Fractional order modeling and control of dissimilar redundant actuating system used in large passenger aircraft

Slmn IJAZ,Lin YAN,Mirz Triq HAMAYUN

aSchool of Automation and Electrical Engineering,Beihang University,Beijing 100083 China

bDepartment of Electrical Engineering,COMSAT Institute of Information Technology,Lahore 54700,Pakistan

1.Introduction

Reliability of aircraft actuating system is of major concern when a new aircraft is designed.Traditional redundant actuating system,i.e.two similar actuators driving a single control surface,cannot meet the reliability requirement of large aircraft.The main reason is that if the fault occurs due to common reason,it may lead to the failure of whole actuating system.Therefore the use of Dissimilar Redundant Actuating System(DRAS)has become a good solution in modern aircraft in order to make the system more reliable.1DRAS used in this research consists of one Hydraulic Actuator(HA)and one Electro Hydraulic Actuator(EHA),and both drive one control surface in order to deliver large torque.Since the driving mechanisms of both actuators are different,failure of whole system due to common cause can be reduced effectively,and eventually the reliability of system increases.

Traditionally,there are two operating modes of DRAS:one is Active/Active(A/A)mode and the other is Active/Passive(A/P)mode.2In A/P mode,the passive channel is isolated from bypass valve.In A/A mode,both actuators drive one control surface simultaneously.Because of their different static and dynamic behavior,they cannot share load equally and often fight against one another to position the load.So a serious force fighting occurs on aileron plane that degrades the system performance and may damage the control surface.

Several control strategies have been proposed in literature to deal with force fighting issue and to achieve precise tracking of control surface,for example,in Refs.3,4,integral action controller is proposed based on the information of position and force feedback and it was deduced that position feedback can reduce the force fighting more effectively as compared to force feedback.The authors5,6proposed states difference feedback controller to deal with force fighting which has been effectively reduced from 500%to 7%of stall load.PID controller,7due to its simple structure,is used to deal with dynamical force equalization.Also fuzzy PI controller is designed in a recent survey8for position and force synchronization of dissimilar redundant actuators,which also reduced force fighting effectively.Adaptive technique is employed9to decouple the HA and EHA,and feed-forward compensator is proposed to match the dynamics of both actuators.All such techniques need more accurate model of system in order to enhance the controller performance.

In recent years,fractional order calculus,which is a generalized version of integral order calculus,has gained more attention due to its accuracy in modeling the dynamics of systems and its simplicity in model structure to represent higher order processes.Identification of fractional order model has gained wide demand in scientific community despite of its difficult task because it requires not only the estimation of model parameters but also the determination of fractional power.Most of identification methods found in the literature are the extension of regular integer order systems,require a priori knowledge of the fractional differentiation orders,and estimate only the model’s parameters.10,11However,very few papers are based on identification of both model parameters and differential power.12On the other hand,fractional order controllers show better transient performance when they are applied to fractional order systems.FOPID controller has been adopted in many applications such as speed control of DC motor,13torsional system’s backlash vibration suppression control,14position tracking control of Electro Hydraulic Servo System(EHSS),15pitch and yaw angle control of Twin Rotor Aerodynamic System(TRAS),16,17motion control of gun control system18and so on.FOPID controller provides robustness,abundant dynamics, fine tracking and low sensitivity to external disturbances as compared to integral order controller.19In practical applications of FOPID controller,tuning of controller parameters is one of the most challenging problem.Closed-loop stability and tracking performance of system are highly in fluenced by setting controller gain to optimal value.To deal with such problem,various optimization techniques have been employed based on time domain19,20and frequency domain constraints.21–23

In this paper,three FOPID controllers are designed for nonlinear DRAS in order to reduce the force fighting and to precisely track the aircraft control surface.EHA/HA system used in A/A mode is under study.Prior to the controller design,fractional order models of both actuators are identified based on the input-output data from nonlinear system in which coefficients are estimated using Recursive Least Square(RLS)algorithm and fractional powers are adjusted using steepest descent algorithm so that square of error is minimized.Based on identified model,two FOPID controllers are designed for each actuator in position feedback con figuration.The force fighting phenomenon is dealt with using third fractional order PI controller that feeds the force compensation signal in the position feedback loop of both actuators.Nelder-Mead(N-M)optimization technique is adopted to tune the controllers’parameters according to the design objectives based on both time and frequency domains.The performance of the proposed scheme is compared with a group of PID controllers that are designed for this system.In order to test the robustness of the proposed scheme,the control surface of aircraft is subjected to the influence of large air load.

