Open access peer-reviewed chapter

# Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

By Tung Lam Nguyen and Minh Duc Duong

Submitted: September 8th 2017Reviewed: October 12th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71657

## Abstract

Considering gantry cable as an elastic string having a distributed mass, we constitute a dynamic model for coupled flexural overhead cranes by using the extended Hamilton principle. Two kinds of nonlinear controllers are proposed based on the Lyapunov stability and its improved version entitled barrier Lyapunov candidate to maintain payload motion in a certain defined range. With such a continuously distributed model, the finite difference method is utilized to numerically simulate the control system. The results show that the controllers work well and the crane system is stabilized.

### Keywords

• finite difference method
• Lyapunov stability
• distributed modeling

## 1. Introduction

Nowadays, cargo transportation plays an important role in many industrial fields. For carrying the cargo in short distance or small area, such as in automotive factories and shipyards, the overhead cranes are naturally applied. To increase productivity, the overhead cranes today are required in high-speed operation. However, the fast motion of overhead cranes usually leads to the large swings of cargo and non-precise movements of trolley and bridge. The faster the cargo transport is, the larger the cargo swings. This makes dangerous and unsafe situation during the operating process. The crane itself and the concerning equipment in the factory can be damaged without proper control strategies.

In recent decades, the control problems of overhead cranes in both theory and practice have attracted many researchers. Various kinds of crane control techniques have been applied from classical methods such as linear control [1], nonlinear control [2, 5, 6], optimal approach [7], adaptive algorithms [8, 9] to modern techniques such as fuzzy logic [3, 4, 10], neural network [11], command shaping [12], and so on.

The abovementioned researches deal with crane motion modeled as pendulum or multi-section pendulum systems. As a result, their dynamics are described as an ordinary differential equation or a system of ordinary differential equations. In practice, the crane rope exhibits a certain degree of flexibility; hence, the equation of motions of the gantry crane with flexible rope is represented by a set of partial differential and ordinary differential equations. In [13, 14, 15], the authors successfully design a controller that can stabilize the system with the rope flexibility. Flexible rope also is considered in [16, 17] where coupled longitudinal-transverse motion and 3D model are investigated.

This chapter accesses the modeling and control of overhead cranes according to the other research direction. We construct a distributed model of overhead cranes in which the mass and the flexibility of payload suspending cable are fully taken into account. We utilize the analytical mechanics including Hamilton principle for constructing such the mathematical model. With the received model, we analyze and design two nonlinear control algorithms based on two versions of Lyapunov stability: one is the so-called traditional Lyapunov function and the other is the so-called barrier Lyapunov. Dissimilar to the preceding study [18, 19] whereas the problem of actuated payload positioning system is considered, the proposed controllers track the trolley to destination precisely while keeping the payload swing small during the transport process and absolutely suppressed at the payload destination with control forces exerted at the trolley end of the system. The quality of control system is investigated by numerical simulation. Since the system dynamics is characterized by a distributed mass model, the finite difference method is applied to simulate the system responses in MATLAB® environment.

The chapter content is structured as follows. Section 2 constructs a distributed mass model of overhead cranes. Section 3 analyzes and designs two nonlinear controllers based on Lyapunov direct theory. The analysis of system stability is included. Section 4 numerically simulates the system responses and analyzes the received results. Finally, the remarks and conclusions are shown in Section 5.

## 2. Distributed mass modeling of overhead cranes

Let us constitute a mathematical model for overhead cranes fully considering the flexibility and mass of cable. In other words, payload handling cable with length L is considered as a distributed mass string with density ρ (kg/m). An overhead crane with its physical features is depicted in Figure 1 . The trolley with mass M (kg) handling the payload m (kg) moves along Ox which can induce the payload swing. The force Fx (N) of motor is created to push the trolley but guaranteeing the payload oscillation as small as possible. The other parameters can be seen in Figures 1 and 2 .

