Optimal Compensator Design using Genetic Algorithm

5th International Conference on Advances in Control and 5th Conference on Optimization of Dynamical Systems 5th International International Conference on Advances Advances in in Control Control and and 5th International Conference on Advances in Control and Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems Available online at www.sciencedirect.com Optimization of Dynamical Systems 5th International Conference on Advances in Control and February February 18-22, 18-22, 2018. 2018. Hyderabad, Hyderabad, India India February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 518–523

Optimal Compensator Design using Optimal Compensator Design using Optimal Compensator Design Optimal Genetic Compensator Design using using Algorithm Algorithm Optimal Genetic Compensator Design using Genetic Algorithm Genetic∗ Algorithm Genetic Algorithm D. Penchalaiah, G. Naresh Kumar, ∗∗ ∗∗ D. Penchalaiah, ∗∗ G. Naresh Kumar, ∗∗

∗∗∗ D. Naresh Murali Mohan Gade, S. E.Kumar, Talole ∗∗∗∗ ∗ G. ∗∗∗ ∗∗ ∗∗∗∗ D. Penchalaiah, Penchalaiah, G. ∗∗∗ Naresh Kumar, ∗∗∗∗ Murali Mohan Gade, S. E. Talole Murali Mohan Gade, S. E. Talole ∗ ∗∗∗ ∗∗∗∗ D. Penchalaiah, G. Naresh Murali Mohan Gade, S. E.Kumar, Talole ∗∗ ∗ ∗∗∗ DRDL, Hyderabad, Scientist, Directorate of Systems, Murali Mohan Gade, S. E. Talole ∗∗∗∗ India ∗ ∗ Scientist, Directorate of Systems, DRDL, Hyderabad, India Directorate of Systems, DRDL, [email protected]) ∗ Scientist,(e-mail: Scientist,(e-mail: Directorate of Systems, DRDL, Hyderabad, Hyderabad, India India [email protected]) ∗∗ [email protected]) ∗ Scientist,(e-mail: Directorate of Systems, DRDL, Hyderabad, India India ∗∗Scientist, Directorate of Systems, DRDL, Hyderabad, [email protected]) ∗∗ Scientist,(e-mail: Directorate of Systems, DRDL, Hyderabad, India Directorate of Systems, DRDL, Hyderabad, India (e-mail: [email protected]) ∗∗ Scientist, (e-mail: [email protected]) Scientist, Directorate of Systems, DRDL, Hyderabad, India (e-mail: [email protected]) ∗∗∗ (e-mail: [email protected]) ∗∗ Scientist, Directorate of Systems, DRDL, Hyderabad, India ∗∗∗Scientist, Directorate of Systems, DRDL, Hyderabad, (e-mail: [email protected]) ∗∗∗ Scientist, Directorate of Systems, DRDL, Hyderabad, India India Directorate of Systems, DRDL, Hyderabad, (e-mail: [email protected]) ∗∗∗ Scientist, (e-mail: [email protected]) Scientist, Directorate of Systems, DRDL, Hyderabad, India India (e-mail: [email protected]) ∗∗∗∗ (e-mail: [email protected]) ∗∗∗Professor, Dept of Aerospace Engineering, DIAT, Pune, India ∗∗∗∗ Scientist, Directorate of Systems, DRDL, DIAT, Hyderabad, India (e-mail: [email protected]) ∗∗∗∗ Professor, Dept of Aerospace Engineering, Pune, India Dept of Aerospace Engineering, DIAT, Pune, (e-mail: [email protected]) ∗∗∗∗ Professor, Professor,(e-mail: Dept [email protected]) Aerospace Engineering, DIAT, Pune, India India (e-mail: [email protected]) (e-mail: [email protected]) ∗∗∗∗ Professor, Dept of Aerospace Engineering, DIAT, Pune, India (e-mail: [email protected]) [email protected]) Abstract: Compensator design (e-mail: in frequency domain is usually an involved and time consuming Abstract: Compensator design in frequency domain is usually an involved and time consuming Abstract: Compensator design in frequency domain is an and task owing to the number of iterations required meet design specifications. The task Abstract: Compensator design in frequencyrequired domain to is usually usually an involved involved and time time consuming consuming task owing to the number of iterations to meet design specifications. The task task owing to the number of iterations required to meet design specifications. The task becomes more cumbersome as the number of design parameters increases. Further, the resulting Abstract: Compensator design in frequency domain is usually an involved and time consuming task owing tocumbersome the numberasoftheiterations required to meet design specifications. The task becomes more number of design parameters increases. Further, the resulting becomes more cumbersome as the number of design parameters increases. Further, the resulting design may not be optimal. Addressing these issues, genetic algorithm is used for optimizing task owing tocumbersome the numberasAddressing oftheiterations togenetic meet algorithm design specifications. The task becomes more numberthese ofrequired design parameters increases. Further, the resulting design may not be optimal. issues, is used for optimizing design may not be optimal. Addressing these issues, genetic algorithm is used for optimizing compensator parameters to as meet prescribed design specifications. The approach uses the search becomes more cumbersome the number of design parameters increases. Further, the resulting design may not be optimal. Addressing these issues, genetic algorithm is used for optimizing compensator parameters to meet prescribed design specifications. The approach the search compensator parameters to meet prescribed design specifications. The approach uses the capability evolutionary to optimally place the poles and theuses compensator. design mayof not be optimal. Addressing these issues, genetic algorithm is of used for optimizing compensator parameters toalgorithms meet prescribed design specifications. Thezeros approach uses the search search capability of evolutionary algorithms to optimally place the poles and zeros of the compensator. capability of evolutionary algorithms to optimally place the poles and zeros of the compensator. It is shownofthat the proposed approach offers better stability margins in comparison with the compensator parameters toalgorithms meet prescribed design specifications. Thezeros approach the search capability evolutionary to optimally place the poles and of theuses compensator. It is shown that the proposed approach offers better stability margins in comparison with the It is shown that the proposed approach offers better stability margins in comparison with the classical method for an unstable servotoplant. Further, it the is shown thatzeros the design iteration time capability of evolutionary algorithms optimally place poles and of the compensator. It is shown that the proposed approach offers better stability margins in comparison with the classical method for an unstable servo plant. Further, it is shown that the design iteration time classical method for an unstable servo plant. Further, it is shown that the design iteration time is significantly less when compared to the classical approach. It is shown that the proposed approach offers better stability margins in comparison with the classical method for an unstable servo plant. Further, it is shown that the design iteration time is significantly less when compared to the classical approach. is significantly less when compared to the classical approach. classical method for an unstable servo plant. Further, it is shown that the design iteration time is significantly less when compared to the classical approach. