# Quadratic Integral Sliding Mode Control for Nonlinear Harmonic Gear Drive Systems with Mismatched Uncertainties

Authors: Runze Ding; Lingfei Xiao
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

For a class of nonlinear harmonic gear drive systems with mismatched uncertainties, a novel robust control method is presented on the basis of quadratic integral sliding mode surface, and the closed-loop system has satisfying performance and strong robustness against mismatched uncertainties and nonlinear disturbances. Considering time-varying nonlinear torques and parameters variations which are caused by nonlinear frictions and backlash, a nonlinear harmonic gear drive system mathematic model is established and the effect of nonlinear parts is compensated during control system design. It is proven that the quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is asymptotic stable robustly. The simulation studies are carried out in comparison with traditional linear sliding mode control and integral sliding mode control, verifying the effectiveness of the proposed method.


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## Body

## 1. Introduction

Harmonic gear is a new type of drive which is realized by elastic deformation movement. Instead of using rigid component, it uses a flexible component to realize the drive. Therefore, it has many special properties which cannot be obtained in other drives. The harmonic gear is composed of three main components, including a circular spline, a flexspline, and a wave generator. Through the action of the wave generator, a movable deformation wave is generated in the flexspline and the flexspline engages with the circular spline to achieve the transmission. The harmonic gear has advantages of large range of transmission ratio, multiteeth gearing, large carrying capacity, high precision, and running smoothly without impact. Because of its own characteristics, harmonic gear is widely applied in aviation control systems, instrumentation, robotics, and other fields [1–4], where drive systems are required to be capable of small volume, light weight, high transmission precision, and small hysteresis.Despite those advantages, many issues will occur during runtime, such as assembly errors, gear wears, and deterioration of working environment. Due to kinetic errors and nonlinear elastic deformations, these issues will cause motion delay, transmission errors, and inaccurate tracking, which leads to the fact that it is very important to focus on angular transmission errors in harmonic gear drive systems.In order to reduce the errors, many researchers have done wide investigations on force analysis and movement analysis. The stress of flexspline in harmonic gear drive is analyzed in [5]. At same time, a harmonic gear drive system dynamic system is established by dynamic equation analysis and least square method in [6]. In [7], the research of theoretical modeling of nonlinear torsional behavior in harmonic gear is carried out. These researches abovementioned focus on harmonic gear drive systems’ dynamic model, analysis of errors in harmonic gear drive systems, or nonlinear frictions model, which illustrates the fact that angular transmission errors of harmonic gear drive system have been widely analyzed and have been mathematically modeled in various conventional studies and make it practical to compensate the error by control methods.However, there are fewer researches on this part and one primary reason for this is harmonic gear drive system, a class of nonlinear systems, which are affected by mismatched uncertainties such as nonlinear torques and working condition deterioration. It is hard to control such nonlinear systems with mismatched uncertainties and nonlinear disturbances. Consequently, researches in this area are developed slowly. Previous research [8] designs a PID controller for a class of harmonic drive system with frictions to reduce transmission errors, and as a result, harmonic gear drive system’s performance is enhanced by error compensation. The results show that using control method to reduce error is feasible but [8] failed to consider mismatched uncertainties and nonlinear disturbances, which are inevitable in practice, while [9] designing a sliding mode controller based on linear sliding mode surface, which has a better performance than PID controller, and taking disturbances into consideration. Though it has its improvements, there are also some limitations in [9] such as ignoring some critical torques in modeling and designing, using linear sliding