The Korean Society of Marine Engineering
[ Original Paper ]
Journal of the Korean Society of Marine Engineering - Vol. 41, No. 8, pp.760-766
ISSN: 2234-7925 (Print) 2234-8352 (Online)
Print publication date 31 Oct 2017
Received 14 Jul 2017 Revised 15 Sep 2017 Accepted 10 Oct 2017
DOI: https://doi.org/10.5916/jkosme.2017.41.8.760

Performance improvement for velocity controller of fish sorting belt conveyor system under bounded disturbance in transient phase

Huy Hung Nguyen1 ; Dae-Hwan Kim2 ; Hak-Kyeong Kim3 ; Sang-Bong Kim
1Department of Mechanical Design Engineering, Pukyong National University, Tel: 051-629-6158 E-mail : nghhung74
2Department of Mechanical Design Engineering, Pukyong National University, Tel: 051-629-6158 kimdh2599@pknu.ac.kr
3Department of Mechanical Design Engineering, Pukyong National University, Tel: 051-629-6158 hakkyeong@pknu.ac.kr

Correspondence to: Department of Mechanical Design Engineering, Pukyong National University, Sinseon-ro 365, Nam-gu, Busan 48547, Korea, E-mail: kimsb@pknu.ac.kr, Tel: 051-629-6158

Copyright © The Korean Society of Marine Engineering
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper proposes a modified-model reference adaptive controller (M-MRAC) for the velocity control of a belt conveyor in a fish sorting system with uncertainty parameters, input saturation, and bounded disturbances. The following aspects are addressed to improve the tracking performance and robustness of the proposed controller for the fish sorting belt conveyor system with bounded disturbances in the transient phase. First the estimated parameters in adaptive laws have smooth variations under bounded external disturbances and a σ-modification is added to the adaptive law for the proposed M-MRAC controller to be robust. Second, an auxiliary error vector is introduced for compensating the error dynamics of the system when the input saturation occurs. Finally, the experimental results of the proposed controller under bounded disturbance and saturated input are better in effectiveness and performance than those of a conventional model reference adaptive controller.

Keywords:

Bounded Disturbance, M-MRAC, σ-modification, Saturated Input, velocity controller

1. Introduction

In a fish sorting system (FSS), captured fishes are transported by a fish pump from a ship into a fish sorting line through a conveyor system. The captured fishes are sorted by the injury rate, which is estimated by using an image processing system. Thus, the conveyor system speed plays a key role in estimating the injury rate of fishes with high accuracy. Accordingly, the velocity of the conveyor system should be controlled with suitable speed to achieve reliable recognition with the image processing system. The conveyor system in a FSS consists of three or more conveyors, the desired velocities of which are defined as trapezoidal velocity profiles [1]. To control the closedloop dynamics of the conveyor system close to the desired velocities, a system model is used to develop a model-based controller. However, the conveyor system has some uncertain parameters, such as a friction factor, belt elastic factor, pulling force, and others, which are unmeasured in the conveyor system.

An adaptive controller was considered for its important ability to deal with system uncertainties without requiring explicit, unknown plant parameter identification [2][3]. A conventional model reference adaptive controller (CMRAC) tuning control parameters directly is one of the main schemes utilized in the field of adaptive control [4]-[6]. Although asymptotic tracking could be achieved in CMRAC systems, the tracking performance in the transient state could be poor [7] because it is impossible to achieve a small deviation of the tracking error in the transient state with an insufficient adaptation rate.

For engineering systems, it should be noted that the control input signal is frequently saturated and has proved to be a source of performance degradation. An input control signal saturation could lead to poor control performance and even closed-loop system instability [8][9] if its effect is not considered in the design of the controller.

In addition, the estimated parameters in the update laws can be varied smoothly in the presence of bounded disturbances. Therefore, certain modifications [10], such as the dead-zone technique, e-modification, or projection operator were utilized to the modify update laws.

This paper proposes a model reference adaptive control approach, in which the reference model is modified by a feedback of a modeling error signal [11], and applied to the velocity control of the conveyor system in a FSS with uncertain parameters, saturated input, and bounded disturbances. In the presence of bounded disturbance, a σ-modification is utilized in the update laws of the proposed controller to eliminate the drift phenomenon of the control parameters. In addition, an auxiliary error vector [12] is employed to compensate for the error dynamics when input saturation occurs. The experimental results verify the effectiveness and the performance of the proposed controller.