Before carrying system modeling, we assume that:

1. Moving masses at the trolley end are symmetrical in X and Y directions.

2. The gantry moving in XY plane and the rope length are unchanged.

3. Friction and external distributed forces are neglected.

4. Longitudinal deformation of the crane rope is negligible.

From this point onward, the argument (z, t) is dropped whenever it is not confusing and (•) t , (•) tt , (•) t , and (•) zz are used to denoted the first and second time and spatial derivatives of (•), respectively. We consider the physical model of an overhead crane as shown in Figure 2 . The tension of the handing cable is of the form

P=gρLz+mE1

With the differential derivation along the cable length L, the potential energy due to the elasticity of cable and gravity is determined by

U=120LPnz2+μz2dz+12EA0L12nz2+μz22+P0E2

where 12EA0L12nz2+μz22is a potential component due to the axial deformation of the cable. The kinetic energy of system includes those of the trolley, payload, and cable motion described by

T=120Lρnt2+μt2dz+12Mnt20t+μt20t+12mnt2Lt+μt2LtE3

With two force components to move trolley and bridge Fx and Fy, the total visual works of system are in the form of

W=Fxn0+Fyμ0E4

Using the generalized form of Hamilton principle, one has the following equation:

H=t1t2δTδU+δWdt=0E5

in which the small variations of kinematic and potential energies, respectively, are described by

δT=δ120Lρnt2+μt2dz+12Mnt20t+μt20t+12mnt2Lt+μt2LtE6
δU=δ120LPnz2+μz2dz+δ12EA0L12nz2+μz22dzE7

and the small derivation of virtual work is written as

δW=Fxδn0t+Fyδμ0tE8

First, one obtains

δ120Lρnt2+μt2dzδ120LPnz2+μz2dzδ12EA0L12nz2+μz22dz

We define Lc as a multivariable function

Lc=12ρnt2+μt212Pnz2+μz212EA.14nz2+μz22=Lct:ntμtnzμzE9

and apply the following property:

δ0LLcdz=0LδLcdz

with

0LδLcdz=0LLcntδnt+Lcμtδμt+Lcnzδnz+LcμzδμzdzE10

We calculate the components of (10) using the expressions of partial integration as follows:

0LLcnzδnzdz=Lcnzδn|L00LLcnzzδndzE11
0LLcμzδμzdz=Lcμzδμ|L00LLcμzzδμdzE12

Inserting (11) and (12) into (10) leads to

0LδLcdz=0LLcntδnt+LcμtδμtLcnzzδnLcμzzδμdz+Lcnzδn|L0+Lcμzδμ|L0

Integrating the abovementioned equation in term of time side by side, one has t1t20LδLcdzdt=t1t20LLcntδnt+LcμtδμtLcnzzδnLcμzzδμdz+Lcnzδn|L0+Lcμzδμ|L0dtdue to Lcnzδnt|t2t1=0. Similarly, one has the following results after a series of calculation

t1t20LLcμtδμtdzdt=t1t20LLcμttδμdzdt

which yields

t1t20LδLcdzdt=t1t20LLcnttδnLcμttδμLcnzzδnLcμzzδμdz+Lcnzδn|0L+Lcμzδμ|0LdtE13

Next, let us calculate

δ12Mnt20t+μt20t+δ12mnt2Lt+μt2Lt

with the below notations

δ12Mnt20t+μt20t=Mnt0tδnt0t+Mμt0tδμt0tE14

and

δ12mnt2Lt+μt2Lt=mntLtδntLt+mμtLtδμtLtE15

Substituting (8), (13), (14), and (15) into (5), one obtains

t1t2{0LLcnttδnLcμttδμLcnzzδnLcμzzδμdz+Lcnzδn|L0+Lcμzδμ|L0+Mnt0tδnt0t+Mμt0tδμt0t+mntLtδntLt+mμtLtδμtLt+Fxδn0t+Fyδμ0tdt=0E16

which is simplified as

t1t2{0LLcnttLcnzzδn+LcμttLcμzzδμdz+LcnzδnLtLcnzδn0t+LcμzδμLtLcμzδμ0tMδn0tntt0tMδμ0tμtt0tmδnLtnttLtmδμLtμttLt+Fxδn0t+Fyδμ0tdt=0E17