The rest of the paper is organized as follows:the nonlinear model of DRAS is presented in Section 2.Section 3 consists of problem formulation and control strategy.Fractional order model identification and fractional power identification algorithms are discussed in Section 4.The overview of FOPID controller design is presented in Section 5 and it also covers the bene fits in term of design freedom and stability region as compared to conventional PID controller.N-M optimization algorithm and performance criteria are explained in Section 6.Simulation results on a nonlinear model are presented in Section 7.Finally concluding remarks are given in Section 8 to show effectiveness of the proposed scheme.

1.1.Mathematical definitions

De finition 1.Grunwald Letnikov approximation24of fractional order system is defined as

where Γ(·)is the Gamma function.

Stability theorem 1.25A fractional order transfer function G(s)=N(sζ)/D(sζ),where N(sζ)and D(sζ)are coprime polynomials,is BIBO stable if and only if

and ∀s∈ C such that R(η)=0

2.Mathematical model of dissimilar redundant actuating system

The parallel driving structure of DRAS is shown in Fig.1,which contains EHA and HA to drive a unique control surface.Due to different dynamics of EHA and HA,they act as a force integrated manner and drive the rudder load together.Each actuator has an independent control channel.The pilot gives the command to flight control computer which drives the motor of EHA and servo valve of HA respectively.In Fig.1 uhand ueare the input to HA and EHA,Fhand Feare the load on the cylinder of HA and EHA,θdis the control surface deflection,xuand xeare output of HA and EHA,current ivis input to the servo valve.This section contains mathematical model of DRAS that includes nonlinear model of both actuators(EHA and HA)and a control surface.A brief description of their components and nonlinear models is presented in the upcoming subsections.

2.1.Mathematical model of EHA

EHA architecture with Fixed Pump and Variable Motor(FPVM)is used in this paper due to its simple structure and higher efficiency.FPVM-EHA module consists of fixed displacement bidirectional pump,a variable speed brushless DC motor,double acting hydraulic cylinder and feedback mechanism as shown in Fig.1.The relationship between input current ieand electromagnetic torque Tedeveloped at motor shaft is represented by torque balance equation26:

where Reand Leare motor resistance and inductance respectively,Kcis coefficient of electromotive force,Textis external load acting on motor,ωeis the motor speed,Beis damping coefficient,Jeis the motor and pump inertia,and Ktis torque constant.The nonlinear relationship between flow Qeand pressure Pe,including inner and outer leakage of EHA pump,can be specifically described as3

where Deis pump displacement,and Ciland Celare internal and external leakage coefficient respectively.The relationship between actuator position xeand flow Qecan be described by

where Aeand Veare the effective area and volume of hydraulic cylinder respectively,Eeis effective bulk modulus,and Ctlis the total leakage factor.

The load force balance equation of piston can be written as

where meand Beare the mass and damping coefficient of hydraulic cylinder rod,Keis connection stiffness,Ffeis velocity dependent friction,and Fdeis the effect of external force acting due to air load.

