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 r (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.

3. Friction and external distributed forces are neglected.

tension of the handing cable is of the form

Figure 2. Physical modeling of overhead crane in OXYZ.

Figure 1. A practical overhead crane.

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

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

http://dx.doi.org/10.5772/intechopen.71657

317

Before carrying system modeling, we assume that:


Nonlinear Control of Flexible Two-Dimensional Overhead Cranes http://dx.doi.org/10.5772/intechopen.71657 317

Figure 1. A practical overhead crane.

The abovementioned researches deal with crane motion modeled as pendulum or multisection 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– 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

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 socalled 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

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

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 r (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

motion and 3D model are investigated.

316 Adaptive Robust Control Systems

shown in Section 5.

be seen in Figures 1 and 2.

simulate the system responses in MATLAB® environment.

2. Distributed mass modeling of overhead cranes

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.

Figure 2. Physical modeling of overhead crane in OXYZ.


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

$$P = \lg[\rho(L - z) + m] \tag{1}$$

Lc <sup>¼</sup> <sup>1</sup> 2 r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � � � <sup>1</sup>

with

ðt2 t1 ðL 0

due to <sup>∂</sup>Lc ∂nz δð Þ nt t2 t1 � � �

which yields

ðt2 t1 ðL 0

δLcdzdt ¼

∂Lc ∂nt

δð Þþ nt

∂Lc ∂μ<sup>t</sup> δ μ<sup>t</sup>

> ðt2 t1

ðL 0 ∂Lc ∂μ<sup>t</sup> δ μ<sup>t</sup> � �dz � �dt ¼ � <sup>ð</sup><sup>t</sup><sup>2</sup>

ðt2 t1

þ ∂Lc ∂nz δnj L <sup>0</sup> þ ∂Lc ∂μ<sup>z</sup> δμj L 0 � dt

ðL 0

� <sup>∂</sup>Lc ∂nt � �

t

and apply the following property:

ðL 0

δLcdz ¼

ðL 0 ∂Lc ∂nz

ðL 0 ∂Lc ∂μ<sup>z</sup> δ μ<sup>z</sup>

ðL 0

þ ∂Lc ∂nz δð Þ n L

∂Lc ∂nt

� � � <sup>∂</sup>Lc ∂nz � �

δ μ� � " #dz (

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

δLcdz ¼

ðL 0

ðL 0

∂Lc ∂nt

2 P n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � � <sup>1</sup>

> δ ðL 0 Lcdz ¼

δð Þþ nt

<sup>δ</sup>ð Þ nz dz <sup>¼</sup> <sup>∂</sup>Lc

� �dz <sup>¼</sup> <sup>∂</sup>Lc

δð Þþ nt

� � � � �

0 þ ∂Lc ∂μ<sup>z</sup>

Integrating the abovementioned equation in term of time side by side, one has Ð<sup>t</sup><sup>2</sup>

z δð Þ� n

∂Lc ∂μ<sup>t</sup> δ μ<sup>t</sup> � � <sup>þ</sup>

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

� � � � �

δ μ� � <sup>L</sup> <sup>0</sup> � ðL 0

� � � � �

� � � <sup>∂</sup>Lc ∂nz � �

> � � � � �

L

0

∂Lc ∂μ<sup>z</sup> � �

z

ðL 0

t

∂Lc ∂μ<sup>t</sup> � �

δμ � <sup>∂</sup>Lc ∂nz � �

δμ " #dz (

z

<sup>δ</sup><sup>n</sup> � <sup>∂</sup>Lc ∂μ<sup>z</sup> � �

z

� <sup>¼</sup> 0. Similarly, one has the following results after a series of calculation

t1

<sup>δ</sup><sup>n</sup> � <sup>∂</sup>Lc ∂μ<sup>t</sup> � �

δ μ� �

δ μ� � " #dz

∂nz δð Þ n L <sup>0</sup> � ðL 0

∂μ<sup>z</sup>

∂Lc ∂μ<sup>t</sup> δ μ<sup>t</sup>

2 EA: 1 <sup>4</sup> <sup>n</sup><sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � �<sup>2</sup> <sup>¼</sup> Lc <sup>t</sup> : nt; <sup>μ</sup><sup>t</sup>

> ðL 0 δLcdz

> > ∂Lc ∂nz

δð Þþ nz

∂Lc ∂nz � �

∂Lc ∂μ<sup>z</sup> � �

> z δð Þ� n

> > þ ∂Lc ∂nz δð Þ n L <sup>0</sup> <sup>þ</sup>

t δ μ� �dz " #dt

z δð Þ n dz

z δ μ� �dz

> ∂Lc ∂μ<sup>z</sup> � �

z

� � � � � t1 Ð L

∂Lc ∂μ<sup>z</sup>

δ μ� �

<sup>0</sup> <sup>δ</sup>Lcdz � �dt <sup>¼</sup>

) dt

(13)

