Robust integral sliding mode control of tower cranes

Abstract

In this study, integral sliding mode control is proposed for tower cranes to ensure precise tracking of the desired position while reducing the oscillations of the payload. The nonlinear robust controller is designed based on high fidelity nonlinear dynamical model, unlike the decoupled or linearized models used in the literature. The advantage of this approach is reducing the model uncertainties resulting in a lower control effort demand that would be required by the sliding mode controller. Moreover, the stability of the under-actuated tower crane system is analyzed using Lyapunov theory to guarantee the practical stability of error dynamics. Experimental results of the proposed control approach are compared with conventional sliding mode control to show its effectiveness and robustness against real system uncertainties.

Keywords

Tower crane under-actuated system coupled dynamics integral sliding mode control

1. Introduction

Cranes have been the subject of research for decades because of their importance in many fields and can be classified into several types; tower cranes, rotary cranes, and overhead cranes. Tower cranes are the main focus of this study for their important role in transporting heavy loads during construction. The transportation process of the payload is required to have high speed, minimum swing, and precision of destination, to raise its efficiency and productivity.

Tower cranes are considered as strong mechanical systems which exhibit complicated nonlinear dynamics. Also, they are under-actuated systems with only two inputs to control four degrees of freedom (DOF). The unactuated DOF are the oscillations of the payloads that are required to be eliminated by the controller. Therefore, the complexity of designing a controller is increased compared with fully actuated systems.

Many researchers have developed different controllers for overhead cranes to handle uncertainties that arise in practical implementations, including adaptive control proposed by Yang and Yang (2007), Cho and Lee (2008), and Le et al. (2012), fuzzy controller by Antic et al. (2012) and Aksjonov et al. (2015). Also, robust sliding mode control (SMC) has been proposed by Vázquez et al. (2015) and Qian and Yi (2016).

In the review of Abdel-Rahman et al. (2003), the proposed controllers for tower cranes are fewer than overhead cranes because of its complexity. The main aim of tower cranes’ controllers is to eliminate the oscillations of the payload that arise during the transportation process and more specifically in fast maneuvers.

Parker et al. (1995) proposed an open-loop optimal control technique using a simplified model for the tower crane. The optimization problem was solved using dynamic programming and the results showed effectiveness in reducing the oscillations but ignored the effect of parameter variations. Golafshani and Aplevich (1995) generated optimal cargo trajectory tracking solutions using an iterative algorithm. In the meanwhile, the oscillation of the payload was significant even at steady state. Besides optimal control, Al-Mousa (2000) proposed a fuzzy logic and time-delayed position feedback controllers.

Most of the previous work is based on the assumption of a friction-less system. In the real system, friction has a strong impact on the dynamics of the tower crane and should be included in the controller design. Therefore, Omar and Nayfeh (2005) used the gain-scheduling control law and a laboratory tower crane by considering the friction factor to enhance the study (Al-Mousa, 2000). The controller gains were scheduled to facilitate the suppression of payload vibration within one cycle at the trolley destination.

Another approach to controlling the tower crane system is by using feedforward control techniques. The input shaping method was used by Vaughan et al. (2010) and Samin et al. (2013) to reduce the system’s sways at response modes. The control input is developed through consideration of the physical and swaying properties of the system. However, these approaches often lack robustness concerning parametric uncertainties (such as varying mass and friction coefficients), external disturbances and could not damp residual swing well.

Various techniques based on closed-loop systems which are known to be less sensitive to parametric uncertainties and external disturbances were proposed such as the robust adaptive control by He et al. (2014) and iterative learning control by He et al. (2018). Elbadawy and Shehata (2015) and Ahmad et al. (2015) tested a closed-loop proportional integral derivative input shaper against external disturbances experimentally.

Böck and Kugi (2014) based the controller on a model that contains explicit differential equations. Its complexity is much lower than that of a full nonlinear model. The path-following controller is combined with the model predictive control and applied into a laboratory tower crane for evaluation. Bariša et al. (2014) decoupled the model into three subsystems, and the cross couplings between the equations were considered as a change in the system’s parameters for each subsystem. The proposed nonlinear model predictive control was designed for each one and validated experimentally for the trolley subsystem. Breuning (2015) linearized the model to apply linear quadratic predictive control. Wu et al. (2016) used a linearized model for the construction of H∞-based adaptive fuzzy control technique which considers uncertainties, time delays, and external disturbances.

The preceding articles commonly developed their controllers based on simplified models, to simplify the control system design. The simplified models ignore nonlinear coupling dynamics by assuming small swing angle changes and the rate of change of the trolley position, and the jib rotation is of the same order of magnitude of the swing angles and their rates. The performance of the controllers based on these simplified or linearized models is degraded in the presence of uncertainties. The desired performance including the elimination of steady-state error and damping the oscillations can be achieved using high control effort. However, when the control demand is high, it will exceed the motors’ capability and the desired performance will not be achieved. Therefore, the current work and the following literature focused on deriving various controllers based on the full nonlinear model without any simplification or linearization.

