Design of Posicast PID control systems using a gravitational search algorithm

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Design of Posicast PID control Systems using a gravitational search algorithm P.B. de Moura Oliveira, E.J. Solteiro Pires, Paulo Novais

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Received date: 30 October 2013 Revised date: 15 December 2014 Accepted date: 28 December 2014 Cite this article as: P.B. de Moura Oliveira, E.J. Solteiro Pires, Paulo Novais, Design of Posicast PID control Systems using a gravitational search algorithm, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2014.12.101 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Design of Posicast PID Control Systems using a Gravitational Search Algorithm P. B. de Moura Oliveira1, E.J. Solteiro Pires2 and Paulo Novais3 1

INESC TEC –UTAD pole UTAD University, Department of Engineering, School of Sciences and Technology, 5001–801 Vila Real, Portugal [email protected] (corresponding author)

Abstract.

2

INESC TEC – UTAD pole UTAD University, Department of Engineering, School of Sciences and Technology, 5001–801 Vila Real, Portugal, [email protected]

3

ALGORITMI Centre Department of Informatics University of Minho, Braga, Portugal [email protected]

In this paper we propose the gravitational search algorithm to design PID

control structures. The controller design is performed considering the objectives of setpoint tracking and disturbance rejection, minimizing the integral of the absolute error criterion. A two-degrees-of-freedom control configuration with a feedforward pre-filter inserted outside the PID feedback loop is used to improve system performance for both design criteria. The pre-filter used is a Posicast controller designed simultaneously with a PID controller. Simulation results are presented which show the proposed technique merit. Keywords: Gravitational Search Algorithm, PID Control, Posicast Control.

1.

Introduction

Controllers based on proportional, integrative and derivative (PID) modes are applied within the majority of industrial control loops. Despite the development of more complex control methodologies, there are several reasons for the success and resilience of PID control, such as simplicity, performance and reliability in a wide range of system dynamics [1]. Thus, the development of new PID control based schemes and design methodologies are relevant research issues. Two classical control system design goals are input reference tracking and disturbance rejection. Optimal PID controller settings for set-point tracking may result in poor disturbance rejection and vice-versa, i.e., optimal disturbance rejection PID settings can result in poor set-point tracking. The design of PID controllers both for set-point tracking and disturbance rejection can be improved using two-degrees-of-freedom (2DOF) configurations [2]. A well known 2DOF configuration uses a feedforward pre-filter applied to the input reference signal

and a PID controller within the feedback loop. The ideal design of such 2DOF controllers requires simultaneous optimization of system response both for set-point tracking and disturbance rejection. The GSA was proposed by Rashedi et al. [3] which reported the advantages of using this algorithm in optimizing a set of benchmark unimodal and multimodal functions. In [3] a comparison was presented between GSA, particle swarm optimization (PSO) and a real genetic algorithm (RGA), showing that GSA outperforms PSO and RGA in a wide and representative set of functions. Since its proposal GSA has been reported successfully in solving several problems [4,5,6,7,8,9], also involving neural networks [18,19,20]. In this paper the gravitational search algorithm (GSA) is proposed to design 2DOF control configuration in which the pre-filter is a Posicast input shaper and the feedback loop uses a PID controller. The GSA is proposed to design simultaneously both the pre-filter controller and the PID controller considering two control objectives: set-point tracking and load disturbance rejection. The PID configuration adopted incorporates a well known anti-windup technique [15]. In terms of the optimization problem considered, a time domain system transient response error based cost function is deployed, in which the objective is not necessarily to reach the controllers optima setting. Indeed, the aim is to achieve a good control performance using a small number of GSA iterations. The remaining of the paper is organized as follows: section 2 reviews the gravitational search algorithm. In section 3 Posicast control is overviewed. In section 4, the PID control designed problem is presented. In section 5 the gravitational search algorithm is proposed to design PID control structures. Section 6 presents simulation results and discussion. Finally, in section 7 some conclusions and perspectives of future work are outlined.