2.2.Mathematical model of HA

HA consists of hydraulic supply,hydraulic servo valve,double acting cylinder and a piston attached to the control surface.The mathematical model representing the oil flow through orifice Qhand pressure Phdelivered to load is represented as27

where Ehis effective bulk modulus,Chlis the total coefficient leakage,and Ahand Vhare the piston area and volume respectively.The major nonlinearity exists between Phand Qhfor an ideal servo valve is described by relationship:

where Psis supply pressure,w is the gradient area of servo valve,Cdis discharge coefficient,ρ is the fluid density and sgn(·)denotes the discontinuous sign function defined as

The relationship between spool displacement xvand the input current ivcan be approximated as 1st order differential equation28:

where kvand τvare gain and time constant of servo valve respectively.The force acting on the piston is derived by 2nd order differential equation:

where Phand Ahare the pressure and area of hydraulic cylinder respectively,mhis the mass attached to the piston,Bhis the damping coefficient,Khis connection stiffness,Ffhis velocity dependent friction force,and Fdhis the effect of external air load.

2.3.Mathematical model of control surface

The control surface is shown on the right part of Fig.1 and is modeled as reduced mass representing the load inertia and equal attachment stiffness.Inputs to the control surface are actuators’displacement(xhand xe)and external force Fext.Outputs are the control surface angle θdand force on both actuators Fdeand Fdh.The mathematical model is given by

where Jdis the equivalent moment of inertia,Fextis the external air load that acts on system under real flight condition.The control surface angle θdcan be linearly approximated as θd≈ xd/rdwhere rdis radial distance of control surface and xdis the control surface displacement.The parameters of the nonlinear model are taken.9

3.Problem formulation and control strategy

Two major problems are taken into account in this research:the first one is the force fighting that occurs due to different driving mechanism and the other is precise tracking of control surface under the influence of external force Fextacting on the control surface due to airload.This section proposed a control strategy to reduce the force fighting and to achieve precise position tracking of both actuators(EHA and HA)and a control surface.Force fighting between two actuators(EHA and HA)can be defined as

where K=Kh=Ke.When γ is equal to zero,it means that force fighting is eliminated.

In order to make sure that the displacement of control surface xcsprecisely tracks the desired input,it is first needed to design a controller in order to make the output of both actuators track the desired inputs at the same time.To do this,two FOPID controllers are designed in the position feedback configuration and are dedicated to maintaining transient performance and preserving robustness.On the other hand,to deal with force fighting and to reduce the effect of coupling,the output of both actuators should be synchronized.This is achieved by adding the third FOPI controller in the path of force compensation signal.The optimal values of three controller parameters are obtained using N-M optimization technique.The controller parameters are tuned according to predefined objective function so that system stability,pursuit and rejection performance should be ensured.A block diagram representation of overall control strategy is shown in Fig.2.

4.Fractional order model identification

This section contains the fractional order model identification procedure for nonlinear DRAS.Since two different actuators(EHA and HA)drive single control surface,their different dynamical behavior leads to cross coupling that results in instability of the system.In order to simplify the identification procedure,coupling is neglected in identification phase.So the both actuators can be identified separately.The control surface is approximated as integral order driven by both actuators simultaneously.This paper contributes fractional order model identification scheme for the class of Signal Input Single Output(SISO)system in which both the system coefficients and fractional power are approximated as well.

4.1.Fractional order system

A general input-output representation of fractional order system is

where aiand bjare non-integer coefficients,naand nbare the number of identifying coefficient sand αiand βjare noninteger positive numbers.y(t)and u(t)are the system output and input.Input-output model in Eq.(16)can be represented in transfer function form as

By using Grunwald-Letnikov approximation Eq.(1),Eq.(16)can be written as

where

Input-output data and system coefficients in regression form are written as

where

The parameter vector θ is identified using RLS algorithm and fractional power(α and β)are estimated using steepest decent algorithm.

4.2.Parameter estimation using RLS algorithm

From Eq.(19),the estimated solution^y(k)is given by structure model:

RLS algorithm29is described below to estimate^θ

where λ is forgotten factor 0＜ λ ＜ 1;κ(k)is adaption gain and its initial value is given by(0 ＜ δ ＜ 1).

