Composite nonlinear feedback controller design for an overhead crane servo system

Abstract

This paper investigates the design of a composite nonlinear feedback (CNF) control law for an overhead crane servo system to improve the transient performance of both displacement tracking of the trolley and anti-sway of the payload. To address the property of underactuation of the overhead crane system, a novel nonlinear function of the CNF control law is specifically proposed to compromise the tracking performance of the trolley and the anti-sway performance of the payload. The performance improvement in both tracking of the trolley and anti-sway of the payload is illustrated with a complete comparison between the CNF control method and the trajectory planning method, which has been proposed in recent literature. The simulation results show that this well-tuned CNF control law can significantly shorten the settling time of the trolley displacement tracking and reduce the sway of the payload.

Keywords

Overhead crane servo system anti-sway control tracking control nonlinear feedback

Introduction

Underactuated systems, which are systems with fewer control input channels than degrees of freedom, have become a focus of research in the control systems area, due to the difficulty in addressing the property of underactuation (Chwa, 2011; Lefeber et al., 2003; Zhang and Tarn, 2003; Zhu et al., 2013). The overhead crane system is a representative underactuated system with strong states coupling and only one input applied to its trolley (Abdel-Rahman et al., 2003). A servo system for the overhead crane system not only needs to drive the trolley to a target position quickly and smoothly, but also needs to keep the swing angle of the payload within a small allowable value. Due to the property of underactuation, the biggest challenge lies in designing a single control input to achieve these objectives simultaneously.

To this end, various innovative control methods have been proposed, including nonlinear coupling control (Fang et al., 2003), input shaping control (Sorensen et al., 2007), emergency braking control (Ma et al., 2010), slide-mode control (Wang et al., 2004), fuzzy adaptive control (Chang, 2007; Liu et al., 2005), fuzzy proprotional–integral–derivative (PID) control (Solihin et al., 2010), a Monte Carlo-based design method (Moustafa et al., 2013; Omar et al., 2011), motion planning adaptive control (Fang et al., 2012) and nested saturation control (Liu et al., 2012). These methods can get good performance in the displacement motion of the trolley, but a payload sway still remains, or there is a long settling time which can be further improved. Recently, Sun et al. (2012) developed a novel kinematic coupling-based off-line trajectory planning method for overhead cranes, which was the first to introduce an anti-swing mechanism into an S-shape reference trajectory, and dramatically improved the payload anti-sway performance. In this paper, the composite nonlinear feedback (CNF) control technique is introduced to design an overhead crane servo system. As illustrated in the paper, the CNF control method can achieve better transient performance than the trajectory planning method proposed in Sun et al. (2012).

The CNF control technique was first proposed by Lin et al. (1998) for a class of second-order linear systems with input saturation to improve the transient performance for the tracking control problem. The CNF control law consists of a linear feedback part and a nonlinear feedback part. Basically, the linear part is designed to yield a closed-loop system with a small damping ratio for a quick response. The nonlinear part is introduced to change the damping ratio of the closed-loop system to reduce the overshoot caused by the linear part. After Lin’s work, Turner et al. (2000) extended the CNF control technique to multivariable systems. Furthermore, Chen et al. (2003) developed CNF control for a more general class of systems with measurement feedback, and it was extended to multivariable systems in He et al. (2005). Then, Lan et al. (2006) extended it to a class of nonlinear systems and linear singular systems (Lin et al., 2011). More recently, it was applied to improve the transient performance of the output regulation problem of linear systems with input saturation (Zhang and Lan, 2013). The CNF control technique has also been successfully applied to a HDD servo system (Chen et al., 2003; Lan et al., 2010), positioning servo systems (Cheng and Peng, 2007; Cheng et al., 2007), and the unmanned aerial vehicle (UAV) helicopter flight control system (Cai et al., 2008).

In this paper, we design a CNF control law for the nonlinear underactuated overhead crane system derived from Sun et al. (2012), such that the transient performance can be further improved. To distinguish it from other applications of CNF control in the literature, our CNF control law is designed for an underactuated system. In addition to the position tracking of the trolley, the swing angle of the payload need to be suppressed simultaneously. Considering such a special requirement, a novel nonlinear function is proposed to improve the performance of the overhead crane system. The form of the nonlinear function smartly utilizes the feedback of both the trolley displacement and the swing angle, and the damping ratio of the closed-loop system can be properly changed with the motions. Specifically, we use two terms in the nonlinear function to compromise between the requirement of quick trolley position tracking and the performance of swing angle suppression. The main purpose of adding the extra term is to allow a compromise between the conflicting performance requirements of trolley position tracking and swing angle suppression, such that a satisfactory overall performance is obtained by the CNF control law. The details of the design of the proposed nonlinear function are presented in Section 3.2. Compared to the trajectory planning method proposed in Sun et al. (2012), the results of simulation experiments show that the CNF control law further improve the performance of both displacement tracking and anti-swaying for the underactuated overhead crane system, and the design procedure of the CNF controller is more straightforward.

The remainder of this paper is organized as follows. Section 2 introduces the dynamic model of the overhead crane system. The design of the CNF control law for an overhead crane system is detailed in Section 3, which includes the construction procedure for the CNF control law, and the designing and tuning method of the proposed nonlinear function. The effectiveness of the proposed method is illustrated by the simulation experiments in Section 4, and specifically, the results are compared with those of the method proposed in Sun et al. (2012) which are outlined in Section 4.1. Section 5 concludes the paper with some remarks.

