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Research article
Robust adaptive integral backstepping control for opto-electronic tracking system based on modified LuGre friction model Fengfa Yue, Xingfei Li∗ The State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China
A R T I C LE I N FO
A B S T R A C T
Index Terms: Robust adaptive control Integral backstepping control LuGre model Nonlinear control Opto-electronic tracking
This paper presents a robust adaptive integral backstepping control strategy with friction compensation for realizing accurate and stable control of opto-electronic tracking system in the presence of nonlinear friction and external disturbance. With the help of integral control term to decrease the steady-state error of the system and combining robust adaptive control approach with the backstepping design method, a novel control method is constructed. Nonlinear modified LuGre observer is designed to estimate friction behavior. Robust adaptive integral backstepping control strategy is developed to compensate the changes in friction behavior and external disturbance of the servo system. The stability of the opto-electronic tracking system is proved by Lyapunov criterion. The performance of robust adaptive integral backstepping controller is verified by the opto-electronic tracking system with modified LuGre model in simulation and practical experiments. Compared to the adaptive integral backstepping sliding mode control method, the root mean square of angle error is reduced by 26.6% when the proposed control method is used. The experiment results demonstrate the effectiveness and robustness of the proposed strategy.
1. Introduction Opto-electronic tracking system has attracted considerable attention recently with applications in military and civil domain [1,2]. Owing to the development of technology and the increasing complexity of optoelectronic tracking system, the design of high performance controller faces a lot of challenges that need to be resolved such as, parameter variation, gyro and sensor noise, friction and external disturbance. As one of the most significant nonlinearities in the opto-electronic tracking servo system, the friction can cause serious control performance deterioration or even instability [3]. Therefore, develop of the controller with satisfactory performance under the condition of nonlinear friction and external disturbance is a very challenging control issue. In order to reduce the influence of friction and achieve high precision motion control, friction compensation has become a research topic of servo control system in recent years. While an adaptive compensator for a friction model occurred in Ref. [4], dynamic friction model had got more attention. The reason is that friction is dynamic [5], due to the elasticity of the contact surface. A novel adaptive finite time sliding mode control [6] with dynamic friction compensation was proposed for high precision tracking of a voice coil motor actuated servo gantry. An adaptive backstepping control [7] with friction compensation was designed for the motor control system. A recursive model free controller ∗
[8] based friction compensation was designed for an X-Y robot. A robust tracking controller [9] with friction compensation was proposed according to a local model approach. An adaptive backstepping method [10] with dynamic friction compensation at both low and high speed motions was designed for improving the tracking accuracy of servo system. To a certain extent, these methods improved the tracking accuracy of the nonlinear servo system. However, the control performance of these methods will be reduced when the disturbance is involved. Adaptive control technique is recognized as a powerful strategy to nonlinear servo system in the presence of friction conditions. The backstepping nonlinear control and the adaptive speed observer [11] were proposed for the induction motor to improve its robustness. A new approach to adaptive robust motion controller combined with disturbance observer [12] was developed for motion control problems. A saturated adaptive robust control [13] was proposed for a nonlinear system with saturated inputs. A new adaptive backstepping technique [14] was present to handle the induction motor rotor resistance tracking problem. A robust direct field oriented control [15] was designed for a three phase induction machine subjected to load disturbances. Adaptive dynamic surface control strategy [16] was designed for a class of strict-feedback nonlinear systems with mismatched parametric uncertainties. Online parameter estimation and adaptive
Corresponding author. E-mail addresses: yueff@tju.edu.cn (F. Yue),
[email protected] (X. Li).
