# Vibration Control of an Axially Moving System with Restricted Input

Authors: Zhijia Zhao; Yonghao Ma; Guiyun Liu; Dachang Zhu; Guilin Wen
Journal: Complexity
 (2019)

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## Abstract

In this study, we consider the global stabilization of an axially moving system under the condition of input saturation nonlinearity and external perturbation. Based on Lyapunov redesign method, observer backstepping, and high-gain observers, an output feedback control strategy with an auxiliary system is constructed to eliminate the input saturation constraint effect and suppress the string system vibration, and a boundary disturbance observer is exploited to cope with the external disturbance. The stability of the controlled system is analyzed and proven based on Lyapunov stability without simplifying or discretizing the infinite dimensional dynamics. The presented simulation results show the effectiveness of the derived control.


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## Body

## 1. Introduction

Axially moving systems are important components in mechanical system and play a significant role in actual production process. However, a few issues exist; nonsmooth input nonlinearities and external perturbations frequently occur and produce severe impacts on system performance. It is worth noting that nonsmooth input nonlinearities containing saturation, backlash, hysteresis, and dead-zone are generally found in industrial control systems, such as mechanical, hydraulic, biomedical, piezoelectric, and physical systems [1–5]. Such nonlinearities usually arise from inherent physical constraints of the dynamical system and constraints in the controller actuators, which are impossible to be eliminated. If the input nonlinearities are ignored in the system model, it is difficult to make the actual axially moving system stabilized. So far, some results associated with how to achieve the control objective for flexible structure systems with the input saturation have been attained [6–8]. In [6], the vibration control and input saturation problem for a vibrating flexible aerial refueling hose with variable lengths are addressed introducing the hyperbolic tangent function and adopting the backstepping approach. In [7], the authors develop the boundary control for a vibrating riser system with mixed input nonlinearities to suppress the deflection and compensate for the input saturation. In [8], the control schemes are constructed for a flexible beam system to suppress the vibration and eliminate the input saturation and output constraint in the presence of disturbances. However, these results do deal with the input saturation issue for stationary flexible systems, but there is little information on how to handle the input restriction for axially moving systems.In recent decades, many achievements regarding vibration control for axially moving systems have been attained, whose dynamics can be mathematically considered to be distributed parameter systems (DPS) with infinite dimensional feature [9–15]. Effective solutions for controlling the DPS mainly include truncation model-based method, and boundary control. Different from truncation model-based method [16–20], which is employed in different ways to extract a finite dimensional subsystem to be controlled while showing robustness to neglecting the remaining infinite dimensional dynamics in the design, boundary control is the implementation of control design based on infinite dimensional system dynamics, which is generally considered to be physically more realistic due to nonintrusive actuation and sensing [21]. For the past few years, the vibration boundary control scheme design for the axially moving system has made great achievement [22–28]. In [22], the deflection of the axially moving string is regulated by the proposed adaptive vibration isolation and the practical experiment illustrates the theoretical results. In [23], an adaptive robust control strategy is constructed for controlling the vibrational offset of an axially moving system in the presence of parameter and disturbance uncertainties. In [24], an iterative learning control scheme is exploited for a stretched flexible string to damp out any string oscillation based on continuous and discrete Lyapunov functions. In [25], the vibration of a translating tensioned beam is exponentially stabilized and effectively suppressed via the choice of a proper Lyapunov function candidate. In [26], a stabilizing control law is derived for a translating tensioned strip to suppress the vibration and the closed-loop system is proven to be exponentially stable. In [27], simultaneous vibration control scheme design and velocity regulation issue are discussed and good stability is attained in the sense of Lyapunov. In [28], the high-gain observer technique and Lyapunov-based observer backstepping method are integrated into the context of boundary control design to generate a stabilizing robust control law for suppressing the deflection of an accelerative belt system. In this article, the axially moving system with the input saturation is studied under the condition of the external disturbance, which makes the control scheme design more complicated and difficult in comparison with previous research.Moreover, in research achievements [22–28], the control schemes are implemented based on the assumption that all the system state signals consisting of the control law can be directly measured by sensors or obtained by algorithms. However, in practice, the measurement noise derived from sensors is completely unavoidable, and its effect will be further magnified in the procedure to obtain the terms of differentiation to time, which would limit the controller in [22–28] implementation. To resolve this issue, the observer backstepping [29] and high-gain observers [30] can be exploited to estimate the unmeasured system states and then an output feedback boundary control is developed to globally stabilize the considered axially moving system.In this study, our interest lies in how to construct an output feedback control for the global stabilization of the axially moving system and simultaneously for the elimination of input saturation nonlinearity effect. Compared with the previous research, the main contributions are summed up as follows.(i)
When there are the unmeasured system states, observer backstepping and high-gain observers are employed to reconstruct the system states and then an output feedback boundary control is generated for vibration reduction of the axially moving system.(ii)
An auxiliary system is introduced to tackle the input saturation nonlinearity and a disturbance observer is exploited to track the external disturbance.(iii)
The uniformly and ultimately bounded stability of the controlled system are analyzed and demonstrated through rigorous Lyapunov analysis without any model reduction.The arrangement of this paper is listed as below. The system dynamics and input saturation function are introduced in Section2. Disturbance observer and boundary control law are constructed for tracking the external disturbance and suppressing the vibration in Section 3, where the stability of the controlled system is analyzed based on Lyapunov stability. Numerical simulations are presented in Section 4 and we reach a conclusion in Section 5.


