Optimal Tracking with Sway Suppression Control for a Gantry Crane System

Abstract This paper presents investigations into the development of hybrid control schemes for input tracking and anti-swaying control of a gantry crane system.. The performances of the

ISSN 1450-216X Vol.33 No.4 (2009), pp.630-641

© EuroJournals Publishing, Inc 2009

http://www.eurojournals.com/ejsr.htm

Optimal Tracking with Sway Suppression Control

for a Gantry Crane System

M.A Ahmad

Control and Instrumentation Research Group (COINS), Faculty of Electrical and Electronics Engineering Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia E-mail: mashraf@ump.edu.my

Tel: +609-5492366; Fax: +609-5492377

R.M.T Raja Ismail

Control and Instrumentation Research Group (COINS) Faculty of Electrical and Electronics Engineering Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia

E-mail: rajamohd@ump.edu.my Tel: +609-5492366; Fax: +609-5492377

M.S Ramli

Control and Instrumentation Research Group (COINS) Faculty of Electrical and Electronics Engineering Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia

E-mail: syakirin@ump.edu.my Tel: +609-5492366; Fax: +609-5492377

N.M Abdul Ghani

Control and Instrumentation Research Group (COINS) Faculty of Electrical and Electronics Engineering Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia

E-mail: normaniha@ump.edu.my Tel: +609-5492366; Fax: +609-5492377

M.A Zawawi

Control and Instrumentation Research Group (COINS) Faculty of Electrical and Electronics Engineering Universiti Malaysia Pahang, Lebuhraya Tun Razak

26300, Kuantan, Pahang, Malaysia

E-mail: mohdanwar@ump.edu.my Tel: +609-5492366; Fax: +609-5492377

Abstract

This paper presents investigations into the development of hybrid control schemes for input tracking and anti-swaying control of a gantry crane system A nonlinear overhead gantry crane system is considered and the dynamic model of the system is derived using the Euler-Lagrange formulation To study the effectiveness of the controllers, initially a Linear Quadratic Regulator (LQR) control is developed for cart position control of gantry crane This is then extended to incorporate input shaper control schemes for anti-swaying control

of the system The positive input shapers with the derivative effects are designed based on the properties of the system Simulation results of the response of the manipulator with the controllers are presented in time and frequency domains The performances of the hybrid control schemes are examined in terms of level of input tracking capability, swing angle reduction, time response specifications and robustness to parameters uncertainty in comparison to the LQR control The effects of derivative order of the input shaper on the performance of the system are investigated Finally, a comparative assessment of the control techniques is presented and discussed

Keywords: Gantry crane, sway control, input shaping, LQR controller

1 Introduction

The main purpose of controlling a gantry crane is transporting the load as fast as possible without causing any excessive swing at the final position However, most of the common gantry crane results

in a swing motion when payload is suddenly stopped after a fast motion [1] The swing motion can be reduced but will be time consuming Moreover, the gantry crane needs a skilful operator to control manually based on his or her experiences to stop the swing immediately at the right position The failure of controlling crane also might cause accident and may harm people and the surrounding

Various attempts in controlling gantry cranes system based on open loop system were proposed For example, open loop time optimal strategies were applied to the crane by many researchers such as discussed in [2,3] They came out with poor results because open loop strategy is sensitive to the system parameters (e.g rope length) and could not compensate for wind disturbances Another open loop control strategies is input shaping [4,5,6] Input shaping is implemented in real time

by convolving the command signal with an impulse sequence The process has the effect of placing zeros at the locations of the flexible poles of the original system An IIR filtering technique related to input shaping has been proposed for controlling suspended payloads [7] Input shaping has been shown

to be effective for controlling oscillation of gantry cranes when the load does not undergo hoisting [8, 9] Experimental results also indicate that shaped commands can be of benefit when the load is hoisted during the motion [10]

On the other hand, feedback control which is well known to be less sensitive to disturbances and parameter variations [11] is also adopted for controlling the gantry crane system Recent work on gantry crane control system was presented by Omar [1] The author had proposed proportional-derivative PD controllers for both position and anti-swing controls Furthermore, a fuzzy-based intelligent gantry crane system has been proposed [12] The proposed fuzzy logic controllers consist of position as well as anti-sway controllers However, most of the feedback control system proposed needs sensors for measuring the cart position as well as the load swing angle In addition, designing the swing angle measurement of the real gantry crane system, in particular, is not an easy task since there

is a hoisting mechanism

This paper presents investigations into the development of hybrid control schemes for input tracking and anti-swaying control of a gantry crane system Hybrid control schemes based on feedforward with LQR controllers are investigated To demonstrate the effectiveness of the proposed

control schemes, initially a LQR controller is developed for cart position control of gantry crane

system This is then extended to incorporate the positive input shapers for swing control of the gantry

crane This paper provides a comparative assessment of the performance of hybrid control schemes

with different derivative order of input shapers

2 Gantry Crane System

The two-dimensional gantry crane system with its payload considered in this work is shown in Figure

