LU Yao,SUN You,LIU Xiaodong,and GAO Bo

1.Beijing Aerospace Automatic Control Institute,Beijing 100854,China;2.National Key Laboratory of Science and Technology on Aerospace Intelligence Control,Beijing 100854,China

Abstract:Aiming at tracking control of a class of innovative control effector(ICE)aircraft with distributed arrays of actuators,this paper proposes a control allocation scheme based on the L′evy flight.Different from the conventional aircraft control allocation problem,the particular characteristic of actuators makes the actuator control command totally subject to integer constraints.In order to tackle this problem, first,the control allocation problem is described as an integer programming problem with two desired objectives.Then considering the requirement of real-time,a metaheuristic algorithm based on the L′evy flight is introduced to tackling this problem.In order to improve the searching efficiency,several targeted and heuristic strategies including variable step length and inherited population initialization according to feedback and so on are designed.Moreover,to prevent the incertitude of the metaheuristic algorithm and ensure the flight stability,a guaranteed control strategy is designed.Finally,a time-varying simulation model is introduced to verifying the effectiveness of the proposed scheme.The contrastive simulation results indicate that the proposed scheme achieves superior tracking performance with appropriate actuator dynamics and computational time,and the improvements for efficiency are active and the parameter settings are reasonable.

Key words: flight control,control allocation,optimization,Le′vy flight,tracking control.

1.Introduction

Many advanced aircraft utilize multiple and redundant effectors to tackle the requirements for control and fault tolerance capability.Thus,an appropriate control allocation scheme is necessary and important for such aircraft to map virtual control commands into actuator control commands.The general control allocation problems of over-actuated mechanical systems and flight control systems have been studied and summarized in plenty of studies[1–5].To solve the problems,many optimization methods,such as the redistributed pseudo-inverse method,daisy chaining method,direct allocation method,simplex method,dynamic optimum-seeking method,linear programming(LP)and quadratic programming(QP)method,have been proposed.Most of the time,these methods are evaluated to be effective in solving the control allocation problems of the aircraft containing conventional effectors,such as rudders,ailerons and elevators.

The characteristic of deployed actuators plays an important role in achieving desired control commands.With the development of technology,many aircraft use heterogeneous actuators for superior stabilization and maneuver control.Different from the conventional effectors,the control models of some heterogeneous actuators,like reaction control system(RCS),are discontinuous.Deploying heterogeneous actuators,a class of morphing aircraft,called innovative control effector(ICE)aircraft,was first presented and investigated by Lockheed Martin Tactical Aircraft Systems[6].The effectors of ICE aircraft are four distributed arrays each of which contains hundreds of shapechange actuators.There are only two states for each actuator,open or closed,i.e.,each on/off-type actuator can merely supply either full or empty control effort(force and moment).The flight control system produces desired control efforts through allocating states of each actuator.Many studies[7–10]proposed control methods for the lateraldirectional dynamics of ICE aircraft.In[7],an adaptive control method considering actuator failure was proposed and it achieved desired system performance.In[8],a faulttolerant control allocation scheme was proposed and it could reject actuator faults and model uncertainties.In[9],a probabilistic robust controller was proposed for achieving the tracking control in the presence of uncertain actuator failures.In[10],an adaptive control scheme with control allocation through updating the number of operating actuators was proposed for ICE aircraft subject to uncertain time-varying uncertainties.Considering the large quantity of actuators,it is unrealistic to regard every actua-tor as a control variable.Therefore,in these studies the actuators are divided into four groups.This simplification of actuator model makes a certain difference between the mathematical model and the practical physical model and reduces the difficulty of getting actuator commands.Obviously,more meticulous division of actuators can make the actuator mathematical model closer to the practical physical model.However,more meticulous division of actuators will increase the number of control variables.Moreover,these studies assume that the active actuators within the same group have no differences in terms of generating force/moments.According to these assumptions,it can be found that each control variable must be an integer.This means the control allocation of actuators can be described as an integer programming problem.