� � � � � L 0

� � � �dz (10)

∂Lc ∂μ<sup>z</sup> δ μ<sup>z</sup>

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

; nz; μ<sup>z</sup> � � (9)

http://dx.doi.org/10.5772/intechopen.71657

(11)

319

(12)

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

$$\mathcal{U} = \frac{1}{2} \int\_0^L P(n\_z^2 + \mu\_z^2) dz + \frac{1}{2} EA \int\_0^L \left[ \frac{1}{2} (n\_z^2 + \mu\_z^2) \right]^2 + P\_0 \tag{2}$$

where <sup>1</sup> <sup>2</sup> EA <sup>Ð</sup> <sup>L</sup> 0 1 <sup>2</sup> n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � � � <sup>2</sup> is 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

$$T = \frac{1}{2} \int\_0^L \rho \left( n\_t^2 + \mu\_t^2 \right) dz + \frac{1}{2} M \{ n\_t^2(0, t) + \mu\_t^2(0, t) \} + \frac{1}{2} m \{ n\_t^2(L, t) + \mu\_t^2(L, t) \} \tag{3}$$

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

$$\mathcal{W} = F\_{\text{x}} n(\mathbf{0}) + F\_{\text{y}} \mu(\mathbf{0}) \tag{4}$$

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

$$H = \int\_{t\_1}^{t\_2} (\delta T - \delta U + \delta W) dt = 0\tag{5}$$

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

$$\delta T = \delta \left[ \frac{1}{2} \int\_0^L \rho \left( n\_t^2 + \mu\_t^2 \right) dz + \frac{1}{2} M \left( n\_t^2(0, t) + \mu\_t^2(0, t) \right) + \frac{1}{2} m \left( n\_t^2(L, t) + \mu\_t^2(L, t) \right) \right] \tag{6}$$

$$
\delta \mathcal{U} = \delta \left[ \frac{1}{2} \int\_0^L P(n\_z^2 + \mu\_z^2) dz \right] + \delta \left\{ \frac{1}{2} EA \int\_0^L \left[ \frac{1}{2} (n\_z^2 + \mu\_z^2) \right]^2 dz \right\} \tag{7}
$$

and the small derivation of virtual work is written as

$$
\delta \delta W = F\_x \delta n(0, t) + F\_y \delta \mu(0, t) \tag{8}
$$

First, one obtains

$$\delta \left[ \frac{1}{2} \int\_0^L \rho \left( n\_t^2 + \mu\_t^2 \right) dz \right] - \delta \left[ \frac{1}{2} \int\_0^L P \left( n\_z^2 + \mu\_z^2 \right) dz \right] - \delta \left\{ \frac{1}{2} EA \int\_0^L \left[ \frac{1}{2} \left( n\_z^2 + \mu\_z^2 \right) \right]^2 dz \right\}$$

We define Lc as a multivariable function

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes http://dx.doi.org/10.5772/intechopen.71657 319

$$L\_c = \frac{1}{2}\rho \left(n\_t^2 + \mu\_t^2\right) - \frac{1}{2}P\left(n\_z^2 + \mu\_z^2\right) - \frac{1}{2}EA.\frac{1}{4}\left(n\_z^2 + \mu\_z^2\right)^2 = L\_c\left(t:n\_t,\mu\_t,n\_z,\mu\_z\right) \tag{9}$$

and apply the following property:

$$\delta \int\_0^L L\_c dz = \int\_0^L \delta L\_c dz$$

with

P ¼ g½ � rð Þþ L � z m (1)

þ P<sup>0</sup> (2)

<sup>t</sup>ð Þ <sup>L</sup>; <sup>t</sup> � � (3)

With the differential derivation along the cable length L, the potential energy due to the elasticity

1 2 EA ðL 0 1 <sup>2</sup> <sup>n</sup><sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � � �<sup>2</sup>

The kinetic energy of system includes those of the trolley, payload, and cable motion described

<sup>t</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � <sup>þ</sup>

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

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

<sup>t</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � <sup>þ</sup>

> <sup>þ</sup> <sup>δ</sup> <sup>1</sup> 2 EA ðL 0 1 <sup>2</sup> <sup>n</sup><sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � � �<sup>2</sup>

<sup>t</sup>ð Þ <sup>L</sup>; <sup>t</sup> � � � �

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

1 2 M n<sup>2</sup>

H ¼ ðt2

t1

1 2 M n<sup>2</sup>

� � � � <sup>2</sup> is a potential component due to the axial deformation of the cable.

