A quantum inspired genetic algorithm for multimodal optimization of wind disturbance alleviation flight control system

Qi BIAN, Brett NENER, Xinmin WANG

a School of Automobile, Chang’an University, Xi’an 710064, China

b Department of Electrical, Electronics, and Computer Engineering, The University of Western Australia, Crawley 6009,WA, Australia

c School of Automation, Northwestern Polytechnical University, Xi’an 710129, China

KEYWORDS

Abstract This paper develops a Quantum-inspired Genetic Algorithm (QGA) to find the sets of optimal parameters for the wind disturbance alleviation Flight Control System (FCS). To search the problem domain more evenly and uniformly,the lattice rule based stratification method is used to create new chromosomes. The chromosomes are coded and updated according to quantuminspired strategies. A niching method is used to ensure every chromosome can converge to its corresponding local minimum in the optimization process. A parallel archive system is adopted to monitor the chromosomes on-line and save all potential feasible solutions in the optimization process. An adaptive search strategy is used to gradually adjust the search domain of each niche to finally approach the local minima.The solutions found by the QGA are compared with some other Multimodal Optimization(MO)algorithms and are tested on the FCS of the Boeing 747 to demonstrate the effectiveness of the proposed algorithm.

1. Introduction

In the stage of cruising, aircraft are inevitably affected by atmospheric turbulence from every direction.1Especially for large transport aircrafts,the non-periodic changes of the aerodynamic forces on wings caused by the turbulence can not only increase the overload of the wings and shorten the fatigue life of the airframe structure,but also deteriorate the flight quality,which affects both ride quality and flight safety.2In order to reduce the influence on aircraft caused by atmospheric disturbance, it is necessary to take measures to alleviate the wingbend overload and the structure vibration. Good wind disturbance alleviation FCS that can maximally reduce the impact from gusts are indispensable for the aircraft design. Further,control saturation in the FCS is the topic of several research programs.Wang et al.3performed comprehensive controllability analysis to evaluate robust adaptive fault-tolerant control with actuator saturation. Qiao et al.4proposed two adaptive anti-saturation controllers for hypersonic vehicles to deal with input saturation and state constraints. Xia and Huo5studied an adaptive fault-tolerant controller for spacecraft rendezvous maneuvers with actuator faults and saturations.

To accomplish a well-designed FCS with qualified performances, control parameter tuning is usually tedious work for the engineers in the design process. Most of the traditional control parameter tuning methods are based on the gradient search strategy such as Gauss-Newton method,BFGS method and their variations.6While for a complex FCS with aerodynamic coupling channels, traditional methods may become ineffectual, especially for multimodal optimization problems where the solution space is a non-convex set with multiple minima. In the past decades, extensive studies in the aircraft optimization fields have been carried out.7Among these studies,the heuristic based optimization methods have become more and more popular and have been successfully applied in many aerospace engineering fields.8,9Li and Duan10proposed a simplified brain storm control parameter optimization algorithm for the F/A-18 automatic carrier landing system. Jia and Yang8developed a novel control parameter optimization method for the attitude tracking control problem of rigid spacecraft based on the genetic algorithm combined with a hyperbolic tangent function. Based on the hybrid neuro fuzzy and artificial bee colony optimization algorithm, Roy and Peyada11showed a new approach to the aerodynamic modeling and parameter estimation. Deng and Duan12presented a pigeon-inspired optimization algorithm to optimize the control parameters in the H-dot autopilot and the approach power compensation system. On this basis, Dou and Duan13studied the Le′vy flight based pigeon-inspired optimization algorithm for the automatic carrier landing system design. Most of the above-mentioned parameter optimization methods performed well and can find a satisfactory solution for the problem.However,most of the time,there is not only one final feasible solution that can satisfy the predefined constraints. Moreover,multiple solutions can also exist because of the non-convex properties of the optimization problem. Thus, it is necessary to explore the hidden properties of the solution domain and find as many of the feasible solutions as possible.

