Nonlinear robustness analysis of flight control laws for highly augmented aircraft

ARTICLE IN PRESS

Control Engineering Practice 15 (2007) 655–662 www.elsevier.com/locate/conengprac

Nonlinear robustness analysis of flight control laws for highly augmented aircraft Prathyush P. Menon, Declan G. Bates, Ian Postlethwaite Control and Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK Received 18 October 2005; accepted 23 November 2006 Available online 2 February 2007

Abstract In this paper, a novel hybrid optimisation algorithm is applied to the problem of analysing the robustness of a nonlinear flight control law in the presence of multiple variations in key aircraft parameters over continuous regions of the flight envelope. The analysis employs a typical nonlinear clearance criterion used by the European aerospace industry together with a simulation model of a high performance aircraft with a full authority control law which is gain scheduled over the flight envelope. The hybrid algorithm incorporates global (differential evolution) and local (sequential quadratic programming) optimisation algorithms to improve convergence properties and reduce computational overheads. The proposed approach is shown to have the potential to significantly improve both the reliability and efficiency of the industrial flight clearance process. r 2006 Elsevier Ltd. All rights reserved. Keywords: Hybrid optimisation; Robustness analysis; Flight control; Nonlinear systems

1. Introduction Modern high performance aircraft is often designed to be naturally unstable or to have reduced natural stability margins due to performance reasons and, therefore, can only be flown by means of a flight control system which provides artificial stability. As the safety of the aircraft is dependent on the controller, it must be proven to the clearance authorities that the controller functions correctly throughout the specified flight envelope in all normal and various failure conditions, and in the presence of all possible parameter variations. In order to do this, industrial flight control engineers have defined a set of clearance criteria which express the various stability and performance requirements for highly augmented aircraft. In the flight clearance process, all violations of these clearance criteria must be found and the worst-case result for each criterion computed. Based on the results of the clearance process, flight restrictions are imposed where necessary. Typically, criteria covering both linear and Corresponding author. Tel.: +44 116 252 2874; fax: +44 116 252 2619.

E-mail addresses: [email protected] (P.P. Menon), [email protected] (D.G. Bates), [email protected] (I. Postlethwaite). 0967-0661/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2006.11.015

nonlinear stability, as well as various handling and performance requirements are used for the purpose of clearance. The clearance criteria can be grouped into four classes: (i) linear stability criteria, (ii) aircraft handling/ pilot induced oscillation (PIO) criteria, (iii) nonlinear stability criteria, and (iv) nonlinear handling criteria. In Gatley, Bates, Hayes, and Postlethwaite (2002), linear robustness analysis methods such as m and the n-gap metric were used to evaluate a linear clearance criterion. This paper, however, focuses on the evaluation of a nonlinear handling criterion, which is described in detail in the next section. More details of other clearance criteria and the industrial flight clearance process can be found in Fielding, Varga, Bennani, and Selier (2002). Note, however, that flight clearance as defined above is essentially a robustness analysis problem (more precisely, it corresponds to the problem of computing lower bounds on worst-case stability/performance robustness). The industrial flight clearance process is a very lengthy and expensive task, particularly for high performance aircraft, where many different combinations of flight parameters (e.g. large variations in mass, inertia, centre of gravity positions, highly nonlinear aerodynamics, aerodynamic tolerances, air data system tolerances,

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structural modes, failure cases, etc.) must be investigated over a large flight envelope—see Fig. 1. Faced with limited time and resources, the current flight clearance process employed by the European aerospace industry uses a gridding approach (Fielding et al., 2002). In this approach, various clearance criteria are evaluated for random samples (Monte-Carlo simulation) or for all combinations of the extreme points of the aircraft’s uncertain parameters at a number of individual flight conditions defined over a gridding of the aircraft’s flight envelope. Clearly, the effort involved in the resulting clearance assessment increases exponentially with the number of uncertain parameters. In addition, there is no guarantee that the worst violation of the clearance criterion has in fact been found, since (a) it is possible that the worst-case combination of uncertain parameters does not lie on the sampled or extreme points, and (b) only a relatively small number of points in the aircraft’s flight envelope can be checked. In this paper, an optimisation-based approach to the flight clearance problem is presented, which avoids the above shortcomings by allowing clearance criteria to be checked for the full range of possible aircraft parameters variations and over the continuous regions of the flight envelope. Previous efforts to apply optimisation methods to nonlinear flight clearance problems are described in Forssell (2003) and Forssell and Hyden (2003). That study applied global optimisation methods such as genetic algorithms (GA) and simulated annealing to a different flight clearance problem at isolated points in the flight envelope. While the results of this study were quite promising, the rather slow convergence of global optimisation methods to the true worst-case result meant that the computational complexity of the proposed approach would be prohibitive in practice. Hybridisation of global algorithms, as described in this paper, significantly improves computation times, so that the clearance of control laws over whole regions of the flight envelope becomes feasible. For a detailed statistical comparison of