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Complex compensator, compensator, Stability margins, Genetic algorithm is significantly less when comparedLead-lag to the classical approach. Keywords: Complex compensator, Lead-lag compensator, Stability margins, Genetic algorithm Keywords: Complex compensator, Lead-lag compensator, Keywords: Complex compensator, Lead-lag compensator, Stability Stability margins, margins, Genetic Genetic algorithm algorithm Keywords: Complex compensator, Lead-lag compensator, Stability margins, Genetic algorithm 1. INTRODUCTION NOMENCLATURE 1. INTRODUCTION INTRODUCTION NOMENCLATURE NOMENCLATURE 1. NOMENCLATURE 1. INTRODUCTION PID controller represents one of the most popular and Parameter Description NOMENCLATURE 1. INTRODUCTION PID controller represents one of of the most most popular and Parameter Description PID controller represents one the popular and widely used design in industrial control systems. However, Parameter Description (s) Plant transfer function G PID controller represents one of the most popular and p Parameter Description widely used design in industrial control systems. However, (s)(s) Plant transfer function G widely used design in industrial control systems. However, the PID controllers susceptible noise popular interference GpLag Plant transfer function Lag Compensator PID controller represents one of theto most and p (s) widely used design inare industrial control systems. However, Parameter Description the PID controllers are susceptible to noise interference (s) Plant transfer function G p G (s) Lag Compensator the PID controllers are susceptible to noise interference and windup effect, in addition tocontrol the difficulties associated GLag (s) Lag Compensator (s) Lead Compensator widely used design in industrial systems. However, Lag the PID controllers are susceptible to noise interference Lead (s) Plant transfer function G and windup effect,(Horng, in addition addition to the difficulties associated Lag Compensator p (s) Lag G (s) Lead Compensator and windup effect, in to the difficulties associated Lead with loop-shaping 2012). Even there exists G (s) Lead Compensator (s)(s) Complex Compensator the PID controllers susceptible tothough noise interference Lead and windup effect,(Horng, in are addition to the difficulties associated c Lag Compensator with loop-shaping 2012). Even though there exists G (s) Lead Compensator Lag Lead G (s) Complex Compensator with loop-shaping (Horng, 2012). Even though there exists c a large number of tuning methods for designing a PID G (s) Complex Compensator K = 1.93 Plant gain and windup effect, in addition to the difficulties associated c with loop-shaping (Horng, 2012). Even though there exists pt G (s) Lead Compensator a large number of tuning methods for designing a PID (s) Complex Compensator Lead cpt = 1.93 K Plant gain a large number of tuning methods for designing a PID controller (Cominos and Munro, 2002; Ogata, 2010; Nise, K = 1.93 gain ω = 25.0 Plant firstCompensator order pole with loop-shaping 2012). Even there pt acontroller large number of(Horng, tuning methods forthough designing aexists PID G (s) Complex (Cominos and Munro, 2002; Ogata, 2010; Nise, Kp1 = 1.93 Plant gain cpt ω = 25.0 Plant first order pole controller (Cominos and Munro, 2002; Ogata, 2010; Nise, p1==0.51 2007), successful tuning of methods the three design parameters, 25.0 Plant first order pole ζω complex pole damping factor a large number of tuning for designing a PID p1 controller (Cominos and Munro, 2002; Ogata, 2010; Nise, p K = 1.93 Plant gain 2007), successful tuning ofgreatly the three three designonparameters, parameters, 25.0 first orderpole poledamping factor pt==0.51 p1 ζζω Plant complex 2007), successful tuning of the design namely KP(Cominos , KI , and KDMunro, depends experience complex pole damping factor ωppp1 =0.51 161.74 Plant natural frequency controller and 2002; Ogata, 2010; Nise, 2007), successful tuning of the three design parameters, np= = 25.0 Plant first order pole namely K , K , and K greatly depends on experience ζω = 0.51 complex pole damping factor P , KI , and D greatly p ω = 161.74 Plant complex pole frequency namely K on experience and understanding of K the plant todepends bedesign controlled. EffecI tuning Dofgreatly 161.74 complex pole natural natural Kpnp Forward path gain 2007), successful the three np namely KP K depends onparameters, experience f== P , KI , and D plant ζω complex pole dampingfrequency factor and understanding of the to be controlled. Effecω =0.51 161.74 Plant Plant natural frequency np K Forward path gain and understanding of the plant to be controlled. Effecf tively, the design exercise turns out to be a trial and error K Forward path gain ω Lag compensator zero namely KPdesign , KI , exercise and greatly on experience f = 161.74 and understanding of K the plantout todepends bebecontrolled. EffecD turns ω Plant complex pole natural frequency tively, the to a trial and error Klgz Forward path gain np f ω Lag compensator zero tively, the design exercise turns out to be a trial and error lgz procedure and non-optimal. ω Lag compensator zero and understanding of theturns plantout to tobebecontrolled. lgz tively, the design exercise a trial andEffecerror Klgp Forward path gainpole procedure and non-optimal. ω Lag compensator zero f lgz ω Lag compensator pole procedure and non-optimal. lgp Lag compensator pole ω Double lead compensator zero1 tively, the design exercise turns out to be a trial and error lgp procedure and non-optimal. ldz1 ω Lag compensator zero More practical alternative is the use of classical leadpole lgz lgp ω Double lead compensator zero1 ldz1 More practical alternative is the use of classical leadzero1 ω Double lead compensator pole1 procedure and non-optimal. ldz1 ldp1 More practical alternative