Le et al. (2013) designed two nonlinear controllers, partial feedback linearization and SMC, based on the full nonlinear model of the tower crane. However, the control effort for these controllers was not discussed, and the controllers were not applied experimentally. Also, the issue of steady-state error was not explicitly addressed that arise in real-life application. The SMC was presented by Utkin et al. (2009) that robustness can be achieved against uncertainties using a discontinuous control technique for several practical systems. Also, it constrains the system motion along the manifolds of reduced dimensionality in the state space.

Sun et al. (2016) has developed an adaptive control against parametric uncertainties using the full nonlinear model without simplifications. Le and Lee (2017) proposed a model reference adaptive-SMC for 4-DOF system, without experimental validation. Adaptive backstepping SMC is proposed for 2-DOF system by Bai and Ren (2018) to adapt the uncertainty due to the environmental interference with the tower crane operation and was tested experimentally.

The limitation of the SMC is being sensitive to noise and uncertainties during the reaching phase because the sliding mode is not realized. Eventually, when the uncertainties are large, the switching gain of SMC needs to be high for robustness. To make the SMC practically realizable, the switching gain should be limited, which results in degrading the performances of crane control. Therefore, integral sliding mode control (ISMC) was proposed by Utkin and Shi (1996), where the system trajectory always starts from the sliding surface and the reaching phase is eliminated.

Xi and Hesketh (2010) proposed ISMC for controlling overhead cranes which were developed for the single input single output system and any cross-coupling between the two motions in the multi input multi output model simply adds to the uncertainties. Integral sliding surface designs were developed for systems with matched and unmatched uncertainties, which might arise for imperfect modeling and external disturbances. Also, Sun et al. (2019) have introduced an integral term into the controller to overcome the steady state error that exists because of uncertainties.

This study presents the development of a robust control scheme for a four-DOF under-actuated tower crane system. The nonlinear controller is based on the high fidelity model without simplifications or linearization. The dynamic model of the system is derived using the Euler–Lagrange formulation with the consideration of friction. The ISMC is proposed to overcome the issue of steady-state error that arises in practical applications. The main aim of the controller is fast and accurate positioning of the payload to the desired final position while damping the oscillations, regardless of uncertainties. The stability of the closed-loop system is provided, including the designed controller and the integral terms. Also, Lyapunov-based stability analysis is used for coupled nonlinear under-actuated dynamics. The robustness of the controller is tested against uncertainties. Finally, a comparative assessment of the proposed controller with the conventional SMC is presented and validated experimentally.

This article is organized as follows: Section 2 introduces the tower crane system and the corresponding nonlinear dynamic model. The controller design process with stability analysis is presented in Section 3. In Section 4, the proposed control is evaluated through simulation and experimental results. Conclusions are drawn in Section 5.

2. Model

Dynamical equations are derived based on the Euler–Lagrange equation from Spong et al. (2006), resulting in four second-order nonlinear differential equations corresponding to four DOF

\frac{d}{d t} (\frac{\partial L}{\partial \dot{\vec{q}}}) - \frac{\partial L}{\partial \vec{q}} = \vec{u} - (B_{\vec{q}} \dot{\vec{q}} + μ_{\vec{q}} sgn (\dot{\vec{q}}))

(1)

where

\vec{q}

represents the joint angle vector including the four DOFs shown in Figure 1

\vec{q} = {[\begin{matrix} q_{1} & q_{2} & q_{3} & q_{4} \end{matrix}]}^{T} = {[\begin{matrix} x & θ & α & β \end{matrix}]}^{T}

. x is the position of the trolley, θ is the rotation of the jib around the z-axis, α is the in-plane oscillation, and β is the out-of-plane oscillation.

B_{\vec{q}} \dot{\vec{q}}

represents the viscous friction and

μ_{\vec{q}} sgn (\dot{\vec{q}})

represents the Coulomb friction. The input vector

\vec{u} = {[\begin{matrix} u_{1} & u_{2} & 0 & 0 \end{matrix}]}^{T}

has only two control inputs which are the trolley driving force (u₁) and the tower rotating torque (u₂) respectively.