4.3.Fractional power estimation algorithm

From Eq.(18),the estimated solution is given as

The estimated parameter vectorobtained from current iteration of parameter estimation step Eq.(22)is used to estimate the fractional powerandFrom Eq.(18),the expression Y(k)and U(k)become

The estimation error using the current parameter value is calculated as

In this context,the estimated values ofandare adjusted using steepest descent algorithm so that[∈(k)]2is minimized.Then the updating equation of

where y(k)is independent of fractional power,so the termis neglected.Substituting Eq.(23)into Eq.(27)and simplifying it,we get

The performance of identification algorithm is evaluated using Best FiTted(BFT)defined as

4.4.Fractional order model of HA and EHA

In this paper,a discrete input-output datum from nonlinear model is obtained for model identification.A multifrequency signal is applied to the input of both actuators.The sampled data of N=4000 samples are taken for model identification and N=[4000;20000]samples are used for model validation.In order to run RLS algorithm,the initial values of variables are assumed as

Also the steepest descent algorithm in Eq.(26)is initialized as follows:

Input signal of amplitude 0.03 sin 2πf1+0.02 sin 2πf2+0.01 sin 2πf3is applied to HA,where f1=0.03 Hz，f2=0.5 Hz and f3=0.01 Hz.Input-output data from the nonlinear HA are applied to RLS algorithm defined in Eqs.(22)–(26).In Eq.(16),substituting na=5,nb=1 and initial value of fractional power is assumed as λ = μ =2.The HA model is identified as

In orderto identify model of EHA,in put signal 0.03 sin 2πf1+0.02 sin 2πf2+0.01 sin 2πf3is applied to nonlinear EHA, where f1=0.1 Hz, f2=0.5 Hz and f3=0.02 Hz.A fractional order model of EHA is given as

The model validation curve is shown in Fig.3 and BFT factor is 91.2%for EHA and 86.45%for HA.The identified models are stable according to the criteria defined in Eqs.(3)and(4).Both models are used as benchmark in the design of FOPID controller.The control surface is modeled as 2nd order transfer function with position of both actuators acting as input and control surface angle serving as output.

5.Fractional order controller

FOPID was first introduced by Podlubny.30The control law of such controller can be described as

where kp，ki,kd∈ R are proportional,integral and differential gains respectively and λ，μ ∈ R+are integral and differential power respectively.andare the fractional integral and differential operator respectively and are defined using Grunwald-Letnikov approximation Eq.(1).

Substitute Eq.(1)into Eq.(34),and the discrete FOPID control law is obtained as

The transfer function representation of FOPID controller is

According to Eqs.(34)and(36),FOPID controller contains five tuning parameters that are kp，ki，kd，λ and μ.As compared to conventional PID controller,FOPID controller has two additional parameters λ and μ.

5.1.More design freedom as compared to PID

In order to obtain analytical expression of fractional order system with FOPID controller,Mittag-Leffler function31is defined as

and its kth derivate is given by

In case of PID α =1，Eq.(37)is exponential function.For half integer n/2,the function Eq.(37)can be represented as

5.2.Stability analysis of fractional order controller

In order to analyze the stability performance of FOPID controller with PID,we consider a closed-loop system with the fractional order plant G(s)Eq.(17)and controller C(s)as shown in Eq.(36),and then closed-loop characteristic equation can be written as

where the coefficients ai，bj，αi，βjare known from model identification algorithm.In case of PID controller,set λ= μ =1,and the stability of closed-loop system can be determined by optimal tuning of three controller parameters(kp，kiand kd),so that all the poles of D(s)lie in left half plane.In case of FOPID,apply stability criteria Eqs.(2)and(3)to closed loop system Eq.(40)and stability region is shown in Fig.4 depending on ζ.It can be clearly seen from Fig.4 that the fractional order system is stable,and with fractional power λ and μ,the stability region of FOPID controller is more than PID.This adds more design freedom to the controller to achieve desired performance and more dynamical behavior can be attained.

Other benefits include steady state error elimination,provision of gain and phase margin,robust behavior to the variation of plant gain and high frequency disturbance.On the other hand,the complexity of tuning controller gain to optimal value increases in case of FOPID controller.

