Hover and Forward Flight of an Autonomous UAV

Transcription

1 Hover and Forward Flight of an Autonomou UAV - uing Optimal Linear Control and Gain Scheduling Mater thei June 27 Group 135b AALBORG UNIVERSITY AAU Department of Control Engineering Intitute of Electronic Sytem

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3 Department of Control Engineering Fredrik Bajer Vej 7C DK-922 Aalborg Ø Phone.: Web: ABSTRACT: TITLE: Hover and forward flight of an Autonomou UAV uing optimal linear control and gain cheduling THEME: Intelligent Autonomou Sytem PROJECT PERIOD: 9 th -1 th emeter September 2 nd 26 - June 7 th 27 PROJECT GROUP: 135b GROUP MEMBERS: Tei Bæk Mad Hammelvang Thoma Bæk Jørgenen SUPERVISOR: Ander la Cour-Harbo NUMBER PRINTED: 5 NUMBER OF PAGES: 135 FINISHED: June 7 th 27 Thi mater thei concern the deign of linear control trategie for making an AUV capable of autonomou flight. The goal i to be able to complete level 1 in the International Aerial Robotic Competition, which involve flight over a ditance of 3 km, a fat a poible. The tarting point for thi thei i taken in the developed nonlinear model of a reconfigured Bergen Indutrial Twin helicopter, where the linear controller are deigned baed on linearization of the nonlinear model. Two different linear control deign method are ued for the controller deign; claic SISO control and optimal LQR control. Furthermore, a gain cheduling approach to perform cheduling between two controller i introduced. A a high level controller a uperviory controller i deigned to handle poition control. Hover control wa deigned uing the two different linear control deign method. The claic SISO controller wa implemented a lead or lag compenator. Furthermore, a forward flight LQR controller wa developed to be able to perform fat forward flight. The gain cheduling approach wa implemented a an oberverbaed gain cheduler framework around the LQR hover and forward flight controller. The LQR hover and forward flight controller, along with the oberver-baed gain cheduling controller, were all teted eparately in three different level 1 pecification to be able to compare them. It wa concluded that the oberver-baed gain cheduling approach wa the mot time efficient control method of the three.

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5 PREFACE Thi mater thei ha been written by group 135b on the pecialization of Intelligent Autonomou Sytem at the Department of Control Engineering, Aalborg Univerity, in the period between September 2 nd 26 and June 7 th June 27. The thei ha been done a a part of the development of an autonomou helicopter baed UAV at the Department of Control Engineering at Aalborg Univerity for future participation in the IARC. The thei conit of three part Helicopter model introduction and control ytem analyi, Control ytem deign and Control ytem tet and concluion, which contain everal chapter each tarting with a hort introduction. Lat in the thei appendice are placed with upplementary ubject and are denoted with capital letter tarting with A. Encloed at the back of thi thei a CD i placed containing MATLAB code, SIMULINK model, C code and a pdf copy of the thei. For viualization in 3D the program GSIM i ued (ee Appendix A). The thei i intended for upervior, examiner, control tudent and other that might have interet in linear control and gain cheduling on an autonomou helicopter. Aalborg Univerity, 27 Tei Bæk Mad Hammelvang Thoma Bæk Jørgenen Reading intruction Reference to literature are done by the Harvard method, where needed pecific page are added, e.g. [Sørenen, 1992, p.45. Figure, equation, and table are numbered conecutively within each chapter. Reference to equation are in addition made in parenthei. To clarify the difference between vector and matrice thee are written with bold lower-cae letter and with bold capital repectively. Variable and ymbol tem, wherever poible, from the ued literature. i

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7 CONTENTS 1 Introduction Prerequiite and Objective Control ytem teting and Concluion I Helicopter model introduction and control ytem analyi 7 2 Helicopter model introduction Reference frame Model overview Input/output relation The feedback ignal Control ytem analyi The phyical environment in level Control ytem overview Superviory controller Linear controller Cloed loop control ytem Cloed loop control ytem uing gain cheduling Control goal and requirement Superviory controller Linear controller II Control ytem deign 23 5 Claic linear control Method overview Controller tructure Controller deign Deign procedure Longitudinal control deign Lateral control deign Vertical control deign Yaw control deign Claic control ytem tet iii

8 iv Content Stationary propertie Dynamical proportie Optimal linear quadratic control Choice of operating point and feedback tate Choice of operating point Choice of control, reference and integral tate The principle of LQR Deign parameter Implementation of reference tate Implementation of integral tate Modification of the LQR control tructure Hover controller deign Analytical weight determination Weight tuning regarding tationary propertie Weight tuning regarding dynamical propertie Comparion of claic and LQR hover control Forward flight controller deign Stationary propertie Dynamical propertie Oberver-baed gain cheduling control Gain cheduling method Direct controller witch Weighted controller witch Controller witch uing oberver-baed gain cheduling Oberver-baed gain cheduling deign Deign of oberver gain F and F Deign of α g function Gain cheduling control tet Superviory control Overview Supervior trategy Control deign Initialization Deciion block Forward flight

9 Content v 8.4 Parameter determination III Control ytem tet and concluion 85 9 Control ytem tet Tet pecification Flight performance in pecification A Flight performance in pecification B Flight performance in pecification C Concluion Future work Acronym 99 Bibliography 99 Appendix 11 A Viualization program Gim 13 B Real tate veru tate etimation 17 C Linearization of the nonlinear model 111 D Calculation of K 115 E Oberver gain matrice 121 F Supervior parameter determination 123

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11 1 INTRODUCTION In everal year the ubject of autonomou flight with helicopter ha been reearched by different project group and a ingle Ph.d. tudent at the Department of Control Engineering at Aalborg Univerity. Two of the tudent project done mot recently are mater thei motivated by the International Aerial Robotic Competition (IARC) 26, organized by the Aociation for Unmanned Vehicle Sytem International (AUVSI [1972). In thi competition an Unmanned Aerial Vehicle (UAV) i required autonomouly to complete a number of level involving different challenge inpired by different miion example. Inpired by the previou work done on autonomou flight and the IARC, thi mater thei focu i on deigning linear control for an UAV making it capable of autonomou flight. For thi purpoe more advanced control method are looked into in order to achieve thi goal. Here the control method gain cheduling i choen becaue of it ability to extend the range at which linear control of a given ytem can be performed. In order to better undertand the motivation behind the IARC, and thereby the bai for making an UAV performing autonomou flight, a hort miion example involving a nuclear diater i given. One of four unit at a nuclear reactor ha exploded, and there are no urvivor at the facility. A afety ditance of three kilometer for the human recue team mut be maintained. One of the remaining three unit till operating at the reactor need to be hutdown manually, a the unit control ytem for automatically hutdown i not operating correctly due ti the exploion. The miion i to have an UAV find the building of the unit till operating and deploy a vehicle to enter the building. The UAV mut carry viual enor in order to find the right building, and obtain picture of the panel gauge and witch poition in the control room, uch that expert can ae the potential for a meltdown of the unit. The reconnaiance miion reult in four level. In level 1 a flight over a ditance of three kilometer with deignated tarting and final waypoint mut be performed. During the flight the UAV mut viit up to four waypoint. In level 2 the UAV mut find a building entry indicated by a image coniting of a black circle with a white cro. In level 3 a vehicle mut be deployed into the building, and image data, with ufficient quality for the judge to obtain the deired reconnaiance information, mut be gathered. Level 4 performance i to complete level 1, level 2 and level 3 within 15 minute. In each level the UAV mut perform autonomou flight. The work done by the aforementioned project group and the Ph.d. tudent reulted i.a. in reconfiguring a Bergen Indutrial Twin model helicopter (Helicopter [2) into an UAV platform by adding enor, on-board computer, ground to helicopter communication and GPS equipment (ee Figure 1.1). A much effort ha been put into thi rather comprehenive tak, and the reult i a fully functional and programmable UAV platform, thi thei take it tarting point here. 1

12 2 Section (a) (b) FIGURE 1.1: The Bergen Indutrial Twin model helicopter before (a) and after (b) being rebuild. The firt of the above mentioned tudent project (Hald et al. [26) concerned modelling of the Bergen helicopter uing firt principle. Some model parameter determination and Linear Quadratic Regulator (LQR) control were carried out a well. However, due to complication with the enor equipment and much emphai on the modelling tak the group never got to tet the developed controller in flight nor participate in the IARC 26. The complication with the enor equipment ha been taken care of by the upervior of thi mater thei Ander la Cour-Harboe and the Ph.d. tudent Morten Bigaard who i alo working with the Bergen helicopter. In the Ph.d. project a nonlinear model of the Bergen helicopter and the neceary enor fuion and etimation have been developed and implemented. Morten Bigaard ha made the enor fuion and etimation and model implementation a well a hi guidance available for the thi thei. The econd tudent project (Holmegaard et al. [26) concerned a navigation ytem enabling the UAV to complete level 1 and 2 in the IARC. The work done by the project group included development of a oftware platform and deign of i.a. navigator, viion ytem, miion control, map generator and optimal path calculator. Tet of the navigation ytem howed the helicopter theoretically able to complete level 1 and 2. However, it i uggeted to optimize the navigator by minimizing the flight time in level 1, by extending the autonomou flying feature of the UAV to encompa fat forward flight alo. Inpired by the uggetion by Holmegaard et al. [26 the overall objective of thi mater thei i tated a: Deign, implementation and tet of a control trategy enabling the UAV to autonomouly complete level 1 in the IARC by employing hover and fat forward flight. Here hover i defined a the UAV being airborne, and the tranlatory and rotational movement of the UAV are cloe to zero. And fat forward flight i defined a the UAV being airborne, and the rotational movement i cloe to zero, where a pecified peed in the ame direction a the heading of the UAV i maintained. The fat forward feature i epecially important, if a future project group at Aalborg Univerity

13 Chapter 1. Introduction 3 reache the point where completion of the IARC, that i achieving level 4, become a goal. Becaue level 2 and 3 involve relaying reconnaiance data, and uch tak involve time-conuming image proceing, the fater level 1 i completed the better. The overall objective i therefore rephraed a: Deign, implementation and tet of different control trategie enabling the UAV to autonomouly complete level 1 in the IARC by employing hover and fat forward flight, uch that the mot efficient control trategy regarding level 1 completion time can be identified. In order to accomplih the overall objective it ha been choen to look at the control ytem in Figure 1.2, from where the control trategie linear control without gain cheduling and linear control with gain cheduling can be een. Baed on the overall objective two linear controller mut be deigned; one for hover and one for fat forward flight. The gain cheduling trategy i then implemented to ue both of thee linear controller. The uperviory controller mut handle the high level control of the ytem. The different block of the control ytem are elaborated on in the following. Level 1 pecification Superviory controller Gain cheduling controller Linear controller Sytem Senor fuion FIGURE 1.2: Illutration of the element contituting the control ytem. The arrow how the data flow direction. 1.1 Prerequiite and Objective Uing a bottom-up approach the block contituting the choen control ytem are treated one by one throughout thi thei divided into a number of objective. Prerequiite for achieving thee objective followed by the actual objective are preented in the following. The order of thee ection alo erve a an outline for the chapter in part I and II of thi thei. Prerequiite A: Sytem and enor fuion A mentioned thi thei take it tarting point in the work done on modelling the reconfigured Bergen helicopter and development of enor fuion and etimation. A nonlinear model (including enor fuion and etimation) implemented in C++ for SIMULINK, able to imulate the motion

14 4 Section 1.1. Prerequiite and Objective of the Bergen helicopter, i available for thi thei. To narrow the focu of thi thei to the control field, thi model i ued for control deign and validation tet of the developed trategie. A model parameter, important for the nonlinear model to reflect the Bergen helicopter preciely, are being determined during the period of thi thei, it ha been choen not to perform tet flight. However, a the complexity of the nonlinear model i aumed to be a high a for reflection of the Bergen helicopter, it i jutified to ue validation baed on imulation for teting developed control trategie. Therefore, when referring to the UAV throughout the thei it refer to the nonlinear model of the Bergen helicopter. An introduction to the nonlinear model and the enor fuion and etimation can be found in Chapter 2. Prerequiite B: Control ytem analyi The control ytem will be analyzed with the purpoe of identifying in- and output of the different block from Figure 1.2. Thi will lead to the identification of the needed reference, feedback ignal and controller output. The analyi can be found in Chapter 3 Prerequiite C: Control goal and requirement It i neceary to determine control goal and requirement for the uperviory controller and linear controller. Thee will be determined with repect to the overall objective tated earlier. Thee goal and requirement can be found in Chapter 4. Objective A: Claic linear control Baed on the available feedback ignal linear control, uing claic Single Input Single Output (SISO) controller, mut be able to tabilize the UAV in hover. It provide a bai for comparion between SISO and Multiple Input Multiple Output (MIMO) control of the ytem. The deign of claic linear control i found in Chapter 5. Objective B: LQR hover control An optimal LQR controller able to tabilize the UAV in hover mut be developed. The performance of thi controller i compared with the performance of the claic linear controller. Furthermore, it will be ued a one of the two controller in the gain cheduling trategy. Becaue the LQR controller i a MIMO controller, and the UAV i a MIMO ytem a well, it i expected to perform better. Objective C: LQR forward flight control In order to facilitate fat forward flight, a econd optimal LQR controller mut be developed for a forward flight operating point. The forward flight controller i expected to perform better than

15 Chapter 1. Introduction 5 the LQR hover controller in the forward flight operating point. The deign of the optimal LQR hover and forward flight controller can be found in Chapter 6. Objective D: Gain cheduling A gain cheduling controller mut be deigned uch it i poible to control the UAV utilizing the two optimal LQR controller, uch that an acceleration from hover to forward flight can be performed. The performance of thi control method i evaluated by comparing it to the performance of the LQR hover and forward flight controller. The deign of gain cheduling control i found in Chapter 7. Objective E: Superviory controller Baed on a given level 1 pecification defining the placement of waypoint, and poition feedback from the UAV, a uperviory controller mut calculate at which peed and heading the UAV mut fly, in order to get through the given level 1 a fat a poible. The complexity of the upervior (hort for uperviory controller) i kept low, a the emphai of thi thei i on linear control and gain cheduling. However, it i an important element in the control ytem in order to obtain a realitic etimate of the level 1 flight time for the Bergen helicopter platform. Deign of the upervior can be found in Chapter Control ytem teting and Concluion When the above lited objective have been reached, the deigned control trategie will be teted with repect to the overall objective and the IARC. Becaue the range of thi thei only pan a far a level 1 of the IARC the deigned control trategie will only be teted with repect to the criteria lited for thi level. The control ytem tet can be found in Chapter 9. Finally the thei will um up the concluded reult obtained from the different objective. The concluion ultimately lead to uggetion for future work to be done regarding the Bergen helicopter. The concluion and future work i found in Chapter 1.

