**1. Introduction**

Underactuated systems [1–9] are now widely applied in modern industry. A crane system is a typical class of underactuated systems with strong state coupling. Due to inertia, when the trolley moves, the unactuated cargo swings back and forth, which affects the transporting efficiency and safety. Therefore, on the one hand, effective controllers are needed to transport the actuated trolley to desired positions. On the other hand, it is also necessary to eliminate residual vibrations of the unactuated cargo. Nevertheless, control problems of crane systems are still non-trivial and challenging since the system is underactuated without enough available control inputs.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In order to tackle control problems of crane systems, various control methods are proposed [10–38]. Specifically, Sun et al. [10, 11] present antiswing controllers to regulate the cargo position to the desired location asymptotically in the presence of ship roll and heave movements for offshore crane systems applied in modern ocean transportation and logistics. Moreover, existing methods also include input shaping [12–15], feedback control [16–28], intelligent control [29–32], and trajectory planning method [33–36]. Specifically, several input shapers are designed to reduce payload swing of bridge crane systems [12–15]. In Ref. [16], an energybased output feedback control scheme is proposed, which achieves both precise trolley positioning and efficient payload swing elimination under control input constraints. In Ref. [17], a payload motion-based control approach is presented in the presence of system parameter uncertainties. In [18–20], non-linear controllers are designed on the basis of partial feedback linearization. In Ref. [21], visual feedback technology is used to achieve the control objective by using two handy cameras. Additionally, sliding mode control strategies are also widely applied to tackle crane system control problems [22–25]. For example, Almutairi and Zribi [22] achieved the asymptotic stability of the closed-loop overhead crane system by proposing a sliding mode control scheme. Xi and Hesketh [23] addressed an integral sliding mode control method for discrete time crane systems with both matched and unmatched uncertainties to ensure the existence of sliding mode in the presence of uncertainties. Based on second-order sliding modes, Bartolin et al. [24] guaranteed a fast and precise payload transferring and swing suppression. Ngo and Hong [25] developed an adaptation law with a varying control gain that transits the system into the designed sliding mode. Moreover, in practical applications, cranes always suffer from unknown or uncertain system parameters (e.g., payload weight changes, varying rope lengths, etc.). Then adaptive control schemes are applied to address these problems [26–28]. Sun et al. [26] addressed the crane antiswing and positioning problem in the presence of payload hoisting/lowering and uncertain parameters with simultaneous payload weight identification. Park et al. [27] proposed an adaptive slidingmode antisway control law with system uncertainties and high-speed hoisting motion. Sun et al. [28] designed an adaptive control scheme to deal with the control problem of tower crane systems with parametric uncertainties without approximating the non-linear dynamics. There are also some intelligent control methods applied in crane systems such as fuzzy control [29, 30], genetic algorithm [31], and neural network [32]. According to the operating experience of real cranes, it is also essential to design suitable trajectories for the system states (positions, velocities, and accelerations). Then, tracking controllers can be used to track the trajectories. In addition to closed-loop control design, many studies also focus on the trajectory planning part and achieve meaningful results [33–36]. Uchiyama et al. [33] generated an S-curve trajectory numerically, which can suppress the residual vibration without measuring it. Sun et al. [34] obtained an analytical three-segment acceleration trajectory. For given transferring task, the proposed trajectory planning method provides a mechanism to determine the parameters to ensure that all the transportation indexes are met. More recently, in Ref. [35], an optimal trajectory is generated with optimal energy consumption by using the proposed optimal planner. There are also antiswing control strategies proposed for double pendulum cranes [37, 38].

may be difficult to tackle by using existing methods. Note that in Ref. [23], integral sliding mode control method is proposed by considering perturbations in unactuated dynamics, but it is only designed for discrete-time systems by estimating the present disturbance signal with its past value. Therefore, in order to derive an effective method to achieve crane control in the presence of unknown persistent (even non-vanishing) perturbations in both unactuated and

**1.** According to whether the perturbation in the unactuated dynamics is vanishing or not, the control problem is stated in two cases. The observer-based robust controller designed in

**2.** By dealing with the unactuated and unknown perturbation as an augmented state variable, an augmented error system is established based on which we design a reduced-order augmented state observer for the crane system to recover the perturbations appearing in

**3.** Together with the observer, by constructing a new sliding manifold, a new observer-based

The proposed controller is applicable to crane systems with unknown persistent perturba-

The rest of this chapter is organized as follows. Section 2 describes the crane dynamics with persistent (even non-vanishing) perturbations and transforms the dynamics into a quasichain-of-integrators form for the convenience of controller design and stability analysis. Also, the control objective is stated in Section 2. Based on the model in Section 2, a reduced-order augmented-state observer and an observer-based control law are developed in Section 3. Then in Section 4, numerical simulation results are included to verify the effectiveness of the

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

̇ <sup>2</sup> sin*θ* = *u* − *f*

*r*

*<sup>r</sup>* + *dx*

denotes the rail friction force expressed

*θ*¨ + *mL* cos*θx*¨ + *mgL* sin*θ* = *dθ*. (2)

, (1)

Robust Control of Crane with Perturbations http://dx.doi.org/10.5772/intechopen.71383 295

actuated dynamics, this chapter proposes an observer-based robust control method.

