# Boundary Control of a Flexible Manipulator Based on a High Order Disturbance Observer with Input Saturation

Authors: Junteng Ma; Dongping Jin; Zhengtao Wei; Ti Chen; Hao Wen
Journal: Shock and Vibration
 (2018)

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## Abstract

This paper studies a new boundary control strategy for a flexible manipulator subject to unknown fast time-varying disturbances. The flexible manipulator essentially is an infinite dimensional continuum. Hence, a continuous function of space and time can be employed to describe the position of such a distributed parameter structure, the motion of which can be described by partial differential equations (PDEs). To cope with fast time-varying external disturbances, a high order disturbance observer is adopted. A control strategy based on such a disturbance observer is proposed for the rest-rest maneuvering of the flexible manipulator. Moreover, a smooth hyperbolic function is included in the controller to satisfy the requirement of input saturation. The stability of the boundary control is analyzed using LaSalle’s invariance principle. Finally, the performance of the presented boundary controller is verified through comparison with that of employing a constant disturbance observer via numerical simulations.


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## Body

## 1. Introduction

At present, flexible manipulators have increasingly wide applications in industrial, agricultural, medical, and aerospace fields. To meet the demand of higher performance, the trend of development of space manipulators is towards lightweight parts, low energy expenditure, and fast movement [1, 2]. Since the 1970s, many studies have been performed on modeling theory and control scheme of flexible manipulators [3]. However, the traditional studies on such distributed parameter structures were based on ordinary differential equations (ODEs) [4, 5]. Although the ODE dynamic model has a simple form and is convenient for controller design, it is difficult to accurately describe the distributed parameter characteristic of the flexible structure, and spillover instability problems may occur [6]. Compared with the ODE model, the PDE model can reflect the dynamic characteristic of the flexible structure more accurately; however, it will increase the difficulty and challenge of controller design.Significant attention has been attracted to the PDE modeling and the controller design of flexible manipulators during recent years; that is, the variational principle is allowed to derive the differential equations [7]. For example, Smith obtained the partial differential solutions via the method of finite difference; this work laid the theoretical foundation for the partial differential control method of flexible manipulators [8]. Ge et al. built the PDE model of a distributed parameter flexible manipulator that avoided the traditional modal truncation error [9]. Jiang et al. studied a boundary controller for a flexible arm applying the PDE robust observer [10]. In addition to the studies of straight manipulators, Liu et al. studied the vibration suppression of curved beams using the PDE model [11].In practice, the performance of the flexible manipulator system is significantly affected by disturbances [12]. To eliminate the influence of disturbances, researchers have developed many solutions, of which the schemes based on disturbance observers are especially effective. The control of the flexible manipulator under distributed disturbances or concentrated disturbances has been discussed in several prior articles. The model of a flexible manipulator was established via PDEs, and an infinite dimensional disturbance observer was applied to estimate the external disturbances in [13]. Morales et al. proposed a nested Generalized Proportional Integral (GPI) controller with a disturbance observer for the tip payload changing [14]. The studies with complex disturbances acting on manipulators are also available. Mahamood presented an adaptive hybrid proportional integral derivative control strategy for the two-link flexible manipulator under step signal, square wave signal, white noise disturbance, and sine wave disturbance [15]. Huang et al. studied the motion of an underwater manipulator and developed a disturbance observer for the restoring and coupling forces [16].Notably, most of the previous studies have investigated the control issue with slowly time-varying disturbances. For example, Chen et al. derived an improved nonlinear disturbance observer that could effectively account for constant disturbances [17]. Yang et al. presented a disturbance observer in the case that the disturbances varied slowly and later defined a lumped disturbance, including model uncertainty, parametric uncertainty, and external disturbances [18]. To optimize trajectory tracking performance, Lee proposed a nonlinear disturbance observer for constant disturbances [19]. In practice, however, the external disturbances are often fast time-varying. To achieve a higher performance, the disturbance observer must be improved. Moreover, it is necessary to consider the condition of input saturation when designing a controller. For example, Liu et al. presented an antiwindup controller for a flexible manipulator in the case of parametric uncertainty, unknown disturbances, and input saturation [20]. Liu et al. investigated the PDE controller of a flexible manipulator with unknown disturbances and input saturation and showed that it worked better than the PD control strategy via numerical simulations [21].Although the boundary control issue of the flexible manipulator has been discussed extensively, there are few studies on the control problem of flexible manipulators described via PDEs with input saturation and fast time-varying disturbances. Thus, the objectives of this research effort can be summarized as follows:(1) a control law with smooth hyperbolic functions is proposed based on the PDE model, and (2) the use of a higher order disturbance observer to compensate for the fast time-varying disturbances to reduce the disturbance effects.The organization of this article is as follows: the flexible manipulator is described using PDEs in Section2; in Section 3, a higher order disturbance observer, is presented to compensate for external disturbances; the control law with input saturation is exhibited in Section 4; numerical simulations are given in Section 5 and a summary of this paper and future perspectives are presented in Section 6.


