CHAPTER 5 AUTOMATIC LANDING SYSTEM

Transcription

1 117 CHAPTER 5 AUTOMATIC LANDING SYSTEM 51 INTRODUCTION The ultimate aim of both military and commercial aviation is allweather operation To achieve this goal, it should be possible to land the aircraft without visual reference to the runway This can be accomplished by a landing system which will guide the aircraft down to a predetermined glide slope and then at a pre-selected altitude to reduce the rate of descent and cause the aircraft to flare out and touch down with an acceptably low rate of descent Blakelock(1991), Ghalia(1993), Crassidis(1993), Cohen(1995), Assi(1997), Iiguni(1998) and Christopher Wilkin(2001) The landing system will enable the UAV to land on a landing field One of the very long term goals of the simulation project is to achieve fully autonomous flight The development of an autonomous landing simulation completes the requirement to begin fully autonomous flight simulation In this chapter, the issues of control algorithm design and operational region identification for a baseline controller is addressed The main focus of the thesis is on the landing operation of an UAV The development and preliminary results of the design of a baseline controller using conventional and backstepping approach for automatic landing of an UAV is presented The control objective is to maintain the aircraft flying along the glide path followed by flare path and touchdown to the desired runway position In this chapter, the final stages of the landing procedure have

2 118 been concentrated The automatic landing system should provide feedback control to allow the aircraft to follow a trajectory in the XYZ space The trajectory, called glide-path, is determined by the position and orientation of the runway and the flight-path angle A typical automatic landing system uses a radio beam directed upward from the ground that can be detected by aircraft sensors to control the landing The control commands to be used are usually three: throttle, pitch and roll command inputs The throttle command is generally left open-loop and it is directly controlled by the pilot or by a separate subsystem Pitch and roll commands usually have some level of feedback control to help the pilot The pilot can modify the command actions using the control stick In this chapter, only longitudinal motion (motion in a vertical plane) is considered during the landing Lateral motion is required primarily to point the aircraft down the runway and it is assumed that most of it is accomplished prior to the landing 52 SEQUENCE OF LANDING SYSTEM The profile of an automatic approach consisting of glide path and flare path is illustrated in Figure 51When the aircraft has descended to 1,500 feet radio altitude, the localizer and glide slope beams are captured The localizer and glide slope beam signals control the aircraft about the roll and pitch axes so that any deviation from the desired glide path is automatically corrected to maintain alignment with the runway At a radio altitude of 330 feet, the aircrafts horizontal stabilizer is automatically repositioned to begin trimming the aircraft to a nose-up attitude The elevators are also deflected to counter the trim and to provide subsequent pitch control in the trimmed aircraft When an altitude is reached at which the landing gear is 45 feet above the ground the flare mode is automatically engaged The flare mode takes

3 119 over pitch attitude control from glide slope and generates a pitch command to bring the aircraft onto a 2 feet/second descent path At the same time, a throttle retard command signal is supplied to the auto throttle system to reduce engine thrust to the limits compatible with the flare path Figure 51 Automatic Landing System 53 GLIDE SLOPE CONTROL SYSTEM The first step in the design of autonomous landing is to define the glide path and flare path geometries kaminer(1990), Ju(2001),Jih-Gau Juang(2001)and Kim(2005) The glide path is defined as a line from some starting point to the end of the runway For this study, a glide path angle of - 25 degrees is used, so the starting point is defined by the LLA (Latitude Longitude Altitude) position of the end of the runway and the desired final approach distance is 23,000 ft in this case A real autonomous landing controller would rely on signals emitted from stations at the runway to maintain the appropriate glide path As a result, the controller becomes more sensitive as the aircraft gets closer to the runway threshold To simulate this relation of the range(r) from the aircraft to the runway, the glide path command signal is applied to calculate the range Figure 52, shows the glide

4 120 path geometry where the commanded height above ground is a function of the range Ebrahimi(1990) H = R*sin(25 o ) H x R 25 o Runway Figure 52 Glide Path Geometry 531 Design Objectives Landing of the aircraft requires a complex combination of e and t to coordinate the speed/pitch/height control The UAV is to be controlled to a height of, approximately 1200 feet and a range of 23,000 feet from the touch down point The UAV is to be guided through the specified glide slope of 25 0 under no wind as well as under specified wind conditions Nho (1999) and Liao (2005) Certain limits are imposed on the dynamic variables, which are: ; ; 20 ( e, a, r ) 20 0 Servo rate 100 o / sec (51) The Elevator deflection ( e) and throttle are the aircraft control input parameters Pitch attitude ( ) and deviations from the glide path (d) are the aircraft control output parameters To simulate, the increasing sensitivity to range, the error signal should be defined as the error in glide slope angle ( ) As shown in Figure 53, the airborne glide path receiver will measure the angle of error between the glide slope and the UAV This angular error will then be converted to longitudinal deviation (d) as

5 121 d R sin( ) (52) The aim is to control the deviation from the glide path ie d; once the condition of d = 0 achieved, the UAV follows the glide slope line Now, d U sin( ) ref and for small angles, (53) d U ( ) ref (54) Figure 53 Geometry of Longitudinal Control Integration of the above equations gives the information about the longitudinal deviations, which is fed to the controller and then the appropriate control input is computed If d is not directly measurable, this can be written as Sin = d/r (55) which is measurable from ILS shown in Figure 54 Figure 54 Range Vs Angular Error

