2. Robust adaptive control system design

#### 2.1. 3D overhead crane system modeling

Figure 1 shows the structure of the 3D overhead crane. The dynamic model of the overhead crane is as follows [26]:

$$M(q)\ddot{q} + \mathbb{C}(q, \dot{q})\dot{q} + g(q) = \tau \tag{1}$$

Mð Þ¼ q

0

BBBBBB@

C qð Þ¼ ; q\_

where

0

Figure 1. Structure of 3D overhead crane.

BBBBBB@

000 0 0 mhl

2

mc þ mh 0 mhl cos θ cosφ 0 mc þ mh þ mx mhl cos θ sinφ

<sup>2</sup> <sup>þ</sup> <sup>J</sup>

�mhl 2

mhl 2

1

CCCA

M qð Þ ; d q€ þ C qð Þ ; q\_; d q\_ þ ɡð Þ¼ q; d D uð Þ þ n qð Þ ; q\_; q€; d; t (6)

�mhl sin θ sin φ mhl sin θ cosφ 0

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768

1

CCCCCCA

(2)

337

(3)

(4)

1

CCCCCCA

<sup>2</sup> sin <sup>2</sup><sup>θ</sup> <sup>þ</sup> <sup>J</sup>

mhl

φ\_ sin θ cos θ

θ\_ sin θ cos θ

�mhlθ\_ cos <sup>θ</sup> sin<sup>φ</sup> � mhlφ\_ sin <sup>θ</sup> cos <sup>φ</sup> mhlθ\_ cos <sup>θ</sup> cos<sup>φ</sup> � mhlφ\_ sin <sup>θ</sup> sin <sup>φ</sup>

, q <sup>¼</sup> ð Þ <sup>x</sup>; <sup>y</sup>; <sup>θ</sup>;<sup>φ</sup> <sup>T</sup>: (5)

mhl cos θ cosφ mhl cos θ sinφ mhl

0 0 �mhlθ\_ sin <sup>θ</sup> cos<sup>φ</sup> � mhlφ\_ cos <sup>θ</sup> sin<sup>φ</sup> 0 0 �mhlθ\_ sin <sup>θ</sup> sin<sup>φ</sup> <sup>þ</sup> mhlφ\_ cos <sup>θ</sup> cos<sup>φ</sup>

φ\_ sin θ cos θ

�mhl sin θ sinφ mhl sin θ cosφ 0

ɡð Þ¼ q

<sup>τ</sup> <sup>¼</sup> ð Þ <sup>u</sup>1; <sup>u</sup>2; <sup>0</sup>; <sup>0</sup> <sup>T</sup>

In considering the system uncertainties, the model (1) is rewritten as the following:

0 0

0

BBB@

0

mhɡl sin θ

where

Figure 1. Structure of 3D overhead crane.

controllers are used for the crane systems to give the good performances with simple construction. However, it is well known that PID controller is sensitive to noises and disturbances. In [4–6], the controllers based on linearized theory are introduced. Also, these controllers cannot guarantee the good performances for the system under condition of uncertain factors. In order to face with system uncertainties, many advanced controllers have been presented such as sliding mode controllers [7–13], fuzzy controllers [14–21], intelligent adaptive strategies [22]

It is well known that, robust adaptive controller is a suitable selection for the systems which are affected by working environment. In [23] an adaptive fuzzy controller is proposed for the overhead crane system to deal with nonlinear disturbances. In this scheme, the fuzzy logic controller is combined with adaptive algorithm to keep stabling for the system as well as to tune the free parameters. The given strategy is simple but robust to the variation of the system parameters (wire length and payload weight) and external disturbances. However, the stability of overall system is not presented. An adaptive sliding-mode anti-sway controller is shown in [24]. The purpose of this scheme is given the good performances for the crane system in the range of high-speed hosting motion. This algorithm includes two parts: sliding-mode controller and fuzzy observer. The first one is to keep the asymptotic stability of the sway dynamic, the other is to cope with the system uncertainties. This algorithm gives the robust anti-sway performance to overhead cranes regardless of hosting velocity and system uncertainties. The stability of the system is proven in analysis and simulation. In [25], a fuzzy sliding-mode control is incorporated with a fuzzy uncertainty observer. By this cooperation, the controller guarantees not only the anti-sway trajectory tracking of the nominal plant but also the robustness to system uncertainties as well as actuator nonlinearity. This scheme guarantees asymp-

In this chapter a robust adaptive controller is introduced for 3D crane system. Firstly, the controller is designed based on the Euler-Lagrange model of the overhead crane system which includes the system uncertainties and external disturbances. Next, by using this controller, the error dynamic of the system is show in the form of state space model. Finally, the simulation is done to verify the effectiveness of the given algorithm. The simulation results show that the proposed controller guarantees the good tracking and no payload swing angle for the crane

Figure 1 shows the structure of the 3D overhead crane. The dynamic model of the overhead

M qð Þq€ þ C qð Þ ; q\_ q\_ þ ɡð Þ¼ q τ (1)

system even under the effect of parameters variation as well as external disturbances.

totic stability and robust performances but it is quite complicated.

2. Robust adaptive control system design

2.1. 3D overhead crane system modeling

crane is as follows [26]:

where

and so on.

336 Adaptive Robust Control Systems

$$\mathbf{M}(q) = \begin{pmatrix} m\_c + m\_h & 0 & m\_hl\cos\theta\cos\varphi & -m\_hl\sin\theta\sin\varphi\\ 0 & m\_c + m\_h + m\_x & m\_hl\cos\theta\sin\varphi & m\_hl\sin\theta\cos\varphi\\ m\_hl\cos\theta\cos\varphi & m\_hl\cos\theta\sin\varphi & m\_hl^2 + I & 0\\ -m\_hl\sin\theta\sin\varphi & m\_hl\sin\theta\cos\varphi & 0 & m\_hl^2\sin^2\theta + I \end{pmatrix} \tag{2}$$
 
$$\mathbf{m}(q) = \begin{pmatrix} 0 & 0 & -m\_hl\dot{\theta}\sin\theta\cos\varphi - m\_hl\dot{\rho}\cos\theta\sin\varphi\\ 0 & 0 & -m\_hl\dot{\theta}\sin\theta\sin\varphi + m\_hl\dot{\rho}\cos\theta\cos\varphi \end{pmatrix} \tag{3}$$

