# An LMI-Based Simple Robust Control for Load Sway Rejection of Rotary Cranes with Double-Pendulum Effect

Authors: Huimin Ouyang; Jian Wang; Guangming Zhang; Lei Mei; Xin Deng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the double-pendulum rotary crane case, the load sway properties become more complicated so that the difficulty of design and analysis of crane control system is increased. Moreover, the change of rope length not only affects the stability of the system, but also leads to the decline of control performance. In order to solve the foregoing problems, a simple model for controller design is obtained by linearizing and decoupling the complex nonlinear model of rotary crane. Then, a linear feedback controller which can furnish robust performance is presented. Finally, numerical simulations verify the effectiveness of the proposed method by comparing with a traditional approach.


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## Body

## 1. Introduction

As a kind of handling robot, the crane systems have the advantages of less executive mechanism and simple structure; therefore, they have been widely used in various production departments [1]. However, load sways caused by the cart or boom motion not only damage goods, reduce production efficiency, but also cause accidents, even casualties. Hence, how to design a control method that can effectively suppress the load sways has been a hot and difficult point in both academic and industrial circles.Until now, various approaches were applied to crane systems [2–26]. Nonetheless, when the hook mass or load shape cannot be ignored, the load swing presents two-stage swing. For this problem, open-loop methods such as input shaping technology [27–32], motion planning method [33, 34], were presented. On the other hand, closed-loop methods were also reported in [35, 36]. Furthermore, an input-shaping-based SIRMs fuzzy approach was reported in [37]. Ouyang et al. reported two novel sliding-mode based approaches and a simple LMI-based robust method [38–40]. A nonlinear quasi-PID scheme and an adaptive nonlinear controller were applied to overhead cranes in [41, 42].In conclusion, although open-loop approaches were introduced in [27–34], accurate crane models were always needed. On the other hand, too complicated algorithms were adopted in [35–42] to achieve good robust or adaptive performance. Moreover, different from the load sways of double-pendulum overhead cranes which were dealt with in the most existing literatures [27–42], the double pendulums in the rotary crane system are both conical pendulums, and the system characteristics are more complex. Hence, even if the similar algorithm shown in [38] is adopted, the method for calculating the controller gain is different. Moreover, as far as we know, there are few related reports on the load sway suppression control of rotary cranes with double-pendulum effect.In order to solve the foregoing problems, a simple model for controller design is obtained by linearizing and decoupling the complex nonlinear model of rotary crane. Next, a linear feedback controller which can furnish robust performance is presented. The controller gains are obtained by using synthesizing a pole placement and a optimal regulator problems [43]. Finally, numerical simulations verify the effectiveness of the proposed method by comparing with a traditional approach.