6 122 Using this relationship, the angular error can be found in terms of the range, the angular error becomes increasingly larger as the range gets smaller At some point, the sensitivity of the glide slope controller exceeds its ability to keep the airplane on the glide path At this point, the flare controller, which is not dependent on the range, must take over control of the airplane ref = 0 Coupler comm o UAV & U Autopilot 573s d 573 R 25 o Figure 55 Glide Slope Control System The block diagram representation of the glide slope control system is shown in Figure 55 The pitch hold autopilot is the basic autopilot mode for this system to descent from a particular height The glide slope controller has a coupler, which contains a PID controller and a lead compensator 54 FLARE PATH CONTROL SYSTEM In Flare path region, the aircraft follows an exponential path During the flare maneuver, the pilot takes transition from flying a straight line to an exponential path to slow the descent rate of the airplane This can be simulated by defining an exponentially decaying flight path and using altitude above ground to generate the error signal to the controller Figure 56 shows the flare path geometry with the intended touchdown zone approximately 500 feet from the runway threshold The equation which governs the idealized exponential flare trajectory is: 0 t h h e (56)

7 123 H H = H e^(-x/ ) 0 x Touchdown ~ 500 ft from threshold Runway Figure 56 Flare Path Geometry The exponential decay constant is a function of the distance x and the distance to the touchdown zone from the threshold must be selected such that the airplane touches down at an acceptable vertical speed The block diagram representation of the flare control system is shown in the Figure 57 The outer loop gives descent command h r The pitch attitude hold autopilot is the basic autopilot mode for this system to descent from flare entry height h 0 hr + - h Coupler (comm) A/C and autopilot h h 1 s h 1 Figure 57 Flare Path Control System Upon main gear touchdown, the elevator must allow the nose of the airplane to move downward to contact the nose gear with the ground and the brakes must be applied Davies and Noury (1982) Because exponentially decaying functions never actually reach zero, the elevator control command must be switched from the flare path to neutral upon main gear touchdown The brakes must be applied smoothly after touchdown otherwise the gear will fail A rate limiter after a switch can be used to accomplish this operation

8 124 With the flight path geometry defined autopilots for pitch, yaw and roll must be designed to fly the airplane autonomously Then controllers for glide slope, flare and directional course must be designed to keep the airplane on the desired flight path Because of the complexity of the problem, it was determined that only pitch control should be used to accomplish the landing Upon successful landing in ideal conditions, lateral controllers can be designed for conditions with wind disturbances 55 NAVIGATION SYSTEM A navigation system is a central element for any automatic landing approach system The purpose of a navigation system is to provide guidance cues to the approach controllers The approach controllers developed in this study are discussed in the following paragraphs At present during a landing approach to a runway, an approach glide path is originated from a point on the center line of the runway A track approach path is generated similarly As stated in the objectives in Chapter 1, the navigation concept developed in this study aims to build on the systems and procedures currently in place, account for future navigation requirements, facilitate the seamless integration of the UAV fleet with the piloted fleet and allow for truly autonomous landing operations On considering the limitations of the scope of this study, the Navigation System presented does not account for any pre navigation requirements The concept developed is presented schematically in Figure 58 Runway and aircraft positions are used to calculate the relative distance and velocity The time to touchdown is computed using the relative distance and velocity A 25 degree approach glide path is generated from that point and the aircrafts vertical deviation from that approach glide path is calculated

9 125 Likewise, an approach track path is generated coincident with the center line of the runway and the aircrafts lateral deviation from that approach track path is calculated The vertical and lateral deviations are the input to the approach controllers Figure 58 Navigation System Data Flow Diagram 551 Time to Touchdown Calculations The Navigation System uses the aircraft and runway earth axes position This is to emulate the GPS reference system and as such the system is conceptually compliant with JPALS requirements The Navigation System is based on predicting the position of the touchdown point at a time in the future, which corresponds to that time at which the aircraft touches down on the center line of the runway This time is a function of distance between the aircraft and the starting position of the runway and their relative velocity

10 126 Figure 59 Time to Touchdown Geometry Time to touchdown t td, is defined as t td d d (57) where the distance between the aircraft and the touchdown point on the runway d is defined as d= (x -x ) 2 +(y -y ) 2 +(z -z ) 2 E E E E E E ac(tdp) ac(tdp) ac(tdp) (58) In this implementation, the rate of change of distance between the aircraft and the touchdown point ḋ is averaged over a period of a half a second This is to reduce the time to touchdown estimates sensitivity to atmospheric disturbances The period of half a second was determined through experiment performed with the simulation

