multi-objective evolutionary …multi-objective evolutionary optimization pid tuning for...

MULTI-OBJECTIVE EVOLUTIONARY OPTIMIZATION PID TUNING FORLONGITUDINAL MOVEMENT OF AN AIRCRAFT.

Garbelini, Matheus∗, Reynoso-Meza, Gilberto†

∗Polytechnic School, Pontifical University of Parana (PUCPR), Curitiba, CO 80215-901 BRA

†PPGEPS, Pontifical University of Parana (PUCPR), Curitiba, CO 80215-901 BRA

Emails: [email protected], [email protected]

Abstract— Longitudinal control tuning is one of the principal tasks in the control system of an aircraft as itspitch movement is affected by different flight conditions and most of the times, aggressive environment. In orderto get a better flight stability, multi-objective evolutionary optimization techniques can be applied to the tuningprocedure in order to depict the trade-off between conflicting objectives. In this paper, we use such techniquesto appreciate the trade-off of the aircraft longitudinal angle response. We compare two different algorithmsin a four-objective problem that lead longitudinal movement control trough a standard proportional-integral-derivative (PID) controller.

Keywords— Multi-objective optimization, PID controller, Longitudinal movement.

1 Introduction

Aerial systems nowadays are demanding a growingcapability in its stabilization control. Both smalland big planes require a controller that is capa-ble of delivering the desired performance underdifferent operation conditions (Duarte-Mermoudet al., 2005). PID controllers are widely usedfor this approach due to its simplicity. Further-more, if multi-objective optimization is used alongto bring the optimum trade-off for the decisionmaker, the results can get very satisfactory whenrelated to the design preferences.

The use of multi-objective optimization evolu-tionary algorithms (MOEAs) corresponds a veryknown category for such multi-objective optimiza-tion techniques (Das et al., 2016). Its use extendsin many different areas where there is the necessityto find the trade-off between the designed problemand its system requirements.

Several control literatures report the com-bined use of a feedback controller with MOEAsfor parameters tuning (Reynoso-Meza et al., 2014,and the references therein). In this context,fuzzy controller and MOEAs was used to aug-ment the flight control for an F16 aircraft in(Stewart et al., 2010), the transition process inflight mode of a tiltwing aircraft in (Holstenand Moormann, 2015), the control of a commer-cial Cessna airplane in (Yamina Boughari andTheel, 2014) and further.

(Duarte-Mermoud et al., 2005) study the im-pact and result of a PID and CMRAC controllersin the pitch of Cessna 182 airplane according toits lifter angle (δe). This paper approach is to di-rectly compare the results from the reference PIDparameters used in the previous paper to the re-sults obtained after the same PID goes throughMOEAs optimization. Further, it can show itsimprovement and viability in real aerial applica-tions generally.

When the multi-objective optimization isunited with the already defined closed loop model,design criteria must be well defined in order tocorrectly get the best performance. For this rea-son, adequate design objectives that can providerobust and meaningful indicators can be challeng-ing to define depending on the complexity of theproblem.

The more clear the objectives are, the easierthe decision-making processes gets. Even though,the algorithms that are used in the evolutionaryprocesses also needs to be fit correctly the prob-lem. In most cases the behavior of a certain al-gorithm is not known for a specific design prob-lem due to the objectives number or complex-ity. Hence, a comparison between different evo-lutionary algorithms needs to be done before pre-ceding to the multi-criteria decision make stage(MCDM). The algorithm trade-off convergence ordivergence rate between different objectives canbe acquired by analyzing the hypervolume graph.

This paper presents a simple multi-objectivestatement and uses reference case parameters toimprove pertinence of the solutions and also tofacilitate the direct comparison between the eval-uated evolutionary algorithms.

The remainder of this paper is organized asfollows: The second section of this paper presentthe background of the multi-objective problem,depicting its model, control loop and the mul-tiobjective problem (MOP) statement. In sec-tion 3 to 5, will be described the fundamentalssteps to implement an optimization problem withmulti-objective approach: The MOP definition,the multi-objective optimization (MOO) processand the MCDM stage. The results are validatedin section 6 and a conclusive view of this work iscommented in section 7.

XIII Simposio Brasileiro de Automacao Inteligente

Porto Alegre – RS, 1o – 4 de Outubro de 2017

ISSN 2175 8905 839

2 Overview

To understand the tuning approach with MOEAs,it’s necessary firstly to understand the require-ments of the aerodynamic system. Many variablesare involved in the design problem, but the prin-cipal challenge is to wisely choose which relevantinformation use in MOEAs decision variables orobjectives.