is the use of classical leadω Lag compensator pole lag compensator. The compensator is usually designed Double lead compensator zero1 lgp ldz1 ω Double lead compensator pole1 More practical alternative is the useis of classical leadldp1 lag compensator. The compensator usually designed pole1 ω Double lead compensator zero2 ldp1 ldz2 lag compensator. The compensator is usually designed ω Double lead compensator zero1 in frequency domain as the frequency domain design pole1 ldz1 ldp1 More practical alternative is the use of classical leadω Double lead compensator zero2 lag compensator. The compensator is usually designed ldz2 in frequency domain as the frequency domain design ω Double lead compensator zero2 pole2 ldz2 ldp2 in frequency domain as the frequency domain design ω Double lead compensator pole1 methodology directly provides the information on stability zero2 ldp1 ldz2 lag compensator. The compensator is usually designed ω Double lead compensator pole2 in frequency domain as the frequency domain design ldp2 methodology provides the information on stability Double lead compensator pole2 Complex compensator zeros natural ω ldp2 n1 methodology directly provides the information on ω Double lead compensator zero2 margins, i.e., directly the phase marginsdomain which are the pole2 ldz2 ldp2 in frequency domain asand thegain frequency design Complex compensator zeros natural ω methodology directly provides the information on stability stability n1 margins, i.e., the phase and gain margins which are the Complex compensator zeros natural ω frequency n1 margins, i.e., the phase and gain margins which are the ωldp2 Double lead compensator pole2 indicators of robustness of the feedback system. The design Complex compensator zeros natural n1 methodology directly provides the information on stability frequency margins, i.e., the phase and gain margins which are the indicators of robustness of the feedback system. The design frequency ζωz1 Complex compensator zeros damping indicators of robustness of the feedback system. The design Complex compensator zeros natural procedure, however, is quite involved and time consuming frequency n1 margins, i.e., the phase and gain margins which are the ζζωz1 Complex compensator zeros damping indicators of robustness of the feedback system. The design procedure, however, is quite involved and time consuming Complex compensator zeros damping natural n2 procedure, is quite involved and time consuming frequency as it needsofhowever, more number of iterations to meet the design ζz1 Complex compensator compensator poles zeros damping z1 indicators robustness of the feedback system. The design ω Complex poles natural procedure, however, is quite involved and time consuming n2 as it needs more number of iterations to meet the design ω Complex compensator natural frequency n2 as it more number of to the ζωz1 Complex compensator poles zeros specifications. The task becomes further complex as the poles damping natural n2 procedure, is quite involved and time consuming frequency as it needs needs however, more number of iterations iterations to meet meet the design design specifications. The task becomes further complex as the frequency ζωz2 Complex compensator poles damping specifications. The task becomes further complex as Complex compensator poles natural number of compensators and their design parameters frequency n2 as it needs more number ofand iterations to meet the design Complex compensator poles poles damping damping specifications. The task becomes further complex as the the z2 number of compensators their design parameters ζζF Complex compensator Fitness function z2 number of compensators and their design parameters frequency increases. the resulting compensator design may ζFz2 Complex compensator poles damping specifications. The task becomes further complex as not the Fitness function number ofAlso compensators and their design parameters increases. Also the resulting compensator design may not F Fitness function J Cost function min increases. Also the resulting compensator design may not ζJz2 Complex compensator poles damping be optimal. Motivated by these issues and in an parameters attempt to F Fitness function number of compensators and their design Cost function increases. Also the resulting compensator design may not min be optimal. Motivated by these issues and in an attempt to Cost function gJ Maximum number of generations min max be optimal. Motivated by these issues and in an attempt to F Fitness function address them, an optimization based approach is proposed J Cost function min increases. Also the resulting compensator design may not ggmax Maximum number of generations be optimal. Motivated by these issuesapproach and in anisattempt to address them, an optimization based proposed Maximum number of generations max address them, an optimization based approach is proposed J Cost function in this work for the design of compensator. To illustrate gmax Maximum number of generations min be this optimal. Motivated by these issuesapproach and in an to address them, an the optimization based isattempt proposed in work for design of compensator. To illustrate in this work for design of compensator. illustrate gmax Maximum number of generations the design philosophy, a pneumatic actuation system used address them, an the optimization approachTo is proposed in this work for the design of based compensator. To illustrate the design philosophy, a pneumatic actuation system used the design philosophy, aa pneumatic actuation system used in flight control applications considered the plant in this work for the design of is compensator. To illustrate the design philosophy, pneumatic actuationas system used in flight control applications is considered as the plant in flight control applications is considered as the plant to be controlled. Since the plant is unstable, a double the design philosophy, a pneumatic actuation system used in flight control applications is considered as the plant to be controlled. Since the plant is unstable, a double to be Since is a in control applications is considered as the plant to flight be controlled. controlled. Since the the plant plant is unstable, unstable, a double double to be by controlled. the reserved. plant is unstable, a double Copyright © 2018, 2018 IFAC 550Hosting 2405-8963 © IFAC (International Federation of Automatic Control) Elsevier Ltd.Since All rights