1 2 m n<sup>2</sup>

W ¼ Fxnð Þþ 0 Fyμð Þ0 (4)

ð Þ δT � δU þ δW dt ¼ 0 (5)

<sup>t</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

dz

( )

dz

(6)

(7)

1 2 m n<sup>2</sup>

( )

δW ¼ Fxδnð Þþ 0; t Fyδμð Þ 0; t (8)

� <sup>δ</sup> <sup>1</sup> 2 EA ðL 0 1 <sup>2</sup> <sup>n</sup><sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � � �<sup>2</sup>

<sup>t</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

of cable and gravity is determined by

where <sup>1</sup>

by

by

<sup>2</sup> EA <sup>Ð</sup> <sup>L</sup> 0 1 <sup>2</sup> n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z

318 Adaptive Robust Control Systems

<sup>T</sup> <sup>¼</sup> <sup>1</sup> 2 ðL 0 r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � �dz <sup>þ</sup>

system are in the form of

<sup>δ</sup><sup>T</sup> <sup>¼</sup> <sup>δ</sup> <sup>1</sup> 2 ðL 0 r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � �dz <sup>þ</sup>

First, one obtains

δ 1 2 ðL 0 r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � �dz � �

<sup>δ</sup><sup>U</sup> <sup>¼</sup> <sup>δ</sup> <sup>1</sup> 2 ðL 0 P n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � �dz � �

We define Lc as a multivariable function

and the small derivation of virtual work is written as

� <sup>δ</sup> <sup>1</sup> 2 ðL 0 P n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � �dz � �

<sup>U</sup> <sup>¼</sup> <sup>1</sup> 2 ðL 0 P n<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � �dz <sup>þ</sup>

$$\int\_{0}^{L} \delta L\_{c} dz = \int\_{0}^{L} \left( \frac{\partial L\_{c}}{\partial n\_{t}} \delta(n\_{t}) + \frac{\partial L\_{c}}{\partial \mu\_{t}} \delta(\mu\_{t}) + \frac{\partial L\_{c}}{\partial n\_{z}} \delta(n\_{z}) + \frac{\partial L\_{c}}{\partial \mu\_{z}} \delta(\mu\_{z}) \right) dz \tag{10}$$

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

$$\int\_{0}^{L} \frac{\partial L\_{c}}{\partial n\_{z}} \delta(n\_{z}) dz = \frac{\partial L\_{c}}{\partial n\_{z}} \delta(n) \left| \frac{L}{0} - \int\_{0}^{L} \left( \frac{\partial L\_{c}}{\partial n\_{z}} \right)\_{z} \delta(n) dz \right| \tag{11}$$

$$\int\_{0}^{L} \frac{\partial L\_c}{\partial \mu\_z} \delta(\mu\_z) dz = \frac{\partial L\_c}{\partial \mu\_z} \delta(\mu) \left| \begin{matrix} L \\ 0 \end{matrix} - \int\_{0}^{L} \left( \frac{\partial L\_c}{\partial \mu\_z} \right)\_z \delta(\mu) dz \right| \tag{12}$$

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

$$\begin{split} \int\_{0}^{L} \delta L\_{c} dz &= \int\_{0}^{L} \left[ \frac{\partial L\_{c}}{\partial n\_{t}} \delta(n\_{t}) + \frac{\partial L\_{c}}{\partial \mu\_{t}} \delta(\mu\_{t}) - \left( \frac{\partial L\_{c}}{\partial n\_{z}} \right)\_{z} \delta(n) - \left( \frac{\partial L\_{c}}{\partial \mu\_{z}} \right)\_{z} \delta(\mu) \right] dz \\ &+ \frac{\partial L\_{c}}{\partial n\_{z}} \delta(n) \left| \frac{L}{0} + \frac{\partial L\_{c}}{\partial \mu\_{z}} \delta(\mu) \right|\_{0}^{L} \end{split}$$