$$\mathbf{C}(q,\dot{q}) = \begin{vmatrix} 0 & 0 & -m\_{\mathrm{h}}l\dot{\theta}\sin\theta\sin\phi + m\_{\mathrm{h}}l\dot{\phi}\cos\theta\cos\phi & m\_{\mathrm{h}}l\dot{\theta}\cos\theta\cos\phi - m\_{\mathrm{h}}l\dot{\phi}\sin\theta\sin\phi\\ 0 & 0 & 0 & -m\_{\mathrm{h}}l^{2}\dot{\varphi}\sin\theta\cos\theta\\ 0 & 0 & m\_{\mathrm{h}}l^{2}\dot{\phi}\sin\theta\cos\theta & m\_{\mathrm{h}}l^{2}\dot{\theta}\sin\theta\cos\theta \end{vmatrix} \tag{3}$$

$$g(q) = \begin{pmatrix} 0 \\ 0 \\ m\_{\hbar} g l \sin \theta \\ 0 \end{pmatrix} \tag{4}$$

$$\boldsymbol{\pi} = \begin{pmatrix} \mu\_1, \mu\_2, 0, 0 \end{pmatrix}^T, \quad \boldsymbol{q} = \begin{pmatrix} \mathbf{x}, \mathbf{y}, \boldsymbol{\theta}, \boldsymbol{\phi} \end{pmatrix}^T. \tag{5}$$

In considering the system uncertainties, the model (1) is rewritten as the following:

$$M(q,d)\ddot{q} + \mathbb{C}(q,\dot{q},d)\dot{q} + g(q,d) = D(u+n(q,\dot{q},\ddot{q},d,t))\tag{6}$$

where

$$\mathbf{D} = \begin{pmatrix} \mathbf{I}\_{2 \times 2} \\ \mathbf{0} \end{pmatrix}, \boldsymbol{\mu} = \begin{pmatrix} \mu\_1, \mu\_2 \end{pmatrix}^T \tag{7}$$

• A3: The relationship between the uncertainty d and the model is linear [27], i.e., the left

M21ð Þ q; d q€<sup>1</sup> þ M22ð Þ q; d q€<sup>2</sup> þ f <sup>2</sup>ð Þ¼ q; q\_; d F2ð Þ q; q\_; q€ d

ð Þ <sup>q</sup>; <sup>d</sup> , C<sup>11</sup> <sup>¼</sup> <sup>C</sup>11ð Þ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> , f <sup>=</sup> <sup>¼</sup> <sup>f</sup>

� �, F<sup>2</sup> <sup>¼</sup> <sup>F</sup><sup>2</sup> <sup>q</sup>; <sup>q</sup>\_; <sup>q</sup>€<sup>1</sup>

<sup>u</sup> <sup>¼</sup> <sup>M</sup><sup>=</sup> <sup>q</sup>€<sup>r</sup> <sup>þ</sup> <sup>K</sup>1<sup>e</sup> <sup>þ</sup> <sup>K</sup>2e\_ � � <sup>þ</sup> <sup>C</sup>11q\_

ð Þ <sup>a</sup> <sup>þ</sup> <sup>1</sup> <sup>a</sup> � � <sup>p</sup> , a <sup>&</sup>gt; 0, and

� �\_ �<sup>1</sup>

� � �

> � � j j <sup>x</sup> <sup>&</sup>lt; δγ a

<sup>q</sup>€<sup>r</sup> <sup>þ</sup> K1<sup>e</sup> <sup>þ</sup> K2e\_ � � <sup>þ</sup> <sup>C</sup>

<sup>O</sup> <sup>¼</sup> <sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>6</sup>

\_ =

� �<sup>T</sup>

F1

<sup>v</sup>\_ <sup>¼</sup> <sup>M</sup><sup>=</sup> <sup>q</sup>; <sup>d</sup>

sðÞ¼ t F1v

<sup>11</sup> ¼ C<sup>11</sup> q; q\_; d

The role of the proposed controller in the system is to adapt to the constant uncertain d and robust with unknown function n(t) so the error e = qr – q1, where qr is the desired value of q1, is

The robust adaptive controller which satisfies the above requirements is obtained by the

� �\_

� �

=

ð Þ¼ q; q\_; d F<sup>1</sup> q; q\_; q€<sup>1</sup>

= ð Þ q; q\_; d

, f \_= ¼ f

<sup>1</sup> þ f

ð Þ K1; K2 x

≤ γ, ∀q is representation of d, and x ¼ colð Þ e; e\_ .

� � (19)

<sup>11</sup>q\_ <sup>1</sup> þ f \_=

þ s þ n (20)

\_

� �d

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768

> <sup>=</sup> q; q\_; d � �\_

<sup>=</sup> <sup>þ</sup> s tð Þ (17)

(15)

339

(16)

(18)

ð Þ q; d q€<sup>1</sup> þ C11ð Þ q; q\_; d q\_<sup>1</sup> þ f

side of Eq. (13) can be expressed as:

(

2.2. Controller design

following theorem.

M<sup>=</sup>

In this part, the following denotations are used:

M

bounded and converges asymptotically to 0.

where K1 <sup>¼</sup> diagð Þ<sup>a</sup> , K2 <sup>¼</sup> diag ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

in which ^d, which satisfies max

M<sup>=</sup>

which can be rewritten as:

<sup>M</sup><sup>=</sup> <sup>¼</sup> <sup>M</sup><sup>=</sup>

\_ <sup>=</sup> <sup>¼</sup> <sup>M</sup><sup>=</sup> <sup>q</sup>; <sup>d</sup>

F<sup>1</sup> ¼ F<sup>1</sup> q; q\_; q€<sup>1</sup>

� �\_ , C \_

Theorem: Consider the system Eq. (13), the following controller:

8 ><

>:

Pn j¼1 m= ij q; d

Proof: Replacing Eq. (18) into Eq. (17), the following is obtained:

<sup>q</sup>€ <sup>þ</sup> C11q\_ <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>=</sup> <sup>¼</sup> <sup>u</sup> <sup>þ</sup> <sup>n</sup> <sup>¼</sup> <sup>M</sup>

� � � �\_

�

1 ≤ i ≤ n

will converge x to the neighborhood of the are O:

The uncertain vector d ∈ R4 includes the unknown constants in the system model and n qð Þ ; q\_; q€; d; t is external disturbance. In the rest of this chapter, n qð Þ ; q\_; q€; d; t is shorten by n(t).