Copyright © 2018 IFAC 550 Copyright 2018 responsibility IFAC 550Control. Peer review© of International Federation of Automatic Copyright ©under 2018 IFAC 550 10.1016/j.ifacol.2018.05.087 Copyright © 2018 IFAC 550

5th International Conference on Advances in Control and Optimization of Dynamical Systems D. Penchalaiah et al. / IFAC PapersOnLine 51-1 (2018) 518–523 February 18-22, 2018. Hyderabad, India

2. STATEMENT OF PROBLEM

Bode Diagram: Uncompensated Plant

Magnitude (dB)

40 20

Gain cross over frequency = 148 rad/s Gain Margin = −9.54 dB

0 −20

−40 −135 Phase (deg)

lead and lag compensator in forward path and a complex compensator in feedback path are chosen to stabilize the plant and to meet the prescribed specifications. The design is cast as an optimization problem which is solved using evolutionary based genetic algorithm. Simulation results are presented to showcase the effectiveness of the design. The remaining paper is organized as follows. The statement of the problem including description of plant model is presented in Section 2. The design of the compensators by casting it as an optimization problem, optimization problem formulation, description of various constraints as well as choice of performance index are presented in Section 3. The simulation results using the proposed formulation are presented in Section 4 and lastly, Section 5 concludes this work.