2.

Gravitational Search Algorithm

The GSA proposed originally [3], is inspired in the natural interaction forces between masses. Accordingly to Newton’s law of gravity, the gravitational force, F, between two particles in the universe can be represented by: F =G

M 1M 2

(1)

R2

where: M1 and M2 are the two particles masses, G is the gravitational constant and R is the distance between the two particles. Newton’s well known second law relates force with acceleration, a, and mass, M, as:

F = Ma ⇔ a =

F M

(2)

Considering a swarm of particles (or population), X, of size s, in which every element represents a potential solution for a given search and optimization problem, moving in a n-dimensional space, with vector x representing the particle position. The force between

particles i and j, for dimension d and iteration t is represented by [3]:

Fijd = G (t )

M i (t ) M j (t ) Rij (t ) + ε

( x dj (t ) − xid (t ))

(3)

where: ε is a small constant, and the gravitational constant can be defined for every iteration by: t β G (t ) = G (t0 ) 0 t

β <1

(4)

with: G(t0) representing the initial gravitational constant, and R representing the Euclidian distance between the two particles. The use of R instead of R2 in (1) was proposed by [3] based on empirical evidence during computational experiments. The total force that acts in each particle i for a certain dimension, d, is evaluated by:

Fid (t ) =

s

∑ ϕ1 j Fij (t )

(5)

j∈K best , j ≠ i

in which, ϕ1j represents an uniform randomly generated number in the interval [0,1] and Kbest is the set of best particles, with size set to k0 (in this case k0=s) at the beginning of the search procedure and decreased linearly over time. The acceleration of mass i, called law of motion [3], is represented by:

Fid (t ) ai = M i (t )

(6)

with Mi representing the inertia mass for particle i, evaluated with:

mi (t )

Mi =

(7)

s

∑ m j (t ) j =1

and: mi (t ) =

fiti (t ) − worst (t ) best (t ) − worst (t )

(8)

where: fit represents the: current fitness value for particle i in iteration t, best, worst represent the population best and worst fitness values in iteration t, respectively. The velocity and position of each particle are updated accordingly to the following equations: vid (t + 1) = ϕ 2 i vid (t ) + aid (t )

(9)

xid (t + 1) = xid (t ) + vid (t + 1)

(10)

where: ϕ2j represents an uniform randomly generated number in the interval [0,1], x and v represent particles position and velocity, respectively.

3.

Posicast Control: An Overview

For some types of system dynamics, optimum set-point tracking can be achieved by using an open-loop control feedforward configuration as illustrated by Figure 1. Feedforward Controller r

Gf

rs

System Gp

y 1 t t

Figure 1. Open-loop feedforward input command shaping example.

The reference input modification in order to improve the system tracking can be implemented by using command shaping techniques [11,12]. The input shaping concept was originally proposed [10] to control underdamped systems. Smith [10] proposed

Posicast control to cancel underdamped complex poles with a feedforward controller. Considering a unit step reference input, the half-cycle Posicast control signal is represented by:

rs (t ) = A1r (t ) + A2 r (t − t1 )

(11)

with:

A1 + A2 = 1

(12)

T t1 = d 2

and: r representing an unit reference input step, rs the shaped signal, A1 and A2 the first and second step amplitudes, Td the undamped time period and t1 the time in which the second step is applied. Equation (11) represented in the Laplace complex domain results in the following shaper transfer function:

G f (s ) = A1 + A2e −t1s = A1 + (1 − A1)e −t1s

(13)

The control signal represented by equation (11) can also be obtained convolving the unit step input with a sequence of two impulses. This is known as a zero-vibration shaper [13], represented in the continuous time-domain as:

ZV (t ) = A1δ (t ) + (1 − A1)δ (t − t1)

(14)

where δ(t) is the Dirac delta function. The amplitude of the first step or impulse is a function of the firth overshoot, Mp: A1 =

1 = 1+ M p

1 −

1+ e

(15)