mode surface and no attention to uncertainties.Sliding mode control (SMC) is a new variable structure robust control method. By choosing the sliding surface and the reaching law, it can make the system response quickly with strong robustness and have insensibility to external interferences and parameters variations. As an important method of nonlinear control, it has been widely researched in recent years.In [10], Xie Jian et al. design a sliding mode controller for hydraulic pump-control-motor system and use simulation results to illustrate that the sliding mode controller has strong anti-interference ability and good tracking performance when compared with PID controller, improving the control accuracy and stability of the system. In [11], by using boundary layer method to restrain the chatting, NE Sadr et al. design a sliding mode controller for missile autopilot and the system has strong robustness to uncertainties and disturbances. Besides, Murat Furat et al. design a second-order integral sliding mode controller for a SISO system in [12], making the system overcome the parameter fluctuation and uncertainties caused by the external load. Furthermore, Azar A T et al. design an adaptive sliding mode controller for the Furuta Pendulum in [13]. In comparison with other controllers, the simulation result shows that the sliding mode control has advantages of fast response and small errors. In addition, Ginoya D et al. design a sliding mode controller using extended disturbance observer for a system with uncertainties in [14] and the closed-loop system has robustness to the uncertainties. Hess RA et al. design a slide mode controller for a nonlinear unmanned aerial vehicle in [15], and the controller has a good performance on nonlinear system.Investigations in [10–15] illustrate that the sliding mode control method finds widel applications on nonlinear system control and has a good performance. Besides, the closed-loop system has insensitivity to disturbance and uncertainties, which means that the system has strong robustness against uncertainties including parameters variations and disturbances. However, the insensitivity to uncertainties is realized only in sliding phase. If system is not on the sliding mode surface, the system will be vulnerable to uncertainties. Additionally, the robustness is only limited to matched uncertainties. The so-called integral sliding mode control (ISMC) is proposed to improve such situation but, unfortunately, the performance of ISMC is conservative.A class of nonlinear harmonic gear drive systems with mismatched uncertainties is studied and the primary contributions of this paper are as follows.(1) A novel quadratic integral sliding mode controller design method is presented on the basis of quadratic integral sliding mode surface. (2) Considering frictions, time-varying nonlinear torques, and parameters variations which are caused by backlash, the influence of nonlinear parts on control system has been studied. The mathematical models of nonlinear torques and frictions are presented and the effect of nonlinear parts is compensated during control system design. (3) In system modeling, take frictions and nonlinear elastic deformation into consideration and the mathematical model of harmonic gear drive system is established. (4) By using Lyapunov stability theory, it is proven that the quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is robustly asymptotic stable. (5) In comparison with the traditional linear sliding mode control (which can be abbreviated as SMC) and the integral sliding mode control (which can be abbreviated as ISMC), the simulation results demonstrated that quadratic integral sliding mode control (which can be abbreviated as QISMC) stabilizes the system rapidly with fast response, short rise time, small overshoot, small error, and strong robustness against the mismatched uncertainties and nonlinear disturbances.The remainder of this paper is organized as follows. Section2 gives details on harmonic gear drive systems modeling and illustrations for backlash and frictions. In Section 3, the quadratic integral sliding surface is presented; furthermore, its corresponding quadratic integral sliding mode controller is designed in Section 4. Section 5 proves that quadratic integral sliding mode surface can be reached in finite time and the closed-loop system is robustly asymptotic stable. The effectiveness of the quadratic integral sliding mode control is verified in comparison with SMC and ISMC in Section 6, and Section 7 draws the conclusions of this paper.