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17 Part I Helicopter model introduction and control ytem analyi 7

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19 2 HELICOPTER MODEL INTRODUCTION To be able to perform atifactory control of the helicopter it i neceary to obtain knowledge about the model. Thi include knowledge about the different reference frame, the general model tructure, and the input and output of the model. 2.1 Reference frame For the purpoe of performing control of the helicopter two reference frame are ued; an Earthfixed reference Frame (EF) and a Body-fixed reference Frame (BF). The xy-plane of the EF i parallel to the urface of the Earth, where the x-axi i pointing north, the y-axi i pointing eat and the z-axi perpendicular to both pointing vertically downward. The origin of the EF i choen arbitrarily but alway with the aforementioned orientation. The EF i ued to decribe the poition and attitude of the helicopter. b y θ e y φ b x ψ e x e z b z FIGURE 2.1: Illutration of the two coordinate ytem ued for control purpoe. The econd reference frame neceary for control purpoe i the BF, which ha it origin in the center of ma of the helicopter, and follow the poition and attitude of the helicopter. The x-axi of the BF i pointing through the noe of the helicopter, the y-axi point through the right ide of the helicopter and the z-axi perpendicular to both and pointing downward through the bottom of the helicopter. It i in the BF that tranlateral and rotational movement of the helicopter are defined. The reference frame are illutrated in Figure 2.1. The illutration wa made by 9

20 1 Section 2.2. Model overview Hald et al. [26, whom ued the two additional frame attached to the main and tail rotor for modeling purpoe. Thee additional frame will not conidered in thi thei. 2.2 Model overview To perform control of the nonlinear model only the input/output relation are taken into conideration. Therefore, the actual model component are looked upon a black boxe, which receive ome actuator input and yield ome output regarding the poition, attitude and movement. The decription of the black boxe will be trictly uperficial, where only the build-up and functionality will be elaborated on. The model ued i divided into three black boxe; the nonlinear helicopter model, the enor emulation, and the enor fuion and etimation. The three block can be een in Figure 2.2. S col e Ξ real e Ξ gp e Ξ et S lat S lon Nonlinear helicopter model e Θ real b Ξreal Senor emulation e Ξgp b F mag ( b Ξ + b g) imu Senor fuion and etimation e Θ et b Ξet S tr b Θ real b Θ imu b Θ et FIGURE 2.2: Block diagram of the model ued in thi thei illutrating the nonlinear helicopter model, the enor emulation, and the enor fuion and etimation. All with related input and output. Nonlinear helicopter model: The nonlinear model of the helicopter i contructed by everal mathematical equation decribing how the helicopter act when given any arbitrary input. Thee mathematical equation include a dynamic model of the actuator mounted on the helicopter, a dynamic model of the flapping motion of the main rotor blade and the tabilizer bar, a calculation of the force and torque generated by and affecting the helicopter and a model of the rigid body dynamic and kinematic, which the force and torque are acting upon. A it can be een from Figure 2.2 the nonlinear model take four input; one for the collective pitch of the main rotor blade S col, one for the lateral cyclic pitch angle of the main rotor S lat, one for the longitudinal cyclic pitch of the main rotor S lon and one defining the reference for the build-in yaw rate control of the tail rotor S tr. Baed on the value of the input the nonlinear model yield four output for control purpoe; the poition given in the EF e Ξ real, the attitude given in the EF e Θ real, the tranlatory velocity given in the BF b Ξreal and the angular velocity given in the BF b Θ real. Senor emulation: The enor emulation block contain model of the different enor mounted on the helicopter. Thee enor model have had meaurement noie added to them, where

21 Chapter 2. Helicopter model introduction 11 the noie ha been determined baed on tet performed on the actual helicopter. Note that the information provided by the enor (both actual and emulation) are not ufficient for control purpoe, hence enor fuion and etimation i neceary. The enor ued for meaurement on the helicopter are a GPS, which meaure the poition e Ξ gp and tranlatory velocity e Ξgp in the EF, a compa that meaure the attitude of the helicopter with repect to the Earth magnetic field b F mag and an IMU, which meaure the the um of the gravitational and helicopter acceleration ( b Ξ + b g) imu and the angular velocitie b Θ imu. Senor fuion and etimation: The enor fuion and etimation block take the information from the enor emulation and utilize them to etimate the poition, attitude and velocitie of the nonlinear model. The etimator i deigned a an Uncented Kalman Filter, which i tatitical calculation method of a random variable undergoing a nonlinear tranformation. The output of the etimator are the etimated tate of the nonlinear model; the poition e Ξ et, the attitude e Θ et, the tranlatory velocity b Ξ et and the angular velocity b Θ et. From the above decription of the model (ued in thi thei) the input and output vector can be determined a [ T u = S col S lat S lon S tr (2.1) [ x = e Ξ T e Θ T b b ΞT Θ T T [ = e x e y e z e φ e θ e ψ b ẋ b ẏ bż T b b b φ θ ψ, (2.2) where S i the input to the ytem and x are the output of the ytem to be controlled. Note that the output vector x i the output of the etimator, and that the ubcript et i removed to eae further ue of the output ignal. 2.3 Input/output relation To eae the deign of different controller for the model decribed above the input/output relation are analyzed further. In addition, to eae the analyi of the relation the effect of cro coupling between model tate are not taken into conideration. Note that the decription below are baed on the BF having the exact ame orientation a the EF. Collective input S col : Thi input control the collective pitch of the main rotor blade, which make the helicopter move vertically. Therefore, thi input affect the vertical poition e z and the vertical velocity b ż. Cyclic input S lat : Thi input control the lateral pitch angle of the main rotor blade making the helicopter move ideway. Thi mean that thi input affect the lateral poition e y, the roll angle of the helicopter e φ, the lateral velocity b ẏ and the roll angular velocity b φ. Cyclic input S lon : Thi input control the longitudinal pitch of the main rotor blade, which make the helicopter move for- or backward. A a reult it i determined that thi in-

22 12 Section 2.4. The feedback ignal put affect the longitudinal poition e x, the pitch angle of the helicopter e θ, the longitudinal tranlatory velocity b ẋ and the pitch angular velocity b θ. Reference input S tr : Thi input et the reference for the build-in yaw rate controller. Thi mean that thi input ultimately affect the yaw angle of the helicopter e ψ and the yaw angular velocity b ψ. The decription of the four input can in aociation with the knowledge gained from the reference frame be ummarized a lited in Table 2.1. Note that the poition and angle given in the Input Tranlatory movement Rotational movement S col > bż > S col < bż < S lat > bẏ < b φ < S lat < bẏ > b φ > S lon > bẋ > b θ < S lon < bẋ < b θ > S tr > b ψ > S tr < b ψ < TABLE 2.1: Overview of the collective, cyclic and reference input to the helicopter decribing the effect of a given input with repect to it ign. Thi effect i een for the ytem initialized in hover and only one input applied at a time. EF are left out becaue the BF doe not have the ame orientation a the EF at all time. 2.4 The feedback ignal It i important to keep in mind that the feedback ignal from the etimator are not perfectly good ignal, hence it may be neceary to deign controller a bit low to be able to handle any form of udden alteration in the ignal. In addition, ome etimated tate are more affected by the meaurement noie from the enor. An example of thi can be een in Figure 2.3, which how the roll, pitch and yaw angle ( e φ, e θ and e ψ repectively) a model tate (to the left) and a etimated tate (to the right) for the nonlinear model trimmed in hover. It i oberved, that the yaw angle (red line) i the leat correct etimate of the three. In addition, the roll angle (blue line) i omewhat affected by the etimation of the yaw angle, which i indicated by the way the roll angle eem to follow change in the yaw angle etimate. In Appendix B graph of all 12 tate are howed; both real and etimated tate for comparion. Having determined the reference frame for control purpoe, and the input and output of

23 Chapter 2. Helicopter model introduction e φreal, e θ real and e ψ real [ rad.5.5 e φ e θ e ψ Time [ e φ et, e θet and e ψet [ rad Time [ FIGURE 2.3: Comparion of the real roll, pitch and yaw angle (left) and the etimated roll, pitch and yaw angle (right). The data i obtained by initializing the ytem in hover. Note that the legend on the left graph i alo applicable on the right graph. the ytem model it i now poible to analyze the control ytem further, which will be done in the following chapter.

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25 3 CONTROL SYSTEM ANALYSIS Thi chapter decribe an analyi of the phyical environment in level 1 of the IARC with regard to the overall goal tated in the introduction. Then an overview of the different control unit are preented, and finally the control ytem elaborated on in thi thei i etablihed. 3.1 The phyical environment in level 1 A the level 1 pecification in the IARC i unknown, the UAV mut be able to handle an infinitely number of different hape pecified by GPS coordinate. Example on flight path are illutrated in Figure 3.1. End point Waypoint 4 End point Waypoint 2 Waypoint 3 End point Start point Waypoint 1 Start point Start point FIGURE 3.1: Three example on flight path the UAV might be ubject to in level 1 of the IARC. The caling of the figure hould be ignored, a the total length of each flight path i known to be three kilometer. From the figure it can be een, that both length on the path egment and the turn angle between the path egment might vary ignificantly from waypoint to waypoint a well a for different level 1 pecification. A decribed in the introduction it i deired to travel the total flight path of level 1 a fat a poible, hence the UAV mut be able to calculate a time-optimizing way of dealing with a given et of path egment and waypoint turn. When conidering optimization regarding the flight path of the UAV, thi can be done by optimizing the flight through the different waypoint defined for a given level 1 pecification with repect to peed and turning method. A likely approach to thi problem i to implement an optimization algorithm in a uperviory controller in thi thei. Note that there are everal different way of dealing with different waypoint with repect to turning, peed and heading. 15

26 16 Section 3.2. Control ytem overview However, ince the focu of thi thei i on deigning linear controller and gain cheduling it i choen only to look at two different type of dealing with waypoint; decelerating to hover and turn, or fly over a waypoint with a given peed and then turn maintaining that ame peed. With the above analyi of the phyical environment of level 1 in mind it i poible to elaborate further on the defined control ytem from the introduction (ee Figure 1.2 on page 3). 3.2 Control ytem overview The general control ytem to be developed in thi thei conit of two main control block; a upervior and a linear controller block (the general control ytem in open loop i illutrated in Figure 3.2). Becaue gain cheduling per definition conit of two or more controller it can not be een a a direct block in the control ytem, but rather a frame work around everal controller, which will be elaborated on later. Superviory Linear e Ξ lvl1 o r u x controller controller Sytem FIGURE 3.2: Block diagram of the general control ytem illutrated in open loop with in- and output of each block. In the following the two controller block from Figure 3.2 will be elaborated on Superviory controller A it can be een from Figure 3.2 the high level control of the control ytem i handled by the upervior. A decribed earlier the upervior mut be able to define the optimal flight path for the UAV baed on the information received about the waypoint poition of level 1 e Ξ lvl1. Furthermore, the upervior mut be able to calculate reference o r for the linear controller, which handle the low level control of the control ytem. Baed on thi information about the reference input and the output of the upervior a more clear definition of the upervior regarding it actual functionality can be determined. Since the upervior handle the planning of the actual flight path for the UAV it i choen to have the upervior handle the entire poition control of the control ytem. In addition, ince the upervior determine the method a to how the UAV mut fly through the different waypoint it i alo neceary for the upervior to know the heading of the UAV. Thi reult in the feedback ignal vector for the upervior to be [ T e Ξ T, (3.1) e ψ where e Ξ i the poition vector given a [ e x e y upervior block (ee Figure 3.3). e z. Thi lead to the in illutration of the

27 Chapter 3. Control ytem analyi 17 [ e Ξ T e Ξ lvl1 e ψ T Superviory o r controller FIGURE 3.3: Illutration of the in- and output for the upervior. Baed on the input the upervior mut be able to calculate reference o r for the linear controller. Since the upervior itelf handle the poition control of the UAV it i not neceary to calculate reference for thee tate. A mentioned in Chapter 2 the total tate vector i [ x = e x e y e z e φ e θ e ψ b ẋ b ẏ bż T b b b φ θ ψ. By removing the poition tate from the tate vector there are a total of nine tate left, which are all deired to be able to control. Therefore, the output reference vector from the upervior mut calculate reference for thee nine tate, which yield the output reference vector [ T o r = re φ re θ re ψ rbẋ rbẏ rbż r b φ r b θ r b ψ. (3.2) Linear controller Having determined the reference input to the linear controller the control tate become [ e Θ T b b ΞT Θ T T [ = e φ e θ e ψ b ẋ b ẏ bż T b b b φ θ ψ. (3.3) It i not neceary to ue all of the feedback tate, which mean that the linear controller may deigned to only utilize ome of the feedback tate and reference. However, a number of four tate are the abolute minimum needed to perform control of the UAV, where thee tate are the tranlateral velocitie and the yaw angle: [ e ψ b ẋ b ẏ bż T. (3.4) A decribed in the introduction to thi thei the linear controller either conit of claic SISO control or MIMO optimal LQR control, where it i neceary to deign one controller for each of the feedback ignal when uing SISO control, and only one controller for all feedback tate when uing LQR control. Baed on the above the actual linear controller block can be defined with repect to input and output (ee Figure 3.4). The linear controller receive the reference ignal o r from the upervior along with the above decribed feedback ignal [ e Θ T b Ξ T b Θ T T a input. The output of the linear controller i then the input vector u to the ytem Cloed loop control ytem Having decribed in- and output of both the upervior and the linear controller the control ytem layout hown in Figure 1.2 on page 3 can be tranformed into the cloed loop control

28 18 Section 3.2. Control ytem overview o r Linear [ e Θ T b Ξ T b Θ T T u controller FIGURE 3.4: Illutration of the in- and output for a linear controller, which i either claic hover, LQR hover or LQR forward flight controller. ytem in Figure 3.5. e Ξ lvl1 Superviory controller o r Claic hover controller LQR hover controller u u Sytem [ e Ξ T e ψ T LQR forward flight controller u [ e Θ T b Ξ T b Θ T T FIGURE 3.5: Illutration of the cloed loop control ytem uing only linear control. The linear controller are deigned eparately and can be utilized one at the time, a indicated by the witch at the ytem input. A it can be een from Figure 3.5 each of the linear controller can be ued one at the time. A for the general illutration of the control ytem, gain cheduling i left out but will be elaborated on in the following ection Cloed loop control ytem uing gain cheduling The gain cheduling approach ued in thi project i baed on Bendten et al. [25. The two LQR controller in Figure 3.5 are inerted in an oberver baed control tructure, and uing a variable α [; 1, the influence of the control ignal from the LQR controller are each weighted a illutrated in Figure 3.6. Having determined the tructure of the control ytem along with the input and output of the different controller block, the requirement and tet pecification for the control ytem can be identified.

29 Chapter 3. Control ytem analyi 19 e Ξ lvl1 Superviory controller o r LQR hover controller LQR forward flight controller 1 α α [ Sytem [ e Θ T b Ξ T b Θ T T e Ξ T e ψ T Gain cheduling controller FIGURE 3.6: Illutration of the gain cheduling approach ued in thi thei. The claic controller block i left out, a it i not ued for the gain cheduling control.

30

31 4 CONTROL GOALS AND REQUIREMENTS Thi chapter decribe how the requirement to the element contituting the control ytem have been etablihed. Hence the overall requirement and etting for the uperviory controller are explained, together with the requirement for the linear controller, which primary concern the validation of the controller. 4.1 Superviory controller To etup requirement to the control ytem element tarting point i taken in the overall objective decribed in the introduction to complete the level 1 of the IARC. A review of the rule aociated with the competition pecifie that an autonomou UAV mut be able to perform a flight of three kilometer viiting up to four waypoint. Whether the waypoint have been viited i determined by judge placed on the ground at each waypoint. If the judge oberve the UAV when looking up the waypoint ha been paed. It i aumed that paing the waypoint within a circle with a radiu of two meter, i ufficient for the judge to oberve the UAV. A the poition control i performed by the upervior, thi i a requirement for thi controller. In addition, the UAV mut initiate and terminate the level 1 flight in hover. Alo a requirement for the upervior. Thee are the only control requirement given in advance and determined by AUVSI [1972. However, to obtain teady hover and forward flight goal and requirement to the linear controller mut be etablihed a well. 4.2 Linear controller The requirement decribed in thi ection concern the control tate in (3.3) and ome overall propertie neceary for the control ytem. The overall objective repeated above, obviouly, reult in the need for a control ytem able to bring the UAV from hover to forward flight and back to hover again. Therefore, the linear controller mut be able to follow the reference calculated by the upervior. In addition, the controller mut feature integral tate to avoid poible teady tate error. The controller mut tabilize the ytem uing the rather noiy tate etimate dicued in Section 2.4 a feedback. The controlled tate can therefore be expected to fluctuate from their repective reference. A a control goal thee fluctuation mut be minimized in order to obtain teady control. 21

32 22 Section 4.2. Linear controller A a requirement to the dynamical propertie the controller mut be able to tabilize the UAV, when the UAV i initialized in a condition not equal to the operating point for the given controller. Thi condition i therefore decribed by offet, which are added to the initial value, in tranlateral velocitie and angle with repect to an operating point. The offet value lited in Table 4.1 have been choen. State: e φ e θ e ψ bẋ bẏ bż Offet value:.5 rad.5 rad.5 rad 2 m 2 m 2 m TABLE 4.1: Offet value pecifying the rquirement to the dynammical propertie of the linear controller. The following um up the requirement etablihed in thi ection. Superviory controller The upervior mut be able convert a given level 1 pecification to reference for ue by the linear controller, uch that the UAV can complete the level 1 path autonomouly. Waypoint mut be paed within a circle with a radiu of 2 m in the xy-plane. The level 1 flight mut be initiated and terminated in hover. Linear controller The linear controller mut feature integral tate to avoid teady tate error. Fluctuation on the controlled tate mut be minimized, in order to obtain teady control. The controller mut be able to tabilize the UAV, when the UAV initialized with the offet value in Table 4.1. Having etablihed the deired control ytem tructure in the previou chapter, and the control goal and requirement above, the next chapter decribe the deign the claic linear controller.

