**2. Problem formulation**

The purpose of this chapter is to propose an effective method to achieve crane control in the presence of persistent (even non-vanishing) perturbations in both unactuated and actuated dynamics. The crane dynamics can be represented by the following equations (shown in **Figure 1**):

$$(M+m)\ddot{x} + mL\ddot{\theta}\cos\theta - mL\,\dot{\theta}^2\sin\theta = u - f\_r + d\_{r'} \tag{1}$$

$$mL^2\ddot{\theta} + mL\cos\theta\ddot{x} + mgL\sin\theta = \,\_0\text{d}\_{\text{f}}\tag{2}$$

The system parameters are defined in **Table 1**, and *f r* denotes the rail friction force expressed as follows:

**Figure 1.** The schematic diagram of an overhead crane system.

$$f\_r = f\_{r0} \tanh\left(\mathbf{\dot{x}}/\mathbf{e}\right) - k\_r \parallel \mathbf{\dot{x}} \parallel \mathbf{\dot{x}}\_r \tag{3}$$

Considering the practical physical constraints, though the exact expressions for the lumped

Before proceeding to describe the control objective, we perform several steps of transformations for the original crane dynamics shown in Eqs. (1) and (2) for the convenience of carrying out controller development and stability analysis in the subsequent section. Considering the fact of *mL* > 0, we divide both sides of Eq. (2) and make some arrangements to obtain the following equation:

cos*<sup>θ</sup>* +

*Ml*, <sup>∣</sup> *<sup>δ</sup><sup>a</sup>* <sup>∣</sup> <sup>≤</sup> ¯¯*δ<sup>a</sup>* <sup>=</sup> ¯¯*d<sup>θ</sup>*

In the view of the explicit expression of Eq. (5), a feedback linearization controller can be

in which *v*(*t*) is a to-be-elaborated auxiliary control input. That is, once we derive the expression of*v*(*t*), the ultimate controller *u*(*t*) can be conveniently obtained according to Eq. (8). By substituting Eq. (8) into Eq. (5), together with Eq. (4), the dynamic Eqs. (1) and (2) can be

cos*<sup>θ</sup>* +

.

*<sup>x</sup>*¨ <sup>=</sup> <sup>−</sup>*<sup>g</sup>* tan*<sup>θ</sup>* <sup>−</sup> *<sup>L</sup>θ*¨ \_\_\_\_\_

*<sup>θ</sup>*¨ <sup>=</sup> *<sup>v</sup>* <sup>+</sup> *<sup>δ</sup><sup>x</sup>* <sup>+</sup> *<sup>δ</sup><sup>a</sup>*

Then one can substitute Eq. (4) into Eq. (1) and make some arrangements to obtain

\_\_\_*<sup>x</sup>*

cos*<sup>θ</sup> v* − *mL θ*

cos*<sup>θ</sup>* (*θ*¨ <sup>−</sup> *<sup>δ</sup><sup>x</sup>* <sup>−</sup> *<sup>δ</sup><sup>a</sup>*) <sup>−</sup> *mL <sup>θ</sup>*

(*t*) and *dθ*(*t*) are unknown, the following assumptions are reasonably

is a priori known. The unactuated perturbation term *dθ*(*t*) is differentiable

, *<sup>i</sup>* <sup>=</sup> 1, 2, <sup>⋯</sup> ,*<sup>n</sup>* <sup>−</sup> 1, where ¯¯*<sup>d</sup> <sup>θ</sup>*

*<sup>d</sup>* \_\_\_\_\_\_\_ *<sup>θ</sup>*

̇ <sup>2</sup> sin*θ* − (*M* + *m*)*g* tan*θ* = *u* − *f*

(*t*) and *δθa*(*t*) are provided as:

(*<sup>M</sup>* <sup>+</sup> *<sup>m</sup>*) \_\_\_\_\_\_\_\_

̇ <sup>2</sup> sin*θ* − (*M* + *m*)*g* tan*θ* + *f*

*<sup>d</sup>* \_\_\_\_\_\_\_ *<sup>θ</sup> mL* cos*θ*,

(*t*) present in the actuated dynamics is bounded as <sup>∣</sup> *dx*

and ¯¯*<sup>d</sup> <sup>i</sup>*

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383

*mL* cos*θ*. (4)

(*<sup>M</sup>* <sup>+</sup> *<sup>m</sup>*) \_\_\_\_\_\_\_\_\_\_\_\_\_ (*<sup>M</sup>* <sup>+</sup> *<sup>m</sup>* sin2 *<sup>θ</sup>*)*mL*2. (6)

*r*

*r*

*MmL*<sup>2</sup> . (7)

, (8)

(9)

, (5)

are priori known

297

perturbation terms *dx*

, where ¯¯*<sup>d</sup> <sup>x</sup>*

up to the *n*-th order; <sup>∣</sup> *<sup>d</sup><sup>θ</sup>* <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>θ</sup>*

constants. It is also assumed that *d<sup>θ</sup>*

**2.1. Crane model transformation**

*<sup>x</sup>*¨ <sup>=</sup> <sup>−</sup>*<sup>g</sup>* tan*<sup>θ</sup>* <sup>−</sup> *<sup>L</sup>θ*¨ \_\_\_\_\_

(*t*) and *δθa*(*t*) are defined as follows:

Based on *Assumption 1*, the upper bounds for *δ<sup>x</sup>*

<sup>∣</sup> *<sup>δ</sup><sup>x</sup>* <sup>∣</sup> <sup>≤</sup> ¯¯*δ<sup>x</sup>* <sup>=</sup> ¯¯*<sup>d</sup>*

re-expressed in the following fashion:

{

*<sup>δ</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup> *dx* cos*<sup>θ</sup>* \_\_\_\_\_\_\_\_\_\_\_ (*<sup>M</sup>* <sup>+</sup> *<sup>m</sup>* sin2 *<sup>θ</sup>*)*L*, *δθ<sup>a</sup>* <sup>=</sup> <sup>−</sup> *<sup>d</sup><sup>θ</sup>*

(*M* + *m* sin2 *θ*)*L* \_\_\_\_\_\_\_\_\_\_\_

(*M* + *m* sin2 *θ*)*L* \_\_\_\_\_\_\_\_\_\_\_

*Assumption 1*: The perturbation term *dx*

, <sup>∣</sup> *<sup>d</sup><sup>θ</sup>* (*i*) (*t*) <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>i</sup>*

> (3) (*t*) ≈ 0.

made.

(*t*) <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>x</sup>*

−

proposed as follows:

*u* = −

where *δ<sup>x</sup>*

where *f r*0 , ϵ, *kr* ∈ *R* are friction parameters, which can be identified by offline experimental tests and data fitting. *dx* (*t*) and *dθ*(*t*) denote the lumped term comprising external perturbations, unmodeled dynamics, the mismatch between the real girder friction and the friction compensation model shown in Eq. (3), and so forth.


**Table 1.** The system parameters of crane systems.

Considering the practical physical constraints, though the exact expressions for the lumped perturbation terms *dx* (*t*) and *dθ*(*t*) are unknown, the following assumptions are reasonably made.

*Assumption 1*: The perturbation term *dx* (*t*) present in the actuated dynamics is bounded as <sup>∣</sup> *dx* (*t*) <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>x</sup>* , where ¯¯*<sup>d</sup> <sup>x</sup>* is a priori known. The unactuated perturbation term *dθ*(*t*) is differentiable up to the *n*-th order; <sup>∣</sup> *<sup>d</sup><sup>θ</sup>* <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>θ</sup>* , <sup>∣</sup> *<sup>d</sup><sup>θ</sup>* (*i*) (*t*) <sup>∣</sup> <sup>≤</sup> ¯¯*<sup>d</sup> <sup>i</sup>* , *<sup>i</sup>* <sup>=</sup> 1, 2, <sup>⋯</sup> ,*<sup>n</sup>* <sup>−</sup> 1, where ¯¯*<sup>d</sup> <sup>θ</sup>* and ¯¯*<sup>d</sup> <sup>i</sup>* are priori known constants. It is also assumed that *d<sup>θ</sup>* (3) (*t*) ≈ 0.

#### **2.1. Crane model transformation**

Before proceeding to describe the control objective, we perform several steps of transformations for the original crane dynamics shown in Eqs. (1) and (2) for the convenience of carrying out controller development and stability analysis in the subsequent section. Considering the fact of *mL* > 0, we divide both sides of Eq. (2) and make some arrangements to obtain the following equation:

$$\ddot{x} = -g\tan\Theta - \frac{L\ddot{\theta}}{\cos\theta} + \frac{d\_{\theta}}{mL\cos\theta}.\tag{4}$$

Then one can substitute Eq. (4) into Eq. (1) and make some arrangements to obtain

$$-\frac{(M+m\sin^2\theta)L}{\cos\theta}(\vec{\theta}-\delta\_x-\delta\_{a\dot{a}})-mL\,\partial^2\sin\theta-(M+m)\text{g}\,\tan\theta=u-f\_r\tag{5}$$

where *δ<sup>x</sup>* (*t*) and *δθa*(*t*) are defined as follows:

*f*

sation model shown in Eq. (3), and so forth.