Consider the following boundaries at x = 0 and x = L:

Lcntt+Lcnzz=0;Lcμtt+Lcμzz=0;LcnzmnttLt=0;LcμzmμttLt=0;LcnzMntt0t+Fx=0;LcμzMμtt0t+Fy=0;E18

Lcntt=ρnttE19a

and

Lcnz=Pnz18EA4nz3+2.2nzμz2E19b

Submitting (18) into (19a) and (19b) in the interval [0, L] of z, one has

ρnttPnzz12EA3nz2nzz+nzzμz2+2nzμzμzz=0E20

and

ρμttPμzz+12EA3μz2μzz+μzznz2+2nzμznzz=0E21

At boundary condition z = L, one obtains

PnzLt+12EAnz3Lt+nzLtμz2Lt+mnttLt=0E22

and

PμzLt+12EAμz3Lt+μzLtnz2Lt+mμttLt=0E23

At boundary condition z = 0, one has

Pnz0t+12EAnz30t+nzμz20tMntt0t+Fx=0E24

and

Pμz0t+12EAμz30t+μz0tnz20tMμtt0t+Fy=0E25

In summary, the dynamic behavior of overhead crane governed a set of six nonlinear partial differential Eqs. (20), (21), (22), (23), (24), and (25), as follows:

ρnttPnzz12EA3nz2nzz+nzzμz2+2nzμzμzz=0ρμttPμzz12EA3μz2μzz+μzznz2+2nzμznzz=0PnzLt12EAnz3Lt+nzLtμz2LtmnttLt=0PμzLt12EAμz3LtμzLtnz2LtmμttLt=0Pnz0t+12EAnz30tnz0tμz20tMntt0t+Fx=0Pμz0t+12EAμz30t+μz0tnz20tMμtt0t+Fy=0

The first and the second equations of the above system of equation represent dynamics of the gantry rope. Boundary conditions at load and trolley ends are given in the third, fourth, fifth, and sixth equations, respectively.

## 3. Lyapunov-based control design

Let us construct two nonlinear controllers using a traditional Lyapunov stability and its advanced version. In the first method, the control law is referred from the negative condition of a Lyapunov candidate V̇0.In the second method, the Lyapunov function is determined so that it satisfies 0 < V ≤ b with b > 0.

### 3.1. Conventional Lyapunov controller

The following theorem points out a nonlinear controller designed based on the second method of Lyapunov stability. The proposed control scheme tracks the outputs of a crane system approach to references asymptotically.

Theorem. Consider a mass distributed model of overhead crane that is described by six partial differential equations: (20) to (25). The following control law composed of two inputs:

Fx=Kanz0t+EA2P0nz30t+nz0tμz20tKpn0tqdn0tμ20t+n20tKdnt0tE26

and

Fy=Kaμz0t+EA2P0μz30t+μz0tnz20tKpμ0tqdμ0tμ20t+n20tKdμt0tE27

pushes all state outputs of dynamic model (20)(25) to reference qd exponentially.

Proof. Define a positive Lyapunov candidate as follows:

V=120Lρnt2+μt2+Pnz2+μz2+EA12nz2+μz22dz+MP02P0+Kant20t+μt20t+12mnt2Lt+μt2Lt+P0Kp2P0+Kan20t+μ20tqd2E28

where P(0) is the tension force of cable at boundary x = 0. Kp and Ka are positive gains.