Model of a two-dimensional overhead crane system

In this paper, the two-dimensional (2D) overhead crane system with a payload from Sun et al. (2012) is considered, as shown in Figure 1. $d$ and $θ$ are the horizontal displacement of the trolley and the swing angle of the payload, respectively, which are regarded as the states and the outputs of the system. $l$ is the length of the rope, $M$ and $m$ are the mass of the trolley and payload respectively, $f_{r}$ is the rail friction and $u_{r}$ is the actuator force. With air resistance being ignored, the nonlinear crane dynamics with constant rope length can be depicted (Sun et al., 2012) as follows

\begin{matrix} (M + m) \overset{\cdot\cdot}{d} + ml \overset{\cdot\cdot}{θ} \cos θ - ml {\overset{\cdot}{θ}}^{2} \sin θ = u \\ m l^{2} \overset{\cdot\cdot}{θ} + ml \overset{\cdot\cdot}{d} \cos θ + mgl \sin θ = 0 \end{matrix}

(1)

where $u = u_{r} - f_{r}$ is the control input to be designed. The form of $f_{r}$ is given by (Sun et al., 2012)

f_{r} (\overset{\cdot}{d}) = f_{r 0} \tanh (\overset{\cdot}{d} / ξ_{r}) - k_{r} | \overset{\cdot}{d} | \overset{\cdot}{d}

(2)

where $f_{r 0}$ , $k_{r} \in ℝ$ are friction-related parameters and $ξ_{r} \in ℝ$ is a static friction coefficient, which can be obtained from off-line experimental data analysis.

Figure 1.

Diagram of an overhead crane system.

Remark 2.1 For the dynamics in equation (1), the following two assumptions derived from Sun et al. (2012) should also be considered:

The rope is massless and inflexible. The rope length is kept constant during a specific transportation process.

During the overall process, the payload is always beneath the trolley, namely $θ (t) \in (- 90 \circ, 90 \circ) .$ ▪

Let $x = {[x_{1} x_{2} x_{3} x_{4}]}^{T} = {[θ \overset{\cdot}{θ} d \overset{\cdot}{d}]}^{T}$ , noting that the approximations

\cos θ \approx 1, \sin θ \approx θ

(3)

hold for constrained trolley acceleration (Sun et al., 2012), and considering the saturation of the actuator force, the linearized crane dynamic model is given by

{\begin{matrix} \overset{\cdot}{x} = Ax + B sat (u), x (0) = x_{0} \\ y = C_{1} x \\ h_{1} = C_{2} x \\ h_{2} = C_{3} x \end{matrix}

(4)

where $y = d$ is the measurement output, $h_{1} = d$ and $h_{2} = θ$ are the controlled outputs referring to the trolley displacement and the swing angle of the payload respectively. sat: $ℝ \to ℝ$ denotes the actuator saturation defined as $sat (u) = sgn (u) min (u_{max}, | u |)$ , where $u_{max}$ is the maximum amplitude of the control input. $A, B, C_{1}, C_{2}$ and $C_{3}$ are given by

\begin{array}{l} A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - \frac{(M + m) g}{M l} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{m g}{M} & 0 & 0 & 0 \end{matrix}], B = [\begin{matrix} 0 \\ - \frac{1}{M l} \\ 0 \\ \frac{1}{M} \end{matrix}], \\ C_{1} = C_{2} = [\begin{matrix} 0 & 0 & 1 & 0 \end{matrix}], C_{3} = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}] \end{array}

(5)

It is not difficult to verify that the system in equation (4) satisfies the following assumptions A1 to A3.

A1 ( $A$ , $B$ ) is stabilizable;

A2 ( $A$ , $B$ , $C_{2}$ ) is invertible and has no zeros at the origin of the complex plane.;

A3 ( $A$ , $C_{1}$ ) is detectable.

Thus, by Chen et al. (2003), we can construct a CNF control law for the overhead crane servo system. We would like to point out that the CNF control law is designed based on the linear dynamic model in equation (4), but it is conducted on the nonlinear dynamic model in equation (1) to examine the effectiveness in improving transient performance in simulation experiments.

Composite nonlinear feedback control for an overhead crane system

In this section, we design a CNF controller for an overhead crane servo system such that the closed-loop system has a desired transient performance. The linear part of the CNF controller can be designed by some classical control methods to yield a quick response of trolley displacement tracking, and the nonlinear part is added to remove the overshoot of the trolley displacement tracking caused by the linear part, and meanwhile to reduce the swing angle of the payload.

Constructing the composite nonlinear feedback control law

To design a CNF control law, we will first design a linear feedback control law for the overhead crane system in equation (4) which will drive the trolley to a target position $r$ asymptotically. The linear-feedback control law is given by

u_{L} = Fx + Gr

(6)

where $r$ is a reference input, $F$ is the feedback gain to be designed, and $G$ is the feedforward gain which can be calculated using equation (7) if $F$ is specified

G = - {[C_{1} {(A + BF)}^{- 1} B]}^{- 1}

(7)

The value of $F$ is selected such that $A + BF$ is Hurwitz which is possible under assumption A1. There are various methods to design the feedback gain $F$ with the assumption that $u$ is unconstrained, such as the $H_{2}$ and $H_{\infty}$ control approaches (Chen, 2000). Specially, in designing the linear-feedback part, a quick response allowing a moderate overshoot is preferred and the overshooting caused by such radical design will be eliminated by the nonlinear part.

Based on the linear feedback gain in equation (6), the nonlinear feedback part of the CNF control law is constructed by

u_{N} = ρ (h) B^{T} P (x - x_{e})

(8)

where $ρ (h)$ is a nonpositive function locally Lipschitz in $h$ , which is used to change the damping ratio of the closed-loop system as the output approaches the reference input $r$ . Here $x_{e}$ is calculated using

\begin{matrix} x_{e} : = - {(A + BF)}^{- 1} BGr \end{matrix}

(9)

and $P > 0$ is the solution of the following Lyapunov equation for some $W > 0$

{(A + BF)}^{T} P + P (A + BF) = - W

(10)

Finally, combining equations (6) and (8), the CNF control law is given by

u = u_{L} + u_{N} = F x + G r + ρ (h) B^{T} P (x - x_{e})

(11)

Rephrased from Chen et al. (2003), the properties of the CNF control law are described by the following theorem which claims that both the linear control law in equation (6) and the CNF control law in equation (11) can solve the tracking control problem. However, the CNF control law can improve the transient performance of the closed-loop system.