Model (1) is rewritten as the following:

$$\underbrace{\begin{pmatrix} M\_{11}(q,d) & M\_{12}(q,d) \\ M\_{21}(q,d) & M\_{22}(q,d) \end{pmatrix}}\_{M(q,d)} \underbrace{\begin{pmatrix} \ddot{q}\_{1} \\ \ddot{q}\_{2} \end{pmatrix}}\_{\ddot{q}} + \underbrace{\begin{pmatrix} \mathsf{C}\_{11}(q,\dot{q},d) & \mathsf{C}\_{12}(q,\dot{q},d) \\ \mathsf{C}\_{21}(q,\dot{q},d) & \mathsf{C}\_{22}(q,\dot{q},d) \end{pmatrix}}\_{\mathsf{C}(q,\dot{q},d)} \underbrace{\begin{pmatrix} \dot{q}\_{1} \\ \dot{q}\_{2} \end{pmatrix}}\_{\dot{q}} + \underbrace{\begin{pmatrix} g\_{1}(q,d) \\ g\_{2}(q,d) \end{pmatrix}}\_{g(q,d)} = \begin{pmatrix} u+n \\ 0 \end{pmatrix} \tag{8}$$

or

$$\begin{cases} M\_{11}(q,d)\ddot{q}\_1 + M\_{12}(q,d)\ddot{q}\_2 + \mathcal{C}\_{11}(q,\dot{q},d)\dot{q}\_1 + f\_1(q,\dot{q},d) = u+n\\ M\_{21}(q,d)\ddot{q}\_1 + M\_{22}(q,d)\ddot{q}\_2 + f\_2(q,\dot{q},d) = 0 \end{cases} \tag{9}$$

where

$$q = \begin{pmatrix} q\_1 \\ q\_2 \end{pmatrix}, q\_1 = (\mathbf{x}, \mathbf{y})^T, \ q\_2 = (\theta, \mathbf{q})^T \tag{10}$$

$$\begin{aligned} f\_1(q, \dot{\boldsymbol{q}}, d) &= M\_{12}(q, d)\ddot{\boldsymbol{q}}\_2 + \mathbb{C}\_{12}(q, \dot{\boldsymbol{q}}, d)\dot{\boldsymbol{q}}\_2 + g\_1(q, d) \\ f\_2(q, \dot{\boldsymbol{q}}, d) &= \mathbb{C}\_{21}(q, \dot{\boldsymbol{q}}, d)\dot{\boldsymbol{q}}\_1 + \mathbb{C}\_{22}(q, \dot{\boldsymbol{q}}, d)\dot{\boldsymbol{q}}\_2 + g\_2(q, d) \end{aligned} \tag{11}$$

Because M(q, d) is positive definite matrix, M11(d, q) and M22(d, q) are invertible. From the second equation of (9), it can be obtained:

$$\ddot{q}\_2 = -M\_{22}(q,d)^{-1} \left[ M\_{21}(q,d)\ddot{q}\_1 + f\_2(q,\dot{q},d) \right] \tag{12}$$

Replacing Eq. (12) into Eq. (9) to get the following:

$$\begin{cases} M^{'}(q,d)\ddot{q}\_{1} + \mathbb{C}\_{11}(q,\dot{q},d)\dot{q}\_{1} + f^{'}(q,\dot{q},d) = u+n\\ M\_{21}(q,d)\ddot{q}\_{1} + M\_{22}(q,d)\ddot{q}\_{2} + f\_{2}(q,\dot{q},d) = 0 \end{cases} \tag{13}$$

where

$$\begin{aligned} M^{'}(q,d) &= M\_{11}(q,d) - M\_{12}(q,d)M\_{22}(q,d)^{-1}M\_{21}(q,d) \\ f^{'}(q,\dot{q},d) &= f\_{1}(q,\dot{q},d) - M\_{12}(q,d)M\_{22}(q,d)^{-1}f\_{2}(q,\dot{q},d) \end{aligned} \tag{14}$$

In this paper, the following assumptions are used:


• A3: The relationship between the uncertainty d and the model is linear [27], i.e., the left side of Eq. (13) can be expressed as:

$$\begin{cases} M^{'}(q,d)\ddot{q}\_{1} + \mathbb{C}\_{11}(q,\dot{q},d)\dot{q}\_{1} + f^{'}(q,\dot{q},d) = F\_{1}(q,\dot{q},\ddot{q}\_{1})d\\ M\_{21}(q,d)\ddot{q}\_{1} + M\_{22}(q,d)\ddot{q}\_{2} + f\_{2}(q,\dot{q},d) = F\_{2}(q,\dot{q},\ddot{q})d \end{cases} \tag{15}$$

#### 2.2. Controller design

<sup>D</sup> <sup>¼</sup> I2�<sup>2</sup> 0 � �

Model (1) is rewritten as the following:

q€1 q€2 !

þ

<sup>q</sup> <sup>¼</sup> <sup>q</sup><sup>1</sup> q2 � �

M21ð Þ q; d q€<sup>1</sup> þ M22ð Þ q; d q€<sup>2</sup> þ f <sup>2</sup>ð Þ¼ q; q\_; d 0


M11ð Þ q; d M12ð Þ q; d M21ð Þ q; d M22ð Þ q; d !

338 Adaptive Robust Control Systems


�

second equation of (9), it can be obtained:

Replacing Eq. (12) into Eq. (9) to get the following:

(

M<sup>=</sup>

In this paper, the following assumptions are used:

• A2: k k nð Þt <sup>∞</sup> ¼ sup<sup>t</sup> j j nð Þt ¼ δ where δ is finite scalar.

ð Þ q; d is quadratic positive definite for all d.

f = M<sup>=</sup>

or

where

where

• A1: M<sup>=</sup>

, u ¼ ð Þ u1; u<sup>2</sup>

q\_1 q\_ 2

þ

ɡ1ð Þ q; d ɡ2ð Þ q; d ! <sup>¼</sup> <sup>u</sup> <sup>þ</sup> <sup>n</sup> 0 !