https://doi.org/10.1016/j.isatra.2018.07.016 Received 17 March 2018; Received in revised form 30 June 2018; Accepted 13 July 2018 0019-0578/ © 2018 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Yue, F., ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.07.016
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2. Problem formulation
control [17] was developed for permanent-magnet synchronous machine. A robust predictive control was designed for permanent-magnet synchronous motors in Ref. [18]. In summary, these adaptive control methods had certain control performance in the application of nonlinear servo systems without friction or external disturbance, or both. The control strategy based on fuzzy logic or neural networks was studied in the past few years. An adaptive fuzzy control [19] was developed for servo control system. The vector control scheme using model reference adaptive systems fuzzy logic observer [20] was designed for induction motor. A fuzzy adaptive output-feedback controller was constructed according to fuzzy logic systems and backstepping approach in Ref. [21]. An adaptive fuzzy finite-time synchronization control was researched in Ref. [22], which present a significant performance in the finite-time control of tele-operation system. An adaptive fuzzy prescribed control [23] was designed for nonlinear systems to resolve the problem of unknown dead-zone inputs. A state-feedback adaptive neural control [24] was introduced using the dynamic surface control technique for nonlinear systems. A robust adaptive neural networks control [25] was designed for solving the position and attitude control of spacecraft. Although these algorithms had achieved some control effect, complex rules and lots of parameters adjustment may reduce the efficiency of them. Up to now, several control methods have been developed for reducing the effect of friction and external disturbance of nonlinear servo system. An adaptive integral backstepping sliding mode (AIBSM) control with friction compensation [1] was designed to achieve accurate and stable control for the opto-electronic tracking system. Sliding mode control (SMC) [26,27] which was based on variable structure control could resolve bounded modeling uncertainties and achieve asymptotic stability performance. A backstepping control and speed estimation method based on sliding mode observer [28] was developed for induction motor. The sliding mode control inevitably has discontinuity and chattering problem, which affects the accuracy of servo control system. Consequently, it is an important issue for the opto-electronic tracking system to develop a high-performance controller which has high control accuracy and high stability performance. A robust adaptive integral backstepping (RAIB) controller with friction compensation is designed for opto-electronic tracking system in this paper. The influence of nonlinear friction and external disturbance of this servo system is decreased by nonlinear modified LuGre observer and robust adaptive backstepping controller. The contributions of this article are given as follows:
The opto-electronic tracking system is a high precision servo control system which contains technology of optics, machine and electronics. It can complete video recording in the process of tracking the target. In this section, system dynamic model is introduced first, and then modified LuGre model is described. 2.1. System dynamic model The model of the opto-electronic tracking system is assumed as follows: (1) motor inductance is neglected. (2) The gimbal is considered rigid. The system dynamic model [29] can be described as below.
θ˙ = ω Jω˙ = Fm − Ff − D Fm = Kt i m im =
(u − K e ω) R
(1)
where J is the total moment of inertia of the motor system, θ and ω are the angular position and the angular speed, respectively. Kt is the motor torque coefficient, Ke is back-EMF coefficient, u is the control input, Fm is the command torque from the drive motor, R is the motor resistance, Ff is the friction force, D is external disturbance. 2.2. Modified LuGre model The friction force Ff is modeled by the LuGre model [8] which describes friction phenomena with friction force variations. This model is given as follows:
Ff = σ0 z + σ1 z˙ + σ2 ω
(2)
ω z g(ω)
(3)
z˙ = ω−
ω 2
σ0 g(ω) = Fc + (Fs − Fc ) e−( ωs )
(4)
where z is the average deflection of bristles, g(ω) represents different friction effects, σ0 is the stiffness coefficient, σ1 is the damping coefficient, σ2 is the viscous friction coefficient, Fc is the Coulomb friction, Fs is the stiction force, ωs is the Stribeck angular speed. The friction parameters will vary with the external environment, so the parameter μ and ν are introduced to correct the LuGre model. The modified LuGre model is defined as below.
1. The nonlinear modified LuGre observer and backstepping controller with friction compensation is used to the opto-electronic tracking system. The advantage of utilizing this method is that it observes and reduces the effect of nonlinear friction and external disturbance in advance. 2. Integral control and robust adaptive control are integrated into backstepping control strategy to strengthen the robustness of the nonlinear servo control system. The stability of the proposed method is guaranteed by the Lyapunov theory. The steady-state error of the system is decreased by integral action. Robust adaptive integral backstepping control strategy used in the opto-electronic tracking system is the most significant characteristic as compared with AIBSM control method [1].
Ff = μ (σ0 z + σ1 z˙ ) + νσ2 ω
(5)
where μ is used to reflect changes the average deflection of bristles, ν is used to reflect changes in the coefficient of viscous friction. They meet 0 < μmin < μ < μmax and 0 < νmin < ν < νmax . This section describes system dynamic model and modified LuGre model. Then, the parameter identification method of LuGre model is introduced in the next section. 3. LuGre model parameter identification The parameter identification [30,31] of LuGre model contains static parameter identification and dynamic parameter identification. Static parameter is estimated by Stribeck curve. Due to the advantages of avoiding local minimum comparing with the least square method under the circumstance of solving nonlinear problems, the genetic algorithm [32] is selected for identifying static and dynamic parameter.
The paper is organized in the following manner. First, in Section 2, the system dynamic model and modified LuGre model are described. The parameter identification method of LuGre model is introduced in Section 3. Then, Section 4 is devoted to develop robust adaptive integral backstepping controller. Simulation and practical experiments are given in Section 5 and Section 6. Finally, the conclusions are given in Section 7.