Actuator dynamics are incorporated into the crane model and presented using equations (2) and (3).

u_{1} = \frac{η_{x} K_{g x} K_{m x}}{R_{a x} r_{x}} G_{a x} U_{x} - \frac{η_{x} k_{g x}^{2} k_{m x}^{2}}{R_{a x} r_{x}^{2}} \dot{x}

(2)

u_{2} = \frac{η_{θ} K_{g θ} K_{m θ}}{R_{a θ}} G_{a θ} U_{θ} - \frac{η_{θ} k_{g θ}^{2} k_{m θ}^{2}}{R_{a θ}} \dot{θ}

(3)

where U_x is x-direction pulse width modulation (PWM) signal and U_θ is θ-direction PWM signal and the rest of the motor parameters with their values are presented in Table 1 and based on Breuning (2015). The dynamic model is represented in the form of

M (q) \ddot{q} + B (q, \dot{q}) + G (q) = u

(4)

where M(q)

\in

R^n×n is the inertia matrix which is a positive definite matrix for L > 0 and its inverse exists,

B (q, \dot{q}) \in R^{n \times 1}

is the Coriolis, centripetal, and friction matrix, and G(q)

\in

R^n×1 is the gravitational force vector.

Table 1.

Motor parameters.

Description	Value in x-direction	Value in θ-direction
Gear ratio	K_gx = 76.84	K_gθ = 275
Motor gearbox and motor efficiency	η_x = 0.36	η_θ = 0.24
Torque constant (Nm/A)	k_mx = 0.032	k_mθ = 0.0195
Radius of pulley	r_x = 0.0375 m	—
Armature resistance	R_ax = 25 Ω	R_aθ = 0.5 Ω
Amplifier gain	G_ax = 15	G_aθ = 12

The DOFs consists of two vectors, actuated states (q_a) and unactuated states (q_u). The actuated states are the trolley position and the jib rotation, whereas the unactuated states are the swing angles of the payload

q_{a} = {[\begin{matrix} x & θ \end{matrix}]}^{T}, q_{u} = {[\begin{matrix} α & β \end{matrix}]}^{T}

(5)

Therefore, equation (4) is arranged into two equations

M_{11} (q) {\ddot{q}}_{a} + M_{12} (q) {\ddot{q}}_{u} + B_{11} (q, \dot{q}) = u_{a}

(6)

M_{21} (q) {\ddot{q}}_{a} + M_{22} (q) {\ddot{q}}_{u} + B_{21} (q, \dot{q}) + G_{2} (q) = 0

(7)

where

\begin{array}{l} M_{11} = [\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}], M_{12} = [\begin{matrix} m_{13} & m_{14} \\ m_{23} & m_{24} \end{matrix}], \\ M_{21} = [\begin{matrix} m_{31} & m_{32} \\ m_{41} & m_{42} \end{matrix}], M_{22} = [\begin{matrix} m_{33} & m_{34} \\ m_{43} & m_{44} \end{matrix}], \\ B_{11} = [\begin{matrix} b_{11} \\ b_{21} \end{matrix}], B_{21} = [\begin{matrix} b_{31} \\ b_{41} \end{matrix}], \\ u_{a} = [\begin{matrix} F_{x} \\ F_{θ} \end{matrix}], G_{2} = [\begin{matrix} g_{1} \\ g_{2} \end{matrix}] \end{array}

The elements of the previous matrices and vectors are presented in Appendix 1. The control inputs have a direct effect on the actuated dynamics equation (6) whereas there is no effect on the unactuated dynamics equation (7). The following reformulation of the equations will allow the control inputs to affect the unactuated dynamics. Equations (6) and (7) are rewritten as follows

{\ddot{q}}_{a} = M_{11}^{- 1} (q) (- M_{12} (q) {\ddot{q}}_{u} - B_{11} (q, \dot{q}) + u_{a})

(8)

{\ddot{q}}_{u} = M_{22}^{- 1} (q) (- M_{21} (q) {\ddot{q}}_{a} - B_{21} (q, \dot{q}) - G_{2} (q))

(9)

Equation (9) is substituted into (6) and (8) is substituted into (7). The resulting equations are rearranged so that the system can be expressed as follows

{\bar{M}}_{1} (q) {\ddot{q}}_{a} + {\bar{B}}_{11} (q, \dot{q}) + G_{1} (q) = u_{a}

(10)

{\bar{M}}_{2} (q) {\ddot{q}}_{u} + {\bar{B}}_{21} (q, \dot{q}) + G_{2} (q) = u_{u}

(11)

where

\begin{array}{l} {\bar{M}}_{1} (q) = M_{11} (q) - M_{12} (q) M_{22} {(q)}^{- 1} M_{21} (q) \\ {\bar{M}}_{2} (q) = M_{22} (q) - M_{21} (q) M_{11} {(q)}^{- 1} M_{12} (q) \\ {\bar{B}}_{11} (q, \dot{q}) = - M_{12} (q) M_{22} {(q)}^{- 1} B_{21} (q, \dot{q}) + B_{11} (q, \dot{q}) \\ {\bar{B}}_{21} (q) = - M_{21} (q) M_{11} {(q)}^{- 1} B_{11} (q, \dot{q}) + B_{21} (q, \dot{q}) \\ G_{1} = - M_{12} (q) M_{22} {(q)}^{- 1} G_{2} (q) \\ u_{u} = - M_{21} (q) M_{11} {(q)}^{- 1} u_{a} \end{array}

The unactuated dynamics in equation (11) are indirectly affected by the control inputs.