## 2. Harmonic Gear Drive System

Harmonic gear drive system (Figure1) is mainly divided into the DC motor system and the harmonic gear system. The DC motor is driven by controller and outputs torques to the wave generator in harmonic gear system; as a result, torques are transmitted by engagement to a flexspline. A mathematical description of harmonic gear drive system will be given in this part.Figure 1
Structure of Harmonic gear drive system.


### 2.1. DC Motor System

Using Kirchhoff’s law of electricity, DC motor system can be presented by the following:(1)ut=iaR+Ldiadt+Kbωmwhere ia is the motor armature current, ut is the motor armature voltage, R is the equivalent resistance, Kb is the back-EMF coefficient, and ωm is the angel speed of motor rotor. L is the motor armature inductance, which is generally ignored due to its tiny value.


### 2.2. Harmonic Gear System

As Figure2 [16] shows, harmonic gear system can be simplified as a typical two-mass system including a circular spline, a flexspline, and a wave generator.Figure 2
Harmonic gear system.Harmonic gear transmission system satisfies following dynamic equations:(2)Jmq¨m=Tm-Ts-FmJlq¨l=Ts-Flwhere qm is the angular displacement of the wave generator, ql is the angular displacement of the flexspline, Tm is the torques on the wave generator, and Ts is the torques on the flexspline. Fm is the equivalent frictions on the wave generator and Fl is the equivalent frictions on the flexspline. Jm is the moment of inertia of the wave generator and Jl is the moment of inertia of the flexspline.The kinetic model of harmonic gear drive system satisfies the following:(3)Ts=Ksqmr-ql(4)Tm=Kmiawhere r is the reduction ratio, Ks is the torsional stiffness of the harmonic gear, and Km is the motor torque coefficient.(1) gives(5)ia=-Kbq˙m+utRSubstituting ia into (4), it would be followed that(6)Tm=-KmKbRq˙m+KmRutSubstituting (3) and (6) into (2), one can obtain harmonic gear drive systems kinetic model:(7)Jmq¨m=KmRut-Ksqmr-ql-KmKbRq˙m+FmJlq¨l=Ksqmr-ql-FlConsidering the frictions model [16], because the angle range in accurate control is small, it is acceptable to choose the average Coulomb's frictions fcm as whole frictions in harmonic gear drive systems.(8)fcm=s1qm2+s2qm+s3The static frictions fsm is about 3.88% larger than fcm; then one can obtain(9)fsm=1.0388×fcmAnd frictions on wave generator Fmx is(10)Fmx=-ψm,q˙m=0,ψm≤fsm-sgnψmfsm,q˙m=0,ψm>fsm-sgnqmfcm,q˙m>0where ψm=-Ksqm/r-ql+Km/Rut.In the same way, frictions on the flexsplineFl(x) would be as follows:(11)Flx=-ψl,q˙l=0,ψl≤fsl-sgnψlfsl,q˙l=0,ψl>fsl-sgnqlfcl,q˙l>0where ψl=1/rKsqm/r-ql.The gears will heat because of frictions in gear drive system. Correspondingly, a gap which is called backlash [17] is designed as the expansion space for gears (shown in Figure 3). These gaps will continue to increase due to wears and frictions caused by repeated stop, variable speed, the working environment, and other reasons, finally leading to motion delay and inaccurate tracking.Figure 3
Backlash.A mathematical expression of backlash is introduced in [18], considering the dead-zone model, and defines the elastic deformation torsion angle Δet caused by backlash as(12)Δet=-Δφ-j,qmtr-qlt-Δφ>j0,qmtr-qlt-Δφ<j-Δφ+j,qmtr-qlt-Δφ<-jwherej is the width of backlash and Δφ is the transmission error caused by nonelastic deformation torsion angle, which can be approximated by random number model. The Δet dead-zone model can be presented by Figure 4.Figure 4
Δ e t dead-zone model.Because ofΔet, a nonlinear torque Tulx=KsΔet is attached to harmonic gear drive systems. In addition, it is inevitable that system parameters and characteristics will drift slowly due to factors such as gear wear and environment. As a result, choose ΔAx as the state parameters variations and ΔB as the control signals variations.Choose state vectors asx1x2x3x4T=qmq˙mqlq˙lT and harmonic gear drive systems can be presented as follows: (13)x˙1=x2x˙2=-φ21x1-φ22x2+φ23x3-Tulx+Tmx+φ2uutx˙3=x4x˙4=φ41x1-φ43x3+Tulx+Tlx+ΔAx+ΔBut(14)φ21=KsrJm,φ22=KmKbJmR,φ23=KsJm,Tmx=FmxJmφ2u=KmJmR,φ41=KsrJl,φ43=KsJl,Tlx=FlxJlIt is shown in (13) and (14) that the system of harmonic gear drive system is a class of nonlinear system with mismatched uncertainties including parameters perturbation and external disturbance.


## 2.1. DC Motor System

Using Kirchhoff’s law of electricity, DC motor system can be presented by the following:(1)ut=iaR+Ldiadt+Kbωmwhere ia is the motor armature current, ut is the motor armature voltage, R is the equivalent resistance, Kb is the back-EMF coefficient, and ωm is the angel speed of motor rotor. L is the motor armature inductance, which is generally ignored due to its tiny value.