the computational complexity of hybrid versus standard global optimisation methods for flight clearance, the reader is referred to Menon, Bates, and Postlethwaite (2005). Initial results of this study were first presented in Menon, Bates, and Postlethwaite (2006). The paper is organised as follows. Section 2 describes the aircraft simulation model, flight control law and clearance criterion used in this study. Section 3 discuss the formulation of the flight clearance problem as an optimisation problem, and describes the novel hybrid algorithm used to solve it. Section 4 summarises the clearance results obtained with the proposed approach, and Section 5 presents some conclusions. 2. Aircraft model and clearance problem 2.1. ADMIRE simulation model and flight control law The aircraft model used in this study is the ADMIRE (Aero-Data Model In a Research Environment) (Forssell, Hovmark, Hyden, & Johansson, 2001), a nonlinear, six degree of freedom simulation model developed by the Swedish Aeronautical Research Institute (FOI) using aerodata obtained from a generic single seated, single engine fighter aircraft with a delta-canard configuration. A schematic of the aircraft is shown in Fig. 2. ADMIRE is augmented with a full authority flight control system and includes engine dynamics and detailed nonlinear actuator models. The model includes a large number of uncertain aerodynamic, actuator, sensor and inertia parameters, whose values, within specified ranges, can be set by the user. The aircraft dynamics are modelled as a set of 12 first order coupled nonlinear differential equations, given as follows: _ ¼ f ðxðtÞ; uðtÞ; DÞ, xðtÞ

(1a)

yðtÞ ¼ hðxðtÞ; uðtÞÞ,

(1b)

7000 6000

Altitude (meter)

5000 WORST CASE FLIGHT CONDITION

4000 3000 2000 1000 0 0

0.2

0.4

0.6 0.8 Mach

Fig. 1. Analysed flight envelope.

1

1.2

1.4

Fig. 2. ADMIRE—aircraft model and control surfaces (not to scale).

ARTICLE IN PRESS P.P. Menon et al. / Control Engineering Practice 15 (2007) 655–662 Table 1 Aircraft model uncertain parameters

90

Parameter

Bound

70

Dmass

½0:1 þ 0:1

e

DI yy DC ma Mach Altitude

80

Description

Variation in aircraft mass from nominal (9100 kg) ½0:075 þ 0:075 Variation in position of centre of mass (m) ½0:05 þ 0:05 Uncertainty in pitching moment due to elevator deflection (1/rad) ½0:1 þ 0:1 Uncertainty in aircraft inertia about y-axis from nominal (81 000 kg m2 ) ½0:05 þ 0:05 Uncertainty in pitching moment due to AoA (1/rad) ½0:4; 0:5 Mach variation ½1000; 4000 Altitude variation (m)

where xðtÞ is the state vector with 12 components, i.e., velocity, angle-of-attack (AoA or a), sideslip angle, and angular rate, attitude and position vectors. D represents the uncertain aircraft parameters—Table 1 shows the uncertain parameters considered in this study. yðtÞ is the output vector, and uðtÞ is the control input vector, whose components are left and right canard, left and right inboard/outboard elevon, leading edge flap and rudder deflection angles, landing gear status (extract/retract), and vertical and horizontal thrust vectoring. The control input is determined by uðtÞ ¼ gðxðtÞ; yREF ðtÞÞ,

