L -based Model Following Control Of An Identi Ed .
L1-based Model Following Control of an IdentifiedHelicopter Model in HoverGiacomo Picardigiacomo.picardi@tuebingen.mpg.deMaster student, University of PisaMax Plank Institute for Biological CyberneticsTübingen, GermanyApril 4, 2016
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Contents1 Introduction1.1 MyCopter project and PAV concept1.2 A brief history of adaptive control .1.3 Literature review and contribution1.4 Outline . . . . . . . . . . . . . . . .11111314152 Helicopter Model Description2.1 Robinson R44 Raven II helicopter . . . . . . . . . . . . . . . . . . . . . . . .2.2 Identification process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171717183 Baseline Controller Design3.1 Goals . . . . . . . . . . . . . . . . . . . . . . . . .3.1.1 PAV reference dynamics . . . . . . . . . .3.2 Baseline controller structure . . . . . . . . . . . .3.3 Performance of the Baseline Controller . . . . . .3.3.1 Uncertainties in the Augmented Helicopter.212121222223.292930303131325 Verification and validation5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33336 Conclusions and Future Work6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4141427 Appendix7.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7.2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4343454 Adaptive Controller Design4.1 Uncertain Plant . . . . . . .4.2 State Predictor . . . . . . .4.3 Adaptation Law . . . . . . .4.4 Control Law . . . . . . . . .4.4.1 Choice of outputs . .4.4.2 Tuning of Γ and C(s).3.
4CONTENTS7.37.2.1 helicopter model.m . . . . . . . . . . . . . . .7.2.2 helicopter model uncertainties ON.m . . . . .7.2.3 pidCL.m . . . . . . . . . . . . . . . . . . . . .7.2.4 pidCL uncertainties ON.m . . . . . . . . . . .7.2.5 Adaptation acc.m . . . . . . . . . . . . . . . .7.2.6 montecarlo acc.m . . . . . . . . . . . . . . . .7.2.7 data analysis acc.m . . . . . . . . . . . . . . .Appendix C . . . . . . . . . . . . . . . . . . . . . . .7.3.1 Augmented helicopter . . . . . . . . . . . . .7.3.2 Augmented helicopter with adaptive controller7.3.3 Adaptive Control Law . . . . . . . . . . . . .7.3.4 State Predictor . . . . . . . . . . . . . . . . .7.3.5 Estimation Law . . . . . . . . . . . . . . . . .45505658606164727273747576
CONTENTSAcknowledgments5
6CONTENTSAbstractThe aim of this study is to augment the uncertain dynamics of the helicopter in order toresemble the dynamics of a new kind of vehicle, the so called Personal Aerial Vehicle. Toachieve this goal a two step procedure is proposed. First, the helicopter model dynamicsis augmented with a PID-based dynamic controller. Such controller implements a modelfollowing on the nominal helicopter model without uncertainties. Then, an L1 adaptivecontroller is designed to restore the nominal responses of the augmented helicopter whenvariations in the identified parameters are considered. The performance of the adaptivecontroller is evaluated via Montecarlo simulations. The results show that the application ofthe adaptive controller to the augmented helicopter dynamics can significantly reduce theeffects of uncertainty due to the identification of the helicopter model. For implementationreasons the adaptive controller was applied to a subset of the outputs of the system. However,the under actuation typical of helicopters makes the tracking of the nominal responses goodalso on the not directly adapted outputs.The thesis is articulated in three main steps. First of all the nominal plant is considered,i.e. no uncertainty is considered in the identified parameters and a baseline controller isdesigned to track the PAV dynamics. When the uncertainty in the values of all identifiedparameters is considered the dynamic augmentation does not perform as desired. Then, anadaptive controller is designed to recover the nominal stability and performance. Finally thedesign is validated with Montecarlo simulations.