33 Part II Control ytem deign 23

34

35 5 CLASSIC LINEAR CONTROL In thi chapter the deign and implementation of a claic linear SISO controller tructure will be decribed. Since there are everal different method of deigning SISO controller the method ued in thi thei will be decribed initially. Note that there will only be deigned a SISO control ytem capable of keeping the UAV in hover. 5.1 Method overview The actual model of the Bergen helicopter i, a mentioned in Chapter 2, nonlinear, why linear control uing SISO controller i rather difficult. Therefore, the nonlinear model i linearized in the hover operating point (no movement), where there will be deigned linear controller for thi pecific flight condition. The performance of the linear SISO controller will ultimately be ued for comparion with the more advanced control method LQR, which will be elaborated on later in thi thei. Since a linearized model of the nonlinear model i available it i poible to deign the controller baed on analyi of the tranfer function of the linearized model obtained from the ytem matrice (ee Appendix C). Baed on the tranfer function, it i poible to identify pole and zero and to analyze the tability of the ytem with repect to the different output, which i done uing root locu plot. The ue of root locu plot to deign the SISO controller lead to the determination of lead or lag compenator, which are approximation of PI and PD controller, but can be jut a effective. The initial deign of the controller will be done baed on the actual model feedback to eae the identification of tability uing the deigned controller. If tability i obtained uing model feedback, then the controller are teted uing the etimator tate a feedback, which help identify the robutne of the deigned controller. 5.2 Controller tructure Becaue the claic control trategy i baed on SISO controller it i neceary to deign everal controller; one for each of the deired output to be controlled. In Chapter 2 it wa determined that the total tate vector available for control purpoe ha a total of 12 tate. Since the overall control trategy decribed in Chapter 3 tate that the poition control of the total control ytem i to be handled by a upervior, only nine tate are left to ue a feedback. In addition, it i not deemed neceary to deign SISO controller for the angular velocitie of the helicopter, which 25

36 26 Section 5.3. Controller deign lead to the ix remaining tate ued for feedback: [ x fb = e Θ T b T [ ΞT = e φ e θ e ψ b ẋ b ẏ bż T. (5.1) Thi reult in a total of ix controller to be deigned in the claic linear control ytem, where the controller tructure i illutrated in Figure 5.1. The controller are denoted C i with i being the indicator for which tate i being controlled. bẋ ref bẏ ref bż ref e ψ ref Cbẋ Cbẏ Cbż Ce ψ e θ ref e φ ref + + Ce θ Ce φ S lon S lat S col S tr Nonlinear helicopter model bẋ bẏ bż e ψ e θ e φ FIGURE 5.1: Illutration of the controller tructure ued for claic linear control of the ytem. A can be een there are implemented cacade coupling for longitudinal and lateral movement. With the tructure of the total claic linear control ytem defined it i poible to deign the different controller. From Figure 5.1 it can be een that Cbẋ and Cbẏ mut be deigned uch that their output are reference to Ce θ and Ce φ repectively. Thi mean that for the purpoe of correct controller deign Ce θ and Ce φ mut be deigned before Cbẋ and Cbẏ. The two remaining controller Cbż and Ce ψ can be deigned without any prerequiite. 5.3 Controller deign The deign of the claic linear controller i divided into four overall deign part; longitudinal control, lateral control, vertical control and yaw control. The controller will be validated a they have been deigned. Furthermore, a complete tet of the total claic linear control ytem will be performed when all the controller have been deigned Deign procedure Throughout thi ection the controller will be deigned baed on reduced verion of the actual linearized tate-pace model of the nonlinear model by common procedure. The reduced tatepace model will be hown a tandard tate-pace model: ẋ i = A i x i + B i u y i = C i x i + D i u,

37 Chapter 5. Claic linear control 27 where i indicating the controlled tate. Furthermore, the tranfer function for the different controlled tate are calculated baed on their tate-pace model uing the following operation: G i = C i (I A i ) 1 B i + D i. (5.2) With the determination of the tranfer function for the control tate the different controller can be deigned uing root locu plot Longitudinal control deign The firt tep in deigning the longitudinal control part i to deign the e θ controller, where the firt tep of the deign procedure i to determine the input/output tranfer function Ge θ = e θ S lon. Baed on the ytem matrice of the linearized model, the tate-pace ytem pecific for e θ i determined a [ e θ b θ = ye θ = [.9992 [ [ e θ b θ [ [ e θ + b θ + S lon S lon From the reduced tate-pace model it can be een that all other tate than e θ and b θ have been decoupled to remove cro coupling, which help eae the deign of the controller. From the tate-pace model and (5.2) it i poible to determine the tranfer function for the ytem Ge θ a Ge θ = ( + 7). (5.3) When analyzing the root locu plot for Ge θ it i een that the root locu of the pole = lie in the right half plane, which indicate that the ytem i untable (ee Figure 5.2(a)). Baed on thi obervation it can be determined that the gain of the e θ controller mut be negative, which will cauethe root locu of the aforementioned pole to be in the left half plane of the -domain (ee Figure 5.2(b)). Ultimately, it i poible to control e θ by the ue of a P-controller. However, the utilization of thi type of controller may caue the ytem to have a teady tate error. The pole and zero of the compenator i determined by uing the MATLAB toolbox iotool, which immediately how the effect of placing pole and zero in the open loop ytem Ce θge θ. Thi reult in a pole-zero placement for the compenator a + 4 Ce θ = K p, e θ + 1, (5.4) which yield the root locu plot of the open loop ytem for e θ a hown in Figure 5.3. From the root locu plot of the open loop ytem, it i poible to determine the actual control gain needed to perform table control. The gain i choen with the intend of having a cloed loop

38 28 Section 5.3. Controller deign Root locu Root locu Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.2: (a) how the root locu plot of Ge θ. (b) how the root locu plot of Ge θ with negative control gain. 25 Root locu 2 15 Imaginary axi Real axi FIGURE 5.3: Root locu plot of the open loop ytem Ce θge θ. From thi plot the controller gain can be determined. teady tate gain (DC gain) of 1 ( db), which reult in a control gain of K p,e θ = 2.54 (5.5) Ce θ = , (5.6)

39 Chapter 5. Claic linear control 29 where the open loop tranfer function for the controlled ytem then become Te θ,ol = 2.54 ( ). (5.7) To determine whether the deigned controller tabilize the ytem or not i done by obervering the bode plot of the open loop tranfer function (ee Figure 5.4(a)). From the bode plot it can be een that the open loop ytem ha a phae margin of approximately 45 and an infinite gain margin, which indicate that the ytem i table in cloed loop. The cloed loop bode plot how that the DC gain i equal to 1 ( db) (ee Figure 5.4(b)). Bode plot Bode plot 5 5 Phae [ Magnitude [db Phae [ Magnitude [db Frequency [ rad (a) Frequency [ rad (b) FIGURE 5.4: (a) how the bode plot of the open loop ytem Te θ,ol. (b) how the bode plot of the cloed loop ytem Te θ,cl. To determine the performance of the deigned controller for the pitch angle, the tep repone of the cloed loop ytem i examined for rie time, ettling time and overhoot (ee Figure 5.5). From the tep repone it can be een that the rie time i approximately.1 with an overhoot of about 2 %, which indicate a fat controller. Furthermore, it can be een that the 2% ettling time for the repone i approximately.5. From the above analyi of the deigned pitch angle controller, it i concluded that it hould be able to perform atifying control of the pitch angle when applied to the nonlinear model. In addition, to tet the performance of the controller the initial pitch angle of the UAV i applied an offet of.5 rad in addition to the hover operating point value of.2 rad. Furthermore, to make ure that cro coupling are not affecting the tet all other tate have been decoupled. The controller i firt teted on the nonlinear model uing the real pitch angle a feedback (ee Figure 5.6(a)). It i een that the controller i able to tabilize e θ in it hover operating point within 2, which i aeed a atifactory. Furthermore, the controller i teted uing the etimated pitch angle a feedback in order to determine it robutne (ee Figure 5.6(b)). A for the previou tet the controller i able to tabilize e θ within 2 econd depite the meaurement noie

40 3 Section 5.3. Controller deign Step repone Amplitude Time [ FIGURE 5.5: Step repone of the controlled ytem for the pitch angle e θ. in the etimated e θ. Baed on thee obervation the pitch angle controller i aeed a having atifactory performance e θ [rad.3.2 e θ [rad Time [ Time [ (a) (b) FIGURE 5.6: Simulation reult of the pitch angle controller ability to tabilize e θ uing (a) model feedback and (b) etimator feedback. With the pitch angle controller deigned it i now poible to deign and tet the longitudinal velocity controller. Becaue the longitudinal velocity controller i the outer controller in the total longitudinal controller it output mut be a e θ reference angle. Therefore, the tranfer function

41 Chapter 5. Claic linear control 31 can initially be identified a Gbẋ = b ẋ e θ. The tate-pace model for b ẋ can then be determined a [ e θ bẋ [ [ [ e θ = bẋ + [ [ e θ 1 bẋ + e θ. ybẋ = 9.82 e θ From the above tate-pace model the tranfer function Gbẋ i given a (ee Eq. (5.2) for calculation method) Gbẋ = (5.8) By analyzing the root locu plot of Gbẋ it can be een that it i neceary to ue a negative control gain in order to enure tability (ee Figure 5.7(a)). The compenator pole and zero are choen uch that + 2 Cbẋ = K p,bẋ + 1, (5.9) which reult in the root locu plot depicted in Figure 5.7(b). From thi root locu plot the control Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.7: (a) how the root locu plot of Gbẋ. (b) how the root locu plot of the open loop ytem CbẋGbẋ. gain i determined a K p,bẋ =.14 (5.1) Cbẋ = , (5.11)

42 32 Section 5.3. Controller deign which yield the open loop ytem Tbẋ,ol = (5.12) In Figure 5.8(a) the bode plot of the open loop ytem Tbẋ,ol i hown, and from the bode plot it can be een that the ytem ha a phae margin of approximately 125 and an infinite gain margin. Thi indicate that the ytem i open loop table and therefore alo cloed loop table. In addition, the tep repone performance of the deigned controller i analyzed (ee Figure 5.8(b)). From the tep repone it can be een that the rie time i 2 and ettling time of about 3.9 indicating a table and fat controller. 4 1 Phae [ Magnitude [db Amplitude Frequency [ rad (a) Time [ (b) FIGURE 5.8: (a) Bode plot of the open loop ytem Teẋ,ol. (b) Step repone of the cloed loop ytem Teẋ,cl. To tet the performance of the controller the initial longitudinal velocity of the nonlinear model i applied an offet of 2 m, where the original initial value in the hover operating point i m. All other tate then e θ and b ẋ are decoupled in order to remove the effect of cro coupling. From the tet imulation uing the real tate of b ẋ (and e θ for inner controller) a feedback it i oberved that the controller i able to tabilize the longitudinal velocity within 6 econd, which i een a being atifactory (ee Figure 5.9(a)). Furthermore, it can be een from the tet imulation uing the etimated tate of b ẋ, that the deigned controller i till able to tabilize the longitudinal velocity within 6 econd depite the meaurement noie. The deigned longitudinal controller i therefore aeed uable for longitudinal velocity control.

43 Chapter 5. Claic linear control bẋ [ m 1 1 bẋ [ m Time [ (a) Time [ (b) FIGURE 5.9: Simulation reult of the longitudinal velocity controller ability to tabilize b ẋ uing (a) model feedback and (b) etimator feedback Lateral control deign The econd part of the claic control deign i the total lateral controller where the firt tep i to deign the roll angle e φ controller. The ytem matrice for e φ are [ e φ b φ = ye φ = [ [ [ e φ b φ [ e φ b φ + + S lat. [ 85.8 S lat From the above tate-pace model and (5.2) the tranfer function Ge φ i calculated a Ge φ = 85.8 ( ). (5.13) From the root locu plot of Ge φ (ee Figure 5.1(a)) it can be een that the controller mut be deigned with a negative control gain in order for the ytem to become table. The compenator pole and zero are choen a Ce φ = K p,e φ + 1, (5.14) which yield the open loop root locu plot a depicted in Figure 5.1(b), from where the control gain i determined a K p,e φ = 3.47 (5.15) Ce φ = (5.16)

44 34 Section 5.3. Controller deign Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.1: (a) how the root locu plot of Ge φ. (b) how the root locu plot of the open loop ytem Ce φge φ. Thi yield the open loop ytem Te φ,ol = 3.47 ( ). (5.17) From the bode plot of the open loop ytem Te φ,ol in Figure 5.11(a), it can be een that the ytem ha a phae margin of 45.1 and an infinite gain margin, which indicate that the ytem i open loop table. Furthermore, it can be een from the tep repone of the cloed loop ytem in Figure 5.11(b), that the rie time of the controlled ytem i approximately.6 econd, and that it ha an overhoot of approximately 2 %. Depite the overhoot it i not conidered an iue becaue the ettling time for the controlled ytem i about.36 econd. Thee obervation indicate that the deigned controller i able to tabilize the roll angle of the UAV. The deigned controller ability to tabilize e φ i teted by applying an offet of.5 rad to the hover operating point angle value of.39 rad. In addition, all other tate are decoupled to remove cro coupling in the nonlinear model. In Figure 5.12(a) the controller i teted uing real tate of e φ a feedback from which it can be een, that the deigned controller i able to tabilize the roll angle within 2 econd. In addition, it can be een from Figure 5.12(b) that the controller, uing the etimated tate of e φ a feedback, i able to tabilize the roll angle within 4 econd, which i aeed a being atifactory. With the roll angle controller deigned it i now poible to deign and tet the lateral velocity controller. Becaue the lateral velocity controller i the outer controller in the lateral control ytem it output mut be a e φ reference angle. Therefore, the tranfer function can initially be identified

45 Chapter 5. Claic linear control 35 Phae [ Magnitude [db Frequency [ rad (a) Amplitude Time [ (b) FIGURE 5.11: (a) Bode plot of the open loop ytem Te φ,ol. (b) Step repone of the cloed loop ytem Te φ,cl. e φ [rad Time [ (a) e φ [rad Time [ (b) FIGURE 5.12: Simulation reult of the roll angle controller ability to tabilize e φ uing (a) model feedback and (b) etimator feedback. a Gbẏ = b ẏ e φ. The tate-pace model for b ẏ can then be identified a [ [ [ [ e φ e φ bẏ = bẏ [ [ e ybẏ = φ 1 bẏ + e φ. e φ

46 36 Section 5.3. Controller deign By uing (5.2) the tranfer function Gbẏ i determined a Gbẏ = (5.18) By oberving the root locu plot of Gbẏ in Figure 5.13(a), it i een that the ytem i table and therefore mut have a poitive control gain. Furthermore, from the root locu plot of Gbẏ the Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.13: (a) how the root locu plot of Gbẏ. (b) how the root locu plot of the open loop ytem CbẏGbẏ. compenator pole and zero are choen a +.23 Cbẏ = K p,bẏ +.63, (5.19) which reult in the open loop root locu plot a een in Figure 5.13(b) from where the controller gain i determined to be The open loop tranfer function then become K p,bẏ =.56 (5.2) Cbẏ = (5.21) Tbẏ,ol = , (5.22) which reult in the bode plot in Figure 5.14(a). From the bode plot the phae margin of the open loop ytem i determined to be approximately 91, and that it ha an infinite gain margin, which indicate that the ytem i open loop table. In addition, analyzing the tep repone of the cloed loop ytem it i oberved that the rie and ettling time are about.1 and.4 econd repectively (ee Figure 5.14(b)). Furthermore, it i noticed that the controlled ytem i over damped.

47 Chapter 5. Claic linear control Phae [ Magnitude [db Amplitude Frequency [ rad (a) Time [ (b) FIGURE 5.14: (a) Bode plot of the open loop ytem Tbẏ,ol. (b) Step repone of the cloed loop ytem Tbẏ,cl. However, the fact that the controlled ytem i over damped i not conidered an iue becaue of the mall rie and ettling time. Baed on thee obervation it i concluded that the deigned controller hould be able to tabilize the lateral velocity of the UAV. The controller i teted by applying an offet to the initial lateral velocity of 2 m, where the hover operating point value i m. In addition, all other tate than e φ and b ẏ have been decoupled to remove cro coupling. Uing the real tate of b ẏ a feedback, it can be een from Figure 5.15(a) that the deigned controller i able to tabilize the lateral velocity within 15 econd, which i rather long compared to previou ettling time. However, it i deemed uable. Furthermore, it can be een from Figure 5.15(b), that the deigned controller, uing the etimated tate of b ẏ a feedback, i able to tabilize the lateral velocity within 6 econd with the rather noiy ignal taken into conideration. Baed on thee obervation the lateral velocity controller i aeed uable Vertical control deign The third part of claic control deign i the vertical velocity control, and it tate-pace ytem i determined a [ [ [ [ e z.9992 e z bż =.91 bż + [ [ e ybż = z 1 bż + S col S col

48 38 Section 5.3. Controller deign bẏ [ m 1 1 bẏ [ m Time [ (a) Time [ (b) FIGURE 5.15: Simulation reult of the lateral velocity controller ability to tabilize b ẏ uing (a) model feedback and (b) etimator feedback. Baed on the above tate-pace ytem the tranfer function Gbż = b ż S col i calculated a Gbż = , (5.23) which reult in the root locu plot a depicted in Figure 5.16(a). From the root locu plot it can be determined that a negative control gain i neceary to tabilize the ytem. Baed on the root Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.16: (a) how the root locu plot of Gbż. (b) how the root locu plot of the open loop ytem CbżGbż.

