3. Lyapunov-based control design

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

#### 3.1. Conventional Lyapunov controller

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

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

$$\begin{aligned} F\_x &= K\_d \left[ n\_z(0, t) + \frac{EA}{2P(0)} \left( n\_z^3(0, t) + n\_z(0, t) \mu\_z^2(0, t) \right) \right] \\ &- K\_p \left( n(0, t) - \frac{q\_d n(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) - K\_d n\_t(0, t) \end{aligned} \tag{26}$$

and

$$\begin{aligned} F\_y &= K\_d \left[ \mu\_z(0, t) + \frac{EA}{2P(0)} \left( \mu\_z^3(0, t) + \mu\_z(0, t) \mu\_z^2(0, t) \right) \right] \\ &- K\_p \left( \mu(0, t) - \frac{q\_d \mu(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) - K\_d \mu\_t(0, t) \end{aligned} \tag{27}$$

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

Proof. Define a positive Lyapunov candidate as follows:

$$\begin{split} V &= \frac{1}{2} \Big| \Big\{ \rho \left( n\_t^2 + \mu\_t^2 \right) + P \left( n\_z^2 + \mu\_z^2 \right) + \text{EA} \left( \frac{1}{2} \left( n\_z^2 + \mu\_z^2 \right) \right)^2 \Big\} dz + \frac{MP(0)}{2(P(0) + K\_d)} \left( n\_t^2(0, t) + \mu\_t^2(0, t) \right) \\ &+ \frac{1}{2} m \left( n\_t^2(L, t) + \mu\_t^2(L, t) \right) + \frac{P(0)K\_p}{2(P(0) + K\_d)} \left( \sqrt{n^2(0, t) + \mu^2(0, t)} - q\_d \right)^2 \end{split} \tag{28}$$

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

$$\begin{aligned} \text{With the notations that} \left| |w|^2 = \begin{bmatrix} \left( n\_t^2 + \mu\_t^2 \right) + \left( n\_z^2 + \mu\_z^2 \right) + \left( n\_z^2 + \mu\_z^2 \right)^2 \\ 0 \end{bmatrix} \right| dz + \left( \mu\_t^2(0, t) + \mu\_t^2(L, t) \right) + \left( \sqrt{n^2(0, t) + \mu^2(0, t)} - q\_d \right)^2. \end{aligned}$$

one has

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

Let us construct two nonlinear controllers using a traditional Lyapunov stability and its advanced version. In the first method, the control law is referred from the negative condition of a Lyapunov candidate V\_ ≤ 0: In the second method, the Lyapunov function is determined so

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

Theorem. Consider a mass distributed model of overhead crane that is described by six partial

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � �

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

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � �

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

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

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

dz þ

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

� �<sup>2</sup>

� Kdntð Þ 0; t

� Kdμ<sup>t</sup>

ð Þ 0; t

MPð Þ0 2ð Þ Pð Þþ 0 Ka

� qd

n2

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

(26)

(27)

(28)

differential equations: (20) to (25). The following control law composed of two inputs:

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

!

� Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

�Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

!

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

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

q

Fx ¼ Ka nzð Þþ 0; t

Fy ¼ Ka μzð Þþ 0; t

Proof. Define a positive Lyapunov candidate as follows:

<sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � <sup>þ</sup> EA <sup>1</sup>

� � � �<sup>2</sup> ( )

Pð Þ0 Kp 2ð Þ Pð Þþ 0 Ka

and sixth equations, respectively.

322 Adaptive Robust Control Systems

that it satisfies 0 < V ≤ b with b > 0.

3.1. Conventional Lyapunov controller

approach to references asymptotically.

and

<sup>V</sup> <sup>¼</sup> <sup>1</sup> 2 ð L

> þ 1 2 m n<sup>2</sup>

0

r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � � <sup>þ</sup> P n<sup>2</sup>

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

3. Lyapunov-based control design

$$\left|K\_{\min} \|\|w\|\right|^2 \le \mathbf{V}(\mathbf{t}) \le K\_{\max} \left||w\|\right|^2$$

with

$$K\_{\min} = \frac{1}{2} \min\left(\rho, P, \frac{EA}{4}, \frac{MP(0)}{P(0) + K\_a}, m, \frac{P(0)K\_p}{P(0) + K\_a}\right)$$

and

$$K\_{\text{max}} = \frac{1}{2} \max\left(\rho, P, \frac{EA}{4}, \frac{MP(0)}{P(0) + K\_a}, m, \frac{P(0)K\_p}{P(0) + K\_a}\right)$$