The integer programming problem is usually an NP-hard problem.Some methods,such as the branch-bound method and the cutting plane method,are usually used to solve the precise optimum solution of the integer programming problem.However,the amount of computation of such conventional methods will quickly increase as the scale of problem expands,which will influence the realtime performance of the flight control system.Meanwhile,the design of the solving method is influenced by the form and number of constraints,which influences the universality of the method for different integer programming problems.Besides the polynomial algorithms and other exact methods,the metaheuristic algorithms have also been widely utilized to solve the integer programming problems in many engineering fields[11–15].In[11],a genetic algorithm based heuristic method was designed for a hybrid system with manufacturing and remanufacturing.In[12],a variable neighborhood search metaheuristic algorithm was presented for the production routing problem.In[13],two different greedy heuristic methods including a populationbased iterated greedy algorithm were proposed for the socalled weighted independent domination problem which is an NP-hard combinatorial optimization problem in graphs.In[14],the particle swarm optimization method was introduced to settling the problem of scheduling energy harvesting roadside units in vehicular ad-hoc networks.In[15],a novel simulated annealing algorithm embedding a mathematical model with an adjustment heuristic was developed for a parallel machine scheduling with time-dependent deterioration and multiple rate-modifying activities.According to the investigations from[16,17],sometimes the metaheuristic algorithms can achieve better solutions in comparison with the heuristic methods.The characteristic of the metaheuristic algorithm is that its amount of computation is mainly influenced by the parameter settings,such as the number of individuals,number of iterations and so on.Therefore,it is easy to control the computational time of algorithm via suitable parameter settings,which makes metaheuristic algorithms appropriate to those problems requiring strict real-time performance rather than a precise optimum solution.

Considering the advantages of metaheuristic algorithms and the strict real-time requirement of flight control,the studies[18,19]utilized a metaheuristic algorithm called cuckoo search algorithm(CSA)to solve the control allocation problem of ICE aircraft.In[18],the adaptive detection probability and amplification factor were designed to enhance the search capability of CSA.In[19],a heuristic population initialization method was designed to improve the quality of the initial population and the search efficiency.These two studies achieved certain effects but their schemes were lacking in the consideration of the contact between neighboring searches and the energy consumption of actuators.Moreover,the population initialization method in[19]and the creation of step length in[18]were not optimized enough.Therefore,further investigations that use metaheuristic algorithms to solve the control allocation problem of ICE aircraft are required.

In this paper,a control allocation scheme is proposed for improving the tracking performance of the flight control system of ICE aircraft.First,the lateral-directional dynamics of ICE aircraft is introduced and the control allocation problem is described as an optimization problem with integer constraints.Next,a scheme based on L′evy flight is proposed for tackling the problem and some components such as population initialization,step length with a form of feedback and so on are designed for efficiency.Following that,a guaranteed solution is designed to ensure the flight stability in case the quality of the searched solution is inferior.Then,the contrast simulations are carried out under a time-varying model of ICE aircraft and the results are discussed.Finally,some conclusions are drawn.

2.Problem formulation

2.1System model description

The wing span of ICE aircraft[9]is illustrated in Fig.1.There are four distributed effector arrays, the uppersurface leading-edge(ULE)array,lower-surface trailingedge(LTE)array,upper-surface trailing-edge(UTE)array,and upper-surface wingtip(UTip)array,on each wing.There are totally 156 actuators on two wings,78 on each wing.The four distributed arrays include 10,22,22 and 24 actuators,respectively.Each actuator can be turned“on”or“off”to generate full or no control effort.

The lateral-directional dynamics of ICE aircraft can be described as follows:

wherex=[Vpr?]T∈R4represents the state vector;V,p,rand?denote the body-axis lateral velocity,roll rate,yaw rate and roll angle,respectively.Assume that all state variables are measurable.A(t)∈R4×4andBv(t)∈R4×2are the system matrices and their Frobenius norms are bounded and the system is controllable.Bvis the full column rank.v(t)=[v1(t)v2(t)]T∈R2represents the ideal control input.