ζπω n ωd

with: ζ representing the damping factor, ωn and ωd representing the undamped and damped natural frequencies, respectively. An alternative Posicast input command shaper proposed by [10] is based in three steps, denoted herein three-step Posicast, and can be represented by:

G fts ( s) = A1 + A2e −t1s + A3e −t 2 s

0 < t1 < t2

(16)

subjected to:

A1 + A2 + A3 = 1 A1 ≥ 1, A2 < 0

(17)

where: A1, A2 and A3 represent the three step amplitudes; t1 and t2 represent the time in which the second and third steps are applied to the system reference input.

4.

PID Control Design: Problem Statement

A general PID control structure for a single-input single-output system can be illustrated using the classical block diagram presented in Figure 2, with: r representing the input command, u the control signal, y the system output, d1 the load disturbance, Gc the PID controller and Gp the system model.

Figure 2. Block diagram of a PID control configuration.

Two of the more relevant control design objectives are set-point tracking and disturbance rejection. The design of PID controllers both for set-point tracking and disturbance rejection can be improved using two-degrees-of-freedom (2DOF) configurations [2]. An example of a 2DOF configuration is presented in Figure 3, using a feedforward controller, Gf, applied to the reference input. This controller is often referred as a pre-filter or an input command shaper. As in this study the input command shaper is a Posicast controller it is denoted as Posicast input command shaper (PICS).

Figure 3. Block diagram of a two-degrees-of-freedom PID control configuration.

The 2DOF controller design can be accomplished using several methodologies. The PID controller can be designed first for achieving good disturbance rejection, and then the PICS can be designed to enhance set-point tracking. The former is a sequential design procedure. However, this can result in low performance. Indeed, both feedforward input shaper and feedback controller should be designed simultaneously. This type of methodology, also called concurrent design [14], is particularly useful when both design objectives are conflicting. In the following section, GSA is proposed as an optimization tool to simultaneously design both PICS and PID controllers.

5.

GSA Design of PID Control Structures

The PID control structure considered in this study, is illustrated in Figure 4. Kd s 1 + Tf s

d

PICS r

+ Kp

e

Gf +

1 + Tf s

-

ur

u Actuator

+

+

System

+

Gp

y

+

Ki s(1 + sTf )

+ -

+ 1/Kp anti-windup

Figure 4. Two-degrees of freedom control configuration with PID control and anti-windup.

The PID controller used in this study is governed by the following equation: s 2 K d + sK p + K i  1   1 + sT s f 

Gc ( s ) =

   

(18)

where: Kp, Ki and Kd represent the proportional, integrative and derivative gains, respectively, and Tf the filter time constant. The actuator saturation limits considered here are: -15≤ u(t) ≤+15. More information about the anti-windup scheme used can be found in [15,16,17]. The GSA is proposed to design the control structures described in the previous section. Both controllers are designed considering the control configuration presented in Figure 4, minimizing the integral of absolute error criterion:

IAE = ∫

T

o

e(t ) dt , with e(t ) = r (t ) − y (t )

(19)

The algorithm used (see Algorithm 1), is based on the original GSA [3], with some minor adaptations. In this case, if the swarm initialization is performed using a totally random procedure, some controller PID gains will make the system unstable. In these unstable cases the IAE value is disproportional high compared with stable cases, which makes the GSA perform badly. Thus, to avoid unstable gains particles incorporating the first population, the swarm is initialized randomly using a candidate interview

procedure. A randomly generated particle is allowed to be part of the initial swarm if it fulfills a predefined minimum IAE threshold. Algorithm 1. Gravitational search algorithm for PID controller design.

t=0 initialize PID controller swarm X(t)

while(!(termination criterion)) evaluate X(t) update G, best, worst, Kbest evaluate particles M and a update particles velocity and position t=t+1 end The gravitational constant is updated using:

 t   G (t ) = G0 1 −  n _ it 

(20)

with n_it representing the total number of iterations. This linear decreasing equation was found better for this application, among other possibilities tested by prior experimentation. Each particle mass and acceleration are evaluated with equations (7) and (6), respectively. Particle velocity and position are updated with (9) and (10), respectively.