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

$$\dot{V} = \left[ \int \left[ \rho\_l n\_{tt} + \mu\_t \mu\_{tt} \right] + P(n\_z n\_{zt} + \mu\_z \mu\_{zt}) + \frac{EA}{2} \left( n\_z^3 n\_{tz} + \mu\_z^3 \mu\_{zt} + n\_z n\_{zt} \mu\_z^2 + \mu\_z \mu\_{zt} \mu\_z^2 \right) \right] dz$$

$$+ \frac{MP(0)}{P(0) + K\_a} \left\{ \begin{array}{c} n\_l(0, t) n\_{lt}(0, t) + \mu\_l(0, t) \mu\_{tl}(0, t) \\\\ + \mu(0, t) \mu\_t(0, t) + n(0, t) n\_l(0, t) \\\\ + m \left[ n\_l(L, t) n\_{tl}(L, t) + \mu\_l(L, t) \mu\_{tl}(L, t) \right] \\\\ - \left( \frac{q\_d \left( \mu(0, t) \mu\_t(0, t) + n(0, t) n\_l(0, t) \right)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) \end{array} \right\} \tag{29}$$

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

$$\begin{aligned} \int\_{0}^{L} \rho \left( n\_{t} n\_{tt} + \mu\_{t} \mu\_{tt} \right) dz &= \int\_{0}^{L} \left\{ n\_{t} \left[ \left( P n\_{z} \right)\_{z} + \frac{1}{2} E A \left( 3 n\_{z}^{2} n\_{zz} + n\_{zz} \mu\_{z}^{2} + 2 n\_{z} \mu\_{z} \mu\_{zz} \right) \right] \\ &+ \mu\_{t} \left[ \left( P \mu\_{z} \right)\_{z} + \frac{1}{2} E A \left( 3 \mu\_{z}^{2} \mu\_{zz} + \mu\_{zz} n\_{z}^{2} + 2 n\_{z} \mu\_{z} n\_{zz} \right) \right] \right\} dz \end{aligned} \tag{30}$$

Using partial integration

$$\int\_{0}^{L} n\_t (Pn\_z)\_z dz = n\_t Pn\_z \left| \begin{matrix} L \\ 0 \end{matrix} - \int\_{0}^{L} Pn\_z n\_{tz} dz \right| $$

and

$$\int\_{0}^{L} \mu\_t (P\mu\_z)\_z dz = \mu\_t P\mu\_z \left| \begin{matrix} L \\ 0 \end{matrix} - \int\_{0}^{L} P\mu\_z \mu\_{tz} dz \right| $$

MPð Þ0 Pð Þþ 0 Ka

<sup>¼</sup> <sup>P</sup>ð Þ<sup>0</sup> Pð Þþ 0 Ka

> Pð Þ0 Pð Þþ 0 Ka

<sup>V</sup>\_ <sup>¼</sup> <sup>P</sup>ð Þ<sup>0</sup> Pð Þþ 0 Ka

þ

exponential stability.

3.2. Barrier Lyapunov controller

derivation of payload position <sup>1</sup>

<sup>V</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> 2 ð L

þ

þ

0

r n<sup>2</sup> <sup>t</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> t � � <sup>þ</sup> P n<sup>2</sup>

MPð Þ0 2ð Þ Pð Þþ 0 Ka

Pð Þ0 Kp 2ð Þ Pð Þþ 0 Ka

Pð Þ0 Pð Þþ 0 Ka

μt

<sup>V</sup>\_ ¼ � <sup>P</sup>ð Þ<sup>0</sup> <sup>K</sup><sup>d</sup> Pð Þþ 0 Ka

> 2 Pð Þ0 Pð Þþ0 Ka

is the so-called barrier Lyapunov V1(t) being in the form of

n2

q

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

μt

þ

ntð Þ 0; t nttð Þþ 0; t μ<sup>t</sup>

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

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

ntð Þ 0; t Fx þ Pð Þ0 nzð Þþ 0; t

ð Þ 0; t Fy þ Pð Þ0 μzð Þþ 0; t

ntð Þ� 0; t Ka nzð Þþ 0; t

ð Þ� 0; t Ka μzð Þþ 0; t

<sup>þ</sup>Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

n2 <sup>t</sup>ð Þ� 0; t

!