## 2. Dynamic Modeling of the Flexible Manipulator

The flexible manipulator of concern is shown in Figure1. In essence, the system is a flexible system that consists of three parts: a motor at the shoulder, a flexible manipulator, and a tip concentrated payload. Because the radius of the motor is extremely small relative to the manipulator, it is ignored in the subsequent analysis.Figure 1
Configuration of a flexible manipulator.The manipulator rotates in the horizontal plane at a low speed, driven by the input torqueu(t) of the motor. The flexible manipulator is subject to small elastic deformation and considered as an Euler-Bernoulli beam. As shown in Figure 1,XOY andxoy are the global frame and the local frame fixed on the body center, respectively; d1(t) and d2(t) denote the input disturbances of the motor and the input disturbances at the tip, respectively; u(t) is the input torque by the motor; F(t) represents the control force acting on the payload; θ(t) is the rotation angle of the motor; and y(x,t) denotes the elastic deflection at the x.For simplicity, the symbols are introduced as follows:(1)θt=θ,∗x=∂∗∂x,∗xx=∂2∗∂x2,∗xxx=∂3∗∂x3,∗xxxx=∂4∗∂x4,∗˙=∂∗∂t,∗¨=∂2∗∂t2,∗3=∂3∗∂t3,∗n=∂n∗∂tn,n=1,…,+∞.To facilitate the analysis, introduce the auxiliary variablez(x,t)=xθ(t)+y(x,t). The PDE dynamic model is established by applying Hamilton’s principle ∫t1t2(δEk-δEp+δWc)dt=0, where δ· is the variation of ·; δEk, δEp, and δWc are the variation of the kinetic energy, the potential energy, and the virtual work, respectively [22]. At an arbitrary moment, the deflection and the rotation angle are zero at the origin; that is, y(0,t)=yx(0,t)=0 such that z(0,t)=y(0,t)=0, zx(0,t)=θ, ∂nz/∂xn=∂ny/∂xn(n≥2).The total kinetic energy of the flexible manipulator is(2)Ek=12Ihθ˙2+12∫0Lρz˙2x,tdx+12mz˙2L,t,where Ih is the rotor inertia, ρ is the mass of per unit length of the flexible manipulator, m is the mass of the tip concentrated payload, and L is the length of the manipulator. The potential energy of the flexible manipulator is(3)Ep=12∫0LEIyxx2x,tdx,where EI is the flexural rigidity.The nonconservative work is given by(4)Wc=ut+d1tθ+Ft+d2tzL,t.Thus, the PDE model of the flexible manipulator reads(5)ρz¨x,t+EIzxxxxx,t=0,(6)Ihθ¨-EIzxx0,t-ut+d1t=0,(7)mz¨L,t-EIzxxxL,t-Ft+d2t=0,(8)zxxL,t=yxxL,t=0,(9)y0,t=yx0,t=0.Two assumptions about the dynamic model are made, as given below.Assumption 1.
The unknown input disturbancesd1 and d2 are bounded; that is, there exist two positive real numbers D1 and D2 such that d1≤D1 and d2≤D2.Assumption 2.
If the total kinetic energy of the flexible manipulator is bounded for∀t∈[0,∞), then both θ˙(t) and z˙(x,t) are bounded for ∀(x,t)∈[0,L]×[0,∞) according to (2) [23].


## 3. Disturbance Observer Based Control

The unknown external disturbances have attracted the attention of many researchers in the field of the control engineering [24, 25]. The disturbances are induced by not only the system environment but also the uncertainty of the control system, such as the system model uncertainty and parametric uncertainty [24]. Generally, it is difficult to measure the external disturbances accurately; however, a disturbance observer may compensate for the defect.