3. Control system design

The antisway controllers are designed based on the nonlinear dynamical model in equations (10) and (11), without any simplification or linearization. The designed controllers are the SMC and ISMC. The stability is demonstrated using Lyapunov’s method.

3.1. Conventional sliding mode control

The main aim of the controller is reaching the desired position of the actuated states while eliminating the unactuated states, increasing the complexity of the control design. The error vectors are defined as follows

e_{a} = q_{a} - q_{a d} = {[\begin{matrix} x - x_{d} & θ - θ_{d} \end{matrix}]}^{T}

(12)

e_{u} = q_{u} - q_{u d} = {[\begin{matrix} α & β \end{matrix}]}^{T}

(13)

which e_a is the tracking error of the actuated states. The desired positions of the trolley and jib are x_d and θ_d respectively. e_u is the error of the unactuated states where the desired swing angles are equal to zero. The sliding surface is defined to be

s = {\dot{e}}_{a} + λ_{1} e_{a} + α {\dot{e}}_{u} + λ_{2} e_{u}

(14)

where λ₁, λ₂ ∈ R⁺ are the positive control parameters affecting the rate in which the states reach the desired position and α is the positive control parameter affecting the rate in which the states will settle, which are described by λ₁ = diag(λ₁₁, λ₁₂), λ₂ = diag(λ₂₁, λ₂₂), and α = diag(α₁, α₂).

The problem of conventional SMC is the error will not coincide with the switching surface in the presence of uncertainties. Hence, steady-state error appears in the system’s output. Therefore, the integral sliding mode is proposed to preserve the performance despite the uncertainties.

3.2. Integral sliding mode control

The dynamical model is updated as follows

M (q) \ddot{q} + B (q, \dot{q}) + G (q) + h (q, \dot{q}, \ddot{q}) = u

(15)

to include the uncertainties term, h ∈ R^n×1. The actual value of

h (q, \dot{q}, \ddot{q})

is unknown, but it is assumed that it is bounded by a known function

‖ h (q, \dot{q}, \ddot{q}) ‖ \leq \bar{h} (q, \dot{q}, \ddot{q})

. Utkin and Shi (1996) defined the switching function as follows

s = \dot{e} + K_{p} e + K_{i} \int_{0}^{t} e (ζ) d ζ

(16)

where K_p, K_i ∈ R⁺ are the proportional and the integral control gains respectively. The system is third-order relative to the variable of interest

\int_{0}^{t} e d t

as explained by Slotine et al. (1991).

The design of the controller is based on Lyapunov’s method by considering

V = \frac{1}{2} s^{T} s

(17)

as a positive definite Lyapunov function, and V(x) is radially unbounded. The sliding manifold for the tower crane system is defined as follows

s = {\dot{e}}_{a} + λ_{1} e_{a} + λ_{2} \int_{0}^{t} e_{a} d t + α {\dot{e}}_{u} + λ_{3} e_{u}

(18)

The Lyapunov function differentiated with respect to time is given as

\dot{V} = s \dot{s}

(19)

The time derivative of the sliding surface is obtained as

\dot{s} = {\ddot{q}}_{a} + λ_{1} {\dot{q}}_{a} + λ_{2} (q_{a} - q_{a d}) + α {\ddot{q}}_{u} + λ_{3} {\dot{q}}_{u}

(20)

where the first and second derivatives of the desired values are set to zero. The equations of motion of system (10) and (11) are substituted into the equation of

\dot{s}

. Also,

\dot{s}

should be in terms of the actuated control input u_a only. Hence, u_u is substituted by −M₂₁(q)M₁₁(q)⁻¹u_a. The resulting equation for

\dot{s}

is substituted in

\dot{V}

\begin{array}{l} \dot{V} = s [u_{a} [{\bar{M}}_{1} {(q)}^{- 1} - α {\bar{M}}_{2} {(q)}^{- 1} M_{21} (q) M_{11} {(q)}^{- 1}] \\ + {\bar{M}}_{1} {(q)}^{- 1} [- {\bar{B}}_{11} (q, \dot{q}) - G_{1} (q)] + λ_{1} {\dot{q}}_{a} + λ_{3} {\dot{q}}_{u} \\ + α [{\bar{M}}_{2} {(q)}^{- 1} [- {\bar{B}}_{21} (q, \dot{q}) - G_{2} (q)]] \\ + λ_{2} (q_{a} - q_{a d})] \end{array}

(21)

In the sliding phase, the time derivative of s on the system trajectories should be made equal to zero to force the states to remain at the desired position. The error dynamics is

\ddot{e} + K_{v} \dot{e} + K_{p} e = 0

(22)

The controller gains K_p and K_v are chosen properly for the characteristic equation (22) is strictly stable. The roots of the characteristic equation are in the left half plane (LHP) which implies that $lim_{t \to \infty} e (t) = 0$ .