## 2. Rotary Crane Dynamics

Figure1 shows the model of a rotary crane with double-pendulum effect [30]: (1)m1+m2l121+θ12θ¨1+m1+m2l12θ1θ2θ¨2+m2l1l21+θ1θ3θ¨3+m2l1l2θ1θ4θ¨4+m1+m2l1LC5-θ1S5θ¨5-m1+m2l12θ2+m2l1l2θ4θ¨6+m1+m2l12θ1θ˙12+m1+m2l12θ1-2m1+m2l12θ˙6θ˙2+m2l1l2θ1θ˙32+m2l1l2θ1θ˙4-2m2l1l2θ˙6θ˙4-m1+m2l1LS5+θ1C5θ˙52-m1+m2l12θ1+m2l1l2θ3+m1+m2l1LS5θ˙62+m1+m2l1gθ1=0(2)m1+m2l12θ1θ2θ¨1+m1+m2l121+θ22θ¨2+m2l1l2θ2θ3θ¨3+m2l1l21+θ2θ4θ¨4-m1+m2l1Lθ2S5θ¨5+m1+m2l12θ1+m1+m2l1LS5+m2l1l2θ3θ¨6+m1+m2l12θ2θ˙1+2m1+m2l12θ˙6θ˙1+m1+m2l12θ2θ˙22+m2l1l2θ2θ˙3+2m2l1l2θ˙6θ˙3+m2l1l2θ2θ˙42+2m1+m2l1LC5θ˙6-m1+m2l1Lθ2C5θ˙5θ˙5-m1+m2l12θ2+m2l1l2θ4θ˙62+m1+m2l1gθ2=0(3)m2l1l21+θ1θ3θ¨1+m2l1l2θ2θ4θ¨2+m2l221+θ32θ¨3+m2l22θ3θ4θ¨4+m2l2LC5-θ3S5θ¨5-m2l22θ4+m2l1l2θ2θ¨6+m2l1l2θ3θ˙12+m2l1l2θ3θ˙2-2m2l1l2θ˙6θ˙2+m2l22θ3θ˙32+m2l22θ3θ˙4-2m2l22θ˙6θ˙4-m2l2LS5+θ3C5θ˙52-m2l22θ3+m2l1l2θ1+m2l2LS5θ˙62+m2l2gθ3=0(4)m2l1l2θ1θ4θ¨1+m2l1l21+θ2θ4θ¨2+m2l22θ3θ4θ¨3+m2l221+θ42θ¨4-m2l2Lθ4S5θ¨5+m2l22θ3+m2l2LS5+m2l1l2θ1θ¨6+m2l1l2θ4θ˙1+2m2l1l2θ˙6θ˙1+m2l1l2θ4θ˙22+m2l22θ4θ˙3+2m2l22θ˙6θ˙3+m2l22θ4θ˙42+2m2l2LC5θ˙6-m2l2Lθ4C5θ˙5θ˙5-m2l1l2θ2+m2l22θ4θ˙62+m2l2gθ4=0(5)m1+m2l1LC5-θ1S5θ¨1-m1+m2l1Lθ2S5θ¨2+m2l2LC5-θ3S5θ¨3-m2l2Lθ4S5θ¨4+m1+m2L2+Iy+J5θ¨5-m1+m2l1Lθ2C5+m2l2Lθ4C5θ¨6-m1+m2l1LS5θ˙12-m1+m2l1LS5θ˙22-m2l2LS5θ˙32-m2l2LS5θ˙42-2m1+m2l1LC5θ˙2+2m2l2LC5θ˙4+m1+m2l1Lθ1C5θ˙6θ˙6-gS512M0L+m1+m2L-12M1L1-12m1+m2L2+Ix-Izsin⁡2θ5θ˙62=γ5-C51θ˙5+f51(6)-m1+m2l12θ2+m2l1l2θ4θ¨1+m1+m2l1LS5+m2l1l2θ3+m1+m2l12θ1θ¨2-m2l1l2θ2+m2l22θ4θ¨3+m2l1l2θ1+m2l22θ3+m2l2LS5θ¨4-m1+m2l1Lθ2C5+m2l2Lθ4S5θ¨5+m1+m2L2S52+m1+m2l12θ12+θ22+m2l22θ32+θ42+Ib+IxS5+IzC5+J6θ¨6+2m1+m2l1Lθ1S5+2m2l2Lθ3S5+2m2l1l2θ1θ3+θ2θ4θ¨6+m1+m2l1Lθ2S5θ˙5+m2l2Lθ4S5θ˙5+2m1+m2l1Lθ1C5θ˙6+2m2l2Lθ3C5θ˙6θ˙5+2m1+m2l12θ1θ˙1+θ2θ˙2+2m2l22θ3θ˙3+θ4θ˙4+2m1+m2l1Lθ˙1S5+2m2l2Lθ˙3S5θ˙6+2m2l1l2θ4θ˙2+θ2θ˙4+θ3θ˙1+θ1θ˙3θ˙6+m1+m2L2+Ix-Izsin⁡2θ5θ˙5θ˙6=γ6-C61θ˙6+f61where M, M1, m1, and m2 are the masses of the boom, the ballast, the hook, and the load, respectively. L, L1, l1, and l2 are the lengths of the boom, the ballast, the rope, and the rigging, respectively. θ1, θ3, θ2, and θ4 are the components of hook angle and load angle in the vertical and horizontal direction of the boom, respectively. θ5 and θ6 are the vertical and horizontal angles of the boom, respectively. In addition, g, Jk+4, γk+4, C(k+4)1 and f(k+4)1 (k=1,2) are the acceleration of gravity, the moment of inertia for boom motion, the system gains, the boom-driven torque, the viscous friction coefficients, and the static friction or Coulomb friction term. In addition, Cj and Sj (j=5,6) denote cos⁡θj and sin⁡θj, respectively. The detailed derivation procedure for (1)-(6) is included in Appendix.Figure 1
Rotary crane model.The dynamics of boom-driven systems are [44, 45] (7)Jk+4θ¨k+4+dk+4=γk+4,k=1,2where dk+4 denotes the disturbance.In order to compensate for the above disturbance, the disturbance observer as shown in Figure2 is designed [43–48]. Then, we have (8)θ¨k+4=vk+4where vk+4 is the control input for vertical and horizontal subsystem, respectively.Figure 2
Proposed disturbance observer.Equations (1)-(4) and (8) are represented as follows: (9)M′Θ¨′+K′Θ′=AkB′vk+4,k=1,2M′=1m2l2m1+m2l1l1l21,K′=gl100gl2,B′=1l11l2,Θ′=θkθk+2where A1=-Lcos⁡θ5f and A2=-Lsin⁡θ5f. θ5f is the desired position of the vertical boom angle.Then, (9) is represented as follows [49]: (10)M′′Ψ¨+K′′Ψ=AkB′′vk+4,k=1,2Ψ=ψkψk+2=H-1θkθk+2,H=1-ωk2m1l2gm1+m21-ωk+22m1l2gm1+m211ωk,k+22=m1+m2l1+l2∓l1+l22m1+m22-4l1l2m1m1+m22m1l112g,where M′′=HTM′H, K′′=HTK′H, and B′′=HTB′.Then, we have(11)X˙=Ak′X+Akbkvk+4,k=1,2Ak′=000100000010000001-ωk2000000-ωk+220000000000,bk=000m1+m2δ-m1+m2δ1X=ψkψk+2θk+4ψ˙kψ˙k+2θ˙k+4where δ=(l1+l2)2(m1+m2)2+2l1l2(m22-m12).From the above equations, the crane system can be divided into two subsystems (i.e., vertical boom motion and horizontal boom motion subsystems). Therefore, we can design feasible and effective controllers for each subsystem, respectively.


