM−3τ.
(20)Ω11′=−P1+Q1+R1+ρ+1R3−Z1,Ω22′=ε−1AdTAd−P2+Q2+R2+ρ+1R4.Proof.
Firstly, by using the delay partitioning idea [25], the lower bound of delay dk can be described as dm=Nτ, where N and τ are integers. Then one can express the time delay dk as dk=dm+δk=Nτ+δk, where δk∈0,dM−Nτ. In the following derivations, one chooses dm=3τ (N=3) to divide dm into three parts.
Now, one chose the Lyapunov-Krasovskii functional as(21)Vk=∑i=17Vik,where
(22)V1k=αTkP100P2αk,V2k=∑i=k−τk−1αTiαTi−ταTi−2τQ100Q2000000000000Q300Q4000000000000Q500Q6αiαi−ταi−2τ,V3k=∑i=k−dMk−1αTiR100R2αi,V4k=∑i=k−dkk−1αTiR300R4αi,V5k=∑j=dM+1−dm∑i=k+jk−1αTiR300R4αi,V6k=τ∑j=−τ−1∑i=k+jk−1ΔxTiZ1Δxi,V7k=dM−dm∑j=−dM−dm−1∑i=k+jk−1ΔxTiZ2Δxi,Δxk=xk+1−xk.
DefiningΔVk=Vk+1−Vk. Then by using the state-space (14), one can get
(23)ΔV1k=xTk+1P1xk+1−xTkP1xk+eTk+1P2ek+1−eTkP2ek=xTkAdTP1Ad−P1xk−2xTkAdTP1B1dKdxk−dk+2xTkAdTP1B1dKdek−dk+xTk−dkKdTB1dTP1B1dKdxk−dk−2xTk−dkKdTB1dTP1B1dKdek−dk+eTk−dkKdTB1dTP1B1dKdek−dk+ekAdTP2Ad−P2−AdTP2LdCd−CdTLdTP2Ad+CdTLdTP2LdCdek,
By using Lemma1, one has
(24)ΔV1k≤xTkAdTP1Ad−P1xk−2xTkAdTP1B1dKdxk−dk+2xTkAdTP1B1dKdek−dk+xTk−dkKdTB1dTP1B1dKdxk−dk−2xTk−dkKdTB1dTP1B1dKdek−dk+eTk−dkKdTB1dTP1B1dKdek−dk+ekAdTP2Ad−P2+εCdTLdTP2P2LdCd+ε−1AdTAd+CdTLdTP2LdCdek,(25)ΔV2k=xTkQ1xk+eTkQ2ek+xTk−τQ3−Q1xk−τ+eTk−τQ4−Q2ek−τ+xTk−2τQ5−Q3xk−2τ+eTk−2τQ6−Q4ek−2τ−xTk−dmQ5xk−dm−eTk−dmQ6ek−dm,(26)ΔV3k=xTkR1xk+eTkR2ek−xTk−dMR1xk−dM−eTk−dMR2ek−dM,(27)ΔV4k=∑i=k+1−dk+1k+1αTiR300R4αi−∑i=k−dkkαTiR300R4αi≤αTkR300R4αk−αTk−dkR300R4αk−dk+∑i=k−dM+1k−dmαTiR300R4αi,(28)ΔV5k=dM−dmαTkR300R4αk−∑i=k−dM+1k−dmαTiR300R4αi.
Defineρ=dM−dm, then from (27) and (28), one can get
(29)ΔV4k+ΔV5k≤xTkρ+1R3xk+eTkρ+1R4ek−xTk−dkR3xk−dk−eTk−dkR4ek−dk.
By using Lemma2, one has
(30)ΔV6k=τ2ΔxTkZ1Δxk−τ∑i=k−τk−1ΔxTiZ1Δxi≤τ2xTkAd−ITZ1Ad−Ixk−2τ2xTkAd−ITZ1B1dKdxk−dk+2τ2xTkAd−ITZ1B1dKdek−dk+τ2xTk−dkKdTB1dTZ1B1dKdxk−dk−2τ2xTk−dkKdTB1dTZ1B1dKdek−dk+τ2eTk−dkKdTB1dTZ1B1dKdek−dk−xkxk−τT−Z1Z1Z1−Z1xkxk−τ.
Similarly, by using Lemma2, one can get
(31)ΔV7k=ρ2ΔxTkZ2Δxk−ρ∑j=k−dkk−dm−1ΔxTiZ2Δxi−ρ∑j=k−dMk−dk−1ΔxTiZ2Δxi≤ρ2xTkAd−ITZ2Ad−Ixk−2ρ2xTkAd−ITZ2B1dKdxk−dk+2ρ2xTkAd−ITZ2B1dKdek−dk+ρ2xTk−dkKdTB1dTZ2B1dKdxk−dk−2ρ2xTk−dkKdTB1dTZ2B1dKdek−dk+ρ2eTk−dkKdTB1dTZ2B1dKdek−dk−xk−dMxk−dkT−Z2Z2Z2−Z2xk−dMxk−dk−xk−dkxk−dmT−Z2Z2Z2−Z2xk−dkxk−dm.
