Introduction
For the specification of the control of an aircraft, other parameters and equipment’s other than the basic aircraft control are required. This is because there are various dynamic performance criteria that need to be factored in the aircraft control system design.
The basic aircraft system requires a navigation and control so that it works efficiently. The basic structure of the aircraft system is identified as below with the various parts performing various functions:
Aircraft system block diagram
The stability augmentation and transient response improvement are provided by the estimation control of the linear feedback block of the system. The forward loop provides shaping of the input for satisfactory input response through a control or compensation block which is still a linear model. The autopilot translates the basic pilot commands into guidance commands for constant heading angle, airspeed, and the bank angle. The guidance block can be extended to include declarative flight management system using inputs from the navigational sensors and algorithms .
Generally, an aircraft is controlled by three main parts which are the elevator, rudder, and the ailerons. Pitch control is achieved by changing the lift on either a forward or backward control surface. Yaw control is achieved by deflecting a flap on the vertical tail called the rudder and the flapped portion of the vertical tail are called the rudder. Roll control is achieved by deflecting small flaps located outboard toward the wing tips in a differential manner .
Aerodynamic control of an aircraft 1
The automatic aircraft control system can be modelled using various methods in control engineering. The basic modelling can be estimated by the standard linear state space model which will have the dynamic matrix A, input matrix B, output matrix C and feed forward matrix D as per the below equations
x=A.x+B.u
y=C.x+D.u
The actuators of the control system can be modelled as a standard second order linear single input single output transfer function given by
GAct(s)=wo2s2+wo. εo.s+wo2
Where the parameters wo is the natural frequency and the damping coefficient is given by εo
The motion of the aircraft can be decomposed into lateral motion and the longitudinal motion during its design. Longitudinal motion is in state space modelling can be seen to have the longitudinal state vector,xlon, the angle of attack, α, the pitch angle θ, elevator deflection ƞ, pitch rate q and the input vector ulon. The thrust level position of the aircraft is also given by δT
xlon= [V α θ q]T
ulon= [ƞ δT]T
Alon=XVZV0MV XaZa0Mα -g.cosγ0-gV0 .cosγ000 XqZq1Mq
This dynamic matrix contains the linearization force elements (XV,ZV), force and moments elements which are a function of the pitch rate (Xq,Zq,Mq) and the force and moments variables influenced by the angle of attack (Xa, Za,Ma). The constants V0 and γ0 refers in this case to the trim point conditions for true airspeed and flight path angle and g is the acceleration due to gravity .
The input matrix Blon will be given by the below:
Blon=XƞZƞ0Mƞ XδTZδT0MδT
The lateral motion contains yaw rate r, sideslip angleβ, roll rate p, and the bank angle∅. There is also parameters rudder deflection and aileron deflections. The linearization moment is given by (Nr,Lr) which are a function of the yaw rate. The angle of sideslip influences the forces and moments (Nβ,Yβ, Lβ) while the role rate will influence the moments and is given by(Np,Lp)
xlat= [r β p ∅ ]T
Alat=Nr-1Lr0 NβYβLβ0 Np0Lp1 0gV000
The reason why baseline controllers are used in the aircraft is to ensure that the aircraft successfully tracks the desired pitch angle and bank angle command signal with a very small error. The automatic flight control system will act as the longitudinal and lateral autopilot providing the necessary elevation and ailerons deflections for the demanded pitch angle given by the reference signal. To design the aircraft control system classically, the system will achieve a linear input/output behaviour of a closed loop system. The flight control system will have two control loops for longitudinal and lateral motion. The whole flight control system will have a pitch angle control and the pitch rate control in the longitudinal block and the roll angle and roll rate in the lateral block
Closed loop system design scheme
Controller scheme
Mathematically, the system controller can be represented by the below control equation for the PI controller
ut=Kpet+0t[Kiet- Kwdsat(t)]dt
In a typical aircraft system, there will be the need for a fault diagnostic system. This will provide some information for subsequently reconfigurable controls that can guarantee system stability. The fault detection and diagnostic (FDD) and diagnostic system will assist in the detection of the breakdown of the sensors, actuators and the flight control computer of the aircraft location accurately and in a timely manner . There are various methods of flight control of the aircraft and this can be flight detection and diagnostic based on the model. This is based on the mathematical modelling of the flight control system in the modern control optimization theory. The hardware redundancy is replaced with the analytical redundancy based on the mathematical model of the system. The residual signal of the system is got by comparing the actual output values and the estimate values. The estimation method can be of state estimation diagnosis, parameter estimation diagnosis and equivalent spatial methods
The second method of fault diagnosis and detection method does not rely on the model. This is mainly of importance since the system is becoming more and more complex thus it’s very difficult to get an accurate mathematical model. This method can be based on signal processing or experience knowledge. For the signal processing model, an analysis of the frequency, amplitude and phase and other output features is based on signal processing techniques. It can use the Fast Fourier Transform (FFT), time domain analysis, spectrum analysis, wavelet analysis and correlation analysis . This method has a disadvantage of being affected by noise and dependency on the result of the signal processing.
