on the use of a global search method and a gradient …

Inverse Problems, Design and Optimization Symposium Miami, Florida,U.S.A., April 16-18, 2007

ON THE USE OF A GLOBAL SEARCH METHOD AND A GRADIENT BASED METHOD FOR THE IDENTIFICATION OF AIRCRAFT LONGITUDINAL

STABILITY AND CONTROL DERIVATIVES

Luiz Carlos Sandoval Góes, Benedito Carlos de Oliveira Maciel and Nei Salis Brasil Neto Department of Mechanical-Aeroeronautical Engineering

Instituto Tecnológico de Aeronáutica Sao José dos Campos, SP, Brazil

[email protected], [email protected], [email protected]

and

Felipe Antonio Chegury Viana and Valder Steffen, Jr School of Mechanical Engineering Federal University of Uberlândia

Uberlândia, MG, Brazil [email protected], [email protected]

ABSTRACT

In this work, two optimization algorithms are investigated to accomplish the parameter identification of the longitudinal motion of a real aircraft by using the Output Error Method. The first algorithm is the nature-inspired algorithm named Life Cycle Model, which is a composed strategy based on other heuristics such as Genetic Algorithms and Particle Swarm Optimization. The second one is the gradient-based technique named Levenberg-Marquardt Algorithm, which is a variant of the Gauss-Newton method. Flight test data, performed with a military aircraft (Xavante AT-26 FAB 4516), were used to feed the Output Error Method. In this context, both optimization algorithms were tested, in solo performance and in a cascade-type approach. Results are reported, aiming to illustrate the success of using the proposed methodology.

INTRODUCTION

One of the most important phases of the modern design and evaluation process concerns modeling and simulation. In order to accomplish this task, the system identification and parameter estimation constitutes a fundamental step. System Identification is a general procedure to match the observed input-output response of a dynamic system by a proper choice of an input-output model and respective physical parameters. From this point of view, the aircraft system

identification or inverse modeling comprises proper choice of aerodynamic models, the development of parameter estimation techniques by optimization of the mismatch error between predicted and real aircraft response and the development of proper tools for integration of the equations of motion within the system simulation and correlated activities [1].

The present contribution reports the identification of longitudinal stability derivatives from flight test data by using the Output Error Method [2,3]. In this scenario, some difficulties that are intrinsic to inverse problems may arise: • an reliable model of the structure is required,

since the results of the identification procedure rely upon the mathematical model used;

• in general, identification problems are ill conditioned from the mathematical point of view. This means that the procedure is sensitive to noise that can contaminate experimental data; and

• experimental data are incomplete either in the spatial sense (responses are available only in a limited number of positions along the structure), as in the time sense (responses are obtained in a given time interval). Consequently, the uniqueness of the solution cannot be assured.

It is well known that the identification by using Output Error Method associated with


classical gradient-based optimization methods is a difficult task due to the existence of local minima in the design space. Moreover, such methods require an initial guess to the solution and it is not possible to assure global convergence. These aspects have motivated the authors of this paper to explore the performance of a gradient-based, namely the Levenberg-Marquardt method (LM) [3], and a heuristic method, namely the LifeCycle Model (LC) [4]. LM is a second order variant of the Gauss-Newton method and LC is a hybrid approach, based on two heuristic methods, namely Genetic Algorithms (GA) and Particle Swarm Optimization (PSO) combined in a evolutionary strategy. The individual performance of both algorithms are tested and then a cascade-type scheme is proposed aiming to take advantage of the global search capabilities of LC and the local search capabilities of LM.

An optimization problem is formulated so that the objective function represents the difference between the measured characteristics of the structure and its model counterpart. From a linear state space model for the longitudinal motion of aircraft, a total number of ten parameters are assumed as being unknown. In this case, the parameters represent the dimensional aerodynamic coefficients of the aircraft.