11 127 Time to touchdown t td, is input to the aircraft motion variables and the predicted touchdown positions ( x, y, z ) and orientation E E E ptd ptd ptd ( ac, ac, ac ) are the outputs ptd ptd ptd The Navigation System presented here is a central element of this study Assuming that GPS is used for position and velocity measurement, this system conceptually satisfies the Joint Precision Approach and Landing System requirements The system can be used by piloted aircraft as well UAVs This facilitates autonomous approaches to landing In this regard, the system satisfies the constraints imposed upon it from the outset of this study and achieves the associated objectives of the present study 56 AUTO THROTTLE APPROACH TRACK CONTROLLER Three approach controllers have been developed which control the aircrafts approach flight path, as generated by the Navigation System and approach speed during a landing approach The general form of an approach controller is presented in Figure 510 The approach controllers consist of three components: (1) An auto throttle (2) An approach track controller and (3) an approach glide path controller An autothrottle controls the aircrafts speed via the throttle A track controller controls lateral deviation from the approach track via ailerons and rudder These two components are common to all three approach controllers developed Each of the three approach controllers differs by the manner in which vertical deviation from the approach glide path is controlled The approach glide path controllers are presented in section 51

12 128 Figure 510 Approach Controller The design and performance of the auto throttle and track controller are presented in the following sections of this chapter Before reviewing these systems an overview of the design process employed in the development of all controllers is presented 561 Design Methodology The design approach is based on standard design methods for classical control systems Classical control systems have been used as they provide a high level of visibility in the design stage The design began with an examination of the performance requirements of each controller along with the constraints on the systems Standard Published requirements for Automatic Landing Systems are not available and as a result, performance requirements were defined for the performance of the track and approach glide path controllers These requirements and constraints are presented in the following sections From the non-linear simulation model, linearised longitudinal and lateral-directional decoupled models were extracted at the design point The

13 129 stability and control characteristics of the linear models were validated against those characteristics mentioned in MIL-STD B The appropriate Stability Augmentation System was implemented in the respective linear models The stability and control characteristics of the augmented aircraft were validated against those characteristics in MIL-STD B The response of the linear models to small control inputs were validated against the response of the non-linear model to the same inputs The architecture of the autothrottle, track controller and the approach glide path controllers were defined using knowledge of the aircrafts behaviour The knowledge of requirements and constraints of each system was gained through the literature review Having defined the architecture, the control system gains were tuned using Simulinks Non-linear Control Design (NCD) tool The NCD tool is an interactive Graphical User Interface (GUI) This tool can be placed in a Simulink model and the signal which is being controlled is attached to it The NCD tool takes the form of a response plot which presents the signal being controlled against time Performance constraints are presented on this response plot These constraints are user defined During this design process, step inputs to the system being developed were used as inputs to the model The tool requires that initial values of the controller gains be defined The NCD tool then runs the simulation using the initial controller gains and compares the response against the constraints If the response is outside the defined constraints, the controller gains are varied and the new response is plotted This process iterates till the response is within the performance constraints

14 130 This tool is very efficient while tuning the controller gains, however care has to be exercised in its use as this tool does not replace good engineering judgement This tool is a mathematical process and the response characteristics are limited to the time period defined in the simulation model Therefore the long term performances of the resultant controller gains were examined before proceeding with the design process The controllers used are Proportional Plus Integral (PI) and Proportional Plus Integral Plus Derivative (PID) controllers The proportional control provides feedback of the error signal The integral term will drive the error to zero and the derivative term will smooth the transient response Having selected the control system gains in this manner, the frequency response characteristics of the controller were examined For all controllers, the minimum acceptable gain margin was defined as 6 db and the minimum acceptable phase margin was defined as 30 degrees These minimums were suggested by McLean(1990) The Military Specification document for Flight Control Systems Design, Installation and Test of Piloted Aircraft, Mil-F-9490 (1992) was not available for reference; however, Kanade (2000) has quoted in Mil-F-9490 a requirement of a minimum phase margin of 45 degrees and minimum gain margin of 6 db These facts were known after the controllers were designed and hence it was decided not to amend the design as a 30 degree phase margin minimum is found to be adequate for the purposes of this study The performance of the controllers was then assessed in the nonlinear simulation model This performance assessment consisted of step responses and the response to atmospheric disturbances The final assessment

15 131 consisted of testing the auto throttle, track controller and each of the approach glide path controllers in the non-linear simulation environment during approach to landing It is an iterative process and several iterations between the linear and the non-linear simulation environments were required until satisfactory performance was achieved 57 AUTOTHROTTLE Accurate control of airspeed during any approach to landing is paramount to a safe and efficient approach This is especially true in the UAV landing environment which is dominated by atmospheric disturbances It should be noted that the speed loop of the pitch stability augmentation system is disengaged when an autothrottle is engaged 571 Performance Criteria Definitive performance requirements for landing based UAVs were unable to be sourced during the course of this study As a result, existing public domain performance requirements for piloted aircraft and UAV have been used in addition to sound engineering judgement Prosser and Wiler (1976) present a first attempt at defining flying qualities requirement for Remotely Piloted Vehicles (RPV), based largely on military specifications for flying qualities of piloted aircraft, MIL-F-8785B Prosser and Wiler suggest that an Auto throttle system should maintain airspeed within ft/sec and that any periodic oscillations within this limit shall not interfere with mission performance Landing specifications include constraints on the way, the approach should be taken (ie, a desired range of values for the aircraft state variables and flight-path) Table 51 shows typical values for a landing approach)