The controller is designed as $u_{a} = \vec{u} + u_{s}$ where the nominal control $\vec{u}$ is defined using the computed torque method and responsible for the performance of the nominal system. The algebraic equation (21) is set to zero and solved with respect to the control input, the resulting control law is presented in equation (23). The nominal control law is designed for all $| α | < π / 2$ and $| β | < π / 2$ . u_s is equal to −K sgn(s) which is the discontinuous part to ensure that the states remain on the sliding surface

\begin{array}{l} u_{a} = {({\bar{M}}_{1} {(q)}^{- 1} - α {\bar{M}}_{2} {(q)}^{- 1} M_{21} (q) M_{11} {(q)}^{- 1})}^{- 1} \\ [{\bar{M}}_{1} {(q)}^{- 1} [{\bar{B}}_{11} (q, \dot{q}) + G_{1} (q)] - λ_{1} {\dot{q}}_{a} - λ_{3} {\dot{q}}_{u} \\ + α [{\bar{M}}_{2} {(q)}^{- 1} [{\bar{B}}_{21} (q, \dot{q}) + G_{2} (q)]] \\ - λ_{2} (q_{a} - q_{a d})] - K sgn (s) \end{array}

(23)

where K = diag(K₁, K₂) ∈ R^n×n is a positive definite diagonal matrix with its diagonal elements satisfying

K_{i} > | h (q, \dot{q}, \ddot{q}) |

Controller equation (23) is substituted into the derivative of Lyapunov function (21)

\begin{array}{l} \dot{V} = s [- K sgn (s) + F] \\ \leq (F - K) | s | < 0 \\ \leq - μ_{0} | s | < 0 \end{array}

(24)

Some terms will not be canceled because of modeling uncertainties, the upper bound is denoted by F. If gain K is designed to be K = F + μ₀, where μ₀ > 0 and chosen to be a diagonal positive definite matrix, then the derivative of the Lyapunov function is ensured to remain negative definite. The time derivative $\dot{V} = s^{T} \dot{s} < 0$ is negative definite. Based on the definition from Khalil (2002), the closed loop system is guaranteed to be globally asymptotically stable. The controller in equation (25) guarantees that the stability of the system trajectories by proper selection of α so that the inverse of the matrix ${\bar{M}}_{1} {(q)}^{- 1} - α {\bar{M}}_{2} {(q)}^{- 1} M_{21} (q) M_{11} {(q)}^{- 1}$ exists.

The main drawback of the high frequency switching control actions in the integral sliding mode is the chattering phenomena due to unmodeled dynamics. The sign function is replaced by a saturation function to avoid chattering in the control input

\begin{array}{l} u_{a} = {({\bar{M}}_{1} {(q)}^{- 1} - α {\bar{M}}_{2} {(q)}^{- 1} M_{21} (q) M_{11} {(q)}^{- 1})}^{- 1} \\ [{\bar{M}}_{1} {(q)}^{- 1} [{\bar{B}}_{11} (q, \dot{q}) + G_{1} (q)] - λ_{1} {\dot{q}}_{a} - λ_{2} (q_{a} - q_{a d}) \\ + α [{\bar{M}}_{2} {(q)}^{- 1} [{\bar{B}}_{21} (q, \dot{q}) + G_{2} (q)]] \\ - λ_{3} {\dot{q}}_{u}] - K sat (s, ϵ) \end{array}

(25)

where

sat (s, ϵ) = {\begin{array}{l} s / ϵ, & for | s / ϵ | \leq 1 \\ sgn (s / ϵ), & for | s / ϵ | > 1 \end{array}

(26)

The proposed control guarantees that the tracking errors converge into a boundary layer whose thickness can be arbitrarily set by a positive constant ϵ.