Based on the principles of quantum mechanics, the hybridizations of quantum computing and evolutionary optimization have been studied for many years and can be generally classified into three categories14:(A)the evolutionary-designed quantum algorithms;(B)the quantum evolutionary algorithms;and(C)the quantum-inspired evolutionary algorithms.In this paper, a quantum evolutionary algorithm is presented to not only tune the control parameters of the FCS but also find as many feasible solutions as possible,which can be used to provide more additional information for the FCS design.Taking advantage of the quantum behaviour,even a single chromosome in the QGA can be in many incompatible states at the same time,while it will collapse to a single state when its state is observed.15Such a mechanism can greatly increase the diversity of the genome and improve the search ability of the whole chromosome colony.To improve the search efficiency,new chromosomes will always be generated according to the integration lattice rule,16which can stratify the search space more evenly and uniformly for the search domain. Currently, for many MO algorithms, the number of final feasible solutions is confined by the total number of the searching particles,which means the maximum number of feasible solutions cannot exceed the population size of the whole colony.On the other hand,considering the computational time,the population size cannot grow limitlessly to enumerate all feasible solutions in a complex optimization problem.Thus,to deal with this problem,a parallel comparing and archiving strategy which is independent from the search group is adopted to compare the new solutions with the old ones on-line,and then save as many as possible of the potential feasible solutions.It is possible that there are many local minima in the problem domain;hence an adaptive search method is used to dynamically adjust the search domain of each niche,which can gradually improve the accuracy and approach the local minima in the optimization process.Finally,all of the discovered feasible solutions are analysed and tested on the FCS of the Boeing 747 which is designed for wind disturbance alleviation purposes to demonstrate the feasibility and effectiveness of the proposed algorithm.

The remainder of this paper is organised as follows. The aircraft model and the longitudinal FCS designed for wind disturbance alleviation are given in Section 2. The QGA based MO method is introduced in Section 3.The statistical analysis and flight simulation are presented in Section 4. Finally, conclusions are discussed in Section 5.

2.Longitudinal flight control system design for wind disturbance alleviation

2.1. DOF rigid model of the Boeing 747

The aircraft model used in this paper was derived from Ref.17,18and.18The model is described in the form of kinematic and dynamic equations which are written in body axes as follows.

The wind model used in this paper was derived from Ref.21.By using band-limited white noise passed through a specific digital filter, the wind signal with Dryden spectral representation is added to the aircraft model as the external turbulence in the simulation. Both the vertical Dryden Power Spectral Densities(PSD)and its realization of the transfer function in the splane are described as follows.

2.2. The longitudinal flight control system design for wind disturbance alleviation

Fig.1 Structure of the longitudinal flight control system.

For different aircraft with their corresponding performance requirements,the wind disturbance alleviation FCS can be carried out in various ways.The most traditional realization is by using stability augment systems with pitch rate and vertical rate feedback to achieve the goals.22The dynamic elements included in the normal digital up-and-away FCS mode of the Boeing 747 used in the cruise phase of the flight are presented in Fig.1.

3. Quantum inspired genetic optimization algorithm for flight control parameter tuning

3.1. Problem formulation

In order to make sure the performance of the FCS satisfies the cruising phase demands, a combination of performance requirements is given in Eq. (7). It can be found from Eq.(7) that the frequency-response criterion (J1), the timehistory envelope criterion (J2), the dynamic time-response criterion (J3) and the steady time-response criterion (J4and J5)are all taken into consideration in the optimization process.Because a lower Jcrepresents better performance of the FCS,the final goal is to find the feasible solutions to minimize Jcas much as possible by tuning the control gains ki（i=1, 2, ...,10）.

where J1is used to optimize the inner stability augment loop with the desired damping and natural frequency. fiare the sampled frequency points that represent the characteristics of the frequency-response curve. ΔG and ΔP are the gain and phase at frequency fi, respectively. J2is used to improve the handling qualities for the pilot so that the controllability curve of the output q at low speed and the output nzat high speed can be balanced.Furthermore both q and nzmust sit on a predefined envelope that satisfies the level-one flying qualities.23J3is used to modify the dynamic performances of the two control channels. PO, RT and ST represent the percentage of overshoot, rise time and settling time of the control channels,respectively. Both J4and J5are used to minimize the steady state errors of the five state variables and the energy costs in the elevator and the engine control channels, respectively.

3.2. Quantum inspired genetic algorithm

Even today,the development of quantum computing is still in its infancy and numerous related studies have been carried out in many engineering fields.24By taking advantage of quantummechanical behaviour,optimization methods can be developed which are very different to traditional methods and exhibit much better performance. In this paper, a QGA algorithm is presented to deal with the FCS parameter optimization problems and to explore the hidden properties of the control gains.

