Modeling and simulation of systems using MATLAB and Simulink Chapter 9

1、MODELLING & SIMULATION OF PITCH CONTROL IN FLIGHT USING SOME NON LINEARITIESINTRODUCTIONNon-linearity is undesirable phenomena in any system. it is present in every system existing in the world.It is to be as less as possible in a system.Unfortunately there is no theory available for validating the

2、non- linear systems.Non-linear properties of a flight control system need to be studied and appropriate actions are to be taken to minimize them.LINEARITY Vs NON-LINEARITYThe most fundamental property of linear-system is the validity of the principle of super position theorem. A non-linear system do

3、es not hold the super position theorem.Linear feedback control systems are idealised model which are made by analyst purely for the simplicity of analysis and design.Linear systems have a wealth of analytical and graphical techniques for design and analysis purpose. Non-linear systems are very diffi

4、cult to treat mathematically and there are no general methods that may be used to solve a wide class of non-linear systems.Super position principleThe principle of super position consists of the following two properties-3(1)Homogenity and (2) additivityTYPES OF NON-LINEARITIESDead-zoneBacklashFricti

5、onRelaySaturation or AmplitudeJump ResonanceFLIGHT CONTROL SYSTEMThe flight control system is designed for automatic stabilization of aircraft attitude about its center of gravity relative to three axis( longitudinal, normal and lateral).Flight altitude hold.Execution of climb, descent and co-ordina

6、ted turns.Automatic dis-engagement and warning of dis-engagement of the AP servo units.Possibility of the pilots prompt interference with the control system of the aircraft flying under the auto pilot by selecting the over ride s/w.PITCH CONTROL IN FLIGHTFCS has control in three channels.1-Roll2-Pit

7、ch3-YawPitch channel comes in operation in control of height of air-craft.NON-LINEARITIES IN FLIGHT CONTROL SYSTEMFlight control system has also some inherent non-linear ties which are following-Backlash in servo reduction gear mechanism.Dead zone in gyroscope.Saturation or amplitude in servo amplif

8、ier.Relay in flight control relay unit.Dead zone in contol stick.Flight control systemThe flight control system (autopilot) is designed for automatic stabilization of the aircraft attitude about its center of gravity.The autopilot provides for The aircraft attitude stabilization about its center of

9、gravity relative to three axis (longitudinal, normal, and lateral).Flight altitude hold.Execution of climb, descend and co-ordinated turns.The elevator auto trimming with indication of availability and direction of force applied to the control column.Automatic disengagement and warning of disengagem

10、ent of the autopilot servo units.Possibility of pilots prompt interference with the control system of the aircraft flying under the autopilot by selecting the over-ride control mode.Possibility of the elevator servo unit disengagement followed by autopilot pitch channel transfer to the synchronizati

11、on mode.Possibility of over powering the autopilot servo units via the aircraft control colunmn.Automatically flying on a great circle course or rhumb line with the help of compass system.Accomplishment the automatic corrective turns through angle of 120 by means of heading selector. For performing

12、above functions autopilot is coupled to a compass system, two vertical gyros, and a flight director system.Flight control system (AVRO- Aircraft)The Autopilot ( Avro aircraft ) provides the following facilities:-Stabilization of the Aircraft in the three axis of pitch, roll and yaw.The ability to ch

13、ange the Aircraft attitude and heading, without disengaging, by operating switches on a remote control.The ability to maintain a constant pressure altitude.The ability to turn on and maintain a pre-selected heading.The ability to make completely automatic approaches, in both the horizontal plane (VO

14、R radial and localizer) and vertical plane (glide path), to an airfield runway. Main components of auto pilot system (AVRO Aircraft )GYRO UNITAMPLIFIER UNIT SERVO MOTORSSPRING STRUT(A) Two Stage Spring Strut (B) Single Stage Spring Strut AUTO TRIM RELAY UNIT FLIGHT PANELCOUPLING UNIT HEADING CONTROL

15、 UNIT HEADING SELECTORV.O.R. FILTER UNITROLL ERROR CUT OUTCONDENSER UNIT Safety switching unit STATIC SENSING UNIT Flight Control System (AN-32 aircraft)The autopilot provides for:The aircraft attitude stabilization about its center of gravity relative to three axis (longitudinal, normal, and latera