## 2. Problem Statement

An axially moving string system with saturated input is depicted in Figure1, where z(x,t) denotes the vibration offset, l, ρ, b, P, and v denote the length, the mass per unit length, the damping coefficient, the tension, and the axial speed of the string, respectively, m and β denote the mass and the damping coefficient of the actuator, respectively, d(t) denotes the external disturbance, u(t) denotes the saturated control input, u0 denotes the control command to be developed, and um denotes the saturation limit.Figure 1
An axially moving string system with saturated input.Remark 1.
Notations are defined as follows:(⋆)(t)=(⋆), (⋆)x=∂(⋆)/∂x, (⋆)t=∂(⋆)/∂t, (⋆)xt=∂2(⋆)/∂x∂t, (⋆)xx=∂2(⋆)/∂x2, and (⋆)tt=∂2(⋆)/∂t2.In this study, the dynamics of the considered string system is presented as follows:(1)ρzttx,t+ρvt+bvzxx,t+2ρvzxtx,t+ρv2-Pzxxx,t+bwtx,t=0,0<x<l(2)z0,t=0(3)Pzxl,t+βztl,t=u-d-mzttl,tand the input saturation function is expressed as (4)u=sgnu0um,u0≥um,u0,u0<um,Assumption 2.
We assume that the axial speedv and external disturbance d are uniformly bounded and there exist two nonnegative constants α1 and α2 such that 0≤v≤α1 and 0≤|d|≤α2, ∀t∈[0,+∞). It is a reasonable assumption as v and d have finite energy and hence are bounded [31].Assumption 3.
We assume that the time derivative of external disturbancedt is uniformly bounded and there exists a nonnegative constant α3 such that 0≤|dt|≤α3, ∀t∈[0,+∞).


## 3. Control Design

Before proceeding further, we first introduce the following coordinate transformation:(5)w1=zl,t(6)w1t=w2=ztl,t(7)w2t=1mu+d-βw2-Pzxl,tIt is evident that (6) contains the state w2=zt(l,t) which is available for feedback in (7) and can be computed exploiting difference algorithms to boundary measured state z(l,t). Nevertheless, the impact of the measured noise arising from sensors is completely unavoidable in practice and would be further magnified in the procedure of differentiation to time, which will make the feedback irrealizable. Fortunately, observer backstepping method [29] can reconstruct the states available for feedback and guarantee both boundedness of the closed-loop states and convergence of the state errors to zero, which can be used to solve the issue aforesaid.Then, we take the place ofw2 in (7) with w^2(8)w^2t=1mu+d-βw^2-Pzxl,tThe definition of the state estimation error w~2 is introduced as (9)w~2=w2-w^2Taking the derivative of (9) and applying (7) and (8) into the consequence lead to (10)w~2t=-βmw~2It can be derived from (10) (11)w~2=w~20e-β/mtSubstituting (9) into (6) gives (12)w1t=w^2+w~2Then, we rewrite the reconstructed states as (13)w1t=w^2+w~2w^2t=1mu+d-βw^2-Pzxl,tw~2t=-βmw~2