## 3. Controller Design Process

### 3.1. Closed-Loop System Representation

The closed-loop control system for rotary crane is(12)X˙=Ak′+Akbkkk+4TX,k=1,2kk+4T=kk+41kk+42kk+43kk+44kk+45kk+46where k(k+4)1, k(k+4)2, k(k+4)3, k(k+4)4, k(k+4)5, and k(k+4)6 are feedback gains. These gains will be calculated via the LMI-based method in this study.


### 3.2. Pole Placement Problem by LMI

In linear control theory, if all eigenvalues of matrixAk′+Akbkkk+4T lie in the left-half of a region shown in Figure 3, the system is stable. Inspired by the literature [43], the following formulations were suggested to solve the pole placement problem in this study: (13)Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2αX1<0Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2βX1>0sin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′Tcos⁡φAk′X1-X1Ak′T+bk′M1-M1Tbk′Tcos⁡φ-Ak′X1+X1Ak′T-bk′M1+M1Tbk′Tsin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′T<0X1>0where M1=kk+4TX1 and bk′=Akbk.Figure 3
Stable region.


### 3.3. Optimal Regulator Problem by LMI

This subsection proposes an LQR method to achieve the optimal control problem. Hence, the following cost functionJ is considered [43]: (14)J=12∫0∞Qk+41/2X2+Rk+41/2vk+4TQk+41/2X2+Rk+41/2vk+4dtwhere Qk+4≥0 and Rk+4>0.Then, inspired by the method presented in [50], the above problem can be converted into the problem of finding X2, Y, and ki such that Trace(Y) is minimal with the following constraints: (15)Ak′X2+bk′M2+X2Ak′T+M2Tbk′T00-I<0YQk+41/2X2+Rk+41/2M2Qk+41/2X2+Rk+41/2M2TX2>0where M2=kk+4TX2.