Integrating the abovementioned equation in term of time side by side, one has Ð<sup>t</sup><sup>2</sup> t1 Ð L <sup>0</sup> <sup>δ</sup>Lcdz � �dt <sup>¼</sup> ðt2 t1 ðL 0 ∂Lc ∂nt δð Þþ nt ∂Lc ∂μ<sup>t</sup> δ μ<sup>t</sup> � � � <sup>∂</sup>Lc ∂nz � � z δð Þ� n ∂Lc ∂μ<sup>z</sup> � � z δ μ� � " #dz ( þ ∂Lc ∂nz δð Þ n L <sup>0</sup> <sup>þ</sup> ∂Lc ∂μ<sup>z</sup> δ μ� � � � � � � � � � � � L 0 ) dt due to <sup>∂</sup>Lc ∂nz δð Þ nt t2 t1 � � � � <sup>¼</sup> 0. Similarly, one has the following results after a series of calculation

$$\int\_{t\_1}^{t\_2} \left[ \int\_0^L \frac{\partial L\_c}{\partial \mu\_t} \delta \left( \mu\_t \right) dz \right] dt = - \int\_{t\_1}^{t\_2} \left[ \int\_0^L \left( \frac{\partial L\_c}{\partial \mu\_t} \right)\_t \delta \left( \mu \right) dz \right] dt$$

which yields

$$\begin{split} \int\_{h}^{\epsilon\_{2}} \int\_{0}^{L} \delta L\_{c} dz dt &= \int\_{\hbar}^{\epsilon\_{2}} \left\{ \int\_{0}^{L} \left[ -\left(\frac{\partial L\_{c}}{\partial n\_{t}}\right)\_{t} \delta n - \left(\frac{\partial L\_{c}}{\partial \mu\_{t}}\right)\_{t} \delta \mu - \left(\frac{\partial L\_{c}}{\partial n\_{z}}\right)\_{z} \delta n - \left(\frac{\partial L\_{c}}{\partial \mu\_{z}}\right)\_{z} \delta \mu \right] dz \right. \\ &\left. + \frac{\partial L\_{c}}{\partial n\_{z}} \delta n \right|\_{0}^{L} + \frac{\partial L\_{c}}{\partial \mu\_{z}} \delta \mu \Big|\_{0}^{L} \right\} dt \end{split} \tag{13}$$

Next, let us calculate

$$
\delta \left[ \frac{1}{2} M(n\_t^2(0, t) + \mu\_t^2(0, t)) \right] + \delta \left[ \frac{1}{2} m \left( n\_t^2(L, t) + \mu\_t^2(L, t) \right) \right],
$$

with the below notations

$$\delta \left[ \frac{1}{2} M(\mu\_t^2(0, t) + \mu\_t^2(0, t)) \right] = M n\_t(0, t) \delta(n\_t(0, t)) + M \mu\_t(0, t) \delta \left( \mu\_t(0, t) \right) \tag{14}$$

∂Lc ∂nz

<sup>r</sup>ntt � ð Þ Pnz <sup>z</sup> � <sup>1</sup>

1 2 EA n<sup>3</sup>

1 2 EA μ<sup>3</sup>

1 2 EA n<sup>3</sup>

1 2 EA μ<sup>3</sup>

<sup>r</sup>ntt � ð Þ Pnz <sup>z</sup> � <sup>1</sup>

rμtt � Pμ<sup>z</sup> � �

�Pnzð Þ� L; t

�Pμzð Þ� L; t

Pnzð Þþ 0; t

Pμzð Þþ 0; t

differential Eqs. (20), (21), (22), (23), (24), and (25), as follows:

2

<sup>z</sup> � <sup>1</sup> 2

> 1 2 EA n<sup>3</sup>

1 2 EA μ<sup>3</sup>

1 2 EA n<sup>3</sup>

1 2 EA μ<sup>3</sup>

EA 3n<sup>2</sup>

EA 3μ<sup>2</sup>

rμtt � Pμ<sup>z</sup> � � <sup>z</sup> þ 1 2

At boundary condition z = L, one obtains

Pnzð Þþ L; t

Pμzð Þþ L; t

Pnzð Þþ 0; t

Pμzð Þþ 0; t

8

>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>:

At boundary condition z = 0, one has

and

and

and

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

2

EA 3n<sup>2</sup>

EA 3μ<sup>2</sup>

<sup>z</sup> ð Þþ <sup>L</sup>; <sup>t</sup> nzð Þ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

<sup>z</sup> ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup>zð Þ <sup>L</sup>; <sup>t</sup> <sup>n</sup><sup>2</sup>

<sup>z</sup> ð Þþ <sup>0</sup>; <sup>t</sup> nzμ<sup>2</sup>

<sup>z</sup> ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup>zð Þ <sup>0</sup>; <sup>t</sup> <sup>n</sup><sup>2</sup>

In summary, the dynamic behavior of overhead crane governed a set of six nonlinear partial

<sup>z</sup>nzz <sup>þ</sup> nzzμ<sup>2</sup>

<sup>z</sup>μzz <sup>þ</sup> <sup>μ</sup>zzn<sup>2</sup>

<sup>z</sup> ð Þþ <sup>L</sup>; <sup>t</sup> nzð Þ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