3.1. Static parameter identification When the angle speed of opto-electronic tracking system keeps constant (z˙ = 0 ) without external disturbance, the total friction Fsf can 2
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3.4. Robust adaptive integral backstepping controller design
be gained by equations (1)–(4). The total friction Fsf is equal to the command torque Fm.
The following equation (15) can be gained by simplifying equation (1).
ω 2
Fsf = ⎛Fc + (Fs − Fc ) e−( ωs ) ⎞ sgn(ω) + σ2 ω = Fm ⎝ ⎠ ⎜
⎟
(6)
Ku KK Jθ¨ = t − t e ω − Ff − D R R
The above four parameters Fc , Fs , ωs , σ2 can be estimated by Stribeck curve which is defined in equation (6) between angle speed and total friction. Supposed the opto-electronic tracking system rotates at a group of invariable angle speeds {ωi}iN= 1, and average command torque is {Fmi}iN= 1. Therefore, Stribeck curve is gained. It is a curve representing the correspondence between friction force and angle speed. Assume that the desired identified parameter vector is as follows:
Q1 = [Fˆc , Fˆs , ωˆ s,σˆ2]
Ku KK Jθ¨ = t − t e ω − μ (σ0 z + σ1 z˙ ) − νσ2 ω − D R R
1 2
(8)
z˙1 = r˙ − θ˙
i=1
(18)
The Lyapunov function is defined as follows:
N
∑ e2 (Q1, ωi)
(17)
z1 = r − θ
where r is desired reference angle. The derivative of the equation (17) is as below.
where Fsf (Q1, ωi ) is the desired friction. The objective function is defined as follows:
S1 =
(16)
The angle tracking error is defined as z1.
(7)
The identification error is defined as below.
e (Q1, ωi ) = Fmi − Fsf (Q1, ωi )
(15)
The following equation (16) is obtained by taking Ff into equation (15).
V (z1) = (9)
1 2 z1 2
(19)
The derivative of Lyapunov function is computed as below.
The static parameters Fc , Fs , ωs and σ2 are identified by minimizing the objective function.
V˙ (z1) = z1 z˙1 = z1 (r˙ − θ˙ )
(20)
The virtual control ρ˙d is chosen as follows:
3.2. Dynamic parameter identification
ρ˙d = r˙ + h1 z1 + h2 ϕ, h1 > 0, h2 > 0
(21)
The identification of dynamic parameters is more difficult than static parameters due to the state vector z which is not measurable. σ00 and σ10 are estimated by the presliding displacement as below.
where the parameters of h1 and h2 are defined as positive constants. The integral term is introduced as below.
ΔF σ00 ≈ Δθ
ϕ= (10)
σ10 ≈ 2 σ00 M − σ2
(11)
(23)
z˙1 is rewritten in the following equation. (24)
The derivative of equation (23) is represented as follows:
(12)
z˙ 2 = r¨ − θ¨ + h1 z˙1 + h2 z1
(25)
Equations (16) and (25) are combined to gain the following equation (26).
(13)
Jz˙ 2 = −
Kt u R
+
Kt K e ˙ θ R
+ Jr¨ + Jh1 z˙1 + Jh2 z1
+ μ (σ0 z + σ1 z˙ ) + νσ2 θ˙ + D
N i=1
(22)
z˙1 = z2 − h1 z1 − h2 ϕ
where θ (ti ) is the angle output of opto-electronic tracking system at ti moment, θ (Q2, ti ) is the angle output of the system model at ti moment. The objective function is defined as follows:
S2 = b1 ∑ e 2 (Q2, ti ) + b2 max{ e (Q2, ti ) }
z1 dt
z2 = ρ˙d − θ˙ = r˙ − θ˙ + h1 z1 + h2 ϕ
The identification error is defined as below.
e (Q2, ti ) = θ (ti ) − θ (Q2, ti )
t
The tracking error is effectively reduced by integral action. Fig. 1 shows the structure diagram of opto-electronic tracking system. z2 is the second error. It is represented in the equation (23).
where M is the mass of the turntable, ΔF and Δθ are the change of command force and angle, respectively. Dynamic parameters σ0 and σ1 are identified by genetic algorithm. The search scope of dynamic parameters is based on the value of σ00 and σ10 . Assume that the desired identified parameter vector is as follows:
Q2 = [σˆ0, σˆ1]
∫0
= −
Kt u R
(
(14)
+
+ μz σ0 −
where b1 and b2 are weight coefficients. The dynamic parameters σ0 and σ1 are identified by minimizing the objective function. In this section, the genetic algorithm is chosen for identifying static and dynamic parameter of LuGre model. Based on the LuGre model parameters and the system model, a robust adaptive integral backstepping controller is proposed for trajectory tracking control of optoelectronic tracking system in the next section.