### 3.1. Disturbance Observer Dynamics

First we defined^ as the estimation of d and then design d^t as (14)d^t=μd-d^where μ>0.Furthermore, the intermediate variable is proposed as(15)ς=d^-μmw^2Differentiating (15) and substituting (8) and (14) into the consequence yields (16)ςt=μPzxl,t+βw^2-u-μd^Then the boundary disturbance observer dynamics is presented as follows (17)d^=ς+μmw^2ςt=μPzxl,t+βw^2-u-μd^The following estimation error is defined (18)d~=d-d^Differentiating (18) and substituting (14), we get (19)d~t=dt-d^t=dt-μd~


### 3.2. Boundary Control

Step 1.
In this step, we first define(20)e=w^2-ωwhere ω is virtual control and e is corresponding error variable.
Then set a Lyapunov function as(21)Vm=Va+Vbwhere (22)Va=12τρ∫0lztx,t+vzxx,t2dx+12τP∫0lzx2x,tdx(23)Vb=ζρ∫0lxzxx,tztx,t+vzxx,tdxwith τ,ζ>0.
Applyingmn≤m2+n2/2 to (23), we have (24)Vb≤ζρl2∫0lzx2x,tdx+ζρl2∫0lztx,t+vzxx,t2dx≤ϱVawhere (25)ϱ=ζρlminτρ,τP
From (24), we get (26)-ϱVa≤Vb≤ϱVa
The proper selection ofζ and τ yields (27)ϱ1=1-ϱ=1-ζρlminτρ,τP>0ϱ2=1+ϱ=1+ζρlminτρ,τP>1Equation (27) shows 0<ϱ<1, and then combining (25) leads to (28)ζ<minτρ,τPρlThen, we further obtain (29)0<ϱ1Va≤Vm≤ϱ2VaDifferentiating Vm, substituting (1), and then using mn≤m2/σ+σn2, σ>0, we have (30)Vmt≤τPv+ζPl-ζρlv22zx2l,t-τvP-ρv22zx20,t+ζρl2w22+τPzxl,tw2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-τb-ζblϖ1∫0lztx,t+vzxx,t2dxwhere ϖ1>0.
Combining (9) and (20), we obtain (31)w2=e+ω+w~2Taking the substitution of (31) into (30), we derive (32)Vmt≤τPv+ζPl-ζρlv22zx2l,t-τvP-ρv22zx20,t+τPzxl,te+ω+w~2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-τb-ζblϖ1∫0lztx,t+vzxx,t2dx+ζρl2e+ω2+ζρl2w~22+ζρle+ωw~2According to (32), the virtual control ω is developed as (33)ω=-c1zxl,twhere c1>0.
Applying (33) to (32) and then using mn≤m2/σ+σn2, σ>0, we get (34)Vmt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t+τPϖ2+ζρl+c1ζρlϖ3w~22-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx+3ζρl2e2-τb-ζblϖ1∫0lztx,t+vzxx,t2dx+τPzxl,tewhere ϖ2,ϖ3>0.Step 2.
In this step, the auxiliary system is developed to cope with the input saturation constraint and the control commandu0 is designed to stabilize e around zero.
Then, we first define the following auxiliary system as(35)ϕt=-c2ϕ-e△u+0.5△u2ϕ+△u,ϕ≥ϕ0,0,ϕ<ϕ0,where c2,ϕ0>0, △u=u-u0, and ϕ is the state of the auxiliary system.
Differentiate (20) and then combine (8) and (33), resulting in (36)et=1mu+d-βe+βc1zxl,t-Pzxl,t+mc1zxtl,tChoose the Lyapunov function candidate as (37)Vn=Vm+m2e2+π2w~22+12ϕ2+12d~2where π>0.
DifferentiatingVn yields (38)Vnt=Vmt+meet+πw~2w~2t+ϕϕt+d~d~tSubstituting (10), (19), (34)-(36) into (38), we obtain (39)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t+τPzxl,te-μ-12d~2+12d~t2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx+3ζρl2e2-τb-ζblϖ1∫0lztx,t+vzxx,t2dx-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-c2-12ϕ2+eu0+d-βe+βc1zxl,t-Pzxl,t+mc1zxtl,tFrom (39), we design the control command u0 as (40)u0=-c3e-d^-βc1zxl,t+Pzxl,t-mc1zxtl,t-τPzxl,t+c4ϕwhere c3,c4>0.
Substituting (40) into (39) and applying mn≤m2+n2/2, we get (41)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t-c3+β-3ζρl2-c42-12e2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-c2-c42-12ϕ2-τb-ζblϖ1∫0lztx,t+vtzxx,t2dx+12d~t2-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-μ-1d~2Combining (20) and (33), the designed control command u0 in (40) is rewritten as (42)u0t=-c3w^2-c1c3zxl,t-d^-βc1zxl,t+Pzxl,t-mc1zxtl,t-τPzxl,t+c4ϕIt is worth mentioning that the boundary state zxt(l,t) in the designed control command (42) can be calculated utilizing a backward difference algorithm to the measured signal zx(l,t) [22]. However, in practice, the measurement noise arising from the sensors will be further magnified in the differential process to obtain zxt(l,t), which will limit the controller (42) implementation. In consequence, to address this issue, high-gain observers are utilized for the estimation of the unmeasured state zxt(l,t).Lemma 4.
