An Anti-Swing Closed-Loop Control Strategy for

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Aug 25, 2018 - To resolve this, a real-time anti-swing closed-loop control ... crane generally includes two parts: fast and accurate positioning .... is the vertical direction angle of the upload, and F(t) is the driving .... Define Az = z − z, σ = B σe, we have ... sampling period was selected as Tm, and the linear part system can be ...
applied sciences Article

An Anti-Swing Closed-Loop Control Strategy for Overhead Cranes Xianghua Ma * and Hanqiu Bao *

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School of Electrical and Electronic Engineering, Shanghai Institute of Technology, Shanghai 201418, China * Correspondence: [email protected] (X.M.); [email protected] (H.B.) Received: 13 July 2018; Accepted: 22 August 2018; Published: 25 August 2018

 

Featured Application: This paper provides a solution to anti-swing control for overhead cranes for outdoor use. This contribution can also be used for other kind of underactuated systems. Abstract: The payload swing of an overhead crane needs to be controlled properly to improve efficiency and avoid accidents. However, the swing angle is usually very difficult to control to zero degrees or for it to even remain within an acceptable range because the overhead crane is a complex nonlinear underactuated system, especially when the actual working environment is accompanied by strong disturbances and great uncertainty. To resolve this, a real-time anti-swing closed-loop control strategy is proposed that considers external disturbances. The swing angle is measured in time and it functions with the load displacement as feedback inputs of the closed-loop system. The nonlinear model of the crane is simplified by a linear system with virtual disturbances, which are estimated by the equivalent input disturbance (EID) method. Both simulation and experimental results for a 2-D overhead crane system are investigated to illustrate the validity of the proposed method. Keywords: overhead crane system; anti-swing closed-loop control; swing angle measurement; equivalent input disturbance

1. Introduction Overhead cranes play an important role as one of the tools of heavy cargo transportation in seaports, steel plants, and other workplaces [1]. As a typical underactuated system, an overhead crane is easily affected by various external disturbances so that the load is frequently swinging in the process of transportation, which seriously affects the positioning accuracy of the load and brings many unsafe possibilities while at the same time of reducing the efficiency of the system [2,3]. Avoiding the swing of the upload and improving safety and efficiency are the main concerns for the study of cranes. Hence, the control objective of an overhead crane generally includes two parts: fast and accurate positioning and effective swing suppression, especially considering external resistance (such as wind resistance, stochastic collision) during the transportation of uploads. Recently, much research has been done to solve the abovementioned problems. The most common and direct control is input shaping, by which suitable trajectories are planned for the trolley by thoroughly analyzing the coupling between the trolley motion and the payload swing [4–8]. A modified composite nonlinear feedback strategy has been proposed to improve the transient performance and eliminate the steady-state errors in path-following control considering the tire force saturations [9]. To simplify the controller design, partial feedback linearization operations were used in [10–12]. As the solution to the existence of uncertainties in the crane system, an adaptive control method was proposed in [13–15]. In addition, more intelligent methods have been followed to increase the robustness of these methods. A particle swarm optimizer in [16] was applied to determine the optimal sequence of control increments in the presence of constraints on input and output variables. The control scheme Appl. Sci. 2018, 8, 1463; doi:10.3390/app8091463