Kt K e ω R ω σ1 g(ω)
+ Jr¨ + Jh1 z˙1 + Jh2 z1
) + (μσ + νσ ) ω + D 1
2
(26)
Due to the uncertain parameters μ , ν and D of the system, they are estimated by the adaptive algorithm. μˆ , νˆ and Dˆ are the estimated values of uncertain parameters μ , ν and D , respectively. The error can be defined as follows: ∼ ∼ = μ − μˆ D = D − Dˆ , ν͠ = ν − νˆ, μ (27) ∼ ∼ is described in the following The time derivative of D , ν͠ and μ equation.
3.3. Robust adaptive integral backstepping controller design and stability analysis
∼ ∼˙ = −μˆ˙ D˙ = −Dˆ˙ , ν˙͠ = −νˆ˙, μ
Robust adaptive integral backstepping controller is developed for opto-electronic tracking system by utilizing backstepping design method.
(28)
Moreover, due to the unmeasured state z of the friction model, a nonlinear observer is needed. Supposing the estimated value of the observer is zˆ , the nonlinear observer of z is defined as below. 3
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Fig. 1. Structure diagram of opto-electronic tracking system.
zˆ˙ = ω −
ω ω ⎞ zˆ + h3 z2 ⎜⎛σ0 − σ1 ⎟ g(ω) g(ω) ⎠ ⎝
KK V˙2 ⎛⎜z1,z2⎟⎞ = −h1 z12 + z1 z2 + h3 z2 ⎜⎛−⎜⎛ tR e ω + Jr¨ + Jh1 z˙1 + Jh2 z1 + ⎝ ⎠ ⎝ ⎝ + h4 z2
(29)
where h3 is a positive constant. The error of state observer z͠ is described in the equation (30).
(
(30)
z͠ = z − zˆ
)
ˆ 2) ω + Dˆ + m sgn(z2) ⎟⎞ + ˆ ˆ σ0 − σ1 g(ωω) + (μσ ˆ 1 + νσ + μz ⎠
(
ω
(31)
)
(
)
∼σ + νσ ͠ 2) ω⎞⎟ ˆ ˆ) σ0 − σ1 g(ωω) + (μ + h3 z2 ⎛⎜ (μz − μz 1 ⎝ ⎠ 1 ∼ 1 ∼∼ 1 ω ͠ ˙͠ − h3 z2 μz͠ σ0 − σ1 g(ω) − η D Dˆ˙ − η μ μ˙ − η νν
3.5. Stability analysis
(
The second Lyapunov function is chosen as follows:
= − h1 z12 + z1 z2 + z2 h3 (−
Kt u R
+
Kt K e ω R
+
1 ∼∼ μ μ˙ η2
+
1 ∼∼ ν ν˙ η3
2
(36)
∼ ω V˙2 ⎛⎜z1,z2⎟⎞ = −h1 z12 − h3 h4 z22 − μ g(ω) z͠ 2 + h3 z2 D − h3 mz2 sgn(z2) ⎝ ⎠
+ Jr¨ + Jh1 z˙1 + Jh2 z1
(
)
∼zˆ) σ − σ ω + (μ ∼σ + νσ ͠ 2) ω⎞⎟ + h3 z2 ⎛⎜ (μz͠ + μ 0 1 g(ω) 1 ⎝ ⎠ 1 ∼ ˆ˙ 1∼ ˙ 1 ω ˙ ͠ˆ − h3 z2 μz͠ σ0 − σ1 g(ω) − η D D − η μ μˆ − η νν
)
))
3
The equation (37) is gained by taking the equation (36) into equation (35).
1 ∼∼ ω + μz σ0 − σ1 g(ω) + (μσ1 + νσ2 ) ω + D⎟⎞ + η D D˙ 1 ⎠ 1 ∼∼ 1 ω ∼ ω ∼ + μz − g(ω) z − h3 z2 σ0 − σ1 g(ω) + η μ μ˙ + η ∼∼ ν ν˙
(
2
∼zˆ ˆ ˆ = μz͠ + μ μz − μz
The derivative of equation (32) is described as below. 1 ∼∼ D D˙ η1
1
The equation (36) is gained by the equations (27) and (30).