1, where x is the horizontal position of the cart, l is the length of the rope, θ is the sway angle of the

rope, M and m is the mass of the cart and payload respectively In this simulation, the cart and payload

can be considered as point masses and are assumed to move in two-dimensional, x-y plane The tension

force that may cause the hoisting rope elongate is also ignored In this study the length of the cart, l =

1.00 m, M = 2.49 kg, m = 1.00 kg and g = 9.81 m/s2 is considered

Figure 1: Description of the gantry crane system

3 Modelling of the Gantry Crane

This section provides a brief description on the modelling of the gantry crane system, as a basis of a

simulation environment for development and assessment of the active sway control techniques The

Euler-Lagrange formulation is considered in characterizing the dynamic behaviour of the crane system

incorporating payload

Considering the motion of the gantry crane system on a two-dimensional plane, the kinetic

energy of the system can thus be formulated as

) cos 2

sin 2 (

2

1 2

1 M x&2 m x&2 l&2 l2θ&2 x l& θ x&lθ& θ

The potential energy of the beam can be formulated as

θ cos

mgl

To obtain a closed-form dynamic model of the gantry crane, the energy expressions in (1) and

(2) are used to formulate the Lagrangian L=T −U Let the generalized forces corresponding to the

generalized displacements q ={x,θ} be F = {F x, } Using Lagrangian’s equation

2 , 1

=

=

∂

∂

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

j F

q

L q

L dt

d

j j j

the equation of motion is obtained as below,

θ θ

θ θ

θ θ

( )

0 sin cos

In order to eliminate the nonlinearity equation in the system, a linear model of gantry crane

system is obtained The linear model of the uncontrolled system can be represented in a state-space

form as shown in equation (6) by assuming the change of rope and sway angle are very small

x y

u x x

C

B A

=

+

=

&

(6) with the vector [ ]T

x x

x= θ & θ& and the matrices A and B are given by

, 1

1 0 0 ,

0 0 ) (

0

0 0 0

1 0 0

0

0 1 0

0

=

=

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

=

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+

−

A

Ml

M Ml

g m

M M

mg

(7)

4 Linear Quadratic Regulator (LQR) Control Scheme

A more common approach in the control of manipulator systems involves the utilization linear

quadratic regulator (LQR) design [13] Such an approach is adopted at this stage of the investigation

here In order to design the LQR controller a linear state-space model of the gantry crane system was

obtained by linearising the equations of motion of the system For a LTI system

Bu Ax

the technique involves choosing a control law u= ψ(x) which stabilizes the origin (i.e., regulates x to

zero) while minimizing the quadratic cost function

∫

∞

+

=

0

) ( ) ( ) ( ) (t Qx t u t Ru t dt x

where Q = Q T ≥ 0 and R = R T > 0 The term “linear-quadratic” refers to the linear system dynamics

and the quadratic cost function

The matrices Q and R are called the state and control penalty matrices, respectively If the

components of Q are chosen large relative to those of R , then deviations of R from zero will be

penalized heavily relative to deviations of u from zero On the other hand, if the components of R are

large relative to those of Q then control effort will be more costly and the state will not converge to

zero as quickly

A famous and somewhat surprising result due to Kalman is that the control law which

minimizes J always takes the form u=ψ(x)=−Kx The optimal regulator for a LTI system with

respect to the quadratic cost function above is always a linear control law With this observation in

mind, the closed-loop system takes the form

x BK A

and the cost function J takes the form

∫

∞

−

− +

=

0

)) ( ( )) ( ( ) ( ) (t Qx t Kx t R Kx t dt x

∞

+

=

0

) ( ) (

) (t Q K RK x t dt x

Assuming that the closed-loop system is internally stable, which is a fundamental requirement

for any feedback controller, the following theorem allows the computation value of the cost function

for a given control gain matrix K

5 Input Shaping Control Schemes

The design objectives of input shaping are to determine the amplitude and time locations of the