16 132 Table 51 Typical Values for a Landing Approach Parameters Angle of attack ( ) Vertical deviation (d v ) Horizontal deviation (dh) Sink rate Flight-path angle Airspeed Pitch angle error ( ) Roll angle error Typical values Between 1015 degrees Maximum 5 feet Maximum 15 feet 2-10 ft/sec Constant between 25 to 35 degrees ft/s Maximum 3 degrees Maximum 10 degrees 572 System Architecture The Autothrottle system architecture is presented in Figure 511the control system gains were optimized using the Simulink NCD tool u s Figure 511 Autothrottle Architecture The control law for the autothrottle controller is defined as k u k u dt (59) pu iu The control system gains are selected as k 659 RPM / ft / s (510) pu k 1564 RPM / ft / s iu

17 Performance Assessment using the Linear Model The open loop frequency response characteristics of the autothrottle are presented in the form of a Bode diagram in Figure 512 The auto throttle control loop is broken along the feedback path The phase margin is <90 degrees at 142 rad/s and the gain margin is infinite Figure 512 Autothrottle Open Loop Bode Diagram The closed loop frequency response characteristics are presented in the form of a Bode diagram in Figure 513 The closed loop bandwidth is 115 rad/s No other loops are active for this response

18 134 Figure 513 Autothrottle Closed Loop Bode Diagram The aircraft autothrottle response to a unit step airspeed demand is presented in Figure 514 All variables presented in Figure 514 are perturbations around the trim flight condition, except for altitude and normal acceleration which are presented in true form The rise time of the response is 114 seconds and the settling time is 245 seconds with no overshoot Figure 514 Autothrottle Response to Unit Step Demand

19 Performance Assessment using the Non-Linear Model The Autothrottle response to a rectangular pulse speed demand is presented in Figure 515 It should be noted that the baseline glide path controller, which is presented in next section, is actively controlling altitude deviations due to velocity change It can be seen that the performance criteria presented in section 571 are achieved The Autothrottle response to continuous moderate turbulence, as defined by Mil-8785C(1986) is generated With the baseline glide path controller controlling altitude deviations is presented in Figure 516 In this case, the airspeed is controlled to within ±01 knots, achieving the performance criteria presented in section 571 It should be noted that engine speed is presented as a perturbation about the trim value, while all other variables are presented in true form Figure 515 Autothrottle Response to Rectangular Pulse Speed Demand

20 136 Figure 516 Autothrottle Response to Continuous Moderate Axial Turbulence 58 TRACK CONTROLLER Flying an approach onto the tight confines of the landing area of runway dictates precise control of the aircrafts lateral position A track controller has been developed which controls the aircraft with respect to lateral position deviation from the desired track 581 Performance Criteria Prosser and Wiler(1976) suggest that a heading hold autopilot should maintain the aircraft in its existing heading when engaged within a static accuracy of ±05 degrees in smooth air When a heading autopilot is used to change heading, the system shall automatically turn through the

21 137 smallest angle to achieve the heading change and the bank angle while turning to the selected heading shall provide satisfactory turn rates and precludes impending stall The aircraft shall not roll in a direction other than the direction required for the aircraft to assume its proper bank angle In addition, the roll-in and roll-out shall be accomplished smoothly with no disturbing variation in roll rate Many of these points are relevant to the operation of the track controller, however further performance requirements were imposed In calm air a lateral position error of 4 feet shall be corrected within five seconds with minimal sideslip during correction within ± 05 feet The value of 4 feet was selected through investigation of the effects of varying lateral turbulence levels on aircraft lateral position Atmospheric disturbances shall be attenuated For steady state operation, track error shall not be greater than ± 05 feet and sideslip angle should be zero 582 System Architecture The track controller contains two components The aircrafts lateral position is controlled via ailerons while the aircrafts sideslip is controlled via the rudder The input to the sideslip controller is measured in sideslip angle The sideslip controller ensures that the aircraft aligns itself with the approach track This ensures that the aircraft does not drift, especially important when close to the runway Ideally the aircraft should touchdown on the runway with zero sideslip Any lateral velocity at touchdown has the effect of inducing a side load on the landing gear The sideslip controller also provides turn coordination

22 138 The control law for the track controller is defined as d k k dt k t d p t t t d dt (511) where the control system gains are selected as k rad / ft p k rad / ft t k 0255 rad / ft d (512) The control law for the sideslip controller is defined as d r k k dt k t d p t t t d dt (513) The control system gains are selected as k p k t k d 372 rad / rad 449 rad / rad 049 rad / rad (514) It was found through testing that to optimize the systems performance for both steady wind and atmospheric disturbance operation, the bank angle demand, d, should be limited to 6 degrees This is implemented for the responses presented in section 584

23 Performance Assessment using the Linear Model The open loop frequency response characteristics of the sideslip controller are presented in the form of a Bode diagram in Figure 517 The control loop is broken at the output of the sideslip sensor The roll Stability Augmentation System loops is closed The phase margin is 885 degrees at 056 rad/s and the gain margin is 184 db at 628 rad/s The closed loop frequency response characteristics of the sideslip controller are presented in the form of a Bode diagram in Figure 518 The roll Stability Augmentation System loops are closed The closed loop bandwidth is 058 rad/s This bandwidth is consistent with the fact that the aircrafts directional dynamics are slower than pitch and roll Figure 517 Sideslip Controller Open Loop Bode Diagram