(8)

(9)

(13)

(14)


, q<sup>2</sup> <sup>¼</sup> ð Þ <sup>θ</sup>;<sup>φ</sup> <sup>T</sup> (10)


!

The uncertain vector d ∈ R4 includes the unknown constants in the system model and n qð Þ ; q\_; q€; d; t is external disturbance. In the rest of this chapter, n qð Þ ; q\_; q€; d; t is shorten by n(t).

> C11ð Þ q; q\_; d C12ð Þ q; q\_; d C21ð Þ q; q\_; d C22ð Þ q; q\_; d !

> |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} C qð Þ ;q\_;d

M11ð Þ q; d q€<sup>1</sup> þ M12ð Þ q; d q€<sup>2</sup> þ C11ð Þ q; q\_; d q\_<sup>1</sup> þ f <sup>1</sup>ð Þ¼ q; q\_; d u þ n

T

<sup>f</sup> <sup>2</sup>ð Þ¼ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> <sup>C</sup>21ð Þ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> <sup>q</sup>\_<sup>1</sup> <sup>þ</sup> <sup>C</sup>22ð Þ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> <sup>q</sup>\_<sup>2</sup> <sup>þ</sup> <sup>ɡ</sup>2ð Þ <sup>q</sup>; <sup>d</sup> (11)

<sup>q</sup>€<sup>2</sup> ¼ �M22ð Þ <sup>q</sup>; <sup>d</sup> �<sup>1</sup> <sup>M</sup>21ð Þ <sup>q</sup>; <sup>d</sup> <sup>q</sup>€<sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>2</sup>ð Þ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> � � (12)

ð Þ¼ q; q\_; d u þ n

M21ð Þ q; d

f <sup>2</sup>ð Þ q; q\_; d

=

, q<sup>1</sup> ¼ ð Þ x; y

f <sup>1</sup>ð Þ¼ q; q\_; d M12ð Þ q; d q€<sup>2</sup> þ C12ð Þ q; q\_; d q\_<sup>2</sup> þ ɡ1ð Þ q; d

Because M(q, d) is positive definite matrix, M11(d, q) and M22(d, q) are invertible. From the

ð Þ q; d q€<sup>1</sup> þ C11ð Þ q; q\_; d q\_<sup>1</sup> þ f

M21ð Þ q; d q€<sup>1</sup> þ M22ð Þ q; d q€<sup>2</sup> þ f <sup>2</sup>ð Þ¼ q; q\_; d 0

ð Þ¼ <sup>q</sup>; <sup>d</sup> <sup>M</sup>11ð Þ� <sup>q</sup>; <sup>d</sup> <sup>M</sup>12ð Þ <sup>q</sup>; <sup>d</sup> <sup>M</sup>22ð Þ <sup>q</sup>; <sup>d</sup> �<sup>1</sup>

ð Þ¼ <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> <sup>f</sup> <sup>1</sup>ð Þ� <sup>q</sup>; <sup>q</sup>\_; <sup>d</sup> <sup>M</sup>12ð Þ <sup>q</sup>; <sup>d</sup> <sup>M</sup>22ð Þ <sup>q</sup>; <sup>d</sup> �<sup>1</sup>

<sup>T</sup> (7)

In this part, the following denotations are used:

$$\begin{aligned} M^{\prime} &= M^{\prime}(q, \boldsymbol{d}), \mathbb{C}\_{11} = \mathbb{C}\_{11}(q, \dot{q}, \boldsymbol{d}), \boldsymbol{f}^{\prime} = \boldsymbol{f}^{\prime}(q, \dot{q}, \boldsymbol{d}) \\ \widehat{M}^{\prime} &= M^{\prime}\left(q, \overset{\frown}{\boldsymbol{d}}\right), \overset{\frown}{\mathbb{C}}\_{11} = \mathbb{C}\_{11}\left(q, \dot{q}, \overset{\frown}{\boldsymbol{d}}\right), \widehat{\boldsymbol{f}}^{\prime} = \boldsymbol{f}^{\prime}\left(q, \dot{q}, \overset{\frown}{\boldsymbol{d}}\right) \\ F\_{1} &= F\_{1}\left(q, \dot{q}, \ddot{q}\_{1}\right), F\_{2} = F\_{2}\left(q, \dot{q}, \ddot{q}\_{1}\right) \end{aligned} \tag{16}$$

The role of the proposed controller in the system is to adapt to the constant uncertain d and robust with unknown function n(t) so the error e = qr – q1, where qr is the desired value of q1, is bounded and converges asymptotically to 0.

The robust adaptive controller which satisfies the above requirements is obtained by the following theorem.

Theorem: Consider the system Eq. (13), the following controller:

$$\mu = M^{\prime} \left[ \ddot{q}\_r + K\_1 \mathbf{e} + K\_2 \dot{\mathbf{e}} \right] + \mathbf{C}\_{11} \dot{q}\_1 + f^{\prime} + \mathbf{s}(t) \tag{17}$$

where K1 <sup>¼</sup> diagð Þ<sup>a</sup> , K2 <sup>¼</sup> diag ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>a</sup> <sup>þ</sup> <sup>1</sup> <sup>a</sup> � � <sup>p</sup> , a <sup>&</sup>gt; 0, and

$$\begin{cases} \dot{\boldsymbol{\sigma}} = \left( \mathbf{M}^{\prime} \left( \boldsymbol{\eta}, \widehat{\mathbf{d}} \right)^{-1} \mathbf{F}\_{1} \right)^{T} (\mathbf{K}\_{1}, \mathbf{K}\_{2}) \mathbf{x} \\ \mathbf{s}(t) = \mathbf{F}\_{1} \boldsymbol{\sigma} \end{cases} \tag{18}$$

in which ^d, which satisfies max 1 ≤ i ≤ n Pn j¼1 m= ij q; d � � � �\_ � � � � ≤ γ, ∀q is representation of d, and x ¼ colð Þ e; e\_ .

will converge x to the neighborhood of the are O:

$$\mathcal{O} = \left\{ \mathfrak{x} \in \mathbb{R}^6 \, \middle| \, |\mathfrak{x}| < \frac{\delta \gamma}{a} \right\} \tag{19}$$

