**4. Simulation verification**

In this section, by using the MATLAB/Simulink software, some simulation results are included to verify the effectiveness of the proposed observer-based robust control method.

For the control objectives of the two cases stated in Eq. (21) and Eq. (22), the simulation is implemented through two groups as follows:

• **Group 1.** The perturbations in the unactuated dynamics are non-vanishing. The perturbation *dθ*(*t*) is set as a constant value *dθ*(*t*) = 1 and a time-varying function *dθ*(*t*) = 0.5 cos(0.1*t*), respectively.

• **Group 2.** The perturbations in the unactuated dynamics are vanishing or negligible. The perturbation *dθ*(*t*) is set as a time-varying function *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>* .

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

306 Adaptive Robust Control Systems

of Eq. (24) that

*e* 3 (*t*), *e*̇ 3

*ϕ*1

ing conclusions:

respectively.

*e*<sup>3</sup> → *e*̇

exponential fashion. Since *rx*

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

(*t*) = *φ*<sup>1</sup>

which is just the result of Eq. (21). In addition, as *r*¨*<sup>x</sup>*

<sup>2</sup> − *δ<sup>u</sup>* → −*δ<sup>u</sup>*

Eq. (16)], it is implied by substituting the result of *e*̇

the unactuated dynamics is vanishing [i.e., *δ<sup>u</sup>* <sup>→</sup> 0, *<sup>δ</sup>*

entire theoretical proof for the theorem is completed.

implemented through two groups as follows:

**4. Simulation verification**

*e*<sup>3</sup> = −*g* tan*θ* → 0, *e*<sup>4</sup> = −*g θ*

2 (*t*) = *e*¨<sup>1</sup>

(*t*) − *rx* → 0, *e*<sup>2</sup>

(*t*) = *e*̇ 1 (*t*) = *φ*̇ 1 (*t*) − *r*̇

(*t*) → 0, *e*¨<sup>2</sup>

exponentially fast, which indicates the cargo motion tracks the planned trajectory *rx*

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

<sup>3</sup> → *e*¨<sup>2</sup> − *δ*

(*t*) tends to *pdx* within *tf*<sup>1</sup>

, *e*̇

tions. Thus, the result of **Case 1** stated in the control objective is proven.

Therefore, in such cases, it is straightforward to indicate from Eq. (53) that

<sup>⇒</sup> *<sup>e</sup>*̇

(*t*) = *e*<sup>1</sup> (3)

> ̇ 1

2

̇ *<sup>u</sup>* → −*δ* ̇ *u* , *e*<sup>4</sup> → *e*̇

(*t*) are convergent to their respective equilibriums drifted by the unactuated perturba-

̇

wherein the conclusions in Eq. (52) have been employed. The results in Eq. (53) indicate that

Subsequently, we proceed to prove the result of **Case 2** where the perturbation term *δθu*(*t*) in

̇

where the definitions in Eq. (10) and Eq. (23) have been used. According to the definition of

lim*<sup>t</sup>*→<sup>∞</sup> *<sup>x</sup>*(*t*) <sup>=</sup> *pdx*, lim*<sup>t</sup>*→<sup>∞</sup> *<sup>x</sup>*̇(*t*) <sup>=</sup> 0. (55)

Collecting up Eqs. (52, 54, 55), the results claimed in Eq. (22) of **Case 2** are hence proven. The

In this section, by using the MATLAB/Simulink software, some simulation results are included

For the control objectives of the two cases stated in Eq. (21) and Eq. (22), the simulation is

• **Group 1.** The perturbations in the unactuated dynamics are non-vanishing. The perturbation *dθ*(*t*) is set as a constant value *dθ*(*t*) = 1 and a time-varying function *dθ*(*t*) = 0.5 cos(0.1*t*),

to verify the effectiveness of the proposed observer-based robust control method.