The QGA works as follows:

Step 1. Defined by a series of column vectors called Qbits,25each initial chromosome ci, i=1, 2, ..., n with d dimensions is created in the following ways.

where Z is the set of integers. Pnis a highly-uniform point set that covers［0, 1）dand later can be projected into the problem domain. ｛c1, c2, ... ,cn｝ are the n chromosomes created.｛v1, v2, ... ,vn｝∈Rdare linearly independent vectors over the real domain R. The ith chromosome cion the jth rank is created according to the direction vector vj: ci=jvjmod j,where each coordinate should be a multiple of 1/n. The vi,jshould follow the rules that nvi,j=a and a=（1, a, a2mod n）. Thus, for any ci, the projection Ld（ci）of Ldis also a lattice.

Step 2. In the search process, by independently observing all of the Q-bits of ci,the corresponding cost value can be calculated using Eq. (7) and assigned to ci. The observation process is described as follows.

where O is the observation operator and r～U（0, 1） is a random variable with uniform distribution. The chromosomes are observed bit-by-bit. Then all of the chromosomes are sorted by their cost values in ascending order and a tournament selection process is implemented to choose the parent chromosomes in the top ρ1percent of the list for quantum crossover and mutation operations described in Eqs. (13)and (14), respectively.27

where cmjand cnjare jth dimensional elements of the two selected chromosomes cmand cn, respectively. Both quantum crossover and mutation operations are carried out on each Q-bit. Then the newly created chromosomes are observed to give levels of their cost values and compared with their parents to reserve only the top ρ2percent chromosomes.

Step 3. The niching method is used to classify the updated chromosomes according to their cost values and Euclidean distances and is described as follows: (A) The reserved chromosomes are first sorted according to their cost values; (B)Then the local minimum centres are selected among all the chromosomes and should satisfy the following conditions: the selected centres should be in the top ρ3percent colony and the Euclidean distances σ1between every two centres should be not less than σ1. (C) All of the eligible centres at the current step are saved to a new archive group Bm=｛b1,b2,...,bm｝,where Bmis updated to eliminate those outdated chromosomes by comparing all of the chromosomes’ cost values and crowding distances. Only the top ρ4percent chromosomes whose crowding distances are larger than σ2can be kept in Bm.

On the other hand, for the search group Pn, all chromosomes are updated by applying the Q-gate which uses a quantum rotation operator G（θ） to adjust the Q-bits’ position in each ci. The update procedure is given as follows.

Step 4. It has been noted in Eq. (9) that all the chromosomes are coded in a binary system with k bits depth. However, the value of k can limit the precision of the search algorithm in a fixed-point number representation system, particularly if k is small. Thus, an adaptive search strategy is developed to adjust the search domain of each niche as follows:

Table 1 Lookup table of sg（）.

Table 1 Lookup table of sg（）.

αlij βlij glhj sg（αlij, βl ij）＞0 ＞0 1 1＞0 ＞0 0 -1＞0 ＜0 1 -1＞0 ＜0 0 1＜0 ＞0 1 -1＜0 ＞0 0 1＜0 ＜0 1 1＜0 ＜0 0 -1

Fig.2 Flowchart of the QGA.

Step 5. Finally, the algorithm stops searching if a predefined maximum iteration number Imaxis reached, or else, it goes back to Step 2 for continued searching. The flowchart of the QGA is presented in Fig.2.

4. Simulation and evaluation

4.1. Simulation settings

In order to test the performance of the proposed algorithm,the QGA is compared with three other MO algorithms including TODE,28NPSO29and MOMCA.30The parameter settings of QGA are given in Table 2. As for TODE, NPSO and MOMCA, the parameter settings are kept the same as presented in their corresponding references. Then the FCSdesigned for wind disturbance alleviation was tested and simulations were carried out many times under different wind conditions.All simulations are performed on a PC with Intel Core i7 processor of 3.6 GHz and 16 GB of RAM.

Table 2 Parameter settings of QGA.

4.2. Results on cost value

Fig.3 compares the convergence speeds of the four algorithms in the first 100 iterations of one simulation. It can be found that, except for MOMCA, all of the three algorithms QGA,TODE and NPSO show fast convergence speeds and enter into stable convergence states in the first 10 iterations. By taking advantage of the lattice stratification strategy, the QGA also has a better initial cost value than the others.