519

−180 Phase cross over frequency = 233 rad/s Phase Margin = −42.0 degrees

−225 −270 100

101 Frequency

102 (rad/s)

103

Fig. 1. Bode Diagram of uncompensated plant

2.1 Plant model In this work, a pneumatic actuation system used in flight vehicle control applications is considered as the plant to be controlled. Each of the four control surfaces is deflected by using an independent and separate actuator. Each actuator is driven by double acting piston and each side of the piston is controlled by an ON-OFF threeway two position solenoid valve, which is normally open. Pulse Width Modulation (PWM) control with a carrier frequency of 100 Hz is used for the control. A notch filter is designed to attenuate 100 Hz signal component entering into the loop due to PWM frequency. In addition, type-2 Chebyshev low pass filter is designed for achieving higher roll-off. Moreover, the pass band ripple of the Chebyshev attenuates 200 Hz harmonic component like an additional notch filter. For the purpose of compensator design, the considered plant consists of the basic servo dynamics, notch filter due to PWM and Chebyshev filter. The pneumatic servo with ON-OFF valve is highly nonlinear and unstable. However, in order to design classical compensator, a linear plant model is needed. To this end, experimental characterization of the servo is carried out to estimate a linear model and the same is given as 2 Kpt ωnp Gp (s) = (1) 2 2 ) (s − ωp1 )(s + 2ζp ωnp s + ωnp The open loop Bode diagram of the uncompensated plant with stability margins is shown in Fig. 1.

path. The lag compensator helps in improving the disturbance rejection ability by enabling use of higher DC gain apart from in achieving the desired GCF. The double lead compensator is used to improve the phase margin at GCF (Nise, 2007). Instead of a single lead stage, the double lead compensator is used as the required phase addition is more due to the unstable nature of the plant. The lag compensator is given as ( ω s + 1) GLag (s) = lgz (2) s ( ωlgp + 1) whereas the double-lead compensator is given as s ( s + 1) ( ωldz2 + 1) GLead (s) = GLead1 ·GLead2 = ωldz1 · s (3) s ( ωldp1 + 1) ( ωldp2 + 1) Further, the complex lead compensator is given by Gc (s) =

s2 2 ωn1

+

s2 2 ωn2

+

2ζz1 ωn1 s 2ζz2 ωn2 s

+1 (4)

+1

and is used in the feedback path for compensating the additional magnitude gain contribution by the lead compensator. The complex lead compensator is selected because the smaller ratio between the high-frequency and low-frequency gain asymptotes of the complex lead compensator offers advantage compared to the double lead compensator with real poles and zeros (Messner et al., 2007). The block diagram of the resulting feedback system is shown in Fig. 2.

2.2 Compensator selection r

The objective of the compensator design is to stabilize the plant and to meet the frequency domain specifications given in Table 1. In order to achieve the objectives, a Table 1. Design specifications Performance parameter GC Frequency (rad/s) Positive Gain margin (dB) Negative Gain margin (dB) Phase margin (deg)

GCF PGM NGM PM

Design goal 60.0 to 95.0 8.0 to 10.0 -8.0 to -10.0 40.0 to 55.0

+

-

Kf

GLag(s)

GLead(s)

Gp(s)

y

Gc(s) Fig. 2. Block diagram of plant and compensators

double lead-lag compensator is used in the forward path along with a complex lead compensator in the feedback 551

With this choice of the compensators, the design objective is to select the poles and zeros of the lag, lead, and complex lead compensator. Additionally, forward path gain, Kf , is also a design parameter. Arriving at appropriate values for the design parameters is an iterative and time consuming

5th International Conference on Advances in Control and 520 D. Penchalaiah et al. / IFAC PapersOnLine 51-1 (2018) 518–523 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India

process and a large number of iterations may be needed to converge to the required specifications. To address the issue, compensator parameters selection process is automated and optimum values of the parameters are obtained using genetic algorithms in this work.

optimum pole-zero location of the compensators (Deb, 2014). The considered cost function consists of violation of desired gain margin, phase margin, and gain crossover frequency from the design specifications. Population size, crossover probability and mutation probability selected for the optimization are tabulated in Table 2.

