Smoczek Jarosław, Szpytko Janusz: Genetic algorithm and pole placement approach to adaptive crane control system. Zastosowanie algorytmu genetycznego i metody lokowania biegunów w adaptacyjnym systemie sterowania suwnicą.

GENETIC ALGORITHM AND POLE PLACEMENT

APPROACH TO ADAPTIVE CRANE CONTROL SYSTEM

ZASTOSOWANIE ALGORYTMU GENETYCZNEGO

I METODY LOKOWANIA BIEGUNÓW

W ADAPTACYJNYM 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 paper is the contribution to an anti-sway crane control problem, which in many of scientific works is solved with using soft computing techniques, like fuzzy logic, artificial neural network, evolutionary algorithms, as well as their hybrids. The indirect adaptive crane control system, which is analyzed in the paper, was based on the genetic algorithm, which is used to estimate the parameters of a known model of the controlled object. Proposed solution allows to adjust the gains of control algorithm with using pole placement method to the actual values of no stationary parameters of crane model, which are the function of rope length and mass of a payload moved by crane.

Keywords: anti-sway crane control, genetic algorithm, pole placement method Streszczenie: Zagadnienie regulacji pozycji ładunku przemieszczanego przez suwnicę jest przedmiotem wielu prac, w których często proponowane są rozwiązania oparte na tak zwanych inteligentnych metodach obliczeniowych. W artykule przedstawione zostało rozwiązanie powyższego problemu oparte na adaptacji pośredniej, w której parametry niestacjonarnego modelu obiektu regulacji estymowane są poprzez zastosowanie algorytmu genetycznego. W oparciu o estymator parametrów modelu suwnicy oraz metodę lokowania biegunów dostrajane są nastawy konwencjonalnych regulatorów proporcjonalno-różniczkujących zastosowanych w układzie regulacji pozycji suwnicy i kąta wychylenia ładunku.

Słowa kluczowe: sterowanie adaptacyjne suwnicą, algorytm genetyczny, lokowanie biegunów

1. Introduction

In automated manufacturing processes the safety, precise and fast transfer of goods realized by automated material handling devices (AMHDs) is required to raise efficiency and productivity of manufacturing process (Szpytko, 2004a; Szpytko 2004b). In those industrial branches where cranes are extensively used the problem of an anti-sway crane control is especially important to speed-up the time of transportation operations. In many of scientific works the problem of load swing suppressing is considered with using unconventional methods called soft computing techniques like fuzzy logic, artificial neural network, evolutionary algorithms, as well as their hybrids. In (Moon et al., 1996) the Mamdani fuzzy controller was applied to solve the problem of time optimal crane control. In (Cho and Lee, 2002) the combination of proportional-derivative (PD) controller of crane mechanisms position and speed, and the fuzzy controller of the load swing was considered for three-dimensional overhead crane. In (Itoh et al., 1993) the measuring method of the load swing based on a camera detector is presented together with fuzzy controller of the load swing, which was used to modify the assumed pattern of acceleration and deceleration of crane’s movement mechanism. The anti-sway crane control problem is also solved with using fuzzy rules-based controllers with Mamdani implications in (Mahfouf et al., 2000; Nalley and Trabia, 2000). However the method of designing of Mamdani type fuzzy controller is rather hardly ever presented in literature, and frequently is based on the heuristic knowledge about controlled process, and formulated in form of rules if-then, which express knowledge about relationships between input and output variables of the system. The alternative solution for Mamdani controller is Takagi-Sugeno-Kang (TSK) fuzzy model in which the rules output is calculated based on the function or singleton specified in rules' conclusion. The solutions of crane control systems based on TSK fuzzy inference systems are proposed in (Kang et al., 1999; Smoczek and Szpytko, 2008; Yi and Yubazaki, 2003). In (Kang et al., 1999) the robust switching control scheme was based on TSK fuzzy system, which switches several controllers, each designated for a different fixed-length of a rope nominal model. The robust crane control system based on TSK controller consisting of knowledge base (KB) with rules, each designed for a given operating point, is presented in (Smoczek, 2010) together with satisfactory results of experiments carried out on the overhead travelling crane with hoisting capacity Q=12,5 tons working in the workshops.

The other approach to crane control system, which was based on artificial neural network is presented in (Acosta et al., 1999; Mendez et al., 1999) Authors successfully carried out simulations with using the neurocontroller, which is a self-tuning system consisting of a conventional controller combined with neural network used to calculate the coefficients of the controller. The control parameters are adjusted using the neural network trained on-line to minimize a quadratic cost function. The combination of neural network and fuzzy logic was used to solve the

crane control problem for example in (Ishide et al., 1993; Smoczek and Szpytko, 2009).