CONTENTS7List of Abbreviations and itude Command Attitude HoldGlobal Positioning SystemIntelligent Flight Control ProgramIntegrated Resilient Aircraft Control ProjectJoint Direct Attack MunitionModel Reference Adaptive ControlPersonal Aerial Transportation SystemsPersonal Air VehicleProportional Integral DerivativePilot Induced OscillationsTranslational Rate Command
8CONTENTSNomenclaturepangular velocity components of helicopter along fuselage x-axis, [rad/s]qangular velocity components of helicopter along fuselage y-axis, [rad/s]rangular velocity components of helicopter along fuselage z-axis, [rad/s]utranslational velocity component of the helicopter along fuselage x-axis, [m/s]vtranslational velocity component of the helicopter along fuselage y-axis, [m/s]wtranslational velocity component of the helicopter along fuselage z-axis, [m/s]xstate vector of helicopter model and (after 3.2) of augmented helicopter modelyoutput vector of helicopter and augmented helicopter modelpBaugcomponent of uncertainty vector in the span of BaugpBumcomponent of uncertainty vector in the span of Bumuadadaptive control inputubslbaseline control inputx̂augmented helicopter state predictionNNumber of Montecarlo simulationsPLyapunov Equation solution from adaptive control lawQLyapunov Equation constant term from adaptive control lawAaugstate transition matrix augmented helicopter modelAhestate transition matrix of helicopter modelBauginput matrix of augmented helicopter modelBheinput matrix of helicopter modelBuminput matrix of unmatched uncertainties of augmented helicopter modelC(s)low pass filter of adaptive control lawCaugoutput transition matrix of augmented helicopter modelCheoutput transition matrix of helicopter modelHm (s)transfer function of nominal augmented helicopter modelHum (s)transfer function between unmatched uncertainties and output of the augmented helicoptMlatTRC reference model from lateral cyclic to lateral translational velocityMlonTRC reference model from longitudinal cyclic to longitudinal translational velocityMcolTRC reference model from collective lever to vertical translational velocityMpedACAH reference model from pedals to yaw rateTssampling time, [s]Tsavesave time in Montecarlo simulations, [s]φRoll Euler angle, [rad]θPitch Euler angle, [rad]ψYaw Euler angle, [rad]ρIdentified parameter vectorδcolCollective pilot input, [grad]δlatLateral cyclic pilot input, [grad]δlonLongitudinal cyclic pilot input, [grad]δpedPedal pilot input, [grad]σ̂ unmatched uncertainties estimation ˆIdentified parameter vector, nominal valuesΓAdaptation Gain
CONTENTS A ρ9State transition matrix perturbation of augmented helicopter modelIdentified parameter variations vector
10CONTENTS
Chapter 1IntroductionThe aim of this study is to augment the uncertain dynamics of the helicopter in order toresemble the dynamics of a new kind of vehicle, the so called Personal Aerial Vehicle. Toachieve this goal a two step procedure is proposed. First, the helicopter model dynamicsis augmented with a PID-based dynamic controller. Such controller implements a modelfollowing on the nominal helicopter model without uncertainties. Then, an L1 adaptivecontroller is designed to restore the nominal responses of the augmented helicopter whenvariations in the identified parameters are considered.The present thesis was driven by two main motivations. On one hand the concept ofPAV played a fundamental role in the definition of the control goals. On the other, recentgrowing interest in adaptive control applications inspired the choice of the control technique.In particular an adaptive control technique based on a reference model was applied. Thisintroduction gives an overview of the two cornerstones of the work. Along with this, theintroduction includes a literature review to understand how the present document contributesto technical progress.1.1MyCopter project and PAV conceptNowadays ground based transportation has come to saturation. Huge jams regularly form inlarge metropolitan areas all over the world. Delays, fuel consumption and pollution resultingfrom this seriously impact the output of global economies. The delays resulting from thiscongestion have been estimated to cost bilions of euros per year in Europe only [1]. Apossible solution could be exploiting the vertical dimension moving part of the commutingtraffic from the ground to the air. The European project myCopter aimed to address thisproblem. The main goal of such a project is well stated by its slogan: enabling technologiesfor personal aerial transportation systems (PATS). Of course such an ambitious goal can onlybe achieved with a structured and complex program. The issues that need to be addressedare not only merely technical. The aims of the project are revolutionary and the social andeconomical implications cannot be neglected.A fundamental block of myCopter project is the definition of the personal air vehicle(PAV) concept. A PAV is a new kind of vehicle whose dynamics is ideal and representthe theoretical reference that a real vehicle should have in order to be easy and safe to fly11