Theorem 3.1 Consider the system given in equation (4), the linear control law in equation (6) and the CNF control law in equation (11). For any real number $δ \in (0, 1)$ , let $c_{δ} > 0$ be the largest positive scalar satisfying the following condition

| Fx | \leq u_{max} (1 - δ) \forall x \in X_{δ} : = {x : x^{T} Px \leq c_{δ}}

(12)

Then, the linear control law in equation (6) is capable of driving the system-controlled output $h (t)$ to track asymptotically a step reference input, $r$ , provided that the initial state $x_{0}$ and $r$ satisfy

{\tilde{x}}_{0} : = (x_{0} - x_{e}) \in X_{d}, | Hr | \leq δ u_{max}

(13)

with $H : [1 - F {(A + BF)}^{- 1} B] G .$ In addition, for any nonpositive function $ρ (h)$ , locally Lipschitz in $h$ , the CNF law in equation (11) is capable of driving the system-controlled output $h (t)$ to track asymptotically the reference input, $r$ , provided that the initial state $x_{0}$ and $r$ satisfy equation (13).

Furthermore, under assumptions A1, A2, and A3, the measurement–feedback CNF control laws can be constructed for the system in equation (4). Specifically, a full-order measurement–feedback control law is given by

\begin{array}{l} {\overset{\cdot}{x}}_{v} = (A + K C_{1}) x_{v} - K y + B sat (u) \\ u = F x_{v} + G r + ρ (y) B^{T} P (x_{v} - x_{e}) \end{array}

(14)

where $F$ and $K$ are selected such that $A + BF$ and $A + KC$ is asymptotically stable. Please refer to Chen et al. (2003) for more details on measurement–feedback CNF control.

Remark 3.1. The CNF control aims to improve the transient performance of the closed-loop system by combining a linear control law and a nonlinear control law. The linear part of the CNF control law is designed to yield a closed-loop system with a small damping ratio for a quick response, while at the same time not exceeding the actuator limit for the desired command input level. The nonlinear part of the CNF control law is used to increase the damping ratio of the closed-loop system as the system output approaches the tracking target, to reduce the overshoot caused by the linear part. If the nonlinear function $ρ (h) = 0$ , the CNF control law will be reduced to a linear control law. Theorem 3.1 claims that any command input that can be asymptotically tracked by the linear control law in equation (6) can also be asymptotically tracked by the CNF control law in equation (11). However, the additional nonlinear term in the CNF control law in equation (11) can be used to improve the tracking performance of the closed-loop system. In fact, without considering the input saturation, the closed-loop system can be rewritten as

\begin{matrix} \overset{\cdot}{\tilde{x}} = (A + BF + ρ (h) B B^{T} P) \tilde{x} \end{matrix}

(15)

where $\tilde{x} = x - x_{e}$ . Defining the auxiliary system $G_{aux} (s)$ as

\begin{matrix} G_{aux} : = C_{aux} {(sI - A_{aux})}^{- 1} B_{aux} : = B^{T} P {(sI - A - BF)}^{- 1} B \end{matrix}

(16)

The closed-loop system in equation (15) can be expressed as a feedback system, as shown in Figure 2. Using the well known classical root-locus theory, the poles of the closed-loop system in equation (15) approach the location of the invariant zeros of $G_{aux} (s)$ as $| ρ (h) |$ becomes larger and larger. It is shown in Chen et al. (2003) that the auxiliary system $G_{aux} (s)$ is stable and invertible with a relative degree equal to $1$ , and is of minimum phase with $n - 1$ stable invariant zeros. It should be noted that the locations of these invariant zeros can be pre-selected by selecting the positive definite matrix $W$ in equation (10). Please refer to Chen and Zheng (1995) for more details on zero assignment technique. Thus, we can improve the transient performance of the closed-loop system with CNF control by carefully designing the linear feedback gain, $F$ , the positive definite matrix, $W$ , and the nonlinear function $ρ (h)$ . Please refer to Chen et al. (2003), He et al. (2005) and Lan et al. (2010) for more information on the design of CNF control laws. It is clear that the nonlinear function $ρ (h)$ needs to be appropriately designed so that the closed-loop system has the desired transient performance. In general, the nonlinear function $ρ (h)$ should satisfy the following two properties: 1) when the controlled output $h$ is far away from the tracking target, $| ρ (h) |$ is small and thus the effect of the nonlinear part on the overall control law is very limited; and 2) when the controlled output $h$ approaches the tracking target, $| ρ (h) |$ will become larger and larger, and the nonlinear part will become effective. However, the specifical nonlinear function $ρ (h)$ need to be designed on a case-by-case basis for various systems. In this paper, to reduce the overshooting of the trolley displacement caused by the linear control, and meanwhile to restrain the swing angle of the payload, we propose a novel nonlinear function $ρ (h)$ accounting for the underactuated property of the overhead crane servo system.▪

Figure 2.

Interpretation of the nonlinear function $ρ$ .