**Table 1.** The system parameters of crane systems.

**Figure 1.** The schematic diagram of an overhead crane system.

where *f r*0

and data fitting. *dx*

296 Adaptive Robust Control Systems

*<sup>r</sup>* = *f*

*M*

Trolley

*<sup>r</sup>*<sup>0</sup> tanh(*x*̇ /ϵ) − *kr* ∣ *x*̇ ∣ *x*̇

unmodeled dynamics, the mismatch between the real girder friction and the friction compen-

**System parameter Physical significant Unit** *M* Trolley mass kg *m* Cargo mass kg *L* Cargo rotation radius m *g* Gravity constant m/s2 *x* Trolley translational displacement m *θ* Cargo rotational angle rad *u* Control input N

, ϵ, *kr* ∈ *R* are friction parameters, which can be identified by offline experimental tests

(*t*) and *dθ*(*t*) denote the lumped term comprising external perturbations,

, (3)

*mg*

Cargo

*x*

*u*

$$
\delta\_{x\_{\chi}\chi\_{\chi}} = \delta\_{\theta\chi^{\prime}} \quad \text{and} \quad \delta\_{\theta\chi^{\prime}} = \delta\_{\theta\chi^{\prime}}.
$$

$$
\delta\_{x} = -\frac{d\_{\chi}\cos\theta}{(M+m\sin^{2}\theta)L^{\prime}}\delta\_{\theta a} = -\frac{d\_{\phi}(M+m)}{(M+m\sin^{2}\theta)mL^{2}}.\tag{6}
$$

Based on *Assumption 1*, the upper bounds for *δ<sup>x</sup>* (*t*) and *δθa*(*t*) are provided as:

$$\parallel \; \delta\_x \mid \le \overline{\delta}\_x = \frac{\overline{d}\_x}{M l^\nu} \mid \; \delta\_{\theta u} \mid \le \overline{\delta}\_{\theta u} = \frac{\overline{d}\_\theta (M+m)}{M m L^2} . \tag{7}$$

In the view of the explicit expression of Eq. (5), a feedback linearization controller can be proposed as follows:

$$u = -\frac{\langle \mathbf{M} + m\sin^2\theta \rangle \mathbf{L}}{\cos\theta} \mathbf{v} - mL\,\partial^2 \sin\theta - (\mathbf{M} + m)\mathbf{g}\,\tan\theta + f\_{\prime} \tag{8}$$

in which *v*(*t*) is a to-be-elaborated auxiliary control input. That is, once we derive the expression of*v*(*t*), the ultimate controller *u*(*t*) can be conveniently obtained according to Eq. (8). By substituting Eq. (8) into Eq. (5), together with Eq. (4), the dynamic Eqs. (1) and (2) can be re-expressed in the following fashion:

Herekspessesed in me ionowung asznou.

$$\begin{cases}
\ddot{x} = -g\tan\theta - \frac{L\ddot{\theta}}{\cos\theta} + \frac{d\_{\theta}}{mL\cos\theta'} \\
\ddot{\theta} = \upsilon + \delta\_{x} + \delta\_{a}.
\end{cases} \tag{9}$$

Further, we define the following coordinate transformations:

$$\begin{aligned} \phi\_1 &= \mathbf{x} + L \ln(\sec \theta + \tan \theta) \\ \phi\_2 &= \mathbf{x} + L \dot{\theta} \sec \theta \\ \phi\_3 &= -g \tan \theta \\ \phi\_4 &= -g \dot{\theta} \sec^2 \theta \end{aligned} \tag{10}$$

**2.2. Control objective**

the following conditions:

where *tf*<sup>1</sup>

controller *u*′

with *tf*<sup>2</sup>

ler *u*′

lim*<sup>t</sup>*→*<sup>t</sup>*

*θ*(*t*) = 0, *θ*

second equation of Eq. (14), that

*δ<sup>u</sup>* = *φ*

*φ*<sup>1</sup>

actions (e.g., lowering) can be taken.

For crane control during the transportation process (between the hoisting and lowering stages), the kernel objective is to transfer the cargo from its initial position to the desired position (destination) and then keep it stationary right above the destination so that further

Hence, the preliminary task is to make the cargo reach the destination by appropriately con-

To make this process smooth enough, instead of set-point control (i.e., directly using *pdx* as the

When there are no external perturbations appearing in the unactuated dynamics (that is, *δθu* ≡ 0 in Eq. (14)), we need also to damp out the cargo swing *θ*(*t*) at the same time, namely,

*θ* → 0 ⇒ *x* → *pdx*. (17)

However, in the case of persistent, non-vanishing perturbations in the unactuated component (i.e., *δθu*(*t*) ≠ 0), there *does not exist any control action* that can completely damp out *θ*(*t*) while keeping the cargo stationary right above the destination. Suppose that there exists such a

> (*t*) = 0, *φ* ̇ 2

being the settling time, then it would follow, by inserting Eq. (18) and Eq. (19) into the

̇

which obviously contradicts with the fact that *δθu* ≠ 0; thus the existence of such a control-

(*t*) is impossible. This fact illustrates the great challenge that will be faced with when

(*t*) to reach*pdx*, and *π<sup>i</sup>*

*φ*<sup>1</sup> = *x* + *L* → *pdx*. (15)

(*i*) ∣ ≤ *π*, *i* = 1, 2, 3, 4, (16)

(*t*) = −*g* tan*θ*(*t*) = 0, (18)

*f*2

<sup>2</sup> − *ϕ*<sup>3</sup> = 0, (20)

, (19)

(*t*) = 0, ∀*t* ≥ *t*

(*t*), respectively.

(*t*), which satisfies

(*i* = 1, 2, 3, 4) stands for the cor-

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383 299

trolling the trolley motion, which can be mathematically depicted as follows:

reference), we want the cargo to follow a smooth time-varying trajectory *rx*

(*t*) = *pdx*, ∣ *rx*

(*t*) that could eliminate the cargo swing, namely,

(*t*) = 0 ⇒ *ϕ*<sup>3</sup>

(*t*) = 0, *φ*<sup>2</sup>

̇

and make the cargo stay stationary at the destination in the sense that

̇ 1

(*t*) = *pdx*, *φ*

*f*1 *rx*

responding upper bound for the *i*-th order derivative for*rx*

denotes the consumed time for *rx*

Then, it is straightforward to obtain the following dynamic equations:

$$\begin{aligned} \dot{\phi}\_1 &= \phi\_{2'} \\ \phi\_2 &= \phi\_3 - h \{\phi\_3\} \phi\_4^2 + \frac{d\_\vartheta}{mL\cos\theta'} \\ \phi\_3 &= \phi\_{4'} \\ \dot{\phi}\_4 &= -g \{v + \delta\_\chi + \delta\_{\rho l}\} \sec^2\theta - 2g \,\partial^2 \sec^2\theta \tan\theta. \end{aligned} \tag{11}$$

In Eq. (11), the function *h*(*φ*<sup>3</sup> ) is with the definition as *h*(*φ*3) <sup>=</sup> *<sup>l</sup> <sup>φ</sup>* \_\_\_\_\_\_\_ <sup>3</sup> (*g*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*<sup>3</sup> 2 ) 1.5 <sup>⇒</sup> <sup>∣</sup> *<sup>h</sup>*(*φ*3) <sup>∣</sup> <sup>≤</sup> 0.004*L*, where the value of the gravity constant is taken as *g* = 9.8 m/s<sup>2</sup> .

For practical applications, the cargo swing is always within 10 degrees, that is, ∣*θ*(*t*) ∣  ≤ *π*/18 rad. In this case, the approximations of sin*θ* ≈ tan *θ* ≈ *θ* and sec*θ* = cos−1*θ* ≈ 1 are valid. In this sense, *φ*<sup>1</sup> (*t*) in Eq. (10) can be approximated as follows:

$$
\phi\_1(t) \approx \mathbf{x} + L \ln(1+\theta) \approx \mathbf{x} + L\theta,\tag{12}
$$

which is right at the horizontal position of the cargo. Also, the cargo swing angular velocity satisfies ∣ *θ* ̇ (*t*) ∣ < < 1 rad/s; considering that the wire length *L*'s order of magnitude is usually 10 m, *h*(*φ*3) *<sup>φ</sup>*<sup>4</sup> <sup>2</sup> = 0.004 *Lg*<sup>2</sup> *θ* ̇ <sup>2</sup> sec<sup>4</sup> *θ* ≈ 0.384*L θ* ̇ <sup>2</sup> sec<sup>4</sup> *<sup>θ</sup>* <sup>≈</sup> 0 holds; hence, *h*(*φ*3) *<sup>φ</sup>*<sup>4</sup> 2 is negligible and can be incorporated as part of the unactuated lumped perturbation *δθu*(*t*) that will be introduced later. For simplicity of denotation, we define

$$
\boldsymbol{\varphi}\_1 = \mathbf{x} + \mathbf{L}\boldsymbol{\theta}, \boldsymbol{\varphi}\_2 = \boldsymbol{\varphi}\_1. \tag{13}
$$

Therefore, the crane dynamics can be described by

$$\begin{cases} \dot{\rho}\_1 = \rho\_{2^\*}\\ \dot{\rho}\_2 = \dot{\phi}\_3 + \delta\_{\theta \omega'}\\ \dot{\rho}\_3 = \dot{\phi}\_{4^\*}\\ \dot{\rho}\_4 = -g \{ \upsilon + \delta\_x + \delta\_{\text{adj}} \} \sec^2 \theta - 2g \, \dot{\theta}^2 \sec^2 \theta \tan \theta \, \end{cases} \tag{14}$$

wherein *δθu*(*t*) represents the unactuated lumped perturbation term mainly consisting of *dθ*/*mL* cos *θ*.

#### **2.2. Control objective**

(10)

(11)

(14)

1.5 <sup>⇒</sup> <sup>∣</sup> *<sup>h</sup>*(*φ*3) <sup>∣</sup> <sup>≤</sup> 0.004*L*, where

is negligible and can be

. (13)

Further, we define the following coordinate transformations:

*φ*<sup>1</sup> = *x* + *L* ln(sec*θ* + tan*θ*) *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>* <sup>+</sup> *<sup>L</sup>θ*˙sec*<sup>θ</sup>*

 *<sup>φ</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup> *<sup>g</sup>* tan*<sup>θ</sup>* 

*φ*<sup>4</sup> = − *gθ*˙ sec2 *θ*

2 +

<sup>4</sup> = −*g*(*v* + *δ<sup>x</sup>* + *δ<sup>a</sup>*) sec2 *θ* − 2*g θ*

*<sup>d</sup>* \_\_\_\_\_\_\_ *<sup>θ</sup> mL* cos*θ*,

) is with the definition as *h*(*φ*3) <sup>=</sup> *<sup>l</sup> <sup>φ</sup>* \_\_\_\_\_\_\_ <sup>3</sup>

For practical applications, the cargo swing is always within 10 degrees, that is, ∣*θ*(*t*) ∣  ≤ *π*/18 rad. In this case, the approximations of sin*θ* ≈ tan *θ* ≈ *θ* and sec*θ* = cos−1*θ* ≈ 1 are valid. In this

which is right at the horizontal position of the cargo. Also, the cargo swing angular velocity

incorporated as part of the unactuated lumped perturbation *δθu*(*t*) that will be introduced

(*t*) ∣ < < 1 rad/s; considering that the wire length *L*'s order of magnitude is usually

̇ <sup>2</sup> sec<sup>4</sup> *<sup>θ</sup>* <sup>≈</sup> 0 holds; hence, *h*(*φ*3) *<sup>φ</sup>*<sup>4</sup>

̇ 1

.