impulses in order to reduce the detrimental effects of system flexibility These parameters are obtained

from the natural frequencies and damping ratios of the system The input shaping process is illustrated

in Figure 2 The corresponding design relations for achieving a zero residual single-mode swaying of a

system and to ensure that the shaped command input produces the same rigid body motion as the

unshaped command yields a two-impulse sequence with parameter as

t1 = 0, t2 =

d

ω

π , A

K

+ 1

1 , A2 =

K

K

+

where

2

ζπ

−

−

= e

(ωn and ζ representing the natural frequency and damping ratio respectively) and t j and A j are the

time location and amplitude of impulse j respectively The robustness of the input shaper to errors in

natural frequencies of the system can be increased by solving the derivatives of the system swaying

equation This yields a four-impulse sequence with parameter as

t 1 = 0, t 2 =

d

ω

π

, t 3 =

d

ω

π

2

, t 4 =

d

ω

π

3

3 2 1

3 3 1

1

K K K

A

+ + +

3 2 2

3 3 1

3

K K K

K A

+ + +

=

3 2

2 3

3 3 1

3

K K K

K A

+ + +

3 2

3 4

3 3

K A

+ + +

where K as in (13)

Figure 2: Illustration of input shaping technique

Time *

A1

A2

Time

Unshaped Input Input Shaper Shaped input

6 Implementation and Result

In this investigation, hybrid control schemes for tracking capability and sway angle suppression of the

gantry crane system are examined Initially, a Linear Quadratic Regulator (LQR) is designed This is

then extended to incorporate input shaping scheme for control of sway angle of the hoisting rope

The tracking performance of the Linear Quadratic Regulator applied to the gantry crane system

was investigated by firstly setting the value of vector K and N which determines the feedback control

law and for elimination of steady state error capability respectively Using the lqr function in the

Matlab, both vector K and N were set as

[700 428 3885 3 8267 1 4066]

=

The natural frequencies were obtained by exciting the gantry crane system with an unshaped reference input under LQR controller The input shapers were designed for pre-processing the trajectory reference input and applied to the system in a closed-loop configuration, as shown in Figure

3

Figure 3: Block diagram of the hybrid control schemes configuration

Gantry Crane System

Output responses

Input shaper

Shaped input

Desired

input

K

+

-N

LQR Controller

6.1 LQR Controller

In this work, the input is applied at the cart of the gantry crane The cart position of the gantry crane is required to follow a trajectory within the range of ± m The first three modes of swing angle 4 frequencies of the system are considered, as these dominate the dynamic of the system

The responses of the gantry crane system to the unshaped trajectory reference input were analyzed in time-domain and frequency domain (spectral density) These results were considered as the system response to the unshaped input under tracking capability and will be used to evaluate the performance of the input shaping techniques The steady-state cart position trajectory of +4 m for the gantry crane was achieved within the rise and settling times and overshoot of 1.372 s, 2.403 s and 0.20

% respectively It is noted that the cart reaches the required position from +4 m to -4 m within 3 s, with little overshoot However, a noticeable amount of swing angle occurs during movement of the cart It is noted from the swing angle response with a maximum residual of ±1.4 rad Moreover, from the PSD of the swing angle response the swaying frequencies are dominated by the first three modes, which are obtained as 0.3925 Hz, 1.177 Hz and 2.06 Hz with magnitude of 33.02 dB, -9.929 dB and -22.86 dB respectively

6.2 Hybrid Controller

In the case of hybrid control schemes, a ZV (two-impulse sequence) and ZVDD (four-impulse sequence) shapers were designed for three modes utilising the properties of the system With the exact natural frequencies of 0.3925 Hz, 1.177 Hz and 2.06 Hz, the time locations and amplitudes of the impulses were obtained by solving equations (13) and (14) For evaluation of robustness, input shapers with error in natural frequencies were also evaluated With the 30% error in natural frequency, the system swaying frequencies were considered at 0.5103 Hz, 1.5301 Hz and 2.678 Hz for the three modes of swaying frequencies Similarly, the amplitudes and time locations of the input shapers with 30% erroneous natural frequencies for both the ZV and ZVDD shapers were calculated For digital implementation of the input shapers, locations of the impulses were selected at the nearest sampling time