Fig.1Con figuration of actuator arrays of ICE aircraft

The control objective is to make the system(1)track the following desired reference model:

whereAm∈R4×4andBm∈R4×1are the predefined constant system matrix and vector of the reference,andc(t)is the reference input signal.Lete=x?xmrepresent the tracking error.Assume that the ideal control inputv(t)can achieve the following target:

This indicates the dynamics of tracking error is asymptotically stable,and the providedAmis a Hurwitz matrix.

2.2Control allocation problem

The ideal control inputv(t)is achieved via the 156 156 actuators distributed on two wings. Drawing lessons from the assumptions in[9],all actuators are divided into eight groups and the following assumptions are made:(i)the control effects of the actuators that belong to the same group are equal;(ii)the actuators on the left wing generate negative control effort and those on the right wing generate positive control effort.The command of actuators is described byn=[n1,...,n8]T.According to the practical situation,nis constrained by the following sets:

whereZ andZ represent the upper and lower boundary ofni,respectively.It is obvious thatnis an integer vector.The control model of actuators can be described as follows:

where(t)∈R2represents the actual control input produced via the actuators,andBu∈R2×8is the mapping matrix which is a full row rank.

Remark 1Compared with[7–10,18,19],the actuators are divided into eight groups in this paper.This modeling is closer to the practical model and can enhance the control accuracy.However,this modeling also increases the number of control variables.

In terms of the above model, the control allocation problem of ICE aircraft can be described as obtaining appropriate actuator commandnto ensure the tracking performance of the closed-loop system.Unfortunately,due to the integer constraints onn,it is difficult to find a solving method from the existing literature investigating aircraft control allocation,and an error between the ideal control inputvand actual control inputvalmost always occurs.In[19],it was concluded that the tracking error of system was related to this error.Then the primary objective of the control allocation problem of ICE aircraft can be represented as the following optimization problem:

where?·?2denotes 2-norm.In addition,another noteworthy objective is to minimize the wear and tear of actuators.In an optimization framework, this secondary objective can be reflected by adding a cost index into(6):

wherenpis the preferred value ofnand it is represented by the last actuator control command in this paper,andψ>0 is a coefficient to reflect the weight between the primary and secondary objectives.

Due to the numerous constraints,the convex quadratic integer programming problem(7)is difficult to be exactly solved within a strict time limit.To solve the optimization problem(7),in this paper,a metaheuristic algorithm with some heuristic ideas is proposed for searching the superior optimum solutionn.Then a cost functionJ(v,n,np)is formulated to assist the searching according to(7):

The value of the cost function will be called fitness value in later chapters.

3.Design of control allocation scheme

Generally,the searching process of a metaheuristic algorithm,the main components of which are the population scale,iteration,renewal manner of the individual and mutation factor,includes two phases:global search phase and local search phase.The strategies and parameter settings of the algorithm should be assorted with the characteristics of different phases.

3.1Renewal manner of the individual

In this paper,the L′evy flight is introduced to updating the individuals.The L′evy flight process[20,21]which has already been identified in a number of animals and insects is a random walk in which the increments are distributed according to a heavy-tailed probability distribution.Some studies[22,23]verified that this process is an optimum search pattern for the case of“no preferred direction”and non-replenishable targets at unknown positions.As an important content,the L′evy flight is introduced to many metaheuristic algorithms and a typical representative of them is the CSA which has been applied to many engineering fields[24].Letdenote thejth solution in thetth generation of the population.The new solutionupdated via Le′vy flight can be described as follows:

whereα∈R8represents the step length vector,?denotes Hadamard multiplication operator,round(·)denotes the rounding operation,L(λ)=[l1(λ),...,l8(λ)]Tis the random step length andli(λ)follows the L′evy distribution[23]:which has an infinite variance with an infinite mean.Hereλis a L′evy flight parameter which influences the magnitude of the random step length.Table 1 shows the statistic|l(λ)|obtained via 10 000 random walks under differentλwith the restriction|l(λ)|?50.