(32)
V˙2 (z1,z2) = z1 z˙1 + h2 ϕz1 + h3 Jz2 z˙ 2 + μz∼∼ z˙ +
)
(35)
1 1 1 1 ∼2 1 ∼2 1 ∼2 h2 ϕ2 + h3 Jz22 + μz∼2 + D + μ + ν 2 2 2 2η1 2η2 2η3
(
)
ω ω 1 ∼∼ 1 ∼∼ 1 ͠ ˙͠ + μz͠ ⎜⎛− g(ω) z͠ − h3 z2 σ0 − σ1 g(ω) ⎟⎞ + η D D˙ + η μ μ˙ + η νν 1 2 3 ⎝ ⎠ ∼ ω = − h1 z12 − h3 h4 z22 − μ g(ω) z͠ 2 + h3 z2 D − h3 mz2 sgn(z2)
(
ω ω ⎞ z˙͠ = − z͠ − h3 z2 ⎜⎛σ0 − σ1 ⎟ g(ω) g( ω) ⎠ ⎝
(
Kt K e ω R
+ Jr¨ + Jh1 z˙1 + Jh2 z1 + μz σ0 − σ1 g(ω) + (μσ1 + νσ2) ω + D⎞⎟ ⎠
The derivative of z͠ is defined the following equation.
V2 (z1,z2) = V1 +
z1 h3
(
3
= −
(33)
h1 z12
∼ ⎛h z +μ 3 2 ⎝
According to the equation (33), the actual control input u is designed as follows:
−
)
h3 h4 z22
−
1
2
ω μ g(ω) z͠ 2
(
( (σ − σ ) zˆ + σ ω) − 0
ω 1 g(ω)
3
∼ + D h3 z2 − 1
1 η2
)−h mz μˆ˙ ⎞ + ν͠ (h z σ ω − νˆ˙ ) ⎠ 1 ˆ˙ D η1
3
3 2 2
2
1 η3
(37)
u=
R ⎛ Kt K e ω ⎜ Kt R
+ Jr¨ + Jh1 z˙1 + Jh2 z1 +
⎝
(
)
z1 h3
The adaptive rate is developed as below.
+ h4 z2
ˆ 2) ω + Dˆ + m sgn(z2) ⎞⎟ ˆ ˆ σ0 − σ1 g(ωω) + (μσ ˆ 1 + νσ + μz ⎠
⎧ Dˆ˙ = η1 h3 z2 ⎪ ω μˆ˙ = η2 h3 z2 σ0 − σ1 g(ω) zˆ + σ1 ω ⎨ ⎪˙ ˆ ⎩ ν = η3 h3 z2 σ2 ω
((
(34)
The equation (35) is gained by combining the equations (33) and (34).
)
)
Then, the equation (39) is simplified by the equation (37). 4
(38)
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ω 2 V˙2 (z1,z2) = −h1 z12 − h3 h4 z22 − μ z͠ − h3 m z2 g(ω)
(39)
where g(ω) > 0 , ω ≥ 0 and m > 0, then the following inequality (40) is obtained.
V˙2 (z1,z2) ≤ 0
(40)
From equations (32) and (40), it is clear that V2 (z1,z2) is positive definite and V˙2 (z1,z2) is negative semi-definite. It can conclude that ∼ ∼ V2 (z1,z2) is bounded, which means that the variables z1, z2 , ψ , z͠ , D , μ ˙ ˙ z , z ∈ L and ν͠ are all bounded. In particular, 1 2 ∞. Since the parameters D , μ and ν are bounded unknown constants. ∼ ∼ From equation (27), it can be seen that the parameter estimates D , μ and ν͠ are also uniformly bounded signals. Furthermore, the angle error z1 = r − θ is bounded and the desired reference angle r is assumed to be bounded, it can be concluded that the angle θ is bounded. Since these parameters z1, r˙ and ψ are bounded, it is seen that ρ˙d in (21) is uniformly bounded. Combining this with the equation z2 = ρ˙d − θ˙ is bounded, it can be seen that the angle speed θ˙ is bounded. From the equation (3), it can be concluded that the variable z is bounded since the angle speed θ˙ = ω is a bounded. It implies that the estimates z͠ is bounded since z͠ = z − zˆ is bounded according to the Lyapunov function. The boundedness of the control input u is evident in the expression in (34). Therefore, it can be concluded that all internal signals are globally uniformly bounded. Furthermore, it can be seen that z˙1 and z˙ 2 are also uniformly bounded from the equations in (18) and (25). In conclusion, z˙1, z˙ 2 ∈ L∞ has been proved. z1, z2 ∈ L2 needs to continue to be proven. To prove it, the variable pmin = min{h1, h3 h4} is selected. The following inequality is obtained according to the equations (39) and (40).