<sup>þ</sup> Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

!

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

EA <sup>2</sup> <sup>μ</sup><sup>3</sup>

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

Pð Þ0 K<sup>d</sup> Pð Þþ 0 Ka

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

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

> Pð Þ<sup>0</sup> Pð Þ<sup>0</sup> þ Ka

ln <sup>k</sup> 2 b1 k2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

!

(35)

þ 1 2

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

� qd

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

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

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

We utilize an improved version of Lyapunov stability to design a control law for overhead cranes. The Lyapunov function is chosen so that its derivative is smaller than a positive constant. By this way, the Lyapunov candidate is selected similar to Eq. (28) but supplementing

> ln <sup>k</sup><sup>2</sup> b1 k2 b1�z<sup>2</sup> 1

<sup>z</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> z � � <sup>þ</sup> EA <sup>1</sup>

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

� �<sup>2</sup>

� � � �<sup>2</sup> ( )dz

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � � �

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � � �

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

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

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

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

(32)

325

(33)

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

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

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

� �. A modified version of Lyapunov candidate

<sup>t</sup>ð Þ 0; t ≤ 0 (34)

þ Fx )

þ Fy )

μ2

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

$$\int\_0^L \frac{EA}{2} n\_z^3 n\_{tz} dz = \int\_0^L \frac{EA}{2} n\_z^3 d(n\_t) = \frac{EA}{2} n\_z^3 n\_t \left| \begin{matrix} L \\ 0 \end{matrix} - \int\_0^L n\_t \frac{EA}{2} 3 n\_z^2 n\_{zz} dz \right|$$

and

$$\int\_0^L \frac{EA}{2} \mu\_z^3 \mu\_{tz} dz = \frac{EA}{2} \mu\_z^3 \mu\_t \left| \begin{matrix} L \\ 0 \end{matrix} - \int\_0^L \mu\_t \frac{EA}{2} 3 \mu\_z^2 \mu\_{zz} dz \right|$$

Then,

$$\int\_{0}^{L} \frac{EA}{2} \left( n\_z n\_{zt} \mu\_z^2 \right) dz = \frac{EA}{2} n\_z \mu\_z^2 n\_t \left| \begin{matrix} L & -EA \\ 0 & -\frac{\pi}{2} \end{matrix} \right|^L n\_t \left( n\_{zz} \mu\_z^2 + 2n\_z \mu\_z \mu\_{zz} \right) dz$$

and

$$\int\_{0}^{L} \frac{EA}{2} \left(\mu\_z \mu\_{zt} n\_z^2\right) dz = \frac{EA}{2} \mu\_z n\_z^2 \mu\_t \left| \begin{matrix} L & EA \\ 0 & -\frac{L}{2} \end{matrix} \right|^L \mu\_t \left(\mu\_{zz} n\_z^2 + 2n\_z \mu\_z n\_{zz}\right) dz$$

The Lyapunov derivative (29) now becomes

$$\begin{aligned} \dot{V} &= n\_t P n\_z \left| \begin{aligned} &L \\ &+\mu\_t P \mu\_z \end{aligned} \right| \frac{L}{0} + \frac{EA}{2} n\_z^3 n\_t \left| \begin{array}{c} \\ &+\frac{EA}{2} \mu\_z^3 \mu\_t \end{array} \right| \frac{L}{0} + \frac{EA}{2} n\_z \mu\_z^2 \left| \begin{array}{c} \\ &+\frac{EA}{2} \mu\_z \mu\_z^2 \end{array} \right| \frac{L}{0} + \frac{EA}{2} \mu\_z^2 \mu\_t^2 \left| \begin{array}{c} \\ &+\frac{EA}{2} \mu\_z \mu\_z^2 \mu\_t \end{array} \right| \end{aligned} \tag{10}$$