The evolutionary algorithm are also frequently applied to solve the both problems concerned the crane movement optimization, anti-sway crane control and crane operations scheduling. The time-optimal control with using genetic algorithm was proposed for unconstrained optimal crane control in (Kimiaghalam et al., 1999). The real-coded genetic algorithm was used to find the desired initial co-states of the system with no-constrains. The objective function was formulated as the minimum cost co-states calculated based on the ability to move the system to the desired state after a given amount of time. In (Nakazono et al., 2007) the anti-sway crane control problem was solved by using the neural controller trained by genetic algorithm. The unconventional method of tree-encoding of chromosomes in genetic algorithm applied in crane control system was presented in (Filipic et al., 1999). The control algorithm was presented in form rules separately for swing increasing and dumping, based on heuristic knowledge about the crane laboratory model. The parameters of controller were modified by genetic algorithm during experiments carried out on the laboratory stand.

The paper presents the solution of indirect adaptive crane control system in which the gains of control system are adjusted based on pole placement method (PPM) according to the parameters of controlled object estimated by using the genetic algorithm (GA).

2. The conception of indirect adaptive crane control system with

genetic algorithm

The problem of anti-sway crane control needs to be solved with using adaptive methods required to adjust the parameters of control algorithm to the changes of controlled object's parameters, that are caused by changes of rope length and mass of a payload, which is suspended on a rope and moved by crane. The classic approach to adaptive control of nonlinear system could be based on the direct adaptive control system, where the parameters of control algorithm are adjusted directly based on the output signals of the controlled object. The genetic algorithm (GA) could be used in this system to optimize the gains of a controller (Fig. 1), which are for example binary encoded in chromosomes of population individuals, however the difficulty is caused by the necessity to formulate of objective function which is essential to evaluate the control quality of a system.

The alternative solution, which could be implemented in a crane control system, is an indirect adaptive pole placement (IAPP) method. This approach was used to create the adaptive crane control system with recursive least square (RLS) algorithm, which was applied to estimate the parameters of controlled object, as well as to elaborate the adaptive crane control system with fuzzy model of controlled object. The both control algorithms presented in (Smoczek, 2010) were tested on the laboratory stand, the laboratory model of an overhead traveling crane, and results of experiments were presented in. The classic RLS learning algorithm

applied in the IAPP control system could be replaced by GA with binary encoded parameters of a controlled object in chromosomes of possible solutions population, and objective function specified as a difference between outputs of an object and their estimated values calculated according to a known model of a controlled object (Fig. 2).

Fig. 1. The direct adaptive crane control system with genetic algorithm (GA) used to change the parameters of control algorithm,

where: x,- the object outputs: crane position and swing angle of a payload,

r

x - reference signal (expected position), u - control signal

Fig. 2. The indirect adaptive pole placement (IAPP) crane control system with genetic algorithm (GA) used to real-time identify the parameters

of a known controlled object model

The linear parametric model of a crane can be assumed as the two discrete transmittances with parameters vary in stochastic way with a rope length and mass of a payload (Smoczek, 2010). The transfer functions (1) and (2) show respectively the relationships between crane velocity X(z) and control signal U(z), and swing of a payload angle (z) and crane speed X(z).

0 0 1 ) ( ) ( ) ( c z d z U z X z G     (1) 0 1 2 0 1 2 ) ( ) ( ) ( a z a z b z b z X z z G      _ (2)

The parameters of the model should be real-time identified in each sample time by genetic algorithm with the selection done based on a fitness calculated for each individual according to the objective function that could be formulated as a difference between measured at the actual time outputs of the crane model and estimated outputs values:

minimum        ) ( ˆ ) ( ) ( ˆ ) ( t t t x t x     (3)

where the estimators are calculated as follows:

                   ) ( ˆ ) ( ˆ ) 1 ( ) 1 ( ) ( ˆ 0 0 t c t d t x t u t x T  

;

                                  ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) ( ˆ 0 1 0 1 t a t a t b t b t t t x t x t T     

(4)

Fig. 3. The closed-loop control system of crane speed and swing angle of a payload implemented to the indirect adaptive pole placement (IAPP) crane control system

The parameters estimated in each sample time by GA are used to calculate the gains of controllers in closed-loop crane control system based on pole placement method (PPM). The control system could be created by using proportional

controller of crane speed and discrete proportional-derivative controller of the swing angle of a payload (Fig. 3).

The gains of controllers are calculated according to the Diophantine equation (5) formulated based on a characteristic equation of closed-loop control system (Fig. 3) and P vector of coefficients of desired characteristic equation derived from the two pairs of assumed stable poles (6).

                                                                                             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 x P (5) where: 0 0 1, , ,q q s

K_Px - the controllers gains,

0 1 2 3,p ,p ,p

p

- the coefficients of desired characteristic equation

determined for expected poles.

         _{ }  _n j _n T_s z_1,2 exp    1 2 (6) where: n

 - nature frequency of a system,

s

T - sample time.