2.1 Longitudinal movement control of a plane

The airplane aerodynamic can be described in dif-ferent ways, but is convenient to use a less com-plex model that keep a certain simplicity and alsoprovides the necessary accuracy about the aero-dynamic movement in a MOP statement. TheFigure 1 depicts a basic plane longitudinal move-ment that needs to be carefully analyzed with thedesigned optimization problem.

Figure 1: Airplane axis movement

When the airplane is flying, its speed and sev-eral external ambient conditions can interfere inits stabilization. Due to the necessity in keepingthe simplicity, these conditions can be abstractedand assumed constant1without compromising themodel accuracy.

In this paper, the airplane is assumed alreadyflying and with its roll and yaw axis in steadystate. By doing this, the MOP can be focused tothe desired task such as pitch axis tunning.

The Airplane can perform the pitch move-ment when the lever is moved along a desired in-clination. The flight control system is responsiblefor the lever positioning system.

As in (Duarte-Mermoud et al., 2005) we usea transfer function P (s) from equation 1 to sim-ulate the response of pitch and also the influenceof mechanical constraints of the lever itself.

P (s) = 347.3544(s+0.05902)(s+2.001)

(s+10)(s2+0.04417s+0.02933)(s2+8.902s+27.79)(1)

2.2 PID controller tuning

The control loop depicted in Figure 2 is responsi-ble for controlling the lever angle O(s) through adesired reference I(s). It comprises a simple PID

1 More information about Cessna plane flight conditionssuch as air humidity, average temperature and pressure,plane average speed and further can be found in (Duarte-Mermoud et al., 2005).

transfer function C(s) that is described in equa-tion 2.

C(s) = kp +kpτis

+ kpτds (2)

where kp is the proportional gain, τi is theintegral time and τd the derivative time.

Figure 2: Control loop

Due to the system behavior dependency ofthe PID controller, the MOP resides in obtainingthe best parameters from C(s) such as propor-tional, derivative and proportional gain in orderto achieve the desired output O(s) performance.Evolutionary multiobjective optimization (EMO)techniques could be useful for the purpose in tun-ing these parameters.

2.3 Multiobjective problem statement

According to (Miettinen, 2012) a MOP with n ob-jectives can be stated such as:

minθJ(θ) = [J1(θ), ..., Jn(θ)] (3)

where θ = [θ1, θ2, . . . , θn] is the decision vec-tor and J(θ) the objective vector subjected to theinequality and equality vectors K(θ) and L(θ) re-spectively (see equation 4 and 5).

K(θ) ≤ 0 (4)

L(θ) = 0 (5)

θ = [θi ≤ θi ≤ θi]; i = [1, 2, . . . , n] (6)

The decision vector θi is limited to the upperand lower bound vector θi and θi respectively.

As in MOPs there are different conflicting ob-jectives, there’s not just a single solution, but in-stead, a set of solutions called Pareto set θP . Theobjective vector J associated with each solutionin Pareto set approximate an optimal Pareto frontJP . All individuals in Pareto front consists of non-dominated solutions (see Figure 3).

An objective vector (Miettinen, 2012) J1(θ1)dominates a second objective vector J(θ2) (de-noted as J(θ2) � J(θ1)) if:

Ji(θ1) ≤ Ji(θ2) ,∀i ∈ [1, 2, . . . , n]

∧Jl(θ

1) < Jl(θ2) : ∃l ∈ [1, 2, . . . , n] (7)

In order to find the best PID parameters forthe longitudinal control, the fundamentals stepsto implement an optimization problem with multi-objective approach are described in the followingsections.



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Figure 3: Pareto front concept

3 Multiobjective problem definition

As in (Duarte-Mermoud et al., 2005), the evalua-tion of the controller performance in the requiredlongitudinal movement of the closed loop (Figure2) is assumed as a unitary step response. In orderto ensure that the PID controller provide bettercharacteristics to the MOP, four objectives J areconsidered in the signal O(s), which are depictedin Figure 4 and described as follows:

• Settling Time JST (θ): Given a differentialfunction F , the settling time at a tolerance∆% is presented in the equation 8.

JST (θ) = lim(a,t)→(|Y (t)−Yref |≤∆%,∞)

F (a, t) (8)

• Rise time JRT (θ): Given an initial time t0and the time tref which the signal Y (t) crossthe reference, the Rise time can be obtainedby the equation 9.

JRT (θ) = tref − t0 (9)

• Overshoot JOV (θ): Indicates the relativeerror of the maximum deviation YOV fromthe reference signal Yref where It’s given bythe equation 10.

JOV (θ) =YOV (θ)− Yref

Yref· 100 (10)

• Maximum value of sensitivity functionJMS(θ): Indicates how sensible a closed loopis in terms of a possible peak variation in adynamic system. It’s given by equation 11.