12CHAPTER 1. INTRODUCTIONFigure 1.1: Logo of European project myCopterby naive pilots. The PAV dynamics were defined by the University of Liverpool [2]. TheFigure 1.2: A PAV conceptexact tuning of the responses was achieved through to a series of experiments involving nonexpert pilots rating the degree of effort required to drive in a simulated environment [1].Implementing a physical vehicle whose dynamics resemble the PAV dynamics can be donein many several ways all of them resulting in different implications. A key feature a PAVshould have is the ability of landing and taking off vertically. This is very important becauseotherwise, with airplane-like aircrafts the ability of door-to-door transportation would getlost and ad hoc infrastructures would be needed. In the last decades a considerable numberof prototypes have been proposed for personal transportation purposes but none has achievedmass production. Among the reasons for this are the costs and excessive amount of trainingrequired to get a license compared to cars. For PAVs to be massively accessible it would benecessary to reduce the cost of training to obtain a PAV license the same way it is possibleto obtain a driving license today [2].As already stated the PAV concept is purely theoretical. A step forward in the directionof the physical implementation consists in the work of Geluardi at the Max Plank Institutefor Biological Cybernetics. The core of his research is the idea of using helicopters as a baseto a PAV. On one hand light civil helicopters feature many properties that a PAV shouldhave, e.g. vertical take off and landing, size and number of seats. On the other hand theyare extremely difficult to control because of their unstable, non minimum phase and highlycoupled behavior. The transition from an helicopter to a PAV can be performed with thedesign of a control system. Such a system is meant to enhance stability and to augment the
1.2. A BRIEF HISTORY OF ADAPTIVE CONTROL13Figure 1.3: Prototypes for personal transportation. From left to right: Terrafugia, NASAPuffin, Moller Skycardynamics of the helicopter in order to resemble the reference developed by the University ofLiverpool. More details on the PAV dynamics will be given in chapter 2 when discussing thecontrol objectives.1.2A brief history of adaptive controlAfter World War II there was a great interest in the development of high performance aircraft.These aircraft operated in a very wide flight envelope, spanning a very large range of speedsand altitudes. In this very diverse conditions the parameters variations are considerable andthe effect of non linearities become non negligible. The first motivation for research in thefield of Adaptive Control was to design a single controller with a mechanism to regulate thevalues of its parameters according to variations in the aircraft parameters. In a few wordsit was an alternative to gain scheduling.After some successful application, suddenly the interest in adaptive control dropped in1967 after a flight test in which the aircraft was destroyed and the pilot killed [10]. Theinstability of the adaptive control system was among the causes of the tragedy.In the last decade there was a renewed interest in adaptive control. This time one of themain motivation for researching and applying adaptive control was the recovery of stabilityand adequate performance in case of failures or external disturbances.The group of Prof. Calise from the Georgia Institute of Technology focused on twomain topics. On one hand he applied adaptive control to exploit the redundancies of controlsurfaces to handle failures and bad atmospheric conditions in the case of piloted aircraft [11].On the other hand, within the Joint Direct Attack Munition (JDAM) project, he proved thatModel Reference Adaptive Control was a viable technique to reduce the importance of anaccurate modeling and thus saving time and money [12].Boing X-45 was an important test bench for adaptive control. Wise and Lavretskyevaluate the performance of MRAC in simulation [13] and L1 adaptive control was appliedin simulation for the case of actuator failures [14].The goal of the Intelligent Flight Control System Project (IFCS) was to optimize performance of aircraft both in nominal and failure conditions. During this project adaptivecontrollers showed a tendency to increase Pilot Induced Oscillations (PIO) [15].One of the most recent project on adaptive control is the Integrated Resilient Aircraft