(2)

where gð; Þ is an industry standard flight control law, which is provided with the ADMIRE model, and yREF ðtÞ is the reference demand that consists of the pilot inputs, i.e. pitch stick demand, roll stick demand, rudder pedal demand, and thrust demand. Eqs. (1) and (2) together represent the closed loop dynamics of the aircraft with the flight control law in the loop. The augmented ADMIRE operational flight envelope is defined up to Mach 1.2 and altitude 6000 m (Forssell et al., 2001). Fig. 1 shows the full flight envelope of the ADMIRE and the boxed subset region represents the continuous region of flight envelope for which clearance results are presented in this paper. The longitudinal control law is gain scheduled with respect to Mach and altitude variations and is designed to ensure robust stability and handling performance over the entire flight envelope. The model also contains rate limiting and saturation blocks (Rundqwist, Stahl-Gunnarsson, & Enhagen, 1997), as well as nonlinear stick shaping elements in its forward path. 2.2. Nonlinear clearance criterion The clearance criterion considered in this study is the AoA limit exceedence criterion (Fielding et al., 2002; Forssell, 2003; Forssell & Hyden, 2003), which aims to assess the effectiveness of the AoA limiting scheme in the flight control system. For this criterion it is required to identify the flight cases where, for a defined pull-up

Pitch Stick (N)

Dxcg DC md

657

60 Rapid Pull 640N/Sec

50 40 30 20 10 0 0

1

2

3

4

5 6 7 Time (sec)

8

9

10

11

Fig. 3. Pitch stick pull command.

manoeuvre, the maximum overshoot occurs in AoA. The corresponding worst-case combination of uncertainties must also be computed. Fig. 3 shows the specified pitch stick command, which corresponds to a rapid pull in longitudinal stick to a defined level (80 N) at a 640 N/s stick rate with stick hold for 10 s. The analysis aims to estimate the clearance criterion (Fielding et al., 2002): amax ¼ maxðaðtÞÞ

for tp10 ðsÞ

(3)

for all possible combinations of aircraft parametric uncertainty. In this paper, numerical optimisation is used to find the flight condition and combination of real parametric uncertainties that gives the maximum value of the criterion defined in (3). The region of the flight envelope considered lies between Mach 0.4 and 0.5 and altitude 1000–4000 m. For the analysis, the aircraft is trimmed straight and level and once the trim is achieved, the pull up manoeuvre is applied and the cost function for the optimisation algorithm is given by Eq. (3). 3. Hybrid optimisation Global optimisation methods based on evolutionary principles are generally accepted as having a high probability of converging to the global or near global solution, if allowed to run for a long enough time with sufficient initial candidate solutions and reasonably correct probabilities for the evolutionary optimisation parameters. The rate of convergence, however, can be very slow, and moreover, there is still no guarantee of convergence to the true global solution. Local optimisation methods, on the other hand, can very rapidly find locally optimal solutions, but the quality of those solutions entirely depends on the starting point chosen for the optimisation routine. In order to try to extract the best from both schemes, several researchers have proposed combining the two approaches (Davis, 1991; Lobo & Goldberg, 1996; Yen, Liao, Randolph, & Lee, 1995). In such hybrid schemes there is the possibility of incorporating domain knowledge, which

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gives them an advantage over a pure blind search based on evolutionary principles. Moreover, during the course of finding the global solution hybrid algorithms provide other local solutions which, if they violate the clearance criterion, might also be of interest, and can be stored for further analysis.

Differential evolution (DE) is a relatively new global optimisation method, introduced by Storn and Price (1997). It belongs to the same class of evolutionary global optimisation techniques as GA (Davis, 1991), but unlike GA it does not require either a selection operator or a particular encoding scheme. Despite its apparent simplicity, the quality of the solutions computed using this approach has been claimed to be often better than that achieved using other evolutionary algorithms, both in terms of accuracy and computational overhead (Storn & Price, 1997). The DE method has recently been applied to several problems in different fields of engineering design, with promising results. In Lampinen and Zelinka (1999), for example, it was applied to find the optimal solution for a mechanical design example formulated as a mixed integer discrete continuous optimisation problem. In Rogalsky, Derksen, and Kocabiyik (2000), the DE method has been applied and compared with other local and global optimisation schemes in an aerodynamic shape optimisation problem for an aerofoil. The DE methodology consists of the following four main steps: (1) random initialisation, (2) mutation, (3) crossover, and (4) evaluation and selection. These steps will be described in detail in the sequel. Random initialisation: Like other evolutionary algorithms, DE works with a fixed number, N p , of potential solution vectors, initially generated at random according to i ¼ 1; 2; . . . ; N p ,