16、l)flight altitude holdExecution of climb descent and coordinated turns with bank up to29 and pitch of 20+20 Livelily the aircraft to bank angles of up to 28+30and pitch angle of up to 20+20The elevator auto trimming witch indication of availability and direction of force applied to the Control colum

17、n Automatic disengagement and warning of disengagement of the autopilot servo units the aileron servo units shall be disengaged at the aileron deflection to angle of 5.5+0.7, or the rudder servo unit at. Automatic disengagement and warning of disengagement of the aileron and rudder servo units at a

18、bank of 32 +2.Possibility of the pilots prompt interference with the control system of the air craft flying under the auto pilot by selecting the override control mode.Possibility of elector servo unit disengagement followed by the autopilot pilot channel transfer to the synchronization mode.Possibi

19、lity of over powering the auto pilot servo units via the air craft control system. Automatic flying on a great circle course or rhumb line with the autopilot compiled to the GMK-IGE compass system.SYSTEM COMPONENTS AN-32 AIRCRAFTControl unit Servo unit Amplifier Altitude ControllerAutopilot to Compa

20、ss system couplerRate gyroAileron servo unit Elevator servo unit Rudder servo unitTrimming actuator Relay unit Trimming control unit Phase sensitive amplifier unit Elevator maximum deflection limit transmitter Aileron maximum defection limit transmitterRudder maximum deflection limit transmitter Hea

21、ding Selector.Auto pilot controllerAP Disengage button COMPONENTS USED IN PITCH CONTROL1-Gyro transmitter (pitch)2-Pitch rate gyro3-Servo amplifier4-Servo motorMovement of control surfacesModes of pitch controlSynchronization modeStabilization modeControl modeAP over-ride control modeSynchronization

22、 & Stabilization modeControl modePitch angle ControlServo motor servo motorT.F. of other componentsBlock diagram of pitch control systemSIMULINK MODEL OF OBTAINED LINEAR FCS(PITCH) SYSTEMSIMULINK MODEL FOR FCS USING SOME NON-LINEARITIEScomparison of performances of linear & non-linear model Effects

23、of dead-zone nonlinearities system performance at different dead zone non-linearities with step inputsystem performance at different dead zone non-linearitieswith sinusoidal as an inputsystem performance at different dead zone non-linearitieswith ramp inputsystem performance at different dead zone n

24、on-linearitieswith constant input Effects of saturation nonlinearitiessystem performance at different saturation nonlinearitieswith Step inputsystem performance at different saturation nonlinearitieswith ramp inputsystem performance at different saturation nonlinearities with sinusodial inputsystem

25、performance at different saturation nonlinearitieswith waveform generator inputsystem performance at different saturation nonlinearitieswith constant inputEffects of backlash non-linearitiessystem performance at different backlash nonlinearitieswith constant inputsystem performance at different back

26、lash nonlinearitieswith step inputsystem performance at different backlash nonlinearitieswith sinusoidal inputsystem performance at different backlash nonlinearitieswith ramp inputsystem performance at different backlash nonlinearitieswith waveform generator inputCumulative Effects of backlash, satu

27、ration, dead zone non-linearities,with step inputsystem performance at different nonlinearitieswith waveform generator input system performance at different nonlinearitieswith ramp inputsystem performance at different nonlinearitieswith sinusodial inputsystem performance at different nonlinearitiesw

28、ith constant inputDesigning a PID controller for Pitch control in Flight with the help of Root Locos Method ( feedback compensation)Designing a feedback system to meet the specifications which are given as belowVelocity error as small as possible ,Damping ratio 0.9 ,Settiling time 10 sec,First,we ca

29、n introduce a proportional controller (a gain ,say A).Open Loop Transfer Function ( O.L.T.F.) of un compensated system is as follows-OLTF=0.24 A s(0.96s2+1.6s+1.64) -(1)Characteristic equation of the un-compensated system is1+0.24 A s (0.96s2+1.6s+1.64) =0 1+0.24 A 0.96 s3 +1.6s2 +1.64s =0-(2)The ro

30、ot locus of eqn (2) can be found as follows-With the help of MATLAB ,num=1;den=0.96 1.6 1.64 0;rlocus(num,den),root locus of the system with out compensator Designing a compensatorIn order for the feedback system to have settling time 10 sec, all the closed loop poles in root locus must lie on the l

31、eft hand side of the vertical line passing through the point (-4 ts = - 4 10 = - 0.4 sec).From the root locus of the system with out compensator as shown in fig 1 ,we see this is not possible for any A0.As a next try we introduce an additional feedback (inner loop).The characteristic equation of the