$$\begin{aligned} \left( \begin{array}{c} \\ &+\frac{P(0)}{P(0)} + K\_d \end{array} \right) \left( n\_l(0,t)\mu\_l(0,t) + \mu\_l(0,t)\mu\_t(0,t) \right) &+ n(n\_l(L,t)\mu\_t(L,t) + \mu\_t(L,t)\mu\_t(L,t)) \\ &+ \frac{P(0)K\_p}{P(0) + K\_d} \left( \mu(0,t)\mu\_t(0,t) + \mu\_t(0,t)\mu\_t(0,t) \right) - \frac{P(0)K\_p}{P(0) + K\_d} \frac{q\_d(\mu(0,t)\mu\_t(0,t) + n(0,t)\mu\_t(0,t))}{\sqrt{\mu^2(0,t) + n^2(0,t)}} \end{aligned} \tag{11}$$

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

$$\begin{aligned} \frac{MP(0)}{P(0) + K\_a} \left( n\_t(0, t) n\_\theta(0, t) + \mu\_t(0, t) \mu\_{tt}(0, t) \right) \\ = \frac{P(0)}{P(0) + K\_a} n\_t(0, t) \left\{ F\_x + \left[ P(0) n\_z(0, t) + \frac{EA}{2} \left( n\_z^3(0, t) + n\_z(0, t) \mu\_z^2(0, t) \right) \right] \right\} \\ + \frac{P(0)}{P(0) + K\_a} \mu\_t(0, t) \left\{ F\_y + \left[ P(0) \mu\_z(0, t) + \frac{EA}{2} \left( \mu\_z^3(0, t) + \mu\_z(0, t) \mu\_z^2(0, t) \right) \right] \right\} \end{aligned} \tag{32}$$

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

$$\begin{split} \dot{V} &= \frac{P(0)}{P(0) + K\_{a}} n\_{t}(0, t) \left\{ -K\_{a} \left[ n\_{z}(0, t) + \frac{EA}{2P(0)} \left( n\_{z}^{3}(0, t) + n\_{z}(0, t) \mu\_{z}^{2}(0, t) \right) \right] \right. \\ &\left. + K\_{p} \left( n(0, t) - \frac{q\_{d} n(0, t)}{\sqrt{\mu^{2}(0, t) + n^{2}(0, t)}} \right) + F\_{x} \right\} \\ &\left. + \frac{P(0)}{P(0) + K\_{a}} \mu\_{t}(0, t) \left\{ -K\_{a} \left[ \mu\_{z}(0, t) + \frac{EA}{2P(0)} \left( \mu\_{z}^{3}(0, t) + \mu\_{z}(0, t) \mu\_{z}^{2}(0, t) \right) \right] \right. \\ &\left. + K\_{p} \left( \mu(0, t) - \frac{q\_{d} \mu(0, t)}{\sqrt{\mu^{2}(0, t) + n^{2}(0, t)}} \right) + F\_{y} \right\} \end{split} \tag{33}$$

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

$$\dot{V} = -\frac{P(0)\mathbf{K}\_d}{P(0) + \mathbf{K}\_d} \mu\_t^2(0, t) - \frac{P(0)\mathbf{K}\_d}{P(0) + \mathbf{K}\_d} \mu\_t^2(0, t) \le 0 \tag{34}$$

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

#### 3.2. Barrier Lyapunov controller

and

324 Adaptive Robust Control Systems

and

Then,

and

<sup>V</sup>\_ <sup>¼</sup> ntPnz

þ

þ

ð L

μ<sup>t</sup> Pμ<sup>z</sup> � � z dz ¼ μ<sup>t</sup>

> ð L

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

<sup>z</sup>μtzdz <sup>¼</sup> EA

<sup>2</sup> nzμ<sup>2</sup> znt L <sup>0</sup> � EA 2 ð L

<sup>2</sup> <sup>μ</sup>zn<sup>2</sup> zμt L <sup>0</sup> � EA 2 ð L

L

0 þ EA <sup>2</sup> <sup>μ</sup><sup>3</sup> zμt � � � � �

ð Þþ <sup>0</sup>; <sup>t</sup> <sup>n</sup>ð Þ <sup>0</sup>; <sup>t</sup> ntð Þ <sup>0</sup>; <sup>t</sup> � � � <sup>P</sup>ð Þ<sup>0</sup> Kp