6.

Simulation Results and Discussion

Three system models were used to test the proposed design methodology: G p1 ( s ) = G p 2 (s) = G p3 ( s ) =

1 2

(1 + s )

1−α s

(s + 1)3 1 4

(1 + s )

e−s

α = 0.5 e− s

(21) (22) (23)

corresponding respectively, to a double pole system with a time delay, a fourth order system with time delay and a non-minimum phase system. These plant models were selected due to their significance to test PID controllers as proposed in [21]. Each swarm particle encodes the PID gains parameters {Kp, Ki, Kd} and the filter constant, Tf, was set to 0.1 sec. The search interval was equal both for initialization and search defined by: 0.1 ≤ Kp, Ki, Kd ≤ 5. The initialization threshold was set to an IAE in the interval of 1000 to 2000. The total number of iterations (n_it=150) was the search termination criterion used. This number was deliberately set low as the aim here is not to achieve the optimal PID settings but good settings in a short evolutionary time period. The value used for the initial gravitational constant was Go=0.5. The scope of this study is not to compare the GSA algorithm to other natural inspired meta-heuristics, as this has been done previously in the context of a wide range of benchmark test functions [3].

However, in terms of the current application, a

comparison between GSA algorithm and a simple Particle Swarm Optimization (PSO) algorithm, showing the evolution of the mean value of the IAE index from a set of 20 runs is presented in Figure 5. The plant considered is represented by (22). The PSO parameters in terms of initial population, swarm size and number of objective function evaluations were the same as used in the GSA. The PSO algorithm cognitive and social constants were set to 2 and the inertia weight, ω, was linearly decreased between 0.9 and 0.4 over the 150 iterations.

700 GSA PSO

650

Average IAE Index

600

550

500

450

400

350

0

50

100

150

Iterations

Figure 5. Comparison between the GSA and PSO algorithm for PID controller design.

For the considered case settings, Figure 5, clearly shows that the GSA convergence rate is faster in an early stage of the run. However, the PSO could be set to a faster convergence rate by reducing the higher limit for the inertia weight (e.g. constant and smaller value). The achieved average fitness value is the same, which indicates that both algorithms converged for the same value in all trial runs. An interesting feature shown in Figure 5 is that the average fitness trend for GSA is more irregular than the PSO. This may prove relevant in escaping search traps such as local optima. The results achieved with the GSA design methodology, considering both a half-cycle Posicast pre-filter (13) and a PID controller (18), are presented in Figures 6, 7 and 8, corresponding to systems (21), (22) and (23), respectively. As it can be observed from these figures, for exposition sake the reference unit step is applied at t=1sec. A unit step load disturbance is applied at t=15sec for Figure 6 and 7 and at t=25sec for Figure 8. The search intervals considered for the half-cycle pre-filter were: 0 ≤ A1 ≤1 and 0 ≤ t1 ≤3sec, and for the PID controller: 0.1 ≤ Kp, Ki, Kd ≤ 5. Equal relevance was given to the objectives of set-point tracking and load disturbance rejection.

y(t)

1.5

1 r(t) y-HCPID y-PID

0.5 0

5

10

15

20

25

30

35

t 15 u-HCPID u-PID

u(t)

10 5 0 -5

0

5

10

15

20

25

30

35

t

Figure 6. The top plot presents the set-point tracking response and load rejection with a unit step perturbation applied at t=15sec for the double pole with time delay system (21). The bottom plot presents the corresponding control signal. In the figure legend HCPID means Half-cycle PID.