The second method based on the experience knowledge and use the symptom-based approach and the qualitative based approach. The symptom-based approach can contain expert systems like the fuzzy reasoning, neural network, robust system control and others
Design of a Flight Control System Using Fuzzy Logic
The methods based on fuzzy theory is used in fault diagnosis aimed at tackling the uncertainty, inaccuracy and process noise of the process itself . It is good especially in time delay, non-linear and time varying systems. The principle of fuzzy logic is entangled in the natural human thought process. The idea of the fuzzy logic is to imitate the control action of a human operator as a collection of the if-then rules. The first part (antecedent) of the rule always specifies the condition under which the rules holds while the second part (consequent) prescribes the corresponding control action. These two parts use vague linguistic terms like small, medium-small, low that represents the operator’s knowledge of the process. The linguistic terms are represented by the fuzzy sets while AND, OR operations are used to combine the linguistic terms
The fuzzy sets- This is an ordered set that associates each value of a variable to its grade of membership. The grades of membership are represented by membership function whose position or shape whether triangular or trapezoidal depend on a particular application
Fuzzy Set Operation- They are performed by logical connectives like AND, OR and NOT.
Fuzzy Logic Control- Using the fuzzy sets and the operation, a fuzzy reasoning system can be designed and it will act as the controller. This control strategy is stored as a series of if-then rules and will represent a static mapping from the input like errors. The computational mechanism of a fuzzy controller of Mamdani type is as below :
-Fuzzification: the membership degrees of the first process variables are computed e.g. small (e), μmedium∆e
-The degree of Fulfilment: This is computed by the first process by using fuzzy logic operators. This degree of fulfilment determines to which degree the ithrule is valid
-Implication: The degree of fulfilment is used here to modify the second part of the corresponding rule accordingly. The operation here represents the if-then implication which is defined as the conjunction operator like in the case of products .
-Aggregation: The scale consequent of all rules are combined into a single fuzzy logic set. The aggregation operator depends on the implication function used for example using a conjunction results in a disjunction operator .
-De-fuzzification: The resulting fuzzy sets are de-fuzzified to yield a crisp value. Methods like the centre of the area, bisector, and middle of maximum (MOM), Largest of Maximum (LOM), or Smallest of Maximum for Mamdani fuzzy logic. Others like weighted average (wtaver) or weighted sum (wtsum) can also be used in the de-fuzzification process
The navigational computer of the automatic flight control system uses fuzzy logic in the control of the aircraft. This can be to control the altitude, sensors, GPS. A fuzzy logic system consists of three main parts which are the fuzzifier, the fuzzy inference engine and the de-fuzzifier. The fuzzifier maps some fuzzy input into some fuzzy sets while the fuzzy inference uses fuzzy IF-THEN rules from a rule base to the reason for the fuzzy output. The de-fuzzifier converts the output back into a crisp value
The fuzzy IF-THEN rules are as shown below in the design of the aircraft control system
Rl: If (x1 is X1l ) AND . AND (xn is Xnl )) THEN y1 isY1l,, yk is Ykl
Where Rl the lth rule is x=(xl1.xn)T ∈U and y=yl1.ykT∈ V are the input and output of the state variables of the controller respectively. U, V ⊂ Realn are the universal discourse of the inputs and output variables respectively, Xl1.XnT ⊂Y1.YkT⊂ V are the labels of the input and output fuzzy sets in linguistic terms and k and n are the numbers of the output and input states respectively
For a multi-input single-output fuzzy logic controller (k=1), that has a single fuzzifier subjected to the triangular membership function, algebraic product for AND operation, product-sum inference and Centroid de-fuzzification method, we get the output as:
yj=l=1M(i=1Nμxil(xi))yjl=1Mi=1Nμxil(xi)
Where M and N represent the total number of rules and the number of the input variables respectively. μxil denote the membership function of the lth input fuzzy set for the ith input variable.