49 Chapter 5. Claic linear control 39 locu plot of Gbż the compenator pole and zero are determined a +.53 Cbż = K p,bż , (5.24) which yield the open loop root locu plot in Figure 5.16(b). The control gain i then determined a which then yield the open loop tranfer function a K p,bż =.17 (5.25) Cbż = , (5.26) Tbż,ol = (5.27) From the bode plot of the open loop ytem Tbż,ol in Figure 5.17(a) it i oberved that the ytem ha a phae margin i approximately 92 and an infinite gain margin, which indicate that the ytem i open loop table and therefore alo cloed loop table. Analyzing the tep repone of Phae [ Magnitude [db Frequency [ rad (a) Amplitude Time [ (b) FIGURE 5.17: (a) Bode plot of the open loop ytem Tbż,ol. (b) Step repone of the cloed loop ytem Tbż,cl. the cloed loop ytem Tbż,cl (ee Figure 5.17(b)) it can be een that the controlled ytem ha a rie time of about.3 econd, a ettling time of about.6 and no overhoot, which indicate a fat controller. Therefore, it can be concluded that the controller hould be able to tabilize the vertical velocity. The controller i teted by applying an offet to the initial vertical velocity of 2 m, where the hover operating point i m. In addition, all other tate have been decoupled to remove cro coupling. Uing the real tate of b ż a feedback it can be oberved that the controller i able to

50 4 Section 5.3. Controller deign.5.5 bż [ m 1 bż [ m Time [ (a) Time [ (b) FIGURE 5.18: Simulation reult of the vertical velocity controller ability to tabilize b ż uing (a) model feedback and (b) etimator feedback. tabilize the vertical velocity within 4 econd (ee Figure 5.18(a)). Furthermore, the controller i teted uing the etimated tate of b ż a feedback, where it can be determined that the controller i alo able to tabilize the vertical velocity here within 4 econd depite meaurement noie (ee Figure 5.18(b)). Baed on the obervation done the vertical velocity controller will be ued Yaw control deign The fourth and lat par of the claic control deign i for the yaw angle and it tate-pace ytem i determined a [ e ψ b ψ = ye ψ = [.9992 [ [ e ψ b ψ [ e ψ b ψ + S tr. + [ S tr From the tate-pace ytem the tranfer function Ge ψ = e ψ S tr i determined a Ge ψ = ( ), (5.28) which reult in the root locu plot een in Figure 5.19(a). From the root locu plot of Ge ψ it i determined that a poitive control gain i neceary. In addition, baed on the root locu plot the compenator pole and zero are choen a Ce ψ = K p,e ψ , (5.29)

51 Chapter 5. Claic linear control Imaginary axi Imaginary axi Real axi Real axi (a) (b) FIGURE 5.19: (a) how the root locu plot of Ge ψ. (b) how the root locu plot of the open loop ytem Ce ψge ψ. which reult in the open loop root locu plot een in Figure 5.19(b). Baed on the root locu plot of the open loop ytem Ce ψge ψ the control gain i determined a which reult in the open loop tranfer function K p,e ψ = 2.53 (5.3) Ce ψ = , (5.31) Te ψ,ol = 2.53 ( ). (5.32) From the bode plot of the open loop ytem Te ψ,ol it can be oberved, that the ytem ha a phae margin of approximately 65 and an infinite gain margin, which indicate that the ytem i open loop table (ee Figure 5.2(a)). Furthermore, from the tep repone of the cloed loop ytem in Figure 5.2(b), it i een that the controlled ytem ha a rie time of about 1 econd and an overhoot of approximately 9 %, but thi overhoot i not conidered an iue ince it i rather mall, and that the ettling time of the controlled ytem i about 4.2 econd. Thi how that the deigned controller hoould be able to tabilize the yaw angle. The yaw angle controller i teted by applying an offet of.5 rad to the initial yaw angle of rad. Furthermore, all other tate are decoupled to remove cro coupling. From Figure 5.21(a) it can be oberved that the controller i able to tabilize the yaw angle uing the real tate of e ψ within 6 econd. In addition, it can be oberved from Figure 5.21(b) that the controller alo i able to tabilize the yaw angle when uing the etimated tate of e ψ a feedback although within 8 econd. It i therefore aeed that the yaw angle controller can be ued.

52 42 Section 5.3. Controller deign Phae [ Magnitude [db Frequency [ rad (a) Amplitude Time [ (b) FIGURE 5.2: (a) Bode plot of the open loop ytem Te ψ,ol. (b) Step repone of the cloed loop ytem Te ψ,cl e ψ [rad.3.2 e ψ [rad Time [ Time [ (a) (b) FIGURE 5.21: Simulation reult of the yaw angle controller ability to tabilize e ψ uing (a) model feedback and (b) etimator feedback. Having deigned all of the neceary SISO controller to maintain the UAV in the hover operating point a more powerful tre tet of the total control ytem will be performed.

53 Chapter 5. Claic linear control Claic control ytem tet In the following the total claic control ytem will be teted for it tationary and dynamical propertie. The individual deigned controller from the previou ection are implemented in the control tructure depicted in Figure 5.1 on page Stationary propertie The total control ytem will be teted for it ability to maintain the UAV in the hover operating point, when the nonlinear model i initialized in thi point. Note that early tet howed ome of the controller had too high control gain, why it i neceary to lower thee gain by applying additional calar to the controller. Having modified control ytem the aforementioned tet i.1 Pitch angle.2 Longitudinal velocity e θ [rad bẋ[ m Roll angle.2 Lateral velocity e φ[rad bẏ [ m Yaw angle.5 Vertical velocity e ψ [rad bż [ m Time [ Time [ FIGURE 5.22: Simulation reult of the total claic control ytem ability to maintain the UAV in the hover operating point. Strong ocillating behavior can be oberved in ome of the controlled tate, which indicate fat tranient repone of their repective controller. performed, where the imulation reult can be een in Figure From the plotted imulation data it can be oberved that the roll and pitch angle exhibit trong ocillating behavior about their repective operating point value, which indicate fat tranient repone of the two con-

54 44 Section 5.4. Claic control ytem tet troller. In contrat to the roll and pitch angle, the yaw angle how more low behavior with repect to the etimator feedback ignal, which ultimately reult in low tracking of the reference ignal. Common for the three angle i that they vary le than.1 rad from their repective operating point value. Obervation of the longitudinal and vertical velocitie alo exhibit ocillating behavior, which, like for the roll and pitch angle, indicate fat tranient repone of the two repective controller. The lateral velocity alo how ign of ocillating behavior, however, not a noticeable a for the other velocitie. All of the velocitie follow their repective operating point value with a maximum variation of.3 m. Baed on the above obervation regarding the control ytem ability to maintain the UAV in the hover operating point, it i concluded that the deigned claic linear control ytem i capable of maintaining the UAV in hover Dynamical proportie The dynamical tet i performed in the exact ame manner a each individual tet of the different controller. However, here all cro coupling are not removed and all controlled output are given an initial offet correponding to the offet ued for each individual controller tet (ee Table 4.1 on page 22 for an overview of the offet). The tet reult can be een in Figure To be able to determine when the UAV can be identified a being in hover the variation value of.1 rad and.3 m, for the angle and velocitie repectively, i ued a limit. From Figure 5.23 it can then be oberved that all controlled tate have ettled within a period of 5 econd except the lateral velocity, which ettle after approximately 1 econd. It i baed on thee obervation concluded that the deigned claic linear control ytem i capable of tabilizing the UAV in the hover operating point and capable of rejecting poible diturbance.

55 Chapter 5. Claic linear control 45.5 Pitch angle 2 Longitudinal velocity e θ [rad bẋ[ m Roll angle 4 Lateral velocity e φ[rad.5 bẏ [ m Yaw angle 2 Vertical velocity e ψ [rad Time [ bż [ m Time [ FIGURE 5.23: Simulation reult of the total claic control ytem ability to tabilize the UAV in the hover operating point when given an initial offet in all of the controlled tate. The total ettling time for the controlled ytem i prolonged due to low lateral velocity control.

56

57 6 OPTIMAL LINEAR QUADRATIC CONTROL Thi chapter decribe the deign and implementation of the LQR hover and forward flight controller. Since the LQR method i a MIMO controller it reult in deign of only two different controller; one for hover and one for forward flight. Before the actual controller deign i performed the choice of operating point and control tate are decribed, followed by the principle of the LQR method. 6.1 Choice of operating point and feedback tate A the firt tep in deigning the hover and forward flight controller the two operating point for linearization of the nonlinear model mut be determined, which i done in the following Choice of operating point A with the claic hover controller the LQR controller deign i baed on linearization of the nonlinear model in a choen operating point. A decribed in the introduction to thi thei two controller, one for hover and one for forward flight, mut be developed uing the LQR method for later ue in gain cheduling. Conequently two operating point mut be determined prior to the deign. The firt operating point for the hover controller i obviouly the hover condition itelf. The econd operating point for the forward flight controller i, however, not given in advance. A tated in the introduction it i deired to complete level 1 in the IARC a fat a poible. An obviou choice of forward flight operating point i therefore the maximum peed of the UAV, which i known to be at leat 92 km h. However, a the ue of gain cheduling provide for the ue of everal forward flight controller for different forward flight peed, the econd operating point i intead choen to be at 1 m to allow the for poibility of implementing everal forward flight controller, where thi thei focue on the uage of two controller Choice of control, reference and integral tate A decribed in Section 3.2 the feedback tate for the linear controller are given a in (6.1). Hence, the upervior mut handle the poition control of the control ytem. [ x = e φ e θ e ψ b ẋ b ẏ bż T b b b φ θ ψ (6.1) 47

58 48 Section 6.2. The principle of LQR The index (repreenting the ytem feedback tate) i ued for the ake of clarity throughout thi chapter, a tate augmentation in ubequent ection involve more indice. At a firt ight the tranlatory velocity tate [ b ẋ b ẏ b ż together with the heading e ψ eem ufficient for controlling the UAV through a given level 1 pecification. However, a the LQR control method utilize weight to determine the influence of tate, it i choen to deign the controller for all nine tate in (6.1). In thi way the larget poible freedom regarding choice of weight i achieved. A decribed in Section 4.1 the LQR controller mut feature both reference and integral tate, which are included in the LQR controller by tate augmentation. A with the ytem tate the reference and integral tate are being weighted, therefore it i choen to include reference and integral tate for all nine tate in (6.1) a well. In the next ection the LQR principle i introduced, and the tate augmentation for including reference and integral tate i explained. 6.2 The principle of LQR The LQR control deign done in thi thei i baed on the procedure in Sørenen [1992, which are decribed in the following. It relie on minimization of the performance index with weight matrice on tate and input. Furthermore, it feature both reference and integral tate, which are needed in thi thei a decribed in the previou ection. The LQR control method i baed on the tate equation from the general dicrete tate-pace decription of the ytem: where x (k) and u(k) are given by (6.1) and (6.4) repectively. x (k + 1) = Φ x (k) + Γ u(k) (6.2) y (k) = H x (k) + P u(k), (6.3) [ T u = S col S lat S lon S tr (6.4) The LQR principal build on determining the optimal input u (k) uch that the performance index given in (6.5) i minimized. I = N 1 k= ( ) x T (k)q 1 x (k) + u T (k)q 2 u(k) + x T (N)Q N x (N) (6.5) The quadratic weight matrice Q 1, Q 2 and Q N weighting the tate x (k), the input u(k) and the final tate x (N) repectively. Thee are all choen along with the time horizon N, therefore the determination of the weight matrice are the actual deign tak in developing an LQR controller. Becaue the weight matrice are choen it reult in a trade-off between achieving mall error in the control tate and mall control ignal. The optimal input u (k) can be found by the control law u (k) = L(k)x (k). (6.6)

59 Chapter 6. Optimal linear quadratic control 49 The proportional matrix L(k) in (6.6) i given a: L(k) = [ Q 2 + Γ T S(k + 1)Γ 1 Γ T S(k + 1)Φ, (6.7) where S(k) atifie the following dicrete time dynamic Ricatti equation: S(k) = Q 1 + Φ T S(k + 1)[Φ Γ L(k). (6.8) It i deired to have a contant feedback gain L() calculated in advance in a cloed loop control ytem a hown in Figure 6.1. Thi i obtained by evaluating (6.7) and (6.8) backward in time, u (k) Sytem x (k) L() FIGURE 6.1: The tructure of the cloed loop ytem for an LQR controller. with S(k) = Q N and k = N 1 initially, until k =. In thi way the performance index i at all time kept N tep ahead in time, where k = denote the current time. Prior to the calculation of the controller L(), the weight matrice Q 1, Q 2 and Q N and the time horizon N mut be choen Deign parameter The ytem having nine tate and four input yield the following dimenion of the weight matrice Q 1 R 9 9 and Q 2 R 4 4. To implify the deign proce it i choen to ue diagonal weight matrice, which reult in a total of 13 weight to be determined. In doing o, each diagonal entry in Q 1 (j, j) and Q 2 (j, j) i interpreted a a meaure of the relative weight of x j (k) and u j (k) repectively. A commonly ued method for chooing the entrie, i baed on a phyical inight in the ytem. If the open loop ytem i ubject to limitation on input and tate, the diagonal entrie can a a tarting point be choen a: Q 1 (j, j) = 1 x 2 j,max Q 2 (j, j) = 1 u 2 j,max. (6.9) In addition, a weight factor can be introduced to weight Q 1 relative to Q 2. The weight on the final tate are choen a Q N = Q 1, which i often the cae when the value of the final tate are of no particular importance (ee Sørenen [1992). The ize of the time horizon N i determined by trial-and-error, by keeping in mind that a large value of N improve the tationary propertie of the control ytem, and that a mall value of N improve the dynamical propertie of the control ytem (ee Sørenen [1992).

60 5 Section 6.2. The principle of LQR Implementation of reference tate The reference tate are implemented in the control deign uch that the controller i able to follow deired et-point. The et-point are provided by the upervior and therefore modeled a contant, yielding the following tate-pace repreentation of the ytem together with the reference: x (k + 1) = Φ x (k) + Γ u(k) (6.1) y (k) = H x (k) + P u(k) (6.11) x r (k + 1) = Φ r x r (k) (6.12) r(k) = H r x r (k), (6.13) where Φ r and H r are 9 9 identity matrice. With the control error given a e(k) = r(k) y (k) = H r x r (k) H x (k), (6.14) the performance index become: I = N 1 k= ( e T (k)q 1e e(k) + u T (k)q 2 u(k) ) + e T (N)Q Ne e(n), (6.15) where Q 1e i the control error weight matrix. To calculate the gain matrice for the tate and reference, the tate vector i augmented to contain both ytem and reference tate: [ T x aug (k) = x T (k) x T r (k). (6.16) The tate-pace repreentation for the augmented tate then become [ x (k + 1) x r (k + 1) = [ Φ Φ r [ e(k) = H The augmented ytem can now be decribed a: [ x (k) x r (k) H r [ x (k) x r (k) + [ Γ u(k) (6.17). (6.18) x aug (k + 1) = Φ aug x aug (k) + Γ aug u(k) (6.19) e(k) = H aug x aug (k). (6.2) The augmented ytem can, a hown for the ytem in (6.2) and (6.3), be ued to calculate the ytem and reference feedback gain depicted in Figure 6.2 for the optimal controller: [ u(k) = L aug ()x aug (k) = L() L r () [ x (k) x r (k). (6.21)

61 Chapter 6. Optimal linear quadratic control 51 x z 1 r (k) u(k) x (k) y(k) L r () + Γ + z 1 H Φ r Φ H r r(k) L() FIGURE 6.2: The tructure of the LQR controller with implemented reference tate Implementation of integral tate A tated in Section 4.1 it i choen to implement integral tate to remove teady tate error. The tate-pace repreentation of the integral i given a x I (k + 1) = x I (k) + e(k) (6.22) e(k) = r(k) y(k). (6.23) With the integral tate the augmented tate vector become [ x aug (k) = x T (k) xt r (k) T xt I (k), (6.24) which yield the new augmented ytem Φ x aug (k + 1) = Φ r x aug(k) + H H r I Γ u(k) (6.25) = Φ aug x aug (k) + Γ aug u(k) (6.26) [ y(k) = H x aug (k) = H y x aug (k) (6.27) [ e(k) = H H r x aug (k) = H e x aug (k) (6.28) [ x I (k) = I x aug (k) = H I x aug (k). (6.29) With the implemented integral tate the performance index become where I = N 1 k= ( ) x T aug (k)q 1x aug (k) + u T aug (k)q 2u aug (k) + x T aug (N)Q Nx aug (N), (6.3) Q 1 = Q N = H T e Q 1eH e + H T I (k)q 1IH I (6.31) Becaue of the implementation of integral action on all feedback tate in the LQR controller, a total of 22 weight parameter for each of the hover and forward flight controller mut be determined. With the augmented ytem given by Φ aug and Γ aug and the deign parameter Q 1, Q 2

62 52 Section 6.2. The principle of LQR and Q N, the optimal controller [ u(k) = L aug ()x aug (k) = L L r L I x (k) x r (k) x I (k), (6.32) with a tructure a depicted in Figure 6.3 can be calculated by evaluating (6.7) and (6.8). z 1 x r (k) u(k) x (k) L r () + + Γ z 1 Φ r L I () Φ H r 1 z 1 x I (k) L() r(k) + + y (k) H FIGURE 6.3: The tructure of the optimal controller with implemented reference and integral tate Modification of the LQR control tructure The current tructure of the LQR controller i deigned to weight the different reference, which the controller mut track. However, early controller deign howed that the gain matrice L r and L are approximately identical with oppoite ign (L r = L). By performing thi ubtitution of L r the following become clear: [ u(k) = L L [ x (k) x r (k) [ = L I I [ x (k) x r (k) = L(x r (k) x (k)) = Le(k). (6.33) From (6.33) it can be een that the modification reult in the controller trying to bring the error e(k) to zero. In addition, only two control gain matrice are needed to be deigned. Becaue the matrice H r and H are identity matrice the controller tructure depicted in Figure 6.4 i obtained. Thi tructure will be ued for implementation and tet of the deigned controller in the nonlinear model in SIMULINK. Having determined the deign method and the actual tructure for implementation of the LQR controller the deign of the hover and forward flight controller can be performed, which i done in the following ection.