24 140 Figure 518 Sideslip Controller Closed Loop Bode Diagram The open loop frequency response characteristics of the lateral position controller are presented in the form of a Bode diagram in Figure 519 The Navigation System is not included in this response Lateral position feedback was used to emulate the Navigation System in the linear simulation model The control loop is open along the lateral position feedback loop The sideslip control loop and yaw Stability Augmentation System loops are closed The phase margin is 144 degrees at 176 rad/s and the gain margin is 49 db at 126 rad/s

25 141 Figure 519 Lateral Position Controller Open Loop Bode Diagram The closed loop frequency response characteristics of lateral position controller are presented in the form of a Bode diagram in Figure 520 The sideslip control loop and yaw Stability Augmentation System loops are closed The closed loop bandwidth is 121 rad/s The aircrafts response to a unit step sideslip demand is presented in Figure 521 The rise time is 65 seconds and the settling time is 8 seconds The roll Stability Augmentation System loops are closed for this response

26 142 Figure 520 Lateral Position Controller Closed Loop Bode Diagram Figure 521 Sideslip Controller Response to a Unit Step Demand The aircrafts response to a unit lateral position demand is presented in Figure 522 The rise time is 72 seconds and the settling time is

27 seconds The effects of adverse aileron yaw are evident from this response plot As with the closed loop frequency response, the Navigation System is not included and the control loop is closed using lateral position The sideslip and yaw Stability Augmentation System loops are closed Figure 522 Lateral Position Controller Responses to a Unit Step Demand 584 Performance Assessment using the Non-Linear Model The track controller response to a rectangular pulse lateral position demand is presented in Figure 523 The track controller response to continuous moderate lateral turbulence is presented in Figure 524 The auto throttle and baseline approach glide path controllers are active for both scenarios The baseline approach glide path controller is presented in Chapter 6 The performance criteria presented in section 581 can be seen to be achieved Considering the inherent atmospheric conditions of the landing environment, control of lateral position deviation in turbulence becomes an important design factor

28 144 Figure 523 Track Controller Responses to Rectangular Pulse Lateral Position Demand Figure 524 Track Controller Responses to Continuous Moderate Turbulence

29 APPROACH GLIDE PATH CONTROLLERS Two different approach glide path controllers have been developed to assess different control strategies for suitability to the aircraft landing task The first or baseline approach glide path controller controls vertical deviation from the approach glide path using elevator control The second approach glide path controller controls vertical deviation from the approach glide path using back stepping design The original intention was to develop a controller which responded to vertical deviations from the approach glide path with pure vertical translation of the aircraft This required a constant pitch attitude to be maintained throughout the approach A controller was developed which maintained a constant pitch attitude through elevator control 510 BASELINE APPROACH GLIDE PATH CONTROLLER In the absence of available published requirements, the baseline approach glide path controller was designed to provide tracking of ±05 feet in steady wind conditions, attenuation of atmospheric disturbances and shall be capable of correcting a 4 foot deviation within 2 seconds 5101 System Architecture The baseline approach for glide path controller architecture is the input to the system is vertical deviation, h is calculated by the Navigation System as presented in section 55 A pitch attitude demand, d is calculated based on Proportional-Plus-Integral-Plus-Derivative (PID) control of vertical deviation

30 146 The control law for the baseline approach glide path controller is defined as dh kpeh k ie h dt k (515) d de dt The control system gains are selected as kpe rad / ft k ie rad / ft k rad / ft de Pitch attitude demand, d, is limited to ± 10 degrees (515a) 5102 Performance Assessment using the Linear Model Even though the navigation system is not included in the frequency response characteristics and step response of the baseline approach glide path controller, it is considered as a comparator, with the altitude demand as a function of predicted touchdown point The system from which the frequency response characteristics and step response have been extracted, where the Navigation System has been replaced by a comparator The open loop frequency response characteristics of the approach glide path controller are presented in the form of a Bode diagram in Figure 525 The control loop is broken along the feedback path The autothrottle loop is closed for this response The phase margin is 302 degrees at 105rad/s and the gain margin is 689 db at 123 rad/s

31 147 Figure 525 Baseline Glide Path Controller Open Loop Bode Diagram The closed loop frequency response characteristics of the baseline approach glide path controller are presented in the form of a Bode diagram in Figure 526 The closed loop bandwidth is 183 rad/s The same additional loops are active for the closed loop response as in the open loop response Figure 526 Baseline Glide Path Controller Closed Loop Bode Diagram