Proof: Replacing Eq. (18) into Eq. (17), the following is obtained:

$$\mathbf{M}^{\prime}\ddot{\boldsymbol{\eta}} + \mathbf{C}\_{11}\dot{\boldsymbol{\eta}}\_{1} + \boldsymbol{f}^{\prime} = \boldsymbol{\mathfrak{u}} + \boldsymbol{\mathfrak{n}} = \widehat{\mathbf{M}}^{\prime} \left[ \ddot{\boldsymbol{\eta}}\_{r} + \mathbf{K}\_{1}\mathbf{e} + \mathbf{K}\_{2}\dot{\boldsymbol{\mathfrak{e}}} \right] + \widehat{\mathbf{C}}\_{11}\dot{\boldsymbol{\eta}}\_{1} + \widehat{\boldsymbol{f}}^{\prime} + \boldsymbol{s} + \boldsymbol{\mathfrak{n}} \tag{20}$$

which can be rewritten as:

$$\left(\mathbf{M}^{\prime} - \widehat{\mathbf{M}}^{\prime}\right)\ddot{\boldsymbol{\eta}} + \left(\mathbf{C}\_{11} - \widehat{\mathbf{C}}\_{11}\right)\dot{\boldsymbol{\eta}}\_{1} + \left(\boldsymbol{f}^{\prime} - \widehat{\boldsymbol{f}}^{\prime}\right) = \widehat{\mathbf{M}}^{\prime}/[\ddot{\mathbf{e}} + \mathbf{K}\_{1}\mathbf{e} + \mathbf{K}\_{2}\dot{\mathbf{e}}] + \mathbf{s} + \mathfrak{n} \tag{21}$$

By using A3, the above equation can be expressed as the following:

$$\mathbf{F}\_1 \left(\mathbf{d} - \widehat{\mathbf{d}}\right) = \widehat{\mathbf{M}}^{\prime} \left[\ddot{\mathbf{e}} + \mathbf{K}\_1 \mathbf{e} + \mathbf{K}\_2 \dot{\mathbf{e}}\right] + \mathbf{s} + \mathfrak{n} \tag{22}$$

Choosing the Lyapunov function as the following:

The derivative of V can be expressed as:

<sup>2</sup> ð Þ <sup>A</sup>ð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> B Fð Þ <sup>1</sup><sup>Δ</sup> � <sup>n</sup>

<sup>2</sup> <sup>A</sup><sup>T</sup><sup>P</sup> <sup>þ</sup> PA � �

<sup>1</sup> 0 0 K<sup>2</sup>

is a symmetric positive definite matrix.

v\_ ¼ ð Þ BF1

0 @

¼ M

¼ M

0 �K1

I3�<sup>3</sup> �K2

<sup>2</sup> � K1

<sup>V</sup>\_ <sup>¼</sup> <sup>1</sup>

¼ 1 2

or

where

By choosing:

then

<sup>þ</sup> <sup>Δ</sup><sup>T</sup>Δ\_

<sup>Q</sup> ¼ � <sup>1</sup>

¼ � <sup>1</sup> 2

<sup>¼</sup> <sup>K</sup><sup>2</sup>

0 @ <sup>V</sup> <sup>¼</sup> <sup>1</sup>

ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup> <sup>A</sup><sup>T</sup><sup>P</sup> <sup>þ</sup> PA � �ð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>Δ</sup><sup>T</sup> ð Þ BF1

<sup>V</sup>\_ ¼ �ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>Qð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>Δ</sup><sup>T</sup> ð Þ BF1

! 2K1K2 K1

<sup>A</sup> <sup>¼</sup> diag <sup>a</sup><sup>2</sup> � �

1

<sup>T</sup>Pð Þ¼ <sup>x</sup> � <sup>x</sup><sup>m</sup>

F1 !<sup>T</sup>

F1 !<sup>T</sup>

\_ <sup>=</sup> � ��<sup>1</sup>

\_ <sup>=</sup> � ��<sup>1</sup>

K1 K2

0

F1

F1 !<sup>T</sup>

1 CA

3 7 5

T

K2

1

<sup>V</sup>\_ ¼ �ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>Qð Þ� <sup>x</sup> � <sup>x</sup><sup>m</sup> ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>PB<sup>n</sup> (33)

M \_ <sup>=</sup> � ��<sup>1</sup>

\_ <sup>=</sup> � ��<sup>1</sup>

0 B@

K1; M

ð Þ K1; K2 ð Þ x � x<sup>m</sup>

2 6 4 þ

" # !

!

<sup>2</sup> ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>Pð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>Δ</sup><sup>T</sup><sup>Δ</sup>

h i

h i (28)

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768

<sup>T</sup>Pð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>P Að Þ ð Þþ <sup>x</sup> � <sup>x</sup><sup>m</sup> B Fð Þ <sup>1</sup><sup>Δ</sup> � <sup>n</sup>

h i � ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>PB<sup>n</sup>

h i � ð Þ <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>T</sup>PB<sup>n</sup> (30)

! 0 I3�<sup>3</sup>

2K1K2 K1

!

K1 K2

Að Þ x � x<sup>m</sup>

�K1 �K2

ð Þ x � x<sup>m</sup>

(29)

341

(31)

(32)

<sup>T</sup>Pð Þ� <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>v</sup>\_

<sup>T</sup>Pð Þ� <sup>x</sup> � <sup>x</sup><sup>m</sup> <sup>v</sup>\_

2K1K2 K1

K1 K2

or

$$\ddot{\mathbf{e}} = -\mathbf{K}\_1 \mathbf{e} - \mathbf{K}\_2 \dot{\mathbf{e}} + \left(\widehat{\mathbf{M}}^{\prime}\right)^{-1} \left[\mathbf{F}\_1 \left(\mathbf{d} - \widehat{\mathbf{d}}\right) - \mathbf{s} - \mathbf{n}\right] \tag{23}$$