̇ <sup>2</sup> sec2 *θ* tan*θ*.

(*g*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*<sup>3</sup> 2 )

(*t*) ≈ *x* + *L* ln(1 + *θ*) ≈ *x* + *L*, (12)

̇ <sup>2</sup> sec2 *θ* tan*θ*,

2

Then, it is straightforward to obtain the following dynamic equations:

<sup>2</sup> = *φ*<sup>3</sup> − *h*(*φ*3) *φ*<sup>4</sup>

*φ* ̇ <sup>1</sup> = *φ*<sup>2</sup> ,

*φ*̇

*φ*̇ <sup>3</sup> = *ϕ*<sup>4</sup> ,

*φ* ̇

the value of the gravity constant is taken as *g* = 9.8 m/s<sup>2</sup>

(*t*) in Eq. (10) can be approximated as follows:

̇ <sup>2</sup> sec<sup>4</sup> *θ* ≈ 0.384*L θ*

*φ*<sup>1</sup> = *x* + *L*, *φ*<sup>2</sup> = *φ*

Therefore, the crane dynamics can be described by

⎧ ⎪ ⎨ ⎪ ⎩

*φ*̇ <sup>1</sup> = *φ*<sup>2</sup> ,

*φ*̇

*φ*̇

<sup>2</sup> = *ϕ*<sup>3</sup> + *δ<sup>u</sup>*

 *<sup>φ</sup>*̇ <sup>3</sup> = *ϕ*<sup>4</sup> ,

,

<sup>4</sup> = −*g*(*v* + *δ<sup>x</sup>* + *δ<sup>a</sup>*) sec2 *θ* − 2*g θ*

wherein *δθu*(*t*) represents the unactuated lumped perturbation term mainly consisting of

In Eq. (11), the function *h*(*φ*<sup>3</sup>

*φ*<sup>1</sup>

<sup>2</sup> = 0.004 *Lg*<sup>2</sup> *θ*

later. For simplicity of denotation, we define

298 Adaptive Robust Control Systems

sense, *φ*<sup>1</sup>

satisfies ∣ *θ*

*dθ*/*mL* cos *θ*.

10 m, *h*(*φ*3) *<sup>φ</sup>*<sup>4</sup>

̇

For crane control during the transportation process (between the hoisting and lowering stages), the kernel objective is to transfer the cargo from its initial position to the desired position (destination) and then keep it stationary right above the destination so that further actions (e.g., lowering) can be taken.

Hence, the preliminary task is to make the cargo reach the destination by appropriately controlling the trolley motion, which can be mathematically depicted as follows:

$$
\varphi\_1 = \propto + L\theta \to p\_{\text{dx}}.\tag{15}
$$

To make this process smooth enough, instead of set-point control (i.e., directly using *pdx* as the reference), we want the cargo to follow a smooth time-varying trajectory *rx* (*t*), which satisfies the following conditions:

$$\lim\_{t \to t\_n} r\_x(t) = |p\_{dx'}| \ |r\_x^{(0)}| \le \pi, i = 1, 2, 3, 4,\tag{16}$$

where *tf*<sup>1</sup> denotes the consumed time for *rx* (*t*) to reach*pdx*, and *π<sup>i</sup>* (*i* = 1, 2, 3, 4) stands for the corresponding upper bound for the *i*-th order derivative for*rx* (*t*), respectively.

When there are no external perturbations appearing in the unactuated dynamics (that is, *δθu* ≡ 0 in Eq. (14)), we need also to damp out the cargo swing *θ*(*t*) at the same time, namely,

$$
\theta \to 0 \text{ } \Rightarrow \text{ x } \to \text{ } p\_{\text{dr}}.\tag{17}
$$

However, in the case of persistent, non-vanishing perturbations in the unactuated component (i.e., *δθu*(*t*) ≠ 0), there *does not exist any control action* that can completely damp out *θ*(*t*) while keeping the cargo stationary right above the destination. Suppose that there exists such a controller *u*′ (*t*) that could eliminate the cargo swing, namely,

$$
\Theta(t) = 0, \dot{\theta}(t) = 0 \Rightarrow \phi\_3(t) = -g \tan \theta(t) = 0,\tag{18}
$$

and make the cargo stay stationary at the destination in the sense that

*f*1

$$
\boldsymbol{\varphi}\_{1}(\mathbf{t}) = \boldsymbol{p}\_{\mathrm{d}\boldsymbol{t}'} \dot{\boldsymbol{\varphi}}\_{1}(\mathbf{t}) = \mathbf{0}, \boldsymbol{\varphi}\_{2}(\mathbf{t}) = \mathbf{0}, \dot{\boldsymbol{\varphi}}\_{2}(\mathbf{t}) = \mathbf{0}, \forall \mathbf{t} \ge \mathbf{t}\_{\mathrm{|}\boldsymbol{\xi}'} \tag{19}
$$

with *tf*<sup>2</sup> being the settling time, then it would follow, by inserting Eq. (18) and Eq. (19) into the second equation of Eq. (14), that

$$
\delta\_{\rho\_0} = \dot{\varphi}\_2 - \phi\_3 = 0,\tag{20}
$$

which obviously contradicts with the fact that *δθu* ≠ 0; thus the existence of such a controller *u*′ (*t*) is impossible. This fact illustrates the great challenge that will be faced with when controlling the crane system in the presence of persistent perturbations in the unactuated dynamics. On the other hand, since *δθu*(*t*) is usually unknown, the control problem becomes even more challenging.

Based on the analysis claimed above, in accordance with the fact whether *δθu* in the unactuated dynamics is vanishing or not, the control objective of this chapter is stated as follows:

• **Case 1.** *Non-vanishing perturbations in the unactuated dynamics.* Drive the unactuated cargo to the desired destination and keep it stationary over the destination thereafter, that is,

$$\lim\_{t \to \ast} \varphi\_1(t) = p\_{\text{d}t'} \lim\_{t \to \ast} \dot{\varphi}\_1(t) = 0. \tag{21}$$

applicable. As a means to achieve the aforementioned control objective, it is required to figure out a suitable strategy that can deal with *δθu*(*t*). Toward this end, before controller development, we will first construct an augmented observer which can recover the lumped perturbation term *δθu*(*t*) appearing in the unactuated dynamics. Then, we treat *δθu*(*t*) as an augmented state variable. The benefit of doing so is that the perturbation observer design procedure would become more concise and clear. By following this line, the augmented error system for

Eq. (24) is established as follows:

⎧

*e*̇ <sup>1</sup> = *e*<sup>2</sup> ,

*e*̇

*e*̇ <sup>3</sup> = *e*<sup>4</sup> ,

*e*̇

*e*̇ <sup>5</sup> = *e*<sup>6</sup>

*e*̇ <sup>6</sup> = *e*<sup>7</sup> ,

⋮ *e*̇ *<sup>n</sup>*+1 = 0, *y* = *e*<sup>1</sup>

<sup>2</sup> = *e*<sup>3</sup> + *e*<sup>5</sup> − *r*¨*<sup>x</sup>*

where we have considered *δθu*(*t*) as an augmented state variable *e*<sup>5</sup>

,

<sup>4</sup> = −*g*(*v* + *δ<sup>x</sup>* + *δ<sup>a</sup>*) sec2 *θ* − 2*g θ*

̇ <sup>2</sup> sec2 *θ* tan*θ*,

(25)

301

(26)

(27)

(*t*) and its derivatives as

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383

(*t*) is regarded as the

(*t*). In order to reduce the computational com-

,

(*t*), ⋯, *en* + 1(*t*), and *y*(*t*) is the corresponding system output signal. In this chapter, the

a reduced-order perturbation observer. For this purpose, consider the following subsystem:

<sup>2</sup> = *e*<sup>3</sup> + *e*<sup>5</sup> − *r*¨*<sup>x</sup>*

new output. It is not difficult to check that the reduced-order augmented-state system shown in Eq. (26) is observable, and the detailed analysis can be found in Appendix A. Based on the structure of Eq. (26), we design the following reduced-order augmented-state observer:

<sup>5</sup> − *r*¨*<sup>x</sup>* − *λ*2(*e*̂

<sup>2</sup> − *y*′ ),

<sup>2</sup> − *y*′

<sup>2</sup> − *y*′ ), ,

<sup>2</sup> − *y*′ ),

),

⎧

*e*̇

*e*̇ <sup>5</sup> = *e*<sup>6</sup> ,

*e*̇ <sup>6</sup> = *e*<sup>7</sup> ,

⋮ *e*̇ *<sup>n</sup>*+1 = 0, *y*' = *e*<sup>2</sup> ,

⎪ ⎨

⎪ ⎩

which is part of the augmented error system shown in Eq. (25), where *y*′

⎧ ⎪ ⎨ ⎪ ⎩

*e*̂ ̇

*e*̂ ̇ <sup>5</sup> = *e*̂

*e* ̇

⋮ *e*̂ ̇

<sup>2</sup> = *e*<sup>3</sup> + *e*̂

*<sup>n</sup>*+1 = −*λn*+1(*e*̂

<sup>6</sup> − *λ*5(*e*̂

̂ 6 <sup>=</sup> *<sup>e</sup>*̂ <sup>7</sup> − *λ*6(*e*̂

(*t*) are measurable, and we merely need to fabricate an observer with

(*t*) and *θθa*(*t*) are unavailable for feedback, we intend to construct

⎪

⎨

⎪

⎩

the aim of recovering the lumped perturbation*e*<sup>5</sup>

*e*6 (*t*), *e*<sup>7</sup>

signals *e*<sup>1</sup>

(*t*), *e*<sup>3</sup>

plexity, noting also that *θ<sup>x</sup>*

(*t*) and *e*<sup>4</sup>

• **Case 2.** *Vanishing or no/negligible perturbations in the unactuated dynamics.* Drive both the trolley and the unactuated cargo to the desired destination, in the sense that

$$\lim\_{t \to \omega} \varrho\_1(t) = p\_{\text{d}t} \lim\_{t \to \omega} \mathbf{x}(t) = p\_{\text{d}t'} \lim\_{t \to \omega} \dot{\vartheta}\_1(t) = 0,\\ \lim\_{t \to \omega} \dot{\mathbf{x}} = 0 \Rightarrow \lim\_{t \to \omega} \theta(t) = 0, \lim\_{t \to \omega} \dot{\theta}(t) = 0. \tag{22}$$