The results presented in Figure 6 for the double pole with time delay system, correspond to the GSA achieved parameters: {A1=0.70; t1=1.98sec.} and {Kp=1.84, Ki=0.98 Kd=1.49} corresponding to the following half-cycle Posicast and PID controller expressions:

G f 1(s ) = 0.7 + 0.3e −1.98s Gc1( s ) =

1.49 s 2 + 1.84 s + 0.98  1    s  1 + 0.1s 

(24) (25)

An IAE=306 was obtained. Figure 7 allows comparing the results obtained with a single controller PID structure with the proposed 2DOF control structure using a half-cycle Posicast and PID controller, both designed using a GSA algorithm. As it can be observed, the results obtained with the 2DOF are superior to the ones with the PID, both in terms of set-point tracking and disturbance rejection, as expected.

1.5

y(t)

1 0.5 r(t) y-HCPID y-PID

0 -0.5

0

5

10

15

20

25

30

35

t 15 u-HCPID u-PID

u(t)

10 5 0 -5

0

5

10

15

20

25

30

35

t

Figure 7. The top plot presents the set-point tracking response and load rejection with a unit step perturbation applied at t=15sec for the non-minimum phase system (22). The bottom plot presents the corresponding control signal. In the figure legend HCPID means Half-cycle PID.

The results presented in Figure 7 for the non-minimum phase system, correspond to the achieved parameters: {A1=0.67; t1=2.04sec} and {Kp=2.64, Ki=1.24 Kd=2.52}. An IAE=270 was obtained. Figure 7 compares the results obtained with a single controller PID structure with the proposed 2DOF control structure with half-cycle Posicast and PID controller, both designed using a GSA algorithm. As it can be observed the results obtained with the 2DOF are superior to the ones obtained with the PID with the prefilter, both in terms of set-point tracking and disturbance rejection. The results presented in Figure 8 for the fourth order with time delay system, correspond to the achieved parameters: {A1=1.0; t1=0.89sec} and {Kp=1.23, Ki=0.41, Kd=1.91}. An IAE=656 was obtained. The results presented in Figure 8 compare the results obtained with a single controller PID structure with the proposed 2DOF control structure with half-cycle Posicast and PID controller, both designed using a GSA algorithm. As it can be observed the results obtained with the 2DOF are totally superimposed with the ones achieved with the PID without half-cycle pre-filter. The justification of the lack of improvement achieved with the 2DOF, is due to the GSA converged value for A1=1.0. This means that the half-cycle reference step is not divided into two parts as intended, and the firth step component is the same as if the pre-filter

was not present. The GSA convergence for this value is due to the fourth order systems dynamics, requiring a more complex pre-filter. To overcome this problem the GSA was used to design a 2DOF with a three-step pre-filter structure, as represented by (16). 2

y(t)

1.5 1 r(t) y-HCPID y-PID

0.5 0

0

5

10

15

20

25 t

30

35

40

45

50

15 u-HCPID u-PID

u(t)

10 5 0 -5

0

5

10

15

20

25 t

30

35

40

45

50

Figure 8. The top plot presents the set-point tracking response and load rejection with a unit step perturbation applied at t=25sec for the fourth-order with time delay system (23). The bottom plot presents the corresponding control signal. In the figure legend HCPID means Half-cycle PID.

For the case in which the pre-filter is a three-step Posicast, the GSA converged to the settings {A1=1.748; A2=-2; A3=0.252; t1=1.36sec; t2=2.34sec} and {Kp=1.48, Ki=0.49 Kd=2.21}, which results in the following controller expressions:

G fts ( s ) = 1.748 − 2e −1.36s + 0.252e −2.34s

(26)

2.21s 2 + 1.48s + 0.49  1    s  1 + 0.1s 

(25)

Gc 4 ( s ) =

As it can be observed by the step response presented in Figure 9, the performance obtained by the 2DOF using a three-step Posicast pre-filter and a PID controller (TSPID)

is improved in comparison with the performance obtained by the same

configuration with the half-cycle pre-filter (HCPID).