### 3.4. Controller Design

In this study, we synthesize the above two problems to obtain the controller by settingX1=X2 and M1=M2 [43]: (16)Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2αX1<0Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2βX1>0sin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′Tcos⁡φAk′X1-X1Ak′T+bk′M1-M1Tbk′Tcos⁡φ-Ak′X1+X1Ak′T-bk′M1+M1Tbk′Tsin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′T<0YQk+41/2X2+Rk+41/2M2Qk+41/2X2+Rk+41/2M2TX1>0


## 3.1. Closed-Loop System Representation

The closed-loop control system for rotary crane is(12)X˙=Ak′+Akbkkk+4TX,k=1,2kk+4T=kk+41kk+42kk+43kk+44kk+45kk+46where k(k+4)1, k(k+4)2, k(k+4)3, k(k+4)4, k(k+4)5, and k(k+4)6 are feedback gains. These gains will be calculated via the LMI-based method in this study.


## 3.2. Pole Placement Problem by LMI

In linear control theory, if all eigenvalues of matrixAk′+Akbkkk+4T lie in the left-half of a region shown in Figure 3, the system is stable. Inspired by the literature [43], the following formulations were suggested to solve the pole placement problem in this study: (13)Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2αX1<0Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2βX1>0sin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′Tcos⁡φAk′X1-X1Ak′T+bk′M1-M1Tbk′Tcos⁡φ-Ak′X1+X1Ak′T-bk′M1+M1Tbk′Tsin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′T<0X1>0where M1=kk+4TX1 and bk′=Akbk.Figure 3
Stable region.


## 3.3. Optimal Regulator Problem by LMI

This subsection proposes an LQR method to achieve the optimal control problem. Hence, the following cost functionJ is considered [43]: (14)J=12∫0∞Qk+41/2X2+Rk+41/2vk+4TQk+41/2X2+Rk+41/2vk+4dtwhere Qk+4≥0 and Rk+4>0.Then, inspired by the method presented in [50], the above problem can be converted into the problem of finding X2, Y, and ki such that Trace(Y) is minimal with the following constraints: (15)Ak′X2+bk′M2+X2Ak′T+M2Tbk′T00-I<0YQk+41/2X2+Rk+41/2M2Qk+41/2X2+Rk+41/2M2TX2>0where M2=kk+4TX2.


## 3.4. Controller Design

In this study, we synthesize the above two problems to obtain the controller by settingX1=X2 and M1=M2 [43]: (16)Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2αX1<0Ak′X1+X1Ak′T+bk′M1+M1Tbk′T+2βX1>0sin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′Tcos⁡φAk′X1-X1Ak′T+bk′M1-M1Tbk′Tcos⁡φ-Ak′X1+X1Ak′T-bk′M1+M1Tbk′Tsin⁡φAk′X1+X1Ak′T+bk′M1+M1Tbk′T<0YQk+41/2X2+Rk+41/2M2Qk+41/2X2+Rk+41/2M2TX1>0