14CHAPTER 1. INTRODUCTIONControl Project (IRAC). The main aim of the project was to investigate the applicability andcompare against each other different adaptive methods and applying them to the GenericTransport Model developed at NASA.Many progresses were made in the field of adaptive control in the last decades. However,many problems remain open at the time of writing. The biggest issue to face is the problemof certification. Up to now, much of the tuning and evaluation is performed using Montecarlosimulation. This technique is time consuming and in some cases does not guarantee anything.The development of metrics to evaluate the performance of adaptive control is a fundamentalstep to be made before adaptive control will be applicable on large scale.1.3Literature review and contributionThe idea of this thesis started from the conclusion of the work of Geluardi [16]. The aimof his work was applying H and µ-synthesis control methods to an identified light civilhelicopter in hover [17] in order to augment its dynamics and resemble the dynamics of a PAV.Excellent results were obtained when no uncertainties in the identified model parameters wereconsidered. The augmented helicopter followed the PAV model in the frequency range ofinterest, i.e. where the model is valid ([0.1, 20]rad/s). However, when variations with respectto identified parameters were considered the proposed control architecture did not behavewell and handling qualities degraded from level 1 to level 2 and 3 of the ADS-33E-PRFstandards [18].To overcome the limitation of robust control law, adaptive control methods can be used.Adaptive controller performs an online estimation of the uncertainty and produces a controlinput to reduce the undesirable deviations of the uncertain system from the nominal behavior[19].The goal of this paper is to design an adaptive control loop for compensating uncertainties of an identified civil light helicopter in hover. The dynamics of the helicopter isfirst augmented with a PID-based controller to resemble translational rate command (TRC)response types of a PAV. Then, an adaptive controller augments the combined dynamics ofan helicopter and PIDs to reject the effects of uncertainties in the identified parameters.Many successful applications of adaptive control can be found in the field of small scalehelicopter control. In the work of Guerreiro et al. [20] L1 -adaptive control theory was usedto provide attitude and velocity stabilization of an autonomous small scale rotorcraft. Theadaptive controller was designed to reject the effects of wind disturbances. In this paper,however, we focus on a full scale manned helicopter and the adaptive controller is designed toreject the effects of identification uncertainties. In the work of Bichlmeier et al. [21] an L1 adaptive controller was designed to augment a preexisting Proportional Integral Derivative(PID) baseline controller on a full scale manned helicopter. The aim of the adaptive controllerwas to maintain handling and flying qualities in situations the baseline controller was notdesigned for or performed poorly. The adaptive controller only compensated for uncertaintiesthat enter the system dynamics through the control channel, i.e. matched uncertainties.Nevertheless the model considered in the present paper is affected by uncertainties thatdo not fall in the matched category, the so called unmatched uncertainties. Few works inliterature refer to adaptive control with unmatched uncertainties. For example in [22] L1 -
1.4. OUTLINE15adaptive control is applied in the case of unmatched uncertainties to the NASA AirStaraircraft. To the best of our knowledge, however, adaptive control has never been applied toan augmented full scale helicopter model to reject the effects of unmatched uncertainties.1.4OutlineThis thesis is organized as follows: Chapter 2 presents the identified helicopter model. First, some specifications on thehelicopter used are given. Then, a quick description of the identification method used isgiven. Finally, the resulting state space model is described along with the uncertaintiesderived from the identification process. In Chapter 3 the design of the baseline dynamic controller is addressed. First, thecontrol goals are stated. Then the baseline controller architecture is presented andthe achievement of the control goals is shown. Finally, the global dynamics of theaugmented system is obtained in order to show how uncertainties affect the augmentedsystem. Chapter 4 describes the adaptive control technique implemented. L1 is presentedvery generally. Then, the theory is revised for our application and the most importantdesign choices are explained. In Chapter 5 the proposed architecture is tested against different parameter realizations. The Montecarlo study performed is described and the results are presented. In Chapter 6 a discussion on the overall work is given together with the conclusion.The most relevant aspect of the work are highlighted and future possibilities discussed. Matlab code and simulink schemes are reported in the the Appendix.