Equation (23) can be written in the state-space form as the following:

$$\begin{aligned} \dot{\mathbf{x}} &= \begin{pmatrix} 0 & \mathbf{I}\_{3 \times 3} \\ -\mathbf{K}\_1 & -\mathbf{K}\_2 \end{pmatrix} \mathbf{x} + \begin{pmatrix} 0 \\ \left(\hat{\mathbf{M}}^{'}\right)^{-1} \end{pmatrix} \begin{bmatrix} \mathbf{F}\_1 \left(\mathbf{d} - \hat{\mathbf{d}}\right) - \mathbf{s} - \mathbf{n} \end{bmatrix} \\ &= \mathbf{A}\mathbf{x} + \mathbf{B} \begin{bmatrix} \mathbf{F}\_1 \left(\mathbf{d} - \hat{\mathbf{d}}\right) - \mathbf{s} - \mathbf{n} \end{bmatrix} \end{aligned} \tag{24}$$

where

$$\mathbf{A} = \begin{pmatrix} \mathbf{e} \\ \dot{\mathbf{e}} \end{pmatrix}, \mathbf{A} = \begin{pmatrix} 0 & \mathbf{I}\_{3 \times 3} \\ -\mathbf{K}\_1 & -\mathbf{K}\_2 \end{pmatrix}, \mathbf{B} = \begin{pmatrix} 0 \\ \left(\hat{\mathbf{M}}'\right)^{-1} \end{pmatrix} \tag{25}$$

Since K1 and K2 are symmetric positive definite matrices, matrix A is stable, it means that all the eigenvalues of A is located in the left side of the complex plane. Consequently, the linear reference model:

$$
\dot{\mathfrak{X}}\_{\mathfrak{m}} = \mathbf{A} \mathfrak{x}\_{\mathfrak{m}} \tag{26}
$$

is stable. Then, xm(t) is bounded and asymptotically converges to zero as t ! ∞ despite the initiative value xm(0).

Next step, it will be shown that, by using the controller Eq. (17) and auxiliary controller Eq. (18), the error (x – xm) is bounded and converges to the neighborhood of the area O defined in Eq. (19).

From Eqs. (24) and (26), the following is obtained:

$$\begin{aligned} \dot{\mathbf{x}} - \dot{\mathbf{x}}\_m &= \mathbf{A}(\mathbf{x} - \mathbf{x}\_m) + \mathbf{B} \left[ \mathbf{F}\_1 \left( \mathbf{d} - \widehat{\mathbf{d}} \right) - \mathbf{s} - \mathfrak{n} \right] \\ &= \mathbf{A}(\mathbf{x} - \mathbf{x}\_m) + \mathbf{B}(\mathbf{F}\_1 \boldsymbol{\Delta} - \mathfrak{n}) \end{aligned} \tag{27}$$

where Δ ¼ d� d \_ �v.

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768 341

Choosing the Lyapunov function as the following:

$$V = \frac{1}{2} \left[ (\mathbf{x} - \mathbf{x}\_m)^T \mathbf{P} (\mathbf{x} - \mathbf{x}\_m) + \boldsymbol{\Delta}^T \boldsymbol{\Delta} \right] \tag{28}$$

The derivative of V can be expressed as:

$$\begin{aligned} \dot{V} &= \frac{1}{2} \left[ \left( \mathbf{A} (\mathbf{x} - \mathbf{x\_m}) + \mathbf{B} (\mathbf{F}\_1 \Delta - \mathbf{n}) \right)^T \mathbf{P} (\mathbf{x} - \mathbf{x\_m}) + (\mathbf{x} - \mathbf{x\_m})^T \mathbf{P} (\mathbf{A} (\mathbf{x} - \mathbf{x\_m}) + \mathbf{B} (\mathbf{F}\_1 \Delta - \mathbf{n})) \right] \\ &+ \Delta^T \dot{\Delta} \\ &= \frac{1}{2} (\mathbf{x} - \mathbf{x\_m})^T \left( \mathbf{A}^T \mathbf{P} + \mathbf{P} \mathbf{A} \right) (\mathbf{x} - \mathbf{x\_m}) + \Delta^T \left[ \left( \mathbf{B} \mathbf{F}\_1 \right)^T \mathbf{P} (\mathbf{x} - \mathbf{x\_m}) - \dot{\boldsymbol{\sigma}} \right] - \left( \mathbf{x} - \mathbf{x\_m} \right)^T \mathbf{P} \mathbf{B} \mathbf{n} \end{aligned} \tag{29}$$

or

<sup>M</sup><sup>=</sup> � <sup>M</sup> \_ <sup>=</sup> � �

340 Adaptive Robust Control Systems

or

where

reference model:

initiative value xm(0).

in Eq. (19).

where Δ ¼ d� d

\_ �v. q€ þ C11 � C

e

x\_ ¼

\_ 11 � �

By using A3, the above equation can be expressed as the following:

€ ¼ �K1e � K2e\_ þ M

Equation (23) can be written in the state-space form as the following:

0 I3�<sup>3</sup>

!

�K1 �K2

¼ Ax þ B F1 d� d

<sup>x</sup> <sup>¼</sup> <sup>e</sup> e\_ � �

From Eqs. (24) and (26), the following is obtained:

F1 d� d � �\_ <sup>q</sup>\_ <sup>1</sup> <sup>þ</sup> <sup>f</sup> <sup>=</sup> � <sup>f</sup> \_= � �

\_ <sup>=</sup> � ��<sup>1</sup>

0

M \_ <sup>=</sup> � ��<sup>1</sup>

� s � n

�K1 �K2 � �

Since K1 and K2 are symmetric positive definite matrices, matrix A is stable, it means that all the eigenvalues of A is located in the left side of the complex plane. Consequently, the linear

is stable. Then, xm(t) is bounded and asymptotically converges to zero as t ! ∞ despite the

Next step, it will be shown that, by using the controller Eq. (17) and auxiliary controller Eq. (18), the error (x – xm) is bounded and converges to the neighborhood of the area O defined

x\_ � x\_<sup>m</sup> ¼ Að Þþ x � x<sup>m</sup> B F1 d� d

¼ Að Þþ x � x<sup>m</sup> B Fð Þ <sup>1</sup>Δ � n

¼ M \_ = e

x þ

h i

, <sup>A</sup> <sup>¼</sup> 0 I3�<sup>3</sup>

� �\_

0 B@ ¼ M \_ = e

F1 d� d � �\_

> 1 CA

, B ¼

� �\_

h i

� s � n

0 @

h i

F1 d� d � �\_

€ þ K1e þ K2 ½ e\_� þ s þ n (21)