V˙2 (z1,z2) = −h1 z12 − h3 h4 z22 ≤ −pmin (z12 + z22) ≤ 0
(41)
The inequality (41) is integrated both sides between 0 and + ∞. Then, the following inequality (42) is gained.
V2 (+∞) − V2 (0) ≤ −pmin ⎜⎛ ⎝
∫0
+∞
z12 dt +
∫0
+∞
z12 dt ⎟⎞ ≤ 0 ⎠
(42)
This means that z1, z2 ∈ L2 . z˙1, z˙ 2 ∈ L∞ has been proved, in this condition, z1 and z2 approach to zero asymptotically utilizing Barbalat's Lemma [33]. Therefore, the angle error z1 converges to zero asymptotically. Through the above derivation process, it can be verified that the closed-loop system is asymptotically stable [7] and the angle error approaches to zero asymptotically by using the Lyapunov function. The debugging strategy of the parameters h1, h2, h3 and h4 is similar to the PID debug strategy. 4. System simulation The performance of robust adaptive integral backstepping controller is verified by constant speed tracking experiment and sinusoidal trajectory tracking experiment with disturbance. Table 1 describes the parameters of the opto-electronic tracking system. Table 1 System parameters. Parameters
Value
Inertia (kgm2) Stiction force (Nm) Coulomb friction (Nm) Viscous friction coefficient (Nms/deg) Stribeck angular velocity (deg /s) Stiffness coefficient (Nm/deg) Damping coefficient (Nms/deg) Torque coefficient (N∙m/A) Back-EMF coefficient(Vs/deg) Motor resistance (Ω)
J = 2.1 Fs = 3.8 Fc = 1.0 σ2 = 0.016 ωs = 1.1 σ0 = 28.3 σ1 = 0.59 Kt = 5.71 Ke = 0.1 R = 5.35
Fig. 2. Constant speed tracking results. (a)–(b)Angle and angle error tracking curve. (c)–(d) Angle speed and angle speed error tracking curve. 5
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Table 2 Comparisons of angle tracking errors. Controller
RAIB
AIBSM
MAX RMS SD
0.0017° 3.413e-004° 3.402e-004°
0.0031° 9.437e-004° 9.429e-004°
the blue lines are tracking result provided by AIBSM + FC method. Fig. 2 (a)–(b) describes the angle and angle error tracking curve. In order to quantify the tracking performance between the two controllers, the maximum (MAX), root mean square (RMS) and standard deviation (SD) of angle tracking errors are analyzed during the whole process of angle tracking, which are indicated in Table 2. It can be seen from Table 2 that the MAX and RMS of angle tracking errors in the proposed RAIB controller are smaller than that of AIBSM controller. The SD angle tracking error of the proposed controller is also less than the AIBSM controller. Fig. 2 (c)–(d) shows the angle speed and angle speed error tracking curve. In the initial tracking stage, the angle speed of the two controllers has a large deviation. Then, the AIBSM controller has a larger angle speed error than RAIB controller at the fifth second. The maximum angle speed error of AIBSM controller is 1.312°/s. However, this value of RAIB controller is 0.149°/s at this moment. Friction and friction observation curve between the two controllers is described in Fig. 3. where red lines are real friction curve, the black lines are friction estimate curve. The proposed friction observer based on RAIB technique still gives a good estimation of the friction as shown in Fig. 3(a). However, the friction observer based on AIBSM controller has a larger deviation as described in Fig. 3(b). The friction observation error curve is shown in Fig. 3 (c). The friction observation error of the RAIB controller reaches a range of −0.203 and 0.208. The corresponding value of the AIBSM controller is between −0.282 and 0.261. Therefore, the friction observation performance of RAIB controller is better than AIBSM controller. The results gained validate that the proposed RAIB method has a powerful strategy to track the angle and angle speed compared to the work described in Ref. [1]. The advantage of this control method lies in the robustness facing to friction at low speed, and can ensure a good friction observation curve.