In order to show the optimization strategy and the approaches, experimental flight test data, performed with a military aircraft (Xavante AT-26 FAB 4516), were used in parameter identification of the stability derivatives. PROBLEM FORMULATION Aircraft Longitudinal Model

In this work the inverse problem formulation is applied to the longitudinal equations of movement of the aircraft, in particular, the short period mode, for which the linear state space equations can be written as:

(1)

( )

1

2

1

,

q

e x

e x

Z Zqq M M

Z be

M b

α α

α

δ

δ

α

δ

+⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥ ⎢⎣ ⎦ ⎣⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥+⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

α+

⎥⎦

and the observation equations are given by:

1

2

1

2

1 0

0 1

0.

0y

y

yy q

be

b

α

δ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= +⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ + ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2)

The previous dynamical equation has ten

unknown parameters that need to be estimated, giving : 10Θ ∈ ℜ

[ ]1 2 1 2, , , , , , , , , Tq q e e x x y yZ Z M M Z M b b b bα α δ δΘ = (3)

Output Error Method

In this section, the parametric identification, in particular the parameter estimation applied to a linear model of the longitudinal motion of an aircraft in space state formulation is presented. The Output Error method, as shown in Fig. 1, is one of the most used estimation methods for aerodynamic parameter estimation from flight test data [5, 6, 7]. It has several desirable statistical properties, including its applicability to nonlinear dynamical systems and the proper accounting of measurements noise [6].

Figure 1. Block diagram of the estimation

procedure. The model is considered to be known, and the

identification process consists in determining the parameter vector Θ , which gives the best prediction of the output signal ( )y t , using an optimization criteria. The attainment of an estimate through optimization of a cost function based on the prediction error of the plant requires, usually, the minimization of a nonlinear function. Thus, LC, LM or a composition of both is used here to estimate the parameters in the model. Therefore, the cost function to be minimized involves the prediction error:

(4) ( ) ( ) ( )ˆ ,e k y k y k= −


where ( )y kˆ

)

is the output prediction based on the actual estimate Θ of the parameter vector . Θ

Consider a dynamic system, identifiable, with a definel model and output . Suppose that

(M Θ y(p y )Θ is the conditional probability

gaussian distribution of the random variable y with dimension m , mean ( )f Θ and covariance

, with dimension m m . R × (p y Θ) is known as the likelihood functional. Goodwin and Payne in [2] attribute this name due to the fact that it is a measure of the probability of occurrence of the observation for a given parameter Θ . Thus, the likelihood functional is:

y

( )( )

( )[ ] [ ] ( )[ ]{22

11

1 .2

1exp , , .

2

nm T

n T Tk

p yR

e k R e k

π−

=

Θ =

− Θ∑ }Θ

Θ

(5)

The Maximum Likelihood Estimate (MLE) is defined as the value of Θ which maximizes this functional, in such a way that the best estimate of

is:

(ˆ ArgMaxp yΘ = Θ) (6)

The maximization of (p y Θ) is equivalent to the minimization of ( )J Θ , which is given by:

( ) ( )[ ] [ ] ( )[ ]{ }1

11 , ,2

n T Tk

J e k R e k−=

Θ = Θ Θ +∑ ln .R (7)

Thus, in the optimization process, ( )J Θ is equivalent to ((ln p y− ))Θ , except for a constant term.

Optimization Strategies

As can be seen in the general framework illustrated by Fig. 1, the Output Error has no specific requirement about the optimization method. Even though most of works in the literature reports the use of classical methods, there is no limitations in using other approaches. In this scenario, three optimizaiton strategies, described in Tab. 1, are explored to solve the identification problem.

LIFE CYCLE MODEL Back to Nature

LifeCycle Model (LC) is a computational tool inspired in the biologic concept of life cycle. From a biology viewpoint, the term is used here

to define the passage through the phases during the life of an individual. Some phases, as sexual maturity, are one-time events, others, as the mating seasons, are re-occurring. Although it does not happen in every case, the transitions between life cycle phases are started by environmental factors or by the necessity to fit to a new condition [4]. The transition process promotes the maturity of an individual and contributes to the adaptation and evolution of its species.

Table 1. Optimization strategies used in the

Output Error scheme. Strategy Optim.