63 Chapter 6. Optimal linear quadratic control 53 x r (k) + + L() u(k) + Γ + z 1 x (k) 1 z 1 x I (k) L I () Φ FIGURE 6.4: The modified tructure of the optimal controller. 6.3 Hover controller deign In the following the deign of the LQR hover controller will be performed. A decribed earlier the actual deign lie in the determination of the weight matrice. Furthermore, it i decribed that if the open loop ytem i ubject to limitation regarding the tate and input, then the weight matrice can be initially be determined baed on thee limitation. Thi will be elaborated further in the following ection Analytical weight determination A decribed in Section the weight can be determined (baed on the limitation) a decribed by (6.9). The limitation in the ytem tate can be determined baed on open loop imulation of the nonlinear model. Becaue the model mut operate in open loop it need to be initialized in an equilibrium point regarding tate and input. Thi equilibrium point i found by linearization of the nonlinear model in a deired operating point (ee Appendix C for information on linearization). Note that it i not poible to operate the nonlinear model in open loop for all of thee equilibria, a the linearization yield only approximated equilibria. The limitation on the different tate are determined a the highet value for which the nonlinear model doe not detabilize. With thi in mind the following tate limitation have been identified by examining different flight velocitie in forward and ideway direction: bẋ max = 33 m with pitch angle: e θ min =.32 rad bẏ max = 2 m with roll angle: e φ max =.5 rad. (6.34) The pitch angle e θ min i found a the lower limit due to it definition. The minimum vertical velocity b ż correponding to upward flight with e θ = e φ = rad i by imilar experiment identified a b ż min = 12 m. The limitation on the yaw angle e ψ i intuitively not poible to determine, a there hould not be a limit on the number of rotation poible to perform. Therefore, it i choen a e ψ max = 1 rad. A with the tranlatory velocitie, it i poible to linearize the nonlinear model to obtain an equilibrium point with a contant rotational velocity b ψ about the z-axi in the BF. The limit ha been identified a b ψ max = π rad correponding to one revolution in two econd. The angular

64 54 Section 6.3. Hover controller deign velocitie b φ and b θ are etimated to be equal to the limit on b ψ yielding b φ max = b θ max = π rad. Having determined the limit on the tate the diagonal weight matrix in (6.35) i obtained for both the error and integral tate. In addition, the weight on the control input are choen a large value relative to the weight on the ytem tate in order to obtain a conervative controller. The diagonal weight matrix for the control ignal in (6.36) i choen. [ Q 1e (j, j) = Q 1I (j, j) = 1 (.5) 2 1 (.32) π 2 1 π 2 1 π 2 = [ (6.35) Q 2 (j, j) = [ (6.36) The time horizon parameter i choen a N = 3, and the LQR hover controller i calculated a decribed in Section 6.2. The deigned controller i teted by imulation on the nonlinear model 5 x 1 3 Pitch angle.2 Longitudinal velocity e θ [rad 5 bẋ [ m Roll angle.5 Lateral velocity e φ [rad.4.3 bẏ [ m Yaw angle.5 Vertical velocity e ψ [rad bż [ m Time [ Time [ FIGURE 6.5: Tet reult for the deigned LQR hover controller ability to tabilize the nonlinear model in hover. The red horizontal line how the hover operating point value. initialized in the hover operating point and with the etimated tate a feedback ignal. The imulation reult i depicted in Figure 6.5, where the hover operating point i illutrated a the red horizontal line. From the imulation of the hover controller it can be oberved that it i capable of tracking

65 Chapter 6. Optimal linear quadratic control 55 the pitch angle et-point with a deviation of le than rad. A imilar obervation can be made for the roll angle, where a deviation of about.1 rad i determined. Taken the etimation ignal and the analytical determined weight matrice into conideration, the performance of the hover controller i evaluated a relatively good becaue of it ability to track the et-point angle. However, the ability to track the pitch and roll angle can be interpreted a a reult of reduced longitudinal and lateral velocity tracking ability. Thi can alo be oberved from Figure 6.5, where the longitudinal and lateral velocitie have a tationary error a they approache negative value over time. Furthermore, it can be oberved that the deigned controller i relatively low taken the time cale into conideration. Thi mean that compenation of diturbance applied to the UAV will happen lowly and may ultimately lead to intability. Therefore, a fater controller mut be deigned. In addition, a the ability to track velocity et-point in longitudinal and lateral direction are deemed more important than the tracking ability of the pitch and roll angle, it i neceary to improve the LQR controller by manually tuning the weight matrice Weight tuning regarding tationary propertie The tuning of the weight matrice i performed iteratively uing trial-and-error during multiple imulation to obtain the weight een in (6.37) (6.39): Q 1e (j, j) = [ (6.37) Q 1I (j, j) = [ (6.38) Q 2 (j, j) = [ (6.39) In addition, the time horizon i changed to N = 5 in order to improve the tationary propertie in the cloed loop. Thi i the final value of N and will therefore not be dicued any further throughout thi thei. A tet imulation uing the new weight matrice i performed with the reult depicted in Figure 6.6. Compared to the reult from Figure 6.5 the roll and pitch angle obtain larger variation, which i a trade-off that lead to better tracking of the longitudinal and lateral velocitie. Thi i explained by the fact, that when the UAV need to gain e.g. longitudinal velocity, thi i done by altering the current pitch angle. It i oberved that the angle e θ, e φ and e ψ varie le than.1 rad from their repective et-point, and the three tranlatory velocitie b ẋ, b ẏ and b ż varie le than.3 m. The ize of thee tationary deviation are etimated a being mall enough for the controller to fulfill their purpoe of keeping the UAV in the condition hover. Furthermore, it can be oberved that the hover controller ha become become a fater controller than the previou, therefore it i decided to ue thi for control of the ytem. Having validated the deigned controller with repect to it tationary propertie, the dynamical propertie of the controller mut be teted. Furthermore, if deemed neceary the controller will be tuned baed on the obervation done by dynamical tet of the cloed loop ytem.

66 56 Section 6.3. Hover controller deign eplacement.4 Pitch angle.5 Longitudinal velocity e θ [rad.2 bẋ [ m Roll angle.5 Lateral velocity e φ [rad bẏ [ m Yaw angle.2 Vertical velocity e ψ [rad bż [ m Time [ Time [ FIGURE 6.6: Tet reult of the improved LQR controller. A it can be oberved the variation of the angle are lager than the previou deigned controller. However, the new controller ha better overall tracking ability and perform fater control Weight tuning regarding dynamical propertie The dynamical propertie of the deigned controller are examined by performing offet tet and analyzing the controller ability to return the UAV to the hover operating point. The tet i performed by initializing the nonlinear model in hover with the offet lited in Table 4.1 on page 22 applied to the initial tate. For thi purpoe the UAV i conidered a being in hover when the angle varie le than.1 rad and the velocitie varie le than.3 m from the hover operating point (value obtained from Figure 6.6). A it i not poible to determine ome maximum time for the controller to tabilize the UAV, it i intead required to ettle the ix control tate equally fat, with a maximum time diperion of 5 econd. The imulation reult i depicted in Figure 6.7. It i oberved that the time for ettling the longitudinal velocity at the tationary limit (indicated by dahed line) i approximately 23 econd, where it only take approximately 6 econd to ettle the pitch angle. An intuitive way for obtaining a lower ettling time for the longitudinal velocity i to increae the correponding weight for bẋ in Q 1e and/or Q 1I. A few trial-and-error imulation with increaed weight for b ẋ quickly

67 eplacement Chapter 6. Optimal linear quadratic control 57 e θ [rad Pitch angle bẋ [ m 2 Longitudinal velocity Roll angle 4 Lateral velocity e φ [rad bẏ [ m Yaw angle 1 Vertical velocity e ψ [rad Time [ bż [ m Time [ FIGURE 6.7: Dynamical tet of the improved LQR hover controller. The large variation in the roll angle caue ignificant variation in the lateral velocity a well. reveal, that thi approach i not the olution. The explanation i found in the evere cro coupling between the pitch angle e θ and the longitudinal velocity b ẋ. From obervation of Figure 6.7 it become clear that although e θ ettle within 6 econd, it doe vary throughout the following 2 econd, which caue b ẋ to ettle relative late. Therefore, the focu mut be turned to tuning the weight for e θ. Similar ocillating behavior i oberved for the roll angle and the lateral and vertical velocitie. Conequently ome of the weight element of the weight matrice in (6.37) and (6.38) have been multiplied by the following factor: Q 1e (1, 1) = 1 Q 1e (2, 2) = 1 Q 1e (3, 3) = 2 Q 1I (1, 1) = 2 Q 1I (2, 2) = 2, (6.4) which reult in the dynamical propertie depicted in Figure 6.8 for the LQR hover controller. It i oberved that the ix control tate ettle in the time period 2 < t < 6, which atifie the requirement of a maximum diperion of 5 econd. In addition, it i oberved that the tationary propertie of the LQR controller have alo improved compared to Figure 6.7. A the LQR hover controller i capable of tabilizing the UAV in the hover operating point

68 58 Section 6.3. Hover controller deign eplacement e θ [rad Pitch angle bẋ [ m 2 1 Longitudinal velocity Roll angle 4 Lateral velocity e φ [rad bẏ [ m Yaw angle 1 Vertical velocity e ψ [rad Time [ bż [ m Time [ FIGURE 6.8: Dynamical tet for the final LQR hover controller. It i een that the control tate ettle fater and with le overhoot with repect to the former controller weight. (given the initial offet value) within 6 econd the tationary and dynamical propertie of the controller are concluded a atifactory for further ue in the gain cheduling controller. In the next ection the claic SISO hover and the optimal LQR hover controller are compared regarding tationary and dynamical propertie Comparion of claic and LQR hover control A decribed in the introduction to thi thei the claic control trategy i applied to be able to compare it with more advanced control trategie, here optimal LQR control. The comparion of the two control trategie i baed on the individual tet performed with repect to tationary and dynamical propertie for each of the two controller. The firt tet performed for each controller concern the tationary propertie. For the LQR controller a new imulation different from the one in Section i performed, a the weight matrice for thi controller were changed during evaluation of the dynamical propertie. Thi new imulation reult i depicted a the blue graph in Figure 6.9 together with the tet reult obtained for the claic controller in Section 5.4 (the green graph).

69 Chapter 6. Optimal linear quadratic control 59 The graph for the yaw angle look much alike which indicate, that the two controller perform equally with repect to the yaw angle, and that the variation to a great extent are a reult of the etimated yaw angle being different from the actual yaw angle for the model. It i oberved that the pitch angle and the longitudinal and vertical velocitie are more reonant for the claic controller, which i an unwanted property for controlling a helicopter. The LQR controller i better than the claic controller for controlling thee, a it maintain the tate cloer to the reference and le reonant. The claic controller eem to control the lateral velocity better than the LQR controller. However, a the main part of the variation of the graph, a with the yaw angle, reult from the etimated tate being different from the real tate, no concluion can be drown regarding which controller i the better for controlling the lateral velocity. It i concluded that the LQR controller overall ha better tationary propertie than the claic controller..1 Pitch angle.2 Longitudinal velocity e θ [rad bẋ [ m Roll angle.2 Lateral velocity e φ [rad bẏ [ m Yaw angle.5 Vertical velocity e ψ [rad bż [ m Time [ Time [ FIGURE 6.9: Comparion of the claic (green) and LQR (blue) controller for hover with repect to the tationary propertie. It i oberved that the pitch angle and the longitudinal and vertical velocitie are more reonant for the claic controller. For both controller tet regarding the dynamical propertie ha been performed a well. The nonlinear model wa initialized with offet in the tate e φ, e θ, e ψ, b ẋ, b ẏ and b ż, and the ability of the controller to ettle thee ix tate to the level hown in Figure 6.9 wa examined in term

70 6 Section 6.4. Forward flight controller deign of time pend. The claic controller ued 1 and the LQR controller 5 6. It i therefore concluded, that the LQR controller ha better dynamical propertie than the claic controller. In the following ection the deign of the LQR forward flight controller i decribed. 6.4 Forward flight controller deign A a tarting point for the deign of the diagonal weight matrice for the forward flight controller, the final weight determined for the hover controller are ued. The forward flight controller i afterward tuned with repect to the tationary propertie uing trial-and-error to obtain the following weight matrice: Q 1e (j, j) = [ (6.41) Q 1I (j, j) = [ (6.42) Q 2 (j, j) = [ (6.43) For tuning both the hover and the forward flight LQR controller ome identified rule of thumb have been exploited in addition to the trial-and-error method. A decribed in the previou ection the trong cro coupling between b ẋ and e θ reulted in changing weight for e θ intead of bẋ for the hover controller. In general, the knowledge on cro coupling in the nonlinear model decribed by the input/output relation divided into four group in Section 2.3 have been ued for weight tuning Stationary propertie A tationary tet imulation of the nonlinear model initialized in the 1 m forward flight operating point and tabilized by the forward flight controller i depicted in Figure 6.1. It i oberved that the lateral velocity deviate from the reference of m with a peak of 1.5 m, which i ignificant compared to the tationary deviation of le than.3 m oberved for the hover controller (ee Section 6.3.2). The explanation i, that ome of the etimated tate ued a feedback for control purpoe have a ignificant deviation from the real tate. In thi cae the deviation on the lateral velocity origin from poor etimate of the yaw angle e ψ et and lateral velocity b ẏ et. Thi i hown in Figure 6.11, where the tet imulation i repeated uing the real tate a feedback. It i oberved from Figure 6.11 that the controller i able to tabilize both e ψ and b ẏ at approximately zero in contrat to their repective etimate. Furthermore, it i oberved that the yaw angle etimate ha a poitive error of up to.25 rad and the lateral velocity a negative error of up to 2.5 m. Another important obervation i the initial increae and decreae in the longitudinal velocity and pitch angle repectively. Thi i due to the initial value of the etimated longitudinal velocity, which at all time initially i zero. Thi lead to an initial error input to the controller of 1 m even though the real error (with repect to the real initial tate) i zero. A thi thei doe not concern enor equipment and enor fuion and etimation, the ignificant etimation error in e ψ et and b ẏ et will not be treated any further in thi thei. A imulation

71 eplacement Chapter 6. Optimal linear quadratic control 61 Pitch angle 11 Longitudinal velocity e θ [rad.1 bẋ [ m Roll angle 2 Lateral velocity e φ [rad bẏ [ m Yaw angle Vertical velocity e ψ [rad bż [ m Time [ Time [ FIGURE 6.1: Stationary imulation reult of the LQR forward flight controller initialized in the forward flight operating point of 1 m..5 Etimated yaw angle 1 x 1 13 Real yaw angle e ψ et [rad e ψ [rad Etimated lateral velocity 1 Real lateral velocity bẏ [ m 2 bẏ [ m Time [ Time [ FIGURE 6.11: Simulation of the forward flight controller in the operating point of 1 m uing the real tate a feedback ignal.