32、 compensated system becomes as-1+A (0.24) 0.96 s3 +1.6s2 +1.64s +0.24sKt =0-(3)Partitioning the characteristic equation and selecting the value of A=5, it gives-Or , 0.96 s3 +1.6s2 +1.64s +1.20= - 0.24 s Kt ,Or , 1 + (0.24 s Kt ) (0.96 s3 +1.6s2 +1.64s +1.20) = 0 -(4)Now plotting root locus for eqn

33、(4)-The root locus of the eqn(4) is shown in fig 2,num=1 0;den=0.96 1.6 1.64 1.20;rlocus(num,den), root locus of the system with compensator The root locus of the system with compensator (fig 2) is analyzed now. The Damping ratio 0.9 line intersects the root locus at two points .Both the points lie

34、on the left hand side of the vertical line passing through ( - 0.4 ).One of these two points is ( - 0.41 + j 1.1).Now putting the value of s =- 0.41 + j 1.1 in eqn (4) ,It gives,Kt = 1.5670 ,So , now selecting the value for Kt = 1.5670 , A = 5.0 (already taken), a new compensated simulink model in c

35、ombination with un-compensated simulink model is developed . Comparison of simulink models (between compensated and un-compensated)Results of Comparison of simulink models (between compensated and un-compensated systems)Stability check (using Routh table )The characteristics eqn of the obtained comp

36、ensated system can be written as follows- 0.96 s3 +1.6s2 +1.64s + 0.24 s Kt +1.20 =0,Putting K=0.24Kt ,0.96 s3 +1.6s2 +(1.64 + K )s +1.20=0 ,Routh table is as under-s3 0.96 (1.64+K)s1 1.6(1.64+K)-1.152 / 1.6 0s0 1.20 0For stable operation , 1.6 (1.64+K)-1.152 / 1.6 0,it gives K - 0.92 ,or, Kt - 3.83

37、.This condition is satisfied by our obtained compensated system as Kt =1.5670.Now we can say that our obtained compensated system is stable.CommentsThe above developed PID controller is developed using feed back compensation .The feed back compensation provides greater stiffness against load disturb

38、ances.Suitable rate gyroscope is available for feed back compensation.From fig 4 , its performance indices are as given below % over shoot = 14 %Designing a PID controller (connected in cascade with the system) for Pitch control in Flight G (s) = 0.24 0.96 s3 +1.6s2 +1.64s , C(s)/R(s) =Kp 0.24 (0.96

39、 s3 +1.6s2 +1.64s ) / (1+ Kp0.24 0.96 s3 +1.6s2 +1.64s ) , = Kp(0.24) / (0.96 s3 +1.6s2 +1.64s+ Kp0.24)The critical gain is determined by Routh array for the characteristics eqn0.96 s3 +1.6s2 +1.64s+ Kp0.24 =0 is as given below-Routh table is as under-s1 0.96Kp(0.24)-(1.6)(1.64)/1.6 0s0 Kp0.24 0For

40、stable operation , 0.96Kp(0.24)-(1.6)(1.64)/1.6 0,it gives KpHence, The critical gain Ker=11.39,To find the frequency of oscillations( Ter)1.6s2 + Kp 0.24 =0 (Auxiliary eqn found from above Routh table)putting Kp=11.9,s = j1.3 rad/sec, Ter =2/1.3 =4.83 sec.Substituting these values in following stan

41、dard eqnsKp =0.6 Ker ,i = 0.5 Ter,d = 0.125 Ter,We get ,Kp=6.834,i =2.46,d =0.60,The Transfer function of developed PID controller may be in the following form-Gc(s) = Kp ( 1 + 1 / is + d s )This PID controller is connected in cascade to the systems transfer function.The PID controller is prepaired

42、on above discussion basis & then simulink model is developed as given in fig below-.Comparing the results with the pre developed PID controller (feedback compensation)- comparing the results with the pre developed PID controller (feedback compensation)CommentsThe above developed PID controller is de

43、veloped using cascade compensation .From fig 6 , its performance indices are as given below % over shoot = 62 %Comparisons Between Both designed PID controllers1. We can easily see from fig 6 (yellow represents design 1 and green represents design 2) that, there is an appreciable difference in % ove