4.2. Sinusoidal trajectory tracking experiment with disturbance The sinusoidal trajectory reference signal is defined as x ref = 2 sin(0.5π t) . Sinusoidal trajectory tracking experiment with external disturbance between RAIB controller and AIBSM controller is described in Figs. 4 and 5. The disturbance D is selected as random signal with zero mean and unity variance as D = rand (1). It is injected into simulation model at t = 10s. The duration is 10 s. In Fig. 4, where green lines are desired sinusoidal, the red lines are tracking result provided by RAIB + FC method, the blue lines are tracking result provided by AIBSM + FC method. Fig. 4 (a)–(b) describes the tracking results of angle and angle error. It can be seen that the RAIB controller follows the sinusoidal trajectory more precisely as compared with the AIBSM controller under the circumstance of external disturbance. The comparisons of the angle tracking errors between the two controllers are shown in Table 3. The MAX angle tracking error of RAIB controller is less than AIBSM controller. The proposed RAIB controller also has better superiority than the AIBSM controller in the RMS and SD of angle tracking errors. The angle speed and angle speed error tracking curve is displayed in Fig. 4(c)–(d). At the beginning of the sinusoidal trajectory tracking, the angle speed of the two controllers has changed rapidly. Then, the maximum angle speed error of AIBSM controller is 0.059°/s during the
Fig. 3. Constant speed tracking result. (a)–(c) Friction and friction observation curve.
In order to test the experiment, the design parameters of RAIB controller are described as follows: h1 = 1325, h2 = 0.01, h3 = 0.0002 and h4 = 1.1. The design parameters of AIBSM controller are the same as the RAIB controller. 4.1. Constant speed tracking experiment The tracking performance between RAIB controller with friction compensation (FC) and AIBSM controller [1] with friction compensation (FC) is shown in Figs. 2 and 3 under the circumstance of constant speed trajectory tracking. The angle speed reference signal is 3°/s. The duration of angle speed is 5 s. In Fig. 2, the green lines are desired angle, the red lines are tracking result provided by RAIB + FC method, 6
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Fig. 5. Sinusoidal trajectory tracking results with disturbance. (a)–(c) Friction and friction observation curve. Table 3 Comparisons of angle tracking errors.
Fig. 4. Sinusoidal trajectory tracking results with disturbance. (a)–(b) Angle and angle error tracking curve.(c)–(d) Angle speed and angle speed error tracking curve. 7
Controller
RAIB
AIBSM
MAX RMS SD
7.754e-004° 6.045e-004° 6.025e-004°
0.0015° 7.661e-004° 7.657e-004°
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Fig. 6. Experimental testing configuration.
Fig. 8. Angle speed reference signal.
angle speed tracking process. However, the corresponding value of RAIB controller is 0.054°/s. Fig. 5 (a)–(c) shows friction and friction observation curve. In the initial stage of friction observation, the friction observation error of AIBSM controller approaches 0.228. However, this value of RAIB controller is close to 0.215. Then, the friction observation error of AIBSM controller is bigger than RAIB controller before the external disturbance is added. Therefore, the friction observation performance of the RAIB controller is better than that of the AIBSM controller. The RAIB controller based on friction compensation has better tracking performance than AIBSM controller according to constant speed tracking experiment and sinusoidal trajectory tracking experiment with disturbance. Therefore, trajectory tracking tests verify that the proposed RAIB controller can be suitable for the opto-electronic tracking system.
stabilized platform and carries coupled device sensor, infrared thermal imager, laser and fiber optic gyro. The pitch and azimuth axes are directly controlled by permanent magnet DC motor without intermediate driven mechanism. The opto-electronic tracking system is composed of control cabinet and outdoor pointing unit. Control cabinet consists of industrial personal computer, motor driver, resolver decoder, transformer and liquid crystal display. Outdoor pointing unit consists of vision camera, infrared thermal imager, laser, bearing base, pitch package, resolver and torque motor. In order to compare the motor angle and angle speed between the two controllers, resolver encoder is coupled with the motor. The power supply of the motor driver is equal to 50 V. This power supply is converted from 220 V to 50 V by transformer.