3.3. Stability analysis

From the control law design, it can be concluded that the closed-loop system is globally asymptotically stable. Moreover, Almutairi and Zribi (2009) explains that the under-actuated system’s states that are on the sliding surface converge to their desired values by satisfying the sufficient conditions on the control gains. The sufficient conditions is derived by combining the sliding surface equation (18) with two unactuated acceleration vector equation (7). Hence, the trajectories on the sliding surface are considered, and equation (18) is set to zero (s = 0) and ${\dot{q}}_{a d} = 0$

\begin{array}{l} {\dot{q}}_{a} = - λ_{1} (q_{a} - q_{a d}) - λ_{2} \int_{0}^{t} (q_{a} - q_{a d}) d t \\ - α {\dot{q}}_{u} - λ_{3} q_{u} \end{array}

(27)

From equations (9), (10), and (25), the unactuated acceleration equation can be written as

\begin{array}{l} {\ddot{q}}_{u} = M_{22}^{- 1} (q) (- M_{21} (q) {\ddot{q}}_{a} - B_{21} (q, \dot{q}) - G_{2} (q)) \\ = M_{22}^{- 1} (q) (- M_{21} (q) [{\bar{M}}_{1} {(q)}^{- 1} (u_{a} - {\bar{B}}_{11} (q, \dot{q}) - G_{1} (q))] \\ - B_{21} (q, \dot{q}) - G_{2} (q)) \\ = M_{22}^{- 1} (q) (- M_{21} (q) [{\bar{M}}_{1} {(q)}^{- 1} (({\bar{M}}_{1} {(q)}^{- 1} \\ {- α {\bar{M}}_{2} {(q)}^{- 1} M_{21} (q) M_{11} {(q)}^{- 1})}^{- 1} [{\bar{M}}_{1} {(q)}^{- 1} [{\bar{B}}_{11} (q, \dot{q}) \\ + G_{1} (q)] - λ_{1} {\dot{q}}_{a} - λ_{2} (q_{a} - q_{a d}) + α [{\bar{M}}_{2} {(q)}^{- 1} [{\bar{B}}_{21} (q, \dot{q}) \\ + G_{2} (q)]] - λ_{3} {\dot{q}}_{u}] - {\bar{B}}_{11} (q, \dot{q}) - G_{1} (q))] \\ - B_{21} (q, \dot{q}) - G_{2} (q)) = h_{1} (x) \end{array}

(28)

Let

x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} q_{u} \\ {\dot{q}}_{u} \\ \int_{0}^{t} (q_{a} - q_{a d}) \\ q_{a} - q_{a d} \end{matrix}]

(29)

Equation (28) is rewritten in terms of x only. Also, equation (27) is rewritten as

{\dot{x}}_{4} = - λ_{1} x_{4} - λ_{2} x_{3} - α x_{2} - λ_{3} x_{1} = h_{2} (x)

(30)

Equations (28)–(30) are combined to get an autonomous system

\dot{x} = [\begin{matrix} x_{2} \\ h_{1} (x) \\ x_{4} \\ h_{2} (x) \end{matrix}] = f (x)

(31)

Linearize the system (31) about the equilibrium point to get $\dot{x}$ = Ax, where

A = \frac{\partial f (x)}{\partial x} |_{x = 0} = [\begin{matrix} 0_{2 \times 2} & I_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} \\ A_{1} & A_{2} & A_{3} & A_{4} \\ 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & I_{2 \times 2} \\ - λ_{3} & - α & - λ_{2} & - λ_{1} \end{matrix}]

(32)

where

\begin{array}{l} A_{1} & = \frac{\partial h_{1} (x)}{\partial x_{1}} = [\begin{matrix} a_{11} & 0 \\ 0 & a_{12} \end{matrix}] \\ A_{2} & = \frac{\partial h_{1} (x)}{\partial x_{2}} = [\begin{matrix} a_{21} & 0 \\ 0 & a_{22} \end{matrix}] \\ A_{3} & = \frac{\partial h_{1} (x)}{\partial x_{3}} = [\begin{matrix} a_{31} & 0 \\ 0 & a_{32} \end{matrix}] \\ A_{4} & = \frac{\partial h_{1} (x)}{\partial x_{4}} = [\begin{matrix} a_{41} & 0 \\ 0 & a_{42} \end{matrix}] \end{array}

(33)