32 148 The aircraft response to a unit step altitude demand is presented in Figure 527 All the variables presented have perturbations about their trim value except for normal acceleration and altitude which are presented in their true form The rise time of the response is 13 seconds and the settling time is 28 seconds with no overshoot The auto throttle control loop was closed for this response Figure 527 Baseline Glide Path Controller Response to Unit Step Demand 511 GLIDE PATH CONTROL DESIGN USING BACKSTEPPING The most popular control scheme for flight control has been based on a conventional Proportional-Integral (PI) controller design method with desired specifications as presented in the previous sections The Lyapunovbased design method has been developed as a next approach for aircraft flight control systems These studies utilized the backstepping method Harkegard(2000) and Sharma (2002) to construct stable nonlinear controllers

33 149 in order to improve the performance of the flight path control systems The design of auto landing controller has been considered as a very challenging problem To achieve successful touchdown, aircrafts must capture the beams emitted from the glide slope and localizer towards the runway The previous sections explained about the designs of glide slope and localizer couplers and track using classical control techniques The stability augmentation of inner loop and path tracking of the outer loop were explained In the final approach, the tracking controller must guide the aircraft to follow the flight path toward the runway However, the guidance and control loops are designed separately in automatic flight control systems The objective of this section is to use the back stepping design procedure to establish a flight path angle control and to achieve the desired glide slope trajectory The back stepping tracking controller is able to guide aircraft for desired trajectory for glide slope It is different from designing guidance and control loops separately for autopilot system The effectiveness of capturing glide slope centreline is illustrated by 6-DOF simulation during the final approach 5111 Backstepping Design Procedure In the final approach, the aircraft is guided to the runway by tracking the glide slope During the glide path interception, the aircraft is guided to approach the runway until the touchdown occurs so that the tracking error angle becomes zero This is detailed in the section 53 t is defined by V ( / Rdt 0 ref ) (516)

34 150 A backstepping design for flight control deals with the longitudinal motion of the aircraft and the control of rigid body dynamics A detailed description of aircrafts small perturbation equations of motion has been explained in chapter 2The assumptions are consistent with initial straight and level flight with constant thrust In this section, an unmanned vehicle model in longitudinal and vertical axes is used The motion equations for longitudinal aircraft with short period mode and phugoid mode are detailed in chapter 2 and rewritten as u XuU X Xqq X X e e Zuu Z Zqq Z Z e e q Muu M Mqq M M e e (517) Where u : Longitudinal velocity (ft/s) q : pitch rate (rad/s) : pitch angle (rad) e : angle of attack deflection (rad) X*, Z*, M * : Stability and control derivatives The parameters X *, Z*, M * : Stability and control derivatives of an unmanned air vehicle flying in a trimmed condition (An altitude of 1500 ft and speed of 120 ft/sec) In this section, the backstepping design is used to derive control laws for the flight path angle tracking The flight control system generates elevator command from the flight path angle command The motion equations for a longitudinal aircraft are described in chapter 2 In this state-space model, the control input e is a function of all state variables ( u,, q, ) However,

35 151 equation (517) is not suitable for designing flight control laws via back stepping Hence, if the surface deflection e on the lift is neglected, the state space model is on correct form for backstepping design In otherwords, the lift contribution of the control surface is neglected, because it is much smaller than the moment contribution The lift force on the surface deflection is intentionally neglected for deriving the longitudinal flight control laws, but the simulation model does not use this assumption Thus the dynamics of aircraft equations without airspeed control can be rewritten as Z Zqq Z q q M Mqq M M e e (518) Using, q and assumed Z q 1,(518) can be rewritten as Z ( Z Z ) q q M Mqq ( M M ) M e e (519) The control objective is to track the flight path angle command for glide slope tracking The design steps for the flight path control by back stepping are given below Step 1: variables as For the flight path angle tracking, we define three new state

36 152 z 1 ref z 2 des (520) z 3 q q des The time derivative of z 1 is z1 ref Z z ( Z Z ) Z 1 ref ref (521) In order for stabilization can be chosen as 1 ( Z Z ) ( Z K z ), des ref 1 1 ref K1 Z (522) Assuming Z Z, Z 0, Z 0 and ref 0,the desired value of can be written as Z 1 K z C z des ref ref (523) Where C Z 1 K, C Now consider, z 2 des C ( C 1) 1 1 ref (524) Then z 1can be written as (assuming Z 1 and Z 0 ) q z1 ( Z K ) z Z z Z [(1 C ) z z ] (525) q Differentiating (524) gives z2 q C 1 q C ( Z z Z Z ) 1 1 ref

37 153 Where q C (526) 1 Z [(1 C ) z z ] (527) Using (525) and (526), the time derivative of is Z [(1 C1) z1 z 2] Z ( q ) (528) Step 2: Define a Lyapunov function candidate as C V F( ) z z 2 (529) where F and C2 are positive definite Using (525),(526) and (528),the time derivative of V1 can be ' V 1 F ( ) C z z z z Where Selecting [ F ' ( ) Z C z C z ] [ z F ' ( ) Z ] q (530) ' ( ) df( ) F d q C z C 3 0 (531) des 3 2 F ' ( ) Z C C 4 0 (532) 4 C (1 C )( C C C ) (533) Substituting (531), (532) and (533) in (14) yields V1 ( C C C ) Z 1 2 C 2 C z 2 (534) For V 1 0 to be negative definite, the following constrain must satisfy