3. The analysis of IAPP crane control system with GA in the Laplace

s-domain

A nonlinear crane dynamic system can be simplified to the linear model consisting of a mass of trolley and/or bridge M and varying parameters: mass of a load m suspended on a rope with length l (Fig. 4).

The motions equations of the system (Fig. 4), derived from Lagrange’s second law type equations are as follows:

           0 sin cos sin cos ) ( 2        mg ml x m F ml ml x m M          (7)

Fig. 4. The two-mass model of a crane (M ) affected by force F=u, and a payload (m) suspended on a rope with length l

The crane dynamic model, derived from motions equations (1) for small values of load swing angle (sin, cos1), can be presented in the form of two continuous transfer functions (2) and (3), which show relations (s)U(s)G₁(s) and X(s)(s)G₂(s). 2 2 1 1 ) ( ) ( ) ( n s Ml s U s s G       (8) 2 2 2 ) ( ) ( ) ( s g ls s s X s G     (9) where: l g M m n         1

 - nature pulsation of the swing angle of a payload,

 

_/ 2

81 ,

9 m s

g - acceleration of gravity.

Owing to the fact, that the system is represented as the two second order oscillation and astatic models connected in series way, the crane control system can be created with using two proportional-derivative (PD) controllers connected in parallel way and placed in crane position and load swing angle feedbacks. The closed-loop control system with conventional PD controllers is developed to the adaptive control scheme by applying the GA to identify the parameters of the transmittance presented in the equation (8), rope length l and mass of a payload m, as well as the algorithm based on PPM, which is used to calculate the gains of PD controllers for the actual values of l and m parameters (Fig. 5).

The gains of PD controllers are calculated for known rope length and mass of a payload values based on Diophantine equation derived from characteristic equation of closed-loop control system and desired polynomial determined for the four assumed stable poles. Assuming all poles equal the natural pulsation of a system

n

s_1,2,3,4  , the Diophantine equation is expressed as follows:

                                                                                                  4 3 2 4 6 4 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 1 0 n n n n P D Px Dx K K K K g g l l Ml l g M m       (10)

Fig. 5. The IAPP crane control system with GA used to identify the parameters of crane model and PPM used to calculate the actual values of gains of the PD controllers

  Dx D P

Px K K K

K , , , for the assumed poles s1,2,3,4 _n

The realization of genetic algorithm was based on the classic binary encoding, single point crossover with probability p_c 90%, mutation realized by bits inversion with probability p_m10%, and tournament selection. The two-dimensional search space is limited by 8-bits resolution of ranges of rope length and mass of a payload values, which are encoded as binary strings in two genes of individual's chromosome.

Fig. 6. The chromosome of individual represented by binary string, where rope length and mass of a payload is encoded with 8-bits resolution in two genes

The fitness function was formulated as: minimum   ˆ( ) ) (s  s  (11)

where estimator of the swing angle ˆ s( ) is calculated based on the equation (8). The adaptive system with GA algorithm used to identify the parameters of crane model represented by continuous transfer functions (8) and (9) was tested in simulations for assumed ranges of rope length and mass of a payload changes:

 

1,4[m]

l and m



10,400



[kg]. The termination condition of genetic algorithm was specified as the five iterations per each 0.01 second of sample time, but the first iteration in the next sample time starts with the population of solutions (individuals) selected in the last iteration of the previous sample time. The examples of results are presented in the figure 7 in form of the object time-responses, crane position and swing of a payload angle.

Fig. 7. The time-responses of controlled object for chosen values of rope length and mass of a payload

In the figure 8 are presented the time courses of estimation of rope length and mass of a payload conducted by genetic algorithm based on input function and output of the controller object.

Fig. 8. The time courses of estimation by GA of crane's model parameters: rope length and mass of a payload

In most cases the GA used in the adaptive control system allows to successfully identify the parameters of crane's model during the fifteen iterations, that is illustrated in the figure 8. This result in to quick adjust the gains of the controllers of closed-loop control system to the estimated parameters of the object and obtain the expected quality of object responses, without overshoot of crane position and with successfully reducing the swing of a payload while expected position has achieved.

4. Conclusions

The crane control problem needs to implement adaptive solutions, owing to the nonlinearity of a controlled object. The solution of anti-sway crane control system could be based on the genetic algorithm, which could be used to optimize directly the gains of controllers. The disadvantage of this approach is the necessity of formulating the objective function to evaluate the performances of the genetic algorithm connected with performances of a control system. The alternative method can be based on the indirect adaptive control system, in which the genetic algorithm is used to estimate the parameters of known model of a system, and performances of the algorithm can be evaluated by comparing the responses of controlled object and its model. The simulations of this solution was successfully conducted in s-domain for the two-dimensional model of a crane.

The research project is financed from the Polish Science budget for the years 2011-2013.

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Dr inż. 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 60 publications, both in Polish and English.

Prof. dr hab. Inż. Janusz SZPYTKO, AGH University of Science and Technology, Faculty of Mechanical Engineering and Robo-tics. 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.