72 62 Section 6.4. Forward flight controller deign over a greater period of time how, that the etimation error doe not become larger, and a the top level uperviory controller provide poition control in the control ytem. Furthermore, the drift effect from the oberved tationary lateral velocity error i aumed not to have noticeable impact on the performance of the control ytem. Baed on thi aumption the deigned LQR forward flight controller it i aeed that it ha atifactory tationary performance in the forward flight operating point. In the following ection the dynamical propertie of the deigned forward flight controller will be evaluated Dynamical propertie The dynamical propertie of the LQR forward flight controller are examined in the ame manner a for the LQR hover controller, but with tarting point in the forward flight operating point. The reult of the tet i depicted in Figure 6.12, where it i oberved that four of the controlled output ettle to their repective reference; e θ, b ẋ, e φ and b ż. Regarding the remaining two control tate e θ [rad.5 Pitch angle bẋ [ m Longitudinal velocity Roll angle 5 Lateral velocity e φ [rad bẏ [ m Yaw angle 5 Vertical velocity e ψ [rad.5 bż [ m Time [ Time [ FIGURE 6.12: Validation tet for the LQR forward flight controller where an initial offet i given on ix of the controlled output; e θ, b ẋ, e φ, bẏ, e ψ and b ż. ( b ẏ and e ψ) the ame behavior a decribed in the previou ection can be oberved. However,

73 Chapter 6. Optimal linear quadratic control 63 ince it ha been concluded that the deviation in b ẏ and e ψ are acceptable thi i overlooked for the dynamical tet a well. However, another important obervation i that of the rather large variation during the firt 15 econd of the different control tate. Thi i mainly due to the initial value of the etimated tate, which are all zero. Conequently the controller compenate on a wrong error of 1 m in the longitudinal velocity, where the actual error on the model tate 2 m. The integral action thereby accumulate a large error, which help explaining the 15 econd of large variation. With the above in mind the deigned LQR forward flight controller i aeed to have atifactory dynamical performance becaue it i capable of returning the UAV to it forward flight operating point. Therefore, thi controller will be ued in the gain cheduling controller, which i treated in the next chapter.

74

75 7 OBSERVER-BASED GAIN SCHEDULING CONTROL Thi chapter explain the deign and implementation of gain cheduling into the control ytem. The implementation of gain cheduling make it poible to witch between everal controller deigned for one or more operating point(). In addition, the ue of gain cheduling enable the helicopter to achieve a more reliable and veratile manoeuvrability when performing level 1 of the IARC. 7.1 Gain cheduling method A mentioned earlier in thi thei the implemented controller are deigned baed on linearization of the nonlinear model in different operating point. Here the operating point have been choen a hover and forward flight of 1 m, and linear controller have been developed for the repective operating point. The challenge i to witch between thee two pecific controller without compromiing overall ytem tability. Three way to approach the problem of witching between controller are; direct witch between controller, gradual tranition uing a controller weight function α g, and an oberverbaed gain cheduling verion of the weighted controller method with improved tability propertie Direct controller witch Direct witch between controller i the mot imple method of gain cheduling. Thi method i executed imply by replacing one controller with another. Conider the optimal LQR controller form u(t) = L e(t) L I x I (t), (7.1) where u(t) i the ytem control ignal, e(t) i the control error and x I (t) i the integral tate with ẋ I (t) = x I (t)+e(t). L and L I are the optimal proportional and integral gain repectively for a given operating point. Now aume that a witch i performed from the firt et of controller (L and L I ) to a econd et of controller (L 1 and L I1 ) at a given time intance t. If L 1 i different from L and/or L I1 i different from L I, which i probably the cae if the controller et are deigned for different operating point, the value of u(t) will be ubject to a momentary change with a ize dependent on the difference between the controller et, unle both e(t) and x I (t) at the time t are both zero. Hence, thi method may caue the control ytem to become untable. 65

76 66 Section 7.1. Gain cheduling method Weighted controller witch A le bumpy tranfer between two controller et are achieved by implementing a weight function α g. The general control law for thi method i u(t) = (α g L 1 + (1 α g L )) e(t) + (α g L I1 + (1 α g )L I )x I (t), (7.2) where α g [; 1. By changing α g gradually from 1, (7.2) will reult in a witch from the firt controller et (L and L I ) to the econd et (L 1 and L I1 ), where a mooth control ignal u(t) (no momentary change) i maintained, a long a e(t) and x I (t) doe not change momentary either. However, it i important to note that thi form of gain cheduling between to et of controller doe not guarantee tability of the control ytem for < α g < 1, Bendten et al. [ Controller witch uing oberver-baed gain cheduling In Bendten et al. [25 a controller contruction that guarantee tability for any α g [; 1 when applied to one linear ytem i introduced. It i claimed, that while it doe not guarantee tability between two operating point, it i till an improvement compared to (7.2). Thi controller contruction i implemented and teted on the nonlinear model of the UAV in thi thei. Starting point i taken in an oberver baed tructure mapping an input ignal u(t) R m to an output ignal y(t) R p, which can be decribed a [ A B H = C D, (7.3) which repreent a tandard tate pace model ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t). (7.4) The above ytem can be extended to a o called two port ytem A B B 2 H = C D D 12, (7.5) C 2 D 21 D 22 which, if interconnected with a zero ytem, yield [ [ [ I A B I H = H = C D = H. (7.6) mapping two input vector to two output vector. The in (7.6) repreent the Redheffer Starproduct, where the calculation method for the interconnection hown in Figure 7.1 between two two port ytem P = A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22, K = A K B K1 B K2 C K1 D K11 D K12 C K2 D K21 D K22

77 Chapter 7. Oberver-baed gain cheduling control 67 i defined a (ee Zhou et al. [1996): [ Ā B S(P, K) = C D, where [ A + B2 R 1 DK11 C 2 B 2 R 1 CK1 Ā = [ B = [ C = [ D = B K1 R 1 C 2 A K + B K1 R 1 D 22 C K1 B 1 + B R 1 2 DK11 D 21 B K1 R 1 D 21 C 1 + D 12 D K11 R 1 C 2 D K21 R 1 C 2 D 11 + D 12 D K11 R 1 D 21 D K21 R 1 D 21 B 2 R 1 DK12 B K2 + B K1 R 1 D 22 D K12 D 12 R 1 CK1 C K2 + D K21 R 1 D 22 C K1 D 21 R 1 DK12 D K22 + D K21 R 1 D 22 D K12 (7.7) (7.8) (7.9) (7.1) R = I D 22 D K11, R = I DK11 D 22. z(t) w(t) P ẑ(t) K ŵ(t) FIGURE 7.1: Illutration of the interconnection between two two port ytem. The above decribed method for interconnection between ytem i combined with the weighted controller witch method to perform oberver-baed gain cheduling between the two LQR controller developed in Chapter 6. In the following ection the matrice aociated with the implementation of the oberver-baed gain cheduling are explained. 7.2 Oberver-baed gain cheduling deign The deign preented throughout thi ection i baed on the work decribed in Bendten et al. [25. It i preented in continuou time, however it can be extended into dicrete time by imple dicretization of the involved matrice. The time index (t) i omitted to eae the notation.

78 68 Section 7.2. Oberver-baed gain cheduling deign For the purpoe of oberver-baed gain cheduling it i neceary to deign oberver for the two operating point; hover, and forward flight of 1 m. The deign of the oberver follow the general oberver deign theory, where any given ytem i implified by linearization, and an oberver feedback gain F i determined uch that the output of the oberver ŷ converge to the enor output y of the ytem a illutrated in Figure 7.2. u Nonlinear helicopter model y F + B + ˆx C ŷ A FIGURE 7.2: Illutration of the general oberver theory with oberver feedback gain. The ytem matrice A and B are determined a decribed in Appendix C for the hover (A and B ) and the forward flight (A 1 and B 1 ) operating point. The output matrice C and C 1 are both identity (I) with the ame dimenion a A and A 1, a all tate can be meaured directly and a full tate oberver i choen. The oberver feedback gain are therefore the only remaining part for implementing the oberver tructure in Figure 7.2. Note that there mut be deigned a pecific oberver for each of the two operating point, which will be elaborated on later. For now the oberver gain are denoted with F and F 1 for the hover and the forward flight operating point repectively. The goal for the oberver deign i to contruct a two port ytem ˆx ˆx I u e q { }} { A B L F C B F B = L I ri L I I C I K ˆx ˆx I e u q, (7.11) with the tructure depicted in Figure 7.3, where r > repreent the integrator factorization and L and L I repreent the LQR hover controller. The LQR controller matrice of dimenion R 4x9 are each extended with three column of zero a feedback gain for the poition error of e x, e y and e z to fit the full tate oberver deign. The augmented oberver tate vector i defined a ˆξ = [ ˆx T ˆx T I T. A it can be een from (7.11) there are two input and two output vector, where K take the input vector e = y y ref and u q, and yield the output vector u and

79 Chapter 7. Oberver-baed gain cheduling control 69 e q = e C ˆx. Notice that the oberver about to be deigned are not oberving the actual tate but the tate error. e u q ri + L I F B ˆx C ê A e q L u FIGURE 7.3: Graphical illutration of the oberver-baed controller given by (7.11) (illutrated in continue time). By performing interconnection between K and a zero ytem it can be hown that the reulting ytem become a tandard oberver-baed controller K with integral action [ [ A B L F C B F I I K = K = L I = K, (7.12) L I which i equal to α g = in K(α g Q) = K (α g Q) (7.13) where K i the total interconnected ytem in Figure 7.4(a). Having determined the tructure for the two port ytem for the firt oberver-baed controller K the next tep i to identify the tructure for the cheduler Q. Becaue Q i the control ytem cheduler it mut by definition contain the econd controller to witch to. Furthermore, it i deired to decouple the firt controller completely when α g = 1 and therefore it i neceary to deign a two port ytem K with the tate vector ξ = [ x T x T I T that, when α g = 1, yield an identity ytem when interconnected with K. By having an identity ytem of K K it i poible to implement any given econd oberverbaed controller K 1 interconnected to K (ee Figure 7.4(b)), uch that replacing Q with K K 1 in (7.13) yield K(α g ( K K 1 )) = K (α g K K1 ). (7.14)

80 7 Section 7.2. Oberver-baed gain cheduling deign u e e q K u q α g u e q K u q e K e 1 u 1 Q α g Q K 1 (a) (b) FIGURE 7.4: (a) how the tandard gain cheduling trategy with an oberverbaed controller K interconnected with a cheduler α g Q. (b) how the deign of the total gain cheduling controller, where Q i defined a K K 1, which yield the total controller a K(α g Q) = K (α g K K1 ). Baed on the tructure of (7.11) K i in Appendix D determined a: K = A F B L I C rl ri L I ri L I I, (7.15) C I uch that the following interconnection A B L F C B B L B F B rl ri L I ri K K F C A F B = L I C L I C + rl ri L I ri L I L I I C C I (7.16) can be made, where it can be een that the interconnection K K truly i an identity ytem, a ˆξ = ξ reult in: u = u 1 (7.17) e 1 = e, (7.18)

81 Chapter 7. Oberver-baed gain cheduling control 71 which mean that the control ignal u 1 from the econd oberver-baed controller K 1 will be the only control ignal affecting the controlled ytem. Likewie, the error e 1 provided to controller K 1 i equal to the actual error of the ytem e. Furthermore it can be een that when the augmented tate vector are identical they will tay that way a long a α g = 1: ˆx = (A F C)ˆx + F e + B u 1 (7.19) ẋ I = rl x r x I + L I e + ru 1 (7.2) x = F C ˆx + A x + F e + B u 1 (7.21) x I = rl x r x I + L I e + ru 1 (7.22) The tructure of the econd oberver baed controller K 1 i the ame a given in (7.12) but deigned in a new operating point. K 1 i then given a K 1 = A 1 B 1 L 1 F 1 C y B 1 F 1 L I1 L 1 I, (7.23) where K 1 ha the augmented tate vector ˆξ 1 = [ ˆx T 1 ˆx T I1 T Deign of oberver gain F and F 1 Having determined the tructure of the oberver-baed gain cheduler and the LQR controller the oberver feedback gain matrice for the two operating point are the next to be deigned. Each of the feedback gain i calculated uing the MATLAB function lqed(), which calculate the Kalman feedback gain matrix F when given the ytem matrix A, proce noie matrix G, output matrix C, oberver etimat covariance matrix Q, meaurement covariance matrix R and the deired ampling time T a input. A i known from Appendix C for both operating point. The etimated tate of the nonlinear model are the oberver meaurement aociated with the covariance matrix R. It i aeed that the meaurement are reliable baed on the tet performed for the LQR controller in Chapter 6, and therefore the covariance matrix R i et mall relative to Q. By imulation with the nonlinear model in open loop and trial-and-error tuning the following matrice are determined: R =.1I Q =.1I G = I (7.24) The two calculated oberver gain F and F 1 can be found in Appendix E). The deigned oberver gain matrice can only be ued for oberver baed gain cheduling approach if the oberver gain are able to tabilize the ytem A uch that A + FC (or A FC) i Hurwitz, which mean that all eigenvalue mut have trictly negative real part (in the dicrete cae the eigenvalue mut be within the unit circle)(bendten et al. [25). By uing the MATLAB function eig() the

82 72 Section 7.2. Oberver-baed gain cheduling deign eigenvalue for the two table ytem A F C and A 1 F 1 C 1 are: eig(a F C ) = [ i i i i eig(a 1 F 1 C 1 ) = [ i i i i i i i i T T A it can be een above all eigenvalue for the two oberver have trictly negative real part, which how that both ytem are Hurwitz and therefore uable for oberver-baed gain cheduling Deign of α g function The cheduling variable α g [; 1 i modeled a a piecewie linear function of the forward flight velocity b ẋ a illutrated in Figure 7.5. A can be een mall value of α g exit for b ẋ < 4 m, 1.8 αg bẋ [ m FIGURE 7.5: Illutration of the choen α g function. and large value of α g (value cloe to one) exit for b ẋ > 6 m. In that way K obtain the mot influence for b ẋ < 4 m and K 1 get the mot influence for b ẋ > 6 m. At a forward flight velocity of b ẋ = 5 m equal influence of the two controller i obtained. Having deigned the α g function, the deign of the oberver-baed gain cheduling controller i concluded. In the following ection the functionality and performance of the controller will be evaluated.

83 Chapter 7. Oberver-baed gain cheduling control Gain cheduling control tet In Section 7.2 the algebraic calculation howed that the interconnected ytem K K i an identity ytem when α g = 1. To determine whether the cheduler can actually be ued for control of the nonlinear model it i implemented and teted with repect to the change of controller, which mean that the gain cheduling ytem mut be teted for the propertie of K K. The property i teted by comparing the two tate vector ˆx and x from K and K repectively. If the interconnection between the two are indeed an identity ytem then ˆx x αg=1. The tet of the gain cheduling ytem i performed by implementing the α g function a a ramp with a lope of.1 and aturation at α g = 1. At the time zero α g = and after 1 econd α g = 1 i obtained. Becaue the cheduling ytem i initialized with α g = the tate oberved by K and K will be different. Therefore, it i expected that when the ramp function become 1 the cheduling ytem will make ˆx x for increaing time t > 1. The reult of the tet of the cheduler can be een in Figure 7.6. It i oberved, that after time t = 1 all oberver error converge to zero (ˆx x ). It i therefore concluded that the cheduler i performing a intended with the ability to witch between the two controller. [ rad Oberver tate error e φ e θ e ψ [ / rad [ m Oberver tate error bẋ bẏ bż b φ b θ b ψ Time [ (a) Time [ (b) FIGURE 7.6: Graph howing the oberver tate error ˆx x for α g going from to 1. (a) how the error in the ytem angle given in the EF. (b) how the error in the ytem velocitie given in the BF. Having determined that the cheduler i able to witch between the two LQR controller the total oberver-baed gain cheduling control ytem can be validated uing the ame approach a with the claic and LQR controller, by initializing the UAV with offet in ix of the control tate. The offet value are the ame a for the tet of the claic and the LQR controller (ee Table 4.1 on page 22). However, it i deired to perform the tet with a mixture of K and K 1 uch that both controller are equally ued. Thi mean that an α g =.5 reulting in a forward

84 74 Section 7.3. Gain cheduling control tet flight velocity of 5 m mut be maintained. The offet tet for the oberver-baed gain cheduling control ytem can be een in Figure Pitch angle 15 Longitudinal velocity e θ [rad bẋ [ m Roll angle 5 Lateral velocity e φ [rad bẏ [ m Yaw angle 2 Vertical velocity e ψ [rad Time [ bż [ m Time [ FIGURE 7.7: Simulation reult of the oberver-baed gain cheduling controller ability to tabilize the UAV at 5 m when given an offet in ix of the controlled tate. It i oberved that rather large variation for all tate during the firt 1 exit. Thee are explained by the etimator problem mentioned in Section 6.4 (applie to the Senor fuion and etimation block preented in Section 2.2) cauing the etimated tate b ẋ et to tart at m. Overall, at time t =, the real tate i 7 m, the etimated tate i b ẋ et = m and the reference i rb ẋ = 5 m. The error baed on the etimated tate een from the controller i therefore 5 m, where the actual error i 2 m, cauing the controller to bring the UAV further away from it reference. Depite the fact that b ẋ et become equal to b ẋ after approximately 2, the large variation continue the firt 1 a a reult of the integral action in the controller accumulating the error. The effort of the controller trying to tabilize the UAV baed on a wrong tate etimate propagate to the other tate a well due to cro coupling in the model. However, the controller tabilize the UAV at the reference afterward. For the final tet of gain cheduling controller, an acceleration from hover to forward flight of 1 m i imulated with the nonlinear model in SIMULINK. The UAV i initialized in hover, and four tep of each 2.5 m at the time 1, 2, 3 and 4 are ued to produce the forward flight

85 Chapter 7. Oberver-baed gain cheduling control 75 reference rbẋ. The longitudinal velocity and the pitch angle from the tet imulation i hown in Figure 7.8. It i oberved that b ẋ ha a mall overhoot at the firt two tep and no overhoot e θ [rad Pitch angle bẋ [ m 1 5 Longitudinal velocity Time [ (a) Time [ (b) FIGURE 7.8: Illutration of; (a) the pitch angle (b) the forward flight velocity (the red line indicate the reference) in a imulation of the gain cheduling controller in cloed loop with the nonlinear model. at the lat two tep, which indicate that different controller are ued a intended. Overall the gain cheduling control ytem follow the reference without teady tate error atifying for ue in the overall control ytem. Having deigned and teted the gain cheduling controller of the overall control ytem, the lat element of the control ytem, the upervior, can be deigned, which i done in the following chapter.