Method Comments

#1 LC

The main explored characteristics are the global search capability and the low level of knowledgement about the search space.

#2 LM

As a classical method, local search capabilities and small number of function evaluations are the main explored adavantages.

#3 LC-LM

The result of a first global search performed by LC is used as an initial guess in the LM method. This cascade approach aims to make use of the main adavantages of both methods.

Take the example of a honeybee swarm. Even

though there exist ten thousand individuals in a colony, there are just three castes of bees: • queens, which produce eggs, • drones or males, which mate with the queen,

and • workers, which are all non-reproducing

females. Table 2 shows the life cycles of each bee type

in a honeybee colony. After this simple example, it is easy to

visualize the concept of life cycle.

Algorithm Description From the implementation point of view,

algorithms such as GA and PSO are heuristics search methods of proven efficiency as optimization tools. LC intends to put together the positive characteristics found in each method and creates a self-adaptive optimization approach. Each individual, as a candidate solution, decides based on its success if it would prefer to belong to


a population of a GA, or to a swarm of a PSO. This means that different heuristic techniques contribute together to form a robust high performance optimization tool. The idea is that complex problems can be conveniently considered from the optimization viewpoint. As can be viewed, the less well-succeeded individuals must change their status in order to improve their fitness. In plain English, the optimization approach does not follow a rigid scheme as proposed in [9], in which various techniques are used sequentially in a cascade-type structure. In other words, it is the mechanism of self-adaptation of the optimization problem that rules the procedure. Figure 2 shows the outline of a basic LC algorithm.

Table 2. Life cycles of bees, measured in days

(adapted from [8]). Queen Drone Worker Egg 3 3 3 Larva 5.5 6.5 6 Cell capped 7.5 10 9 Pupa 8 14.5 12 Period of development 16 24 21

Start of fertility ≈ 23 ≈ 38 N/A

Take the example of a LC with just GA and

PSO as heuristics. According to Fig. 2, the algorithm is initialized with a set of particles of a PSO swarm, which can turn into GA individuals, and then, according to their performance, return to particles again, and so on. A LifeCycle individual switches its stage when there is no fitness improvement for more than a previously defined number of iterations. This is a parameter that can be adjusted according to the problem.

Algorithm Parameters

Since the algorithm is composed by various heuristics, it is necessary to set the parameters of every heuristic used in the LifeCycle. Nevertheless, there is a parameter inherent to the LifeCycle, namely the number of iterations that represents a stage of the LifeCycle. The authors, to improve the algorithm performance, have introduced this parameter differently from what is usually done, by simply fixing this number as in [4]. This number is called stage interval. At the end of each stage interval, the less well-succeeded

individuals must change their stage aiming at improving their fitness.

More detailed information about LC can be found in [4, 10].

Figure 2. Basic scheme for LC.

Genetic Algorithms

GA is an optimization algorithm based on Darwin's theory of survival and evolution of species. The algorithm starts from a population of random individuals, viewed as candidate solutions to the problem. During the evolutionary process, each individual of the population is evaluated, reflecting its adaptation capability to the environment. Some of the individuals of the population are preserved while others are discarded. This process mimics the natural selection in the Darwinism. The remained group of individuals is paired in order to generate new individuals to replace the worst ones in the population, which are discarded in the selection process. Finally, some of them can be submitted to mutation, and as a consequence, the chromosomes of these individuals are altered. The entire process is repeated until a satisfactory solution is found. The outline of a basic GA is as follows: 1. Define the GA parameters (population size,

selection method, crossover method, mutation rate, etc.).

2. Create an initial population, randomly distributed throughout the design space (other distributions can be performed).

3. Evaluate the objective function and take it as a fitness measure of each individual.


4. Select mates to the crossover; this mimics the natural selection.

5. Reproduce and replace the worst individuals in the population by the offspring.

6. Mutate, to avoid premature convergence (other parts of the design space are explored).

7. Go to step 3 and repeat until the stop criterion is achieved.

Although the initial proposed GA was dedicated to discrete variables, nowadays, improvements are available to deal with discrete and continuous variables, see [11, 12] for more details.