2. System Modeling

A typical conveyor system in an FSS, as shown in Figure 1, consists of an on-loading conveyor (1st conveyor), a camera conveyor (2nd conveyor) to examine moving fish boxes using an image processing system, and a transition conveyor (3rd conveyor). Each conveyor consists of a mechanical subsystem and an electrical subsystem. The simplified model of the mechanical subsystem of the ith conveyor of the FSS is shown in Figure 2 (i = 1, 2, 3).

Figure 1:

Typical conveyor system in a FSS

Figure 2:

Simplified model of the ith conveyor of FSS

In Figure 2, Ji1 and Ji2 are the moments of inertia of the driving and driven rollers, ωi1 and ωi2 are the angular velocities of the driving and driven roller, fi1 and fi2 are friction coefficients of bearings inside the driving and driven rollers, and Di1 and Di2 are diameters of the driving and driven rollers, respectively.

To simplify the mechanical subsystem modeling, Assumption 1 is proposed as follows:

•  Assumption 1
√  The connection between the motor shaft and driving roller is rigid and short.
√  Belt slippage on the rollers is negligible.
√  Fish box slippage on the belt is negligible.

The electrical subsystem is used to drive the mechanical subsystem. The inverter with a DC voltage input controls the induction motor to generate sufficient torque to drive the mechanical subsystem, as shown in Figure 2.

Under the above assumptions, the ith mechanical driven system can be expressed by the following:

τi=Jiω˙i1+fiωi1+τdit(1) 

where Ji = Ji1 + Ji2, fi = fi1 + fi2, τdi(t) is a bounded external disturbance torque and τi is the sufficient torque to drive the mechanical subsystem of the ith conveyor, given as follows:

τi=kiui*(2) 

where ki is an amplifier gain and ui* is the DC voltage input of the ith inverter to create the desired torque τi. ui* is defined as a saturated control input of the ith conveyor as follows:

ui*=ui minfor ui<ui min u0for ui maxuui maxui maxfor ui>ui max(3) 

where ui* is the designed control input for the ith conveyor by the proposed controller, and uimin and uimax are the limited thresholds of the ith designed control inputs.

A dynamic FSS based on Equation (1) - Equation (3) can be expressed in the state space as follows:

x˙=Ax+Bu*-dt(4) 

where x = [ω11 ω21 ω31]T is an angular velocity output vector of the FSS measured by encoders attached to the driving rollers, ωi1 is the angular velocity of the driving roller of the ith conveyor, u*=satu=u1* u2* u3*T is a saturated control input vector, d(t/)=[d1 d2 d3]T is a bounded external disturbance vector with di=Tditki, and the unknown constant matrices A, B∈ℜ3×3 and a known matrix Bm∈ℜ3×3 are given as follows: A=a11000a22000a33, B=b11000b22000b33 with aii=-fiJi and bii=-kiJi.


3. Controller Design

The control objective is to determine a designed control input vector u = [u1 u2 u3]T for a modified model reference adaptive system with saturated inputs and bounded disturbance, such that the angular velocity output vector tracks the output vector of a reference model.

A modified reference model used for its output vector to asymptotically track a trapezoidal-type reference input vector r is chosen as follows:

x˙m=Amxm+Bmr+r˙+λe(5) 
e=x-xm(6) 

where λ > 0 is an error feedback gain, xm = [ωm1 ωm2 ωm3]T the angular velocity output vector of the modified reference model, ωmi is the ith reference angular velocity, e is a modeling error vector, r = [r1 r2 r3]T is the reference angular velocity input vector, and Am,Bm∈ℜ3×3 are given as follows:

Am=am1000am2000am3,Bm=bm1000bm2000bm3

where amiv, bmi are reference model parameters chosen to satisfy Assumptions 2 and 3 as follows.

•  Assumption 2: Given a known Hurwitz matrix Am∈ℜ3×3 and a known matrix Bm∈ℜ3×3 of full rank, there exists an unknown control gain matrix K∈ℜ3×3 and an unknown positive definite diagonal constant matrix Λ∈ℜ3×3 such that the following equations hold:

A=Am-BKB=BmΛ(7) 

•  Assumption 3: A positive symmetric definite matrix P = PT > 0 is the solution of the following Lyapunov equation:

AmTP+PAm=-Qm(8) 

where Qm is a positive definite matrix.