Table 1Statistics of|l(λ)|under different λ

According to the statistics,assuming that the parameterα=[1,...,1]T,the appropriate interval of the parameterλis[2.3,2.8]in the global search which needs a larger random step length,and the appropriate interval of the parameterλis[1.8,2.3)in the local search which needs a smaller random step length.The search mode is determined by the comparison betweenJ()and a parameterη:ifJ()>η,then the search mode is regarded as a global search andλis chosen asλg;ifJ()?η,then the search mode is regarded as a local search andλis chosen asλl.

The design of parametersαandλshould be coordinated with each other.The parameterαinfluences the change ofto.It should be noticed that the influences onvcreated by different actuator groups are different.Considering this,a feedback system is established for designing the parameterα.Let

According to the comparison betweenBiand Δv,considering the above-mentioned magnitude of the random step length under different selections ofλ,the following rule forαiis established:

Under this rule,the matching parametersλgandλlare chosen asλg=2.5 andλl=2.0,respectively.It should be noted that this rule is simple and eclectic. Obviously, the rule can be established more perfect via more meticulous analysis, like the branch-bound method or the cutting plane method.However,that will increase the computation load and take up more storage space of the onboard software.Therefore,a degree of trade-off is unavoidable.

Remark 2Lu et al.[18]also designed an adaptive rule of the step length vector.Compared with the rule in[18],the rule(11)considers the mapping matrixBu,i.e.,the possible control effect of each group.This is conducive to the search directions of individuals and improves the search efficiency.

3.2Population initialization

A high-quality initial population is propitious to get a final superior solution.In[19],a population initialization method was proposed,but it is unstable as the problem scale expands and the energy wear of actuators is omitted.Therefore,an improved population initialization strategy needs to be designed.

Considering the secondary objective in(7),the last actuator control inputnpcan be selected as a part of the current initial population because this can ensure that the second half of the cost function(8)is zero.Letσdenote the proportion of the number ofnpto the number of individuals in population,?=J(v(t?1),np,np(t?1))denote the fitness value of the best searched solution in the(t?1)th generation.Considering thatBuis constant,it can be inferred that if both?v(t)?v(t?1)?2and?are small,npis probably a superior solution in the current period.Under this circumstance,the selected value ofσshould be larger;contrarily, the selected value ofσshould be smaller.Based on this,the parameterσis designed as follows:

where?>0 is a predetermined criterion.LetNrepresent the number of individuals of population;the final number ofnpin the initial population is fix(σN)where fix(·)denotes rounding the element to the nearest integer towards zero.

The other initial individuals the number of which is represented byMare created via the Tent map.The Tent map[25]which is related to Devaney chaos can stably create qualified individuals from the possible space according to the following formula:

where?denotes a random number within the interval of(0,1).When initializing population,a random qualified individualnis created and mapped into the interval of chaotic variables viaAfter this,2Miterations are carried out via(13)and then reduce the chaotic sequence into the constraints(4)viaFollowing that,calculate the fitness values of all 2Malternatives and selectMsuperior individuals into the initial population.Then all the initial individuals are determined.

Remark 3Compared with the population initialization method designed in[19],the proposed strategy considers the connection between the present ideal control inputv(t)and the last ideal control inputv(t?1),and then creates the initial population according to the quality of the last searched optimum individual.Moreover,all the initial individuals created via the proposed strategy must be within the constrained sets(4).However,the initial individuals created via the method in[19]are likely to go beyond the boundaries of the constrained sets.

3.3Individual qualification

In the searching process,some generated individuals may escape the constraints(4).To tackle this problem,when a new individualis generated via(9),the following measure is adopted:

Then the updated individualis fully qualified.