<sup>z</sup>d nð Þ¼ <sup>t</sup>

<sup>2</sup> <sup>μ</sup><sup>3</sup> zμt L <sup>0</sup> � ð L

> � � � � � �

> > � � � � � �

ð Þ <sup>0</sup>; <sup>t</sup> <sup>μ</sup>ttð Þ <sup>0</sup>; <sup>t</sup> � � <sup>þ</sup> m ntð Þ <sup>L</sup>; <sup>t</sup> nttð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>t</sup>

0

Pμ<sup>z</sup> L <sup>0</sup> � ð L

� � � � � �

EA <sup>2</sup> <sup>n</sup><sup>3</sup> znt L <sup>0</sup> � ð L

� � � � � � 0

� � � � � �

0 μt EA <sup>2</sup> <sup>3</sup>μ<sup>2</sup>

0

0

L

0 þ EA <sup>2</sup> nzμ<sup>2</sup> znt � � � � �

Pð Þþ 0 Ka

nt nzzμ<sup>2</sup>

<sup>μ</sup><sup>t</sup> <sup>μ</sup>zzn<sup>2</sup>

Pμzμtzdz,

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

<sup>z</sup>μzzdz

<sup>z</sup> þ 2nzμzμzz � �dz

<sup>z</sup> þ 2nzμznzz � �dz

L

L

0

(31)

0 þ EA <sup>2</sup> <sup>μ</sup>zn<sup>2</sup> zμt � � � � �

qd μð Þ 0; t μ<sup>t</sup>

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

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

<sup>z</sup>nzzdz

0

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

<sup>z</sup>ntzdz ¼

ð L

EA <sup>2</sup> <sup>μ</sup><sup>3</sup>

0

<sup>2</sup> nznztμ<sup>2</sup> z � �dz <sup>¼</sup> EA

<sup>2</sup> <sup>μ</sup>zμztn<sup>2</sup> z � �dz <sup>¼</sup> EA

> � � � � � �

L

0 þ EA <sup>2</sup> <sup>n</sup><sup>3</sup> znt � � � � �

ntð Þ 0; t nttð Þþ 0; t μ<sup>t</sup>

The Lyapunov derivative (29) now becomes

μð Þ 0; t μ<sup>t</sup>

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

ð L

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

0

ð L

EA

0

ð L

EA

0

L

� � � � � �

MPð Þ0 Pð Þþ 0 Ka

Pð Þ0 Kp Pð Þþ 0 Ka

0 þ μ<sup>t</sup> Pμ<sup>z</sup> We utilize an improved version of Lyapunov stability to design a control law for overhead cranes. The Lyapunov function is chosen so that its derivative is smaller than a positive constant. By this way, the Lyapunov candidate is selected similar to Eq. (28) but supplementing derivation of payload position <sup>1</sup> 2 Pð Þ0 Pð Þþ0 Ka ln <sup>k</sup><sup>2</sup> b1 k2 b1�z<sup>2</sup> 1 � �. A modified version of Lyapunov candidate is the so-called barrier Lyapunov V1(t) being in the form of

$$\begin{split} V\_{1} &= \frac{1}{2} \Bigg[ \left\{ \rho \left( n\_{t}^{2} + \mu\_{t}^{2} \right) + P \left( n\_{z}^{2} + \mu\_{z}^{2} \right) + \mathrm{EA} \left( \frac{1}{2} \left( n\_{z}^{2} + \mu\_{z}^{2} \right) \right)^{2} \right\} \Bigg] \mathrm{d}z \\ &+ \frac{M P(0)}{2(P(0) + K\_{4})} \left( n\_{t}^{2}(0, t) + \mu\_{t}^{2}(0, t) \right) + \frac{1}{2} m \left( n\_{t}^{2}(L, t) + \mu\_{t}^{2}(L, t) \right) \\ &+ \frac{P(0) K\_{p}}{2(P(0) + K\_{4})} \left( \sqrt{n^{2}(0, t) + \mu^{2}(0, t)} - q\_{d} \right)^{2} + \frac{1}{2} \frac{P\_{(0)}}{P\_{(0)} + K\_{4}} \ln \left( \frac{k\_{b1}^{2}}{k\_{b1}^{2} - z\_{1}^{2}} \right) \end{split} \tag{35}$$