44、r shoots of the outputs for the given step input. 2. Rise time in design 2 is better. 3. There is no remarkable difference in settling times of both the designs. Conclusion From the above discussion it is concluded that design 1 is the desired design.Design of P, I, D, PD, PI, PID, Fuzzy controllers

45、Design of P, I, D, PD, PI, PID, Fuzzy controllers and their use on linear as well as non linear system is carried out .The compared results of application of these controllers(for linear as well as non linear system) are shown after each model.From fig 1 to fig 7 it is clear that the performance of

46、linear system can be controlled and improved by different controllers i.e. P, I, PD, PI, PID, Fuzzy controller .The blue line curves show linear system and green line curves show non linear system .But the performance of non linear system can not be controlled and improved by different controllers i

47、.e. P, I, PD, PI, PID controllers . Only Fuzzy controller is able to control the non linear system.The results of application of Fuzzy controller on both the systems is satisfactory.Proportional controllerThe proportional mode adjusts the output signal in direct proportion to the controller input (w

48、hich is the error signal) . The adjustable parameter is the controller gain.A proportional controller reduces error but does not eliminate it(unless the process has naturally integrating properties) ie an offset between actual and the desired value will normally exist .Results of application of Prop

49、ortional controller Integrational controllerResults of application of Integrational controller Derivative controllerResults of application of Derivative controller Proportional-Integral ControllerThe additional integral mode (often referred to as reset ) corrects for any reset (error) that may occur

50、 between the desired value (setpoint) and the process out put automatically. The adjustable parameter to be specified is the integral time of the controller.V(s) / e(s) = Kc1+1/Ti(s)Results of application of Proportional-Integral ControllerProportional -Derivative ControllerResults of application of

51、 Proportional -Derivative ControllerProportional-Integral-Derivative (PID)ControllerThe PID controller is the most popular feedback controller used within the process industries. It has been successfully used over 50 is a robust easily understood algorithm that can provide excellent control performa

52、nce despite the varied dynamic characteristics of process plant. 9The mathematical representation of the PID Controller is as below-V(s) / e(s) = Kc1+1/Ti(s) + Td(s)Results of application of Proportional- integration-Derivative (PID) Controller Design of fuzzy controller Results of application of Fu

53、zzy Controller Results of Different controllers (P, PD, PI PID, Fuzzy controllers) connected at a place for linear systemResults of Different controllers (P, PD, PI, PID , Fuzzy controllers) with some nonlinearities connected at a placeResults of Different controllers (P, PD, PI, PID , Fuzzy control

54、lers) for Increased dead zone nonlinearities connected at a placeConclusions & future scopeIt is quite clear from the results that the nonlinearities are affecting the performance of flight control system (FCS). Flight control system (FCS) simulink model has been evolved using some applicable nonlin

55、earities, which is not performing in a desired way, as it is responding to a distorted output for step input , ramp input, constant input.Two different controllers have been developed based on feed back method and cascade methodes.Comparison between these two controllers carried out, feed back metho

56、d controller is most desirable.Different controllers have been developed i.e. proportional ,derivative, integral, proportional derivative (PD), proportional- integral (PI), proportional- integral-derivative (PID), Fuzzy controller.These controllers are used on both modles (linear as well as non line

57、ar model).It can be seen from the results that all the developed controllers (including PID controller )are unable to control the non linear system.It can be seen from the results that the Fuzzy controller is not only controlling the non linear system, but also giving the desired result.This work ma

58、y be further extended as follows Adaptive / Neuro- fuzzy controller may be developed .On-line tunning of Fuzzy Logic Controller (FLC) may be done using Genetic Algorithm.Yow and Roll control can be done with Fuzzy /Neuro Fuzzy controllers.References1 HS-748 (Avro) maintenance manual.2 AN-32 aircraft

59、 maintenance manual3 Dr Sushil Das Gupta,“ control system theory” Khanna publishers ,Delhi,1975. 4E. H. Mamdani and S. Assilian , “ An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller”, Int. journal Man-Machine Studies, UK, 1975. 5 D. McLean , “Globally Stable Nonlinear Flight Contro

60、l System”, IEE PROC, Vol 130, Pt. D, No. 3,England., MAY 1983.6 Benjamin C. Kuo , “Automatic control systems” Prentice Hall of India new Delhi ,1990.7James C. Bezdek , “Fuzzy Models What are they and Why ?”, IEEE Transactions on Fuzzy Systems,Vol. 1. No.1, February 1993.8 Jan Jantzen “Design Of Fuzz