Fig.3 Comparison of cost values of four algorithms.

Then 50 simulations on each algorithm were carried out and statistical analyses of the final cost values are presented in Table 3. It can be found that both QGA and TODE have lower average cost values and 100 percent success rates.Moreover, the computation stability of the proposed algorithm is better than the others and has the lowest standard deviation.The cost value distribution of QGA is shown in Fig.4, from which it can be found that in the 50 simulations,the estimated probability density of the final cost values exhibit a normal distribution characteristic with low variation, which once again demonstrates the computation stability of the proposed algorithm.

The average computational time of the four algorithms are shown in Fig.5. It can be found that there are roughly linear positive correlations between the iterations and computational time for all of the algorithms. The proposed algorithm shows the shortest computational time in four algorithms to reach the same iterations. Besides, it can be found that the computational efficiency of the proposed algorithm is more obvious as the iterations increase.

Fig.4 Distribution and probability density of final cost values.

Fig.5 Average computational time of four algorithms.

Table 3 Comparison of the four algorithms in 50 simulations.

4.3. Results on multiple solutions

When one optimization is completed, all of the feasible solutions found by QGA are stored in Bm. Fig.6 shows the statistical analysis of the ten control gains in Bm. It can be found that the 1st,3rd,5th,6th and 10th control gains have relatively larger standard deviations than the rest, which means such control gains have wider selectable ranges when designing the FCS. The maximum and minimum values of each control gain also give suggested limits to tune these parameters.

In this paper, we not only want to acquire many feasible solutions for the FCS design, but also to explore the hidden properties of the solution space.In order to analyse a solution space with 10 dimensions, statistical data of the multiple solutions are presented in Figs.7 and 8.By sorting the cost values of the chromosomes, the final multiple solutions are clustered in 22 niches and are listed in ascending order in Fig.7. Fig.8 shows the niche size and dispersion of each niche,which represent the total number of chromosomes in each niche and the average Euclidean distances between the surrounding chromosomes and the niche centres, respectively. Figs. 7 and 8 illustrate that the niche size and the corresponding cost value distribution are decreasing with the centre cost value increasing, which means that more feasible solutions are located on the bottom of the solution space and several large niches exist there;whereas with the centre cost value increasing,only a few small niches exist on the upper side of the solution space where the feasible solutions are more difficult to find.

Fig.6 Statistical analysis of ten control gains.

Fig.7 Cost values analysis of solutions in 22 niches.

Fig.8 Sizes and dispersions of 22 niches.

Fig.9 Forward speed and vertical speed.

Fig.10 Pitch rate and pitch angle.

Fig.11 Vertical acceleration and vertical displacement.

Table 4 Standard deviation of each state variable.

4.4. Results from flight simulation

Finally, using the optimised control gains, the FCS is tested under three different wind conditions: turbulence intensities set to light, moderate and severe, separately. The deviations of the six state-variables around their equilibrium points in the simulations are shown in Figs.9-11,and the corresponding standard deviations, which represent the wind alleviation degrees, are given in Table 4. From Figs. 9-11 and Table 4 it can be found that the optimized FCS is working stably during the whole landing process and all state variables are confined within reasonable ranges in all wind conditions,demonstrating the feasibility of the proposed method.

5. Conclusions

In this paper,an effective QGA is developed for the FCS optimization problems in the field of wind disturbance alleviation.A niching method combined with a parallel archive system is used to explore every potential feasible solution in the optimization process. Meanwhile, an adaptive search strategy is adopted to gradually adjust the search domain and finally approach the local minima. Thus the proposed multimodal optimization algorithm can not only optimize the parameters of the FCS based on the predefined cost function,but also find as many as possible feasible solutions for the FCS design.The comparisons of the proposed algorithm against three other MO algorithms demonstrate its accuracy and computation stability. The flight simulation tests under different wind conditions verify the robustness of the optimized FCS and the effectiveness of the QGA.Moreover,by analyzing the final multiple solutions, the hidden properties of the problem domain have been explored and statistical data of the ten control gains are also presented as additional information for the FCS parameter tuning.For future research,the multi-objective based multimodal optimization problems should be studied and the relationships between different cost functions should also be clarified.