Mostly in a common aircraft control system, three fuzzy logic controller for navigation are designed and they control the heading, the altitude and the airspeed. The additional routing subsystem controls which waypoint is the next one and it might be the position, the altitude, speed and the altitude information of the place. The inputs to the heading subsystem are the current position of the aircraft in longitude and latitude in GPS format, the current roll angles from the .sensors by the flight computer and the next waypoint position
Navigation Computer Design 1 from
The two classes of the fuzzy controllers are the position type i.e. PD fuzzy and the velocity type, PI fuzzy logic circuit. The position type fuzzy logic generates the control input (u) from the error (e) and the error rate (∆e).The velocity type generates the incremental control input (∆u) from the error and the error rate .
The PI fuzzy logic controller has two inputs, the error e (t) and the change of error∆e(t).
The control signal is given by u (t) =u (t-1) +∆u(t)
The three fuzzy logic system in the automatic aircraft system are the roll angle fuzzy logic in the heading subsystem, the throttle fuzzy logic in the speed subsystem and the elevator fuzzy logic in the altitude subsystem.
The throttle fuzzy logic has the following inputs, the difference between the desired speed and the actual speed called the speed error and its rate of change
The following gives the block diagram of the throttle fuzzy logic using either PI, PD or PID controllers
PD and PI fuzzy logic controller respect
PID type fuzzy logic controller
The elevator control has two inputs, the altitude error and its derivative. The control output block is the elevator responsible for the aircraft going up or down. PI type fuzzy controllers are chosen since the aircraft system is highly non-linear and an inference can be made between the controlled parameters to predict the required change rather than the exact value
Since the PI controller has a limitation of poor performance in transient response and the PD controller cannot remove the steady state error, a combination of the two to form a PID fuzzy controller can be used. While using the PID fuzzy controller, the triangular membership functions can be used to for each input to the fuzzy logic and simple rules tables are defined taking into consideration the specialist knowledge and experience. The output function can be represented by a chosen number of membership functions and the spacing between them chosen in a given range for simulation.
Landing- For the landing part of the automatic control aircraft system, the following parameters have to be controlled and they include; the approach velocity, heading and heading offset, vertical velocity, flare, altitude and alignment of the aircraft. The controller will track the predetermined path trajectory for a safe landing.
The desired downward velocity is proportional to the square of the air vehicle height. For a further decrease in altitude, the required downward velocity diminishes thus the aircraft controller will make the aircraft descend from altitude promptly but touch down very gently and avoid damage. The two state variables for this simulation using fuzzy logic are the height above the ground, h, vertical velocity, v, and the control output will be the force that alters the height and velocity when applied to the aircraft. The equations used are derived from Newton’s laws of motion where a mass of the aircraft, m, moving at a velocity, v, has a momentum p=mv. If no external force is applied, the mass will continue in the same direction but if a force is applied, will result in some changes in velocity .
The equations of the landing mechanism can be depicted as
vi+1=vi+fi
hi+1=hi+vi(1)
Where vi+1 is the new velocity and vi is the old velocity same to the heights.
The membership functions for velocity, height and output table can be used to design the controller using fuzzy logic as
Sample membership for control force 1
Sample membership table for height 1
Sample membership table for velocity 1
Using fuzzy logic, all membership functions are defined for state variables as in the tables. The membership function for the control output are also defined and the rules defined and summarised in the table as below. The initial conditions are defined as the vertical height of the aircraft and initial velocity and then any simulation can be done while updating the state variables for each cycle found from the following equations
vi+1=vi+1+fi
hi+1=hi+vi
The Braking Control System of the Aircraft- The control system is supposed to stop the aircraft safely as fast as possible. The control target will be to maintain the friction coefficient between the tyre and the road within a safe range to avoid wheels blockage and preserve the lateral stability a reduced stopping distance.The antilock braking system is a design problem due to nonlinearities and the uncertainties in the process.