Remark 3.2. The closed-loop system in equation (15) has two pairs of poles, and they are a function of $ρ (h)$ . Thus, we can change the damping ratio of the closed-loop system by selecting an appropriate nonlinear function $ρ (h)$ . For the overhead crane system investigated in Sun et al. (2012), the damping ratio corresponding to each pair of poles is shown in Figure 3. The damping ratio $ξ_{1}$ of the first pair of poles refers to the closed loop with trolley displacement output, while the damping ratio $ξ_{2}$ of the second pair of poles refers to the closed loop with payload swing angle output. It is clear that when $| ρ (h) | > 5$ , the damping ratios of the closed-loop system increase with $| ρ (h) |$ . Therefore, we can design an appropriate $ρ (h)$ which varies in the range $| ρ (h) | > 5$ . Especially in $| ρ (h) | \in [5, 35]$ , a smaller value of $| ρ (h) |$ results in a less damped system, while a larger value of $| ρ (h) |$ results in a more damped system.

Figure 3.

Damping ratio of each pair of poles of the closed-loop system in equation (15).

Design of the nonlinear function

Once the linear feedback gain $F$ and the positive definite matrix $W$ are fixed, the nonlinear function in the CNF control law determines the performance of the closed-loop system (Lan and Chen, 2007). The difficulty in developing CNF control, lies in designing an appropriate nonlinear function $ρ (h)$ to obtain the desired transient performance. As shown in Theorem 3.1, to guarantee the stability of the closed-loop system, the only requirement on the nonlinear function $ρ (h)$ is $ρ (h) \leq 0$ for all $r$ and $h$ . It is clear that the selection of $ρ (h)$ is not unique and there are various versions of $| ρ (h) |$ as listed in (Lan and Chen, 2007). For a tracking control problem, the nonlinear function $ρ (h)$ is in general chosen as a function of the track error $e = h - r$ as proposed in Lan et al. (2010), which is available for feedback in most situations. However, the overhead crane servo system involves both trolley displacement tracking and payload anti-sway. Therefore, to address both the tracking performance and the anti-sway performance, we design a novel form of the nonlinear function as follows

ρ (h) = ρ_{1} (h_{1}) + ρ_{2} (h_{2})

(17)

ρ_{1} (h_{1}) = - β_{1} e^{- α_{0} α_{1} | h_{1} - r |}

(18)

ρ_{2} (h_{2}) = - β_{2} e^{- α_{s} α_{2} h_{2}}

(19)

where $r$ is the tracking target position of the trolley, $h_{1}$ is the trolley displacement, $h_{2}$ is the swing angle of the payload, and $α_{1}, α_{2}, β_{1}, β_{2} > 0$ are constant parameters to be tuned. $α_{0}$ is determined by $r$ and the initial position of the trolley, $h_{1} (0)$

α_{0} = {\begin{matrix} {| h_{1} (0) - r |}^{- 1}, h_{1} (0) \neq r \\ 1, h_{1} (0) = r \end{matrix}

(20)

and is used to scale the initial tracking error of the trolley position (Lan and Chen, 2007). The parameter $α_{s}$ is introduced to judge the moving direction of the trolley, which ensures the nonlinear function $ρ_{2} (h_{2})$ appropriately influences the transient performance when the trolley moves in either a positive or negative direction

α_{s} = sgn (r - h_{1} (0))

(21)

If only the tracking performance of the trolley displacement is considered, a nonlinear function in the form of $ρ (h) = ρ_{1} (h_{1})$ can be applied to improve the transient performance (Lan et al., 2010). It is clear that if the controlled output $h_{1}$ is far away from the reference point, that is, when $| h_{1} - r |$ is large, $| ρ (h) |$ is small, and vice versa. As shown in Remark 3.2, the damping ratio of the closed-loop system increased with increasing $| ρ (h) |$ . Thus, as the trolley approaches the tracking target, the damping ratio is increased to reduce the overshoot caused by the linear feedback part.

In this paper, an extra part $ρ_{2} (h_{2})$ is added to $ρ (h)$ such that the damping ratio of the closed-loop system can also be greatly influenced by the response of the swing angle of the payload, and it can be used to reduce the payload sway, whilst not increasing the settling time of the trolley significantly. To explain the operating principle of the function $ρ_{2} (h_{2})$ , we divide the payload sway movement into following two cases and discuss the change of the nonlinear function $ρ_{2} (h_{2})$ respectively. Assume $α_{s} = 1$ , that is, the trolley moves to the tracking target $r$ in a positive direction.

Case 1 $h_{2} = θ < 0$ as shown in Figure 4(a). In this case, the trolley drags the payload, so the slow movement of the trolley (to decrease ${\overset{\cdot}{h}}_{1}$ ) tends to decrease the swing angle $h_{2}$ . Thus, in order to reduce $h_{2}$ , we try to increase $ρ (h)$ to raise the damping ratio of the closed-loop system by increasing $| ρ_{2} (h_{2}) |$ . Since $h_{2} < 0$ , $- α_{s} α_{2} h_{2} > 0$ , $| ρ_{2} (h_{2}) |$ can increase exponentially as $| h_{2} |$ increases.

Figure 4.

Two cases of payload movement.

To be specific, if $\frac{d | h_{2} |}{d t} > 0$ , the payload tends to move away from the axis. As $| h_{2} |$ increases, $| ρ_{2} (h_{2}) |$ increases exponentially to a large value so that $| ρ (h) |$ increases, which exponentially raises the damping ratio of the closed-loop system. Then ${\overset{\cdot}{h}}_{1}$ will drop strikingly to decrease $\frac{d | h_{2} |}{d t}$ , which will help improve the payload anti-sway performance.

While if $\frac{d | h_{2} |}{d t} < 0$ , the payload tends to go back to the axis. As $| h_{2} |$ decreases, $| ρ_{2} (h_{2}) |$ decreases exponentially so that $| ρ (h) |$ decreases, which exponentially reduces the damping ratio of the closed-loop system. Then ${\overset{\cdot}{h}}_{1}$ will increase strikingly, which will help recover the displacement tracking performance.