## 2.2. Harmonic Gear System

As Figure2 [16] shows, harmonic gear system can be simplified as a typical two-mass system including a circular spline, a flexspline, and a wave generator.Figure 2
Harmonic gear system.Harmonic gear transmission system satisfies following dynamic equations:(2)Jmq¨m=Tm-Ts-FmJlq¨l=Ts-Flwhere qm is the angular displacement of the wave generator, ql is the angular displacement of the flexspline, Tm is the torques on the wave generator, and Ts is the torques on the flexspline. Fm is the equivalent frictions on the wave generator and Fl is the equivalent frictions on the flexspline. Jm is the moment of inertia of the wave generator and Jl is the moment of inertia of the flexspline.The kinetic model of harmonic gear drive system satisfies the following:(3)Ts=Ksqmr-ql(4)Tm=Kmiawhere r is the reduction ratio, Ks is the torsional stiffness of the harmonic gear, and Km is the motor torque coefficient.(1) gives(5)ia=-Kbq˙m+utRSubstituting ia into (4), it would be followed that(6)Tm=-KmKbRq˙m+KmRutSubstituting (3) and (6) into (2), one can obtain harmonic gear drive systems kinetic model:(7)Jmq¨m=KmRut-Ksqmr-ql-KmKbRq˙m+FmJlq¨l=Ksqmr-ql-FlConsidering the frictions model [16], because the angle range in accurate control is small, it is acceptable to choose the average Coulomb's frictions fcm as whole frictions in harmonic gear drive systems.(8)fcm=s1qm2+s2qm+s3The static frictions fsm is about 3.88% larger than fcm; then one can obtain(9)fsm=1.0388×fcmAnd frictions on wave generator Fmx is(10)Fmx=-ψm,q˙m=0,ψm≤fsm-sgnψmfsm,q˙m=0,ψm>fsm-sgnqmfcm,q˙m>0where ψm=-Ksqm/r-ql+Km/Rut.In the same way, frictions on the flexsplineFl(x) would be as follows:(11)Flx=-ψl,q˙l=0,ψl≤fsl-sgnψlfsl,q˙l=0,ψl>fsl-sgnqlfcl,q˙l>0where ψl=1/rKsqm/r-ql.The gears will heat because of frictions in gear drive system. Correspondingly, a gap which is called backlash [17] is designed as the expansion space for gears (shown in Figure 3). These gaps will continue to increase due to wears and frictions caused by repeated stop, variable speed, the working environment, and other reasons, finally leading to motion delay and inaccurate tracking.Figure 3
Backlash.A mathematical expression of backlash is introduced in [18], considering the dead-zone model, and defines the elastic deformation torsion angle Δet caused by backlash as(12)Δet=-Δφ-j,qmtr-qlt-Δφ>j0,qmtr-qlt-Δφ<j-Δφ+j,qmtr-qlt-Δφ<-jwherej is the width of backlash and Δφ is the transmission error caused by nonelastic deformation torsion angle, which can be approximated by random number model. The Δet dead-zone model can be presented by Figure 4.Figure 4
Δ e t dead-zone model.Because ofΔet, a nonlinear torque Tulx=KsΔet is attached to harmonic gear drive systems. In addition, it is inevitable that system parameters and characteristics will drift slowly due to factors such as gear wear and environment. As a result, choose ΔAx as the state parameters variations and ΔB as the control signals variations.Choose state vectors asx1x2x3x4T=qmq˙mqlq˙lT and harmonic gear drive systems can be presented as follows: (13)x˙1=x2x˙2=-φ21x1-φ22x2+φ23x3-Tulx+Tmx+φ2uutx˙3=x4x˙4=φ41x1-φ43x3+Tulx+Tlx+ΔAx+ΔBut(14)φ21=KsrJm,φ22=KmKbJmR,φ23=KsJm,Tmx=FmxJmφ2u=KmJmR,φ41=KsrJl,φ43=KsJl,Tlx=FlxJlIt is shown in (13) and (14) that the system of harmonic gear drive system is a class of nonlinear system with mismatched uncertainties including parameters perturbation and external disturbance.