Substituting Equation (7) into Equation (4), adding and subtracting Bmr, r˙ yields

x˙=AmX+Bmr+r˙+BmΛu*-ur(9) 

where Φ = Λ−1, Ω=(Bm/)-1, and ur is an ideal control input vector as follows:

ur=Kx+Φr+Ωr˙+dt(10) 

The first-order time derivative of e is given by

e˙=Am-λIe+BmΛu*-ur(11) 

If u* = ur, e˙=Am-λIe. Because Am and (Am-λI) are Hurwitz matrices, it can be concluded that e→0 as t →∞. This implies that the plant in Equation (4) can asymptotically track the reference model in Equation (5). However, the ideal control input vector cannot be implemented, because the matrices K, Φ, and Ω, and the disturbance vector d(t) are unknown. Therefore, a designed control input vector u is chosen as an estimate of ur in the following form:

u=K^x+Φ^r+Ω^r˙+d^t(12) 

where K^,Φ^,Ω^ are estimations of unknown control gain matrices K,Φ,Ω and d^∈ℜ3 is an estimated vector of an unknown constant vector d¯, which is the averaged-value vector of d(t) in Equation (4).

A saturated input error vector is defined as

u=u-u*(13) 

From Equation (10) - Equation (13), the first-order time derivative of e is given as

e˙=Am-λIe+BmΛK~x+Φ~r+Ω~r˙+d~+BmΛd¯-d-Δu(14) 

where K~=K^-K, Φ~=Φ^-Φ, Ω~=Ω^, and d~=d^-d.

To remove the effect of the saturated input, an auxiliary error vector eΔ is defined as

e˙Δ=Am-λIeΔ-K^ΔΔu(15) 

where (AmλI) is a stable Hurwitz matrix and K^ΔR3×3 is the adaptable parameter matrix.

Therefore, a new error vector is defined as follows:

eu=e-e(16) 

From Equation (14) ~ Equation (16), the first-order time derivative of eu is given as

e˙u=Am-λIeu+BmΛK~x+Φ~r+Ω~r˙+d~+K~∆u+BmΛd¯-d(17) 

where K~=K^-BmΛ.

The control gains K^,Φ^,Ω^,d^ in Equation (12) are estimated by update laws based on a σ-modification that will be designed in Theorem 1.

Theorem 1: An M-MRAC system of Equation (4) is stable as long as a designed control input vector of Equation (12) is given and update laws using a σ-modification are given as

K^˙=-σK^-γ1BmTPeuxTΦ^˙=-σΦ^-γ1BmTPeurTΩ^˙=-σΩ^-γ1BmTPeur˙T(18) 
K^˙Δ=-σK^Δ-γ2PeuΔuT, and(19) 
d^˙=-σd^-γ3BmTPeu(20) 

where γ1, γ2, γ3 > 0 are the adaptation rates, and σ is a design parameter.

[Proof of Theorem 1]: A candidate Lyapunov function is chosen to analyze the stability of the system as follows:

Vt=euTPeu+1γ1traceK~TΛK~+Φ~TΛΦ~+Φ~TΛΦ~+1γ2traceK~ΔTK~Δ+1γ3traced~Td~Λ0(21) 

The first-order time derivative of V(t) is given as

V˙=-euTQm+2λPeu+2traceK~TΛBmTPeuxT+Φ~TΛBmTPeurT+Ω~TΛBmTPeur˙T+K~ΔTPeuΔuT+2d~TΛBmTPeu+2λ1traceK~TΛK^˙+Φ~TΛΦ^˙+Ω~TΛΩ^˙ +2λ1traceK~TΛK^˙+Φ~TΛΦ^˙+Ω~TΛΩ^˙+2λ2traceK~ΔTK^˙Δ+2λ3traced~ΔTd^˙Δ+2euTPBmΛd¯-d(22) 

Using Equation (18) - Equation (20) yields

V˙=-euTQm+2λPeu-2σγ1traceK~TΛK^+Φ~TΛΦ^+Ω~TΛΩ^-2σγ2traceK~ΔTK^Δ-2σγ3traced~Td^Λ+2euTPBmΛd--d(23) 

Using the Rayleigh principle, Equation (23) can be written as follows:

V˙-a1eu2+2eud*-2σγ1λminΛK~F2+Φ~F2+Ω~F2-2σγ2λminΛK~ΔF2-2σγ3λminΛd~2-a1eu2+2eud*+c(24) 

where a1=λminQm+2λminPλ>0,d*=PBmΛd-d¯l, .F is the Frobenius norm and c is given as follows:

c=-2σγ1λminΛK~F2+Φ~F2+Ω~F2-2σγ2λminΛK~ΔF2-2σγ3λminΛd~2(25) 