2

y(t)

1.5 1 r(t) y-HCPID y-TSPID

0.5 0

0

5

10

15

20

25 t

30

35

40

45

50

20 u-HCPID u-TSPID

u(t)

10 0 -10 -20

0

5

10

15

20

25 t

30

35

40

45

50

Figure 9. The top plot presented the set-point tracking response and load rejection with a unit step perturbation applied at t=25sec for the fourth-order with time delay system (23). The bottom plot presents the corresponding control signal. In the legend, HCPID means Half-cycle PID and TSPID means Threestep PID.

7.

Conclusions

The GSA was proposed to optimize PID control structures using the integral of absolute error criterion. Two control configurations were addressed: i) classical feedback loop with PID controller for set-point tracking ii) 2DOF configuration using a feedforward Posicast input command shaper, placed outside the feedback PID loop. Both Posicast parameters and PID gains were designed simultaneously, both for the objectives of setpoint tracking and disturbance rejection. The same relevance was given to both objectives in the IAE based cost function. The results presented show that GSA can conveniently design both the input shaper and PID controller in the proposed 2DOF configuration. The results achieved with the use of the proposed 2DOF control configuration, with controllers design with the GSA, clearly improve the results of the corresponding PID loop configuration without the pre-filter. Further research will explore the proposed technique for other process dynamics and considering multi-objective GSA formulations of the proposed control problem.

References [1] Åström K J and Hägglund T (2001), The Future of PID Control. Control Engineering Practice, Vol. 9, 11, pp. 1163-1175 [2] Araki M and Taguchi H (2003), Two-Degree-of-Freedom PID Controllers. International Journal of Control Automation, and Systems Vol. 1, No. 4, pp. 401-411. [3] Rashedi E, Nezamabadi-pour H and Saryazdi S (2009), GSA: A Gravitational Search Algorithm, Information Sciences, 179, pp. 2232-2248. [4] Precup R E, David R C, Petriu E M, Preitl S and Răda M C (2011), Gravitational Search Algorithms in Fuzzy Control Systems Tuning, Preprints of the 18th IFAC World Congress, pp. 13624- 13629 [5] Khajehzadeh M, Raihan Taha M, El-Shafie A and Eslami M (2011), Search for critical failure surface in slope stability analysis by gravitational search algorithm, International Journal of the Physical Sciences Vol. 6 (21), pp. 5012-5021, Academic Journals. [6] Duman S, Sonmez Y, Guvenc U. and Yorukeren N (2011), Application of Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problem, IEEE Symposium Innovations in Intelligent Systems and Applications (INISTA), pp. 519-523. [7] Khajehzadeh M, Taha M R, El-Shafie A and Eslami M (2012), A modified gravitational searchalgorithm for slope stability analysis, Engineering Applications of Artificial Intelligence, Vol. 25, pp. 1589–1597. [8] Chatterjee A and Mahanti G K (2010), Comparative Performance of Gravitational Search Algorithm and Modified Particle Swarm Optimization Algorithm for Synthesis of Thinned Scanned Concentric Ring Array Antenna, Progress In Electromagnetics Research B, Vol. 25, pp. 331-348. [9] Roy P K (2013), Solution of unit commitment problem using gravitational search algorithm, Electrical Power and Energy Systems, Vol. 53, pp. 85–94. [10] Smith O J M (1957), Posicast Control of Damped Oscillatory Systems, Proc. IRE, Vol. 45, No. 9, pp. 1249-1255. [11] Huey J R, Sorensen K L and Singhose W E (2008), Useful applications of closed-loop signal shaping controllers, Control Engineering Practice, Vol. 16, pp. 836–846. [12] Tuttle T D (1997), Creatig Time-Optimal Commands for Linear Systems, PhD Thesis, MIT. [13] Singer NC and Seering W P (1990), Preshaping command inputs to reduce system vibration, Journal of Dynamic Systems Measurement and Control, Vol. 112 (3), pp. 76-82. [14] Chang P H and Park J (2001), A concurrent design of input shaping technique and a robust control for high-speed/high-precision control of a chip mounter. Control Engineering Practice 9 (2001) 1279–1285 [15] Hanus K M and Henrotte J L, (1987), Conditioning technique, a general anti-windup and bumpless transfer method. Automatica, Vol.23, No. 6, pp. 729-739. [16] De Moura Oliveira P B and Vrančić D., (2012), Underdamped Second-Order Systems Overshoot Control, IFAC Conference on Advances in PID Control, Brescia, PID’12, March 28-30, pp. 6.