With the notations that w2=0Lnt2+μt2+nz2+μz2+nz2+μz22dz+nt20t+μt20t+nt2Lt+μt2Lt+n20t+μ20tqd2,.

one has

Kminw2VtKmaxw2

with

Kmin=12minρPEA4MP0P0+KamP0KpP0+Ka

and

Kmax=12maxρPEA4MP0P0+KamP0KpP0+Ka

Differentiating Lyapunov function (28) with respect to time, one obtains

V̇=0Lρntntt+μtμtt+Pnznzt+μzμzt+EA2nz3ntz+μz3μzt+nznztμz2+μzμztμz2dz+MP0P0+Kant0tntt0t+μt0tμtt0t+μ0tμt0t+n0tnt0t+mntLtnttLt+μtLtμttLtqdμ0tμt0t+n0tnt0tμ20t+n20tE29

Let us calculate the components of Lyapunov derivative (29). We refer from (20) and (21) that

0Lρntntt+μtμttdz=0LntPnzz+12EA3nz2nzz+nzzμz2+2nzμzμzz+μtPμzz+12EA3μz2μzz+μzznz2+2nzμznzzdzE30

Using partial integration

0LntPnzzdz=ntPnz|L00LPnzntzdz

and

0LμtPμzzdz=μtPμz|L00LPμzμtzdz,

one obtains the following components of (30) as follows:

0LEA2nz3ntzdz=0LEA2nz3dnt=EA2nz3nt|L00LntEA23nz2nzzdz

and

0LEA2μz3μtzdz=EA2μz3μt|L00LμtEA23μz2μzzdz

Then,

0LEA2nznztμz2dz=EA2nzμz2nt|L0EA20Lntnzzμz2+2nzμzμzzdz

and

0LEA2μzμztnz2dz=EA2μznz2μt|L0EA20Lμtμzznz2+2nzμznzzdz

The Lyapunov derivative (29) now becomes

V̇=ntPnz|L0+μtPμz|L0+EA2nz3nt|L0+EA2μz3μt|L0+EA2nzμz2nt|L0+EA2μznz2μt|L0+MP0P0+Kant0tntt0t+μt0tμtt0t+mntLtnttLt+μtLtμttLt+P0KpP0+Kaμ0tμt0t+n0tnt0tP0KpP0+Kaqdμ0tμt0t+n0tnt0tμ20t+n20tE31

Additionally, modification of (24) and (25) yields

MP0P0+Kant0tntt0t+μt0tμtt0t=P0P0+Kant0tFx+P0nz0t+EA2nz30t+nz0tμz20t+P0P0+Kaμt0tFy+P0μz0t+EA2μz30t+μz0tnz20tE32

Submitting (32) into (31) with a series of calculation, we obtain

V̇=P0P0+Kant0tKanz0t+EA2P0nz30t+nz0tμz20t+Kpn0tqdn0tμ20t+n20t+Fx+P0P0+Kaμt0tKaμz0t+EA2P0μz30t+μz0tnz20t+Kpμ0tqdμ0tμ20t+n20t+FyE33

Substituting the control law (26) and (27) into (33) leads the Lyapunov function to

V̇=P0KdP0+Kant20tP0KdP0+Kaμt20t0E34

With the negative definition of expression (34), we can conclude that the system is now exponential stability.

### 3.2. Barrier Lyapunov controller

We utilize an improved version of Lyapunov stability to design a control law for overhead cranes. The Lyapunov function is chosen so that its derivative is smaller than a positive constant. By this way, the Lyapunov candidate is selected similar to Eq. (28) but supplementing derivation of payload position 12P0P0+Kalnkb12kb12z12. A modified version of Lyapunov candidate is the so-called barrier Lyapunov V 1(t) being in the form of

V1=120Lρnt2+μt2+Pnz2+μz2+EA12nz2+μz22dz+MP02P0+Kant20t+μt20t+12mnt2Lt+μt2Lt+P0Kp2P0+Kan20t+μ20tqd2+12P0P0+Kalnkb12kb12z12E35

where z1=n2Lt+μ2Ltn20t+μ20tis relative position of payload in comparison with that of trolley. k b1 is a positive gain satisfying condition k b1 > |z 1|. The modification of (35) leads to

V̇1=P0P0+Kant0tFxKanz0t+EA2.P0nz30t+nz0tμz20t+Kpn0tqdn0tμ20t+n20t+P0P0+Kaμt0tFyKaμz0t+EA2.P0μz30t+μz0tnz20t+Kpμ0tqdμ0tμ20t+n20t+P0P0+Kaz1z1tkb12z12E36