V˙t in Equation (24) is negative semi-definite outside the compact set

H=eu,K~,Φ~,Ω~,K~Δ,d~:a1eu2-2eud*>c(26) 

This implies that eu,K^,Φ^,Ω^,K^Δ, and d^, and d^ are bounded from Equations (21), (24), and (26), and 0 eu → as t →∞ by Barbalat's lemma. Hence, eeΔ and e is bounded if and only if eΔ is also bounded. The boundedness eΔ is proven as follows:

A candidate of Lyapunov function is chosen as

W=eΔTPeΔ0(27) 

Using Equations (15) and (27), the first-order time derivative of W is given as

W˙=-eTQm+2λPeΔ-2eΔTPK^ΔΔu-e2a1+λ3λ2W+λ3(28) 

where λ3=2-eΔTPK^TΔu0 and λ2=a1λmaxP>0.

By using the Gronwall–Bellman Inequality, Equation (28) implies that

WWe0-λ3λ2exp-λ2t+λ3λ2(29) 

Using Equations (27) and (29), the following are obtained:

limteΔTPeΔλ3λ2(30) 
λ3λ2limteΔTPeΔλminPlimte2, and(31) 
limteλ3λ2λminP(32) 

It can be proven that eΔ is also bounded.

E.O.D

The block diagram of the proposed controller is shown in Figure 3.

Figure 3:

Block diagram of the proposed controller

A comparison of M-MRAC with CMRAC is given in Table 1.

Comparison of M-MRAC with CMRAC


4. Experimental Results

To evaluate the effectiveness of the proposed controller (MMRAC) with σ -modification and compare it with the CMRAC, an experiment was carried out under the following conditions.

The initial values of the state variables and the controller inputs were set to zero. The input voltages of the inverters considered as control inputs of the proposed controller can vary in range from u1min = u2min = u3min = 0 V to u1max = u2max = u3max = 5 V.

The parameters of the modified model reference system was given by am1 = am2 = am3 = −30 and bm1 = bm2 = bm3 = 30. The error feedback gain was chosen as λ = 10, the fixed controller gains were chosen as γ2 = 3.3, γ3 = 1.3, σ = 0.03, and the positive symmetric definite matrix was chosen as P = diag([10-8 10-8 10-8]). The reference inputs for the conveyor plant were the angular velocity inputs and are given in Figure 4.

Figure 4:

Velocity profiles of all conveyors

To demonstrate the effectiveness of the proposed controller, the following three cases were considered.

Case 1: The adaptation rates of both the CMRAC and MMRAC are set to γ1 = 1.67. It can be seen that both the output of the proposed M-MRAC x1(t) and the output of the CMRAC x1M(t) for the 1st conveyor track the reference input r1(t), as shown in Figure 5. However, the output amplitude of the CMRAC varies more strongly than that of the proposed MMRAC. High-frequency oscillations are generated in the control input signal u1M(t) of the CMRAC, whereas the control input signal u1(t) of the proposed M-MRAC remains nearly unchanged, as shown in Figure 6, when the angular velocity output reaches 42.1 rad/s in Figure 5.

Figure 5:

Output of the CMRAC and M-MRAC for the 1st conveyor

Figure 6:

Control input of the CMRAC and M-MRAC for the 1st conveyor

Case 2: The adaptation rates of both the CMRAC and MMRAC are set to γ1 = 10. The angular velocity output x2(t) of the proposed M-MRAC for the 2nd conveyor tracks the reference input r2(t) with a small error (from −1.1 rad/s to +1.3 rad/s), whereas the angular velocity output x2M(t) of the CMRAC for the 2nd conveyor tracks the reference input r2(t) with an error from −2.3 rad/s to +2.5 rad/s in Figure 7. The control input u2M(t) of the CMRAC also oscillates with higher frequency and amplitude than the control input u2(t) of the proposed M-MRAC, as shown in Figure 8. The maximum and minimum values of the control input of the CMRAC are 4.1 V and 3.72 V, respectively, and the average value of the control input is 3.88 V. The control input for the proposed M-MRAC varies more slowly than that for the CMRAC. Therefore, the performance of the proposed M-MRAC is better than that of the CMRAC in this case.