<sup>z</sup> ð Þ� <sup>L</sup>; <sup>t</sup> <sup>μ</sup>zð Þ <sup>L</sup>; <sup>t</sup> <sup>n</sup><sup>2</sup>

<sup>z</sup> ð Þ� <sup>0</sup>; <sup>t</sup> nzð Þ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

<sup>z</sup> ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup>zð Þ <sup>0</sup>; <sup>t</sup> <sup>n</sup><sup>2</sup>

<sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμzμzz � � <sup>¼</sup> <sup>0</sup>

<sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμznzz � � <sup>¼</sup> <sup>0</sup>

<sup>z</sup> ð Þ <sup>L</sup>; <sup>t</sup> � � � mnttð Þ¼ <sup>L</sup>; <sup>t</sup> <sup>0</sup>

<sup>z</sup> ð Þ <sup>L</sup>; <sup>t</sup> � � � <sup>m</sup>μttð Þ¼ <sup>L</sup>; <sup>t</sup> <sup>0</sup>

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � Mnttð Þþ <sup>0</sup>; <sup>t</sup> Fx <sup>¼</sup> <sup>0</sup>

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � <sup>M</sup>μttð Þþ <sup>0</sup>; <sup>t</sup> Fy <sup>¼</sup> <sup>0</sup>

¼ �Pnz � <sup>1</sup>

8

EA 4n<sup>3</sup>

<sup>z</sup>nzz <sup>þ</sup> nzzμ<sup>2</sup>

<sup>z</sup>μzz <sup>þ</sup> <sup>μ</sup>zzn<sup>2</sup>

<sup>z</sup> <sup>þ</sup> <sup>2</sup>:2nzμ<sup>2</sup>

z

<sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμznzz � � � � <sup>¼</sup> <sup>0</sup> (21)

<sup>z</sup> ð Þ <sup>L</sup>; <sup>t</sup> � � <sup>þ</sup> mnttð Þ¼ <sup>L</sup>; <sup>t</sup> <sup>0</sup> (22)

<sup>z</sup> ð Þ <sup>L</sup>; <sup>t</sup> � � <sup>þ</sup> <sup>m</sup>μttð Þ¼ <sup>L</sup>; <sup>t</sup> <sup>0</sup> (23)

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � Mnttð Þþ <sup>0</sup>; <sup>t</sup> Fx <sup>¼</sup> <sup>0</sup> (24)

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � <sup>M</sup>μttð Þþ <sup>0</sup>; <sup>t</sup> Fy <sup>¼</sup> <sup>0</sup> (25)

� � (19b)

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321

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

<sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμzμzz � � <sup>¼</sup> <sup>0</sup> (20)

and

$$\delta \left[ \frac{1}{2} m \left( n\_t^2(L, t) + \mu\_t^2(L, t) \right) \right] = m n\_t(L, t) \delta(n\_t(L, t)) + m \mu\_t(L, t) \delta(\mu\_t(L, t)) \tag{15}$$

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

$$\begin{aligned} \int\_{t\_1}^{t\_2} \left\{ \int\_0^L \left[ -\left(\frac{\partial L\_c}{\partial n\_t}\right)\_t \delta(n) - \left(\frac{\partial L\_c}{\partial \mu\_t}\right)\_t \delta(\mu) - \left(\frac{\partial L\_c}{\partial n\_z}\right)\_z \delta(n) - \left(\frac{\partial L\_c}{\partial \mu\_z}\right)\_z \delta(\mu) \right] dz \right. \\ \left. + \left. \frac{\partial L\_c}{\partial n\_z} \delta(n) \right|\_0^L + \left. \frac{\partial L\_c}{\partial \mu\_z} \delta(\mu) \right|\_0^L + \left. M n\_t(0, t) \delta(n\_t(0, t)) + M \mu\_t(0, t) \delta\left(\mu\_t(0, t)\right) \right. \\ \left. + m n\_t(L, t) \delta(n\_t(L, t)) + m \mu\_t(L, t) \delta\left(\mu\_t(L, t)\right) + F\_x \delta n(0, t) + F\_y \delta\mu(0, t) \right] d\_l = 0 \end{aligned} \tag{16}$$