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input and output variables. The control scheme was successfully tested on a laboratory-scale overhead crane for different constraints and operating conditions. An error tracking control method was used in [17], for which the error trajectories of the trolley and the payload swing can be prespecified. Some other new complexity control methods have also been put forward to guarantee Sci. 2018, 8, 1463 2 of 12 fast andAppl. accurate positioning and effective swing suppression, such as passivity-based control schemes [18–21] and sliding mode control (SMC) methods [22–24]. Regarding unknown inputs, the tested on overhead crane for different constraints and are operating state andwas thesuccessfully output vectors ofa alaboratory-scale system can be reconstructed [25]. Also relevant the complete conditions. An error tracking control method was used in [17], for which the error trajectories of the Lyapunov-based stability analysis in [26], genetic-algorithm-based control [27], and fuzzy logic-based trolley and the payload swing can be prespecified. Some other new complexity control methods have methodsalso [28–30]. been put forward to guarantee fast and accurate positioning and effective swing suppression, However, most of the above are based on some harsh assumptions, for[22–24]. example, the such as passivity-based controlmethods schemes [18–21] and sliding mode control (SMC) methods initial payload swing angleinputs, should zero, accurate real-time anglecan ofbe the payload swing Regarding unknown thebe state andthe the output vectors of a system reconstructed [25]. can be Also relevant are the complete Lyapunov-based stability analysis in [26], genetic-algorithm-based known a priori, or velocity sensor measurements can be taken without noise. In fact, the payload control [27], and fuzzy logic-based methods [28–30]. easily swings within an unacceptable range in actual working environments, where some external However, most of the above methods are based on some harsh assumptions, for example, the disturbances such as wind resistance and stochastic collision frequently occur. Therefore, these initial payload swing angle should be zero, the accurate real-time angle of the payload swing can be above methods are weak in disturbance suppression and poor for real because they known a priori, or velocity sensor measurements can be taken without noise.applications In fact, the payload belong toeasily open-loop control closed-loop controlenvironments, system withwhere the feedback of real-time swings within an structures. unacceptableA range in actual working some external disturbances such as wind resistance and stochastic collision frequently occur. Therefore, these above angle measurement values of the payload swing is the most effective way to tackle the anti-swing are weak in disturbance suppression and poor for real applications because they belong problem.methods Further, overhead cranes are usually linearly modeled, which ignores their nonlinear to open-loop control structures. A closed-loop control system with the feedback of real-time angle characteristics and thus cannot guarantee a crane’s practical performance. Therefore, an anti-swing measurement values of the payload swing is the most effective way to tackle the anti-swing problem. closed-loop control strategy isusually proposed this paper. This their control system has two Further, overhead cranes are linearlyin modeled, which ignores nonlinear characteristics andfeedback quantities ofcannot trolley movinga crane’s displacement and swing angle, which are bothclosed-loop measured in real time. thus guarantee practical performance. Therefore, an anti-swing control strategycharacteristics is proposed in this paper. This control system has two quantities of trolley moving The nonlinear hidden in the crane system arefeedback transformed into an equivalent input displacement and swing angle, which are both measured in real time. The nonlinear characteristics disturbance, which can be estimated by a disturbance predictor. hidden in the crane system are transformed into an equivalent input disturbance, which can be Thisestimated paper consists of the following four parts: a simplified model of a 2-D overhead crane by a disturbance predictor. system is given in Section controller is described in Section The control This paper consists2. of The the following four design parts: a simplified modelin of adetail 2-D overhead crane 3. system strategy is given verified by simulation and experiments in Section 4. The conclusion presented in Section 5. in Section 2. The controller design is described in detail in Section 3. Theiscontrol strategy is verified by simulation and experiments in Section 4. The conclusion is presented in Section 5.

2. Mathematical Model

2. Mathematical Model

Usually,Usually, an overhead crane consists ofofaahoist, load,and andtransport transport trolley, as shown in1.Figure 1. an overhead crane consists hoist, load, trolley, as shown in Figure Its corresponding two-dimensional simplified model is shown in Figure 2. Its corresponding two-dimensional simplified physical physical model is shown in Figure 2.

Figure 1. A1. real crane with hoist Figure A real cranediagram diagram with hoist andand load.load.

Appl. Sci. 2018, 8, 1463 Appl. Sci. 2018, 8, x

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x

θ m

Figure 2. Two-dimensional plane model of an overhead crane. Figure 2. Two-dimensional plane model of an overhead crane.

Assumingthat thatthe thelength length hoisting rope doeschange not change transportation the l not Assuming of of hoisting rope l does duringduring transportation and the and friction . .. ̇ = 𝑙 ̈ = 0, μ = 0, the dynamic friction between the trolley and the platform μ is negligible, that is 𝑙 between the trolley and the platform µ is negligible, that is l = l = 0, µ = 0, the dynamic equation of equation of the 2-D overhead systemasisfollows: obtained as follows: the 2-D overhead crane systemcrane is obtained

"

#" .. # "  #" 0 0 . # "mgl sin   #  "0  #  2 ml 2 ml cos 0   mgl sin θ   ml ml cos θ 0 θ      0.   θ.  + + = ..  ml  sin  0 0 F ( t ) ml cos  M  m x x            ml cos θ M + m 0 F (t) x −ml θ sin θ 0 x