### 3.1. Constant Disturbance Observer

Considering the boundary conditions (6) and (7), if the measurements of the state variables θ˙ and z˙(L,t) are available and the initial state conditions θ˙0 and z˙0 are known, then the system can be presented as [17](10)θ¨=1IhEIyxx0,t+ut+1Ihd1t,θ˙0=θ˙0,z¨L,t=1mEIyxxxL,t+Ft+1md2t,z˙L,0=z˙0.Theorem 3.
Given two sets of constantsKi>0 and Kj>0(i,j=1,…,n), the constant disturbance observer can be expressed as [17](11)d^1=KiIhθ˙-γ1,γ˙1=EIyxx0,t+ut+d^1,d^2=Kjmz˙L,t-γ2,γ˙2=EIyxxxL,t+Ft+d^2,where γ1 and γ2 are the auxiliary variables with γ1(0)=Ihθ˙0 and γ2(0)=mz˙0; d^1 and d^2 are the estimations of d1 and d2, respectively. Denoting d~1=d1-d^1 and d~2=d2-d^2, the disturbance estimation errors d~1 and d~2 converge to zero exponentially.Proof.
According to the hypothesisd˙1=0 and d˙2=0, one has d~˙1+Kid~1=d˙1=0, d~˙2+Kjd~2=d˙2=0 from (11), the solutions of which are d~1=d~1(0)e-Kit, d~2=d~2(0)e-Kjt. Therefore, the disturbance estimation errors d~1 and d~2 converge to zero exponentially.


### 3.2. High Order Disturbance Observer

For the fast time-varying disturbances, it is expected that the performance of such constant disturbance observer will deteriorate because the assumption of the slowly varying characteristic on the disturbance is no longer valid [26]. To address the problem, a high order disturbance observer is developed by modeling the unknown time-varying disturbance as [27](12)dt=∑k=0qDktk,where Dk(k∈[0,q]) are unknown constants. Without the loss of generality, the external disturbance approximated in this work as a quadratic function in time such that(13)dt=D0+D1t+D2t2.Denotingd~=d-d^, where d^ and d~ are the disturbance estimation and the disturbance tracking error, respectively, the disturbance estimation is constructed as(14)d^t=λ0h0t+λ1h1t+λ2h2t,where h˙0(t)=d-d^=d~ and hk(t)=∫0thk-1(τ)dτ for k=1,2,3. Accordingly, one obtains(15)d^1=λ0∫0td~1τdτ+λ1∫0t∫0δd~1τdδdτ+λ2∫0t∫0σ∫0δd~1τdσdδdτ,d^2=λ0∫0td~2τdτ+λ1∫0t∫0δd~2τdδdτ+λ2∫0t∫0σ∫0δd~2τdσdδdτ.From (15) and dhk(t)/dt=hk-1 for k=1,…,3, it can be derived that(16)d^13=λ0d~¨1+λ1d~˙1+λ2d~1,d^23=λ0d~¨2+λ1d~˙2+λ2d~2.Substitutingd~1=d1-d^1, d~2=d2-d^2 into (16) yields(17)d13=d~13+λ0d~¨1+λ1d~˙1+λ2d~1,d23=d~23+λ0d~¨2+λ1d~˙2+λ2d~2.From (13), it can be seen that d1(3)=0 and d2(3)=0; thus, one has(18)d~13+λ0d~¨1+λ1d~˙1+λ2d~1=d13=0,d~23+λ0d~¨2+λ1d~˙2+λ2d~2=d23=0.Consider the characteristic polynomial(19)Qs=s3+λ0s2+λ1s+λ2which makes the disturbance error convergent to zero through the Hurwitz stability criterion, if the coefficients of Q(s)=0 are chosen properly such that λ0, λ1, λ2>0 and λ0λ1>λ2.The auxiliary variablesκ˙1 and κ˙2 are defined as(20)κ˙1=EIyxx0,t+u+d^1,κ˙2=EIyxxxL,t+F+d^2,where κ10=Ihθ˙0 and κ2(0)=mz˙0.Combining (6) and (7) with (20), one obtains(21)Ihθ¨-κ˙1=Ihθ¨-EIyxx0,t+u+d^1=d1-d^1=d~1,mz¨L,t-κ˙2=mz¨L,t-EIyxxxL,t+F+d^2=d2-d^2=d~2.Integrating both sides from 0 to t yields(22)Ihθ˙-κ1=∫0td~1dτ,mz˙L,t-κ2=∫0td~2dτ.By observing (15), (20), and (22), the high order disturbance observers for the flexible manipulator can be formulated as follows:(23)d^1=λ0Ihθ˙-κ1+λ1∫0tIhθ˙-κ1dτ+λ2∫0t∫0δIhθ˙-κ1dδdτ,κ˙1=EIyxx0,t+u+d^1,d^2=λ0mz˙L,t-κ2+λ1∫0tmz˙L,t-κ2dτ+λ2∫0t∫0δmz˙L,t-κ2dδdτ,κ˙2=EIyxxxL,t+F+d^2,where λ0, λ1, λ2>0 and λ0λ1>λ2.