which is simplified as

$$\begin{pmatrix} \int\_{t\_1}^{t\_2} \left\{ \int\_0^L \left[ \left( -\left(\frac{\partial L\_c}{\partial n\_t}\right)\_t - \left(\frac{\partial L\_c}{\partial n\_z}\right)\_z \right) \delta(n) + \left[ -\left(\frac{\partial L\_c}{\partial \mu\_t}\right)\_t - \left(\frac{\partial L\_c}{\partial \mu\_z}\right)\_z \right] \delta(\mu) \right\} dz \\ + \frac{\partial L\_c}{\partial n\_z} \delta n(L,t) - \frac{\partial L\_c}{\partial n\_z} \delta n(0,t) + \frac{\partial L\_c}{\partial \mu\_z} \delta\mu(L,t) - \frac{\partial L\_c}{\partial \mu\_z} \delta\mu(0,t) \\ - M\delta n(0,t)n\_{\mathrm{ff}}(0,t) - M\delta\mu(0,t)\mu\_{\mathrm{ff}}(0,t) - \mathrm{m}\delta n(L,t)n\_{\mathrm{ff}}(L,t) \\ - \mathrm{m}\delta\mu(L,t)\mu\_{\mathrm{ff}}(L,t) + F\_x \delta n(0,t) + F\_y \delta\mu(0,t) \right\} dt \end{pmatrix} = 0$$

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

$$\begin{aligned} \left(\frac{\partial L\_c}{\partial n\_t}\right)\_t + \left(\frac{\partial L\_c}{\partial n\_z}\right)\_z &= 0; \ \left(\frac{\partial L\_c}{\partial \mu\_t}\right)\_t + \left(\frac{\partial L\_c}{\partial \mu\_z}\right)\_z = 0; \ \frac{\partial L\_c}{\partial n\_z} - m n\_{\text{ft}}(L, t) = 0;\\ \frac{\partial L\_c}{\partial \mu\_z} - m \mu\_{\text{ft}}(L, t) &= 0; \ \frac{\partial L\_c}{\partial n\_z} - M n\_{\text{ft}}(0, t) + F\_x = 0; \ \frac{\partial L\_c}{\partial \mu\_z} - M \mu\_{\text{ft}}(0, t) + F\_y = 0; \end{aligned} \tag{18}$$

which leads to

$$
\left(\frac{\partial L\_c}{\partial n\_t}\right)\_t = \rho n\_{tt} \tag{19a}
$$

and

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes http://dx.doi.org/10.5772/intechopen.71657 321

$$\frac{\partial L\_{\text{c}}}{\partial n\_{z}} = -Pn\_{z} - \frac{1}{8}EA\left(4n\_{z}^{3} + 2.2n\_{z}\mu\_{z}^{2}\right) \tag{19b}$$

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

$$
\rho u\_{\rm th} - \left( P n\_z \right)\_z - \frac{1}{2} E A \left( 3 n\_z^2 n\_{zz} + n\_{zz} \mu\_z^2 + 2 n\_z \mu\_z \mu\_{zz} \right) = 0 \tag{20}
$$

and

Next, let us calculate

320 Adaptive Robust Control Systems

with the below notations

and

δ 1 2 M n<sup>2</sup>

δ 1 2 m n<sup>2</sup>

ðt2 t1

þ ∂Lc ∂nz δð Þ n L 0 þ ∂Lc ∂μ<sup>z</sup> δ μ � �

which is simplified as

0

BBBBBBBBB@

ðt2 t1

þ ∂Lc ∂nz

∂Lc ∂nt � �

∂Lc ∂μ<sup>z</sup>

which leads to

and

t þ

ðL 0

(

ðL 0

> � � � � �

(

δ 1 2 M n<sup>2</sup>

<sup>t</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � � �

<sup>t</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>L</sup>; <sup>t</sup> � � � �

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

t δð Þ� n

t

∂Lc ∂nz

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

z ¼ 0;

∂Lc ∂nz � �

� mμttð Þ¼ L; t 0;

� �

� � � � �

� <sup>∂</sup>Lc ∂nz � �

δnð Þþ 0; t

�mδμð Þ L; t μttð Þþ L; t Fxδnð Þþ 0; t Fyδμð Þ 0; t

∂Lc ∂nz L 0

∂Lc ∂μ<sup>t</sup> � �

ð Þ L; t δ μ<sup>t</sup>

z

�Mδnð Þ 0; t nttð Þ� 0; t Mδμð Þ 0; t μttð Þ� 0; t mδn Lð Þ ; t nttð Þ L; t

∂Lc ∂μ<sup>t</sup> � �

t þ

∂Lc ∂μ<sup>z</sup>

δð Þþ � n

∂Lc ∂μ<sup>z</sup> � �

� Mnttð Þþ 0; t Fx ¼ 0;

t

∂Lc ∂nt � � z ¼ 0;

� � ( )