Suppose that a system outputy(t) and its first m derivatives are bounded such that |y(n)|<Yn with positive constants Yn, n=1,⋯,m; then the following linear system is considered [30] (43)ϵηit=ηi+1,i=1,⋯,m-1,ϵηmt=-λ¯1ηm-λ¯2ηm-1-⋯-λ¯m-1η2-η1+x1,where ϵ is any small positive constant and the parameters λ¯i, for i=1,⋯,m-1, are chosen such that the polynomial sm+λ¯1sm-1+...+λ¯m-1s+1 is Hurwitz. Then, we have the following property: (44)χn=ηnϵn-1-x1n-1=-ϵψn,n=1,⋯,m-1where ψ=ηm+λ¯1ηm-1+⋯+λ¯m-1η1 with ψ(n) denoting the nth derivative of ψ. In addition, there exist positive constants T and hn such that ∀t>T; we have χn≤ϵhn, n=1,⋯,m, where · denotes the standard Euclidean norm.It is evidently obtained from Lemma4 that ηn+1/ϵn converges to x1(n), which is the nth derivative of x1; that is to say, χn converges to zero due to the high-gain 1/ϵ provided that x1 and its nth derivatives are bounded. As a consequence, it is proper to choose ηn+1/ϵn as an observer to estimate the output signals up to the mth order derivative.Definex1=zx(l,t) and x2=zxt(l,t). We consider the observer for the system with m=2 and then design the estimate of the state x2 as (45)x^2=η2ϵwhere the dynamics of η2 and the error x~2 are defined as (46)ϵη1t=η2ϵη2t=-λ¯1η2-η1+x1x~2=x^2-x2It follows from (45) and (46) that the designed control command (40) is updated as (47)u0=-c3e-d^-βc1+τP-Px1-mc1x^2+c4ϕSubstituting (47) into (39) and applying mn≤m2+n2/2, we have (48)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t-c3+β-3ζρl2-c42-12-mc1ϖ4e2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-c2-c42-12ϕ2-τb-ζblϖ1∫0lztx,t+vtzxx,t2dx+12d~t2+mc1ϖ4x~22-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-μ-1d~2where ϖ4>0.The parametersc1, c2, c3, c4, τ, ζ, π, μ, ϖ1, ϖ2, ϖ3, and ϖ4 are chosen such that (49)ζ<minτρ,τTρlP-ρv2≥0τc1P-ζρlc12-τPvt+ζPl-ζρlv2t2-τPϖ2-c1ζρlϖ3≥0λ1=τb-ζblϖ1>0λ2=ζP2-ζρ2α12-ζblϖ1>0λ3=c3+β-3ζρl2-c42-12-mc1ϖ4>0λ4=πβm-τPϖ2-ζρl-c1ζρlϖ3>0λ5=c2-c42-12>0λ6=μ-1>0Substituting (49) into (48) gives (50)Vnt≤-λ1∫0lztx,t+vzxx,t2dx-λ2∫0lzx2x,tdx-λ3e2-λ4w~22-λ5ϖ2-λ6d~2+γwhere γ=1/2α32+mc1ϖ4ϵ2h22<+∞.It can be derived from (50) (51)Vnt≤-ε1Va+m2e2+π2w~22+12ϕ2+12d~2+γwhere ε1=min(2λ1/τρ,2λ2/τP,2λ3/m,2λ4/π,2λ5,2λ6).Combining (21), (22), (29), and (37) yields (52)0<ε2Va+m2e2+π2w~22+12ϕ2+12d~2≤Vn≤ε3Va+m2e2+π2w~22+12ϕ2+12d~2where ε2=min(ϱ1,1)>0 and ε3=max(ϱ2,1)>0.Combining (51) and (52), we further obtain (53)Vnt≤-εVn+γwhere ε=(ε1/ε3).Then, we rewrite the designed control commandu0 in (47) as (54)u0=-c3w^2-d^-c1c3+βc1+τP-Px1-mc1x^2+c4ϕTheorem 5.
Given the considered system described by (1), (2), and (13), under the designed disturbance observer (17), control command (54), and Assumptions 2 and 3, provided that the initial conditions are bounded and the design parameters c1, c2, c3, c4, τ, ζ, π, μ, ϖ1, ϖ2, ϖ3, and ϖ4 are chosen such that the constraints specified in (49) hold, then the close-loop system signals z(x,t) and ϕ(t) are uniformly bounded.Proof.
Integrating (53) from 0 to t gives (55)Vn≤Vn0-γεe-εt+γε≤V0e-εt+γε∈L∞The combination of (52), Va, and Lemma 2 in [3] yields (56)τP2lz2x,t≤τP2∫0Lzx2x,tdx≤Va≤1ε2Vn(57)12ϕ2t≤1ε2VnSubstituting (55) into (56) and (57), respectively, leads to (58)zx,t≤2lτε2PVn0e-εt+γε,∀x,t∈0,l×0,+∞(59)ϕt≤2ε2Vn0e-εt+γε,∀t∈0,+∞From the aforementioned analysis, we further obtain (60)limt→∞zx,t≤2lγτε2εP,∀x∈0,l(61)limt→∞ϖt≤2γε2εThus, the proof is completed.Remark 6.
In this paper, the controller and observer design proceed on the basis of the infinite dimensional partial differential dynamics; hence there is absence of control spillover issue. In future studies, we will exploit modal method to conduct neural network or learning based control design for achieving the transient performance regulation [32–44].