## 3. Quadratic Integral Sliding Mode Design

In this part, a quadratic integral sliding mode surface is designed to compensate the influence of mismatched uncertainties and a detailed analysis of system in sliding mode is given to verify the effectiveness of proposed method.Consider an arbitrary nonlinear system with mismatched uncertainties as follows:(15)x˙=fx,t+Δfx,t+gx,t+Δgx,tuwhere x∈Rn is state vector, u∈R is control signal, and fx,t=f1x,tf2x,t…f4x,tT are nonlinear dynamic vectors. gx,t=g1x,tg2x,t…g4x,tT are nonlinear control gain vectors, Δfx,t=Δf1x,tΔf2x,t…Δf4x,tT are parameters perturbations and external disturbances, and Δgx,t=Δg1x,tΔg2x,t…Δg4x,tT are parameters perturbations of control gain. Here, Δfx,t+Δgx,tu are uncertainties.Suppose that there exists a control lawu0, which is designed in a desired way, stabilizing nominal system (16), with a given accuracy (the nominal system is the system without uncertainties).(16)x-˙=fx,t+gx,tu0Considering the nonlinear system with mismatched uncertainties (15), the uncertainties are unknown, but it is acceptable to assume that the uncertainties are bounded. Consequently, assumptions are introduced as follows.Assumption 1.
Δ f x , t ≤ ξ 1 x + ξ 0 ≤ f x , t, with the scalars ξ0>0,ξ1>0; here· stands for Euclidean norm.Assumption 2.
Δ g x , t ≤ ζ 1 x + ζ 0 ≤ g x , t, with the scalars ζ0>0,ζ1>0; here· stands for Euclidean norm.The quadratic integral sliding mode control law will be designed as(17)u=u0+u1where u0 stands for the abovementioned control law of nominal system, which is responsible for the performance of the nominal system and satisfiesAssumption3.Assumption 3.
u 0 ≤ β 0 + β 1 x, with the scalars β0>0,β1>0.Whileu1 stands for discontinuous control action that compensates the mismatched uncertainties, based on quadratic integral sliding mode surface, the definition of u1 will be introduced in (32).In order to compensate the effect of mismatched uncertainties, the quadratic integral sliding mode surface is designed as(18)s=12xTtxt-xt0Txt0-∫t0txTtfx,t+gx,tu-bxu1dtHere xt0 is the initial values of the state vector and b(x)∈R is demonstrated in (19)(19)bx=σ+ζ0x+ζ1x2+xTtgx,twith the arbitrary scalar σ>0.The sliding mode surface (18) is in a quadratic form; therefore, (18) is a quadratic integral sliding mode surface. Assuming that the quadratic integral sliding mode surface s=0 can be reached, it leads to(20)12xTtxt-xt0Txt0-∫t0txTtfx,t+gx,tu-bxu1dt=0and the following can be obtained:(21)s=∫t0txTtΔfx,t+Δgx,tu+bxu1dt=0If the quadratic integral sliding mode surface (21) is reached and remains there, then (22)s˙=xTtΔfx,t+Δgx,tu+bxu1=0One can obtain the equivalent control of u1:(23)u1eq=-xTtΔfx,t+Δgx,tu0xtTΔgx,t+bxSubstituting u1eq into (15), then the system in sliding mode will be(24)x˙=fx,t+gx,tu0+△fx,tInΓ+△gx,tu0InΓwhere Γ=1-xTtgx,t+△gx,t/xT△g(x,t)+b(x) and Γ could be written into(25)Γ=1-xTtgx,t+xTt△gx,txTt△gx,t+bxIt is evident that(26)Γ=bx-xTtgx,tbx+xTt△gx,tConsider that(27)-xTtgx,t≤-xTtgx,t≤xTtgx,t-ζ1x2-ζ0x≤xTt△gx,t≤ζ1x2+ζ0xTakingAssumption2 and (19) into consideration, it would follow that(28)bx-xTtgx,t≤σ+ζ1x2+ζ0xσ+xTtgx,t≤xTt△gx,t+bxThen(29)Γ≤σ+ζ1x2+ζ0xσ+xTtgx,t≤1Besides(30)0<σ≤bx-xTtgx,t0<σ≤bx-xTt△gx,tOne can obtain the conclusion(31)0<Γ≤1It is obvious that the effect of uncertainties is compensated. Equation (31) indicates that if the quadratic integral sliding mode surface is reached and remains there, the effect of uncertainties Δfx,t+Δgx,tu0 can be compensated, achieving robustness to mismatched uncertainties. Besides, the system will be on sliding mode surface from any initial state; consequently, the system is robust to mismatched uncertainties from any initial state.