Case 2 $h_{2} = θ > 0$ as shown in Figure 4(b). In this case, the payload drags the trolley, so the quick movement of the trolley (to increase ${\overset{\cdot}{h}}_{1}$ ) tends to decrease the swing angle $h_{2}$ . Thus, we try to reduce the damping ratio of the closed-loop system by decreasing $| ρ_{2} (h_{2}) |$ . Since $h_{2} > 0$ , $- α_{s} α_{2} h_{2} < 0$ , thus $| ρ_{2} (h_{2}) |$ can decrease exponentially if $| h_{2} |$ increases.

To be specific, if $\frac{d | h_{2} |}{d t} > 0$ , the payload tends to move away from the axis. As $| h_{2} |$ increases, $| ρ_{2} (h_{2}) |$ decreases exponentially to a small value so that $| ρ (h) |$ decreases, which reduces the damping ratio of the closed-loop system. Then ${\overset{\cdot}{h}}_{1}$ will increase strikingly whilst $\frac{d | h_{2} |}{d t}$ decreases, which will improve both the displacement tracking performance and the payload anti-sway performance.

While if $\frac{d | h_{2} |}{d t} < 0$ , the payload tends to go back to the axis. As $| h_{2} |$ decreases, $| ρ_{2} (h_{2}) |$ increases, and meanwhile $| ρ_{1} (h_{1}) |$ increases when the trolley is approaching its target, so that $| ρ (h) |$ increases, which raises the damping ratio of the closed-loop system. Then ${\overset{\cdot}{h}}_{1}$ will decrease to eliminate the overshoot of the displacement tracking whilst $| h_{2} |$ continues to decrease.

In Case 1, since $- α_{s} α_{2} h_{2} > 0$ , as $| h_{2} |$ increases, $| ρ_{2} (h_{2}) |$ appears to have a more dramatic affect on changing the value of $| ρ (h) |$ than $ρ_{1} (h_{1})$ . Therefore, in Case 1 the change of $| ρ (y) |$ is more sensitive to the change of $| h_{2} |$ than the change of $| h_{1} - r |$ . Thus, the increase of the swing angle $| h_{2} |$ is restrained in priori to the change of trolley position tracking error $| h_{1} - r |$ . However, in Case 2, $ρ_{2} (h_{2})$ has a much weaker affect in changing the value of $| ρ (h) |$ since $- α_{s} α_{2} h_{2} < 0$ . In this case, as $| h_{2} |$ increases, $| ρ (h) |$ decreases to reduce the damping ratio. Thus ${\overset{\cdot}{h}}_{1}$ increases when $| h_{2} |$ increases, which not only improves the displacement tracking performance but also improves the anti-sway performance. Also, $| ρ_{1} (h_{1}) |$ will increase when the trolley is approaching the target position, to reduce the overshoot of the displacement tracking.

By introducing $ρ_{2} (h_{2})$ , $ρ (h)$ obtains a larger range variation so that it can produce either a larger damping ratio to retrain the swing angle of the payload, or a smaller one to force the trolley to approach the tracking target. Similarly, when the trolley moves in a negative direction ( $α_{s} = - 1$ ), $ρ_{2} (h_{2})$ plays the same role as described above.

Parameter tuning

In this subsection, a simple method is given to tune the parameters $α_{1}$ , $α_{2}$ , $β_{1}$ and $β_{2}$ of the proposed nonlinear function in equation (17) such that the closed-loop system has a desired transient performance. By Theorem 3.1, the closed-loop system is asymptotically stable for any $α_{1}, α_{2}, β_{1}, β_{2} > 0$ . To determine the value of $β = β_{1} + β_{2}$ , let us consider the closed-loop steady-state system in equation (15) consisting of equations (4) and (11) with the proposed nonlinear function in equation (17), i.e.

\overset{\cdot}{\tilde{x}} = (A + B F + ρ B B^{T} P) \tilde{x}

where $\tilde{x} = x - x_{e}$ . By Theorem 3.1, we can obtain $lim_{t \to \infty} h_{1} (t) = r$ and $lim_{t \to \infty} h_{2} (t) = 0$ , implies which that

\lim_{t \to \infty} ρ (h (t)) = - (β_{1} + β_{2}) = - β

(22)

Thus, the steady-state system is given by

\begin{matrix} \overset{\cdot}{\tilde{x}} = (A + BF - β B B^{T} P) \tilde{x}, β = β_{1} + β_{2} \end{matrix}

(23)

As shown in Remark 3.2, we can select $β$ such that the closed-loop steady-state system in equation (23) has a desired damping ratio. Then, $β$ is allotted to $β_{1}$ and $β_{2}$ proportionally according to the desired performance preference. Please note that overweighting on $ρ_{2} (h_{2})$ can damage the tracking performance, especially the settling time of the trolley displacement tracking. Therefore, we need to balance the parameters $β_{1}$ and $β_{2}$ . Once $β_{1}$ and $β_{2}$ are fixed, the other parameters $α_{1}$ and $α_{2}$ can be simply tuned by trial and error.

Illustrative examples

In this section, we design a CNF control law with the proposed nonlinear function in equation (17) for the overhead crane system in Sun et al. (2012) of which the parameters are listed below

\begin{array}{l} M = 7.00 k g, m = 1.025 k g, l = 0.70 m, \\ g = 9.8 m / s^{2}, r = 0.6 m \end{array}

After a short review of the trajectory planning control design method in Sun et al. (2012), we will design a state feedback CNF control law and an output feedback CNF control law for the same plant. Then, both the proposed CNF control law and the trajectory planning method (Sun et al., 2012) will be applied to the nonlinear dynamic model in equation (1). Finally, the results of three simulation experiments are compared between these two control algorithms.

