SMOCZEK Jarosław, SZPYTKO Janusz: Pole placement approach to crane control problem. Zastosowanie metody lokowania biegunów w systemie sterowania suwnicą.

POLE PLACEMENT APPROACH TO CRANE

CONTROL PROBLEM

ZASTOSOWANIE METODY LOKOWANIA

BIEGUNÓW W SYSTEMIE STEROWANIA SUWNICĄ

Jarosław Smoczek 1, Janusz Szpytko 2 (1, 2) AGH University of Science and Technology

Faculty of Mechanical Engineering and Robotics al. Mickiewicza 30, 30-059 Kraków

E-mail: (1) smoczek@agh.edu.pl, (2) szpytko@agh.edu.pl

Abstract: The time and positioning accuracy of transportation operations realized by material handling systems are more and more significant problem in automated manufacturing processes, as well as the exploitation quality, safety and reliability, which can be met by implementing automation systems and improving control quality of material handling devices. The presented problem of anti-sway control system was solved using pole placement method employed to the time-discrete closed loop crane control system, which can be used in adaptive solutions. The proposed solution is based on the assumption, that crane nonlinear system is considered as a linear model with varying parameters.

Keywords: overhead traveling crane, anti-sway control, pole placement

Streszczenie: W zautomatyzowanych procesach produkcyjnych coraz istotniejszą rolę odgrywa zarówno czas i dokładność zadań realizowanych przez urządzenia transportowe, jak również ich niezawodność i jakość eksploatacji. Coraz wyższe wymagania stawiane systemom i urządzeniom transportu technologicznego, w tym również suwnicom, spełnione być mogą poprzez wdrażanie zautomatyzowanych rozwiązań i poprawę jakości sterowania. W artykule przedstawiono metodę projektowania systemu regulacji pozycji i prędkości suwnicy pomostowej oraz kąta wychylenia przemieszczanego przez urządzenie ładunku, opartą na metodzie lokowania biegunów oraz przyjętej do celów identyfikacji strukturze modelu dynamiki obiektu regulacji.

Słowa kluczowe: suwnica pomostowa, regulacja tłumienia wahań ładunku, lokowanie biegunów

1. Introduction

The material handling systems are the key part of a manufacturing cycle in which they support technological and storage operations by handle materials within the manufacturing departments and production halls. The Large-Dimensional Rail-Mounted Handling Devices (WSUT) (Szpytko, 2004; Szpytko, 2006) is the class of the material handling devices to which belong the cranes (overhead traveling cranes, gantry, portal, railway and other types of cranes) that are used in many industrial branches, especially in heavy industry, metallurgical, shipbuilding, aircraft, machine-building and armaments industries, etc. In automated industrial processes the higher and higher requirements are put on time and precision of transportation tasks realized by material handling devices like overhead traveling cranes, as well as on ensuring suitable device exploitation quality, safety and reliability. From the transport means safety and reliability point of view is always important to reduce overloads that arise during transient states of crane’s power transmission systems working, which are caused by non-uniform loading, that leads to the crane’s bridge beveling, and affects unfavorable on exploitation of a wheel-rail system of the overhead traveling cranes. Those requirements can be met by applying automated solutions of material handling systems and control quality improvement. In many manufacturing processes where the transportation operations are realized by cranes the safety and precise transfer of materials is required with minimizing the load oscillations and the operation time. In the non-automatic systems the resulting performance depends on the human operator experience and capability which can be unreliable.

The problem of positioning a payload shifted by a crane is considered in many of scientific works, where is addressed an solved using different algorithms. In many of science works the problem of a load swing suppressing is considered using optimal control theory (Al.-Garni, et al., 1995; Auernig, and Troger, 1987; Hamalainen, et al., 1995). The problem of optimal path planning of a payload moved by crane is generally solved by minimizing the assumed function, which corresponds to the swing angle and its derivative, or to the energy consumption, that leads to obtain the values of control variables which transfer the nonlinear dynamic system from the initial state to the final state. In (Bartolini, et al., 2002; Boustany, and d’Andrea-Novel, 1968) the problem of moving a suspended load using a crane was addressed and solved using the feedback linearization. In other scientific works the considered problem is solved using poles assignment (Lew, and Halder, 2003; Hicar, and Ritok, 2006), Linear Quadratic