## 3.1. Disturbance Observer Dynamics

First we defined^ as the estimation of d and then design d^t as (14)d^t=μd-d^where μ>0.Furthermore, the intermediate variable is proposed as(15)ς=d^-μmw^2Differentiating (15) and substituting (8) and (14) into the consequence yields (16)ςt=μPzxl,t+βw^2-u-μd^Then the boundary disturbance observer dynamics is presented as follows (17)d^=ς+μmw^2ςt=μPzxl,t+βw^2-u-μd^The following estimation error is defined (18)d~=d-d^Differentiating (18) and substituting (14), we get (19)d~t=dt-d^t=dt-μd~


## 3.2. Boundary Control

Step 1.
In this step, we first define(20)e=w^2-ωwhere ω is virtual control and e is corresponding error variable.
Then set a Lyapunov function as(21)Vm=Va+Vbwhere (22)Va=12τρ∫0lztx,t+vzxx,t2dx+12τP∫0lzx2x,tdx(23)Vb=ζρ∫0lxzxx,tztx,t+vzxx,tdxwith τ,ζ>0.
Applyingmn≤m2+n2/2 to (23), we have (24)Vb≤ζρl2∫0lzx2x,tdx+ζρl2∫0lztx,t+vzxx,t2dx≤ϱVawhere (25)ϱ=ζρlminτρ,τP
From (24), we get (26)-ϱVa≤Vb≤ϱVa
The proper selection ofζ and τ yields (27)ϱ1=1-ϱ=1-ζρlminτρ,τP>0ϱ2=1+ϱ=1+ζρlminτρ,τP>1Equation (27) shows 0<ϱ<1, and then combining (25) leads to (28)ζ<minτρ,τPρlThen, we further obtain (29)0<ϱ1Va≤Vm≤ϱ2VaDifferentiating Vm, substituting (1), and then using mn≤m2/σ+σn2, σ>0, we have (30)Vmt≤τPv+ζPl-ζρlv22zx2l,t-τvP-ρv22zx20,t+ζρl2w22+τPzxl,tw2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-τb-ζblϖ1∫0lztx,t+vzxx,t2dxwhere ϖ1>0.
Combining (9) and (20), we obtain (31)w2=e+ω+w~2Taking the substitution of (31) into (30), we derive (32)Vmt≤τPv+ζPl-ζρlv22zx2l,t-τvP-ρv22zx20,t+τPzxl,te+ω+w~2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-τb-ζblϖ1∫0lztx,t+vzxx,t2dx+ζρl2e+ω2+ζρl2w~22+ζρle+ωw~2According to (32), the virtual control ω is developed as (33)ω=-c1zxl,twhere c1>0.
Applying (33) to (32) and then using mn≤m2/σ+σn2, σ>0, we get (34)Vmt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t+τPϖ2+ζρl+c1ζρlϖ3w~22-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx+3ζρl2e2-τb-ζblϖ1∫0lztx,t+vzxx,t2dx+τPzxl,tewhere ϖ2,ϖ3>0.Step 2.
In this step, the auxiliary system is developed to cope with the input saturation constraint and the control commandu0 is designed to stabilize e around zero.
Then, we first define the following auxiliary system as(35)ϕt=-c2ϕ-e△u+0.5△u2ϕ+△u,ϕ≥ϕ0,0,ϕ<ϕ0,where c2,ϕ0>0, △u=u-u0, and ϕ is the state of the auxiliary system.
Differentiate (20) and then combine (8) and (33), resulting in (36)et=1mu+d-βe+βc1zxl,t-Pzxl,t+mc1zxtl,tChoose the Lyapunov function candidate as (37)Vn=Vm+m2e2+π2w~22+12ϕ2+12d~2where π>0.