Review of the trajectory planning method

In Sun et al. (2012), a kinematic coupling-based off-line trajectory planning method was proposed for the overhead cranes in equation (1). Specifically, an anti-swing mechanism was introduced into an S-shape reference trajectory based on rigorous analysis for the coupling behaviour between the payload and the trolley in equation (24). After that, the combined trajectory was further tuned through the novel iterative learning strategy in equation (25), which guarantees accurate trolley positioning. The algorithm proposed in Sun et al. (2012) is summarized as follows

\begin{matrix} {\overset{\cdot\cdot}{x}}_{d} (k, t) = {\overset{\cdot\cdot}{x}}_{r} (k, t) + Γ {\overset{\cdot\cdot}{x}}_{a} (k, t), \\ x_{r} (k, t) = \frac{p_{r} (k)}{2} + \frac{1}{k_{2}} \ln [\frac{\cosh (k_{1} t - ε)}{\cosh (k_{1} t - ε - k_{2} p_{r} (k))}]), \\ {\overset{\cdot\cdot}{x}}_{a} (k, t) = \frac{- g \sin θ (t) + al \overset{\cdot}{θ} (t) + b (\overset{\cdot}{θ} (t) + a θ)}{\cos θ (t)}, \\ l \overset{\cdot\cdot}{θ} (t) + {\overset{\cdot\cdot}{x}}_{d} (k, t) + g θ (t) = 0 . \end{matrix}

(24)

I (k) = - \frac{Γ a b}{g + Γ a b} p_{r} (k), e (k) = p_{d} - p_{r} (k) - I (k), p_{r} (k + 1) = p_{r} (k) + Γ e (k)

(25)

$x_{d} (k, t)$ can be solved iteratively based on the above equations (24) and (25) for $k = 1, 2, \dots$ . After some iterations, $x_{d} (k, t)$ is used as the planned trajectory of the trolley displacement in Sun et al. (2012). The parameters in equations (24) and (25) are given by

a = b = 50, ε = 3.5, k_{1} = 1.2, k_{2} = 0.48, p_{d} = r = 0.6 m

Γ = 0.015, η = 3.0, p_{r} (1) = r = 0.6 m

In the simulation section in Sun et al. (2012), first let $k = 8$ and obtain the trajectory $x_{d} (8, t)$ . Using the planned trajectory $x_{d} (8, t)$ , a proportional–derivative (PD) controller is designed in the form of

u_{PD} = - k_{p} e_{tr} - k_{d} {\overset{\cdot}{e}}_{tr}

(26)

where $k_{p}$ , $k_{d} \in ℝ^{+}$ are positive gains and the tracking error $e_{tr} (t)$ is defined as $e_{tr} (t) = d (t) - x_{d} (8, t)$ . The control gains are selected as $k_{p} = 145$ and $k_{d} = 50$ . And when the friction compensation term $f_{r}$ is considered, the actuator force is of the following form (see eqn (67) in Sun et al. (2012))

u_{rPD} = - k_{p} e_{tr} - k_{d} {\overset{\cdot}{e}}_{tr} + f_{r}

(27)

where $f_{r}$ is given by equation (2) with $f_{r 0} = 4.4$ , $ξ_{r} = 0.01$ and $k_{r} = - 0.5$ .

State feedback composite nonlinear feedback control

The CNF control law for the overhead gantry system in equation (1) is designed based on the linearized dynamic model in equation (4) with

\begin{matrix} A = [\begin{matrix} 0 & 1 & 0 & 0 \\ - 16.0500 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1.4350 & 0 & 0 & 0 \end{matrix}], B = [\begin{matrix} 0 \\ - 0.2041 \\ 0 \\ 0.1429 \end{matrix}], \\ C_{1} = C_{2} = [\begin{matrix} 0 & 0 & 1 & 0 \end{matrix}], C_{3} = [\begin{matrix} 1 & 0 & 0 & 0 \end{matrix}] \end{matrix}

First, the linear feedback part in equation (6) of the CNF control law is considered. $F$ is selected such that $(A + BF)$ is a stable matrix, and the system $\overset{\cdot}{x} = (A + B F) x$ has a small damping ratio. Such an $F$ is given by

F = [229.3378 - 30.3685 - 120.0000 - 96.0468]

(28)

It is clear that $A + BF$ is stable with two pairs of poles $- 2.3820 \pm 7.5743 i$ and $- 1.3796 \pm 1.3796 i$ with damping ratios of the system $0.3000$ and $0.7071$ respectively. The feedforward gain $G$ is calculated by equation (7) which yields

G = - {[C_{1} {(A + BF)}^{- 1} B]}^{- 1} = 120.0000

(29)

Next, we choose $W$ of the Lyapunov equation in equation (10) to be a diagonal positive matrix with diagonal elements being $10^{3}$ , $10^{3}$ , $1$ and $1$ respectively. Consequently, solving equation (10), we obtain

\begin{matrix} P {= 10}^{3} \times [\begin{matrix} 7.1446 & - 0.9924 & - 1.9444 & - 1.8386 \\ - 0.9924 & 0.3635 & 0.5161 & 0.4058 \\ - 1.9444 & 0.5161 & 0.9318 & 0.7376 \\ - 1.8386 & 0.4058 & 0.7376 & 0.6339 \end{matrix}] \end{matrix}

(30)

Thus, we can have the CNF control law in the form of

u = F x + G r + ρ (h) B^{T} P (x - x_{e})

(31)

with the proposed nonlinear function in equation (17) consisting of equations (18) and (19), and $x_{e} = - {(A + BF)}^{- 1} BGr$ . Both the trolley displacement and the swing angle of the payload are fedback by the nonlinear function $ρ (h)$ .