When the aircraft is braking or accelerating, the tractive forces Ff, Frl, Frr developed by the road on the tire are proportional to the normal forces Z1 and Z21=Z2r=Z2 of the road acting on the tire . The parameters Ff, Frl, Frr is the front, left rear and the right rear tractive forces respectively. Parameters φ is the road adhesion coefficient and is a constant while φl, φr are a function of the wheel slip α and depend as a parameter on the aircraft velocity v and the road condition c like whether dry or wet.
The equations are as given below
Ff=φl, Z1 , Frl=φl, Z2, Frr=φr, Z2 and φl, ≅φl, a,v,c, φr, ≅φl, a,v,c
Forces developed during airplane braking 1
Using Newton’s second law of motion, along the horizontal axis, the moments about the contact points A, B of the tire and the front wheel dynamics are given by
-mv=wI+wrZ2+ φZ1-F+D
-mvh+2Z2A-mga+aL+F-Dh=0, -mvh+2Z1A-mgA-a-A-aL+F-Dh=0
-Iwl+MN+φ1Z2R=0, -Iwr+MN+φrZ2R=0,
D=ρSCDv2/2, L=ρSCLv2/2,
Where the components m is the mass of the aircraft, F is the thrust force, D is the drag, L is the lift force,
φ is the air density, h is the height of the aircraft sprung, A is the distance between the front and the rear wheel axle, b is the distance from the center of gravity to landing/rear gears axle. I is the moment of inertia of each rear wheel, wr and wl are the angular velocity of the right and left wheel gear respectively, S is the wing area, g is the acceleration due to gravity and CL is the lift coefficient and CD is the drag coefficient
Solving the above equations for Z1 and Z2 and performing numerical integration, the wheel slips will be given as below;
αl=v-wlRv
αr=v-wrRv
The fuzzy logic of this automatic braking system of the aircraft is based on the detection and inference of the slip ratio α and road label ‘l’.
Modelling of the Roll Control of an Automatic Aircraft System- The roll control plays a very important role in the aircraft’s operations because degradation might cause flight control failures quickly. Integral feedback is needed to cope with the various structure disturbances although it consequently degrades dynamic performance and debases stability margins. Because the roll channel has the fastest response in the airframe, the characteristics of the actuators like the rudder is a very key factor. The bandwidth of the actuator needs to exceed 3 times that of the roll airframe when a classical PID controller is used. Without this, the stability margin and the disturbance attenuation ability of the system will not be guaranteed simultaneously .
The structure of the fuzzy logic control system of the automatic roll system is shown as in the block diagram below
Structure of the fuzzy logic based controller
The value of error and its change are expressed as ekand ∆e(k) respectively.The inputs to the system are defined as the below form the equations
ek=rk-y(k)
∆ek=ek-e(k-1)
Where r(k) is the reference input, y(k) is the output input and k is the sampling step. The crisp inputs e(k) and ∆ek are converted into fuzzy membership subsets. The subsets are defined as zero (Z), positive (P) and negative (N) .
The roll control system can also be designed as a self-tuning fuzzy logic controller where the performance will be improved by tuning the ranges of the values of the fuzzy subsets of the error. The change in error is directly used in the fuzzy controller. The symbols of the gains are G1 and G2 for the error (e) and change in error ∆e
Self-Tuning fuzzy roll controller
A sample simulation model for the roll control system can be implemented with the various parameters as in the below figure for a given roll control transfer function.