3. COMPENSATOR DESIGN USING GA

Table 2. GA operators used for optimization

Genetic algorithms proposed by Holland (Holland, 1975) are numerical optimization algorithms inspired by natural selection and natural genetics. These algorithms follow the principles of Charles Darwin theory of survival of the fittest. Due to its good performance in optimization, GA has been regarded as a function optimizer. The GA techniques differ from more traditional search algorithms in that they work with a number of candidate solutions rather than one candidate solution (Binitha et al., 2012). The GAs have been widely used for tuning of PID controllers and have shown excellent performance over the classification methods for PID controllers (Chakraborty et al., 2013; Altınten et al., 2008; Sadasivarao and Chidambaram, 2006; Yusoff et al., 2015; Patra et al., 2012; Aly, 2011). Application of GA for design of controller for a stable plant in frequency-domain can be found in Yang et al. (1991) whereas in Horng (2012), GAs are used for controller design with tuning in time-domain. However, GAs are rarely applied for lead-lag compensator design/tuning. In Bourmistrova and Khantsis (2009), an evolutionary design methodology for design of a control law and to obtain the corresponding control parameters using genetic programming is presented. It allows to solve the problems automatically even when little or no knowledge about the plant is available. However, it does not guaranty inherently robust solution in terms of both performance and unmodeled dynamics. The present plant of interest is highly unstable and design needs to be carried out in frequency domain to ensure the specified robustness margins. Genetic algorithms have potential for global optimization, compared to the gradient based methods. The gradient based methods have tendency to get trapped into local minima whereas the Genetic algorithms (GAs) are global, parallel, random search and optimization methods and work on the evolutionary principles by Darwin (1859). They work with a population of potential solutions to a problem. Each individual within the population represents a particular solution to the problem, generally expressed in some form of genetic code. The population gets evolved, over generations to produce better solution to the problem. The basic operators involved in the process are selection, ˇ crossover, and mutation (Yusoff et al., 2015; Situm and Cikovi´c, 2014). In present work, the random search capability of genetic algorithms is used to locate poles and zeros of the compensator optimally for meeting the given specifications. The number of design parameters consists of one forward gain (Kf ), two lag compensator parameters (ωlgz and ωlgp ), four double lead compensator parameters (ωldz1 , ωldp1 , ωldz2 , and ωldp2 ) and four complex compensator parameters (ωn1 , ωn2 , ωz1 , and ωz2 ). Thus, total 11 design parameters involved in the design need to be selected/found out. For the purpose, a simple binary coded genetic algorithm based optimization is carried out for 552

GA parameter selection operator crossover operator mutation operator design variables population size cross over probability mutation probability

selection roulette-wheel single point bit-wise 6 50 0.8 0.003

3.1 Cost function The cost function, being a measure of violation of design specifications, is taken as 2 2 2 2 + Jpgm + Jngm + Jpm (5) Jmin = Jgcf Component of the objective value tends to zero when respective values lies within the specified bound given as Jgcf =

0

for

60 <

Jpm =

0

for

40 <

GCF ≤ PM ≤

95 55

(6)

Jgm = 0 for 8 < |GM | ≤ 10 From (5), it can be seen that the penalty is more when the performance deviates from the design specifications. Since the desired GCF is between 60 to 95 rad/s, any deviation from the desired value is penalized as if(GCF < 60)

then

Jgcf = (60 − GCF )2

if(P M < 40)

then

Jpm = (40 − P M )2

(7) if(GCF > 95) then Jgcf = (95 − GCF )2 In order to ensure the required damping for the servo system, a phase margin of 40 to 55 degree is desirable and any deviation from this is penalized as (8) if(P M > 55) then Jpm = (55 − P M )2 Typically for unstable systems, both positive and negative gain margins play vital role in the stability analysis (Kadam, 2009). The positive gain margin indicates margin available for increasing the overall loop gain whereas negative gain margin shows available margin for reducing it. Since, the considered plant is unstable as can be seen in Fig. 1, it is essential to keep both the margins within the specified bounds. The specified gain margin is between 8 to 10 dB as lower than 8 dB makes the system less robust whereas above 10 dB leads to a sluggish system. Hence, taking this aspect into account, any deviation of both positive and negative gain margins from these limits are penalized as

if(P GM > 10) then

Jpgm = (8 − P GM )2

Jpgm = (10 − P GM )2

(9)

if(N GM > −8) then

Jngm = (−8 − N GM )2

(10)

if(P GM < 8) then

if(N GM < −10) then

Jngm = (−10 − N GM )2

5th International Conference on Advances in Control and Optimization of Dynamical Systems D. Penchalaiah et al. / IFAC PapersOnLine 51-1 (2018) 518–523 February 18-22, 2018. Hyderabad, India