Regulator (LQR) (Benhidjeb, abd Gissinger, 1995; Lew, and Halder, 2003), gain scheduling (Corriga, et al., 1998), Lyapunov-equivalence-based observer (Giua, et al., 1999), as well as unconventional methods, mostly based on fuzzy logic (Benhidjeb, and Gissinger, 1995; Cho, and Lee, 2002; Mahfouf, et al., 2000).

In this paper the problem of crane control system was addressed and solved using pole placement method. The nonlinear crane system was presented in a form of the linear model with varying parameters, which correspond to the values changes of rope length and masses of a load. The known model of a crane dynamic allows to employ the pole placement method to solve the problem of determining the parameters of time-discrete closed-loop crane control system.

2. Identification of a crane dynamic system

On the assumption that a crane dynamic system can be considered as a linear model with varying parameters, depended on values of rope length l and mass of a load, denoted as m₂, the model of a crane dynamic can be formulated as the relationship between output signals (crane position x and speed x, and the load swing ) and input function u.

) ( ) ( 1 ) ( U z G z z z z X   _x   (1) ) ( ) ( ) (z U z G z X   _x (2) ) ( ) ( ) (z X z G_ z   (3)

The assumed structure of crane dynamic model is shown in the figure 1.

Fig. 1 The assumed model of a crane dynamic consisting of two models expressed in a form of discrete transfer functions G_x(z) and G_(z)

The model of controlled object consists of two sub-models which are expressed as the discrete transfer functions G_x(z) and G_(z). In presented in the figure 1 model the swing of a load influence on a crane speed was omitted for simplicity. Consequently the parametric model of a controlled object consists of two models that present relationship between crane speed

x and input function u (transfer function G_x(z)), as well as the load swing

 and crane velocity x (transfer function G_(z)).

0 0 ) ( ) ( ) ( c z d z C z D z G_x     (4) 0 1 2 0 1 ) ( ) ( ) ( a z a z b z b z A z B z G       (5)

The presented crane dynamic models can be identified separately, using off-line methods, e.g. output error (OE) method, or on-off-line recursive least squares (RLS) algorithm, depending on the assumed adaptive control algorithm.

3. The closed loop speed and anti-sway crane control system

The time-discrete crane control system can be built based on the identified parametric model of a crane dynamic presented in a form of time-discrete transmittances (4) and (5). In the figure 2 the anti-sway and crane speed control system was presented with two parallel controllers of the crane speed R_x(z) and the load swing angle R_(z).

Fig. 2 The time-discrete crane speed and the load swing angle control system x P x z K R( )  (6) 0 0 1 ) ( ) ( ) ( s z q z q z S z Q z R      (7)

The transfer function G_C(z) of the closed loop control system is formulated as follows: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( z S z D z A K z Q z D z B z S z C z A z D z B z S K z X z z G x P x P d C                    (8)

For the characteristic equation of closed loop control system transmittance (8), and desired characteristic equation, denoted as P(z), the Diophantine equation is formulated as:

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (z C z S z B z D z Q z K A z D z S z P z A       _P    (9)

where the P(z) is the n4 order equation (10).

0 1 2 2 3 3 4 1 0 ) (z z p z z p z p z p z p P n i i i n        



  (10)

A vector of characteristic equation coefficients



p₃,p₂,p₁,p₀



T, is derived based on two the same pairs desired poles (11), depended on dimensionless dumping coefficient  and pulsation ₀ of the load swing in the closed loop control system.

           _{  }  _{0 0} 2 ₀ 4 , 3 , 2 , 1 exp j 1 T z  (11)

The unknown vector of controllers gains



K_Px,q₁,q₀,s₀



T is derived from equations system (12) determined based on Diophantine equation (9).





