Then we begin to tune the parameters of the CNF controller. First, as shown in Remark 3.2, an appropriate $ρ (h)$ varies in the range $| ρ (h) | > 5$ , especially in $| ρ (h) | \in [5, 35]$ . Checking Figure 3 and letting $β$ in equation (23) equal $16$ ( $β = 16$ ), the damping ratios of the system are $4.3533$ and $0.7319$ corresponding to the poles of $A + BF + β B B^{T} P$ , which are $s_{1} = - 1.0097$ , $s_{2} = - 74.5065$ and $s_{3, 4} = - 1.3109 \pm 1.2205 i$ . Allot $β$ to $β_{1}$ and $β_{2}$ with $β_{1} = β_{2} = 8$ . Then, $α_{1}$ and $α_{2}$ can be simply tuned by trial and error as $α_{1} = 0.4, α_{2} = 60$ . Finally the nonlinear function for the CNF control law in equation (31) is given by

ρ (h) = - 8 e^{- 0.4 α_{0} | h_{1} - r |} - 8 e^{- 60 α_{s} h_{2}}

(32)

In addition, if $α_{1} = 0$ and $α_{2} = 0$ , the CNF control law in equation (31) is reduced to a well-damped linear control law

\begin{matrix} u = Fx + Gr - β B^{T} P (x - x_{e}) \end{matrix}

(33)

where $β = β_{1} + β_{2}$ . Actually, equation (33) is the steady-state form of the CNF control law in equation (31) for $h_{1} (\infty) = r$ and $h_{2} (\infty) = 0$ .

Output feedback composite nonlinear feedback control

For practical applications, the state of the payload swing angle $θ$ and $\overset{\cdot}{θ}$ usually cannot be measured for feedback control. Therefore, for the case that only the trolley displacement $y = d$ can be measured and available to feed back, we design an output feedback CNF control law. It is clear that $(A, C_{1})$ is observable, we can design an overhead crane servo system with a full-order measurement-feedback CNF control law as follows

\begin{matrix} {\overset{\cdot}{x}}_{v} & = (A + K C_{1}) x_{v} - Ky + B sat (u) \\ u & = F x_{v} + Gr + ρ (y) B^{T} P (x_{v} - x_{e}) \\ ρ (y) & = - β_{1} e^{- α_{0} α_{1} | y - r |} - β_{2} e^{- α_{s} α_{2} C_{3} x_{v}} \end{matrix}

(34)

where $x_{e}$ , $β_{1}$ , $β_{2}$ , $α_{1}$ , $α_{2}$ , $F$ , $G$ , and $P$ are the same as the state feedback CNF control law in equation (31). And $K$ is given by

K = {[5 5 - 15 - 15]}^{T}

(35)

Moreover, considering the nonlinear rail friction, we insert a compensation term from equation (2) as in Sun et al. (2012), and the actuator force $u_{r}$ is given by

u_{r} = u + f_{r} = F x_{v} + Gr + ρ (y) B^{T} P (x_{v} - x_{e}) + f_{r}

(36)

where $f_{r}$ is given by equation (2) with $f_{r 0} = 4.4$ , $ξ_{r} = 0.01$ , and $k_{r} = - 0.5$ .

Simulation experiments

To further illustrate the transient performance of the proposed CNF control law, the results of three different simulation experiments are compared with the results given in Sun et al. (2012). To validate the function of the proposed CNF control, although the CNF control laws above are designed based on the linearized dynamic model in equation (4), they will be applied to the nonlinear dynamic equation in equation (1). That is, both the proposed CNF control law and the trajectory planning method (Sun et al., 2012) will be applied to the nonlinear dynamic model in equation (1).

Simulation Experiment 1: compare the transient performance of the overhead crane system under the state feedback CNF control law in equation (31), the planned trajectory $x_{d} (8, t)$ in Sun et al. (2012), and the CNF reduced linear control law in equation (33).

The results are shown in Figure 5 and the quantified results are listed in Table 1. As described in Sun et al. (2012), when using the planned trajectory $x_{d} (8, t)$ , the trolley reaches the tracking target position in $6 s (5.3700 s)$ without overshoot while the maximum swing amplitude is less than $1.6 \circ (1.5891 \circ)$ and it decays to $0$ soon after the trolley stops $(5.8260 s)$ , but the final position of the trolley is 0.5992 m and there remains a slight steady-state error. By contrast, using the state feedback CNF control law, also without overshoot, the trolley reaches the target position in only 4.4740 s and meanwhile the maximum swing amplitude is restrained to $1.2255 \circ$ . The payload swing angle decays to $0$ in 4.9940 s and the final position is as accurate as the target position, 0.6 m. Thus, the nonlinear overhead crane system in equation (1), using the state feedback CNF control law in equation (31), shows better transient performance both in trolley displacement tracking and in payload anti-sway, than that using the trajectory planning method (Sun et al., 2012). In addition, using the CNF reduced linear control law in equation (33) results in the largest $| θ |_{max}$ $(2.0141 \circ$ ) among the three methods and a longer settling time of the trolley displacement tracking than the CNF control law in equation (31), which indicatesthe nonlinear function $ρ (h)$ in equation (32) plays a significant role in improving the performance of both displacement tracking and payload anti-sway.

Figure 5.

Performance comparison: trolley displacement and swing angle.

Table 1.

Transient performance of the overhead crane system in simulation experiment 1. $t_{s 1}$ and $t_{s 2}$ denote the settling time of displacement tracking and anti-sway respectively.