16CHAPTER 1. INTRODUCTION
Chapter 2Helicopter Model Description2.1Robinson R44 Raven II helicopterThe helicopter model used in this thesis is an identified model of a Robinson R44 Raven II(see fig. ?)in hover. The Robinson R44 Raven II is a four-seat light helicopter producedFigure 2.1: Robinson R44 Raven II.by Robinson Helicopter Company. It was chosen because is affordable and has limiteddimension. Those characteristic perfectly fit the idea of myCopter project of promoting abrand new transportation system. Fig. ? shows the dimensions of the rotorcraft.2.2Identification processThe design of a dynamic controller requires a mathematical model of the system. In thiswork the mathematical model of hovering dynamics of the helicopter was obtained via identification techniques [17].The identification process consists of taking measurements of inputsand outputs of the system and using such data in combination with some knowledge of thesystem dynamics to derive a mathematical model.In our case the piloted inputs from cyclic stick (δlat , δlon ), collective lever (δcol ) andpedals (δped ) were recorded by four optical sensors directly mounted on the commands. Theoutputs considered are taken from two Global Positioning System antennas and an InertialMeasurement Unit. Data were collected while performing several piloted frequency sweeps17
18CHAPTER 2. HELICOPTER MODEL DESCRIPTIONFigure 2.2: Robinson R44 Raven II, dimensions in inches.and doublets around the hover trim condition [16]. After a first non parametric analysis atransfer function model was implemented to take into account fundamental dynamics, orderof the system and level of coupling.The identified parameters ρ of the helicopter model are not known exactly. Each identifiedparameter can be written in the form:ρi ρ̄i ρi(2.1)where ρ̄i represents the nominal value and ρi is the unknown perturbation of the i-thidentified parameter, with i 1,2,.28. The expected standard deviation of ρi are defined by Cramer-Rao bounds (CR) computed during the identification. Specifically, ρi [ 3CRi , 3CRi ] with 99% of probability [24].2.3Model descriptionThe mathematical model resulting from the identification method [17] is a state space formmodel. This model considers pilot inputs from cyclic stick (δlat and δlon ), collective lever(δcol ) and tail rotor pedals (δped ) ?. The outputs considered are the translational velocitiesFigure 2.3: Pilot commands of an helicopterin body frame u, v, w, the angular rates in body frame p, q, r and the roll and pitch angles
2.3. MODEL DESCRIPTION19φ, θ. The model considered in highly realistic because it takes into account rotor-fuselagecoupling dynamics such as flapping, inflow and lead lag. The resulting dynamics is describedin eq. 2.2.ẋ Ahe (ρ̄ ρ) Bhe (ρ̄ ρ)u(2.2)y Che xwhere x R17 is the helicopter state vector, u R4 is the input vector from the pilot andy R8 is the output vector. The matrices Ahe , Bhe and Che are of appropriate dimensions.It is important to note that Ahe and Bhe are not exactly known because depend on theuncertain identified parameters ρ.The dynamics of the identified helicopter are non minimum phase, unstable and fullycoupled among the four control axis.