3.4Other rules

Besides the above strategies,there are some other rules used to improve the searching efficiency.First,to avoid the algorithm falling into local optimum,a targeted rule similar to mutation operation is adopted:if the quality of one individual cannot be improved in five iterations,this individual will be abandoned and replaced with a new individual.Second,to save the searching time,the following rule is adopted:if the fitness value of one individual is smaller than the criterion parameterχ,then stop the searching process and employ this individual as the final searched optimum solution.

3.5Guaranteed solution of the control allocation scheme

The general shortcoming of metaheuristic algorithms is that the performance of the final searched solution cannot be ensured,i.e.,the quality of the final searched solution is unstable.For a flight control system,the stability of the closed-system must be absolutely ensured.Considering this,the following guaranteed solution is used:

where(·)+denotes pseudo-inverse.θis a real-valued con-trol effect vector and an exact solution becauseBuis a full row rank matrix.Then the actual actuator commands can be obtained via rounding.This is an obvious and natural approach which could be somewhat similar to the method in[7]and it could be straightforward to implement.Letθ=round().Through appropriate design of the ideal control inputv(t)and deployment of actuators,the control capability can be enough to ensureθmeeting the constraints(4).Then addθinto the initial population as the guaranteed solution.

According to the above-mentioned strategies,the control system for ICE aircraft can be illustrated in Fig.2.

Fig.2Control system structure of ICE aircraft

3.6Stability analysis

Generally,due to the constraints(6),an error almost always occurs betweenv(t)andv(t).In terms of(3),it can be concluded that ifv(t)acts on the system,there exists a global Lyapunov functionW=eTPein the closed-loop system making˙W=?eTQe,wherePis a constant positive definite symmetric matrix and?Q=?(ATmP+PAm)is a corresponding negative definite matrix.Letδ=v(t)?v(t).The dynamics of the tracking errorecan be described as follows whenv(t)acts on the system:

Consider the alternate Lyapunov functionW=eTPe.In terms of(16)the time derivative ofWcan be written as

Letnbestrepresent the final searched solution.Becauseθis an individual in the initial population,and thereforeJ(nbest)?J(θ),i.e.,

Let Δθ=θ?θ.Then the following inequality yields

It is not hard to see that when0.This implies that the tracking erroreis bounded.

3.7Specific searching process

For an individual,its evolutionary process in one period can be illustrated in Fig.3 wheretmaxrepresents the maximum of iterations.After all individuals are finished to be updated,the final searched optimum solutionnbestwhich has the best fitness value within the whole searching process is obtained.Then compare(nbest)with(θ)and the final actuator control command is determined.

3.8Time complexity of algorithm

The time complexity of a metaheuristic algorithm is an important index reflecting the real-time performance of the algorithm.Especially for the control allocation of a flight vehicle which needs on-line calculations,it influences the practicality of the algorithm severely.Generally,the time complexity of a metaheuristic algorithm is evaluated via the number of the comparisons of the fitness values between two individuals,or the maximum number of itera-tions[26].It can be described asO(f(σ1,σ2,...)),wheref(·)represents an auxiliary function,andσ1,σ2,...represent the factors influencing the computational time of the algorithm.LetNrepresent the number of individuals in population.Referring to Fig.3,considering the possible most complex iterative process,there are four main factors influencing the computation time of the proposed algorithm as follows:

Fig.3Evolutionary process of an individual

(i)one comparison between?andfrom(12);

(ii)2Niterations according to(13)for creating an original initial population andN(N?1)/2 comparisons for selecting the superior individuals and putting them into the final initial population;

(iii)tmaxiterations and four comparisons(between

J()andη,andδ,andJ(),holding times and five mentioned in other rules)in every iteration for each individual;

(iv)tmaxcomparisons betweentandtmax.