Performance Control method	Displacement tracking			Payload anti-sway
	Overshoot	$t_{s 1}$ (s) ( $2 %$ )	Final Position (m)	$\| θ \|_{max}$ $(\circ)$	$t_{s 2}$ (s) ( $θ < 0.1 \circ$ )	Residual swing $(\circ)$
State feedback CNF law in equation (31)	0	4.4740	0.6000	1.2255	4.9940	$\approx 0$
Planned trajectory $x_{d} (8, t)$	0	5.3700	0.5992	1.5891	5.8260	$< 0.1$
CNF reduced linear law in equation (33)	0	4.6530	0.6000	2.0141	4.4400	$\approx 0$

In addition, to further illustrate the influence of the proposed nonlinear function $ρ (h)$ in equation (32) on the closed-loop system, the trajectories of $| ρ (t) |$ , $| ρ_{1} (t) |$ and $| ρ_{2} (t) |$ are drawn in Figure 6. It is shown that $| ρ (t) |$ varies in the interval $[11.3842, 34.3418]$ . According to the tendency shown in Figure 3, in this interval the damping ratio of the closed-loop system rises as $| ρ (t) |$ increases. It is clear that, at first, $ρ_{2} (h_{2})$ plays a leading role and it increases dramatically to increase the damping ratio to restrain the payload sway. After $ρ_{2} (h_{2})$ decreases, $ρ_{1} (h_{1})$ smoothly increases to guarantee the accurate tracking of trolley displacement without overshoot.

Figure 6.

Trajectory of the nonlinear function.

Simulation Experiment 2: compare the transient performance of the overhead crane system using the output feedback CNF control law in equation (34), and the PD control law in equation (26) proposed in Sun et al. (2012). Here the actuator force $u_{r}$ is limited in the range $[- 8 N, 8 N]$ . To guarantee that the actuator force (36) (with the friction compensation) will not exceed this range, we conservatively design the output feedback CNF control law $u$ in equation (34) in the interval $[- 2 N, 2 N]$ .

The results are shown in Figure 7 and Table 2. First, for the performance of the trolley displacement tracking, both control laws produce no overshoot, but the final positions of trolley are 0.6000 m and 0.5992 m respectively, which means that when using the PD control law there still remains a slight steady-state error. Furthermore, the settling time using the proposed CNF control law (4.6210 s) is over 0.7 s shorter than that obtained using the PD control law (5.3480 s). Then, for the performance of the trolley displacement tracking, there exists almost no residual swing for both control laws. When the PD controller is applied, the maximum payload swing angle is $1.6209 \circ$ and the settling time is 5.7840 s. By contrast, when using the proposed CNF control law, the maximum payload swing angle is reduced to only $1.2061 \circ$ and the settling time is shortened to 5.1410 s. The results provide a clear picture of the benefits of the proposed CNF control strategy, not only in restricting swinging of the payload, but also in achieving tracking quickly and smoothly.

Figure 7.

Performance comparison: trolley displacement and swing angle.

Table 2.

Transient performance of the overhead crane system in simulation experiment 2.

Performance Control method	Displacement tracking			Payload anti-sway
	Overshoot	$t_{s 1}$ (s) ( $2 %$ )	Final position (m)	$\| θ \|_{max}$ ( $\circ$ )	$t_{s 2}$ (s) ( $θ < 0.1 \circ$ )	Residual swing ( $\circ$ )
Output feedback CNF law in equation (34)	0	4.6210	0.6000	1.2061	5.1410	$\approx 0$
PD control law in equation (26)	0	5.3480	0.5992	1.6209	5.7840	$\approx 0$

Moreover, the control inputs and the actuator forces (with friction compensation) using both methods are shown in Figure 8. It is shown that both control laws vary in about the same range.

Figure 8.

The trajectories of control inputs.

Simulation Experiment 3: compare the robustness validation of actuator force using the current feedback CNF controller in equation (36) and the PD controller in equation (27) proposed in Sun et al. (2012). To test the robustness of both methods against payload variations, we utilized these two control methods with payloads of different mass.

The results are shown in Figure 9 and Table 3. It is shown that when the payload is increased from $m = 1.525 kg$ to $m = 2.025 kg$ , both controllers (27) and (36) can retain their performance, which indicates that both controllers are robust to an allowable variation of the mass of payload.

Figure 9.

Performance comparison with payloads of different mass.

Table 3.

Transient performance of the overhead crane system in simulation experiment 3.

Performance Control method	Displacement tracking			Payload anti-sway
	Over- shoot	$t_{s 1}$ (s) ( $2 %$ )	Final position (m)	$\| θ \|_{max}$ $(\circ)$	$t_{s 2}$ (s) ( $θ < 0.1 \circ$ )	Residual swing ( $\circ$ )
CNF law in equation (36) ( $m = 1.525 k g$ )	0	4.5500	0.6000	1.2125	5.1180	$\approx 0$
PD law in equation (27) ( $m = 1.525 k g$ )	0	5.3470	0.5992	1.6242	5.7690	$\approx 0$
CNF law in equation (36) ( $m = 2.025 k g$ )	0	4.4820	0.6000	1.2240	5.0960	$\approx 0$
PD law in equation (27) ( $m = 2.025 k g$ )	0	5.3460	0.5960	1.6274	5.7510	$< 0.1$

In summary, the results of the simulation experiments 1–3 illustrate that using the proposed CNF control law, the overhead crane system obtains better transient performance than using the trajectory planning method proposed in Sun et al. (2012). In addition, the design process of CNF controller is more straightforward.

Conclusion

In this paper, a novel nonlinear function in a CNF control law is proposed to improve the transient performance for an underactuated overhead crane system, considering both trolley displacement tracking and anti-sway control. The motivation of the proposed nonlinear function is specifically introduced and explained with simulation experiments. The proposed strategy is applied to design a CNF control law for a small-scale overhead crane servo system. The results of simulation experiments on the CNF controller show an improvement in transient performance. With the novel nonlinear function, not only is the swing angle of the payload restrained much more, but the performance of trolley displacement tracking is also improved.

Footnotes

Funding

The work was partially supported by National Nature Science Foundation of China (grant number 61074004), and the Research Fund for the Doctoral Program of Higher Education (grant number 20110121110017).

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