To achieve the control objective, together with Eq. (16), let the following error signals be defined:

$$e\_1 = \ \rho\_1 - r\_{s'} \\ e\_2 = \ \rho\_2 - \dot{r}\_{s'} \\ e\_3 = \ \phi\_{s'} \\ e\_4 = \phi\_{s'} \tag{23}$$

Thus, we are led to the following open-loop error system:

$$\begin{cases} \dot{e}\_1 = e\_{2'} \\ \dot{e}\_2 = e\_3 + \delta\_{\theta u} - \ddot{r}\_{x'} \\ \dot{e}\_3 = e\_{4'} \\ \dot{e}\_4 = -g(v + \delta\_x + \delta\_{\theta u})\sec^2\theta - 2g\,\dot{\theta}^2\sec^2\theta\tan\theta \end{cases} \tag{24}$$

which is the basis for the observer-controller design and analysis in the section that follows.

#### **3. Main results**

In order to achieve the control objective claimed in the previous section, we will propose a perturbation observer-based robust control scheme. More precisely, to deal with the unactuated unknown persistent perturbations, an augmented-state observer will be constructed. Then, we will present a novel robust control law, which can achieve superior control performance and provide the corresponding theoretical stability analysis.

#### **3.1. Observer design**

The fact that the perturbation term *δθu*(*t*) is *unactuated and unknown* brings much difficulty for the controller design and analysis and it makes traditional robust control methods not applicable. As a means to achieve the aforementioned control objective, it is required to figure out a suitable strategy that can deal with *δθu*(*t*). Toward this end, before controller development, we will first construct an augmented observer which can recover the lumped perturbation term *δθu*(*t*) appearing in the unactuated dynamics. Then, we treat *δθu*(*t*) as an augmented state variable. The benefit of doing so is that the perturbation observer design procedure would become more concise and clear. By following this line, the augmented error system for Eq. (24) is established as follows:

controlling the crane system in the presence of persistent perturbations in the unactuated dynamics. On the other hand, since *δθu*(*t*) is usually unknown, the control problem becomes

Based on the analysis claimed above, in accordance with the fact whether *δθu* in the unactuated dynamics is vanishing or not, the control objective of this chapter is stated as follows:

• **Case 1.** *Non-vanishing perturbations in the unactuated dynamics.* Drive the unactuated cargo to the desired destination and keep it stationary over the destination thereafter, that is,

(*t*) <sup>=</sup> *pdx*, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>φ</sup>*

ley and the unactuated cargo to the desired destination, in the sense that

̇ 1

• **Case 2.** *Vanishing or no/negligible perturbations in the unactuated dynamics.* Drive both the trol-

To achieve the control objective, together with Eq. (16), let the following error signals be

*x* , *e*<sup>3</sup> = *ϕ*<sup>3</sup>

, *e*<sup>2</sup> = *φ*<sup>2</sup> − *r*̇

,

<sup>4</sup> = −*g*(*v* + *δ<sup>x</sup>* + *δ<sup>a</sup>*) sec2 *θ* − 2*g θ*

which is the basis for the observer-controller design and analysis in the section that follows.

In order to achieve the control objective claimed in the previous section, we will propose a perturbation observer-based robust control scheme. More precisely, to deal with the unactuated unknown persistent perturbations, an augmented-state observer will be constructed. Then, we will present a novel robust control law, which can achieve superior control perfor-

The fact that the perturbation term *δθu*(*t*) is *unactuated and unknown* brings much difficulty for the controller design and analysis and it makes traditional robust control methods not

̇ 1

(*t*) <sup>=</sup> 0, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>x</sup>*̇ <sup>=</sup> <sup>0</sup> <sup>⇒</sup> lim*<sup>t</sup>*→<sup>∞</sup> *<sup>θ</sup>*(*t*) <sup>=</sup> 0, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>θ</sup>*

, *e*<sup>4</sup> = *ϕ*<sup>4</sup>

̇ <sup>2</sup> sec2 *θ* tan*θ*,

(*t*) = 0. (21)

̇

. (23)

(*t*) = 0. (22)

(24)

even more challenging.

300 Adaptive Robust Control Systems

lim*<sup>t</sup>*→<sup>∞</sup> *φ*<sup>1</sup>

defined:

**3. Main results**

**3.1. Observer design**

lim*<sup>t</sup>*→<sup>∞</sup> *φ*<sup>1</sup>

*e*<sup>1</sup> = *φ*<sup>1</sup> − *rx*

(*t*) <sup>=</sup> *pdx*, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>x</sup>*(*t*) <sup>=</sup> *pdx*, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>φ</sup>*

Thus, we are led to the following open-loop error system:

<sup>3</sup> = *e*<sup>4</sup> ,

<sup>2</sup> = *e*<sup>3</sup> + *δ<sup>u</sup>* − *r*¨*<sup>x</sup>*

mance and provide the corresponding theoretical stability analysis.

*<sup>e</sup>*̇

⎧ ⎪ ⎨ ⎪ ⎩

*e*̇ <sup>1</sup> = *e*<sup>2</sup> ,

*e*̇

*e*̇

$$\begin{aligned} \begin{cases} e\_1 &= e\_{\varphi} \\ e\_2 &= e\_3 + e\_5 - \ddot{r}\_{\varphi} \\ e\_3 &= e\_{\varphi} \\ e\_4 &= -g(\upsilon + \delta\_{\dot{x}} + \delta\_{\dot{u}}) \sec^2 \theta - 2g \,\theta^2 \sec^2 \theta \tan \theta, \\ e\_5 &= e\_{\varphi} \\ e\_6 &= e\_{\varphi} \\ \vdots \\ e\_{n+1} &= 0, \\ y &= e\_1 \end{cases} \end{aligned} \tag{25}$$

where we have considered *δθu*(*t*) as an augmented state variable *e*<sup>5</sup> (*t*) and its derivatives as *e*6 (*t*), *e*<sup>7</sup> (*t*), ⋯, *en* + 1(*t*), and *y*(*t*) is the corresponding system output signal. In this chapter, the signals *e*<sup>1</sup> (*t*), *e*<sup>3</sup> (*t*) and *e*<sup>4</sup> (*t*) are measurable, and we merely need to fabricate an observer with the aim of recovering the lumped perturbation*e*<sup>5</sup> (*t*). In order to reduce the computational complexity, noting also that *θ<sup>x</sup>* (*t*) and *θθa*(*t*) are unavailable for feedback, we intend to construct a reduced-order perturbation observer. For this purpose, consider the following subsystem:

$$\begin{cases} e\_{\,\,2} = e\_{\,\,3} + e\_{\,\,5} - \ddot{r}\_{\,\,\nu'}\\ e\_{\,\,5} = e\_{\,\,6'}\\ e\_{\,\,6} = e\_{\,\,\nu'}\\ \vdots\\ e\_{\,\,n+1} = 0,\\ y^{\,\,'} = e\_{\,\,2'} \end{cases} \tag{26}$$

which is part of the augmented error system shown in Eq. (25), where *y*′ (*t*) is regarded as the new output. It is not difficult to check that the reduced-order augmented-state system shown in Eq. (26) is observable, and the detailed analysis can be found in Appendix A. Based on the structure of Eq. (26), we design the following reduced-order augmented-state observer:

$$\begin{cases} \dot{\mathbf{e}}\_{2} = \mathbf{e}\_{3} + \hat{\mathbf{e}}\_{5} - \vec{\mathbf{r}}\_{\times} - \lambda\_{2} \langle \hat{\mathbf{e}}\_{2} - \mathbf{y}' \rangle, \\ \dot{\mathbf{e}}\_{5} = \hat{\mathbf{e}}\_{6} - \lambda\_{5} \langle \hat{\mathbf{e}}\_{2} - \mathbf{y}' \rangle, \\ \dot{\mathbf{e}}\_{6} = \hat{\mathbf{e}}\_{7} - \lambda\_{6} \langle \hat{\mathbf{e}}\_{2} - \mathbf{y}' \rangle, \\ \vdots \\ \dot{\mathbf{e}}\_{n+1} = -\lambda\_{n+1} \langle \hat{\mathbf{e}}\_{2} - \mathbf{y}' \rangle. \end{cases} \tag{27}$$

where *λ*<sup>2</sup> , *λ*<sup>5</sup> , *λ*<sup>6</sup> , ⋯, *λ<sup>n</sup>* + 1 denote the observer gains. Define the following error signals:

$$
\xi\_{\parallel} = \hat{e}\_{\parallel} - e\_{\rho} \text{ i } = \text{~2,5,6,} \cdots \text{;} \text{n} + 1,\tag{28}
$$

and denote the corresponding error vector by

$$\xi(t) = \begin{bmatrix} \xi\_2(t) \ \xi\_2(t) \ \xi\_6(t) \ \cdots \ \xi\_{n+1}(t) \end{bmatrix}^\top. \tag{29}$$

Then, one can subtract Eq. (26) from Eq. (27) to derive the following observer error system:

$$
\dot{\xi} = \mathsf{T}\,\mathsf{L}\,\xi.\tag{30}
$$

exp(*Ωt*) = *Γ* exp(*Λt*)*Γ*<sup>−</sup><sup>1</sup> ⇒ *ξ* = *Γ* exp(*Λt*)*Γ*<sup>−</sup><sup>1</sup> *ξ*(0). (35)

‖*<sup>m</sup>*<sup>∞</sup>

expression for *Ω*. Further, with the aid of such software as MATLAB, it is easy to calculate