86

87 8 SUPERVISORY CONTROL Thi chapter decribe the deign of the uperviory controller ued for providing reference to the low level linear controller. Different algorithm ued for calculating the reference baed on imple rule on how to handle the approach to and turning at waypoint are decribed. A number of parameter aociated with threhold for the different deciion are determined a decribed lat in the chapter. 8.1 Overview The purpoe of the upervior i to provide poition control for the control ytem. Baed on poition and heading it mut be able to generate reference for the LQR and gain cheduling control unit, uch that the UAV i able to complete level 1 in the IARC a decribed in ection 3.1. Thee reference are created by rule on how to handle all poible ituation given the actual poition and heading of the UAV compared to the target waypoint (the next waypoint to be reached). The following in- and output are thu identified: Input: Level { 1 pecification: } e Ξ lvl1 = [ e x e wp,1 y e wp,1 z wp,1,..., [ e x e wp,n y e wp,n z wp,n Input: Feedback control ignal: e Ξ = [ e x e y e z and e ψ Output: Reference ignal: o r = [re φ re θ re ψ reẋ reẏ reż r b φ r b θ r b ψ In the IARC the waypoint are given a [ e x e wp y e wp z wp, where the e z wp coordinate i included uch that the UAV can be controlled to maintain a certain altitude. The lat feedback ignal e ψ i ued in flight cloe to a waypoint if a hover turn mut be performed, which mean that the UAV top above a waypoint, then turn toward the next waypoint and finally continue toward the next waypoint. Before the deign of control rule the trategy of the upervior i decribed. 8.2 Supervior trategy Given a level 1 pecification: { e Ξ lvl1 = [ e x e wp1 y e wp1 z wp1, [ e x e wp2 y e wp2 z wp2, [ e x e wp3 y e wp3 z wp3 } and auming that the UAV i initialized in hover with the altitude of the firt waypoint, a imple control trategy repeated for all waypoint can be decribed a: 1. Turn toward target waypoint. 77

88 78 Section 8.3. Control deign 2. Fly to target waypoint. 3. Update target waypoint. 4. Repeat for n waypoint. The challenge lie in identifying appropriate rate limitation for the calculated output reference in order to obtain a teady flight. The rate limitation are epecially important when paing a waypoint and thereby updating the target waypoint, which, without rate limitation, would induce unwanted reference change in term of tep. In addition, a uitable turn method mut be identified for each waypoint to further enure a teady flight, but alo in ome ituation allow the UAV to pa waypoint without topping to reduce the total flight time in level Control deign The control trategy decribed above i illutrated by the flowchart in Figure 8.1. In the following Initialization. Forward flight toward target waypoint. Ha target waypoint been reached? no ye I target waypoint the final waypoint? no Set target waypoint to the next waypoint. ye Level 1 completed. FIGURE 8.1: Illutration of the overall upervior control trategy. The UAV i initialized in hover with the altitude equal to that of the firt waypoint with a heading e ψ = rad. ection the different deciion and action block will be elaborated on. A number of contant will for the ake of clarity be referred to only by ymbol. The exact ize of thee are determined a decribed in ection 8.4 on page 82.

89 Chapter 8. Superviory control Initialization A depicted in Figure 8.2 the control trategy i baed on two path yielding two poible turning method; A Hover turn where path (a) i followed, and a Forward flight turn where path (b) i followed. [ e x wp,i+1 e y wp,i+1 Path (b) Path (a) e ψ turn,i e v r = v r,medium [ e x wp,i e y wp,i [ e x e y e v r = v r,fat e v r = v r,low R vr,low FIGURE 8.2: Illutration of factor and approach aociated with calculation of the velocity reference e v r and the identification of turn method. Here hown for the waypoint [ e x wp,i e y wp,i. Hover turn: A hover turn conit of the following action: 1. Decelerate toward waypoint e Ξ wp,i until it i reached. 2. Hover and turn toward e Ξ wp,i Accelerate toward e Ξ wp,i+1 following path (a). In the hover approach a velocity reference e v r coniting of a longitudinal (reẋ) and a lateral (reẏ) component i et to the ize v r,low, when a certain ditance R vr,low (indicated by the circle) ha been reached. Due to the rate limit on the reference, the hift from v r,fat to v r,low reult in the deceleration referred to above. A e v r i mapped from the EF to the BF and then provided to the LQR controller, it i expected that the lateral reference component rbẏ in the BF i cloe to zero when approaching the waypoint having a velocity reference v r,low. Auming that the LQR controller i capable of keeping the controlled tate at the reference, no ideway movement of the UAV can be expected. The hover and turn action in point 2. above i performed by keeping e v r at zero, from the time a waypoint i reached, until the heading of the UAV become equal to the heading reference.

90 8 Section 8.3. Control deign Forward flight turn: A forward flight turn conit of the following action: 1. Decelerate at R vr,low to v r,medium toward waypoint e Ξ wp,i until it i reached. 2. Turning with v r,medium until the direction of e Ξ wp,i+1 i reached. 3. Accelerate to v r,fat toward e Ξ wp,i+1 following path (b). The idea with the forward flight turn i to avoid a deceleration to v r,low and hereby optimize the time conumption while performing a turn. The helicopter make the deceleration at R vr,low to v r,medium and keep the heading until e Ξ wp,i i reached. At the waypoint a new velocity reference vector e v r pointing in the direction of e Ξ wp,i+1 i provided to the controller immediately. If the next waypoint do not have the ame heading a the previou waypoint, there can occur ome ideway flight, hereof a velocity in b ẏ. The UAV will keep on turning until the heading e ψ ha reached the reference re ψ, and hereafter the helicopter tart to accelerate to the maximum velocity of the given controller. Chooing turn method: The upervior mut decide once for each waypoint to perform either a hover turn or a forward flight turn. Thi deciion will be baed on: 1. the ditance l wp,i i+1 between e Ξ wp,i and e Ξ wp,i+1 and, 2. the turn angle e ψ turn,i meaured between the continuation of a virtual traight line between e Ξ wp,i 1 and e Ξ wp,i and another traight line between e Ξ wp,i and e Ξ wp,i+1 according to Figure 8.2. The idea i, that if the ditance l wp,i i+1 i large enough and the turn angle e ψ turn,i i not too large, a forward flight turn i performed in order to reduce the total flight time of the level. Appropriate contant limit mut therefore be determined by imulation, uch that the following two tatement can be teted by the upervior: if l wp,i i+1 < l turn,hover then perform hover turn ele if e ψ turn,i > e ψ turn,hover then perform hover turn, where l turn,hover i the limit on the ditance and e ψ turn,hover the limit on the turn angle Deciion block In the firt deciion block it mut be determined whether the current waypoint e Ξ wp,i ha been reached. For thi purpoe a circle of radiu R vr,hover i defined in the xy-plane. When the UAV enter thi circle the given waypoint ha been reached. The econd deciion keep track of the target waypoint, and terminate when the lat waypoint ha been reached. If the waypoint i not the lat, the target waypoint i et to the next, which i done in the ucceive action block.

91 Chapter 8. Superviory control Forward flight The action block Forward flight toward the target waypoint handle the following tak: 1. Calculate heading reference re ψ toward the target waypoint. 2. Calculate velocity reference e v r = [reẋ reẏ. In addition, the forward flight action block contain a control of the altitude of the UAV, which imply increae or decreae the velocity reference reż when the difference between the deired altitude e z wp,i and the feedback control ignal e z exceed a certain threhold. Furthermore, it contain rate limit control on all the calculated reference [re ψ reẋ reẏ reż. Due to the implicity of thee control, they will not be decribed any further. Calculation of heading reference Auming that the UAV ha the bet flying propertie in forward direction, it i deired to keep the heading given by e ψ toward the target waypoint at all time. Thi i done by calculating re ψ continuouly during flight, rather than once at each waypoint. A can be een from Figure 8.3 thi i eaily done by the tangent calculation: re ψ = π 2 arctan(e x wp,i e x e y wp,i e y ), (8.1) with the pecial ituation: e y wp,i e y = (8.2) e y wp,i e y < (8.3) If (8.2) i valid the calculation will fail, a dividing by zero i not poible. If (8.3) i valid the reult will be wrong, becaue tangent only evaluate to poitive value. x { e x wp,i, e y wp,i } re ψ { e x, e y} y FIGURE 8.3: Illutration of the frame ued to calculate the heading reference re ψ. The frame ha the ame orientation a the EF with an offet equal to the poition of the UAV.

92 82 Section 8.4. Parameter determination Calculation of velocity reference When the ize of the velocity reference e v r ha been et to either v r,low or v r,fat, the velocity reference reẋ and reẏ can be calculated (here for v r,low ) a: reẋ = in( π 2 re ψ) v r,low reẏ = co( π 2 re ψ) v r,low (8.4a) (8.4b) 8.4 Parameter determination To finih the deign of the upervior the different parameter have to be determined. The parameter determination will be done baed on knowledge of the ytem and the rule aociated with the IARC. A decribed in Section 4.1 on page 21 the UAV mut pa a waypoint o a judge can ee it from the ground by looking traight up and therefore R vr,hover i choen to 2 m. Furthermore, the velocity reference v r,low, ued for a teady approach to a hover turn, i choen to 1 m. Baed on the determination of v r,low, the parameter v r,rate limit, v r,fat and R vr,hover for each control trategy can be determined by experimentation (ee Appendix F). Then by having determined the different value of v r,fat for the control trategie, the medium velocity reference v r,medium i choen uch, that it i equal to half of the maximum velocity reference. The different value of the determined upervior parameter can be een in Table 8.1. The two remaining parameter l turn,hover Parameter Hover Forward flight Gain cheduling Unit R vr,hover [m R vr,low [m [ v r,low m [ v r,fat m [ v r,medium m [ v r,rate limit ±5 ±5 ±5 m 2 l turn,hover [m e ψ turn,hover ±π ±π ±π [rad TABLE 8.1: Lit of parameter value ued by the upervior together with each of the three controller. and e ψ turn,hover, which decide whether a hover turn mut be performed or not, are determined by imulation tet. The imulation how that the UAV i capable of performing a 18 turn uing the forward flight turn approach by decelerating to the medium velocity reference v r,medium. Therefore, the implementation of the maximum turn angle e ψ turn,hover i omitted from the upervior. Having determined that a hover turn i only neceary to define baed on the ditance between two waypoint, the ditance l turn,hover can be determined a the ditance between two waypoint, involving a 18 turn, where a forward flight turn i a fat a a hover turn (ee Figure

93 Chapter 8. Superviory control (a) for illutration). Baed on tet imulation it i determined that if the ditance between two waypoint i greater than 1 m a forward flight turn i the mot effective turning method uing the hover or the gain cheduling controller (15 m in cae of forward flight controller). Another example of the advantage of uing forward flight turn i depicted in Figure 8.4(b). Here a 45 turn mut be made with a ditance of 3 m to the next waypoint. From the tet it i oberved that performing a forward flight turn i 5 econd fater than performing a hover turn. Having determined all of the neceary parameter for the upervior it i poible to tet the performance of the overall control ytem, which will be decribed in the following chapter e x 3 e x e 1 2 y 1 1 e 2 3 y (a) (b) FIGURE 8.4: Illutration for determination of l turn,hover where the blue line i a hover turn and the red line in a forward flight turn - (a) 18 turn where both the completion time are 62. (b) 45 turn where the completion time are 7 and 65 for the forward flight turn and hover turn, repectively.

94

95 Part III Control ytem tet and concluion 85

96

97 9 CONTROL SYSTEM TEST Thi chapter decribe the concluive tet performed on the three control trategie; LQR hover control, LQR forward flight control and oberver-baed gain cheduling control. Three tet regarding different level 1 pecification for the IARC are performed for each control trategy, uch that the mot time efficient trategy can be identified. In addition the performance of the three control trategie i evaluated for the lat level 1 pecification. 9.1 Tet pecification Starting point for the control ytem tet i taken in the overall objective decribed in the introduction, hence the purpoe i to identify the mot efficient control trategy regarding completion time of level 1 of the IARC. The control trategie to be teted are; LQR hover control, LQR forward flight control and gain cheduling control. The three controller are each teted together with the upervior. To identify the bet overall control trategy, it i deired to tet the control ytem in different ituation regarding the level 1 pecification of the IARC, uch that the bet overall control trategy can be identified. The three different level 1 pecification (A, B and C) are depicted in Figure 9.1. Each pecification ha a length of 3 km and contain a tarting and end End point End point and waypoint 1 and 3 Waypoint 1 Waypoint 3 Waypoint 2 End point Starting point Starting point and waypoint 2 and 4 Starting point Waypoint 4 Level 1 pec. A Level 1 pec. B Level 1 pec. C FIGURE 9.1: Illutration of the three level 1 pecification, all with a total flight length of 3 km ued for performing the control ytem tet. A ha no waypoint, B ha four waypoint placed uch that a 18 turn mut be performed at each waypoint, and C alo ha four waypoint placed uch that everal different turn type are obtained. point, and up to 4 waypoint puruant to the IARC rule. 87

98 88 Section 9.2. Flight performance in pecification A Level 1 pec. A: Thi pecification i aumed to be the leat time demanding of the three, a it imply conit of a tarting and an end point placed 3 km from each other. The ability to accelerate and decelerate together with the maximum flight velocity are the deciive factor for identifying the bet control trategy for A. Level 1 pec. B: Thi pecification i aumed to be the mot time demanding pecification, a it beide tarting and end point conit of four waypoint placed uch that a 18 turn ha to be performed at each waypoint. The ability to accelerate, decelerate and turn will be the deciive factor in thi tet. Level 1 pec. C: The lat pecification alo contain four waypoint, but the waypoint are placed uch that different turn angle are obtained. Waypoint 1 and 2 are intentionally placed uch that the upervior i forced to decide for a hover turn at waypoint 1. Graph of the ix control tate will alo be preented for Level 1 pec. C, and the performance of the three control trategie regarding the tate will be dicued. In the following ection the tet reult are dicued. To eae the reading the abbreviation LQR i left out when referring to the LQR hover and/or the LQR forward flight controller. 9.2 Flight performance in pecification A A decribed above the ability to accelerate and decelerate and the maximum forward flight velocity are the deciive factor in A. From Table 8.1 on page 82 it can be een that the gain cheduling controller ha the highet forward flight velocity reference v r,fat and it i therefore expected that uing the gain cheduling trategy to complete A take le time than with the hover or the forward flight controller. The reulting flight path in the xy-plane i depicted in Figure 9.2 where the blue, red and cyan line are the hover, forward flight and gain cheduling trategy repectively. It i oberved that with the hover control trategy the UAV deviate more from the optimal path, indicated by the traight line, than with the other trategie. Thi i becaue the flight mainly conit of forward flight with maximum velocity, which mean that the hover controller i far from the operating point for which it i developed cauing it to drift in the e y direction. The flight route for the gain cheduling and the forward flight trategie are much alike. Thi i becaue the gain cheduling controller decouple the hover controller at high velocitie, uch that only the control ignal from the forward flight controller are ued in the oberver contruction. The completion time for each control trategy i lited in Table 9.1. A expected the gain cheduling trategy ha the lowet completion time.