Thus,the time complexity of the proposed algorithm can be represented asO((4N+1)tmax+(N2+3N)/2+1).This indicates that the computational time of the proposed algorithm can be controlled via choosing appropriateNandtmaxwith the considerations of the performance of the onboard processor.

4.Simulation results

In this section,the proposed control allocation scheme is applied to the lateral dynamic model of ICE aircraft to verify its performance.In order to reduce the difficulty of design of the ideal control law,the lateral-directional dynamics of ICE aircraft(1)are described as follows:

whereAandBvare constant matrices which come from the flight condition at Mach 0.6 and 4 572 m[9,19]:

The desired reference model in(2)is given as follows[9,19]:

Then the ideal control lawv(t)can be designed as

where the gain matricesK1andK2are designed as

The expression ofL(t)is

It is easy to verify that the control law(23)can make the closed-loop system achieve the target(3)in terms of the Lyapunov theory.

Remark 4The particular configuration of Δ(t)in(22)can make the closed-loop system achieve the target(3)purely.In terms of the form of,if the head or end of the diagonal of Δ(t)is not zero,the tracking errorecannot meet the target(3)due to an error relating to the termwhereIdenotes identity.

The mapping matrixBuis chosen as

The coefficientψin(8)is chosen asψ=0.000 25.The parameter?in(12)is chosen as=1.The parametersλgandλlare chosen asλg=2.5 andλl=2.0,respectively.The parameterηis chosen asη=2.5.The criterion parameterχis chosen asχ=0.05.The parameter?in(13)is chosen as?=0.45.The constraints onnare shown in Table 2.

Table 2Constraints on n

According to Table 2,the number of the possible combinations of actuator states is about 2.08 billion.Although the exact optimum solution of the optimization problem(8)can be obtained by some mathematical programming solvers,certainly the solving process cannot meet the realtime requirement for the flight system.Then the proposed scheme is used to achieve it.The number of population and maximum number of iterations are chosen as 10 and 50,respectively.

First,the searching capability of the proposed scheme in one period is studied via two experiments.The initial conditions are shown in Table 3.

Table 3Initial conditions of two experiments

Corresponding to the strategy(12),the initial condition of experiment 1 simulates the situation of a larger?v(t)?v(t?1)?2and the initial condition of experiment2 simulates the situation of a smaller?v(t)?v(t?1)?2.For comparison,another scheme proposed in[19]is also introduced to solve the optimization problem.To avoid contingency,each experiment is carried out 1 000 times.It should be noted that in these two experiments,the criterion parameterχis set toχ=0 for avoiding its influence.

Firstly,the quality of the initial population is studied.The contrastive scheme is shown in Section 3.3.1 in[19].There are 10 000 initial individuals produced by each scheme within 1 000 times runs.Their fitness values are calculated and the statistical results are shown in Table 4.

Table 4Fitness values of initial population

Table 4 shows the comparison of the indices while the mean and standard deviation can indicate the quality of initial population. In terms of the results, the quality of the initial population produced by the proposed scheme outperforms that produced by the contrastive scheme.The quality of the initial individuals produced by the contrastive scheme is worse because lots of individuals go beyond the constraints(4)when they are originally created.After they are qualified via(14),their qualities cannot be ensured as planned.It also can be found that the mean of fitness values in experiment 2 is smaller than that in experiment 1.This is because in each experiment 2,according to the strategy(12), five high-quality initial individuals are inherited fromnp,and other decentralized five are preferentially selected from double five alternatives created via(13).In each experiment 1,all the ten initial individuals are preferentially selected from double ten alternatives created via(13).

Secondly,the searching capability of the proposed strategies on the parameterλand step lengthαand the qualities of the final searched optimum solutions are stu-died.The proposed scheme adopts targetedαandλ,and the contrastive scheme adopts fixedαandλ.For eliminating the influence of the quality of initial population,both the two schemes adopt the initial population created via the proposed scheme in this paper.The fitness values of the 1 000 final searched optimum solutions are calculated and the statistical results are shown in Table 5.