Particle Swarm Optimization

PSO was introduced by Kennedy and Eberhart in [13]. The method reflects the social behavior of some bird species. The main idea of the PSO is to reproduce the search procedure used by a swarm of birds when looking for food or nest. This is achieved by modeling the flying using a velocity vector, somewhat analogous to a search direction. The velocity vector considers a contribution of the current velocity and other two portions referred to the knowledge of the particle itself and of the swarm, respectively, about the search space. In such a way, the velocity vector is used to update the position of each particle in the swarm. The outline of a basic PSO is as follows: 1. Define the PSO parameters (swarm size,

inertia value, "trust" parameters, etc.). 2. Create an initial swarm. 3. Update the velocity vector for each particle. 4. Update the position for each particle. 5. Go to step 3 and repeat until the stop criterion

is achieved. PSO is comprehensively presented in [13, 14].

LEVENBERG-MARQUARDT OPTIMIZATION METHOD

The Levenberg-Marquardt method is a second

order technique based on the Gauss-Newton method. This method, although complex, is suitable for a quadratic cost function, and is expected to converge quickly. First, we approximate (

( ) ( )

( ) ( )

( ) ( )[ ]( )21.

2

L L L

T TL L

TL L L

J J

J

J

Θ

Θ

Θ ≅ Θ +

Θ−Θ ∇ Θ +

Θ−Θ ∇ Θ Θ−Θ

(8)

The optimization condition is obtained when:

(9) ( ) 0.J ∗Θ∇ Θ =

Applying Eq. (9) to Eq. (8):

( ) ( )

( ) ( )[ ]2 0,

L L

TL L

J J

J

Θ Θ

Θ

∇ Θ = ∇ Θ +

Θ −Θ ∇ Θ = (10)

∗which can be used to find the optimum, Θ , through the recursion:

( )[ ] ( )121 .T

i i i iJ J−+ Θ ΘΘ = Θ − ∇ Θ ∇ Θ (11)

The complexity in the calculation of the Hessian matrix, ( )2

iJΘ∇ Θ , in Eq. 11, is avoided through the Gauss-Newton method, which uses the approximation:

(12) ( ) ( )[ ] ( )[ ]12

1ˆˆ ˆ ,n T

kJ y R y

−Θ Θ Θ=

⎡ ⎤∇ Θ ≈ ∇ Θ ∇ Θ⎢ ⎥⎣ ⎦∑where the terms involving the second derivative are discarded and the gradient of the estimated output, , is called sensibility function. ( )yΘ∇ Θ

LM is an extension of the Gauss-Newton algorithm [3]. The idea is to modify Eq. (12) to:

(13) ( ) ( )[ ] ( )[ ]12

1ˆˆ ˆ

,

n Tk

J y R y

Iλ

−Θ Θ Θ=

⎡ ⎤∇ Θ ≈ ∇ Θ ∇ Θ +⎢ ⎥⎣ ⎦∑

where the inversion of the matrix is not performed explicitly, but solving by Singular Value Decomposition (SVD) the following expression:

( )[ ] ( )2 ˆ^ .TJ I JλΘ Θ∇ Θ + ΔΘ = ∇ Θ (14)

The addition of the term λ in Eq. (14) solves the problem of ill-conditioning of the matrix. The Levenberg-Marquardt algorithm can be interpreted in the following manner: for small values of it behaves like the Gauss-Newton algorithm, while for high values of λ it behaves like the steepest gradient algorithm.

I

λ

)J Θ by a parabolic function under the condition (retaining only

the 3 first Taylor series terms):

( )ΘLJ LΘ


RESULTS The identification of longitudinal stability

derivatives from flight test data were performed using a set of experimental flight data performed with a military aircraft, namely the Xavante (AT-26 FAB 4516), as illustrated in Fig. 3. Table 3 reports the geometric and inertial parameters of the aircraft. Figure 4 shows the elevator deflection used as input in the identification procedure. The flight test data acquisition system installed on the aircraft is composed by an inertial sensor, an anemometric boom, a GPS sensor, a surface position sensor; and a data acquisition and record unity.