Denoting(32)ξTk=xTk,eTk,xTk−dk,eTk−dk,xTk−τ,eTk−τ,xTk−2τ,eTk−2τ,xTk−dm,eTk−dm,xTk−dM,eTk−dM.
Then, from (23), (24), (25), (26), (27), (28), (29), (30), and (31), one has
(33)ΔVk=∑i=17ΔVik≤ξTkΩξk,where
(34)Ω=Ω110Ω13Ω14Z10000000∗Ω220000000000∗∗Ω33Ω340000Z20Z20∗∗∗Ω4400000000∗∗∗∗Q3−Q1−Z10000000∗∗∗∗∗Q4−Q2000000∗∗∗∗∗∗Q5−Q300000∗∗∗∗∗∗∗Q6−Q40000∗∗∗∗∗∗∗∗−Q5−Z2000∗∗∗∗∗∗∗∗∗−Q600∗∗∗∗∗∗∗∗∗∗−R1−Z20∗∗∗∗∗∗∗∗∗∗∗−R2,with
(35)Ω11=AdTP1Ad−P1+Q1+R1+ρ+1R3+τ2Ad−ITZ1Ad−I−Z1+ρ2Ad−ITZ2Ad−I,Ω13=−AdTP1B1dKd−Ad−ITZ1B1dKd−Ad−ITZ2B1dKd,Ω14=AdTP1B1dKd+Ad−ITZ1B1dKd+Ad−ITZ2B1dKd,Ω22=AdTP2Ad+εCdTLdTP2P2LdCd+ε−1AdTAd+CdTLdTP2LdCd−P2+Q2+R2+ρ+1R4,Ω33=KdTB1dTP1B1dKd−R3+KdTB1dTZ1B1dKd+KdTB1dTZ2B1dKd−2Z2,Ω34=−KdTB1dTP1B1dKd−KdTB1dTZ1B1dKd−KdTB1dTZ2B1dKd,Ω44=KdTB1dTP1B1dKd+KdTB1dTZ1B1dKd+KdTB1dTZ2B1dKd−R4.
Then, the delay-dependent stability condition for system (14) is Ω<0.
DefiningLdT=VP2−1, then the condition Ω<0 is equivalent to the inequality (19) by using the Schur complement (Lemma 3). This completes the proof.Remark 1.
In the Lyapunov-Krasovskii functionalVk, the summation term V4k and double summation term V5k use the same matrix variables R3 and R4. On one hand, the number of matrices that need to be solved is reduced, and the total amount of calculation is reduced. On the other hand, in the process of solving the forward difference ΔV4k and ΔV5k, the terms
(36)+∑i=k−dM+1k−dmαTiR300R4αi,
and(37)−∑i=k−dM+1k−dmαTiR300R4αi,
cancelled each other, which simplifies the final result of Theorem1.Remark 2.
By introducing the partitioned delayτ into the double summation term V6k, the time delay information of the system is fully utilized, and it is expected that the results obtained in this paper are less conservative.Remark 3.
The double summation termV7k introduces the time delay bound information dm and dM. During the process of solving the forward difference ΔV7k, the term −ρ∑j=k−dMk−dm−1ΔxTiZ2Δxi is divided into two parts: −ρ∑j=k−dkk−dm−1ΔxTiZ2Δxi and −ρ∑j=k−dMk−dk−1ΔxTiZ2Δxi, then the time delay information dk is increased, which is expected to reduce the possible conservatism. This can be further proved by the following examples.