(*t*) = *x*(*t*) + *Lθ*(*t*) given in Eq. (13), the results in Eq. (52) and Eq. (54) directly yield the follow-

(*t*), *r x* (3) *<sup>x</sup>* → 0

[see Eq. (16)], it is easily shown that

(*t*) = 0, (52)

(*t*) → 0 into the second and third equations

<sup>3</sup> → −*δ* ̇ *u*

*<sup>u</sup>* <sup>→</sup> 0] or negligible [i.e., *δ<sup>u</sup>*

sec2 *θ* → 0 ⇒ *θ* = 0, *θ*

(*t*) <sup>→</sup> 0 as *t* → 0 by definition [see

, (53)

(*t*) = 0, *δ* ̇ *u* (*t*) = 0].

̇ = 0, (54)

(*t*) <sup>→</sup> 0, (51)

(*t*) in an

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> , the controller parameters as*λ*<sup>2</sup>  = 10, *λ*<sup>5</sup>  = 30, *λ*<sup>6</sup>  = 55, *λ*<sup>7</sup>  = 25, *α* = 2, *β* = 1, *γ* = 0.2, *ε* = 0.01, *ku*  = 60, *ka*  = 0.1, and the to-be-tracked trajectory in Eq. (16) as *rx* (*t*) = 3.5, the simulation results are obtained and are shown in **Figures 2**–**4**.

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

**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

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

and the unactuated cargo are driven to the desired destination, when there are no/negligible

To sum-up, the simulation results indicate that the proposed observer-based robust controller can achieve robust control in the presence of uncertainties or external perturbations, which is

Considering unknown persistent perturbations in unactuated dynamics, this chapter designs an observer-based robust control method for underactuated crane systems. Specifically, a reduced-order augmented-state observer is designed to recover the lumped perturbation terms in unactuated dynamics. Further, based on the observer, a new sliding manifold is constructed to improve the robust performance of the control system. Then, the state variables are made to stay on the manifold by applying a designed robust control law in the presence of non-vanishing perturbations in unactuated dynamics. Finally, the convergence is proved in this chapter theoretically by using Lyapunov control theories. Moreover, the proposed observer-based robust controller is verified to be effective and robust by numerical

<sup>y</sup>′ <sup>=</sup> *<sup>C</sup>*

0 1 0 ⋯ 0 0

 0 1 ⋯ 0 0 0 0 ⋯ 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 1 0 0 0 ⋯ 0 1 0 0 ⋯ 0 0

*C* = (1 0 0 ⋯ 0 0).

Then, considering the observability criteria, we can first derive the observable matrix *Ψ* as

2 , *e*̇ 5 , *e*̇ 6 , ⋯ ,*e*̇ *n*+1) ⊤

, 

⎟

⎠

⎞

(56)

(57)

. The system parameter

in the unactuated dynamics.

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383 309

or vanishing perturbations *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>*

consistent with the theoretical analysis.

The system shown in Eq. (26) can be rewritten as follows:

where the system variable vector *χ*(*t*) is defined as <sup>=</sup> (*e*̇

matrix *A* ∈ *R*(*n* − 2) × (*n* − 2) and *C* ∈ *R*(*n* − 2) are

*<sup>A</sup>* <sup>=</sup>

̇ <sup>=</sup> *<sup>A</sup>* <sup>+</sup> (*e*<sup>3</sup> <sup>−</sup> *<sup>r</sup>*¨*<sup>x</sup>* <sup>0</sup> <sup>0</sup> <sup>⋯</sup> <sup>0</sup> 0)⊤,

⎛

⎜

⎝

**5. Concluding remarks**

simulation results.

**Appendix**

follows:

**Figure 3.** The simulation results of the proposed controller when *dθ*(*t*) = 0.5 cos(0.1*t*) (solid line – simulation results, dash line – desired trajectory).

**Figure 4.** The simulation results of the proposed controller when *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>* (solid line – simulation results, dash line – desired trajectory).

and the unactuated cargo are driven to the desired destination, when there are no/negligible or vanishing perturbations *dθ*(*t*) = 1.5*e*<sup>−</sup>*<sup>t</sup>* in the unactuated dynamics.

To sum-up, the simulation results indicate that the proposed observer-based robust controller can achieve robust control in the presence of uncertainties or external perturbations, which is consistent with the theoretical analysis.