(23)

(24)

(27)

€ þ K1e þ K2 ½ e\_� þ s þ n (22)

� s � n

1

x\_<sup>m</sup> ¼ Ax<sup>m</sup> (26)

A (25)

� s � n

h i

0

M \_ <sup>=</sup> � ��<sup>1</sup>

$$\dot{V} = -(\mathbf{x} - \mathbf{x}\_m)^T \mathbf{Q} (\mathbf{x} - \mathbf{x}\_m) + \boldsymbol{\Delta}^T \left[ (\mathbf{B} \mathbf{F}\_1)^T \mathbf{P} (\mathbf{x} - \mathbf{x}\_m) - \dot{\mathbf{z}} \right] - (\mathbf{x} - \mathbf{x}\_m)^T \mathbf{P} \mathbf{B} \mathbf{u} \tag{30}$$

where

$$\begin{aligned} \mathbf{Q} &= -\frac{1}{2} \left( \mathbf{A}^T \mathbf{P} + \mathbf{P} \mathbf{A} \right) \\ &= -\frac{1}{2} \left[ \begin{pmatrix} 0 & -\mathbf{K}\_1 \\ \mathbf{I}\_{3 \times 3} & -\mathbf{K}\_2 \end{pmatrix} \begin{pmatrix} 2\mathbf{K}\_1 \mathbf{K}\_2 & \mathbf{K}\_1 \\ \mathbf{K}\_1 & \mathbf{K}\_2 \end{pmatrix} + \begin{pmatrix} 2\mathbf{K}\_1 \mathbf{K}\_2 & \mathbf{K}\_1 \\ \mathbf{K}\_1 & \mathbf{K}\_2 \end{pmatrix} \begin{pmatrix} 0 & \mathbf{I}\_{3 \times 3} \\ -\mathbf{K}\_1 & -\mathbf{K}\_2 \end{pmatrix} \right] \\ &= \begin{pmatrix} \mathbf{K}\_1^2 & 0 \\ 0 & \mathbf{K}\_2^2 - \mathbf{K}\_1 \end{pmatrix} = \text{diag}\{a^2\} \end{aligned} \tag{31}$$

is a symmetric positive definite matrix.

By choosing:

$$\begin{split} \dot{\boldsymbol{\sigma}} &= (\mathbf{B} \mathbf{F}\_1)^T \mathbf{P} (\mathbf{x} - \mathbf{x}\_m) = \left[ \begin{pmatrix} 0 \\ \left( \boldsymbol{\hat{\mathbf{M}}}^{'} \right)^{-1} \mathbf{F}\_1 \end{pmatrix} \right]^T \begin{pmatrix} 2 \mathbf{K}\_1 \mathbf{K}\_2 & \mathbf{K}\_1 \\ \mathbf{K}\_1 & \mathbf{K}\_2 \end{pmatrix} (\mathbf{x} - \mathbf{x}\_m) \\ &= \left( \left( \left( \boldsymbol{\hat{\mathbf{M}}}^{'} \right)^{-1} \mathbf{F}\_1 \right)^T \mathbf{K}\_1, \left( \left( \left( \boldsymbol{\hat{\mathbf{M}}}^{'} \right)^{-1} \mathbf{F}\_1 \right)^T \mathbf{K}\_2 \right) (\mathbf{x} - \mathbf{x}\_m) \\ &= \left( \left( \left( \boldsymbol{\hat{\mathbf{M}}}^{'} \right)^{-1} \mathbf{F}\_1 \right)^T (\mathbf{K}\_1, \mathbf{K}\_2) (\mathbf{x} - \mathbf{x}\_m) \end{split} \tag{32} \right. \end{split} \tag{33}$$

then

$$\dot{V} = -(\mathbf{x} - \mathbf{x}\_m)^T \mathbf{Q} (\mathbf{x} - \mathbf{x}\_m) - (\mathbf{x} - \mathbf{x}\_m)^T \mathbf{P} \mathbf{B} \mathbf{u} \tag{33}$$

Both Eqs. (32) and (33) are always feasible with any initial values of xm. For the simplicity, the initial value of xm is chosen at xm(0) = 0. Consequently, this leads to the following:

$$
\dot{\boldsymbol{\sigma}} = \left( \left( \stackrel{\frown}{\mathbf{M}}^{\prime} \right)^{-1} \mathbf{F}\_1 \right)^{\mathrm{T}} (\mathbf{K}\_1, \mathbf{K}\_2) \mathbf{x} \tag{34}
$$

and

$$\begin{split} \dot{V} &= -\mathbf{x}^T \mathbf{Q} \mathbf{x} - \mathbf{x}^T \mathbf{P} \mathbf{B} \boldsymbol{\mathfrak{u}} = -a^2 |\mathbf{x}|^2 - \mathbf{x}^T \mathbf{P} \mathbf{B} \boldsymbol{\mathfrak{u}} \\ &\le -a^2 |\mathbf{x}|^2 + ||\mathbf{P} \mathbf{B}|| \delta |\mathbf{x}| \le a[-a|\mathbf{x}| + \gamma \delta] |\mathbf{x}| \end{split} \tag{35}$$

This implies that as γδ <sup>a</sup> <sup>&</sup>lt; j j <sup>x</sup> , i.e. when <sup>x</sup>(t) is steel on the outside of the area <sup>O</sup>, <sup>V</sup>\_ <sup>&</sup>lt; 0 so the change of j j xð Þt is monotonous decrease. This completely proves that by using the proposed controller, the trajectory x will converge to the neighborhood of the area O.

#### 3. Simulation verification

In order to verify the effectiveness of the proposed controller, a simulation is setup based on the MATLAB/Simulink tool. The parameters of the overhead crane system are as follow:

$$m\_c = 10 \text{ kg}, m\_h = 10 \text{ kg}, m\_x = 5 \text{ kg}, l = 1.2 \text{ m}, g = 9.8 \text{ m/s}^2.$$

The simulation is carried out under the three cases:

Case 1: The system parameters are nominal, no input disturbances.