The main contribution of this chapter is as follows:

the unactuated dynamics.

**2. Problem formulation**

*mL*<sup>2</sup>

(*M* + *m*)*x*¨ + *mLθ*¨ cos*θ* − *mL θ*

The system parameters are defined in **Table 1**, and *f*

in **Figure 1**):

as follows:

sliding mode controller is developed.

this chapter can achieve the control objectives for both cases.

tions in the unactuated dynamics and achieves robust control effectively.

proposed controller. Section 5 summarizes the entire work of this chapter.

However, most of the existing methods for underactuated crane systems tackle the control problem without considering the perturbations in the unactuated dynamics. In practical applications, perturbations widely exist in both actuated and unactuated dynamics, which may be difficult to tackle by using existing methods. Note that in Ref. [23], integral sliding mode control method is proposed by considering perturbations in unactuated dynamics, but it is only designed for discrete-time systems by estimating the present disturbance signal with its past value. Therefore, in order to derive an effective method to achieve crane control in the presence of unknown persistent (even non-vanishing) perturbations in both unactuated and actuated dynamics, this chapter proposes an observer-based robust control method.

The main contribution of this chapter is as follows:

In order to tackle control problems of crane systems, various control methods are proposed [10–38]. Specifically, Sun et al. [10, 11] present antiswing controllers to regulate the cargo position to the desired location asymptotically in the presence of ship roll and heave movements for offshore crane systems applied in modern ocean transportation and logistics. Moreover, existing methods also include input shaping [12–15], feedback control [16–28], intelligent control [29–32], and trajectory planning method [33–36]. Specifically, several input shapers are designed to reduce payload swing of bridge crane systems [12–15]. In Ref. [16], an energybased output feedback control scheme is proposed, which achieves both precise trolley positioning and efficient payload swing elimination under control input constraints. In Ref. [17], a payload motion-based control approach is presented in the presence of system parameter uncertainties. In [18–20], non-linear controllers are designed on the basis of partial feedback linearization. In Ref. [21], visual feedback technology is used to achieve the control objective by using two handy cameras. Additionally, sliding mode control strategies are also widely applied to tackle crane system control problems [22–25]. For example, Almutairi and Zribi [22] achieved the asymptotic stability of the closed-loop overhead crane system by proposing a sliding mode control scheme. Xi and Hesketh [23] addressed an integral sliding mode control method for discrete time crane systems with both matched and unmatched uncertainties to ensure the existence of sliding mode in the presence of uncertainties. Based on second-order sliding modes, Bartolin et al. [24] guaranteed a fast and precise payload transferring and swing suppression. Ngo and Hong [25] developed an adaptation law with a varying control gain that transits the system into the designed sliding mode. Moreover, in practical applications, cranes always suffer from unknown or uncertain system parameters (e.g., payload weight changes, varying rope lengths, etc.). Then adaptive control schemes are applied to address these problems [26–28]. Sun et al. [26] addressed the crane antiswing and positioning problem in the presence of payload hoisting/lowering and uncertain parameters with simultaneous payload weight identification. Park et al. [27] proposed an adaptive slidingmode antisway control law with system uncertainties and high-speed hoisting motion. Sun et al. [28] designed an adaptive control scheme to deal with the control problem of tower crane systems with parametric uncertainties without approximating the non-linear dynamics. There are also some intelligent control methods applied in crane systems such as fuzzy control [29, 30], genetic algorithm [31], and neural network [32]. According to the operating experience of real cranes, it is also essential to design suitable trajectories for the system states (positions, velocities, and accelerations). Then, tracking controllers can be used to track the trajectories. In addition to closed-loop control design, many studies also focus on the trajectory planning part and achieve meaningful results [33–36]. Uchiyama et al. [33] generated an S-curve trajectory numerically, which can suppress the residual vibration without measuring it. Sun et al. [34] obtained an analytical three-segment acceleration trajectory. For given transferring task, the proposed trajectory planning method provides a mechanism to determine the parameters to ensure that all the transportation indexes are met. More recently, in Ref. [35], an optimal trajectory is generated with optimal energy consumption by using the proposed optimal planner. There are also antiswing control strategies proposed for double pendulum cranes [37, 38]. However, most of the existing methods for underactuated crane systems tackle the control problem without considering the perturbations in the unactuated dynamics. In practical applications, perturbations widely exist in both actuated and unactuated dynamics, which

294 Adaptive Robust Control Systems


The proposed controller is applicable to crane systems with unknown persistent perturbations in the unactuated dynamics and achieves robust control effectively.

The rest of this chapter is organized as follows. Section 2 describes the crane dynamics with persistent (even non-vanishing) perturbations and transforms the dynamics into a quasichain-of-integrators form for the convenience of controller design and stability analysis. Also, the control objective is stated in Section 2. Based on the model in Section 2, a reduced-order augmented-state observer and an observer-based control law are developed in Section 3. Then in Section 4, numerical simulation results are included to verify the effectiveness of the proposed controller. Section 5 summarizes the entire work of this chapter.