DifferentiatingVn yields (38)Vnt=Vmt+meet+πw~2w~2t+ϕϕt+d~d~tSubstituting (10), (19), (34)-(36) into (38), we obtain (39)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t+τPzxl,te-μ-12d~2+12d~t2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx+3ζρl2e2-τb-ζblϖ1∫0lztx,t+vzxx,t2dx-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-c2-12ϕ2+eu0+d-βe+βc1zxl,t-Pzxl,t+mc1zxtl,tFrom (39), we design the control command u0 as (40)u0=-c3e-d^-βc1zxl,t+Pzxl,t-mc1zxtl,t-τPzxl,t+c4ϕwhere c3,c4>0.
Substituting (40) into (39) and applying mn≤m2+n2/2, we get (41)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t-c3+β-3ζρl2-c42-12e2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-c2-c42-12ϕ2-τb-ζblϖ1∫0lztx,t+vtzxx,t2dx+12d~t2-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-μ-1d~2Combining (20) and (33), the designed control command u0 in (40) is rewritten as (42)u0t=-c3w^2-c1c3zxl,t-d^-βc1zxl,t+Pzxl,t-mc1zxtl,t-τPzxl,t+c4ϕIt is worth mentioning that the boundary state zxt(l,t) in the designed control command (42) can be calculated utilizing a backward difference algorithm to the measured signal zx(l,t) [22]. However, in practice, the measurement noise arising from the sensors will be further magnified in the differential process to obtain zxt(l,t), which will limit the controller (42) implementation. In consequence, to address this issue, high-gain observers are utilized for the estimation of the unmeasured state zxt(l,t).Lemma 4.
Suppose that a system outputy(t) and its first m derivatives are bounded such that |y(n)|<Yn with positive constants Yn, n=1,⋯,m; then the following linear system is considered [30] (43)ϵηit=ηi+1,i=1,⋯,m-1,ϵηmt=-λ¯1ηm-λ¯2ηm-1-⋯-λ¯m-1η2-η1+x1,where ϵ is any small positive constant and the parameters λ¯i, for i=1,⋯,m-1, are chosen such that the polynomial sm+λ¯1sm-1+...+λ¯m-1s+1 is Hurwitz. Then, we have the following property: (44)χn=ηnϵn-1-x1n-1=-ϵψn,n=1,⋯,m-1where ψ=ηm+λ¯1ηm-1+⋯+λ¯m-1η1 with ψ(n) denoting the nth derivative of ψ. In addition, there exist positive constants T and hn such that ∀t>T; we have χn≤ϵhn, n=1,⋯,m, where · denotes the standard Euclidean norm.It is evidently obtained from Lemma4 that ηn+1/ϵn converges to x1(n), which is the nth derivative of x1; that is to say, χn converges to zero due to the high-gain 1/ϵ provided that x1 and its nth derivatives are bounded. As a consequence, it is proper to choose ηn+1/ϵn as an observer to estimate the output signals up to the mth order derivative.Definex1=zx(l,t) and x2=zxt(l,t). We consider the observer for the system with m=2 and then design the estimate of the state