## 3.1. Constant Disturbance Observer

Considering the boundary conditions (6) and (7), if the measurements of the state variables θ˙ and z˙(L,t) are available and the initial state conditions θ˙0 and z˙0 are known, then the system can be presented as [17](10)θ¨=1IhEIyxx0,t+ut+1Ihd1t,θ˙0=θ˙0,z¨L,t=1mEIyxxxL,t+Ft+1md2t,z˙L,0=z˙0.Theorem 3.
Given two sets of constantsKi>0 and Kj>0(i,j=1,…,n), the constant disturbance observer can be expressed as [17](11)d^1=KiIhθ˙-γ1,γ˙1=EIyxx0,t+ut+d^1,d^2=Kjmz˙L,t-γ2,γ˙2=EIyxxxL,t+Ft+d^2,where γ1 and γ2 are the auxiliary variables with γ1(0)=Ihθ˙0 and γ2(0)=mz˙0; d^1 and d^2 are the estimations of d1 and d2, respectively. Denoting d~1=d1-d^1 and d~2=d2-d^2, the disturbance estimation errors d~1 and d~2 converge to zero exponentially.Proof.
According to the hypothesisd˙1=0 and d˙2=0, one has d~˙1+Kid~1=d˙1=0, d~˙2+Kjd~2=d˙2=0 from (11), the solutions of which are d~1=d~1(0)e-Kit, d~2=d~2(0)e-Kjt. Therefore, the disturbance estimation errors d~1 and d~2 converge to zero exponentially.


## 3.2. High Order Disturbance Observer

For the fast time-varying disturbances, it is expected that the performance of such constant disturbance observer will deteriorate because the assumption of the slowly varying characteristic on the disturbance is no longer valid [26]. To address the problem, a high order disturbance observer is developed by modeling the unknown time-varying disturbance as [27](12)dt=∑k=0qDktk,where Dk(k∈[0,q]) are unknown constants. Without the loss of generality, the external disturbance approximated in this work as a quadratic function in time such that(13)dt=D0+D1t+D2t2.Denotingd~=d-d^, where d^ and d~ are the disturbance estimation and the disturbance tracking error, respectively, the disturbance estimation is constructed as(14)d^t=λ0h0t+λ1h1t+λ2h2t,where h˙0(t)=d-d^=d~ and hk(t)=∫0thk-1(τ)dτ for k=1,2,3. Accordingly, one obtains(15)d^1=λ0∫0td~1τdτ+λ1∫0t∫0δd~1τdδdτ+λ2∫0t∫0σ∫0δd~1τdσdδdτ,d^2=λ0∫0td~2τdτ+λ1∫0t∫0δd~2τdδdτ+λ2∫0t∫0σ∫0δd~2τdσdδdτ.From (15) and dhk(t)/dt=hk-1 for k=1,…,3, it can be derived that(16)d^13=λ0d~¨1+λ1d~˙1+λ2d~1,d^23=λ0d~¨2+λ1d~˙2+λ2d~2.Substitutingd~1=d1-d^1, d~2=d2-d^2 into (16) yields(17)d13=d~13+λ0d~¨1+λ1d~˙1+λ2d~1,d23=d~23+λ0d~¨2+λ1d~˙2+λ2d~2.From (13), it can be seen that d1(3)=0 and d2(3)=0; thus, one has(18)d~13+λ0d~¨1+λ1d~˙1+λ2d~1=d13=0,d~23+λ0d~¨2+λ1d~˙2+λ2d~2=d23=0.Consider the characteristic polynomial(19)Qs=s3+λ0s2+λ1s+λ2which makes the disturbance error convergent to zero through the Hurwitz stability criterion, if the coefficients of Q(s)=0 are chosen properly such that λ0, λ1, λ2>0 and λ0λ1>λ2.The auxiliary variablesκ˙1 and κ˙2 are defined as(20)κ˙1=EIyxx0,t+u+d^1,κ˙2=EIyxxxL,t+F+d^2,where κ10=Ihθ˙0 and κ2(0)=mz˙0.Combining (6) and (7) with (20), one obtains(21)Ihθ¨-κ˙1=Ihθ¨-EIyxx0,t+u+d^1=d1-d^1=d~1,mz¨L,t-κ˙2=mz¨L,t-EIyxxxL,t+F+d^2=d2-d^2=d~2.Integrating both sides from 0 to t yields(22)Ihθ˙-κ1=∫0td~1dτ,mz˙L,t-κ2=∫0td~2dτ.By observing (15), (20), and (22), the high order disturbance observers for the flexible manipulator can be formulated as follows:(23)d^1=λ0Ihθ˙-κ1+λ1∫0tIhθ˙-κ1dτ+λ2∫0t∫0δIhθ˙-κ1dδdτ,κ˙1=EIyxx0,t+u+d^1,d^2=λ0mz˙L,t-κ2+λ1∫0tmz˙L,t-κ2dτ+λ2∫0t∫0δmz˙L,t-κ2dδdτ,κ˙2=EIyxxxL,t+F+d^2,where λ0, λ1, λ2>0 and λ0λ1>λ2.