## 4. Results and Discussion

### 4.1. Simulation Conditions

In order to evaluate the control performance of the proposed controller, a cycloid-like motion trajectory was applied to the crane system [43]: (17)θk+4d=θk+4f-θk+40tts-12πsin⁡2πtts+xk+40t∈0,tsθk+4ft∈ts,tfwhere θ(k+4)f, x(k+4)0, tr, and tf are the desired boom angle, the initial boom angle, the reaching time, and the finish time. These parameters were selected as θ50=0[deg], θ60=0[deg], θ5f=30[deg], θ6f=45[deg], tr=3[s], and tf=10[s].Parameters as shown in Figure3 were selected as 80, 1, and 30, respectively. We also set Q5=diag{150,15,50,315,15,50}, Q6=diag{50,150,50,300,150,50}, and R5=R6=1 in (16). The controller gains for the decoupled linear system were obtained as follows by the method proposed in Section 3 with the rotary crane parameters shown in Table 1, when the rope length varies from 0.25[m] to from 0.75[m]. (18)k5T=-39.65124.0810.61-17.12-8.7514.73(19)k6T=-61.92210.7810.77-27.01-12.9914.45Using the relationship shown in (10), the gains for the original rotary crane are(20)k5′T=103.08-17.8210.61137.15-3.1114.73(21)k6′T=177.99-27.3010.77231.19-4.1014.45Table 1
Rotary crane model.
M 0[kg] 0.86 M 1[kg] 2.03 m 1[kg] 2.00  m 2[kg] 0.56 L[m] 0.65 L 1[m] 0.19  l 1[m] 0.50 l 2[m] 0.30 g[m/s2] 9.80  J 5[kgm2] 0.56 J 6[kgm2] 0.65 I b[Nm] 0.38  C 51[Nm/(rad/s)] 1.57 C 61[Nm/(rad/s)] 1.39 f 51[Nm] 3.91  f 61[Nm] 3.58 – – – –


### 4.2. Stability Analysis

We conducted several numerical to explain the robust stability of the proposed control system when the rope lengthl1 varies from 0.25[m] to from 0.75[m]. The initial poles of the vertical boom motion subsystem were set to p11=-1.04[1/s], p21=-1.45+9.07j[1/s], p31=-1.45-9.07j[1/s], p41=-4.80+6.92j[1/s], p51=-4.80-6.92j[1/s], and p61=-6.07[1/s], and the initial poles of the horizontal boom motion subsystem were set to p12=-1.51+9.54j[1/s], p22=-1.51-9.45j[1/s], p32=-4.96+6.30j[1/s], p42=-4.96-6.30j[1/s], p52=-7.62+2.64j[1/s], and p62=-7.62-2.64j[1/s]. It is confirmed that the crane system can be stabilized by the controller from the simulation results in Figure 4.Figure 4
Poles placement.
(a)
Original figure for vertical boom motion subsystem (b)
Enlarged view for vertical boom motion subsystem (c)
Original figure for horizontal boom motion subsystem (d)
Enlarged view for horizontal boom motion subsystem


### 4.3. Simulation Results

We conducted several simulations based on the linear model under different rope length, firstly. The results for the vertical subsystem are shown in Figure5. Although there are transient errors and slight overshoot in these figures, the vertical boom motion converges to the desired position rapidly. Meanwhile, there is almost no residual modal swaying vibration. On the other hand, the results for the horizontal subsystem are shown in Figure 6. Almost the same results were obtained as those in Figure 5. The results based on the original crane system for the terms of the load sways θ1, θ2, θ3, θ4, the tracking error for vertical boom angle e5, the tracking errors for horizontal boom angle e6, the vertical boom angle θ5, the horizontal boom angle θ6, and the control inputs v5 and v6 are shown in Figure 7. Because of the effect of the ignored nonlinearity of the crane model, the residual swaying angles are bigger than those in Figures 5 and 6.Figure 5
Simulations under the decoupled linear crane model in vertical boom motion subsystem.Figure 6
Simulations under the decoupled linear crane model in horizontal boom motion subsystem.Figure 7
Simulations under the original crane model with different rope length.For verifying that the presented controller also has robustness to the rigging length variation, we conducted simulations with the gains in (20) and (21) by set the l2 as l2=0.20[m] and l2=0.40[m]. By comparing the results in Figures 7 and 8, it is found that although the second sway angle of the horizontal direction θ4 increases with the increase of the rigging length, it still remains in a reasonable range.Figure 8
Simulations under the original crane model with different rigging length.