38 154 ( C C C ) Z 1 0 (535) Under the condition Z 0 and C1 1the constrains on C1, C3 and C4 must hold ( C C C ) C (1 C )( C C C ) (536) C 3 C Step 3: This step intends to determine the control law z q q q C z (537) 3 des 3 2 Using (526) the time derivative of z3 can be z3 q C ( q C ) (538) 3 1 Substituting (519) and (520) in (538) yields z3 q C z 3 2 C C z C z M e e (539) Where C 1 11 C 1 C 3 Z M C M C ( M ) 22 3 q C (540) 3 C M 33 q C M 3 Summarize the differential equations of (, z2, z3) as follows

39 155 Z C Z z Z z z 2 C C z z (541) z 3 C C z C z M e e To derive the control rules, choose a Lyapunov function candidate as V C 5 V 1 2 z 3 (542) where C 5 is a positive definite Using (530) and (539),the time derivative of V2 can be V 2 C [( ) C C C Z C C z C z z z ] z [ C C z C z M e] e C [( ) C C C Z C C z ] z [( 5 ) ( 5 ) ] 3 C 11 C 4 C C 22 C z 2 C 33 z 3 M e e (543) Let C 5 C, so that 22 V 2 z [( ) ] 3 C 11 C 4 C 5 C 33 z 3 M e e In order for V 2 to be negative definite, choosing M ( ) e e C 11 C 4 C 5 C 6 z 3 (544) (545) Substituting (545) into (543) yields V 2 C [( ) ] ( ) 2 5 C C C Z C C z C C z Selecting 2 C5 C 3, the following differential equation is obtained (546)

40 156 C ( ) 2 22 C 5 M C 3 M q C 3 C 3 M C M 3 q (547) The constrain on C3 must hold C M M 1 3 q (548) For V 2 to be a negative, the constrain on C5 and C6 must hold C C C C (549) 6 33 Using (520) and (524), (527) can be written as Z [(1 C ) z z ] Z ( ) (550) Substituting (537) and (550) into (545), the control law is of the form (assuming ref 0 ) e M 1 [( ) ( ) ( ) ] e C 11 C 4 C 5 Z C 6 q C 3 C 6 C 1 C 3 C 6 ref C 3 C 6 ref M 1 [ K ( ) ] 5 q K K K K e ref 3 ref (551) Where K 1 C C C K 2 ( C 11 C C ) Z 4 5 K 3 C C C K 4 C C 3 6 K 5 C 6 (552)

41 System Architecture The control law (551) using the backstepping design is shown in Figure 528The control scheme includes the cascaded control structure with the feed forward path The resulting controller is parameterized by five parameters ( K,,,, 1 K 2 K 3 K 4 K 5 ), each of which includes the parameters ( C,,,, 5, 1 C 2 C 3 C 4 C C 6 ) and the relative aero-coefficients The design specifications are: the flight path angle tracking error converges to zero for the glide slope tracking and angle of attack goes to zero for the runway approach This automatic landing system would control the aircraft to descend along the glide-slope and then takes an exponential path of the flare until touchdown Figure 528 Glide Slope Architecture using Backstepping 5113 Simulation Results This section presents the simulation results of backstepping the flight path controller Figure 529 depicts the simulated results of backstepping flight path controller in a level flight trimmed condition (at

42 158 speed of 120 ft/sec and altitude of 1500 ft)the parameters used in backstepping controller designs are: C 1 =159, C 2 =365,C 3 =12, C 4 =25, C 5 =144, C 6 =068 The parameters tuned by design specifications and constraints mentioned in the section 571 The simulation results show that the flight path controllers via backstepping provides good step tracking on ref (-25 o ), pitch angle of 25 o and maintains zero angle of attack, although there is a little overshoot in the flight path angle However, the flare control would take over the glidepath control to guide the aircraft before touchdown Figure 529 Response of Glide Scope Control System 512 FLARE PATH CONTROL SYSTEM The final phase of the landing is the transition from the glide slope to the actual touchdown, generally referred to as the Flare In this region, the aircraft follows an exponential path During the flare maneuver, the flight path angle of the aircraft has to be changed from 25 degree to the positive value, which is recommended for touchdown In other words, during the flare

43 159 maneuver the control system must control the height of the aircrafts CG and its rate of change such that the resulting trajectory corresponds to the idealized exponential path while at the same time causing the aircraft to rotate for touchdown Mc Lean (1990) follows: h cmd where, Flare altitude and the altitude rate commands are defined as h exp x x 0 0 h h cmd cmd TD0 0 (552) h h h h 0 TD0 0 TD0 h cmd is the altitude command (ft) x cmd is the negative of the ground range to the predicted touchdown point (ft) h 0 is the flare starting altitude (ft) t 0 is the flare starting time (sec) h TD is the nominal touchdown vertical speed (ft/sec) x is the value of x cmd cmd at beginning of flare (ft) 0 x is the value of x cmdtd cmd at touchdown (ft) h ( t ) h cmd 0 0 (553) x x h exp cmd cmd0 h V 0 0 s h G (554) cmd s h h 0 TD 0 V G is the ground speed (ft/sec)