Simulink model for roll control fuzzy logic system and the self-tuning fuzzy logic system 1
Design of Flight Control System Using Robust Control
This modelling scheme is used to gain robustness in the modelling of uncertainties and parameter variations of the aircraft. This problem can be relayed to the aircraft control system by designing an automatic robust control landing system and an enhanced high angle of attack control system. If a robust control system is designed, it will have global stability within the known functional bounds. This probabilistic control uses randomized algorithms for with polynomial complexity to characterize system robustness
The flight model of a robust control system can be modelled from the high-incidence research model of the aircraft. The aircraft can be termed stable in both the lateral and the longitudinal direction although the combination of the angle of attack and the control surface deflection causes the aircraft to be unstable. The robustness of the automatic aircraft control system is quantified by the evaluation of the Monte Carlo effects of the plant parameter uncertainty and in maximized by searching for the control design parameter with genetic algorithms
Robustness is the probability that the stability of the automatic aircraft control system will fall within the acceptable bounds of commonly used design metrics. The probabilistic analysis and design in the method of quantifying the robustness and synthesizing a robust controller. The Monte Carlo evaluation will provide the estimates over the plant parameter space in terms of the settling time, violation, instability and others. The genetic algorithms will be used over the design parameter space
The probabilistic analysis and design of the automatic aircraft control system are designed to satisfy the design constraints and minimize the cost function of the design. The genetic algorithm operation is acted upon by evaluation, selection, crossover and mutation and a feasible control system designed will emerge when the genetic search converges to the global optimum in the design space Q and contains all possible design parameter strings.
Sequential probabilistic analysis and design algorithm
The block diagram of a dynamically scheduled robust controller
The general procedure of a robust system design of the automatic aircraft control system is by first considering parametric uncertainty in a space Q given by the following relationship q∈Q. Then a stochastic robust control law structure parameterized by the design parameter vector d. The corresponding robustness metrics are defined and corresponding stochastic robustness cost function formulated. For every candidate vector of the design parameter, a Monte Carlo simulations are used to evaluate the probability of the defined metric violation. Then lastly optimization algorithms like genetic algorithms are used to search the design parameter space to minimize the stochastic robustness cost function. The design procedure can be broken into several stages the first one being the fixing the initial state at the origin and putting the disturbance of the system as a unit impulse thus the effect of the system parameter on the closed-loop system is the only one that will be estimated. The second stage can be to change the system parameters and the initial conditions using a random number generator to specify the initial area of interest. The last stage can be to add the variation of the magnitude of the disturbance within the interval of interest .
Because the Monte Carlo estimation in the search of the control design parameters leads to a variation from the true value, the difference is noise in the evaluation of the cost function. Also, since the cost function is not convex, it leads to the use of the genetic algorithm as the best option for the search of the for the control design parameters. This is done by assuming that the design parameters are chromosomes of organisms trying to compete to survive in the environment specified by a cost J from generation to generation. The operations in each generation include evaluation, selection, crossover and mutation. The initial population is formed randomly by generating a number of chromosomes. Then each chromosome is evaluated by the Monte Carlo simulation, and the best-fitted chromosomes are selected to survive the next generation. A chromosome that will have a lower cost will translate to it having a higher fitness. The chromosomes selected are then paired randomly and subjected to crossover with a general probability of (0.6, 1.0) by swapping tails of the pairs with a binary number sequence. Then each chromosome may be mutated with a very low probability mostly < 0.1 and then altered from 1 to 0 or from 0 to 1. Thus, the uncertainty on control law design and robustness can be shown by this method