Applying the following inequality

z1tKnt20t+μt20t

or

z1z1tz1z1t=z1z1tkb1Knt20t+μt20t

with K being positive constant leads to

P0P0+Kaz1z1tkb12z12P0P0+Ka1kb12z12kb1Knt20t+μt20tE37

Inserting (37) into (36) yields

V̇1P0nt0tP0+KaFxKanz0t+EA2.P0nz30t+nz0tμz20t+Kpn0tqdn0tμ20t+n20t+Kpμ0tqdμ0tμ20t+n20t+P0μt0tP0+KaFyKaμz0t+EA2P0μz30t+μz0tnz20t+P0kb1KP0+Kakb12z12nt20t+μt20tE38

Inserting the following inequality

nt20t+μt20tnt0t+μt0t

or

nt20t+μt20tnt0tsgnnt0t+μt0tsgnμt0t

into (38), one obtains

V̇1P0P0+Kant0tFxKanz0t+EA2P0nz30t+nz0tμz20t+Kpn0tqdn0tμ20t+n20t+1kb12z12kb1Ksgnnt0t+P0P0+Kaμt0tFyKaμz0t+EA2.P0μz30t+μz0tnz20t+Kpμ0tqdμ0tμ20t+n20t+1kb12z12kb1Ksgnμt0tE39

To force the Lyapunov differentiation being negative, the control law with two components is structured as

Fx=Kanz0t+EA2.P0nz30t+nz0tμz20tKdnt0tKpn0tqdn0tμ20t+n20t1kb12z12kb1Ksgnnt0tE40

and

Fy=Kaμz0t+EA2.P0μz30t+μz0tnz20tKdμt0tKpμ0tqdμ0tμ20t+n20t1kb12z12kb1Ksgnμt0tE41

which leads the Eq. (31) to

V̇1P0KdP0+Kant20t+μt20t0E42

for every positive gains K d  > 0 and Ka  > 0. This implies that V ≤ V(0). Hence, the system is now asymptotical stability.

## 4. Simulation and results

Consider the case that only the trolley motion is activated, we numerically simulate the distributed system dynamics (20)(25) driven by either conventional Lyapunov-based input or barrier Lyapunov-based law. The finite difference method is applied for programing the control system in MATLAB environment. The system parameters used in simulation are composed of

m=5kg;M=1kg;L=3,6,9m;Ka=200;Kp=5;Kd=42;

The simulation results are depicted in Figures 3 6 . Trolley and payload approach to destination qd  = 2 m precisely and speedy without maximum overshoots. The payload swing stays in a small region during the transient state and absolutely suppressed at steady state (or payload destination). However, the longer length of cable is, the lager the payload swings. The system responses show the robustness in the face of parametric uncertainty. Despite the large variation of cable length, the system responses still kept consistency as shown in Figures 3 5 . It can be seen from Figure 6 that with the application of the barrier Lyapunov function, payload fluctuation is controlled in an area defined by kb . Because the motion of the trolley in X and Y directions is forced to travel the same distance to reach the desired location, system responses in X and Y directions are similar.

## 5. Conclusions

The dynamic model of overhead crane with distributed mass and elasticity of handling cable is formulated using the extended Hamilton’s principle. Based on the model, we successfully analyzed and designed two nonlinear robust controllers using two versions of Lyapunov candidate functions. The first can steer the payload to the desired location, while the second can maintain payload fluctuation in a defined span. The proposed controllers well stabilize all system responses despite the large variation of cable length and payload weight. Enhancing for 3D motion with carrying rope length will be proposed in the future studies.

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Tung Lam Nguyen and Minh Duc Duong (December 20th 2017). Nonlinear Control of Flexible Two-Dimensional Overhead Cranes, Adaptive Robust Control Systems, Le Anh Tuan, IntechOpen, DOI: 10.5772/intechopen.71657. Available from:

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