x2 as (45)x^2=η2ϵwhere the dynamics of η2 and the error x~2 are defined as (46)ϵη1t=η2ϵη2t=-λ¯1η2-η1+x1x~2=x^2-x2It follows from (45) and (46) that the designed control command (40) is updated as (47)u0=-c3e-d^-βc1+τP-Px1-mc1x^2+c4ϕSubstituting (47) into (39) and applying mn≤m2+n2/2, we have (48)Vnt≤-τc1P-ζρlc12-τPv+ζPl-ζρlv22-τPϖ2-c1ζρlϖ3zx2l,t-τvP-ρv22zx20,t-c3+β-3ζρl2-c42-12-mc1ϖ4e2-ζP2-ζρ2α12-ζblϖ1∫0lzx2x,tdx-c2-c42-12ϕ2-τb-ζblϖ1∫0lztx,t+vtzxx,t2dx+12d~t2+mc1ϖ4x~22-πβm-τPϖ2-ζρl-c1ζρlϖ3w~22-μ-1d~2where ϖ4>0.The parametersc1, c2, c3, c4, τ, ζ, π, μ, ϖ1, ϖ2, ϖ3, and ϖ4 are chosen such that (49)ζ<minτρ,τTρlP-ρv2≥0τc1P-ζρlc12-τPvt+ζPl-ζρlv2t2-τPϖ2-c1ζρlϖ3≥0λ1=τb-ζblϖ1>0λ2=ζP2-ζρ2α12-ζblϖ1>0λ3=c3+β-3ζρl2-c42-12-mc1ϖ4>0λ4=πβm-τPϖ2-ζρl-c1ζρlϖ3>0λ5=c2-c42-12>0λ6=μ-1>0Substituting (49) into (48) gives (50)Vnt≤-λ1∫0lztx,t+vzxx,t2dx-λ2∫0lzx2x,tdx-λ3e2-λ4w~22-λ5ϖ2-λ6d~2+γwhere γ=1/2α32+mc1ϖ4ϵ2h22<+∞.It can be derived from (50) (51)Vnt≤-ε1Va+m2e2+π2w~22+12ϕ2+12d~2+γwhere ε1=min(2λ1/τρ,2λ2/τP,2λ3/m,2λ4/π,2λ5,2λ6).Combining (21), (22), (29), and (37) yields (52)0<ε2Va+m2e2+π2w~22+12ϕ2+12d~2≤Vn≤ε3Va+m2e2+π2w~22+12ϕ2+12d~2where ε2=min(ϱ1,1)>0 and ε3=max(ϱ2,1)>0.Combining (51) and (52), we further obtain (53)Vnt≤-εVn+γwhere ε=(ε1/ε3).Then, we rewrite the designed control commandu0 in (47) as (54)u0=-c3w^2-d^-c1c3+βc1+τP-Px1-mc1x^2+c4ϕTheorem 5.
Given the considered system described by (1), (2), and (13), under the designed disturbance observer (17), control command (54), and Assumptions 2 and 3, provided that the initial conditions are bounded and the design parameters c1, c2, c3, c4, τ, ζ, π, μ, ϖ1, ϖ2, ϖ3, and ϖ4 are chosen such that the constraints specified in (49) hold, then the close-loop system signals z(x,t) and ϕ(t) are uniformly bounded.Proof.
Integrating (53) from 0 to t gives (55)Vn≤Vn0-γεe-εt+γε≤V0e-εt+γε∈L∞The combination of (52), Va, and Lemma 2 in [3] yields (56)τP2lz2x,t≤τP2∫0Lzx2x,tdx≤Va≤1ε2Vn(57)12ϕ2t≤1ε2VnSubstituting (55) into (56) and (57), respectively, leads to (58)zx,t≤2lτε2PVn0e-εt+γε,∀x,t∈0,l×0,+∞(59)ϕt≤2ε2Vn0e-εt+γε,∀t∈0,+∞From the aforementioned analysis, we further obtain (60)limt→∞zx,t≤2lγτε2εP,∀x∈0,l(61)limt→∞ϖt≤2γε2εThus, the proof is completed.Remark 6.
In this paper, the controller and observer design proceed on the basis of the infinite dimensional partial differential dynamics; hence there is absence of control spillover issue. In future studies, we will exploit modal method to conduct neural network or learning based control design for achieving the transient performance regulation [32–44].