(4)

where xU and xL are the upper and lower bounds of the parameters of the solution vector and ri is a vector of random numbers in the range ½0 1. N p is fixed at 25 in the current study. Each xi consists of elements ðx1i ; x2i ; . . . ; xdi Þ. The fitness of each of these N p solution vectors is evaluated using the cost function given in Eq. (3). Mutation: The scaled difference vector F m Dij between two random solution vectors xi and xj is added to another randomly selected solution vector xk to generate the new mutated solution vector x¯ Gþ1 , i.e., n ¼ xG x¯ Gþ1 n k þ F m Dij ;

G Dij ¼ xG i  xj ,

Global Solution

FmDij Dij x kG

3.1. Differential evolution

xi ¼ xL þ ri ðxU  xL Þ;

x2

(5)

where F m is the mutation scale factor, a real valued number in the range ½0; 1, (fixed at 0.8 in this study), and G represents the iteration number. Fig. 4 shows a simple two dimensional example of the mutation operation used in the DE scheme. The difference vector Dij determines the search direction and F m determines the step size in that direction

x- G+1 n Dij x Gi x Gi x1 Fig. 4. DE mutation strategy.

from the point xk . For more details, the reader is referred to Menon et al. (2005), Storn and Price (1997). Crossover: During crossover, each element of the nth solution vector of the new iteration, xGþ1 , is reproduced from the mutant vector x¯ Gþ1 and a chosen parent n individual xG n as given in Eq. (6), where j ¼ 1; . . . ; d and n ¼ 1; . . . ; N p . Note that x¯ Gþ1 has elements ðx¯ Gþ1 n 1n ; Gþ1 Gþ1 G G G x¯ 2n ; . . . ; x¯ dn Þ and xn has elements ðxG 1n ; x2n ; . . . ; xdn Þ. ( G xjn if a generated random number4rc ; xGþ1 ¼ Gþ1 jn x¯ jn otherwise; (6) rc 2 ½0; 1 is the crossover factor, which is fixed at 0.8 in the present study. Evaluation and selection: After crossover, the fitness of the new candidate xnGþ1 is evaluated using the cost function in Eq. (3). If the new candidate xGþ1 has a better fitness n Gþ1 than the parent candidate xG , then x is selected to n n become part of the next iteration. Otherwise xG n is selected and subsequently identified as xGþ1 . n Termination criterion: Many different termination criteria can be employed. In the present study, an adaptive termination criterion is used that is dependent on improvement in the solution accuracy over a finite number of successive generations. The algorithm terminates the search if there is no improvement on the best solution achieved (above a defined accuracy level, here chosen as 106 ) for 20 successive iterations. 3.2. Hybrid DE In Rogalsky and Derksen (2000), the conventional DE methodology was augmented by combining it with a downhill simplex local optimisation scheme. This hybrid scheme was applied to an aerofoil shape optimisation problem and was found to significantly improve the convergence properties of the method. At each iteration,

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local optimisation was applied to the best individual in a current random set. In this study, a slightly different hybridisation scheme is used for the analysis problem. The local optimisation used in the present study is a method based on sequential quadratic programming (SQP). In particular, the implementation provided in Optimization Toolbox User’s Guide, Version 2 (2000) (the function ‘‘fmincon’’) was used to find the constrained minimum of a scalar function of several variables starting from some initial estimate. A medium scale optimisation scheme is chosen where the gradients are estimated by the function itself using the finite difference method. The hybrid algorithm begins with a set of random initial solution vectors in the search space. To exploit the capability of global optimisation methods to escape from local optima, only the DE algorithm is allowed to run for an initial number of iterations, fixed at 12 in the present study. The improvement in best fitness obtained is updated at each iteration. After the first 12 iterations, if at any iteration the DE provides no improvement in best fitness the local optimisation scheme is selected. Note that in this scheme a random solution vector from the current solution set is chosen as the initial guess for the local optimisation scheme. This is in contrast to the algorithm proposed in Rogalsky and Derksen (2000) which used the best current solution vector as the initial guess. Interestingly, significantly better results have been observed in terms of convergence to the global solution by using a random solution from the current set as the initial guess for the local algorithm. The explanation for the success of this seemingly counter-intuitive strategy seems to be that by randomly picking an individual for subsequent local improvement, greater diversity is maintained in the set of candidate solutions. This provides a better balance between the local and global optimisation strategies and hence Table 2 Hybrid differential evolution algorithm—pseudocode (1) Initialise random candidate solutions in search space (2) Evaluate fitness of each solution and choose best fitness (3) Apply DE for a fixed number of initial iterations; update best fitness value in each iteration (4) While termination criteria not satisfied (a) Calculate the improvement in best fitness (i) If Improvement in best fitness (ii) Continue DE (iii) else (iv) Choose a random solution from current set, say X 0 (v) Apply local optimisation with X 0 as initial point (termination occurs when exceeding the defined maximum number of function evaluations) (vi) If Improvement in best fitness (vii) Replace X 0 with the new solution (viii) else (ix) Keep X 0 in the set (x) end (xi) end (5) end of While

