Meccanica e scienze avanzate dell ingegneria

Transcription

1 Alma Mater Stdiorm Università di Bologna DOTTORATO DI RICERCA IN Meccanica e scienze avanzate dell ingegneria Ciclo XIII Settore/i scientifico-disciplinare/i di afferenza: ING-IND/3 Meccanica del volo MODEL PREDICTIVE FLIGHT CONTROL SYSTEMS FOR ROTORCRAFT UAS Presentata da: Colatti Stefano Coordinatore Dottorato Relatore Prof Franco Persiani Prof. Franco Persiani Eng. Jan-Floris Boer Esame finale anno 211

2 Table of contents List of figres...3 List of tables...5 Abstract Introdction Helicopter models Non-linear model Main rotor Hiller bar Tail rotor Control chain Fselage Engine Validation Linearized models Model strctre Model linearization procedre Model validation Simlation models Prediction model Model ncertainties Model predictive control MPC basics Stability Feasibility Controller implementation Open loop optimal problem soltion Reference tracing Feasibility garantee Virtal reference Infinite horizon...4 1

3 4.3.3 Soft constraints Offset-free control Integral action State observer Stability and Control Agmentation System SCAS control strctre Simlation reslts Trajectory Tracing System System architectre Simlation reslts Path Following System Conclsion and otloo...71 Bibliography

4 List of figres figre 1.1: UniBo RUAS...8 figre 2.1: Main rotor and Hiller bar hb...12 figre 2.2: Hiller bar...13 figre 2.3: PWM to pitch data...14 figre 2.4: fselage aerodynamic data...15 figre 2.5: Typical power spectral density of the non linear model otpts...19 figre 2.6 Comparison between linear and nonlinear models...21 figre 2.7: Simlation model scheme...21 figre 2.8: rotor momentms acting on the fselage...24 figre 3.1 MPC base strctre...26 figre 3.2 MPC optimization...27 figre 3.3 terminal set and feasibility...3 figre 4.1: performances as fnction of prediction horizon...39 figre 4.2: virtal set-point as fnction of prediction horizon...4 figre 4.3: inner stabilizer loop...42 figre 4.4: infinite horizon performances...43 figre 4.5: comparison between soft and hard state constraints...45 figre 4.6: Integral action architectre scheme...47 figre 4.7: effects of the integral action

5 figre 4.8: effect of the anti wind-p system...49 figre 4.9: Monte Carlo simlation...5 figre 4.1: state observer control architectre...51 figre 4.11: Monte Carlo simlation...52 figre 5.1: longitdinal accelerations...56 figre 5.2: lateral accelerations...57 figre 5.3: Monte Carlo Simlation...58 figre 6.1: Trajectory tracing system architectre...61 figre 6.2: longitdinal reposition...62 figre 6.3: Lateral reposition manevers...63 figre 6.4: Constant vs. nown reference, NED position...64 figre 6.5: Constant vs. nown reference, body speeds...64 figre 6.6: constant vs. nown reference, attitde angles...65 figre 6.7: constant vs. nown reference, controls displacement...65 figre 6.8: Monte Carlo simlation...66 figre 6.9: Time to obtain the control action...67 figre 7.1: Path following principle...69 figre 7.2: path following vs. trajectory tracing...69 figre 7.3: speed time plot...7 figre 7.4: attitde time plot...7 4

6 List of tables Table 2.1: model and real data comparison...16 Table 4.1: anti wind-p limit variables...48 Table 6.1: SCAS constraints

7 Abstract Constraints are widely present in the flight control problems: actators satrations or flight envelope limitations are only some eamples of that. The ability of Model Predictive Control MPC) of dealing with the constraints joined with the increased comptational power of modern calclators maes this approach attractive also for fast dynamics systems sch as agile air vehicles. This PhD thesis presents the reslts, achieved at the Aerospace Engineering Department of the University of Bologna in collaboration with the Dtch National Aerospace Laboratories NLR), concerning the development of a model predictive control system for small scale rotorcraft UAS. Several different predictive architectres have been evalated and tested by means of simlation, as a reslt of this analysis the most promising one has been sed to implement three different control systems: a Stability and Control Agmentation System, a trajectory tracing and a path following system. The systems have been compared with a corresponding baseline controller and showed several advantages in terms of performance, stability and robstness. 6