JMS(θref ) = max0≤s=jw<∞

| 1

1 + C(s) · P (s)| (11)

With these definitions, the objective vectorJ(θ) can be defined such as:

J(θ) = [JST (θ), JRT (θ), JOV (θ), JMS(θ)] (12)

In which the decision variables comprehendsthe PID parameters θ = [kp, τi, τd].

Figure 4: Rise time, settling time and overshootconcepts

3.1 Constraints

Due to the known behavior of theses variables incontrol literature, it is possible to easily determi-nate bounds to them. As Kp and τd can affectthe behavior of the system, they must generallybe lower.

The decision vector θ is limited to decisionbounds θ and θ that are given by:

θ = [5, 100, 2]

θ = [0, 0, 0] (13)

Thus the decision vector become:

θ = [0 ≤ kp ≤ 5, 0 ≤ τi ≤ 100, 0 ≤ τd ≤ 2] (14)

As the objectives are meaningful values fromthe problem perspective, it is a common practiceto adopt objectives constraints in order to ensureperformance and to match design requirements ofthe MOP. These constraints are proposed as fol-lows:

• Settling time must be ≤ 0.75 seconds, givena tolerance condition of 2%.

• Rise time must be ≤ 0.5 seconds.

• Overshoot must be ≤ 4%.

• Ms must be held between an absolute valueof 1.2 and 1.8 to allow solutions with a slightlyweak, robust or nearly strong PID controller.

4 MOO algorithm

A multi-objective optimization algorithm (MOO)must offer to the designer a good quality set ofsolutions and be very descriptive, allowing moreflexibility in the decision process. Such require-ments is presented as challenges that the algo-rithm must overcome (Reynoso-Meza, 2009):

• To avoid earlier convergence in an optimal lo-cal space, losing generalization of the Paretofront JP .



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• To guarantee the diversity of solutions inorder to improve the representation of thePareto front JP .

In this paper, Spherical pruning multi-objective optimization differential evolution (sp-MODE) and Indicator based multi-objective opti-mization differential evolution (ib-MODE) are dis-cussed.

4.1 sp-MODE

sp-MODE is a Differential Evolution (DE) basedfrom (Reynoso-Meza, 2009), which is a stochasticalgorithm. Its main characteristics are discussedas follows:

• To improve the convergence rate, the best so-lutions participate trough the entire evolutionprocess.

• Spherical pruning is used in the objectivespace to maintain diversity of solutions, re-duction of cardinality of Pareto set θP andto avoid non-dominated solutions to be lostin the evolution process.

Due to MOP approach, sp-MODE uses justtwo operators from DE: Mutation and Crossover1.

The main steps of sp-Mode are described inalgorithm 1.

4.2 ib-MODE

Ib-MODE is DE based in the indicator-based evo-lutionary algorithm (IBEA). The algorithm takein account preferences of DM to drive the environ-mental and mating selection (Thiele et al., 2009).Its main characteristics are:

• Uses a reference point as a pertinence mecha-nism to focus in the DM’s solution space pref-erence and to generate a local approximationof Pareto optimal set θP .

• Uses an indicator I (Zitzler E., 2004) to gen-erate a fitness value in order to select the de-sired individuals from a population θ.

The main steps of the ib-MODE are describedin algorithm 2.

5 Multi-criteria decision-making stage

Given that in this case, there are four objec-tives, further tool are required in the multi-criteriadecision making stage (MCDM) of a MOOD.As it’s not possible to directly visualize a four-dimensional data, there is are alternatives suchas parallel coordinates2 (Inselberg and Dimsdale,

1More information about mutation and crossover oper-ators can be seen on (Storn and Price, 1997).

2The tool used can be found on http://www.xdat.org/

1 Initial population P (0) is initialized withrandom N individuals selected fromsearching space θ;

2 Evaluate P (0);3 Search for non-dominated solutions in

P (0) to get D(0);4 Apply spherical pruning in D(0) to get

A(0) and store the solutions;5 for i=1:max. gen. or convergence reached.

do6 Random select subpopulation of NS(i)

individuals with solutions in P (i) andA(i);

7 Apply mutation and crossoveroperations on NS(i) to get theoffspring O(i) (Fixing boundaryviolations if needed);

8 Evaluate O(i);9 if child < parent then

10 parent = child;11 end12 Apply dominance on A(i) ∪O(i) to get

D(i);13 Apply spherical pruning on D(i) to get

A(i+ 1);14 Store A(i+ 1);

15 end16 Solution in A with the lower |J | is

proposed as the single run of sp-MODE.;

1 Initial population P (0) initialized with sizeα and generation count m = 0;

2 Calculate fitness value of all point inPareto set θP using indicator I;

3 for i=1:size of population ≤ α do4 Choose a point θ∗ in θP ;5 Remove θ∗ from the population;6 Update fitness value of the remaining

individuals using indicator I;

7 end8 while m < max. gen. or convergence

criterion do9 Perform binary tournament selection

with replacement on θP in order tofill the temporary pool θ′P ;

10 Apply recombination and mutationoperators to the mating pool θ′P andadd the resulting offspring to θP .Increment the generator count m;

11 end12 set A to points in θP that represent

non-dominated solutions;



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1991), which will be used to better depict thetrade-off between the solutions in both optimiza-tion algorithms.