## 4. Quadratic Integral Sliding Mode Controller Design

The design of the abovementioned quadratic integral sliding mode controller will be illustrated in this part. The structure of the quadratic integral sliding mode controller is shown in Figure5.Figure 5
Structure of control system for harmonic gear drive systems.As was mentioned earlier, the quadratic integral sliding mode control law will be designed asu=u0+u1. Then u1 is the discontinuous control action that compensates the mismatched uncertainties, with definition as(32)u1=-b-1xλ0+λ1xs+η0+η1xsgnswhere λ0, λ1, η0, and η1∈R are scalars defined as follows:(33)λ0≥ε1ζ1x2+ζ0x+σσ(34)λ1≥0(35)η0≥ε2ζ1x2+ζ0x+σσ(36)η1≥σδ-1ξ0+ζ0β0+ξ1+ζ0β1+ζ1β0x+ζ1β1x2The two approaching parameters ε1 and ε2 in (33) and (35) are arbitrary positive scalars, and δ=σ+ζ0x+ζ1x2-1.The sliding mode control will inevitably cause chattering due to switching time lag, spatial lag, and inertia effect. Chattering will affect the control precision and reduce systems control quality, in more serious cases, and even destroy control elements. In most investigations, the sign function is replaced with the saturation function to reduce the chattering. To some degree, this method can weaken the chattering, but the saturation function is a discontinuous function, making the result conservative. In this paper, the sign function is replaced by the hyperbolic tangent function, which is continuous and smooth.Hyperbolic tangent function is shown as follows:(37)tanhsε=es/ε-e-s/εes/ε+e-s/εwhere s stands for the quadratic integral sliding mode and ε>0 is a positive scalar, which decides the change rate of hyperbolic tangent inflection point.


## 5. Harmonic Gear Drive Systems Stability

The reachability of the sliding mode surface and the stability of closed-loop system under quadratic sliding mode control will be proved in this part.Reachability of the Sliding Mode.Derivation of the quadratic integral sliding mode surface (18) along time and substituting (15) into (18), differential of (18) would be (38)s˙=xTtΔfx,t+Δgx,tu+bxu1By using the quadratic sliding mode control law (33)-(36), the following equations can be obtained:(39)ss˙=xTtΔfx,ts+sxTtΔgx,tu+sbx-b-1xλ0+λ1xs+η0+η1xsgns=xTtΔfx,ts+sxTtΔgx,tu-λ0+λ1xs2-η0+η1xswhere(40)sxTtΔgx,tu=sxTtΔgx,tu0-b-1xλ0+λ1xs+η0+η1xsgns=sxTtΔgx,tu0-xTtΔgx,tb-1xλ0+λ1xs2+η0+η1xsand then(41)ss˙=xTtΔfx,ts+sxTtΔgx,tu0-xTtΔgx,tb-1xλ0+λ1xs2+η0+η1xs-λ0+λ1xs2+η0+η1xs=xTtΔfx,ts+sxTtΔgx,tu0-1+xTtΔgx,tb-1xλ0+λ1xs2+η0+η1xsBecause ofAssumption1,Assumption2, andAssumption3, the inequation above will be(42)ss˙≤ξ1x+ξ0+ζ1x+ζ0β0+β1xxs+xTtΔgx,tb-1xλ0+λ1xs2+η0+η1xs-λ0+λ1xs2+η0+η1xs≤ξ1x+ξ0+ζ1x+ζ0β0+β1xxs+xζ1x+ζ0b-1x-1λ0+λ1xs2+η0+η1xsConsidering (19), it would lead to(43)ss˙≤ξ0+ζ0β0+ξ1+ζ0β1+ζ1β0x+ζ1β1x2x·s-σ+xTtgxbxλ0+λ1xs2-σ+xTtgxbxη0+η1xsConsidering (33) to (36), then ss˙≤-ε1s2-ε2s.In consequence, the quadratic integral sliding mode surface (18) can be reached in finite time and remains there.This completes the proof.⬜Stability of the Closed-Loop System in Sliding Mode.Select the Lyapunov function as(44)Vx=12xTtxt(45)V˙x=xTtx˙t=xTtfx,t+Δfx,t+gx,tu+Δgx,tuThe quadratic integral sliding mode surface (18) has been proved to be reachable. If the system reaches the surface and remains there, then (46)xTtΔfx,t+Δgx,tu+bxu1=0Substituting (46) into (45), one can obtain(47)V˙x=xTtfx,t+gx,tu0+bxu1When (18) is reached, then u1=0 and (47) will be(48)V˙x=xTtfx,t+gx,tu0It can be seen that if the nominal system (16) can be stabilized by u0, then V˙(x)<0. In other words, if u0 is designed to stabilized the nominal system (16), then the closed-loop nonlinear system with mismatched system is asymptotically stable under the quadratic integral sliding mode control law u.This completes the proof.⬜In conclusion, it is proved that the quadratic sliding mode surface (18) is reachable and the closed-loop nonlinear system with mismatched uncertainties in sliding mode is asymptotically stable with robustness to mismatched uncertainties. The system’s dynamic performance is depended on u0, and various control theories (such as LQR, H-infinity control) could be applied in u0 designing according to different requirements.