## 4. Controller Design and Analysis

### 4.1. Control Scheme

The objective of the scheme is to drive the angle of the motor at the shoulder to the desired value and realize the vibration suppression of the elastic beam simultaneously. By employing the high order disturbance observer, the control laws with input saturation are constructed as follows:(24)u=-α1l1tanh⁡l1e-α2l2tanh⁡l2e˙-ksgn⁡e˙-d^1,F=-α3l3tanh⁡l3z˙L,t-ksgn⁡z˙L,t-d^2,where α1, α2, α3, l1, l2, and l3>0; k≥max⁡d~1max,d~2max; e=θ-θd, e˙=θ˙, e¨=θ¨, and θd is the desired angle.Theorem 4.
The control laws are given by (24) with the high order disturbance observer.
(1) The boundary control is asymptotically stable: forx∈[0,L] one has θ→θd, θ˙→0, y(x,t)→0, y˙(x,t)→0, z˙(L,t)→0, d~1→0, d~2→0 when t→∞.
(2) The inputs are bounded, such as(25)u≤α1l1+α2l2+k+D-1,F≤α3l3+k+D-2.Proof.
(1) See Appendix for the details of the proof.
(2) For arbitraryx, the domain of the hyperbolic tangent function y=tanh⁡(x) is -1,1. For x increasing from 0 to +∞, y increases very rapidly at the beginning, and finally y→1. According to the property of the odd function tanh⁡(x), y→-1 for x decreasing from 0 to -∞. Therefore, the hyperbolic tangent function can well simulate the process of input saturation, and the absolute value tanh⁡(x)≤1.
Based on the above analysis, one has-α1l1tanh⁡(l1e)≤-α1l1=α1l1, -α2l2tanh⁡l2e≤-α2l2=α2l2, and -α3l3tanh⁡[l3z˙(L,t)]≤-α3l3=α3l3. As is known, ksgn⁡(e˙)≤k, d^1≤D-1, and d^2≤D-2; therefore,(26)u=-α1l1tanh⁡l1e-α2l2tanh⁡l2e˙-ksgn⁡e˙-d^1≤α1l1+α2l2+k+D-1,F=-α3l3tanh⁡l3z˙L,t-ksgn⁡z˙L,t-d^2≤α3l3+k+D-2,where D-1 and D-2 denote the boundaries of disturbance estimations. The above u and F can satisfy the condition of input saturation by changing the parameters α1, α2, α3, l1, l2, l3, and k.