To achieve robust control in the presence of uncertainties or external perturbations, we will develop a new observer-based sliding mode controller. The fundamental idea of the sliding mode control method is to construct a sliding manifold (surface) on which the system state is convergent and then develop a suitable control law that renders the state reaches the manifold within finite time. Traditionally, the key step is constructing an appropriate sliding surface,

However, the major drawback of most currently available sliding mode control methods is that they are merely capable of tackling uncertainties or perturbations in the actuated part, and when uncertainties or perturbations are present in the unactuated component, their performance will degrade significantly and even become unstable. To illustrate this point, we will show some brief analysis for the conventional sliding mode control approach. More precisely, for the open-loop error system shown in Eq. (24), one will design the conventional

⋅ ‖(0)‖<sup>2</sup>

‖(0)‖<sup>2</sup> <sup>≤</sup> (*<sup>n</sup>* <sup>−</sup> 2) exp(*λ*max *<sup>t</sup>*) <sup>⋅</sup> ‖*Γ*‖*<sup>m</sup>*<sup>∞</sup>

<sup>⋅</sup> ‖*Γ*<sup>−</sup>1‖*<sup>m</sup>*<sup>∞</sup>

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383

⋅ ‖*ξ*(0)‖<sup>2</sup>

, (38)

 + *γs*<sup>3</sup>

, *i* . *e* . ,*ζ*(*t*) = 0,

*f* 3

 ≤ ‖*A*‖*m* ⋅ ‖*x*‖*v*, where *A* ∈ *R <sup>n</sup>* × *<sup>n</sup>*

 = 0 is

and

, *λ*<sup>5</sup> , *λ*<sup>6</sup>

denotes the Euclidean norm, ‖⋅‖*<sup>m</sup>*∞ represents the *m*∞-

<sup>⋅</sup> ‖*Γ*<sup>−</sup>1‖*<sup>m</sup>*<sup>∞</sup>

(*t*)∣, as shown in Eq. (37), can be computed without

. It is further implied from Eq. (36) that

⋅ ‖(0)‖<sup>2</sup>

. (37)

, ⋯, *λ<sup>n</sup>* + 1 and the

, (36)

303

Taking the Euclidean norm for both sides of Eq. (35), we are led to the following results:

‖‖<sup>2</sup> = ‖*Γ* exp(*t*) *Γ*<sup>−</sup><sup>1</sup> (0)‖<sup>2</sup> ≤ ‖*Γ* exp(*t*) *Γ*<sup>−</sup><sup>1</sup>

<sup>⋅</sup> ‖*Γ*<sup>−</sup>1‖*<sup>m</sup>*<sup>∞</sup>

‖*Γ*‖*<sup>m</sup>*<sup>∞</sup> ⋅ ‖*Γ*−1‖*<sup>m</sup>*∞; hence, the bound for ∣*ξ<sup>i</sup>*

**3.2. Controller development and stability analysis**

sliding manifold, denoted by *ζ*(*t*) in the following fashion:

*ζ* = *e*<sup>1</sup> + *α e*<sup>2</sup> + *β e*<sup>3</sup> + *γ e*<sup>4</sup>

where *λ*max = maxi = 1, 2, ⋯, n - 2{*λ*<sup>i</sup>

norm for matrices1

∣ *ξ<sup>i</sup>*

difficulty.

*ζ* ̇

1

2

(*t*) = 0, ∀*t* ≥ *t*

*f* 3

A matrix norm ‖⋅‖*<sup>m</sup>* ∈ *R<sup>n</sup>* × *<sup>n</sup>*

For a square matrix *A* = (*aij*)*<sup>n</sup>* × *<sup>n</sup>* ∈ *R<sup>n</sup>* × *<sup>n</sup>*

<sup>≤</sup> ‖*Γ*‖*<sup>m</sup>*<sup>∞</sup>

}, ‖⋅‖<sup>2</sup>

, which are compatible norms2

Using the pole assignment technique, one can derive the values for *λ*<sup>2</sup>

(*t*) <sup>∣</sup> <sup>≤</sup> ‖ *<sup>ξ</sup>*‖<sup>2</sup> <sup>≤</sup> ¯¯*<sup>ξ</sup>* <sup>≜</sup> (*<sup>n</sup>* <sup>−</sup> 2) exp(*λ*max *<sup>t</sup>*) <sup>⋅</sup> ‖*Γ*‖*<sup>m</sup>*<sup>∞</sup>

and the corresponding controller can usually be obtained straightforwardly.

where *α*, *β*, and *γ* are sliding slopes chosen such that the polynomial 1 + *αs* + *βs*<sup>2</sup>

drives the system state variables to *ζ*(*t*) such that *ζ*(*t*) = 0 after certain finite time *t*

is said to be compatible with a vector norm ‖⋅‖*<sup>v</sup>* ∈ *R<sup>n</sup>*

*x* ∈ *R*. It is not difficult to verify that the *<sup>m</sup>*∞-norm for matrices is compatible with the Euclidean norm for vectors.

Hurwitz, with *s* being the complex variable. It is not difficult to design a control law that

. By recursively using the first three equations in Eq. (24) and regrouping the

if ‖*Ax*‖*<sup>v</sup>*

, ‖*A*‖*m*∞ = *n*max*<sup>i</sup>*, *<sup>j</sup>* ∣ *aij*∣ is defined as the *m*∞-norm for *A*.

<sup>⋅</sup> ‖exp(*t*)‖*<sup>m</sup>*<sup>∞</sup>

where Ω ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) is defined as:

$$
\boldsymbol{\Omega} = \begin{pmatrix}
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
\end{pmatrix} \tag{31}
$$

As stated previously, the system shown in Eq. (26) is observable. Hence, without difficulty, we are admitted to choose a proper set of *λ*<sup>2</sup> , *λ*<sup>5</sup> , *λ*<sup>6</sup> , ⋯, *λ<sup>n</sup>* + 1 conveniently via pole placement, such that Ω is a Hurwitz matrix with *the eigenvalues' real parts being different from each other.* In this sense,

$$
\xi\_{\;i} = \hat{e}\_{\;i} - e\_{\;i} \to 0,\\
i = 2, 5, 6, \cdots, n+1,\tag{32}
$$

exponentially fast, which indicates that the designed perturbation observer shown in Eq. (27) can online recover the perturbations.

In addition, it can be obtained from Eq. (30) that the trajectories of the observer error signals are represented by

$$
\xi \equiv \exp(\Omega t) \xi(0). \tag{33}
$$

Since we have rendered, by proper pole placement, that the poles (i.e., the eigenvalues of Ω) of the closed-loop system shown in Eq. (30) have different negative real parts, there exists an invertible matrix *Γ* ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) that can transform Ω into a diagonal matrix, that is,

$$
\Gamma^{-1}\Box\Gamma = \Lambda,\tag{34}
$$

where *Λ* = diag {*λ*<sup>1</sup> , *λ*<sup>2</sup> , ⋯, *λ*n - 2} with *λ<sup>i</sup>* , *i* = 1, 2, ⋯, *n* − 2 being the (*n* − 2) eigenvalues of *Γ*. Therefore, we can rewrite the exponential matrix exp(Ω*t*) and *ξ*(*t*) as [39]

$$\exp(\Omega t) = \Gamma \exp(\Lambda t) \Gamma^{-1} \Rightarrow \xi = \Gamma \exp(\Lambda t) \Gamma^{-1} \xi(0). \tag{35}$$

Taking the Euclidean norm for both sides of Eq. (35), we are led to the following results:

$$\begin{aligned} \text{Taking the Euclidean norm for both sides of Eq. (35), we are led to the following results:}\\ \|\{\xi\|\}\_2 &= \left\|\Gamma \exp(\Lambda t)\Gamma^{-1}\xi(0)\right\|\_2 \le \left\|\Gamma \exp(\Lambda t)\Gamma^{-1}\right\|\_{\mathfrak{u}\_-} \cdot \left\|\xi(0)\right\|\_2\\ &\le \left\|\Gamma\right\|\_{\mathfrak{u}\_-} \cdot \left\|\Gamma^{-1}\right\|\_{\mathfrak{u}\_-} \cdot \left\|\exp(\Lambda t)\right\|\_{\mathfrak{u}\_-} \left\|\xi(0)\right\|\_2 \le (n-2) \exp(\Lambda\_{\text{max}}t) \cdot \left\|\Gamma\right\|\_{\mathfrak{u}\_-} \cdot \left\|\xi(0)\right\|\_{\mathfrak{u}\_-} \end{aligned} \tag{36}$$

where *λ*max = maxi = 1, 2, ⋯, n - 2{*λ*<sup>i</sup> }, ‖⋅‖<sup>2</sup> denotes the Euclidean norm, ‖⋅‖*<sup>m</sup>*∞ represents the *m*∞ norm for matrices1 , which are compatible norms2 . It is further implied from Eq. (36) that

$$\parallel \ \xi(\mathfrak{t}) \mid \leq \parallel \ \xi \|\_{2} \leq \overline{\xi} \ \triangleq \ \mbox{( $n-2$ )} \ \mbox{exp}\{\lambda\_{\max} t\} \cdot \left\Vert \Gamma \right\Vert\_{n\_{\ast}} \cdot \left\Vert \Gamma^{-1} \right\Vert\_{n\_{\ast}} \cdot \left\Vert \xi(\mathfrak{0}) \right\Vert\_{2}. \tag{37}$$

Using the pole assignment technique, one can derive the values for *λ*<sup>2</sup> , *λ*<sup>5</sup> , *λ*<sup>6</sup> , ⋯, *λ<sup>n</sup>* + 1 and the expression for *Ω*. Further, with the aid of such software as MATLAB, it is easy to calculate ‖*Γ*‖*<sup>m</sup>*<sup>∞</sup> ⋅ ‖*Γ*−1‖*<sup>m</sup>*∞; hence, the bound for ∣*ξ<sup>i</sup>* (*t*)∣, as shown in Eq. (37), can be computed without difficulty.