δμð Þ� L; t

t δ μ

� <sup>∂</sup>Lc ∂nt � �

þmntð Þ L; t δð Þþ ntð Þ L; t mμ<sup>t</sup>

� <sup>∂</sup>Lc ∂nt � �

δn Lð Þ� ; t

<sup>t</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � � �

<sup>þ</sup> <sup>δ</sup> <sup>1</sup> 2 m n<sup>2</sup>

¼ Mntð Þ 0; t δð Þþ ntð Þ 0; t Mμ<sup>t</sup>

¼ mntð Þ L; t δð Þþ ntð Þ L; t mμ<sup>t</sup>

� � � <sup>∂</sup>Lc ∂nz � �

þ Mntð Þ 0; t δð Þþ ntð Þ 0; t Mμ<sup>t</sup>

∂Lc ∂μ<sup>t</sup> � �

> ∂Lc ∂μ<sup>z</sup>

� dt

� � " #

z δð Þ� n

ð Þ <sup>L</sup>; <sup>t</sup> � � <sup>þ</sup> Fxδnð Þþ <sup>0</sup>; <sup>t</sup> Fyδμð Þ <sup>0</sup>; <sup>t</sup>

t

∂Lc ∂nz

> ∂Lc ∂μ<sup>z</sup>

" #

� <sup>∂</sup>Lc ∂μ<sup>z</sup> � �

δμð Þ 0; t

<sup>t</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>L</sup>; <sup>t</sup> � � � �

ð Þ 0; t δ μ<sup>t</sup>

ð Þ L; t δ μ<sup>t</sup>

∂Lc ∂μ<sup>z</sup> � �

ð Þ 0; t δ μ<sup>t</sup>

z

� mnttð Þ¼ L; t 0;

� Mμttð Þþ 0; t Fy ¼ 0;

¼ rntt (19a)

δ μ

z δ μ

ð Þ <sup>0</sup>; <sup>t</sup> � �

� dt ¼ 0

dz

1

CCCCCCCCCA

¼ 0 (17)

(18)

ð Þ <sup>0</sup>; <sup>t</sup> � � (14)

ð Þ <sup>L</sup>; <sup>t</sup> � � (15)

dz

(16)

$$
\rho \mu\_{tt} - \left[ \left( P \mu\_z \right)\_z + \frac{1}{2} EA \left( 3 \mu\_z^2 \mu\_{zz} + \mu\_{zz} n\_z^2 + 2 n\_z \mu\_z n\_{\varpi} \right) \right] = 0 \tag{21}
$$

At boundary condition z = L, one obtains

$$\frac{1}{2}P n\_z(L, t) + \frac{1}{2} EA \left( n\_z^3(L, t) + n\_z(L, t) \mu\_z^2(L, t) \right) + m m\_{\text{fl}}(L, t) = 0 \tag{22}$$

and

$$P\mu\_z(L,t) + \frac{1}{2}EA\left(\mu\_z^3(L,t) + \mu\_z(L,t)n\_z^2(L,t)\right) + m\mu\_{tt}(L,t) = 0\tag{23}$$

At boundary condition z = 0, one has

$$\frac{1}{2}P n\_z(0,t) + \frac{1}{2}EA\left(n\_z^3(0,t) + n\_z\mu\_z^2(0,t)\right) - M n\_{lt}(0,t) + F\_x = 0\tag{24}$$

and

$$P\mu\_z(0,t) + \frac{1}{2}EA\{\mu\_z^3(0,t) + \mu\_z(0,t)n\_z^2(0,t)\} - M\mu\_{tt}(0,t) + F\_y = 0\tag{25}$$

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:

$$\begin{cases} \rho n\_{tt} - (\mathcal{P}n\_{z})\_{z} - \frac{1}{2} EA \left( 3n\_{z}^{2} n\_{zx} + n\_{zx} \mu\_{z}^{2} + 2n\_{z} \mu\_{z} \mu\_{zx} \right) = 0 \\\\ \rho \mu\_{tt} - \left( P\mu\_{z} \right)\_{z} - \frac{1}{2} EA \left( 3 \mu\_{z}^{2} \mu\_{zx} + \mu\_{zx} n\_{z}^{2} + 2 n\_{z} \mu\_{z} n\_{zx} \right) = 0 \\\\ - P n\_{z}(L, t) - \frac{1}{2} EA \left( n\_{z}^{3}(L, t) + n\_{z}(L, t) \mu\_{z}^{2}(L, t) \right) - m n\_{tt}(L, t) = 0 \\\\ - P \mu\_{z}(L, t) - \frac{1}{2} EA \left( \mu\_{z}^{3}(L, t) - \mu\_{z}(L, t) n\_{z}^{2}(L, t) \right) - m \mu\_{tt}(L, t) = 0 \\\\ P n\_{z}(0, t) + \frac{1}{2} EA \left( n\_{z}^{3}(0, t) - n\_{z}(0, t) \mu\_{z}^{2}(0, t) \right) - M n\_{tt}(0, t) + F\_{x} = 0 \\\\ P \mu\_{z}(0, t) + \frac{1}{2} EA \left( \mu\_{z}^{3}(0, t) + \mu\_{z}(0, t) n\_{z}^{2}(0, t) \right) - M \mu\_{tt}(0, t) + F\_{y} = 0 \end{cases}$$

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.