### 4.4. Comparative Simulation

In order to confirm the proposed method can provide superior robust performance, some comparative simulations were conducted by a conventional state feedback controller. A dominant poles placement method was applied to calculate the controller gains which are as follows:(22)k5′′T=-253.276.4211.12-292.64-41.28-47.10(23)k6′′T=-460.0614.7819.26-499.43-62.48-81.58The results are shown in Figure 9. Although the load sway angles reduced completely under condition l1=0.25[m] or l1=0.50[m], the load sway angles cannot be reduced by using gains in (22) and (23) under condition l1=0.75[m]. Meanwhile, the quantitative analyses between the proposed method and the existing one are in Tables 2–4. It is indicated that the presented method can furnish preferable robust performance than the traditional one from these figures and tables.Table 2
Quantified results under conditionl1=0.25[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.25 3.84  Conventional controller 3.08 3.84  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.20 1.70  Conventional controller 1.92 1.82  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.49 2.14  Conventional controller 2.61 2.56Table 3
Quantified results under conditionl1=0.50[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.31 3.82  Conventional controller 3.12 3.82  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.31 2.01  Conventional controller 2.10 2.10  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.64 2.53  Conventional controller 2.75 2.94Table 4
Quantified results under conditionl1=0.75[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.36 3.81  Conventional controller 3.10 3.75  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.40 2.31  Conventional controller 3.01 3.00  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.76 2.81  Conventional controller 4.21 5.14Figure 9
Conventional controller results.


## 4.1. Simulation Conditions

In order to evaluate the control performance of the proposed controller, a cycloid-like motion trajectory was applied to the crane system [43]: (17)θk+4d=θk+4f-θk+40tts-12πsin⁡2πtts+xk+40t∈0,tsθk+4ft∈ts,tfwhere θ(k+4)f, x(k+4)0, tr, and tf are the desired boom angle, the initial boom angle, the reaching time, and the finish time. These parameters were selected as θ50=0[deg], θ60=0[deg], θ5f=30[deg], θ6f=45[deg], tr=3[s], and tf=10[s].Parameters as shown in Figure3 were selected as 80, 1, and 30, respectively. We also set Q5=diag{150,15,50,315,15,50}, Q6=diag{50,150,50,300,150,50}, and R5=R6=1 in (16). The controller gains for the decoupled linear system were obtained as follows by the method proposed in Section 3 with the rotary crane parameters shown in Table 1, when the rope length varies from 0.25[m] to from 0.75[m]. (18)k5T=-39.65124.0810.61-17.12-8.7514.73(19)k6T=-61.92210.7810.77-27.01-12.9914.45Using the relationship shown in (10), the gains for the original rotary crane are(20)k5′T=103.08-17.8210.61137.15-3.1114.73(21)k6′T=177.99-27.3010.77231.19-4.1014.45Table 1
Rotary crane model.
M 0[kg] 0.86 M 1[kg] 2.03 m 1[kg] 2.00  m 2[kg] 0.56 L[m] 0.65 L 1[m] 0.19  l 1[m] 0.50 l 2[m] 0.30 g[m/s2] 9.80  J 5[kgm2] 0.56 J 6[kgm2] 0.65 I b[Nm] 0.38  C 51[Nm/(rad/s)] 1.57 C 61[Nm/(rad/s)] 1.39 f 51[Nm] 3.91  f 61[Nm] 3.58 – – – –


## 4.2. Stability Analysis

We conducted several numerical to explain the robust stability of the proposed control system when the rope lengthl1 varies from 0.25[m] to from 0.75[m]. The initial poles of the vertical boom motion subsystem were set to p11=-1.04[1/s], p21=-1.45+9.07j[1/s], p31=-1.45-9.07j[1/s], p41=-4.80+6.92j[1/s], p51=-4.80-6.92j[1/s], and p61=-6.07[1/s], and the initial poles of the horizontal boom motion subsystem were set to p12=-1.51+9.54j[1/s], p22=-1.51-9.45j[1/s], p32=-4.96+6.30j[1/s], p42=-4.96-6.30j[1/s], p52=-7.62+2.64j[1/s], and p62=-7.62-2.64j[1/s]. It is confirmed that the crane system can be stabilized by the controller from the simulation results in Figure 4.Figure 4
Poles placement.
(a)
Original figure for vertical boom motion subsystem (b)
Enlarged view for vertical boom motion subsystem (c)
Original figure for horizontal boom motion subsystem (d)
Enlarged view for horizontal boom motion subsystem