where <sup>z</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup><sup>2</sup>ð Þþ <sup>L</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>ð Þ <sup>L</sup>; <sup>t</sup> <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>n</sup><sup>2</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>p</sup> is relative position of payload in comparison with that of trolley. kb<sup>1</sup> is a positive gain satisfying condition kb<sup>1</sup> > |z1|. The modification of (35) leads to

$$\begin{split} \dot{V}\_{1} &= \frac{P(0)}{P(0) + K\_{a}} n\_{t}(0, t) \left\{ F\_{x} - K\_{a} \left[ n\_{z}(0, t) + \frac{EA}{2.P(0)} \left( n\_{z}^{3}(0, t) + n\_{z}(0, t) \mu\_{z}^{2}(0, t) \right) \right] \right. \\ &\left. + K\_{p} \left( n(0, t) - \frac{q\_{d} n(0, t)}{\sqrt{\mu^{2}(0, t) + n^{2}(0, t)}} \right) \right\} \\ &\left. + \frac{P(0)}{P(0) + K\_{a}} \mu\_{t}(0, t) \left\{ F\_{y} - K\_{a} \left[ \mu\_{z}(0, t) + \frac{EA}{2.P(0)} \left( \mu\_{z}^{3}(0, t) + \mu\_{z}(0, t) \mu\_{z}^{2}(0, t) \right) \right] \right. \\ &\left. + K\_{p} \left( \mu(0, t) - \frac{q\_{d} \mu(0, t)}{\sqrt{\mu^{2}(0, t) + n^{2}(0, t)}} \right) \right\} + \frac{P(0)}{P(0) + K\_{a} k\_{b1}^{2} - z\_{1}^{2}} \end{split} \tag{36}$$

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ntð Þ 0; t Fx � Ka nzð Þþ 0; t

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

ð Þ 0; t Fy � Ka μzð Þþ 0; t

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

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

!

� Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

!

� Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>V</sup>\_ <sup>1</sup> <sup>≤</sup> � <sup>P</sup>ð Þ<sup>0</sup> <sup>K</sup><sup>d</sup>

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

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

Pð Þþ 0 Ka

<sup>t</sup>ð Þ 0; t

≤ ntð Þ 0; t sgn ð Þþ ntð Þ 0; t μ<sup>t</sup>

≤ j j ntð Þ 0; t þ μ<sup>t</sup>

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

1 k 2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

1 k 2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

> � <sup>1</sup> k2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

� <sup>1</sup> k 2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

þ

þ

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

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

n2

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

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

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

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

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

ð Þ 0; t sgn μ<sup>t</sup>

Nonlinear Control of Flexible Two-Dimensional Overhead Cranes

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

kb1K sgn ð Þ ntð Þ 0; t

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

ð Þ <sup>0</sup>; <sup>t</sup> � �)

kb1K sgn ð Þ ntð Þ 0; t

ð Þ 0; t

ð Þ <sup>0</sup>; <sup>t</sup> � �

kb1K sgn μ<sup>t</sup>

<sup>t</sup>ð Þ <sup>0</sup>; <sup>t</sup> � � <sup>≤</sup> <sup>0</sup> (42)

kb1K sgn μ<sup>t</sup>

ð Þ <sup>0</sup>; <sup>t</sup> � �

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

)

(39)

327

(40)

(41)

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

n2

<sup>t</sup>ð Þ 0; t

<sup>þ</sup> Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

μt

Fx ¼ Ka nzð Þþ 0; t

Fy ¼ Ka μzð Þþ 0; t

which leads the Eq. (31) to

asymptotical stability.

!

!

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

n2

Pð Þ0 Pð Þþ 0 Ka

Pð Þ0 Pð Þþ 0 Ka

q

into (38), one obtains

V\_ <sup>1</sup> ≤

þ

structured as

and

or

Applying the following inequality

$$\left| (z\_1)\_t \right| \le K \left| \sqrt{n\_t^2(0, t) + \mu\_t^2(0, t)} \right| $$

or

$$\left|z\_1(z\_1)\_t \le \left|z\_1(z\_1)\_t\right| = |z\_1| \left| (z\_1)\_t \right| \le k\_{b1} K \sqrt{n\_t^2(0,t) + \mu\_t^2(0,t)}$$

with K being positive constant leads to

$$\frac{P(0)}{P(0) + K\_d} \frac{z\_1(z\_1)\_t}{k\_{b1}^2 - z\_1^2} \le \frac{P(0)}{P(0) + K\_d} \frac{1}{k\_{b1}^2 - z\_1^2} k\_{b1} K \sqrt{n\_t^2(0, t) + \mu\_t^2(0, t)}\tag{37}$$