Case 2: The system parameters are variation (150%), no input disturbances.

Case 3: The system parameters are nominal, existing input disturbances.

The destination positions for all cases are 1.5 m for x-axis and 2 m for y-axis, the controller gains are as follow:

$$K\_1 = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}, \ K\_2 = \begin{bmatrix} 5.48 & 0 \\ 0 & 5.48 \end{bmatrix} \tag{36}$$

negligible, and the payload swing quickly disappears as the trolleys finish their movements. In the Figure 3 (Case 2), the system parameters are 150% variation but the results are nearly unchanged, i.e. the transient time is less than 5 sec, the maximum swing angle is smaller than

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768 343

Figure 2. Simulation result of the robust adaptive controller for the case of certain system parameters.

Figure 4 is the waveform of the system under the condition of existing the external disturbances. In this case, a sinusoidal with amplitude of 2 degree is added into the inputs. The

0.3 deg. and it is kept almost zero at the steady state.

system responses are little oscillation but it is insignificant.

The simulation results are shown in Figures 2–4. In each figure, the a part is result for the x-axis and the b part is for y-axis. In addition, from top to bottom are the waveforms of trolley position, payload swing angle, and control signal, respectively.

It can be seen from Figure 2 that, in the case of system is certainty (Case 1), the trolley reaches the destination point after 3 sec in x-axis and 4 sec in y-axis, the steady state errors are

Both Eqs. (32) and (33) are always feasible with any initial values of xm. For the simplicity, the

F1 !<sup>T</sup>

j j <sup>x</sup> <sup>2</sup> <sup>þ</sup> k k PB <sup>δ</sup>j j <sup>x</sup> <sup>≤</sup> <sup>a</sup>½ � �aj j <sup>x</sup> <sup>þ</sup> γδ j j <sup>x</sup>

change of j j xð Þt is monotonous decrease. This completely proves that by using the proposed

In order to verify the effectiveness of the proposed controller, a simulation is setup based on the MATLAB/Simulink tool. The parameters of the overhead crane system are as follow:

The destination positions for all cases are 1.5 m for x-axis and 2 m for y-axis, the controller

The simulation results are shown in Figures 2–4. In each figure, the a part is result for the x-axis and the b part is for y-axis. In addition, from top to bottom are the waveforms of trolley

It can be seen from Figure 2 that, in the case of system is certainty (Case 1), the trolley reaches the destination point after 3 sec in x-axis and 4 sec in y-axis, the steady state errors are

, K<sup>2</sup> <sup>¼</sup> <sup>5</sup>:48 0

0 5:48 � �

.

ð Þ K1; K2 x (34)

(35)

(36)

j j <sup>x</sup> <sup>2</sup> � <sup>x</sup><sup>T</sup>PB<sup>n</sup>

<sup>a</sup> <sup>&</sup>lt; j j <sup>x</sup> , i.e. when <sup>x</sup>(t) is steel on the outside of the area <sup>O</sup>, <sup>V</sup>\_ <sup>&</sup>lt; 0 so the

initial value of xm is chosen at xm(0) = 0. Consequently, this leads to the following:

<sup>V</sup>\_ ¼ �x<sup>T</sup>Q<sup>x</sup> � <sup>x</sup><sup>T</sup>PB<sup>n</sup> ¼ �a<sup>2</sup>

\_ <sup>=</sup> � ��<sup>1</sup>

v\_ ¼ M

controller, the trajectory x will converge to the neighborhood of the area O.

<sup>≤</sup> � <sup>a</sup><sup>2</sup>

mc = 10 kg, mh = 10 kg, mx = 5 kg, l = 1.2 m, g = 9.8 m/s<sup>2</sup>

Case 1: The system parameters are nominal, no input disturbances.

Case 2: The system parameters are variation (150%), no input disturbances.

Case 3: The system parameters are nominal, existing input disturbances.

<sup>K</sup><sup>1</sup> <sup>¼</sup> 5 0 0 5 � �

position, payload swing angle, and control signal, respectively.

The simulation is carried out under the three cases:

and

This implies that as γδ

342 Adaptive Robust Control Systems

gains are as follow:

3. Simulation verification

Figure 2. Simulation result of the robust adaptive controller for the case of certain system parameters.

negligible, and the payload swing quickly disappears as the trolleys finish their movements. In the Figure 3 (Case 2), the system parameters are 150% variation but the results are nearly unchanged, i.e. the transient time is less than 5 sec, the maximum swing angle is smaller than 0.3 deg. and it is kept almost zero at the steady state.

Figure 4 is the waveform of the system under the condition of existing the external disturbances. In this case, a sinusoidal with amplitude of 2 degree is added into the inputs. The system responses are little oscillation but it is insignificant.

Figure 3. Simulation result of the robust adaptive controller for the case of 150% variation system parameters.

In Table 1, θmax, φmax, θss, and φss are maximum and steady state values of θ and φ, respectively. From the above results is can be seen that the proposed controller gives a good performance under various conditions of working. It has the ability to adapt with the uncertainties of the system such as the variation of the trolley mass, load mass, and cable length. Moreover, this controller is also robust to the external disturbance.

Figure 4. Simulation result of the robust adaptive controller for the case of existing external disturbances.

Case 1 0.2 0 0.3 0 3 4 Case 2 0.2 0 0.25 0 5 5 Case 3 0.2 0 0.25 0 4 5

Table 1. Summarize the results for all cases.

θmax (deg) θss (deg) ϕmax (deg) ϕss (deg) x-axis settling time (sec) y-axis settling time (sec)

Robust Adaptive Control of 3D Overhead Crane System http://dx.doi.org/10.5772/intechopen.72768 345

Figure 4. Simulation result of the robust adaptive controller for the case of existing external disturbances.


Table 1. Summarize the results for all cases.

In Table 1, θmax, φmax, θss, and φss are maximum and steady state values of θ and φ, respectively. From the above results is can be seen that the proposed controller gives a good performance under various conditions of working. It has the ability to adapt with the uncertainties of the system such as the variation of the trolley mass, load mass, and cable length. Moreover,

Figure 3. Simulation result of the robust adaptive controller for the case of 150% variation system parameters.

this controller is also robust to the external disturbance.

344 Adaptive Robust Control Systems