20CHAPTER 2. HELICOPTER MODEL DESCRIPTION
Chapter 3Baseline Controller DesignIn this chapter the helicopter dynamics in eq. 2.2 are augmented with a dynamic controllerin order to follow the PAV reference model. No uncertainty in the identified parameter vectoris considered, i.e. ρ ρ̄.3.1GoalsThe goal of the baseline controller is to augment the dynamics of the helicopter and to followthe reference model of the PAV.The PAV reference model was defined during MyCopter Project by the University ofLiverpool [23]. These reference dynamics follow the guidelines for the definition of level onehandling qualities defined in the ADS-33E-PRF [18], but are modified to be more suitablefor non expert pilots. The definition of the reference dynamics was done with extensiveexperiments with naive pilots in a simulator. The study showed that non pilot performedbetter when Translational Rate Command (TRC) response type are used for the lateral,longitudinal and vertical translational degrees of freedom. A Rate Command Attitude Hold(RCAH) response type is implemented in yaw.3.1.1PAV reference dynamicsThe PAV reference dynamics is expressed in terms of transfer functions. Following theguidelines of ADS-33E-PRF in order to ensure handling qualities of level 1 the augmentedhelicopter must respond like a first order system to a step command with appropriate risingtime. The corresponding transfer functions are:Mlat vrefδlatMlon urefδlon1.4,1.25s 1[ m/s]deg 1.4,0.7s 1[ m/s]degMcol wrefδcol 0.502,0.25s 1[ m/s]degMped rrefδped 1,0.82s 1[ rad/s]deg (3.1)21
223.2CHAPTER 3. BASELINE CONTROLLER DESIGNBaseline controller structureA model following approach was implemented to achieve response types selected as referencedynamics for each axis (Model reference dynamics [M]) (see eq.3.1).δlatδlonδcolδpedMlatMlonMcolMpedvref uref wref rref eveuewerP D1P D2P ID33P ID44pref qref epeqP ID11P ID22u1u2u3HelicopterM odelu4vuwpqrθφFigure 3.1: Scheme of the helicopter augmented with PID-based controller and referencedynamicsTo achieve handling qualities requirements of PAV, the helicopter dynamics was augmented with a PID-based controller as shown in figure ?. A multi loop controller architecture is implemented on the lateral and longitudinal axes. On the outer loop two PDcontrollers provide roll and pitch rate references (pref ), qref expressed in terms of the error on lateral and longitudinal velocities (ev , eu ). The inner loop tracks the angular ratereference and stabilizes the dynamics.A single loop controller architecture is implemented on the vertical and yaw axis. P ID33and P ID44 define the inputs of the helicopter in terms of the error on vertical velocity andyaw rate (ew , er ), respectively. Tuning of the baseline controller was done to minimize thedifference between the augmented helicopter dynamics and the model reference over theP IDs parameters in the range of frequencies where the identified model is valid ([0.1,20]rad/s). However, the proportional gain of P ID44 was manually incremented to better decouple vertical velocity and yaw rate.3.3Performance of the Baseline ControllerThe presented control architecture achieves adequate model tracking in the frequency rangeof interest as shown in fig. 3.2a-3.2d.To investigate whether the resulting augmented dynamic system satisfies handling qualities requirements of ADS-33E-PRF standard [18], responses to a step command must beconsidered.