[17] De Moura Oliveira P. B., Damir Vrančić e J. Boaventura Cunha, (2012), “Posicast PID Control of Oscilatory Systems”, Controlo' 2012- 10th Portuguese Conference on Automatic Control, pp. 27-32, July 16-18, Madeira, Portugal. [18] Mirjalili S. A., Hashim S. Z. M., Sardroudi H. M. (2012), “Training feedforward neural networks using hybrid particle swarm optimization and gravitational search algorithm”, Applied Mathematics and Computation, 218, pp. 11125–11137. [19] Xu Bao-Chang and Zhang Ying-Ying, (2014), “An Improved Gravitational Search Algorithm for Dynamic Neural Network Identification”, International Journal of Automation and Computing, 11(4), August , pp. 434-440. [20] Fedorovici L. O., Precup R. E., Dragan F., David R. Codrut, Purcaru C. (2012), “Embedding Gravitational Search Algorithms in Convolutional Neural Networks for OCR Applications”, 7th IEEE International Symposium on Applied Computational lntelligence and Informatics, May 24-26, Timisoara, Romania, pp. 125-130. [21] Åström, K.J., Hägglund, T.: Benhmark systems for PID Control. In: Digital Control- Past, present, and future of PID Control, Elsevier (2000).

Paulo Moura Oliveira received the Electrical Engineering degree in 1991, from the UTAD University, Portugal, MSc in Industrial Control Systems in 1994 and PhD in Control Engineering in 1998, both from Salford University, Manchester in the UK. He is an Associated Professor with Habilitation at the Engineering Department of UTAD University and a researcher at the INESC TEC institute. Currently, he is the the director of the Master and Phd Courses in Electrical and Computers Engineering in UTAD. His research interests are focused on the fields of control engineering, intelligent control, PID control, control education, evolutionary and natural inspired algorithms for single and multiple objective optimization problem solving. He is author in more than 100 peer-reviewed scientific publications.

E. J. Solteiro Pires received the degree in Electrical Engineering at the University of Coimbra, in 1993. He pursued post graduate studies and obtained, in 1999, an MSc degree in Electrical and Computer Engineering at the University of Oporto. In 2006, he graduated with a PhD degree at UTAD University. Since 2006 he works as an Assistant Professor at the Engineering Department of UTAD University. His main research interests are in evolutionary computation, multi-objective problems, fractional calculus, and diffusion of innovation.

Paulo Novais is an Associate Professor of Computer Sciences at the Department of Informatics, in the University of Minho, Braga, Portugal and researcher at the CCTC (Computer Science and Technology Center) in which he is the coordinator of the Intelligent Systems Lab (http://islab.di.uminho.pt/). From the same university he received a PhD in Computer Sciences in 2003 and his Habilitation in Computer Sciences (Agregação ramo de conhecimento em Informática) in 2011.

He is the director of the Master in Informatics Engineering, Master in Informatics and Deputy Director and co-founder of the Master in Law and Informatics at the University of Minho. He develops scientific research in the field of Artificial Intelligence, namely Knowledge Representation and Reasoning, Machine Learning and Multi-Agent Systems, with applications to the areas of Law and Ambient Intelligence. He has leaded and participated in several research projects sponsored by Portuguese and European public and private Institutions and has supervised several PhD and MSc students. He is the co-author of over 100 book chapters, journal papers, conference and workshop papers and books. He is Vice-president of APPIA, the Portuguese Association for Artificial Intelligence.

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