Figure 3. Xavante AT-26 FAB 4516.

Table 3. Xavante’s geometric and inertial

parameters. Symbol Value Unit

0m 4210 kg yI 6500 2.kg m c 1.90 m b − m S 20.62 2m

Figure 4. Elevator deflection.

Table 4 gives the setup for LifeCycle used in

the identification tests.

Table 4. LifeCycle parameters used in experimental identification.

Parameter Search space Zα 5− 0 qZ 0 5 Mα 5− 0 qM 5− 0 eZδ 0 5 eMδ 10− 0

1xb 0.1− 0.1

2xb 0.1− 0.1

1yb 0.1− 0.1

2yb 0.1− 0.1 Population size 100 Number of iterations 100 Stage interval 5

Table 5 reports a set of identification results

obtained following the three different optimization strategies (as presented in Tab. 1). For those cases in which LM were used,

refers to an estimative of the relative standard deviation, calculated from the Fischer information matrix.

StdDev

For strategy #1, the behavior of the LC along the iterations can be observed in Fig. 5. The transitions due to its self-adaptation skills can be seen in Fig. 5(a). Figure 5(b) shows which heuristics is conducting the optimization process at a given iteration. Also for this strategy, Fig. 6 demonstrates the prediction capacity of the LC method.

As an observation for strategy #2, the LM algorithm was tested using two different initial guesses. The first and well-succeeded test is shown in 5. However, in the second test, starting from , the LM algorithm was able to find any solution, i.e., the algorithm finished without convergence. Even thougth x seems unusual, it demonstrates what happens in the case of an improper initial guess.

0 0=x

0 0=

For strategy #3, the behavior of the LM stage along the iterations can be observed in Fig. 8. Also for this strategy, Fig. 7 demonstrates the prediction capacity of the LC-LM approach.


(a) Angle of attack.

(b) Pitch angular rate.

(a) LC self-adaptation.

(b) LC stage’s performance.

Figure 5. LC evolution and performance. Figure 7. LC-LM identification results.

(a) Angle of attack.

(b) Pitch angular rate.

Figure 8. LM evolution.

CONCLUSION

This paper presented an identification

procedure to determine the longitudinal stability and control derivatives of a military aircraft. The tested inverse problem solvers are based on LC and LM methods. In the present contribution, GA and PSO represented the phases of the LC algorithm. Finally, the experimental investigation illustrated the possibility of using the present technique in real world environment. The results are very encouraging in the sense that more complex models, embracing non-linearities, will be analyzed in further research.

Figure 6. LC identification results.


Table 5. Identification results. LC LM LC-LM

Result Initial guess Result StdDev StdDev Result Zα 1.3033− 3.5312% − 3.53% 1.6042− 0.7013− 0.7013 qZ 6.7593% 0.2308 6.76% 0.3233 0.5121 0.2308 Mα 2.3040− 1.6707% 2.3695− 1.67% 4.2477− 2.3698− qM 1.9528% 1.1700− 1.95% 1.77− 2.6133− 1.1756−

eZδ 10.4686% 0.3842 10.47% 2.0564 0.7249 0.3841 eMδ 8.6597− 0.8962% 7.1643− 0.90% 7.0266− 7.1638−

1xb 16.1390% 0.0093− 16.14% 0.0206 0.0385 0.0093−

2xb 1.0783% 0.2037 1.078% 0.0899 0.1649 0.2037

1yb 0.8209% 0.0774 0.82% 0.0903 0.0741 0.0774

2yb 5.8122% 0.0182 5.81% 0.0857 0.1532 0.0182 J 82.2279 10−× 111.1746 10−× 111.1745 10−×

Iterations 100 8 7 Evaluations 10000 1104 966

ACKNOWLEDGMENTS

The authors are thankful to the Brazilian

Research Agencies, CNPq and FAPESP, for the Ph.D. and research scholarships. Dr. Valder Steffen also acknowledges CNPq for the partial financing (Proc. 470346/2006-0).

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