5. System implementation and experimental results
5.2. Practical experiments
5.1. System implementation
The proposed RAIB controller based on friction compensation is applied in the opto-electronic tracking system. When the angle speed reference signal of the opto-electronic tracking system is 3°/s, tracking curve of angle speed reference signal, angle error and angle speed error are described in Figs. 8 and 9. The results in Fig. 9 (a)–(b) describe the tracking performance between the proposed RAIB controller and AIBSM controller. The comparison results of actual angle tracking errors are displayed in Table 4. Due to the RAIB control strategy, angle tracking error is effectively decreased. RMS of actual angle tracking error is 0.158° utilizing the AIBSM method. However, it reduces to 0.116° utilizing the proposed strategy. The angle tracking accuracy improves by 26.6% utilizing the proposed strategy. It is seen that the MAX and SD of the actual angle tracking errors in the proposed RAIB controller is also smaller than that of AIBSM controller. It can be seen from Fig. 9 that the angle speed error of RAIB controller reaches a stable range of −0.416°/s and 0.463°/s. The corresponding value of AIBSM controller is about between −0.482°/s and 0.375°/s. As the roughness of the azimuth contact surface is different, the angle speed may increase in some places. Although the maximum angle speed error of RAIB controller is a little larger than AIBSM controller, the angle error of RAIB controller is much smaller than AIBSM controller. The tracking performance between the two controllers is again assessed by angle response experiment. Fig. 10(a)–(b) show the angle speed and angle tracking curve. The starting point of the angle response is at t = 19.5 s. The angle speed of RAIB controller is stable in the range of −1.772°/s and 5.313°/ s. This value of AIBSM controller is stable between −1.880°/s and 4.236°/s. Moreover, the maximum angle speed of the RAIB controller is
The effectiveness of the proposed RAIB controller based on friction compensation for the opto-electronic tracking system illustrated in Fig. 6 was tested experimentally. The configuration of the experimental system is composed essentially of permanent magnet DC motor whose parameters are present in Table 1, motor driver and an industrial personal computer. The industrial personal computer consists of image capture card, digital I/O card and serial port card. Fig. 7 shows the opto-electronic tracking system. The system is a two degree of freedom
Fig. 7. The opto-electronic tracking system. 8
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Fig. 9. Practical experiment tracking results. (a)–(b) Angle speed error and angle error tracking curve.
Fig. 10. Practical experiment tracking results with angle response. (a)–(b) Angle speed and angle tracking curve.
Table 4 Comparisons of actual angle tracking errors. Controller
RAIB
AIBSM
Maximum Root mean square Standard deviation
0.318° 0.116° 0.112°
0.334° 0.158° 0.163°
controller has been evaluated on the opto-electronic tracking system in simulation and practical experiments using different angle trajectories. The experimental results of the opto-electronic tracking system show that system tracking accuracy and stability can be improved by the proposed RAIB control method. Future work will be focused on adaptive controller design based on friction compensation using neural network approximations and experimental verification on the opto-electronic tracking system.
greater than that of AIBSM controller, so it has a faster response speed. The maximum angle and the regulating time of RAIB controller are 2.922° and 4.8 s, respectively. These corresponding values of AIBSM controller is 2.930° and 10.3 s, respectively. Therefore, the experimental results of angle response show that the angle provided by the proposed strategy presents a smaller regulating time and a faster response speed compared to the control method presented in Ref. [1]. The comparison between simulation and the experimental results of the control strategies shows that the proposed RAIB control approach can obtain high-performance tracking control for the opto-electronic tracking system under the circumstance of nonlinear friction and external disturbance.
Acknowledgements This work is supported by the Foundation of the National Natural Science Foundation of China (No. 61427810, 61503283 and 61733012), and Natural Science Foundation of Tianjin (No. 16JCZDJC30100). References [1] Yue FF, Li XF, Chen C, Tan WB. Adaptive integral backstepping sliding mode control for opto-electronic tracking system based on modified LuGre friction model. Int J Syst Sci 2017;48:3374–81. [2] Yue FF, Li XF. Improved kernelized correlation filter algorithm and application in the optoelectronic tracking system. Int J Adv Rob Syst 2018;15:1–10. [3] Naso D, Cupertino F, Turchiano B. Precise position control of tubular linear motors with neural networks and composite learning. Contr Eng Pract 2010;18:515–22. [4] Lee TH, Tan KK, Huang SN. Adaptive friction compensation with a dynamical friction model. IEEE/ASME Trans Mechatron 2011;16:133–40. [5] Huang SN, Tan KK. Intelligent friction modeling and compensation using neural network approximations. IEEE Trans Ind Electron 2012;59:3342–9. [6] Zhang YM, Yan P, Zhang Z. High precision tracking control of a servo gantry with dynamic friction compensation. ISA Trans 2016;62:349–435. [7] Tan YL, Chang J, Tan HL. Adaptive backstepping control and friction compensation for AC servo with inertia and load uncertainties. IEEE Trans Ind Electron 2003;50:944–52.
6. Conclusions A robust adaptive integral backstepping control method based on friction compensation is proposed for opto-electronic tracking system to achieve accurate and stable tracking control under the circumstance of friction and external disturbance. The effect of friction and external disturbance is reduced by modified LuGre observer and robust adaptive backstepping controller with friction compensation. The steady-state error of the system is reduced by integral action. The stability analysis of the system is demonstrated by Lyapunov criterion. The proposed 9
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