where

\begin{array}{l} a_{11} = \frac{- (L m_{2} (g + λ_{11} λ_{31}))}{(m_{2} L (L - α_{1}) + J_{2})} \\ a_{12} = \frac{- (L m_{2} (g + x_{d} λ_{12} λ_{32}))}{(J_{2} + L^{2} m_{2} - L x_{d} α_{2} m_{2})} \\ a_{21} = \frac{- (B_{a} - L λ_{31} m_{2} + L α_{1} λ_{11} m_{2})}{(m_{2} L^{2} - α_{1} m_{2} L + J_{2})} \\ a_{22} = \frac{- (B_{b} - L x_{d} λ_{32} m_{2} + L x_{d} α_{2} λ_{12} m_{2})}{(J_{2} + L^{2} m_{2} - L x_{d} α_{2} m_{2})} \\ a_{31} = \frac{- (L λ_{11} λ_{21} m_{2})}{(m_{2} L^{2} - α_{1} m_{2} L + J_{2})} \\ a_{32} = \frac{- (L λ_{12} λ_{22} m_{2} x_{d})}{(J_{2} + L^{2} m_{2} - L x_{d} α_{2} m_{2})} \\ a_{41} = \frac{(L m_{2} (- λ_{11}^{2} + λ_{21}))}{(m_{2} L^{2} - α_{1} m_{2} L + J_{2})} \\ a_{42} = \frac{(L m_{2} x_{d} (- λ_{12}^{2} + λ_{22}))}{(J_{2} + L^{2} m_{2} - L x_{d} α_{2} m_{2})} \end{array}

(34)

The stability of the linearized system $\dot{x}$ = Ax is guaranteed when the linearized state matrix A should have its eigenvalues in the LHP of the complex plane. Using the Routh–Hurwitz criterion, the characteristic equation of the linearized system is calculated by det(sI − A) = 0. The analysis gives sufficient conditions shown in equation (35). The parameters α, λ₁, λ₂, and λ₃ are chosen to satisfy the following conditions

\begin{array}{l} L^{2} λ_{11} m_{2} + B_{a} + λ_{11} J_{2} > λ_{31} L m_{2} \\ L^{2} λ_{12} m_{2} + B_{b} + λ_{12} J_{2} > λ_{32} L m_{2} x_{d} \\ L^{2} m_{2} + J_{2} > α_{1} L m_{2} \\ L^{2} m_{2} + J_{2} > α_{2} L m_{2} x_{d} \end{array}

(35)

Because the linearized system is proven to be asymptotically stable, x converges to zero asymptotically. Therefore, from equation (29), q_a converges to q_ad and q_u, ${\dot{q}}_{u}$ converges to zero as t goes to infinity. From equation (27), ${\dot{q}}_{a}$ converges to zero as t goes to infinity. Hence, the integral sliding mode controller in equation (25) guarantees the asymptotic stability of the tower crane system. The sufficient conditions imposed by equation (35) guarantee that q_a, ${\dot{q}}_{a}$ , q_u, and ${\dot{q}}_{u}$ reach their desired values as t goes to infinity.

4. Results and discussion

The controllers are implemented using MATLAB/Simulink environment. The solver used is discrete: ode4 (Runge–Kutta algorithm) with a sampling time of 1 ms. The results presented are for the simultaneous motion where the trolley translation and the jib rotation are both excited at the same time. During all simulations and experiments, (1) the length is kept constant with a value of 0.5 m, (2) The desired position for the trolley is 0.3 m, (3) the desired position for the jib is 135°, and (5) the simulation time is 30 s.

The crane’s parameters used in the model equations are given in Table 2. Altaf (2010) estimated the parameters using the prediction error method. Other parameters such as friction coefficients and the inertia are estimated using an offline identification parameter estimation tool in MATLAB via the sum square method based on the structure of the model.

Table 2.

Model parameters.

Description	Value
Mass of the trolley	m₁ = 0.7 kg
Mass of the payload	m₂ = 0.32 kg
Coulomb friction coefficient in x-direction	μ_x = 0.36 Nm/s
Coulomb friction coefficient in θ-direction	μ_θ = 0.9 Nm/s
Viscous friction coefficient in x-direction	B_x = 34 Nm/s
Viscous friction coefficient in θ-direction	B_θ = 18 Nm/s
Viscous friction coefficient in α-direction	B_α = 0.0015 Nm/s
Viscous friction coefficient in β-direction	B_β = 0.0015 Nm/s
Jib’s moment of inertia	J₁ = 4 kgm²
Load’s moment of inertia	J₂ = 0.02 kgm²
Gravitational constant	g = 9.81 m/s²

4.1. Robustness of ISMC

During simulations, the uncertainties are represented by the friction model. In other words, the controllers are based on the model without the friction model. The controllers are tested at two different values of the boundary layers. When uncertainties are added to the system, the SMC results at the boundary layer equal to 2 show that the trolley position has a steady-state error of 0.02 m and the jib rotation has a steady-state error of 8°. Therefore, the boundary layer has been reduced to 0.5. As shown from Figures 2 and 3, the steady-state error has been successfully eliminated. However, Figures 4 and 5 show that with a smaller boundary layer, the control effort demand has been increased.