                            0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 2 0 1 0 1 1 0 0 0 1 0 1 0 3 0 0 0 1 p d a s K q b d s c a p s a a d K q b q b d s c a a c a p s a d K q b d s c a c a a p d K s c a x P x P x P x P     (12)

The equations system (12) can be presented as:

P Q D B S D A S C A  K_Px       (13) where:

A, B, C, D - the matrixes composed of the controlled object

model’s parameters,               0 0 1 1 , s q q S

Q - the vectors consist of the load swing controller gains,



_{p p p p}



T 0 1 2 3, , , 

P - the vector of characteristic equation P(z) coefficients. The equation (13) is expanded in the expression (14).

                                                                                              0 1 2 3 0 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 1 0 1 p p p p q q b b b b s a a a a K d s c c a a a a a a P (14)

The nonlinear equations (14) lead to obtain two vectors of roots, expected gains of the controllers. The Diophantine equation (13) determined for desired poles of closed loop control system (Fig. 2) under consideration allows to solve the problem of crane speed and the load swing control system, which can be next expended by adding the proportional controller of crane position.

4. The closed loop position and anti-sway crane control system

In the figure 3 is presented the crane position and the load swing angle control system with proportional controllers of crane position and speed

Px

K and K_Px respectively, and the load swing angle time-discrete controller R_(z).

Fig. 3 The time-discrete crane position and the load swing control system For the characteristic equation of closed loop control system transmittance the Diophantine equation is formulated as:





) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 z P z S z D z A K K z z S z D z A K z Q z D z B z S z C z A z x P Px x P                                 (15)

The P(z) is the n5 order desired characteristic equation of the closed loop control system derived for desired poles:



0 0



5 0 2 0 0 4 , 3 , 2 , 1 exp j 1 T , z exp T z _             _{  }   (16)

The Diophantine equation can be formulated as the expression (17)



z-1

 

 ACSK_PxADSBQD



K_PxK_PxzADSP (17) which can be expanded to the expression (18):

                                                                                                                                                                                                 0 1 2 3 4 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 p p p p p q q b b b b b b b b s a a a a K s a a a a a a a a K d s c c a a a a a a a a a a a a Px x P  (18)

The nonlinear equation (17) leads to obtain three vectors of roots, in form of controllers gains of the crane position and the load swing angle closed loop control system (Fig. 3).

The considered time-discrete crane control systems, formulated for output error models of identified controlled object, as well as the method of gains determining based on pole placement (PP), can be used to built adaptive control systems.

5. Simulations and experimental results

The anti-sway crane control system (Fig. 3), with proportional controllers of crane position and speed, and discrete controller of load swing corresponded to the proportional-derivative controller, and the method of gains determining presented in the previous stage, based on pole placement

method (PPM), were tested during simulations and experiments. The researches object was the laboratory overhead traveling crane with hoisting capacity equals Q=150 [kg]. The control system was simulated using model of crane dynamic, formulated as the transfer functions (4) and (5), identified using output error method based on data measured during experiments carried out on the laboratory object. The examples of simulations results are presented in the figures 47, where performances of time-discrete control systems with parameters determined for chosen rope length values

] [ } 7 , 1 ; 2 , 1 ; 7 , 0 { m

l  and fixed mass of a load m ₂ 10[kg] are compared. The controllers were designed for poles determined for known pulsation of closed loop control system, and assumed value of dumping coefficient 1. The expected position of a crane and shifted load was x_d 1 m[ ]. The result of simulations confirm obtaining the expected aims, lack of output signals oscillations and overshoots after setting time, depended of course on the value of rope length. The performances of crane control systems, designed for models identified for fixed rope length and mass of a load, confirm also correctness of presented method of gains determining.