659

reduces the chance of obtaining sub-optimal solutions. When the local scheme is chosen, the optimisation starts from the given initial condition and continues until it either converges or reaches a defined maximum number of cost function evaluations. If there is an improvement in the quality of the solution on termination of the local optimisation algorithm, the solution obtained from the local optimisation is used to replace the initial solution in the current set of solution vectors. The algorithm is simple, and tries to search for the global optimum in a ‘‘greedy’’ way, demanding improvement in the achieved optimum value at each iteration. The termination criterion for the hybrid algorithm is as previously defined, i.e. the algorithm terminates when the improvement in best fitness is less than 1e6 successively for a certain number of iterations, here fixed at 20. A pseudocode for the hybrid DE algorithm is given in Table 2. 4. Flight clearance results A continuous region of the flight envelope, with Mach varying from 0.4 to 0.5 and altitude varying from 1000 to 4000 m, was analysed in this study. In the simulation model, care was taken to update the scope of each variable globally, when the different flight conditions are considered. As simultaneous variations of two aerodynamic uncertainties DCmde and DCma are considered, the actual allowable uncertainty bounds for these parameters are scaled down by a factor of 62% in accordance with current industrial practice (Forssell et al., 2001). The optimisation begins with 25 randomly generated values for the flight conditions and associated uncertain parameters, to give a set of candidate solution vectors. About each of these flight conditions, the aircraft is trimmed straight and level. Once trim is achieved, the model is simulated for 10 s with the pull up manoeuvre defined in Section 2.1 and the objective function is evaluated using Eq. (3). To investigate the consistency and repeatability of the results, 12 different trials with the HDE algorithm (each with different randomly generated initial solution vectors) were conducted. The optimisation algorithm performance over these trials is shown in Fig. 5. From Fig. 5 it is observed that the algorithm visits other potential local solution points during the course of tracking the global solution. Figs. 6–8 show the worst-case values of Mach, altitude, a, and D computed in each of the 12 different trials, together with the required number of simulations. The statistics of the results from the different trials are given in Table 3. The trial which required the maximum number of simulations (third trial in Fig. 7) took 2 h and 50 min on a Pentium 4 machine. The results of all trials are extremely consistent, providing a high level of confidence that the true worst-case values of all parameters are in fact being identified. Note also that the worst-case values of the aerodynamic uncertainties DCmde and DCma are not on their bounds, and thus would not have been found by a standard grid-based analysis. For these parameters, the standard

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660 75

ΔXcg

0.05 0 −0.05

Δmass

α(t) (deg)

70

65

yy

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

ΔI

0

Number of Simulations

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6 8 Number of Trials

10

12

0.1 0.05 0 −0.05 −0.1

60

55

0

0.1 0.05 0 −0.05 −0.1

Fig. 5. Optimisation performance.

Fig. 7. Trail results—configuration uncertainties.

Mach

0.5 0.45 2

4

6

8

10

12

0

2

4

6

8

10

12

ΔCmα

80 70 60 50 0

No. of Simulations

ΔCmδe

0 4000 3000 2000 1000

α(t) (deg)

Altitude

0.4

2

4

6

8

10

12

0.03 0.02 0.01 0 −0.01 −0.02 −0.03

2

4

6 8 Number of Trials

10

12

Fig. 6. Trial results—Mach, altitude, aðtÞ, and no. of simulations.

deviation value is also slightly higher when compared to that of other parameters. This issue was investigated by doing a single parameter sensitivity analysis with the other parameter values fixed at their worst-case values. The results are shown in Figs. 9 and 10. From these figures it can be observed that a range of DCma and DCmde values provide the maximum value of a, i.e. DCma values from 0 to 0:02 and DCmde values from 0.02 to 0.04 all return almost exactly the same value of the cost function. This explains the slightly larger variation in the values of these parameters over different trials. Note also the highly nonlinear dependence of a on the values of these parameters over their full range of variation. To further evaluate the performance of the DE algorithm, 12 trials of the local optimisation scheme

2

4

0

2

4

6

8

10

12

6 8 Number of Trials

10

12

0.03 0.02 0.01 0 −0.01 −0.02 −0.03

4000 2000 0 0

0

Fig. 8. Trail results—aerodynamic uncertainties.