## 6. Numerical Simulation Results

We adopt the parameters of the haptic interface in literatures [3, 5, 30]: m=0.5 kg and b=0.1 N·s/m. Supposing the sampling period is T=0.001 s.According to expression (3), the continuous plant of the system is given by
(38)ẋ=010−0.2x+02u+02f,y=10x.With(39)A=010−0.2,B=02,C=10.According to (6), the corresponding discretized state space matrixes are
(40)Ad=10.9999×10−300.9998,Bd=0.9999×10−60.1999×10−2,Cd=10.The special case ofdk=0 is firstly considered here. For comparison with the passivity conditions in previous literatures, we define two dimensionless parameters α=Be/KeT and β=b/Be. The passivity and stability regions in the α−β plane with dk=0 are shown in Figure 3. Where the dotted line is the passivity bound provided by the literature [3, 30], the dashed line is the passivity bound by the literature [5], and the solid line shows the stability bound. From Figure 3, we can conclude that the passivity domain is only a small region contained in the stability domain.Figure 3
Passivity and stability regions in theα–β plane (without delay).To further testify the influence of the human operator model, the stability regions of the system with different operator models are given in Figure2. Parameters [11, 14] for the human operator model are shown in Table 1. The curve 0 in Figure 2 is the stability region for the system without the human operator model for dk=0, and curves 1 to 6 denotes the stability regions for different operator parameters in Table 1. As can be seen from Figure 2, the stability region for the system without considering the human operator model is the smallest one. That is to say, the human operator can actually make the system more stable.Table 1
Parameters for different operator models [11, 14].
Model
mh (kg)
bh (N·s/m)
kh (N/m)
1
17.51
175.12
175.12
2
4.54
6.83
12.5
3
2.0
2.0
10.0
4
1.0
12.6
39.5
5
0.8
5.5
568
6
0.15
0.5
7.0We now add the time-varying delaydk into the haptic system. Parameters of the feedback control law (virtual wall) in expression (10) are selected as Ke=760, Be=40 (α≈52 and β=0.0025). Supposing that the lower delay bound is dm=3τ. According to the main results of Theorem 1 in this paper, we can see that system (14) is asymptotically stable for dM≤15. The allowable maximum dM for different τ is presented in Table 2.Table 2
Allowable maximumdM for different τ.
dm
3
6
9
τ
1
2
3
dM
6
10
15The observer gains for different cases are as follows:(41)Ld1=2.6511748T,dm=3,τ=1,dM=6,(42)Ld2=2.9962220T,dm=6,τ=2,dM=10,Ld3=3.71173442T,dm=9,τ=3,dM=15.Introducing the maximum delaydM=15, the real and observed states (displacement q and velocity v) of the haptic interface system under the sinusoidal input of ft are shown in Figures 4 and 5. In Figure 5, curve 0 denotes the real velocity and curves 1 to 3 denote the observed velocity with the observer gain Ld1 to Ld3, respectively. From Figures 4 and 5, we can see that the minimum state estimate errors are obtained by using the observer gains Ld3 in (42). Thus, the observer gains Ld3 will be used in the following simulations.Figure 4
Real and observed positionq of the haptic device.Figure 5
Real and observed velocityv of the haptic device.Figure6 shows the stability regions in the B–K parametric plane with different dM. As can be seen from Figure 6, the maximum time delay is closely related to the stability of the system. The larger the upper delay bound dM is, the smaller is the stability region.Figure 6
Stability regions in theB–K parametric plane.Table3 lists the maximum stiffness Ke and the corresponding damping coefficient Be under different dM. As can be seen from Figure 3, Table 3, and Figure 6, the allowable virtual wall parameters are far greater than the passive ones.Table 3
MaximumKe for different dM.
dM
6
10
15
Ke (N/m)
5830
2320
1097
Be (N·s/m)
70
40
30To further testify advantages of the stability criterion proposed in this paper, we also use other methods in existing literatures for the above plant (40). Table 4 lists the allowable maximum delay upper bounds dM for different dm. From Table 4, we can see that the results by using the approach proposed in this paper are obviously superior to those of the existing literatures, so the advantages and less conservativeness of the stability criterion given in this paper are further proved.Table 4
MaximumdM for different dm.
dm
3
6
9
12
Ibrir et al. [28]
5
8
11
13
Zhang et al. [19]
6
8
12
13
Dang et al. [15]
6
9
13
14
Theorem1
6
10
15
15


## 7. Conclusions

This paper presents the robust stability analysis based on LMI for force feedback haptic interface systems with uncertain delay. By using the delay partitioning approach and augmented vector composed of states and state estimate errors, a novel less conservative stability criterion is obtained based on LMI. Through a concrete haptic example, it can be concluded that the passive condition is only a small region contained in the stability domain, and the maximum time delay has a great effect on the stability of the system. By comparing with the results available in the existing literatures, the advantages and less conservativeness of the stability criterion given in this paper are further proved.



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