3.2 Fitness function Fitness function indicates the goodness of every individual population with respect to all others. Proper selection of the fitness function plays a key role in convergence to an optimum solution. The fitness function is selected for maximization as it is naturally compatible with the genetic based search algorithms (Deb, 2004). The maximum value of the fitness function becomes one when all the performance parameters are within the design constraints as is obvious from 1.0 (11) F = 1.0 + Jmin 3.3 Bounds on design parameters Forward gain and lag compensator parameter bounds are chosen to achieve required gain crossover frequency and disturbance rejection. The parameter bounds for the double lead compensator with real poles and zeros are chosen to give maximum phase lead in the vicinity of the gain crossover frequency. The complex phase lead compensator bounds are chosen to improve positive gain margin and to extend the phase crossover frequency. Accordingly, the bounds on individual parameters are considered as given in Table 3. Table 3. Bounds on design parameters Parameter Kf ωlgz ωlgp ωldz1 ωldp1 ωldz2 ωldp2 ωn1 ζz1 ωn2 ζz2

Units rad/s rad/s rad/s rad/s rad/s rad/s rad/s rad/s -

Lower 12.0 10.0 1.0 40.0 90.0 60.0 90.0 180.0 0.1 500.0 0.5

Upper 25.0 20.0 5.0 65.0 110.0 80.0 120.0 250.0 0.2 600.0 1.0

randomly for each design variable based on the allowable range bracket as described in Table 3. This process is continued for 500 generations for the optimum parameter convergence and in every generation, the best solution is retained. The mean and maximum fitness of the populations evaluated over number of generations are given in Fig. 3 from where it can be noticed that after 25 generations, the fitness reached to its peak value and the best solution is selected from this generation. Hence, for this class of problems, around 50 to 100 generations are sufficient to give global optimum. Figures 4 to 8 depict the initial and final (500th) generation population statistics. The forward gain statistics is given in Fig. 4 and it can be seen that it has reached to around 13 from very wide spread of 12 to 25. Similarly, Fig. 5 shows the lag compensator pole-zero statistical values and the transfer function corresponding to the best solution is obtained as ( s + 1) (12) GLag (s) = 16.4 s ( 2.34 + 1) Figures 6 and 7 show the double lead compensator polezero statistics and the resulting lead compensator corresponding to the best solution out of 25000 (50×500) solutions is obtained as s ( s + 1) ( 78.5 + 1) GLead (s) = 44.9 · s (13) s ( 102.2 + 1) ( 93.6 + 1) The statistics of complex compensator parameters is shown in Fig. 8 and the resulting optimum compensator obtained from the proposed optimization formulation is Gc (s) =

s2 218.22 s2 546.02

+ +

2(0.19) 218.2 s 2(0.57) 546.0 s

+1

(14)

+1

1 0.9 0.8

Fitness value

S. No. 1 2 3 4 5 6 7 8 9 10 11

521

3.4 Genetic algorithmic steps

0.7

Maximum Average

0.6 0.5 0.4 0.3

After the selection of the required GA parameters, the algorithm for optimization procedure consists of the following steps (Deb, 2004; Patel, 2012; Mantri and Kulkarni, 2013; Yusoff et al., 2015):

0

As stated earlier, total 11 design variable are optimized with 50 population size. Initial population is generated 553

100

150

200

250

300

No.of generations

350

400

30

450

Initial Final

25 20 15 10 5 0 12

4. SIMULATION RESULTS

50

Fig. 3. Fitness function variation

No.of cases

(1) Choose the GA operators (refer: Table 2) (2) Initialize random population of strings and set maximum number of generations (gmax ) (3) Evaluate the population based on the fitness function, retain the best solution (4) If number of generations ≥ gmax , terminate (5) Perform reproduction on population using roulettewheel selection (6) Perform crossover on random pair of strings (7) Perform mutation on strings (8) Evaluate strings in new population, increase the generation number and go to step 4.

0.2

14

16

18

20

22

Population statistics: forward gain, Kf

Fig. 4. Forward gain (Kf ) population statistics

24

26

5th International Conference on Advances in Control and 522 D. Penchalaiah et al. / IFAC PapersOnLine 51-1 (2018) 518–523 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India 40

20 10 0 10

12

14

16

18

10 1.5

2

2.5

3

3.5

4

Lag compensator pole, ω

4.5

(rad/s)

240

45

50

55

60

30

Magnitude (dB)

10 95 100 105 Lead compensator pole1, ωldp1 (rad/s)

10 70

Lead compensator zero2, ω

75

ldz2

20

(rad/s)

80

Initial Final

15 10 5 100

105

110

0.8

1

z2

GM = −9.54 dB

0 NGM = −8.4 dB −20 −40

PGM = 8.38 dB

Uncompensated Compensated

−180

115

Lead compensator pole2, ωldp2 (rad/s)