## 4. Simulations

In this section, the finite difference method is exploited to simulate the performance of the proposed control in MATLAB software, which can provide a straightforward and accurate process to resolve partial differential equations with two independent variables. The parameters of the considered system are given asρ=1.0kg/m, m=5.0kg, l=1.0m, c=1.0Ns/m2, β=1.0Ns/m, P=100N, and v(t)=2+sin(t). The initial conditions of the considered system are represented as z(x,0)=zt(x,0)=0. The external disturbance is described as d(t)=sin⁡(t)+3sin⁡(3t)+5sin⁡(5t).Figure2 displays the spatiotemporal response of the string system under free vibration; namely, u(t)=0, in the presence of external disturbance and input saturation. Figure 3 depicts the spatial time representation of the string system under the proposed control with the choice of the control design parameters c1=c2=c3=100, c4=10, μ=100, τ=0.1, ϕ0=0.001, um=2, ϵ=0.005, and λ¯1=10. Figures 4 and 5 show the vibrational offset of the considered string system examined at x=1.0m and x=0.5m for controlled and uncontrolled response, respectively. Figure 6 displays the time responses of the external disturbance tracking. The time histories of the designed control command u0(t) and saturated control input u(t) are, respectively, shown in Figures 7 and 8.Figure 2
Deflection of the string without control.Figure 3
Deflection of the string with the proposed control.Figure 4
Two-dimensional deflection of the string without control.Figure 5
Two-dimensional deflection of the string with control.Figure 6
External disturbance tracking.Figure 7
Designed control commandu0(t).Figure 8
Saturated control inputu(t).We draw a conclusion from Figures2–8 that the vibrational deflection of the string system is significantly suppressed when acting on the designed control, the estimated disturbance can better and faster track the actual disturbance, the designed control command u0(t) is mainly used for rejecting the external disturbance, and the restricted control input u(t) is saturated in the domain [-2, 2] due to the existence of input saturation.


## 5. Conclusion

In this paper, the vibration boundary control issue for an axial string system subject to input restriction and external disturbance has been studied. The output feedback control has been developed to damp the vibration and stabilize the system around the equilibrium position synthesizing observer backstepping and high-gain observers. Besides, the auxiliary system and disturbance observer have been introduced to compensate the nonlinear input saturation effect and reject the external disturbance. Based on the rigorous Lyapunov analysis, the stability of the controlled system has been demonstrated without resorting to model reduction method. Numerical simulations have been conducted to clarify the performance of the proposed control approach. In the future, we will concentrate on the output and input restriction problem for axially moving systems and applying this control approach to the actual system is also an interesting topic.



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*Source: 2386435-2019-01-02.xml*