6.Parameter optimization of FOPID controller

6.1.Nelder-Mead optimization algorithm

This section includes the brief description of N-M optimization technique that is applied to tune the parameters of FOPID controller.N-M technique was first proposed by Nelder and Mead.32It is simplex based algorithm to find local minima and has the closest resemblance to Particle Swarm Optimization(PSO)and Differential Evolution(DE).33N-M optimization is considered as fast algorithm to search local minima and is applicable to multi-dimensional optimization problems.Despite this,this technique has many other bene fits:it is easy to code,fast algorithm to search local minima,applicable to multi-dimensional optimization and it requires less space for its implementation in real time applications.The convergence of N-M algorithm depends on reflection coefficient Υ,contraction coefficient ψ and expansion coefficient ς.For low dimensional problem,the N-M optimization providesf aster convergence.However,in 5-dimensional problem,the additional parameters can be restricted to limit range of search space.It improves the convergence speed to tune controller parameters to optimal value.N-M optimization technique has been used in various other control problems to find optimal parameters.34,35The N-M algorithm is shown in Fig.5.

6.2.Performance criteria

In this paper,a performance criterion is formulated to find effective solution of FOPID controller design for DRAS system.Several performance criteria in time domain(Integrated Absolute Error(IAE),Integrated Square Error(ISE),Integrated Time Absolute Error(ITAE))or in frequency domain(gain margin GM,phase margin PM)are available to evaluate the performance of the proposed scheme.The criteria in this paper include both the transient domain specifications(i.e.settling time Ts,steady state error σe,overshoot Os,ITAE)and frequency domain parameters PMincluding the coupling term between two actuators(i.e.the term

The objective function is described by introducing weighted sum approach on each performance parameter.The proposed performance criterion is thus defined as

where ¯h ∈ {kp，ki，kd，λ，μ}denotes the controller parameters.The proposed performance criterion Eq.(41)is composed of six terms and each of them is evaluated by weighting factors wi.Increase in wiof specific term results in improvement on that term at the expense of degrades on other criteria.In order to ensure stability,an approach using penalty function(see Ref.36)is employed to ensure stability.The value of fitness function of each particle of N-M is determined by evaluating function denoted by h(¯hi)as

The penalty for the individual¯hiis calculated by means of the penalty function p(¯hi)given by

If the individual¯hidoes not satisfy stability,it is penalized with large positive constant p1.Otherwise,the individual¯hiis not penalized and is feasible.

7.Controller parameter setting and simulation results

This section contains the FOPID controller parameter setting,implementation and simulation results for DRAS.The parameters of each controller are tuned in such a way that the objective function defined in Eq.(42)is minimized.

7.1.Parameter setting

This subsection explains the parameter setting of FOPID and PID controllers for DRAS according to the proposed control strategy shown in Fig.2.The criterion includes the design of two separate controllers for each actuator,and 3rd controller deals with force fighting and keeps the actuator dynamics in similar pursuit.Fractional order models identified in Eqs.(32)and(33)are used as benchmark for controller design.N-M optimization technique is opted here to find optimal values of controller parameters based on specified objective function defined in Eq.(43).The algorithm provides pattern search approach with k+1 parameters in n-dimensional space where n is the number of controller parameters.The algorithm is initialized by assigning the value to each weight given as w1=100,w2=1,w3=1,w4=20,w5=50 and w6=1.In order to reduce the time of optimization,the controller gains and fractional powers are bounded as kp=ki=kd=105and λ = μ =2.The standard value of Υ ，ψ，ς and ρ are taken as 1,1/2,–1/2 and 1.Number of trials performed is 30.The optimal values of FOPID and PID controller parameters are shown in Tables 1-3.FOPID controller is designed and implemented using FOMCON toolbox in MATLAB.37This toolboxallows us to simulate and analyze FOPID controllers easily via its functions.The stability of the proposed controller is veri fied in closed loop according to the stability criteria given in Eqs.(3)and(4).