The system responses of the gantry crane system to the shaped trajectory input with exact natural frequencies using LQR control with ZV and ZVDD shapers are shown in Figure 4 Table 1 summarises the levels of swaying reduction of the system responses at the first three modes in comparison to the LQR control Higher levels of swaying reduction were obtained using LQR control with ZVDD shaper as compared to the case with ZV shaper However, with ZVDD shaper, the system response is slower Hence, it is evidenced that the speed of the system response reduces with the increase in number of impulse sequence The corresponding rise time, setting time and overshoot of the cart trajectory response using LQR control with ZV and ZVDD shapers with exact natural frequencies

is depicted in Table 1 It is noted that a slower cart trajectory response with less overshoot, as compared to the LQR control, was achieved

To examine the robustness of the shapers, the shapers with 30% error in swaying frequencies were designed and implemented to the gantry crane system The analysis from Figure 5 shows that the swaying angles of the system were considerable reduced as compared to the system with LQR control However, the level of swaying reduction is slightly less than the case with exact natural frequencies Table 1 summarises the levels of swaying reduction with erroneous natural frequencies in comparison

to the LQR control The time response specifications of the cart trajectory with error in natural frequencies are summarised in Table 1 It is noted that the response is slightly faster for the shaped input with error in natural frequencies than the case with exact frequencies However, the overshoot of the response is slightly higher than the case with exact frequencies Significant swaying reduction was achieved for the overall response of the system to the shaped input with 30% error in natural frequencies, and hence proved the robustness of the input shapers

Figure 4: Response of the gantry crane with exact natural frequencies

(a): Cart trajectory

-5 -4 -3 -2 -1 0 1 2 3 4 5

Time (s)

LQR-ZV Shaper LQR-ZVDD Shaper

(b): Sway angle

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Time (s)

LQR-ZV Shaper LQR-ZVDD Shaper

(c): Power spectral density

-100 -80 -60 -40 -20 0 20 40

Frequency (Hz)

LQR LQR-ZV Shaper LQR-ZVDD Shaper

Figure 5: Response of the gantry crane with erroneous natural frequencies

(a): Cart trajectory

-5 -4 -3 -2 -1 0 1 2 3 4 5

Time (s)

LQR-ZV Shaper LQR-ZVDD Shaper

(b): Sway angle

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Time (s)

LQR-ZV Shaper LQR-ZVDD Shaper

(c): Power spectral density

-100 -80 -60 -40 -20 0 20 40

Frequency Hz

LQR LQR-ZV Shaper LQR-ZVDD Shaper

6.3 Comparative Performance Assessment

By comparing the results presented in Table 1, it is noted that the higher performance in the reduction

of swaying of the system is achieved using LQR control with ZVDD shaper This is observed and compared to the LQR control with ZV shaper at the first three modes of swaying frequency For comparative assessment, the levels of swaying reduction of the hoisting rope using LQR control with both ZV and ZVDD shapers are shown with the bar graphs in Figure 6 The result shows that, highest level of swaying reduction is achieved in hybrid control schemes using the ZVDD shaper, followed by the ZV shaper for all modes of swaying frequency Therefore, it can be concluded that the LQR control with ZVDD shapers provide better performance in swaying reduction effect as compared to the LQR control with ZV shapers in overall

Comparisons of the specifications of the cart trajectory responses of hybrid control schemes using both ZV and ZVDD shapers are summarised in Figure 7 for the rise times and settling times It is noted that the differences in rise times of the cart trajectory response for the LQR control with ZV and ZVDD shapers are negligibly small However, the settling time of the cart trajectory response using the LQR control with ZV shaper is faster than the case using the ZVDD shaper It shows that, by incorporating more number of impulses in hybrid control schemes resulted in a slower response

Comparison of the results shown in Table 1 for the shaping techniques with error in natural frequencies reveals that the higher robustness to parameter uncertainty is achieved with the LQR control with ZVDD shaper For both case of the ZV and ZVDD shapers, errors in natural frequencies can successfully be handled This is revealed by comparing the magnitude level of swaying frequency

of the system in Figure 6 Comparisons of the cart trajectory response using LQR control with ZV and ZVDD shapers with erroneous natural frequencies are summarised in Figure 7 The results show a similar pattern as the case with exact natural frequencies The system response with ZVDD shaper provides slightly slower responses than the ZV shaper

Figure 6: Level of swaying reduction with exact and erroneous natural frequencies using ZV and ZVDD

shapers

0 10 20 30 40 50 60 70

Mode of Vibration

LQR-ZVDD Shaper

LQR-ZV Shaper (error) LQR-ZVDD Shaper (error)