Table 5Fitness values of final searched best solutions

To further verify the effectiveness of the proposed scheme,a statisticξis added to the experiments.In the beginning of one experiment,its initial value is set to zero;if an updated individual is proved to be superior to its origin,i.e.,,thenξ=ξ+1.It can be considered that the statisticξindicates the number of effective random walks.The results ofξare illustrated in Table 6.

Table 6Results of statistic ξ

It can be observed that in both experiments,the numbers of effective random walks under the proposed scheme are obviously superior to those under the contrastive scheme.This can verify that the searching efficiency of the proposed scheme is higher than that of the contrastive scheme.

Next,the rationality of the selection of the parametersλgandλlis examined.In order to verify this,under the initial condition of experiment 1,another four settings of parametersλgandλl(λg=2.8,λl=2.0;λg=2.5,λl=2.3;λg=2.3,λl=2.0 andλg=2.5,λl=1.8),are put into the search and their final searched results are obtained and shown in Table 7.

Table 7Final searching results under different settings of λgand λl

It can be observed that the best result is surely obtained under the selection ofλg=2.5 andλl=2.0.

In terms of the above results, it can be concluded that the improvements of the proposed scheme are surely working and conducive to the acquirement of high-quality final solutions.

Second,the proposed scheme is applied to the simulative flight process(21)for verifying its performance.The fixed-step size of the simulation is set to 0.02 s and the length of the flight process is set to 20 s.The initial flight states and control inputs are set tox(0)=04×1,n(0)=08×1,v(0)=02×1,?(0)=10.Meanwhile,for comparison,the contrastive scheme in[19]is introduced and applied to the identical model of the flight process under identical initial conditions.

Firstly,the performances of the ideal control law(23)and the guaranteed solutionθ(t)are applied to the system.The simulation results are illustrated in Fig.4.

It can be observed that the actual flight states can track the desired reference model exactly under the ideal control law(23).It can also be observed that the tracking performance under the guaranteed solutionθ(t)is stable but inferior.Therefore,the guaranteed solutionθ(t)canbe utilized to ensure the flight stability when the quality of the searched solution obtained via the metaheuristic algorithm is poor but a further control allocation scheme is necessary for ICE aircraft to improve the tracking performance.

Fig.4State responses under ideal control law and guaranteed solution

Secondly,the proposed control allocation scheme and the contrastive scheme are applied to the flight system.The simulation results under the two schemes are illustrated in Fig.5–Fig.7 and Table 8 and Table 9.

Fig.5State responses and tracking errors under the proposed scheme and the contrastive scheme

Fig.5 illustrates the state responses and tracking errors of the closed-loop system.In Fig.5(b),the labels of theyaxis denote the absolute values of elements of the tracking errore.To avoid contingency,1 000times simulative flight processes under identical initial conditions are carried out and the averages of the accumulated tracking errors of all the 1 000 sample points in one flight process are counted.The statistical results are shown in Table 8.

Table 8Accumulated errors per flight process

From Fig.5 and Table 8,it can be observed that the tracking performance under the proposed scheme is superior to that under the contrastive scheme.Considering the previous results from Table 5–Table 7,this result is normal due to the stronger searching capability the proposed scheme has.

Fig.6 illustrates the commands of actuators under the proposed and contrastive schemes.It can be obviously observed that the changes of the states of actuators under the contrastive scheme are more drastic than those under the proposed scheme.To avoid contingency,the averages of number of state switches of eight actuator groups from the above-mentioned 1 000 times simulative flight processes are counted.The statistical results are shown in Table 9.