Table 1 FOPID parameters tuned using N-M.

Table 2 PID parameters tuned using N-M.

Table 3 PID parameters tuned using high gain.

7.2.Simulation result

FOPID controllers designed in previous subsection are implemented on nonlinear model of DRAS according to the control strategy defined in Fig.2.In actual system,only the position of actuators is available at the output so the other states are assumed to be available in this paper.Major nonlinearities like nonlinear pressure flow relationship,valve opening area variation,and nonlinearity of check valve are included in nonlinear model of DRAS,while those having less impact on system dynamics are neglected.N-M technique is opted to tune controller parameters,which provides pattern search approach with N+1 dimensional shape,where N is the number of controller parameters.The two input signals of magnitude 0.03 m are applied to both actuators.To test the controller according to real flight condition, an external load of Fext=15KNsin(2πf)is applied to the input of control surface.In order to check the robustness of the proposed scheme,two disturbance input signals function of system states(103sinxh;103sinxe)are applied at control input channel of both actuators.The performance of the proposed controller is compared with a set of PID controllers in which one is tuned using N-M technique and the others using the high gain.The performance comparison is shown in Table 4.

7.2.1.Tracking performance analysis

Simulation result in Fig.6(a)shows tracking performance of control surface at commanded signal 0.01u(t-1)+0.02u(t-6)+0.03u(t-13)that was applied at input channels of both actuators.No external load is applied at this stage.Simulation result clearly shows the precise position tracking of control surface θdat commanded signal.Moreover,FOPID controller is able to quickly settle to steady state value at each step as compared to optimal PID controllers,whereas high gain PID controller deteriorates the system behavior and possesses little overshoot.

In Fig.6(b),a step signal is applied to input of both actuators.The external air load,a sinusoidal signal of amplitude 15 kN,is applied at input channel of control surface.Simulation results on nonlinear system clearly show that FOPID controller is more capable to settle control surface to its set point value while the optimal PID and high gain PID controllers sustain some oscillation in the presence of airload.Two disturbance signals 1000 sin(xh，xe)function of system states are applied at control input channels of both actuators.Simulation result in Fig.6(c)clearly shows the capability of FOPIDcontroller to sustain angular position even in the presence of disturbance,whereas the system output is distorted more in case of optimal PID and high gain PID controllers.

Table 4 Performance specifications.

7.2.2.Force fighting analysis

Simulation results in Fig.7 show the force equalization curve when both actuators operate simultaneously.The step command is applied to both actuators and force fighting signal γ is analyzed according to Eq.(15).Simulation results show that the behavior of both controllers is almost the same in transient phase,but FOPID controller settles more quickly close to equilibrium point as compared to PID controller,whereas high gain PID controller possesses some oscillation before settling to steady state value.

7.2.3.Comparison of control effort

Finally,the comparison of control effort provided by the controllers to control the position of both actuators is shown in Fig.8.It can be clearly seen that FOPID controller requires less control efforts to achieve the desired performance as compared to optimal PID controller,whereas the PID controller with high gain requires high control energy as compared to others.The negative value of control effort in Fig.8(a)in case of PID with high gain is due to deceleration effect.

8.Conclusions

This paper focuses on the design of FOPID controllers to control the angular position of aircraft control surface driven by two dissimilar actuators.The force fighting issue due to different system dynamics has also been addressed in this paper.The fractional order model of DRAS is identified first using RLS algorithm which gives better representation of system dynamics as compared to integer order model.Three FOPID controllers are designed based on identified fractional order model.The objective function based on both time domain and frequency domain constraints is developed.N-M optimization technique is employed to optimally tune controller parameters based on the objective function.The performance of the proposed strategy is validated on nonlinear model of DRAS.Simulation results on nonlinear system demonstrate the effectiveness of the proposed control scheme as compared to optimal tuned PID and high gain PID controllers.Moreover,FOPID controller provides better tracking of control surface,effectively reduces force fighting,and shows robustness in the presence of external disturbances.

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