## 6. Simulation Studies

What deserved our attention is that the quadratic integral sliding mode control could be applied to the uncertain systems with linear time-invariant nominal plants, wherefx,t=Ax, gx,t=B, A is the system matrix, and B is the input matrix.The harmonic gear drive system parameters [6], which are shown in Table 1, are used in following simulations. Substituting parameters into harmonic gear drive systems model (13), then the system will be (49)x˙1=x2x˙2=-9.897×106x1-69.9855x2+7.918×108x3+148.188utx˙3=x4x˙4=2.8723×105x1-2.297×107x3+ΔAx+ΔBut+Tux(50)ΔAx=0.1x2-9.897×105x1-6.99855x2+7.918×107x30.1x42.8723×104x1-2.297×106x3,Tux=0Tmx-Tulx0Tlx+Tulx,ΔB=07.409400,(51)Tmx=16.82×10-4-ψm,x˙1=0,ψm≤fsm-sgnψmfsm,x˙1=0,ψm>fsm-sgnqmfcm,x˙1>0,ψm=-5.4×105x180-x3+0.092ut,fcm=1.5738×10-6x12-3.7901×10-4x1+0.0720,fsm=1.0388×fcm.(52)Tlx=12.35×10-2-ψl,x˙3=0,ψl≤fsl-sgnψlfsl,x˙3=0ψl>fsl-sgnqlfcl,x˙3>0,ψl=6.75×103x180-x3,fcl=1.5738×10-6x32-3.7901×10-4x3+0.0720,fsl=1.0388×fcl.(53)Tulx=5.4×105Δet,Δet=-Δφ-j,x180-x3-Δφ>j0,x180-x3-Δφ<j-Δφ+j,x180-x3-Δφ<-j,j=0.5mm.Table 1
Simulation parameters.
Parameter Unit Value R Ω 5.6 K m N⋅m⋅A-1 0.517 K e V⋅s⋅rad-1 0.517 Jm Kg⋅m2 6.82 × 1 0 - 4 J l K g ⋅ m 2 2.35 × 1 0 - 2 r  80 K s Nm⋅rad-1 5.4 × 1 0 5 T s < 235 N m 8.8 × 1 0 5 ( 235 N m ≤ T s < 843 N m ) 9.8 × 1 0 5 ( T s ≥ 843 N m )Choosing the initial state vector’s condition asx0=000.050, σ=1, ζ0=0,ζ1=0,λ0=60,λ0=0,η0=0,andη0=4+x, suppose that the system is not affected by nonlinear mismatched uncertainties. Define the angular transmission errors as e=qm/r-ql. u0=CB-1-CAx+μ˙, μ is an auxiliary variable, and μ=Cx=221.070.0833-1.5151x, μ˙=-5μ.Simulation results in comparison with SMC and ISMC, shown in Table2 and Figures 6–12.Table 2
Simulation data without uncertainties.
Steady-state Maximum overshoot Stable time Control values Error values Maximum Control values errorrad r a d s ∑ u i 2 V 2 ∑ e i 2 r a d 2 m a x u i V QISMC 0.0042 0.0472 0.0572 1.6307 × 1 0 12 0.1555 2.6428 × 1 0 5 ISMC 0.0024 0.0809 0.0741 2.2879 × 1 0 12 0.2256 4.7746 × 1 0 5 SMC 0.0070 0.0537 0.0827 1.6250 × 1 0 12 0.1978 2.6783 × 1 0 5Figure 6
Angular transmission errore without uncertainties.Figure 7
State vectorx1 without uncertainties.Figure 8
State vectorx2 without uncertainties.Figure 9
State vectorx3 without uncertainties.Figure 10
State vectorx4 without uncertainties.Figure 11
Control signalu without uncertainties.Figure 12
Comparison of control signalu without uncertainties.As the simulation results indicated, when choosing SMC, the system has acceptable overshoot but has a larger steady state error and much longer stable time. When choosing ISMC, the system rises rapidly with a fewer steady state error but has larger control values and maximum overshoot, which will inevitably cause impact to the system.When choosing QISMC, the system will steady rapidly with an acceptable steady state error. In comparison with SMC, control values rise a little and the relative increment is0.2%~4%. The system has minimum overshoot, minimum stable time, and minimum error values, which satisfy the requirements of high precision, fast response, and smooth operation.Considering that the system is affected by nonlinear mismatched uncertainties, simulation results are presented in Tables3 and 4 and Figures 13–19.Table 3
Simulation data with uncertainties.