0u, [dB]v, [dB]3.3. PERFORMANCE OF THE BASELINE CONTROLLER 50100101phase, [deg]phase, [deg] 10010 4 10 3 10 2 10 10 100 200 410 310 2 11010ω, [rad/s]100200 20 40 6010 4 10 3 10 2 10 110 2000 20 40 60 8010 4 10 3 10 2 10 1ω, [rad/s](c) From: δcol To: w101100101100101100101r, [dB](b) From: δlon To: u100101phase, [deg]phase, [deg]w, [dB](a) From: δlat To: v 20 30 40 5010 4 10 3 10 2 10 11000 10010 4 10 3 10 2 10 1ω, [rad/s]1231001010 20 40 60 8010 4 10 3 10 2 10 10 50 10010 4 10 3 10 2 10 1ω, [rad/s](d) From: δped To: rFigure 3.2: Bode plot comparison. Dashed: Reference Model, Solid: Augmented Helicopter.Figures 3.3a-3.3d show that the augmented helicopter behaves approximately like a firstorder system, with rising time very close to the reference model in every channel and negligible overshoot in response 3.3c. Therefore the ADS-33E-Prf requirements are satisfied.However, the responses v to δlat and u to δlon of the augmented helicopter present a delaydue to the non minimum phase of the helicopter dynamics.3.3.1Uncertainties in the Augmented HelicopterAfter the dynamic augmentation with PID controllers and reference dynamics M, all uncertainties only affect the state transition matrix of the system. The proof can be found inAppendix A. The resulting augmented system can be written in the form:ẋaug Aaug (ρ̄ ρ)xaug Baug uaugy Caug xaug(3.2)where xaug R31 , uaug [vcmd , ucmd , wcmd , rcmd ] R4 and y R8 , Aaug R31x31 , Baug R31x4 , Caug R8x31 . From now on we will refer to the state and input vectors of theaugmented helicopter as x and u, respectively, to reduce the use of subscripts.
24CHAPTER 3. BASELINE CONTROLLER DESIGN0.61.5w, [m/s]v, [m/s]10.50.40.2002468100120(a) From: δlat To: v0.511.522.5(b) From: δcol To: w1.21.510.8r, [m/s]u, [m/s]10.50.60.40.2002468101200(c) From: δlon To: u2468(d) From: δped To: rFigure 3.3: Step responses comparison. Dashed: Reference Model, Solid: Augmented Helicopter.To apply the adaptive control technique presented in the following chapter, the systemin 3.2 must be rewritten in a convenient form. First, the nominal dynamics are separatedfrom the perturbation as follows:ẋ (Aaug (ρ̄) A(ρ̄ ρ))x Baug uy Caug x(3.3)where A is unknown and accounts for the uncertainty due to ρ. Then, the vector Ax canbe written as the sum of two vectors: the component of Ax along the span of Baug (matcheduncertainties) and along its orthogonal complement Bum (unmatcheduncertainties).Here TBum R31x27 is a constant matrix such that BaugBum 0 and Baug Bum has full rank.The two components can be expressed as: pBaug Baug Baug Ax, pBum Bum Bum Ax.(3.4)
3.3. PERFORMANCE OF THE BASELINE CONTROLLER25where superscript indicates the Moore-Penrose pseudoinverse. As proven in Appendix A,in our case the component of A along span(Baug ) is null. Thus, the resulting form of theaugmented helicopter dynamics 3.2 is:ẋ Aaug x Baug u pBumy Caug x(3.5)When the considered unmatched uncertainties are included, the baseline controller presentedin Figure ? fails to track the reference model. Figures ? - ? show Bode diagrams of (u,v, w, r) from (δlat , δlon , δcol , δped ) respectively for 50 random parameter variations. Clearlythe responses are different for different uncertain parameters realizations. In particular theuncertainty is larger when an input on the cyclic is considered. This makes the handlingqualities of the augmented helicopter degrade. To overcome this limitation, an L1 adaptivecontroller was designed.
26CHAPTER 3. BASELINE CONTROLLER DESIGNv, [dB]0 20 4010 310 210 110010110 210 1ω, [rad/s]100101[deg]0 100 200 30010 3(a) From: δlat To: vu, [dB]200 2010 310 210 110010110 210 1ω, [rad/s]100101[deg]0 100 200 30010 3(b) From: δlon To: u
3.3. PERFORMANCE OF THE BASELINE CONTROLLE
April 4, 2016. 2. Contents 1 Introduction11 . Pu n, Moller Skycar dynamics of the helicopter in order to resemble the reference developed by the University of Liverpool. More details on the PAV dynamics will be g
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