Rigid Body Equations of Motion for an Aircraft- The dynamic equations of the motion of an aircraft is combined with the wind and body axes to be represented as below:
V=Fuxm-gsinγ
α=q-qwcosβ-pcosαtanβ- rsinαtanβ
β=rw- psinα-rcosα
γ=qwrcosφ- rwsinφ
φ=pw+(qwsinφ+ rw cosφ)tanγ
ψ =qwsinφ+ rw cosφcos γ
q= 1Iyy [M+ Ixy r2- p2+Izz-Ixx rp]
p r= [Ixx -Ixz-IxzIzz]-1 [L+ Ixzpq+(Iyy- Izz)qrN- Ixzqr+(Ixx -Iyy )pq]
Where
qw= -FwzmV- gV cosγcosφ
rw= -FwymV- gV cosγsinφ
pw= pcosαcosβ+q-αsinβ+rsinαcosβ
LMN=LAMANA+ LTMTNT
FwxFwyFwz= -DSL+ TwxTwyTwz
Where the parameters are defined by
V – is the flight path velocity
α- is the angle of attack
β- Is the sideslip angle
γ , φ , ψ- are the wind axis Euler angles
p, q,r – are the body axis angular rates
pw, qw, rw, - are the wind axis angular rates
L,M ,N – are the body axis total rolling, pitching, and yawning moments
LA, MA, NA-the body-axis aerodynamic moments
D, S, L – drag, side and lift forces in the wind axis
LT, MT, NT-body-axis moments due to engine thrust
Fwx, Fwy, Fwz-wind axis total forces
Twx, Twy, Twz- wind axis thrust
M is the quotient of airspeed V and the local speed of sound α
And also the transformation matrix from the body to the wind axis is given by
LWB= [ cosαcosβsinβsinαcosβ-cosαsinβcosβ-sinαsinβ-sinα0cosα]
Aerodynamics- The aerodynamics forces and moments (FxA,FyA,FzA) and (LA,MA,NA)are represented in terms of nondimensional aerodynamics coefficients (CX,CY,CZ) and (Cl,Cm,Cn) as follows where ρ is the air density, S is the aircraft wings platform area, b denotes the span and c denotes the mean aerodynamic chord
FxA=12ρV2SCXFyA=12ρV2SCYFzA=12ρV2SCZLA=12ρV2SbClMA=12ρV2SbCmNA=12ρV2SbCn
The aerodynamic moments and the force coefficients are highly nonlinear functions of the angle of attack α, the sideslip angle β, airspeed V and the angular rates p,q,r and the control deflections
The stochastic robustness of the closed loop system of the automatic aircraft control system will characterise the probability of the system to have an unstable performance when subjected to parametric uncertainties. The probability function will be given as:
Pd=QI[Hq,C(d)]pr(q)dq
And the estimate based on N samples is given by using Monte Carlo simulations with pr(q)
Pd=1Nk=1NI[Hqk, C(d)]
The design cost function that will judge the fitness of each parameter found from genetic algorithm is give by
Jd=j=0MwjPjd
Where wj is the weight assigned to the probability that the design metric Pj is violated. The design cost and the plant model of all the design metric probabilities are determined by the designer . Where the parameters H is the plant, C is the application specific controller , q is a vector of varying plant parameters in space Q with distribution pr(q), d is the design parameter vector for the controller
The robust system is designed by augmenting a system that tracks the pilot commands with the response for handling quality across the various flight envelopes and uncertain aerodynamic parameters. The following are the aircraft commands that are to be controlled by the automatic robust flight control system using the high-incidence research model. Pilot commands should control the response using the lateral stick deflection, velocity vector roll rate and longitudinal stick deflection commands pitch rate. Lastly, the pilot commands should have the rudder pedal deflection commands sideslip angle and the lever deflection command as velocity vector airspeed
The design envelope will have a defined range of Mach number, angle of attack, sideslip and the altitude ranges
The modelling of the errors in the automatic control flight system can be done and the and the ranges of the variation of the error recorded. These variations can be used mainly to assess the aerodynamic uncertainties in the assessment of the nonlinear time responses assuming the uncertainties take a uniform distribution. The specified angle of attack should fall within the specified limits of the design and the normal acceleration of the aircraft within the specified limits also with the specified overshoot.
Sample probability design robustness metrics for an aircraft
The controller structure design
The sample force diagrams of an automatic missile system of the aircraft using robust modelling
-The non-linear inversion design is used in this method. The design utilises two separate systems one defined as the fast dynamic system and the other one is the slow dynamic system. The assumptions can be that the dynamic angular rates is faster than the angles of attack of the sideslip.The slow dynamic are derived from direct pilot inputs and the throttle engine velocity from the inversion of the velocity dynamics. For the fast dynamics design, the parameters are chosen specifically through the inversion of a first order differentiation of angular velocities and are defined in terms of the derived dynamics of angular rates .