With the notations that ∣j j w

<sup>t</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>

<sup>þ</sup> <sup>n</sup><sup>2</sup>

with

and

<sup>V</sup>\_ <sup>¼</sup> ð L

þ

MPð Þ0 Pð Þþ 0 Ka

> ð L

> > 0

Using partial integration

0

r ntntt þ μ<sup>t</sup>

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

r ntntt þ μ<sup>t</sup> <sup>μ</sup>tt � �dz <sup>¼</sup>

one has

2 ¼ ð L

<sup>K</sup>min <sup>¼</sup> <sup>1</sup> 2

<sup>K</sup>max <sup>¼</sup> <sup>1</sup> 2

<sup>μ</sup>tt � � <sup>þ</sup> P nznzt <sup>þ</sup> <sup>μ</sup>zμzt � � � <sup>þ</sup>

þμð Þ 0; t μ<sup>t</sup>

� qd <sup>μ</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>t</sup>

ntð Þ 0; t nttð Þþ 0; t μ<sup>t</sup>

þm ntð Þ L; t nttð Þþ L; t μ<sup>t</sup>

ð L

0

þμ<sup>t</sup> Pμ<sup>z</sup> � � <sup>z</sup> þ 1 2

ð L

0

<sup>t</sup>ð Þ <sup>L</sup>; <sup>t</sup> � � <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0

n2 <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � � <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>n</sup><sup>2</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>p</sup> � qd � �<sup>2</sup>

Kmin∣j j w

min r; P;

max r; P;

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

EA

EA

<sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � <sup>þ</sup> <sup>n</sup><sup>2</sup>

, .

<sup>2</sup> <sup>≤</sup> V tð Þ <sup>≤</sup>Kmax∣j j <sup>w</sup>

� �

� �

<sup>z</sup>ntz <sup>þ</sup> <sup>μ</sup><sup>3</sup>

9

>>>>>>>>>>>>=

>>>>>>>>>>>>;

<sup>z</sup>μzz <sup>þ</sup> <sup>μ</sup>zzn<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμznzz � � � ��dz

Pnzntzdz

<sup>z</sup>nzz <sup>þ</sup> nzzμ<sup>2</sup> <sup>z</sup> <sup>þ</sup> <sup>2</sup>nzμzμzz � � � � �

<sup>4</sup> ; MPð Þ<sup>0</sup> Pð Þþ 0 Ka

<sup>4</sup> ; MPð Þ<sup>0</sup> Pð Þþ 0 Ka

> EA <sup>2</sup> <sup>n</sup><sup>3</sup>

ð Þ 0; t μttð Þ 0; t

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

1 2

EA 3μ<sup>2</sup>

L <sup>0</sup> � ð L

� � � � � �

0

EA 3n<sup>2</sup>

ð Þþ 0; t nð Þ 0; t ntð Þ 0; t

ð Þ <sup>L</sup>; <sup>t</sup> <sup>μ</sup>ttð Þ <sup>L</sup>; <sup>t</sup> � �

ð Þþ <sup>0</sup>; <sup>t</sup> <sup>n</sup>ð Þ <sup>0</sup>; <sup>t</sup> ntð Þ <sup>0</sup>; <sup>t</sup> � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>μ</sup><sup>2</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>n</sup><sup>2</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>p</sup> !

nt ð Þ Pnz <sup>z</sup> þ

ntð Þ Pnz <sup>z</sup>dz ¼ ntPnz

� �<sup>2</sup> n odz <sup>þ</sup> <sup>n</sup><sup>2</sup>

<sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

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2

; <sup>m</sup>; <sup>P</sup>ð Þ<sup>0</sup> Kp Pð Þþ 0 Ka

; <sup>m</sup>; <sup>P</sup>ð Þ<sup>0</sup> Kp Pð Þþ 0 Ka

<sup>z</sup>μzt <sup>þ</sup> nznztμ<sup>2</sup>

� ��dz

<sup>z</sup> <sup>þ</sup> <sup>μ</sup>zμztμ<sup>2</sup>

z

(29)

(30)

<sup>t</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup> <sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � 323