44 160 V U cos( ) G 0 h ( t ) h cmd 0 0 (555) (556) s h V 0 G h h 0 TD 0 (557) h x x ln TD cmd cmd s TD 0 h 0 (558) h t h TD TD (559) 5121 System Architecture The flare controller design would be similar to the glide slope controller The outer loop gives the descent command h r The pitch hold autopilot is the basic autopilot mode for this system to descent from flare entry height h 0 The second lead network in the coupler is used to obtain a higher value of coupler sensitivity, thus preventing the aircraft from flying into the runway too soon 5122 Simulation Results The response of flare path control system is shown in Figure 530 At a height of 15 meters (50ft) the flare maneuver starts which results in the nose being lifted up and reducing the vertical speed of the aircraft During this time the control law has to be continuously adjusted It can be seen that the performance criteria presented in section 571 are achieved

45 161 Figure 530 Response of Flare Path Control System 513 COMPARISON OF PID AND BACKSTEPPING The response of various parameters obtained by using PID and Backstepping controllers were compared in Figure 531 The variations in angle of attack, pitch, pitch rate and height during the flare path were compared It is evident from Figure 531 that the sink rate is better when using Backstepping procedure The variations in flight path angle and tracking angle error during glide slope were also compared

46 162 Figure 531 Comparison Responses of Flare Path and Glide Slope Control using PID and Bckstepping 514 DISCUSSIONS AND CONCLUSION The glide slope controller performs well in updrafts and downdrafts as shown in Figure 532 In the updraft case, the aircraft virtually does not leave the desired path until the flare command and in the downdraft case, the aircraft is still established on the glide path at a range of around 10,000 ft 1,400 1,400 1,200 1,200 1,000 1, ,000 20,000 15,000 10,000 5, ,000 25,000 20,000 15,000 10,000 5,000 0 Distance (ft) Distance (ft) Figure 532 Glide Slope Response with Updraft and Downdraft of 20 ft/s

47 163 It is impossible to start the simulation at the flare engagement point So the method of switching control from the glide slope to the flare path had a profound effect on the ability to design the flare controller Blakelock (1991) suggests that the flare controller should be the same as the glide slope controller with an additional lead network After completing the glide slope controller an attempt for the flare controller was made Blakelock (1991) also suggests that the sink rate should be controlled This is because the flare path command is an exponential function for which the derivative is a constant multiplied by an instantaneous altitude The constant is the inverse of the exponential decay time constant Then, assuming that the aircraft will touch down in four or five time constants, the appropriate time constant for the function can be determined The following equations show this relationship for the flare path command H = H e 0 t 1 H = H e 0 t (560) (561) 1 H = H (562) Considering Blakelocks (1991) suggestions, the aircraft sink rate is used for the flare command and along with the glide slope controller in the initial attempt The range (which will be discussed later in chapter 6) at glide slope/flare switch is selected as 2,000 ft This is because the glide slope controller proves to perform well to a range less than 2,000 ft and the aircraft is just below 100 ft altitude above ground level at a range of 2,000 ft The initial attempt is to switch the signals at the instant, the range becomes equal

48 164 to or less than 2,000 ft A problem is immediately evident; ie, the aircraft became amazingly unstable the moment the command signal switch Upon further evaluation of Blakelocks methods, it was determined that a time dependent function is not appropriate as required and the aircraft must be at the proper position for flare at a certain time step in the simulation The geometry of the glide path and flare are both dependent on the position of the runway threshold Defining the flare path command in terms of the distance of the runway seemed much more appropriate because the distance of the runway is always known, regardless of the simulation time step The path is then defined as: H = H e 0 x (563) where x is the distance from the glide slope engagement point determined by subtracting the instantaneous LLA from the initial LLA, converting the latitude and longitude differences in terms of feet and solving for x by the Pythagorean Theorem Because the starting point of the flare command signal coincides with the glide path command signal, the value of the decay constant is found to be 5,500 This corresponds to touchdown in approximately five decay constants Having the flare path defined in terms of the distance from the runway, the altitude above ground level is used as feedback to generate the error signal Again, the glide slope controller is evaluated for use in the flare, but not suitable as the command signal is no longer dependent on the sink rate By experiment, it was observed that with a very small gain, the aircraft could be made to flare very well and touch down at any point desired With a flare command gain of , the aircraft touches down approximately

49 ft after the runway threshold The only problem with this is that it requires ideal conditions to work as there is no integrator in the compensator An additional problem in the flare controller design is the method of switching from glide slope to flare command signals With the switch at a range of 2,000 ft, the aircraft would not be made stable when a switch in the simulation caused adverse affects on the varying input signals It was determined by experiment that constants in Simulink creates problems in solving the algebraic loop and cause the simulation to produce erroneous results To compensate for the effect of switching from glide to flare these step blocks were used with the step value equal to that of the desired constant and the step time equal to the first time step of simulation (001 s) Even with this correction, stability problems are still evident at the switch of control commands To compensate for the sudden switch in commands, a blending function was developed to smoothen the effect of the switch which will be discussed later in chapter 6 This function blends the signals over range values of 5,000 ft to 3,000 ft These values were selected to ensure that the aircraft would be established under the flare controller before reaching the desired switch range of 2,000 ft