Controller design using fast and slow dynamics
Slow dynamics design
It deals with the design parameters based on the force equations and the kinematic equations for velocity roll rate. The following equations will characterise the equation of the design of the robust slow dynamic systems
TxwTywTzw = LWB TxTyTz = LWB 2FE00 = 2FEcosαcosβ-2FEcosαsinβ-2FEsinα
And
FwxFwyFwz= -D+2FEcosαcosβ-S-2FEcosαsinβ-L-2FEsinα
The wind axis load factors will be given by
nwx=Fwxmg = -D+2FEcosαcosβmg
nwy=Fwymg = - S-2FEcosαsinβmg
nwz=Fwzmg = -L-2FEsinαmg
qw=-g(cosγcosφ+nwz)V
rw=gcosγcosφ+nwyV
Then in the design setting the values of the α and β to zero we ontain the following equations
pwc=(pcosα+rsinα)
And β=- rcosα + pwc tan α + gVnwy+cosγcosφcosα
V= 2FEcosαcosβ-Dm – g sin γ
VI = 0t[VT-(Vtrim+ VC)]dT
βI = 0t(β(T)-βc)dT
And the augmented state vectors of the slow dynamics equation is given by
xs = [VI V βI β]T
And the dynamic model of the slow dynamic system is given by
VIVβIβI=V-(Vtrim+Vc)-2ᶓvwV[V-(Vtrim+Vc)]-VIwV2β-βc-2εβ wβ(β-βc)wβ2βI
Where the valuesεv,,wV, εβ and wβ are the damping ratio, frequency for velocity dynamics, desired damping ratio of the sideslip and frequency for velocity dynamics of the sideslip angles respectively
Fast dynamics
This is computed by inverting the dynamics of the moment equation of the aircraft to derive a vector of control surface given by the angular rates. The new state variables can be written as below just as for the slow dynamics. The integral compensation will work to reduce the steady state error of the pitch rate command response
qI = 0t[qT- qC]dT
βI = 0t(β(T)-βc)dT
And the augmented state vectors of the fast dynamics is given by
xf = [p r qI q]T
And the dynamic model for the angular rates is given by
p=-wpp-pc
r=-wr(r-rc)
q=-2εqwqq-qc-wq2qI
And the dynamic model of the slow dynamic system is given by
The aerodynamic moment of this fast moving system is given by the following and the moments are still non-linear in our case
LANAMA=-LTNTMT+-Ixzpq+(Izz- Iyy)qrIxzqr+(Iyy- Ixx)qprp(Ixx -Izz)+Ixz(p2-r2)+Ixx-Ixz0-IxzIzz000Iyy -wp(p-pc)-wr(r-rc)-2εqwqq-qc-wq2qI
The following are the sample formulated robust metrics for an automatic aircraft control system which can be used in the implementation of the simulation of the designed controller
Sample metric for robust simulation design
The gap between genetic algorithm found from the global variables can be estimated using the statistical estimation theory. The genetic algorithm can be used to find the design parameters vectors of the robust automatic control system. A sample set of parameters can be given by
d=[ 0.42 1.046, 2.872.]
The performance of the closed-loop control system of the automatic aircraft control system can be treated as the set of maneuvers of the aircraft. A sample set of maneuvers can be defined as in the table below
Maneuvers at a set flight conditions
The various performance metrics like the pitch rate, velocity vector roll rate, airspeed, and sideslip angles can be plotted using Matlab from the given set of conditions
Fault tolerance
Modelling of a fault tolerance system in the robust control system is very important especially for guided military aircraft in a civilian environment. Assuming a constant gravitational force at a given altitude and airspeed of the aircraft, the non-linear longitudinal of motion is described as below
mα VT= FTsinα-LAα,δ+mVTq
q=M(α,δ)Iyy
Where the propulsion force is given byFT, the angle of attack is α and the pitch rate is q.The aerodynamics lift force is LA, the pitching moment is given by, M, are functions of α and the control surface angular deflection, δ. The aircraft airspeed is given by, VT, mass is given by m and the pitch axis moment of inertia is given by Iyy
The equations obtained can be summarised as below assuming that propulsion force is zero and lift force change and pitching moment are relatively linear.
α=ZαVTα + ZδVTδ+q
q=Mαα+Mδδ
Where Zα is the acceleration and Mα is the moment stability derivatives constant relative to α
X
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