(1) (1)

where M, m is the mass of the trolley and the hoist, respectively, x(t) is the horizontal displacement, where M, m is the mass of the trolley and the hoist, respectively, x(t) is the horizontal displacement, #θ(t) θ(t) is the vertical direction angle of the upload, and F(t) is the" driving force. m m 11 12 m12  angle  ml 2of theml cos   and F(t) is the driving force. M(x, θ ) =  m direction is the vertical upload, = M(x, ) =  11 = m21 m22 is the inertial matrix of the system (1).   "  m21 m22 #  ml cos  M  m  ml 2 ml cos θ Known from [31], although 2-D of underactuated is the inertial amatrix the system (1).system cannot be fully state-feedback ml cos θ M + m linearized, x can be implemented as a system input, and the driven state quantity part [ x, x ] can Known from [31], although a 2-D underactuated system cannot be fully state-feedback linearized, .. . linearized. Inspiredas by this, the model 2-Dstate overhead crane transformed by be implemented a system input, and of thethe driven quantity partcan [ x, xbe be linearized. xbecan ] can homeomorphism coordinate Inspired by this, the model transformation: of the 2-D overhead crane can be transformed by homeomorphism coordinate transformation:   z1  x   ( ) + α(θ )   z1 = x∂L   L ( xθ, ) ) . . ( x,   z2z2= =mm1111xx+ m m1212xx . Γ  (2) ∂ x x (2)    z = θ 3     z3 .  z4 = θ

 z4  

where L(x, θ) is the Lagrange equation of the 2-D overhead crane system. Simultaneously from (1), (2) is obtained: where L(x, θ) is the Lagrange equation of the 2-D overhead crane system. Simultaneously from (1), (2) is obtained: . z = f (z) + Bu

z  f ( z )  Bu

where: where:

(3) (3)

dα(θ ) m11 z2 +( − ) z 4 ] ζ = Γ −1 ( z ) zm2 11 d dθ ( ) mm1112 f1 ( z )  [ 1 . .( T ∂M( x,θ ) ) z4 ]  1 ( z ) m−12 mgl sin θ ] ζ =Γ−1 (z) f 2 (z) = m [ 11[ x, θ ] d 2 ∂x T f (z) = [ 1f 1 (z), f 2T(z)M , z4(, x0,] ) fB2 (= z ) [0,[0, 0,[ x1,]T ]  mgl sin  ]  1 ( z ) 2 x f 1 (z) = [

. . T

fζ( = z ) [x,[θ,f1x, ( zθ),] f 2 ( z ), z4 , 0]T The value of the costate coefficient B  [0, 0, 0,1]T α(θ ) is set 0 in this paper to ensure the conciseness of the transformed system structure. It is provedT later that the controllability and observability of the system   x,value , x,  ]of α(θ). is not affected by the choice of[the The value of the costate coefficient  ( ) is set 0 in this paper to ensure the conciseness of the Define: transformed system structure. It is proved σ =later f (z)that − Azthe controllability and observability of the (4) system is not affected by the choice of the value of 𝛼(θ). Define:

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  f ( z )  Az

Appl. Sci. 2018, 8, 1463

(4)

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The system (1) is equivalent to the following mathematical model:

The system (1) is equivalent to the following mathematical  Bu   model:  z  Az



.

where:

where:

 a11  a f ( z ) A |z 0   a2111 z  0 ∂ f (z)  a A= |z=0 =  21 ∂z  00

a12 a22 a12 0 a22 00

(5)

y  Cz

z= Az + Bu + σ y = Cz

(

(5)

a13 a14   a23 a24  fi ( z ) |z 0 , i  1, 2, j  1, 2,3, 4 a13 a14  , aij   0 1  z  j a23 a24   , aij = ∂ f i (z) |z=0 , i = 1, 2, j = 1, 2, 3, 4  ∂z j 00 10  

0 0 #0 0 0 0 0 0 0 0  0 1 0 0 1 0  From mathematical description (8), a conclusion can be made that the nonlinear crane system (1) canFrom be equivalent to a linear combination of a linearcan subsystem nonlinear term. The nonlinear mathematical description (8), a conclusion be madeand thatathe nonlinear crane system (1) term σ can be treated as acombination virtual disturbance. For the linearand subsystem, it can be The controlled and can be equivalent to a linear of a linear subsystem a nonlinear term. nonlinear observed only as if the following inequalityFor is satisfied: term σ can ifbeand treated a virtual disturbance. the linear subsystem, it can be controlled and observed if and only if the following inequality is satisfied: " 1 C  1 C = 0 0

 m (mgl sin  )

(mgl sin  ) 

1

0    12   x , 0  0  111 ( x  0) x θ ) − ∂(mgl sin θ ) |  6= 0, − m 1211∂ ( mgl sin mm