The results of ISMC shows that there is no steady-state error in the position of the trolley and the jib. Also, the control effort demand is low compared with the SMC. Therefore, ISMC shows better results than conventional SMC. These results include reducing the control effort demand from 16 N to 8.6 N for the trolley and 30 N to 17.6 N for the jib rotation. Also, It shows that the effects of the uncertainties are suppressed by the proposed ISMC. The rise time for the two controllers using different boundary layers are almost equal to 2.5 s, so they are capable of fast maneuvers. The alpha oscillations are damped after 18 s and the beta oscillations are damped after 6.8 s. Therefore, the controllers have successfully damped the oscillations of the payload at the destination as shown in Figures 6 and 7.

4.2. Experimental validation

The SMC and ISMC are implemented for the real setup. The unmodeled uncertainties are the difference between the real system and the derived full nonlinear model.

4.3. Hardware

The test bench is a laboratory INTECO tower crane shown in Figure 8. The feedback for the position of the states (x, θ, α, and β) are measured by high-resolution encoders. All the experiments are conducted in real time where the Simulink model is converted into C code to get the required signals using a built-in C compiler toolbox in MATLAB. The interface board sends the output signals to the actuators, which control the crane’s movement. Then the controllers’ outputs are plotted against each other. The initial trolley position is set to 0.22 m whereas the initial jib position and the initial payload swings are set to zero.

The experimental results of the SMC results with the boundary layer are equal to 2, show that the trolley position has a steady-state error of 0.03 m and the jib rotation have an error of 2°. When the boundary layer is reduced to 0.5 as shown from Figures 9 and 10, the steady-state error is not eliminated for the trolley but successfully eliminated for the jib rotation. However, Figures 11 and 12 show that with a smaller boundary layer, the control effort demand has been increased. The results of ISMC shows that there is no steady-state error in the position of the trolley and the jib. In addition, the control effort demand is reduced from 7 to 4 N for the trolley input and from 20 to 10 N. There is no overshoot in any of the experiments. Also, the residual oscillations in the swing angles are very small ranging between almost −0.5 and 0.5°. As shown in Figures 13 and 14, they are hardly observed in reality.

5. Conclusion

Nonlinear robust controllers are implemented for precise tracking of the desired position while damping the payload oscillations in fast maneuvers. These controllers are derived based on the nonlinear dynamical model without any linearization or decoupling to reduce the model uncertainties. The problem with a high value of model uncertainties is that the switching gain must be high for robustness. However, the switching gain must be limited to implement the SMC on the real system, resulting in the steady-state error existence in the system’s outputs. Therefore, the ISMC is proposed which enhances the performance of the tower crane system against uncertainties. The simulation and experimental results of the ISMC show that the steady-state error was eliminated while maintaining a low control effort demand, compared with the conventional SMC.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ayman A El-Badawy

Appendix 1

Coefficients of M(q) matrix are

\begin{array}{l} m_{11} = m_{1} + m_{2} \\ m_{12} = m_{13} = m_{14} = m_{21} = 0 \\ m_{22} = J_{1} + \frac{3}{4} L^{2} m_{2} + m_{1} x^{2} + m_{2} x^{2} - \frac{1}{4} L^{2} m_{2} \cos (2 α) - \frac{1}{2} L^{2} m_{2} \cos {(α)}^{2} \cos (2 β) \\ m_{23} = m_{32} = m_{34} = m_{41} = m_{43} = 0 \\ m_{24} = L m_{2} x \cos (α) \cos (β) \\ m_{31} = L m_{2} \cos (α) \\ m_{33} = J_{2} + L^{2} m_{2} \\ m_{42} = L m_{2} x \cos (α) \cos (β) \\ m_{44} = J_{2} + L^{2} m_{2} \cos {(α)}^{2} \end{array}

Coefficients of B(q) matrix are

\begin{array}{l} b_{11} = B_{x} \dot{x} + μ_{x} sgn (\dot{x}) \\ b_{21} = B_{θ} \dot{θ} + μ_{θ} sgn (\dot{θ}) + 2 m_{1} x \dot{x} \dot{θ} + 2 m_{2} x \dot{x} \dot{θ} + 2 L m_{2} x \dot{θ} \dot{α} \cos (α) \\ b_{31} = B_{α} \dot{α} - L m_{2} x {\dot{θ}}^{2} \cos (α) - 2 L^{2} m_{2} \dot{θ} \dot{β} \cos {(α)}^{2} \cos (β) \\ b_{41} = 2 L m_{2} \dot{x} \dot{θ} \cos (α) \cos (β) + 2 L^{2} m_{2} \dot{θ} \dot{α} \cos {(α)}^{2} \cos (β) + B_{β} \dot{β} \end{array}

Coefficients of G(q) are

\begin{array}{l} g_{1} = g L m_{2} \cos (β) \sin (α) \\ g_{2} = g L m_{2} \cos (α) \sin (β) \end{array}

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