#### **3.2. Controller development and stability analysis**

where *λ*<sup>2</sup>

this sense,

are represented by

where *Λ* = diag {*λ*<sup>1</sup>

, *λ*<sup>5</sup> , *λ*<sup>6</sup>

302 Adaptive Robust Control Systems

*ξ<sup>i</sup>* = *e*̂

*ξ*(*t*) = [*ξ*<sup>2</sup>

*ξ*

where Ω ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) is defined as:

Ω =

*ξ<sup>i</sup>* = *e*̂

can online recover the perturbations.

, *λ*<sup>2</sup>

, ⋯, *λ*n - 2} with *λ<sup>i</sup>*

fore, we can rewrite the exponential matrix exp(Ω*t*) and *ξ*(*t*) as [39]

we are admitted to choose a proper set of *λ*<sup>2</sup>

and denote the corresponding error vector by

, ⋯, *λ<sup>n</sup>* + 1 denote the observer gains. Define the following error signals:

, *i* = 2, 5, 6, ⋯,*n* + 1, (28)

*̇* = Ω *ξ*, (30)

<sup>⊤</sup>. (29)

. (31)

, ⋯, *λ<sup>n</sup>* + 1 conveniently via pole placement,

*<sup>i</sup>* − *ei*

(*t*) *ξ*<sup>5</sup>

⎛

⎜

⎝

(*t*) *ξ*<sup>6</sup>

Then, one can subtract Eq. (26) from Eq. (27) to derive the following observer error system:

−*λ*<sup>2</sup> 1 0 ⋯ 0 0

−*λ*<sup>5</sup> 0 1 ⋯ 0 0 −*λ*<sup>6</sup> 0 0 ⋱ 0 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ −*λ<sup>n</sup>* 0 0 ⋯ 0 1 −*λ<sup>n</sup>*+1 0 0 ⋯ 0 0

As stated previously, the system shown in Eq. (26) is observable. Hence, without difficulty,

, *λ*<sup>5</sup> , *λ*<sup>6</sup>

such that Ω is a Hurwitz matrix with *the eigenvalues' real parts being different from each other.* In

exponentially fast, which indicates that the designed perturbation observer shown in Eq. (27)

In addition, it can be obtained from Eq. (30) that the trajectories of the observer error signals

*ξ* = exp(Ω*t*)(0). (33)

Since we have rendered, by proper pole placement, that the poles (i.e., the eigenvalues of Ω) of the closed-loop system shown in Eq. (30) have different negative real parts, there exists an

*Γ*<sup>−</sup><sup>1</sup> Ω = *Λ*, (34)

invertible matrix *Γ* ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) that can transform Ω into a diagonal matrix, that is,

(*t*) ⋯ *ξ<sup>n</sup>*+1

(*t*) ]

⎞

⎟

⎠

*<sup>i</sup>* − *ei* → 0, *i* = 2, 5, 6, ⋯,*n* + 1, (32)

, *i* = 1, 2, ⋯, *n* − 2 being the (*n* − 2) eigenvalues of *Γ*. There-

To achieve robust control in the presence of uncertainties or external perturbations, we will develop a new observer-based sliding mode controller. The fundamental idea of the sliding mode control method is to construct a sliding manifold (surface) on which the system state is convergent and then develop a suitable control law that renders the state reaches the manifold within finite time. Traditionally, the key step is constructing an appropriate sliding surface, and the corresponding controller can usually be obtained straightforwardly.

However, the major drawback of most currently available sliding mode control methods is that they are merely capable of tackling uncertainties or perturbations in the actuated part, and when uncertainties or perturbations are present in the unactuated component, their performance will degrade significantly and even become unstable. To illustrate this point, we will show some brief analysis for the conventional sliding mode control approach. More precisely, for the open-loop error system shown in Eq. (24), one will design the conventional sliding manifold, denoted by *ζ*(*t*) in the following fashion:

$$
\zeta = e\_1 + \alpha \, e\_2 + \beta \, e\_3 + \gamma \, e\_{4'} \tag{38}
$$

where *α*, *β*, and *γ* are sliding slopes chosen such that the polynomial 1 + *αs* + *βs*<sup>2</sup>  + *γs*<sup>3</sup>  = 0 is Hurwitz, with *s* being the complex variable. It is not difficult to design a control law that drives the system state variables to *ζ*(*t*) such that *ζ*(*t*) = 0 after certain finite time *t f* 3 , *i* . *e* . ,*ζ*(*t*) = 0, *ζ* ̇ (*t*) = 0, ∀*t* ≥ *t f* . By recursively using the first three equations in Eq. (24) and regrouping the

3

<sup>1</sup> For a square matrix *A* = (*aij*)*<sup>n</sup>* × *<sup>n</sup>* ∈ *R<sup>n</sup>* × *<sup>n</sup>* , ‖*A*‖*m*∞ = *n*max*<sup>i</sup>*, *<sup>j</sup>* ∣ *aij*∣ is defined as the *m*∞-norm for *A*.

<sup>2</sup> A matrix norm ‖⋅‖*<sup>m</sup>* ∈ *R<sup>n</sup>* × *<sup>n</sup>* is said to be compatible with a vector norm ‖⋅‖*<sup>v</sup>* ∈ *R<sup>n</sup>* if ‖*Ax*‖*<sup>v</sup>*  ≤ ‖*A*‖*m* ⋅ ‖*x*‖*v*, where *A* ∈ *R <sup>n</sup>* × *<sup>n</sup>* and *x* ∈ *R*. It is not difficult to verify that the *<sup>m</sup>*∞-norm for matrices is compatible with the Euclidean norm for vectors.

resulting terms, one can derive from *ζ*(*t*) = 0 and Eq. (38) that the state variable *e*<sup>1</sup> (*t*) is dominated by the following dynamics on the sliding manifold:

$$
\dot{e}\_1 + \alpha \,\dot{e}\_1 + \beta \,\ddot{e}\_1 + \gamma \,e^{(3)}\_1 = -\beta (\ddot{r}\_x - \delta\_{\alpha}) - \gamma \left(r\_x^{(3)} - \delta\_{\alpha}\right). \tag{39}
$$

denote the standard sign function. We can further substitute Eq. (43) into Eq. (8) to obtain the

<sup>5</sup> − *r*¨*x*) − *γ*[*rx*

<sup>+</sup> *<sup>β</sup>* [*e*<sup>4</sup> <sup>+</sup> *<sup>e</sup>*̂

+ *ku* sign(ϵ)

*r* . (46)

The main results for the proposed control scheme are summarized by the theorem that follows. *Theorem 1.* The designed control law shown in Eq. (46), together with the reduced-order augmented-state observer shown in Eq. (27), can achieve the control objective claimed by **Case 1** in the case of unactuated persistent non-vanishing disturbances or **Case 2** if the unactuated

*Proof:* Consider *V*(*t*) defined in Eq. (41) as a Lyapunov function candidate, and its time deriva-

5) − *β ku* sign(*ε*) − *g* sec2 *θ*[*ku* sign(*ε*) + *δ<sup>x</sup>* + *δ<sup>a</sup>*]

upon the use of the relationship in Eq. (28). Then, the following results are straightforward

<sup>≤</sup> <sup>−</sup>(*<sup>β</sup> ku* <sup>−</sup> *<sup>α</sup>*|*ξ*<sup>5</sup> |) <sup>∣</sup> *<sup>ε</sup>* <sup>∣</sup> <sup>−</sup>*g* sec 2 *<sup>θ</sup>*(*ka* <sup>−</sup> (|*δ<sup>x</sup>* |+|*δ<sup>a</sup>*|)) <sup>∣</sup> *<sup>ε</sup>* <sup>∣</sup>

2*V*, (49)

where the gain conditions shown in Eq. (44) have been utilized. The conclusion of Eq. (49) indicates that *V*(*t*), and hence *ε*(*t*), converges to zero in finite time. Further, on the sliding manifold where *ε*(*t*) = 0, the system state variables satisfy the following dynamic equation array:

As by pole assignment, the matrix Ω ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) is Hurwitz, and *α*, *β* and *γ* also render

*<sup>ξ</sup>*

<sup>1</sup> + *β e*¨<sup>1</sup> + *γ e*<sup>1</sup>

*V*̇ = −*β ku* ∣ *ε* ∣ −<sup>5</sup> *ε* + *g* sec2 *θ*[*ku* |*ε*|+(*δ<sup>x</sup>* + *δ<sup>a</sup>*)*ε*]

\_\_\_

*e*<sup>1</sup> + *α e*̇

= <sup>−</sup><sup>5</sup> <sup>−</sup> *<sup>β</sup> ku* sign(*ε*) <sup>−</sup> *<sup>g</sup>* sec2 *<sup>θ</sup>*[*ku* sign(*ε*) <sup>+</sup> *<sup>δ</sup><sup>x</sup>* <sup>+</sup> *<sup>δ</sup><sup>a</sup>*], (48)

(3) = − <sup>5</sup> − <sup>6</sup>

 = 0 Hurwitz; it is clearly seen that the entire closed system Eq. (50) is exponen-

,

̇ <sup>=</sup> <sup>Ω</sup> *<sup>ξ</sup>.* (50)

(4) − *e*̂ <sup>7</sup> + *λ*6(*e*̂

]+*g* sec2 *θ*[2 *θ*

<sup>2</sup> − *e*2)]

̇ <sup>2</sup> tan*<sup>θ</sup>* <sup>−</sup> *ka* sign(ϵ)]}

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383 305

. (47)

(*t*) in Eq. (42) and regrouping the common terms,

ultimate control law as follows:

− *mL θ*

disturbances are vanishing/negligible.

*V*̇ = *εε*̇

By inserting Eq. (43) into the expression of *ε*̇

after the substitution of Eq. (48) into Eq. (47):

≤ −(*β ku* − *α*|*ξ*<sup>5</sup> |) √

one can obtain the following equation:

*<sup>ε</sup>*̇ <sup>=</sup> *<sup>α</sup>*(*e*<sup>5</sup> <sup>−</sup> *<sup>e</sup>*̂

{

tially stable at the equilibrium point, and hence

 + *γs*<sup>3</sup>

tive is given by

(*M* + *m* sin2 *θ*)*L* \_\_\_\_\_\_\_\_\_\_\_

<sup>6</sup> − *λ*5(*e*̂

*<sup>g</sup>* sec*<sup>θ</sup>* <sup>⋅</sup> {*e*<sup>2</sup> <sup>+</sup> *<sup>α</sup>*(*e*<sup>3</sup> <sup>+</sup> *<sup>e</sup>*̂

<sup>2</sup> − *e*2) − *rx*

̇ <sup>2</sup> sin*θ* − (*M* + *m*)*g* tan*θ* + *f*

(3)

*u* = −

1 + *αs* + *βs*<sup>2</sup>

Clearly, if the perturbation terms *δ<sup>u</sup>* (*t*), *δ* ̇ *u* (*t*) appearing in the unactuated dynamics are nonvanishing, *e*<sup>1</sup> (*t*) will never tend to zero.