Inserting (37) into (36) yields

$$\begin{split} \dot{V}\_{1} &\leq \frac{P(0)n\_{t}(0,t)}{P(0)+K\_{a}} \left\{ F\_{x} - K\_{a} \left[ n\_{z}(0,t) + \frac{EA}{2.P(0)} \left( n\_{z}^{3}(0,t) + n\_{z}(0,t) \mu\_{z}^{2}(0,t) \right) \right] \right. \\ &\left. + K\_{p} \left( n(0,t) - \frac{q\_{d}n(0,t)}{\sqrt{\mu^{2}(0,t) + n^{2}(0,t)}} \right) \right\} + K\_{p} \left( \mu(0,t) - \frac{q\_{d}\mu(0,t)}{\sqrt{\mu^{2}(0,t) + n^{2}(0,t)}} \right) \right\} \\ &\quad + \frac{P(0)\mu\_{t}(0,t)}{P(0)+K\_{a}} \left\{ F\_{y} - K\_{a} \left[ \mu\_{z}(0,t) + \frac{EA}{2P(0)} \left( \mu\_{z}^{3}(0,t) + \mu\_{z}(0,t)n\_{z}^{2}(0,t) \right) \right] \right. \\ &\left. + \frac{P(0)k\_{b1}K}{(P(0)+K\_{a})(k\_{b1}^{2} - z\_{1}^{2})} \sqrt{n\_{t}^{2}(0,t) + \mu\_{t}^{2}(0,t)} \end{split} \tag{38}$$

Inserting the following inequality

$$
\sqrt{n\_t^2(0,t) + \mu\_t^2(0,t)} \le |n\_t(0,t)| + \left|\mu\_t(0,t)\right|^2
$$

or

where <sup>z</sup><sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

326 Adaptive Robust Control Systems

<sup>V</sup>\_ <sup>1</sup> <sup>¼</sup> <sup>P</sup>ð Þ<sup>0</sup> Pð Þþ 0 Ka

þ

Pð Þ0 Pð Þþ 0 Ka

Applying the following inequality

with K being positive constant leads to

Pð Þ0 ntð Þ 0; t Pð Þþ 0 Ka

Inserting (37) into (36) yields

V\_ <sup>1</sup> ≤

þ

þ

Pð Þ0 μ<sup>t</sup>

Inserting the following inequality

ð Þ 0; t Pð Þþ 0 Ka

Pð Þ0 kb1K ð Þ <sup>P</sup>ð Þþ <sup>0</sup> Ka <sup>k</sup><sup>2</sup>

Pð Þ0 Pð Þþ 0 Ka

(35) leads to

or

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

<sup>þ</sup> Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>þ</sup> Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

z1ð Þ z<sup>1</sup> <sup>t</sup> ≤ z1ð Þ z<sup>1</sup> <sup>t</sup> � � �

> z1ð Þ z<sup>1</sup> <sup>t</sup> k 2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1 ≤

<sup>þ</sup> Kp <sup>n</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdnð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1 � �

Fx � Ka nzð Þþ 0; t

!)

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

Fy � Ka μzð Þþ 0; t

q

n2

μt

ntð Þ 0; t Fx � Ka nzð Þþ 0; t

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

ð Þ 0; t Fy � Ka μzð Þþ 0; t

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

!)

!)

ð Þ z<sup>1</sup> <sup>t</sup> � � � � ≤K

ison with that of trolley. kb<sup>1</sup> is a positive gain satisfying condition kb<sup>1</sup> > |z1|. The modification of

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

� ≤ kb1K

1 k 2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1 kb1K

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

<sup>t</sup>ð Þ 0; t

EA <sup>2</sup>:Pð Þ<sup>0</sup> <sup>n</sup><sup>3</sup>

EA <sup>2</sup>Pð Þ<sup>0</sup> <sup>μ</sup><sup>3</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

n2

� � q � �

� ¼ j j z<sup>1</sup> ð Þ z<sup>1</sup> <sup>t</sup> � � �

Pð Þ0 Pð Þþ 0 Ka

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

Pð Þ0 Pð Þþ 0 Ka

<sup>t</sup>ð Þ 0; t

n2

q

� � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>t</sup>ð Þ 0; t

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>t</sup>ð Þ 0; t

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

!)