## 4.3. Simulation Results

We conducted several simulations based on the linear model under different rope length, firstly. The results for the vertical subsystem are shown in Figure5. Although there are transient errors and slight overshoot in these figures, the vertical boom motion converges to the desired position rapidly. Meanwhile, there is almost no residual modal swaying vibration. On the other hand, the results for the horizontal subsystem are shown in Figure 6. Almost the same results were obtained as those in Figure 5. The results based on the original crane system for the terms of the load sways θ1, θ2, θ3, θ4, the tracking error for vertical boom angle e5, the tracking errors for horizontal boom angle e6, the vertical boom angle θ5, the horizontal boom angle θ6, and the control inputs v5 and v6 are shown in Figure 7. Because of the effect of the ignored nonlinearity of the crane model, the residual swaying angles are bigger than those in Figures 5 and 6.Figure 5
Simulations under the decoupled linear crane model in vertical boom motion subsystem.Figure 6
Simulations under the decoupled linear crane model in horizontal boom motion subsystem.Figure 7
Simulations under the original crane model with different rope length.For verifying that the presented controller also has robustness to the rigging length variation, we conducted simulations with the gains in (20) and (21) by set the l2 as l2=0.20[m] and l2=0.40[m]. By comparing the results in Figures 7 and 8, it is found that although the second sway angle of the horizontal direction θ4 increases with the increase of the rigging length, it still remains in a reasonable range.Figure 8
Simulations under the original crane model with different rigging length.


## 4.4. Comparative Simulation

In order to confirm the proposed method can provide superior robust performance, some comparative simulations were conducted by a conventional state feedback controller. A dominant poles placement method was applied to calculate the controller gains which are as follows:(22)k5′′T=-253.276.4211.12-292.64-41.28-47.10(23)k6′′T=-460.0614.7819.26-499.43-62.48-81.58The results are shown in Figure 9. Although the load sway angles reduced completely under condition l1=0.25[m] or l1=0.50[m], the load sway angles cannot be reduced by using gains in (22) and (23) under condition l1=0.75[m]. Meanwhile, the quantitative analyses between the proposed method and the existing one are in Tables 2–4. It is indicated that the presented method can furnish preferable robust performance than the traditional one from these figures and tables.Table 2
Quantified results under conditionl1=0.25[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.25 3.84  Conventional controller 3.08 3.84  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.20 1.70  Conventional controller 1.92 1.82  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.49 2.14  Conventional controller 2.61 2.56Table 3
Quantified results under conditionl1=0.50[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.31 3.82  Conventional controller 3.12 3.82  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.31 2.01  Conventional controller 2.10 2.10  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.64 2.53  Conventional controller 2.75 2.94Table 4
Quantified results under conditionl1=0.75[m].
Control method The maximum value of tracking errore5max[deg] The maximum value of tracking errore6max[deg]  Proposed controller 3.36 3.81  Conventional controller 3.10 3.75  The maximum value of hook sway angleθ1max[deg] The maximum value of hook sway angleθ2max[deg]  Proposed controller 2.40 2.31  Conventional controller 3.01 3.00  The maximum value of load sway angleθ3max[deg] The maximum value of load sway angleθ4max[deg]  Proposed controller 2.76 2.81  Conventional controller 4.21 5.14Figure 9
Conventional controller results.


## 5. Conclusion

In order to achieve the robust control for a rotary crane with double-pendulum effect, a simple model for controller design is obtained by linearizing and decoupling the complex nonlinear model of rotary crane. Then, a linear feedback controller which can furnish robust performance was presented. Finally, numerical simulations verified the effectiveness of the proposed method by comparing with a traditional approach.



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*Source: 3128356-2019-02-11.xml*