6 Proposal validation

The simulations and Pareto front approximationswere carried by Matlab and performed in a stan-dard personal computer with Intel Core i5-4210U,1.7GHz, 8GB RAM.

As the MOEAS adds stochasticity, 51 runswere performed on both MOP algorithms. In or-der to evaluate the Pareto front approximationperformance of the algorithms, the median valueof hypervolume indicator (The Nadir point is usedalong with the objectives bounds as its reference)is generated and depicted in Figure 5. Fromthis distribution is possible to affirm that that ib-MODE is preferable for this MOP as its conver-gence is less variable than sp-MODE.

The parallel coordinate of the Pareto front ap-proximation from both algorithms is also shown inFigure 7, in which is possible to notice that the so-lutions on ib-MODE have more quality due to itsconvergence. For this reason, ib-MODE Paretofront approximation will be chosen to select thedesired controller structure and the following cri-teria are adopted:

• Those controllers with JST (θ) ≥ 0.73 are fil-tered and excluded from the analyses as theyare very near to the objective constraint andcould represent risk solutions.

• Those controllers with JOV (θ) ≥ 1.5 are dis-carded, since their performance is outsideDM’s preferences.

• Those controllers with JMS(θ) ≥ 1.76 are alsodiscarded, since the MOP needs less aggres-sive controllers.

After this constraints, a total of 4 possible so-lutions are chosen to be evaluated. To help in fil-tering the remain solution, a parallel coordinatesplot is generated with an additional axis contain-ing the integral derivative time Td, which can in-fluence the sensitivity of the closed loop systemdue to noise issues, thus, needed to be low. Bydoing this, it’s possible to acquire the preferablesolution.

Such solution response is shown in Figure6. For further analysis, the response of a moreaggressive controller with an objective vectorJ(θagr) = [0.75, 0.194, 0.455, 1.799] and θagr =[4.0, 17.819, 0.130] is also included.

Given the reference controller θref =[2.400, 5.5996, 0.1750] with J(θref ) =[0.750, 0.510, 3.900, 1.514] as described in (Duarte-Mermoud et al., 2005), the preferable solutionshows the following decision variables θS andobjective vector J(θS):

θS = [3.987, 10.328, 0.144]

J(θS) = [0.729, 0.213, 1.404, 1.760] (15)

Even tough in this paper we select one prefer-able solution, it can be argued that all the finalsolutions: preferable, reference and aggressive ac-complish the design requirements. The DM canchose which is adequate and if is necessary, eventhe aggressive solution can be considered in a spe-cific situation in which the aircraft wouldn’t besusceptible to an aggressive environment.

Figure 5: Hypervolume’s distribution.

7 Conclusion and future works

The fundamentals steps to implement a MOPhave shown their usefulness in the controller tun-ing applications such as the control of longitudinalmovement of an aircraft. They allow the possibil-ity to appreciate the trade-offs between usual ob-jectives such as settling time, rise time and over-shoot. The MCDM stage fitted well when morethan 3 objectives are involved and allows flexibil-ity to the DM.

It is important to mention that even tough weused 4 objectives, rise time and overshoot wereobserved to be not-conflicting objectives, whichcould reduce the MOP to a 3 objectives prob-lem, thus reducing the complexity and computa-tion time of the problem. Furthermore, in pro-posal validation, although sp-mode showed to bemore viable than ib-mode for PID tunning in thisproblem, there are MOPs with different cost func-tions and plant models that ib-mode can performbetter than the former. Thus, a comparative ap-proach must be made for each case.

Future works involve the use of new heuristicsand changes in the multi-objective optimizationalgorithm that could offer better Pareto approx-imation. Furthermore, in order to better reflectthe designer’s preference, a restructuration of thedesign objectives could be made.

Acknowledgements

This work was partially supported by the Con-selho Nacional do Desenvolvimento Cientıfico



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Figure 6: Step response results.

Figure 7: Parallel coordinate of the Pareto front approximation.

e tecnologico of Brazil (CNPq) by fellowshipsPIBIC-104678/2015-1 and BJT-304804/2014-2.

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