99 Chapter 9. Control ytem tet Hover controller Forward flight controller Gain cheduling controller e y [m Starting point End point e x[m FIGURE 9.2: Illutration of the flight path from the imulation of A. It i oberved that the flight path uing forward flight and gain cheduling trategie are much alike, and that the hover trategy caue the UAV to drift with repect to the optimal path between tarting and end point. Strategy Hover Forward flight Gain cheduling Completion time TABLE 9.1: Completion time for the three flight path of the UAV in level 1 pecification A. 9.3 Flight performance in pecification B A mentioned earlier pecification B conit of four waypoint placed on top of tarting and end point (ee Figure 9.1), which reult in the flight path being placed on top of each other a well. Conequently only two part of the reulting flight path are preented (ee Figure 9.3(a) and 9.3(b)). It i oberved that the forward flight and the gain cheduling trategie, a wa alo the cae in pecification A, reult in imilar flight path. The ame explanation a before applie to the flight in pecification B. Notice that forward flight turn with the velocity reference v r,medium lited in Table 8.1 are performed for all waypoint, a the ditance between each waypoint i ufficient for the UAV to get back on track after each turn. Conequently the gain cheduling trategy primarily ue the forward flight controller. From Figure 9.3(a) it i oberved that the UAV turn right when uing by the forward flight and the gain cheduling trategie and left for the hover trategy. The right turn i performed becaue the UAV approache waypoint 1 from left and the upervior therefore chooe to perform a right turn a it i the hortet (the oppoite way for the hover trategy). A given in Table 8.1, the ditance R vr,low, indicating when a given controller mut tart decelerating toward a waypoint, varie between 37 m and 62 m, and a the ditance between each

100 9 Section 9.3. Flight performance in pecification B Waypoint 1 6 Waypoint e x[m 3 e x[m Waypoint 2 Waypoint e y [m e y [m (a) (b) FIGURE 9.3: The part from waypoint 1 to 2 and waypoint 2 to 3 of the flight path from the imulation of pecification B. It i oberved that the flight path uing forward flight (red) and gain cheduling (cyan) trategie are imilar cauing the UAV to make right turn at the waypoint, where left turn are performed for the hover trategy (blue). waypoint in pecification B i 6 m the tet flight primarily conit of flight with maximum velocity. The higher maximum velocity for the forward flight and the gain cheduling trategy, with repect to the hover trategy, therefore reult in maller completion time a lited in Table 9.2. Looking at the plit time for waypoint 1 and 2 for the gain cheduling controller, it can be hown that it take approximately 4 to perform a 18 turn. 1 i ubtracted from both time mark a the UAV initiate in hover, where the velocity when leaving waypoint 1 after completing the turn i approximately 5 m yielding t 18 = (91 1) 2 (44 1) = 4. The ame reult of 4 can obtained for the hover and the forward flight trategie, which indicate that the difference in the completion time follow from the ability of the trategie to accelerate and decelerate, and not from the ability to turn.

101 Chapter 9. Control ytem tet 91 Strategy Waypoint 1 Waypoint 2 Waypoint 3 Waypoint 4 End Hover Forward flight Gain cheduling TABLE 9.2: Completion time for the UAV in pecification B with plit time at each waypoint. 9.4 Flight performance in pecification C The flight path of the UAV flight in level 1 pecification C are depicted in Figure 9.4 for hover (blue), forward flight (red) and gain cheduling (cyan) trategie repectively. The tet reult do 1 8 Waypoint 1 Waypoint 2 Waypoint 3 6 e x[m 4 End 2 Start Waypoint e y [m FIGURE 9.4: Illutration of the flight path from the imulation of pecification C for hover (blue), forward flight (red) and gain cheduling (cyan) trategie repectively. not reveal any ignificantly difference between the flight path for the three control trategie due to the cale, however, the completion time lited in Table 9.3 tate, that uing the gain cheduling control trategy i the mot time efficient trategy. The tet flight in pecification B wa aumed to be the mot time demanding, however, the completion time for the tet flight in pecification B and C how that for the hover and forward flight trategie the flight in pecification C i more time demanding. Thi i due to the hover turn preent at waypoint 1. A expected the flight uing

102 92 Section 9.4. Flight performance in pecification C Strategy Hover control Forward flight control Gain cheduling control Completion time TABLE 9.3: Completion time for the UAV for the imulation of pecification C. the gain cheduling trategy ue the leat time, which again i related to the higher maximum velocity. In the following the performance of the three control trategie are compared more thorough by invetigating angle and tranlateral velocity tate of the UAV. The ix tate are depicted in Figure 9.5, 9.6 and 9.7 for the hover, forward flight and gain cheduling trategy repectively..5 Pitch angle 15 Longitudinal velocity e θ [rad bẋ [ m Roll angle 2 Lateral velocity e φ [rad bẏ [ m e ψ [rad Yaw angle Time [ bż [ m Vertical velocity Time [ FIGURE 9.5: Illutration of ix of the control tate found by imulation of the UAV flying trough pecification C uing the hover trategy. For all control trategie it i oberved that the pitch angle become negative when accelerating in the longitudinal direction (and poitive for deceleration), but with the forward flight trategy (ee Figure 9.6) the UAV ha larger peak in the pitch angle, than with hover or gain cheduling trategy indicating a more aggreive controller. Furthermore, thi indicate that the gain cheduling controller mixe the hover an forward flight controller when b ẋ change in the interval [; 1 m a intended, a it ha maller pitch angle variation than the forward flight trategy

103 Chapter 9. Control ytem tet 93 and greater variation than the hover trategy. A imilar obervation applie to the roll angle graph, where the forward flight trategy caue higher and more reonant variation than with the other trategie. The hover trategy affect the gain cheduling controller, uch that a more teady control i obtained. Overall it i concluded, from the whole control ytem tet, that the gain cheduling control trategy i the mot time efficient control trategy regarding flight in each of the three level 1 pecification dicued. In addition, the gain cheduling trategy i hown able to combine the two LQR controller, in order benefit from the performance of the hover controller at low velocitie and the forward flight controller at high velocitie..5 Pitch angle 15 Longitudinal velocity e θ [rad bẋ [ m Roll angle 2 Lateral velocity e φ [rad bẏ [ m e ψ [rad Yaw angle Time [ bż [ m Vertical velocity Time [ FIGURE 9.6: Illutration of ix of the control tate found by imulation of the UAV flying trough pecification C uing the forward flight trategy.

104 94 Section 9.4. Flight performance in pecification C.5 Pitch angle 15 Longitudinal velocity e θ [rad bẋ [ m Roll angle 2 Lateral velocity e φ [rad bẏ [ m e ψ [rad Yaw angle Time [ bż [ m Vertical velocity Time [ FIGURE 9.7: Illutration of ix of the control tate found by imulation of the UAV flying trough pecification C uing the gain cheduling trategy.

105 1 CONCLUSION The overall goal of thi mater thei wa to deign control trategie uch that the Bergen Indutrial Twin helicopter i able to perform autonomou flight. The goal of the autonomou flight wa to make the Unmanned Aerial Vehicle capable of competing in the International Aerial Robotic Competition and complete level 1 of thi competition a fat a poible. Therefore, the overall objective for thi thei wa determined to be: Deign, implementation and tet of different control trategie enabling the UAV to autonomouly complete level 1 in the IARC by employing hover and fat forward flight, uch that the mot efficient control trategy regarding level 1 completion time can be identified. To accommodate the overall objective variou ubobjective were defined a decribed in Section 1.1. Thee objective will be ummarized in the following and concluded on eparately followed by an overall concluion of the achievement of thi thei. Objective A - Claic linear control: It wa deired to deign claical SISO controller for the UAV to be able to compare thi method to the optimal LQR method. The controller were deigned baed on tate-pace matrice obtained by linearization of the nonlinear model in the hover operating point, and implemented a lead and lag compenator. Each of the deigned controller were teted eparately and were all found to be able to tabilize their individual control tate. After all of the SISO controller were deigned the claic control ytem wa teted, where the nonlinear model wa initialized in hover with an offet applied to each of the controlled tate. It could be concluded that the deigned claic control ytem wa able to tabilize the nonlinear model with repect to all of the controlled tate. Objective B - LQR hover control: Beide the claical SISO control ytem it wa deired to utilize more advanced control method. For thi purpoe the MIMO control method optimal LQR control wa ued. The LQR hover control wa, a the claic hover control, baed on the linearized ytem matrice, and by identifying weight matrice for the controlled tate. It wa choen to perform control on a total of nine tate to be able to perform more table control. The main iue regarding the deign wa to determine the weight matrice ued in the performance function. Initial value ued in the weight matrice wa determined by identifying tate limitation of the nonlinear model in open loop. Uing thee determined weight matrice an initial tet wa performed to examine the performance of the LQR hover controller, where the tet howed that the deigned controller wa not able to tabilize the UAV Therefore, the weight matrice were altered. 95

106 96 Section The final deign of the LQR hover controller wa teted on the nonlinear model initialized in hover with an offet applied to the controlled tate a done for claic linear control. The tet howed that the deigned controller wa able to tabilize all of the controlled tate of the nonlinear model in a teady fahion and therefore aeed a performing atifactory. Furthermore, it wa concluded that the LQR hover controller performed more teady control than the claic linear controller, and wa able to tabilize the UAV fater. Objective C - LQR forward flight control: A forward flight LQR controller wa developed for an operating point in 1 m. The deign of the thi controller took tarting point in the weight matrice determined for the LQR hover controller. The LQR forward flight controller wa teted a the LQR hover controller, however, the nonlinear model wa initialized in the 1 m operating point and then applied an offet in the controlled tate. The tet howed that the LQR forward flight controller wa able to tabilize the nonlinear model in the operating point. The tet performed for the forward flight controller howed the etimator to yield faulty etimate for the yaw angle and lateral velocity, which compromied the tet of the controller. However, due to the controller ability to tabilize the UAV depite the faulty etimate, it wa aeed that thi fault wa acceptable becaue it did not accumulate over time. Objective D - Gain cheduling: It wa deired to optimize the change from hover to forward flight, why it wa decided to deign gain cheduling between the two LQR controller. Thi wa done by deigning an oberver-baed gain cheduling controller to obtain bumple tranfer. Since the two LQR controller already were deigned only the oberver-baed gain cheduler needed to be deigned. For thi purpoe two oberver were deigned baed on tandard oberver deign theory. To determine tability of the oberver baed gain cheduling controller a tet wa performed like for the LQR controller. The nonlinear model wa initialized in a 5 m and then applied an offet in the controlled tate. Thi operating point wa choen becaue both LQR controller were equally ued by the cheduler at thi velocity. The tet howed that the oberver baed gain cheduling control ytem wa able to tabilize the nonlinear model for thi etting. Objective E - Superviory controller: To obtain poition and heading control a uperviory controller wa developed. Thi high level controller wa deigned, uch that it could calculate reference for the linear controller baed on the poition and heading of the UAV. The upervior wa able to choe between two different waypoint turning method; hover turn and forward flight turn, baed on the waypoint poition pecification.

107 Chapter 1. Concluion 97 The tet of the upervior wa performed a part of the control ytem tet, where it wa concluded that it wa able to handle variou turn angle, and that it wa able to minimize drift in the lateral direction at high velocitie. To identify the mot time efficient control trategy three different control ytem tet were performed for each of the three control trategie; LQR hover, LQR forward flight and gain cheduling. From thee tet it wa identified that the gain cheduling trategy wa the mot time efficient of the three. The deciive factor were that the gain cheduling controller wa able to benefit from the performance propertie of both the LQR hover and forward flight controller. In addition, the gain cheduling controller wa identified a having the highet maximum velocity. 1.1 Future work The work done in thi thei qualifie a a bai for future work on control of the reconfigured Bergen Indutrial Twin helicopter, a utilization of linear control howed good performance with repect to control of the nonlinear model. However, becaue the nonlinear model ued in thi thei i of an older verion, future controller deign mut be baed on the preent nonlinear model. Further optimization of the gain cheduling approach can advantageouly be performed by adding everal additional controller in different operating point. In addition, a further analyi of the cheduling function α g could help optimize the benefit gained from each controller implemented in the gain cheduler. Implementing the α g cheduler function into the upervior could alo help optimize the ue of the gain cheduling approach. Thi implementation could be ued by the upervior to choe between everal additional rate limit a e.g. velocitie. A natural control approach to the Bergen Indutrial Twin helicopter i to develop nonlinear control, which could be advantageou becaue of the complexity of the nonlinear model.

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109 ACRONYMS BF EF IARC LQR Body-fixed reference Frame Earth-fixed reference Frame International Aerial Robotic Competition Linear Quadratic Regulator MIMO Multiple Input Multiple Output SISO UAV Single Input Single Output Unmanned Aerial Vehicle 99

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111 BIBLIOGRAPHY AUVSI. Aociation for unmanned vehicle ytem international, URL January 19 th 27. J. D. Bendten, J. Stoutrup, and K. Trangbæk. Bumple tranfer between oberver-baed gain cheduled controller. International Journal of Control, 78(7):491 54, 25. M. Bigaard. Ph.d tudent, 27. URL November 6 th 26. U. Hald, M. Heelbæk, and M. Siegumfeldt. Nonlinear modeling and optimal control of a miniature autonomou helicopter. Mater thei, Aalborg Univerity - Department of Control Engineering, 26. URL B. R. Helicopter. Weite for bergen r/c helicopter, 2. URL January 19 th 27. J. Holmegaard, C. Jenen, and S. Lænner. Development and navigation of an autonomou uav. Mater thei, Aalborg Univerity - Department of Control Engineering, 26. URL P. K. Jenen, M. Y. Hunddahl, H. Hanen, A. M. Iveren, L. S. Lantow, and J. H. Pederen. Development and application of a protocol for viualiing imulation uing a 3d engine. Article, Aalborg Univerity - Department of Control Engineering, 26. O. Sørenen. Optimal regulering, 2. udgave. Afdeling for procekontrol, Aalborg Univeritet, K. Zhou, J. Doyle, and K. Glover. Robut and Optimal Control. Prentice-Hall, New Jerey, USA, ISBN

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113 APPENDIX A VISUALIZATION PROGRAM GSIM Thi appendix decribe how the viualization program GSIM ued in thi thei for viualization of the motion of the controlled UAV work. The program ha epecially been ued when deigning the LQR controller in thi thei. A.1 Functionality of Gim The viualization program GSIM i a ditributed ytem created by Jenen et al. [26 capable of viualizing 2D graph imulation from SIMULINK in a 3D environment. The program i able to ue multiple hot, where one hot can run the imulation and another hot run the 3D viualization program. The program include a SIMULINK block 3D Viualization Sink (ee Figure A.1), which mut be defined with pecific information of i.a. what 3D object i to be viualized and the IP of the target hot running the 3D viualization program. Thi SIMULINK block can handle object with ix degree of freedom, e.g. the UAV in thi thei, which reult in ix input needed for the viualization program: The poition of the UAV given in the EF e Ξ T = [ e x e y e z T. The attitude of the UAV given in the EF e Θ T = [ e φ e θ e ψ T. 3D Viualization Sink Object name FIGURE A.1: Illutration of the SIMULINK block 3D Viualization Sink. The 3D Viualization Sink block end the information received from the actual imulation to the 3D viualization program, which render the information into 3D. In Figure A.2 the 3D viualization program i hown with no data received from the 3D Viualization Sink block in SIMULINK. Furthermore, it can be een that the program how the hot IP, which the 3D Viualization Sink block mut target. In Figure A.3 the 3D viualization program i hown with data received from SIMULINK, where it can be een that the program ha rendered a 3D helicopter model, which i being moved around and rotated with repect to the data received from the actual imulation. In addition, it 13

114 14 A.1. Functionality of Gim FIGURE A.2: Screenhot of the 3D viualization program with no imulation data received. The program how the IP of the program hot that mut be targeted by the 3D Viualization Sink block in SIMULINK. FIGURE A.3: Screenhot of the 3D viualization program with imulation data received regarding the poition and attitude of the UAV. In the top right corner tab are hown; one for object info and one for camera control.