Table 3 Worst-case parameter values Parameter

Maximum

Minimum

Std. Dev

Mach Altitude DXcg Dmass DI yy DCmde DCma aðtÞ No. of simulations

0.4531 3045 +0.075 +0.1 0.1 +0.0278 +0.0138 74.1223 5141

0.4504 3027 +0.0728 +0.0988 0.0997 +0.0151 +0.0015 74.0851 2827

6.9582e4 5 6.571e4 4.036e4 1.0184e4 3.4e3 3.1e3 0.0108 692.95

‘‘fmincon’’ with a set of initial points generated by randomly sampling the parameter space were undertaken. To ensure a fair comparison the local search with random

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76 74

α(t) (deg)

72

1 2 3 4 5 6 7 8 9 10 11 12

70 68 66 64 −0.04 −0.03 −0.02 −0.01

0 ΔCmδe

0.01

0.02

0.03

0.04

Fig. 9. DCmde sensitivities.

661

approach, it can be argued that the greater number of simulations is a responsible price to be paid for securing far more reliable clearance results, since now continuous regions as opposed to isolated points in the flight envelope can now be cleared. Secondly, the optimisationbased approach finds a higher value of AoA (74:1 ) than the gridding method (71:5 ). The fact that the true worstcase value of AoA occurs in the interior of the flight envelope ‘‘box’’, as shown in Fig. 1, shows clearly the limitations of the standard gridding approach. Thirdly, for a true comparison, the number of simulations required by the gridding approach should be multiplied by the number of grid points chosen within a particular region of the flight envelope. Finally, the number of simulations required by the gridding approach will of course increase exponentially with the number of uncertain parameters considered—note that a total of 20 uncertain parameters in longitudinal and lateral/directional axes are defined in Fielding et al. (2002).

75

5. Conclusions 70

α(t) (deg)

65

60

55

50

1 2 3 4 5 6 7 8 9 10 11 12

45 −0.04 −0.03 −0.02 −0.01

0 ΔCmα

0.01

0.02

0.03

0.04

Fig. 10. DCma sensitivities.

initial guesses was allowed to run for 3 h, slightly longer than the maximum time required for the HDE algorithm over the 12 trials. The maximum value of AoA obtained over the 12 trials of the local algorithm with random initial guesses was 65:0081 , which is almost 10 lower than the maximum value found by the HDE algorithm. For the purposes of further comparison, an analysis based on the standard gridding approach was carried out at the vertices of the flight envelope region considered. The region represented by Mach ½0:4; 0:5 and altitude ½1000; 4000 is thus considered as a single box obtained from a gridding of the overall flight envelope. The maximum cost function value obtained by checking all extreme points of the uncertain parameters at the corners of this box was 71:5 . Though this approach took significantly fewer simulations (128) than that required by the optimisation-based

In this study, a hybrid global/local optimisation algorithm was developed and applied to the problem of analysing the robustness of a nonlinear flight control law over continuous regions of the flight envelope. The analysis employs a typical nonlinear clearance criterion used by the European aerospace industry together with a simulation model of a high performance aircraft with a full authority control law which is gain scheduled over the flight envelope. The flexibility and efficiency of hybridised global optimisation methods was exploited to allow for clearance of the flight control law over continuous regions of the flight envelope in the presence of multiple continuous variations in key aircraft parameters. The proposed approach avoids the principal limitations of the current industrial flight clearance process, namely that only a small number of random samples or combinations of extreme variations in uncertain parameters are checked over a gridding of isolated points in the aircraft flight envelope. It should therefore offer the potential to significantly improve both the reliability and efficiency of the industrial flight clearance process for highly augmented aircraft.

Acknowledgements This work was carried out under EPSRC research Grant GR/S61874/01. The authors are grateful to the industrial and academic members of GARTEUR FM-AG11 Action Group on ‘‘New Analysis Techniques for the Clearance of Flight Control Laws’’, and GARTEUR FM-AG17 Action Group on ‘‘Nonlinear Analysis and Synthesis Techniques for Aircraft Control’’ for many helpful discussions during the course of this research.

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