It can be observed that the results are similar to those in Fig.6.This is because the contrastive scheme does not utilize former searched results to conduct the population initialization in the new searching process,i.e.,its searching process is independent in each sample point.For the proposed scheme,the strategy on population initialization decreases the probability of drastic change when the last searching result is superior.From Fig.5(a)–Fig.5(d)and Fig.6(a),it can be noticed that the changes of the actuator states are relatively smooth at those sample points which are far from the switching points ofc(t)(5 s,10 s,15 s).This is because the indices?and?v(t)?v(t?1)?2are usually smaller at those smooth intervals.Thus,the situation of their initial individuals is similar to the abovementioned experiment 2 in Table 4,i.e.,a part of the initial individuals are inherited from the lastnpwith high qualities.

Table 9Average of number of actuator state switches per sample point

Fig.7 illustrates the comparison between the ideal control inputv(t)and actual actuator control inputBun(t)from(5)under the proposed control allocation scheme.In Fig.7,the labels of they-axisv(1)andv(2)denote the first and last elements of the two-dimensional column vectorv(t),respectively.This result can further show the effectiveness of the proposed scheme.Moreover,it can be observed that the control inputv(2)at 15 s is larger than that at 5 s under the samec(t).This is because the disturbance Δ(15)is larger than Δ(5).

In addition,the average of the statistical elapsed time of one simulative process under the proposed scheme is about 3.178 s with the setting thatχ=0 and an Intel Core i7-4790 CPU@3.60 GHz.That is,one single searching process at one sample point takes up about 6.356 ms on average.Meanwhile,the statistical maximum of computational time at one sample point is 6.562 ms.This computational speed is acceptable to most flight vehicle control systems.

In terms of the characteristic of the metaheuristic algorithm,although the qualities of solutions obtained via the proposed scheme are uncertain in limit time in theory,the condition(θ)(n)for replacingnwithθmentioned is rarely reached due to the inferior(θ)which can be observed from Fig.4.To simulate the uncertainty and further verify the effectiveness of the proposed guaranteed strategy,an artificial setting that uses the guaranteed solutionθas the actuator control command without searching every 0.1 s is put into the simulative flight process.The simulation result is illustrated in Fig.8.

It can be observed that the closed-loop system is stable and the tracking performance is between the result under the proposed scheme from Fig.5(e)–Fig.5(h)and the result under pure guaranteed solution from Fig.4.Therefore,the effectiveness of the proposed guaranteed strategy is further verified.

Fig.6Actuator control command

Fig.7Comparison between ideal control input and actuator control input under the proposed scheme

Fig.8State responses and tracking errors under the proposed scheme with discontinuous θ

Remark 5According to the stability analysis and simulation results,it can be known that the tracking errorecan be ensured bounded in theory,and due to the guaranteed solutionθ,the flight states of ICE aircraft can still keep stable even if the searching processes are inefficient.Moreover,it is verified in simulation that the computing time of the proposed method can be acceptable to most flight control systems.Therefore,it can be concluded that the utility of the approach to ICE aircraft is better.

5.Conclusions and future work

A control allocation scheme for tracking control of ICE aircraft is proposed.In this scheme,two desired objectives,tracking accuracy and energy consumption of actuators,are considered and represented as an optimization problem.Following that,the L′evy flight is introduced for exploring the optimum solution.Meanwhile,some improvements in step length,population initialization and other heuristic strategies are proposed for searching efficiency.To prevent possible low-quality solutions obtained via the proposed scheme,a guaranteed strategy is proposed for ensuring flight stability.To verify the effectiveness of the proposed scheme,a time-varying ICE aircraft model is introduced and some contrastive simulations are carried out.The simulation results show that the proposed control allocation scheme can achieve superior tracking performance with a low-level wear and tear of actuators and the proposed improvements are effective.The study of this paper indicates that for an optimization problem with requirement of real-time,metaheuristic algorithms are especially noteworthy tools to tackle it.

Future studies are planned to be classified in two directions.One is to find and summarize the connection between the flight dynamics and the actions of actuators thoroughly for supplying heuristic searching directions.The other direction is to improve the search efficiency from the metaheuristic algorithm itself,including designing appropriate computing strategies on the parameters of the algorithm,cancelling some unnecessary comparison operations,and so on.