As indicated from the above-mentioned analysis, to make sliding mode control applicable to crane systems with *unknown persistent perturbations* in the unactuated component, it is needed to construct a new sliding manifold to improve the robust performance of the control system. To do so, on the basis of the designed perturbation observer in the previous subsection, we design the following sliding manifold that will be used in the subsequent controller development:

$$
\varepsilon = e\_1 + \alpha \, e\_2 + \beta \left( e\_3 + \hat{e}\_5 - \vec{r}\_\times \right) + \gamma \left( e\_4 + \hat{e}\_6 - r\_\times^{(3)} \right). \tag{40}
$$

where *α*, *β*, *γ* are defined in Eq. (38) and *e*̂ 5 (*t*), *<sup>e</sup>*̂ 6 (*t*) are the observer-recovered signals for the lumped perturbation term [see Eq. (27)]. Before giving the expression for the auxiliary "control input" *v*(*t*), we first construct the following non-negative scalar function *V*(*t*):

$$V = \frac{1}{2}\varepsilon^2 = \frac{1}{2}\left[\varepsilon\_1 + \alpha \,\,\varepsilon\_2 + \beta \{\varepsilon\_3 + \hat{e}\_5 - \vec{r}\_\times\} + \gamma \,\,e\_4 \{\varepsilon\_4 + \hat{e}\_6 - r\_\times^3\}\right]^2. \tag{41}$$

The derivative of *ε*(*t*) with regard to time can be obtained as follows:

$$\begin{aligned} \text{The derivative of } \varepsilon(t) \text{ with regard to time can be obtained as follows:}\\ \varepsilon &= \varepsilon\_2 + a(\varepsilon\_3 + \varepsilon\_5 - \overline{r}\_\varepsilon) + \beta \left[\varepsilon\_4 + \widehat{e}\_\varepsilon - \lambda\_5(\widehat{e}\_z - \varepsilon\_2) - r\_\varepsilon^{(3)}\right] - \gamma \xi \{\upsilon + \delta\_\upsilon + \delta\_\alpha\} \sec^2 \theta \\ &- \gamma \left[2\mathfrak{g} \cdot \partial^2 \sec^2 \theta \,\tan \theta - \widehat{e}\_\gamma + \lambda\_\delta(\widehat{e}\_z - \varepsilon\_2) + r\_\chi^{(4)}\right] \end{aligned} \tag{42}$$

where Eq. (25) and Eq. (27) have been employed for implications. Then, in view of the structure of Eq. (42), *v*(*t*) is developed in the following fashion:

$$\begin{aligned} \left| v(t) \right> &\text{is developed in the following fashion:}\\ \left| v \right> &= \frac{1}{\mathcal{G}\mathcal{Y}\sec^{2}\theta} \cdot \left\{ e\_{2} + \alpha \{ e\_{3} + \hat{e}\_{5} - \check{r}\_{\times} \} - \gamma \left[ r\_{\times}^{(4)} - \hat{e}\_{7} + \lambda\_{\rm s} \{ \hat{e}\_{2} - e\_{2} \} \right] \\ &+ \beta \left[ e\_{4} + \hat{e}\_{6} - \lambda\_{\rm s} \{ \hat{e}\_{2} - e\_{2} \} - r\_{\times}^{(3)} + k\_{\rm u} \text{sign}(\varepsilon) \right] \right\} - 2 \,\partial^{2} \tan \theta + k\_{\rm s} \text{sign}(\varepsilon) \end{aligned} \tag{43}$$

where

$$k\_{\iota} \ge \overline{\delta}\_{\iota} + \overline{\delta}\_{\iota^{\rho}} k\_{\iota} > \frac{a}{\beta} \overline{\xi}. \tag{44}$$

are positive control gains [see Eqs. (7) and (37) for the definitions of ¯¯*<sup>δ</sup> <sup>x</sup>* ,¯¯*δ <sup>θ</sup>* , and ¯¯*ξ*], and

$$\begin{array}{c} \text{true} \text{ Assume common game (sc. cit. (\"\mu\nu\}\text{) nor the minimum of } v\_{\star}, v\_{\sigma'} \text{ must } \epsilon\_{\lambda} \text{ minus the maximum of } v\_{\star} \text{ and } \epsilon\_{\lambda}, \text{ respectively.}\\\\ \text{sign}(\star) = \begin{cases} \star/\text{ } | \; \star \mid \text{ } \star \mid \text{ } \star \text{ } 0, \\ 0, & \star = 0, \end{cases} \end{array} \tag{45}$$

denote the standard sign function. We can further substitute Eq. (43) into Eq. (8) to obtain the ultimate control law as follows:

resulting terms, one can derive from *ζ*(*t*) = 0 and Eq. (38) that the state variable *e*<sup>1</sup>

(3) = −*β*(*r*¨*<sup>x</sup>* − *δ<sup>u</sup>*) − *γ*(*rx*

As indicated from the above-mentioned analysis, to make sliding mode control applicable to crane systems with *unknown persistent perturbations* in the unactuated component, it is needed to construct a new sliding manifold to improve the robust performance of the control system. To do so, on the basis of the designed perturbation observer in the previous subsection, we design the following sliding manifold that will be used in the subsequent controller

^

lumped perturbation term [see Eq. (27)]. Before giving the expression for the auxiliary "con-

<sup>2</sup> − *e*2) − *rx*

where Eq. (25) and Eq. (27) have been employed for implications. Then, in view of the struc-

<sup>5</sup> − *r*¨*x*) − *γ*[*rx*

+ *ku* sign(*ε*)

*δθ*, *ku* > \_\_ *α β* ¯¯

⋆ /∣ ⋆ ∣, ⋆ ≠ 0,

+ *<sup>β</sup>*[*e*<sup>4</sup> <sup>+</sup> *<sup>e</sup>*̂

(3)

*δ<sup>x</sup>* + ¯¯

<sup>−</sup>*γ*[2*<sup>g</sup> <sup>θ</sup>*

<sup>2</sup> − *e*2) + *rx*

(3)

(4) − *e*̂ <sup>7</sup> + *λ*6(*e*̂

]} − 2 *θ*

(4)

5 (*t*), *<sup>e</sup>*̂ 6

trol input" *v*(*t*), we first construct the following non-negative scalar function *V*(*t*):

<sup>2</sup> [*e*<sup>1</sup> + *α e*<sup>2</sup> + *β*(*e*<sup>3</sup> + *e*̂

<sup>6</sup> − *λ*5(*e*̂

<sup>2</sup> − *e*2) − *rx*

are positive control gains [see Eqs. (7) and (37) for the definitions of ¯¯*<sup>δ</sup> <sup>x</sup>*

<sup>7</sup> + *λ*6(*e*̂

<sup>5</sup> − *r*¨*x*) + *γ*(*e*<sup>4</sup> + *e*

^ <sup>6</sup> − *rx* (3)

<sup>5</sup> − *r*¨*x*) + *γ e*4(*e*<sup>4</sup> + *e*̂

] − *g*(*v* + *δ<sup>x</sup>* + *δ<sup>a</sup>*) sec2 *θ*

<sup>2</sup> − *e*2)]

̇ <sup>2</sup> tan*θ* + *ka* sign(*ε*)

*ξ*. (44)

, and ¯¯*ξ*], and

,¯¯*δ <sup>θ</sup>*

0, <sup>⋆</sup> <sup>=</sup> 0, (45)

(3) − *δ* ̇

(*t*) appearing in the unactuated dynamics are non-

<sup>1</sup> + *β e*¨<sup>1</sup> + *γ e*<sup>1</sup>

(*t*), *δ* ̇ *u*

nated by the following dynamics on the sliding manifold:

(*t*) will never tend to zero.

*ε* = *e*<sup>1</sup> + *α e*<sup>2</sup> + *β*(*e*<sup>3</sup> + *e*

<sup>2</sup> *<sup>ε</sup>* <sup>2</sup> <sup>=</sup> \_\_1

ture of Eq. (42), *v*(*t*) is developed in the following fashion:

*<sup>g</sup>* sec2 *<sup>θ</sup>* <sup>⋅</sup> {*e*<sup>2</sup> <sup>+</sup> *<sup>α</sup>*(*e*<sup>3</sup> <sup>+</sup> *<sup>e</sup>*̂

<sup>6</sup> − *λ*5(*e*̂

̇ <sup>2</sup> sec2 *θ* tan*θ* − *e*̂

*v* = \_\_\_\_\_\_\_ <sup>1</sup>

*ka* ≥ ¯¯

sign(⋆) <sup>=</sup> {

The derivative of *ε*(*t*) with regard to time can be obtained as follows:

where *α*, *β*, *γ* are defined in Eq. (38) and *e*̂

*e*<sup>1</sup> + *α e*̇

304 Adaptive Robust Control Systems

vanishing, *e*<sup>1</sup>

development:

where

*V* = \_\_1

*ε*̇ = *e*<sup>2</sup> + *α*(*e*<sup>3</sup> + *e*<sup>5</sup> − *r*¨*x*) + *β*[*e*<sup>4</sup> + *e*̂

Clearly, if the perturbation terms *δ<sup>u</sup>*

(*t*) is domi-

*<sup>u</sup>*). (39)

). (40)

. (41)

(43)

(*t*) are the observer-recovered signals for the

], (42)

<sup>6</sup> − *rx* (3 )] 2

*u* = − (*M* + *m* sin2 *θ*)*L* \_\_\_\_\_\_\_\_\_\_\_ *<sup>g</sup>* sec*<sup>θ</sup>* <sup>⋅</sup> {*e*<sup>2</sup> <sup>+</sup> *<sup>α</sup>*(*e*<sup>3</sup> <sup>+</sup> *<sup>e</sup>*̂ <sup>5</sup> − *r*¨*x*) − *γ*[*rx* (4) − *e*̂ <sup>7</sup> + *λ*6(*e*̂ <sup>2</sup> − *e*2)] <sup>+</sup> *<sup>β</sup>* [*e*<sup>4</sup> <sup>+</sup> *<sup>e</sup>*̂ <sup>6</sup> − *λ*5(*e*̂ <sup>2</sup> − *e*2) − *rx* (3) + *ku* sign(ϵ) ]+*g* sec2 *θ*[2 *θ* ̇ <sup>2</sup> tan*<sup>θ</sup>* <sup>−</sup> *ka* sign(ϵ)]} − *mL θ* ̇ <sup>2</sup> sin*θ* − (*M* + *m*)*g* tan*θ* + *f r* . (46)

The main results for the proposed control scheme are summarized by the theorem that follows.