Steady-state error Maximum overshoot Stable time Control values Error values Maximum Control values r a d r a d s ∑ u i 2 ( V 2 ) ∑ e i 2 r a d 2 m a x u i V QISMC 0.0046 0.0538 0.0608 1.6206 × 1 0 12 0.1609 2.6428 × 1 0 5 ISMC 0.0033 0.0972 0.0846 2.2279 × 1 0 12 0.2408 4.7746 × 1 0 5 SMC 0.0087 0.0634 0.0901 1.6183 × 1 0 12 0.2192 2.6783 × 1 0 5Table 4
System deterioration.
Steady-state error Maximum overshoot Stable time Error values QISMC 9.52 % 13.98 % 6.29 % 3.78 % ISMC 37.50 % 20.14 % 11.42 % 6.8 % SMC 12.42 % 13.66 % 8.94 % 10.8 %Figure 13
Angular transmission errore with uncertainties.Figure 14
State vectorx1 with uncertainties.Figure 15
State vectorx2 with uncertainties.Figure 16
State vectorx3 with uncertainties.Figure 17
State vectorx4 with uncertainties.Figure 18
Control signalu with uncertainties.Figure 19
Comparison of control signalu with uncertainties.The simulation results without uncertainties (in Table2) are compared with simulation results with uncertainties (in Table 3), and then system deterioration caused by uncertainties is demonstrated in Table 4, which will indicate system robustness against mismatched uncertainties.As shown in results, when system is affected by uncertainties, ISMC has the most serious deterioration and simulation curves cannot keep smooth. Despite the fact that SMC can still stabilize the system, the control system quality will not be accepted after deterioration. In comparison with SMC and ISMC, QISMC can stabilized the system with a series advantages such as fast response, fewer rise time, fewer overshoot, and fewer small error. The relative control value increment lies in0.2%~5.3%. Simulation results illustrate that QISMC has strong robustness to nonlinear mismatched uncertainties and disturbance.


## 7. Conclusions

In order to compensate harmonic gear transmission errors by control method, this paper studies quadratic integral sliding mode control for a class of nonlinear harmonic gear drive systems with mismatched uncertainties, where a new quadratic integral sliding mode controller design method based on quadratic integral sliding mode surface is presented. Considering the nonlinear torques which are caused by backlash and frictions, the models of nonlinear frictions and torques are presented and the influence of nonlinear parts is compensated during control system design. In system modeling, frictions and nonlinear elastic deformations are taken into consideration and then the mathematical model of harmonic gear drive system is established. By using Lyapunov stability theory, it is proven that the quadratic integral sliding mode can be reached in finite time and closed-loop systems are asymptotic stable robustly. Simulation studies are carried out and the results verify the effectiveness of the proposed method. Compared with the traditional linear sliding mode control (SMC) and the integral sliding mode control (ISMC), the quadratic integral sliding mode control stabilizes the system rapidly with shorter rise time, lower overshoot, smaller errors, and strong robustness against mismatched uncertainties and nonlinear disturbances. The results indicate that QISMC has a good performance with strong robustness on a class of nonlinear system with mismatched uncertainties.



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*Source: 2372305-2018-07-22.xml*