Fig. 4. The crane position for ] [ } 7 , 1 ; 2 , 1 ; 7 , 0 { m l  and m ₂ 10[kg]

Fig. 5. The load swing for ] [ } 7 , 1 ; 2 , 1 ; 7 , 0 { m l  and m ₂ 10[kg]

Fig. 6. The crane position for ] [ } 7 , 1 ; 2 , 1 ; 7 , 0 { m l  and m ₂ 50[kg]

Fig. 7. The load swing for ] [ } 7 , 1 ; 2 , 1 ; 7 , 0 { m l  and m ₂ 50[kg]

In the figures 811, the results of simulations carried out using two models, determined for l 0,7[m] and m ₂ 10[kg], and for l 1,2[m] and m ₂ 70[kg], are presented. The performances of control systems (Fig. 3), determined for parameters of those models, are compared with results obtained using controllers designed for parameters of crane dynamic model identified for rope length

] [ 7 , 1 m

l  and mass of a load m ₂ 10[kg]. The results presented in the figures 811 confirm the lack of robustness of control system against the changes of controlled system parameters.

The figures 12 and 13 present the example of experimental results obtained on the laboratory object using the control system (Fig. 3) realized based on the PC with Matlab software (RTW tools) and I/O board (PCI-1710HG control-measurement card manufactured by Advantech firm). The experiment was conducted for constant values of rope length l 0,7[m] and mass of a load m ₂ 30[kg]. The performances of the same control system were compared, with controllers gains determined for assumed in experiment parameters (l 0,7[m], m ₂ 30[kg]), and next with gains derived based on the model identified for l 1,7[m],

] [ 10

2 kg

m  . The figure 13 present the load swing in a form of deviation of the load from vertical symmetry axis of the rope drum (calculated as a product of rope length l and the load swing angle ).

The experimental results confirm conclusions obtained from simulations. The control system presented in the figure 3, with gains derived for different parameters (l 1,7[m], m ₂ 10[kg]) is not robustness against changes of those parameters (l 0,7[m], m ₂ 30[kg]): the oscillations are reduced to the expected tolerance 0,02 [m] at about 10 seconds (Fig. 13), while the expected setting time was about 7 seconds. The properly derived controllers ( for l 0,7[m], m ₂ 30[kg]) reduce the oscillations of load swing about the desired range just at 3-4 seconds, and next are definitely suppressed at the setting time, 7 seconds.

Fig. 8 The crane position, simulation for l 0,7[m] and m ₂ 10[kg]

Fig. 9 The load swing, simulation for ] [ 7 , 0 m l  and m ₂ 10[kg]

Fig. 10 The crane position, simulation for l 1,2[m] and m ₂ 70[kg]

Fig. 11 The load swing, simulation for ] [ 2 , 1 m l  and m ₂ 70[kg]

Fig. 12 The crane position, experiment for l 0,7[m] and m ₂ 30[kg]

Fig. 13 The load swing, experiment for l 0,7[m] and m ₂ 30[kg] 6. Final remarks

The proposed method of conventional anti-sway crane control system designing allows to realize automated transportation operations of a crane, with expected precision and setting time, which is more and more significant problem in automated manufacturing processes. The automation of material handling systems is required owing to higher and higher demands put not only on the accuracy, efficiency and productivity, but also on the exploitation quality, safety and reliability of transportation devices. The presented non-adaptive solution of crane speed or position, and anti-sway control systems, as well as methods of controlled object identification and control system designing can be used in adaptive approaches to a crane control problem, for example in the gain scheduling system with a set of controllers derived for fixed values of rope length and mass of a load, or using indirect adaptive pole placement system with real-time identification of a crane dynamic model.

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PhD. Eng. Jarosław Smoczek, AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics. Specialist in designing and exploitation of transport systems and devices, automatics, monitoring and diagnostics. Author or co-author of more then 40 publications, both in Polish and English.

Prof. Janusz Szpytko DSc. Eng., AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics. Specialist in designing and exploitation of transport systems and devices, automatics, safety and reliability, monitoring and diagnostics, decision making systems, telematics. Author or co-author of more then 300 publications, both in Polish and English. Member of: STST KT PAN, TC IFAC, SEFI, ISPE, PTD, PTB, PSRA, ISA, SITPH and others. Visiting professor at the universities in: UK, France, Canada, Italy, Greece, Canada, Laos. Coordinator and member of several R&D projects both national and international. Organizer and member of several scientific and programme committees of international and national conferences and symposiums.