PM= 42 deg

−225 −270

PM= −42 deg

−315 0

10

1

10 Frequency (rad/s)

20

2

3

10

10

NGM = −8.36 dB PGM = 14.0 dB

0 −20

NGM = −8.4 dB PGM = 8.38 dB

Classical GA

−40

120

PM = 48 deg

−180 PM = 42 deg −270 −360 −1 10

95

0.6

Poles damping, ζ

−90 Phase (deg)

65

0 0.4

−135

Magnitude (dB)

20

0 60

600

(rad/s)

10

Fig. 9. Bode Diagram: Compensated & uncompensated Plant

Initial Final

30

550

Poles ω

110

Fig. 6. Population statistics of lead compensator-1 40

0.25

20

20

−360 −1 10

0 90

0.2

−90

Initial Final

20

0.15

Zeros damping, ζ

Fig. 8. Population statistics of complex compensator

65

Lead compensator zero1, ωldz1 (rad/s)

5

z1

5

n2

10 0 40

10

30

0 500

Initial Final

20

15

0 0.1

260

(rad/s)

10

5

Phase (deg)

No.of cases

30

No.of cases

220

Zeros ω

No.of cases

No.of cases

No.of cases

20

Fig. 5. Population statistics of lag compensator

No.of cases

200

n1

Initial Final

lgp

No.of cases

10

15

30

0 90

20

0 180

40

0 1

30

20

Lag compensator zero, ωlgz (rad/s)

20

Initial Final

No.of cases

Initial Final

No.of cases

No.of cases

30

0

10

1

10 Frequency (rad/s)

2

10

3

10

Fig. 7. Population statistics of lead compensator-2

Fig. 10. Bode Diagram: Classical compensator & GA compensator plant

Figure 9 depicts the Bode diagram of the uncompensated as well as compensated plant. From the plot, it can be seen that the optimum compensator design has resulted

into overall system stability. Further, the design has met all the specifications such as gain crossover frequency, gain margin, and phase margin given in Table 4.

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5th International Conference on Advances in Control and Optimization of Dynamical Systems D. Penchalaiah et al. / IFAC PapersOnLine 51-1 (2018) 518–523 February 18-22, 2018. Hyderabad, India

Next, the performance of the GA based compensator is compared with the one obtained using classical approach. In classical approach, more than 10 iterations are needed for reaching to the final solution. The compensator performance obtained by classical method is compared with the performance obtained using GA compensator and the performance in terms of the Bode plots is shown in Fig. 10. The comparative performance results are tabulated in Table 4 from where it can be seen that while the GA results met all the specifications, the positive gain margin using classical tuning is more than design specification (14 dB against of 8 to 10 dB). Also reduction in 14 dB PGM could not be achieved even with several iterations in classical approach. The main disadvantage of this higher value is the degradation of agility of system. Lastly, noise attenuation at high frequency (200 rad/s) is better with GA compensator. Table 4. Stability margins: Compensated plant GCF (rad/s) PGM (dB) NGM (dB) PM (deg)

Design goal 60.0 to 95.0 8.0 to 10.0 -8.0 to -10.0 40.0 to 55.0

GA 91.4 8.38 -8.40 42.0

Classical 88.4 14.0 -8.36 48.0

5. CONCLUSIONS Genetic algorithm based lead-lag compensator design in frequency domain is carried out for an unstable plant. It is shown that the proposed formulation represents a viable solution for design of compensator to stabilize the plant and to ensure all the robustness performance specifications required for the considered servo system. Efficacy of the formulation is compared with classical approach and it is found that the GA based design offers superior performance in terms of robustness characteristics, high frequency noise attenuation and design time cycle. In the design, total 11 parameters are optimized with 50 population size and the optimal compensator has met all the design specifications within 25 generations. ACKNOWLEDGEMENTS The authors thank competent authorities of Defence Research and Development Laboratory (DRDL), Hyderabad for granting the permission to publish this piece of work REFERENCES Altınten, A., Ketevanlio˘ glu, F., Erdo˘ gan, S., Hapo˘glu, H., and Alpbaz, M. (2008). Self-tuning pid control of jacketed batch polystyrene reactor using genetic algorithm. Chemical Engineering Journal, 138(1), 490–497. Aly, A.A. (2011). Pid parameters optimization using genetic algorithm technique for electrohydraulic servo control system. Intelligent Control and Automation, 2(02), 69. Binitha, S., Sathya, S.S., et al. (2012). A survey of bio inspired optimization algorithms. International Journal of Soft Computing and Engineering, 2(2), 137–151. Bourmistrova, A. and Khantsis, S. (2009). Flight control system design optimisation via genetic programming. In Aerial Vehicles. InTech. 555

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