## 4.1. Control Scheme

The objective of the scheme is to drive the angle of the motor at the shoulder to the desired value and realize the vibration suppression of the elastic beam simultaneously. By employing the high order disturbance observer, the control laws with input saturation are constructed as follows:(24)u=-α1l1tanh⁡l1e-α2l2tanh⁡l2e˙-ksgn⁡e˙-d^1,F=-α3l3tanh⁡l3z˙L,t-ksgn⁡z˙L,t-d^2,where α1, α2, α3, l1, l2, and l3>0; k≥max⁡d~1max,d~2max; e=θ-θd, e˙=θ˙, e¨=θ¨, and θd is the desired angle.Theorem 4.
The control laws are given by (24) with the high order disturbance observer.
(1) The boundary control is asymptotically stable: forx∈[0,L] one has θ→θd, θ˙→0, y(x,t)→0, y˙(x,t)→0, z˙(L,t)→0, d~1→0, d~2→0 when t→∞.
(2) The inputs are bounded, such as(25)u≤α1l1+α2l2+k+D-1,F≤α3l3+k+D-2.Proof.
(1) See Appendix for the details of the proof.
(2) For arbitraryx, the domain of the hyperbolic tangent function y=tanh⁡(x) is -1,1. For x increasing from 0 to +∞, y increases very rapidly at the beginning, and finally y→1. According to the property of the odd function tanh⁡(x), y→-1 for x decreasing from 0 to -∞. Therefore, the hyperbolic tangent function can well simulate the process of input saturation, and the absolute value tanh⁡(x)≤1.
Based on the above analysis, one has-α1l1tanh⁡(l1e)≤-α1l1=α1l1, -α2l2tanh⁡l2e≤-α2l2=α2l2, and -α3l3tanh⁡[l3z˙(L,t)]≤-α3l3=α3l3. As is known, ksgn⁡(e˙)≤k, d^1≤D-1, and d^2≤D-2; therefore,(26)u=-α1l1tanh⁡l1e-α2l2tanh⁡l2e˙-ksgn⁡e˙-d^1≤α1l1+α2l2+k+D-1,F=-α3l3tanh⁡l3z˙L,t-ksgn⁡z˙L,t-d^2≤α3l3+k+D-2,where D-1 and D-2 denote the boundaries of disturbance estimations. The above u and F can satisfy the condition of input saturation by changing the parameters α1, α2, α3, l1, l2, l3, and k.


## 5. Numerical Simulation Examples

Numerical simulations are presented in this section to verify the boundary control based on the high order disturbance observer with input saturation. For comparison purposes, the simulations under the control scheme employing the constant disturbance observer are also given in this section. The physical parameters of the manipulator are listed in Table1.Table 1
Physical parameters of the manipulator.
Parameter
Physical significance
Value
E
I (N⋅m2)
Flexural rigidity of the beam
2
ρ
(
k
g
/
m
)
Mass of per unit length
0.2
I
h (kg⋅m2)
Inertia of the motor
0.5
L
(
m
)
Length of the flexible beam
1
m
(
k
g
)
Mass at the tip
2The aim of all the simulations is to drive the angle of the motor to the desired value; that is,θd=0.5rad, without residual vibration of the elastic beam. The external disturbances are hybrid wave signals.The parameters in the control scheme proposed in this paper are chosen asα1=80, α2=10, α3=100, l1=0.4, l2=1, l3=0.2, k=0.075, λ0=120, λ1=60, and λ2=60.In the control strategy employing the constant disturbance observer, the parameters are chosen asα1=80, α2=10, α3=100, l1=0.4, l2=1, l3=0.2, Ki=5, and Kj=5.As indicated in Figure2(a), under the proposed control scheme, the angle of the motor arrives at the specified location in approximately 12 seconds and later stops steadily. The response curves in Figure 2(a) are quite smooth in shape. However, under the control scheme employing the constant disturbance observer, Figure 2(b) shows that there exist significant positioning errors, even in the last part of the angular response.Figure 2
Responses in motor angle and angular velocity for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
(a)
(b)Figure3 depicts the deflections at the tip and at the middle of the flexible manipulator. As shown in Figure 3(a), the amplitude of the elastic vibration under the proposed control law with the high order observer gradually decreases to zero in 12 seconds. By contrast, under the control scheme employing the constant disturbance observer, the residual vibration of the flexible manipulator does not vanish, even after a long time.Figure 3
Deflections at the tip and at the middle for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
(a)
(b)The disturbance estimates by the two types of disturbance observers are compared in Figures4 and 5 (zoom-in view). The simulation results indicate that the high order disturbance observer outperforms the constant one in terms of the disturbance estimation errors, especially for the disturbances of fast time-varying or transient. The constant disturbance observer leads to a relatively poor performance in the scenario and is suitable to estimate the slow time-varying disturbances or smooth signals.Figure 4
Disturbance estimates for (a) the boundary control scheme with the high order observer and (b) the control scheme employing the constant disturbance observer.
(a)
(b)Figure 5
Comparison of the disturbance estimations.


## 6. Conclusions

The motion of the flexible manipulator was described via PDEs to overcome the problems caused by model truncation, and a high order disturbance observer was proposed thereafter to estimate the external disturbances for counteracting the disturbance effects. The physical requirement of input saturation was considered in the proposed control law using smooth hyperbolic functions. The stability of the boundary control system was demonstrated using LaSalle’s invariance principle. Finally, numerical simulations illustrated that the boundary controller works notably well. By contrast with the constant disturbance observer, the high order disturbance observer can accurately estimate the fast time-varying disturbances or the transient signals. In the future, the vibration suppression of multilink flexible manipulators will be discussed.



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