Figure 7:

Output of the CMRAC and the M-MRAC for the 2nd conveyor

Figure 8:

Control input of the CMRAC and M-MRAC for the 2nd conveyor

Case 3: Similarly, the adaptation rates of both the CMRAC and M-MRAC are set to γ1 = 6.67. The angular velocity output x3(t) of the proposed M-MRAC for the 3rd conveyor also tracks the reference input r3(t) better than the angular velocity output x3M(t) of the CMRAC, as shown in Figure 9. Because the reference input r3(t) is a step type, the control input signal amplitudes of both the proposed M-MRAC and the CMRAC are large (15.5 V and 12.5 V, respectively), as shown in Figure 10. Therefore, the saturated control input u3*(t) is set to u3max = 5 V, and the angular velocity outputs of both the proposed MMRAC and the CMRAC reach 90.3 rad/s in Figure 9.

Figure 9:

Output of the CMRAC and the M−MRAC for the 3rd conveyor

Figure 10:

Control input of the CMRAC and the M−MRAC for the 3rd conveyor

The tracking performance of the proposed M-MRAC controller versus the CMRAC is given in Table 2.

Modeling error of the CMRAC and M−MRAC


5. Conclusion

A modified-model reference adaptive controller for belt conveyors in an FSS with uncertainty parameters, input saturation, and bounded disturbances was proposed. The feedback of the modeling error signal in the proposed MMRAC controller obtained a smaller modeling error than that in the CMRAC. The tracking performance of the proposed M−MRAC showed a better improvement in both the transient and asymptotic states than that of the CMRAC, and the highfrequency elements in the control input signals were reduced in the proposed M−MRAC when the adaptation rate was increased. The error dynamics under the input saturation were compensated by the auxiliary output error. The experimental results showed that the proposed M-MRAC became more effective than the CMRAC when the adaptation rate was large and the error feedback gain was suitably selected.

Acknowledgments

This research was a part of the project titled “Localization of unloading automation system related to Korean type of fish pump,” funded by the Ministry of Oceans and Fisheries, Korea.

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Figure 1:

Figure 1:
Typical conveyor system in a FSS

Figure 2:

Figure 2:
Simplified model of the ith conveyor of FSS

Figure 3:

Figure 3:
Block diagram of the proposed controller

Figure 4:

Figure 4:
Velocity profiles of all conveyors

Figure 5:

Figure 5:
Output of the CMRAC and M-MRAC for the 1st conveyor

Figure 6:

Figure 6:
Control input of the CMRAC and M-MRAC for the 1st conveyor

Figure 7:

Figure 7:
Output of the CMRAC and the M-MRAC for the 2nd conveyor

Figure 8:

Figure 8:
Control input of the CMRAC and M-MRAC for the 2nd conveyor

Figure 9:

Figure 9:
Output of the CMRAC and the M−MRAC for the 3rd conveyor

Figure 10:

Figure 10:
Control input of the CMRAC and the M−MRAC for the 3rd conveyor

Table 1:

Comparison of M-MRAC with CMRAC

CMRAC M−MRAC
Ref. model x˙m=Amxm+Bmr+r˙ x˙m=Amxm+Bmr+r˙+λe
Update laws K^˙=-γeTPBmx.Φ^˙=-γeTPBmr.Ω^˙=-γeTPBmr˙.d^˙=-γeTPBm. K^˙=-σK^-γeTPBmx.Φ^˙=-σΦ^-γeTPBmr.Ω^˙=-σΩ^-γeTPBmr˙.d^˙=-σd^-γeTPBm.

Table 2:

Modeling error of the CMRAC and M−MRAC

Parameters CMRAC M−MRAC
Case 1 Output (rad/s) 42.1± 1.9
2.1
42.1± 0.9
0.5
Control input (V) 2.3V ± 0.3 2.3V± 0.04
0.07
Adaptation rate 1.67 1.67
Modeling error 5% 2.1%
Case 2 Output (rad/s) 70 ± 2.5
3.5
70± 1.3
1.1
Control input (V) 3.9V ± 0.2
0.3
3.9V ± 0.1
0.16
Adaptation rate 10 10
Modeling error 5% 1.86%
Case 3 Output (rad/s) 70 ± 2 70 ± 0.6
0.8
Control input (V) 3.9V ± 0.12
0.24
3.9V ± 0.04
0.07
Adaptation rate 6.67 6.67
Modeling error 2.9% 1.14%