*Theorem 1.* The designed control law shown in Eq. (46), together with the reduced-order augmented-state observer shown in Eq. (27), can achieve the control objective claimed by **Case 1** in the case of unactuated persistent non-vanishing disturbances or **Case 2** if the unactuated disturbances are vanishing/negligible.

*Proof:* Consider *V*(*t*) defined in Eq. (41) as a Lyapunov function candidate, and its time derivative is given by

$$
\dot{V} = \varepsilon \dot{\varepsilon}.\tag{47}
$$

By inserting Eq. (43) into the expression of *ε*̇ (*t*) in Eq. (42) and regrouping the common terms, one can obtain the following equation:

$$\begin{aligned} \text{Once can obtain the following equation:}\\ \begin{aligned} \varepsilon &= \alpha (\varepsilon\_{\natural} - \hat{e}\_{\natural}) - \beta \, k\_{\natural} \text{sign}(\varepsilon) - \text{gy} \, \text{sec}^{2} \, \theta \left[ k\_{\natural} \text{sign}(\varepsilon) + \delta\_{\natural} + \delta\_{\text{sa}} \right] \\ &= -\alpha \sharp\_{\natural} - \beta \, k\_{\natural} \text{sign}(\varepsilon) - \text{gy} \, \text{sec}^{2} \, \theta \left[ k\_{\natural} \text{sign}(\varepsilon) + \delta\_{\natural} + \delta\_{\text{sa}} \right] \end{aligned} \tag{48}$$

upon the use of the relationship in Eq. (28). Then, the following results are straightforward after the substitution of Eq. (48) into Eq. (47):

$$\begin{split} \dot{V} &= -\beta \, k\_u \parallel \varepsilon \, \parallel - a \varepsilon\_5 \, \varepsilon + \text{gy} \, \text{sec}^2 \, \mathcal{O} \left[ k\_u \parallel \varepsilon \, \parallel + (\delta\_x + \delta\_{\theta u}) \varepsilon \right] \\ &\leq - \left( \beta \, k\_u - a \, \vert \, \varepsilon\_5 \, \vert \, \right) \parallel \varepsilon \, \parallel - \text{gy} \, \text{sec}^2 \, \mathcal{O} \left( k\_u - \left( \lfloor \delta\_x \rfloor + \lfloor \delta\_{\theta u} \rfloor \right) \right) \parallel \varepsilon \, \mid \\ &\leq - \left( \beta \, k\_u - a \, \vert \, \varepsilon\_5 \, \vert \right) \sqrt{2V} \, \end{split} \tag{49}$$

where the gain conditions shown in Eq. (44) have been utilized. The conclusion of Eq. (49) indicates that *V*(*t*), and hence *ε*(*t*), converges to zero in finite time. Further, on the sliding manifold where *ε*(*t*) = 0, the system state variables satisfy the following dynamic equation array:

 $\varepsilon\_{\nu}$  are the  $\varepsilon\_{\nu}$ -norm,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ ,  $\varepsilon\_{\nu}$ , 
$$\begin{cases} e\_{1} + \alpha \, \varepsilon\_{i} + \beta \, \overline{e}\_{1} + \gamma \, e\_{1}^{\varepsilon\_{0}} = -\beta \overline{e}\_{5} - \gamma \overline{e}\_{8'} \\ \xi = \Omega \ \overline{\xi}. \end{cases} \tag{50}$$

As by pole assignment, the matrix Ω ∈ *R*(*n* − 2) × (*<sup>n</sup>* − 2) is Hurwitz, and *α*, *β* and *γ* also render 1 + *αs* + *βs*<sup>2</sup>  + *γs*<sup>3</sup>  = 0 Hurwitz; it is clearly seen that the entire closed system Eq. (50) is exponentially stable at the equilibrium point, and hence

$$\begin{aligned} \dot{e}\_1(t) &= \dot{\varrho}\_1(t) - r\_x \to 0, \dot{e}\_2(t) = \dot{e}\_1(t) = \dot{\varrho}\_1(t) - r\_x \to 0 \\ &\Rightarrow \dot{e}\_2(t) = \ddot{e}\_1(t) \to 0, \ddot{e}\_2(t) = \dot{e}\_1^{\text{(3)}}(t) \to 0, \end{aligned} \tag{51}$$

exponentially fast, which indicates the cargo motion tracks the planned trajectory *rx* (*t*) in an exponential fashion. Since *rx* (*t*) tends to *pdx* within *tf*<sup>1</sup> [see Eq. (16)], it is easily shown that

$$\lim\_{l\to\ast} \varphi\_l(t) = p\_{d\ast} \lim\_{l\to\ast} \dot{\varphi}\_l(t) = 0,\tag{52}$$

• **Group 2.** The perturbations in the unactuated dynamics are vanishing or negligible. The

**Figures 2** and **3** show the simulation results of **Group 1** where the solid lines denote the simulation results and the dash lines denote the desired trajectories. In **Figure 2**, the perturbation *dθ*(*t*) is set as a constant value *dθ*(*t*) = 1, and in **Figure 3**, the perturbation is set as a time-varying function *dθ*(*t*) = 0.5 cos(0.1*t*). It can be seen from **Figures 2** and **3** that when there exist persistent (non-vanishing) perturbations in the unactuated dynamics, by applying the proposed controller, the unactuated cargo is driven to the desired destination and is kept stationary. Therefore, the objectives stated in **Case 1** [see Eq. (21)] are achieved effectively. By dealing with the robust control for crane systems when the perturbations are non-vanishing,

**Figure 4** shows the results of **Group 2**. It is clear that the proposed observer-based robust control method can achieve the objectives stated in **Case 2** [see Eq. (22)] that both the trolley

0 10 20 30 40 50

0 10 20 30 40 50

0 10 20 30 40 50

0 10 20 30 40 50

**Figure 2.** The simulation results of the proposed controller when *dθ*(*t*) = 1 (solid line – simulation results, dash line –

Time[s]

For all the cases, by setting the system parameters as *M* = 6kg, *m* = 2.5kg, *L* = 1.2 m, *g* = 9.8m/s<sup>2</sup>

 = 55, *λ*<sup>7</sup>

.

 = 25, *α* = 2, *β* = 1, *γ* = 0.2, *ε* = 0.01, *ku*

(*t*) = 3.5, the simulation results are obtained and

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383

, the

307

 = 0.1,

 = 60, *ka*

perturbation *dθ*(*t*) is set as a time-varying function *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>*

 = 30, *λ*<sup>6</sup>

the results of **Group 1** validate the robustness of the presented controller.

 = 10, *λ*<sup>5</sup>

and the to-be-tracked trajectory in Eq. (16) as *rx*

controller parameters as*λ*<sup>2</sup>

are shown in **Figures 2**–**4**.

0

x

x[m]

θ

> u

desired trajectory).

[N*·*m]

[rad]

+ Lθ [m]

4

0

10

0

−10

200

−200

0

2

2

4

which is just the result of Eq. (21). In addition, as *r*¨*<sup>x</sup>* (*t*), *r x* (3) (*t*) <sup>→</sup> 0 as *t* → 0 by definition [see Eq. (16)], it is implied by substituting the result of *e*̇ 2 (*t*) → 0 into the second and third equations of Eq. (24) that

$$
\varepsilon\_3 \to \bar{e}\_2 - \delta\_{\theta u} \to -\delta\_{\theta u'} \bar{e}\_3 \to \bar{e}\_2 - \delta\_{\theta u} \to -\delta\_{\theta u'} \bar{e}\_4 \to \bar{e}\_3 \to -\delta\_{\theta u'} \tag{53}
$$

wherein the conclusions in Eq. (52) have been employed. The results in Eq. (53) indicate that *e* 3 (*t*), *e*̇ 3 (*t*) are convergent to their respective equilibriums drifted by the unactuated perturbations. Thus, the result of **Case 1** stated in the control objective is proven.

Subsequently, we proceed to prove the result of **Case 2** where the perturbation term *δθu*(*t*) in the unactuated dynamics is vanishing [i.e., *δ<sup>u</sup>* <sup>→</sup> 0, *<sup>δ</sup>* ̇ *<sup>u</sup>* <sup>→</sup> 0] or negligible [i.e., *δ<sup>u</sup>* (*t*) = 0, *δ* ̇ *u* (*t*) = 0]. Therefore, in such cases, it is straightforward to indicate from Eq. (53) that

$$e\_3 = -g \tan \theta \to 0,\\ e\_4 = -g \, \dot{\theta} \sec^2 \theta \to 0 \, \Rightarrow \, \theta = 0,\\ \dot{\theta} = 0,\tag{54}$$

where the definitions in Eq. (10) and Eq. (23) have been used. According to the definition of *ϕ*1 (*t*) = *x*(*t*) + *Lθ*(*t*) given in Eq. (13), the results in Eq. (52) and Eq. (54) directly yield the following conclusions:

$$\lim\_{t \to \ast} \mathbf{x}(t) = p\_{dv'} \lim\_{t \to \ast} \dot{\mathbf{x}}(t) = 0. \tag{55}$$

Collecting up Eqs. (52, 54, 55), the results claimed in Eq. (22) of **Case 2** are hence proven. The entire theoretical proof for the theorem is completed.