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

<sup>t</sup>ð Þþ 0; t μ<sup>2</sup>

n2

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

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

<sup>þ</sup>Kp <sup>μ</sup>ð Þ� <sup>0</sup>; <sup>t</sup> qdμð Þ <sup>0</sup>; <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q

<sup>z</sup> ð Þ <sup>0</sup>; <sup>t</sup> � � � � �

<sup>n</sup><sup>2</sup>ð Þþ <sup>0</sup>; <sup>t</sup> <sup>μ</sup><sup>2</sup>ð Þ <sup>0</sup>; <sup>t</sup> <sup>p</sup> is relative position of payload in compar-

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

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

(36)

(37)

(38)

z1ð Þ z<sup>1</sup> <sup>t</sup> k2 <sup>b</sup><sup>1</sup> � <sup>z</sup><sup>2</sup> 1

$$
\sqrt{n\_t^2(0,t) + \mu\_t^2(0,t)} \le n\_t(0,t) \operatorname{sgn}\left(n\_t(0,t)\right) + \mu\_t(0,t) \operatorname{sgn}\left(\mu\_t(0,t)\right),
$$

into (38), one obtains

$$\begin{aligned} \dot{V}\_1 &\le \frac{P(0)}{P(0) + K\_a} n\_l(0, t) \left\{ F\_x - K\_a \left[ n\_z(0, t) + \frac{EA}{2P(0)} \left( n\_z^3(0, t) + n\_z(0, t) \mu\_z^2(0, t) \right) \right] \right. \\\\ &\left. + K\_p \left( n(0, t) - \frac{q\_d n(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) + \frac{1}{k\_{b1}^2 - z\_1^2} k\_{b1} K \operatorname{sgn} \left( n\_l(0, t) \right) \right\} \\\\ &\quad + \frac{P(0)}{P(0) + K\_a} \mu\_t(0, t) \left\{ F\_y - K\_a \left[ \mu\_z(0, t) + \frac{EA}{2P(0)} \left( \mu\_z^3(0, t) + \mu\_z(0, t) \mu\_z^2(0, t) \right) \right] \right. \\\\ &\left. + K\_p \left( \mu(0, t) - \frac{q\_d \mu(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) + \frac{1}{k\_{b1}^2 - z\_1^2} k\_{b1} K \operatorname{sgn} \left( \mu\_t(0, t) \right) \right\} \end{aligned} \tag{39}$$

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

$$\begin{aligned} F\_x &= K\_d \left[ n\_z(0, t) + \frac{EA}{2.P(0)} \left( n\_z^3(0, t) + n\_z(0, t) \mu\_z^2(0, t) \right) \right] - K\_d n\_t(0, t) \\ &- K\_p \left( n(0, t) - \frac{q\_d n(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) - \frac{1}{k\_{b1}^2 - z\_1^2} k\_{b1} K \operatorname{sgn} \left( n\_t(0, t) \right) \end{aligned} \tag{40}$$

and

$$\begin{aligned} F\_y &= K\_d \left[ \mu\_z(0, t) + \frac{EA}{2.P(0)} \left( \mu\_z^3(0, t) + \mu\_z(0, t) n\_z^2(0, t) \right) \right] - K\_d \mu\_t(0, t) \\ &- K\_p \left( \mu(0, t) - \frac{q\_d \mu(0, t)}{\sqrt{\mu^2(0, t) + n^2(0, t)}} \right) - \frac{1}{k\_{b1}^2 - z\_1^2} k\_{b1} K \operatorname{sgn} \left( \mu\_t(0, t) \right) \end{aligned} \tag{41}$$

which leads the Eq. (31) to

$$\dot{V}\_1 \le -\frac{P(0)\mathcal{K}\_d}{P(0) + \mathcal{K}\_d} \left( n\_t^2(0, t) + \mu\_t^2(0, t) \right) \le 0 \tag{42}$$

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