## 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:

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

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

Friction and external distributed forces are neglected.

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}*, and (•)*

_{t}*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*

_{zz}**Figure 2**. The tension of the handing cable is of the form

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

where

With two force components to move trolley and bridge F_{x} and F_{y}, the total visual works of system are in the form of

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

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

and the small derivation of virtual work is written as

First, one obtains

We define *L _{c}
* as a multivariable function

and apply the following property:

with

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

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

Integrating the abovementioned equation in term of time side by side, one has

which yields

Next, let us calculate

with the below notations

and

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

(16) |

which is simplified as

(17) |

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

(18) |

which leads to

and

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

and

At boundary condition z = L, one obtains

and

At boundary condition z = 0, one has

and

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:

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
*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:

and

pushes all state outputs of dynamic model (20)–(25) to reference *q _{d}
* exponentially.

**Proof.** Define a positive Lyapunov candidate as follows:

(28) |

where *P*(0) is the tension force of cable at boundary x = 0. *K _{p}
* and

*K*are positive gains.

_{a}With the notations that

one has

with

and

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

(29) |

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

Using partial integration

and

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

and

Then,

and

The Lyapunov derivative (29) now becomes

(31) |

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

(32) |

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

(33) |

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

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
*V*
_{1}(*t*) being in the form of

(35) |

where
*k*
_{
b1} is a positive gain satisfying condition *k*
_{
b1} > |*z*
_{1}|. The modification of (35) leads to

(36) |

Applying the following inequality

or

with K being positive constant leads to

Inserting (37) into (36) yields

(38) |

Inserting the following inequality

or

into (38), one obtains

(39) |

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

and

which leads the Eq. (31) to

for every positive gains K*
_{d}
* > 0 and

*K*> 0. This implies that

_{a}*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

The simulation results are depicted in
**Figures 3**
–
**6**
. Trolley and payload approach to destination *q _{d}
* = 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

*k*. 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.

_{b}## 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.