8 1 Introdction Unmanned Aerial Systems UAS) has been widely sed in the military field in the past decade. Several mission profiles sch as srveillance, reconnaissance and, more recently, attac have been committed to these systems obtaining nmeros advantages in terms of hman safety, cost redction and wor rate efficiency. The same advantages cold be eploited in the civil maret where UAS can be employed into a even wider range of tass sch as: law enforcement traffic control weather monitoring aerial photography.. The great potential of UAS for the civil maret and the interest demonstrated by indstry indced, in 21, the Eropean Commnity to sponsor the UAS development program CAPECON 1, to attempt to ic-start a civil UAS indstry in Erope and try to fill the gap with the United States. Its main goal was to provide Eropean indstry with detailed design and manfactre now-how on safe cost effective and commercially viable civil UAS. 1 Civil av APplications & Economic effectivity of potential CONfigration soltions 7

9 After a srvey on indstrial needs and the development of formal reqirements five fied wing and two rotary wing architectres were defined in order to cover all the possible mission reqirements. As part of this effort the University of Bologna focsed its research on the conventional helicopter configration and developed the UniBo Rotorcraft Unmanned Aerial System RUAS). The aim was bilding a technological demonstrator for the national indstries potentially interested in the nmanned systems and to be sed inside the niversity as platform for research in innovative navigation and control laws or for Hman Machine Interface stdies. figre 1.1: UniBo RUAS The system, shown in figre 1.1, is bilt arond a modified Hirobo Eagle II 6 hobby helicopter which was modified to accommodate the avionics hardware, eqipped with a more powerfl engine, longer fiberglass blades, both for the main and the tail rotor, and a longer tail boom. The new main rotor is a 2 blades see-saw type rotor with a diameter of 1.84 m. The rotorcraft is eqipped with Bell-Hiller stabilizer bar, which agments servo torqe with aerodynamic moment to change the blades cyclic pitch and adds lagged rate feedbac to improve the helicopter handling qalities. The helicopter total mass is abot 11.2 g and the engine has a maimm otpt power of 3hp. The avionics bo has been accommodated in the ndercarriage and contains: a National Instrments CompactRIO system which performs both the tas of Atopilot and Flight Management System FMS). 8

10 a crossbow NAV42 GPS-aided Attitde and Heading Reference System AHRS) to provide the position, speed and attitde of the rotorcraft an ltrasonic sensor sed to measre the altitde above the grond of the vehicle a data lin system to transmit and receive information from the grond station. The system architectre has been validated by means of hardware in the loop testing, flight testing and the development of a baseline nested loop PID atomatic controller. For more details on the evoltion of the project refer to [1], [2], [3]. In the frame of this project this thesis has the aim of presenting the reslts achieved by the athor in the field of the Flight Control Systems FCS) for small scale rotorcraft UAS. In particlar the development of an innovative FCS based on the Model Predictive Control MPC) theory is presented. The development of a control algorithm for helicopters is a rather challenging tas de to the instability of the system dynamics and the strong cross-copling between all the ais. Moreover, as for the fied wing aircrafts, several constraints have to be taen into accont designing the control laws; actator satrations, maimm attitde angles, limit load factors and speed constraints are only few eamples of it. In the last decade several different control approaches have been sed to tacle the helicopter flight control problem, from simple PID nested loop controllers [4] to more comple and elaborate architectres based on optimal [5] robst [6] and non linear techniqes [7]. As normally done in the majority of indstrial applications the respect of constraints is addressed only dring the calibration phase together with the other performance reqirements introdcing three main disadvantages: the reslt of the calibration is always a compromise between different reqirements in particlar when aggressive manoevring is reqired, the calibration process is complicated and time demanding, 9

11 there is no garantee that the constraints will be respected and that stability maintained. Conversely model predictive control offers the possibility of eplicitly introdce the presence of constraints into the controller formlation and garantees that them will be respected for all the possible states configration. This has the advantage of simplifying the calibration of the system since the respect of constraints has been already taen care of. Moreover stability of the constrained closed loop system can be mathematically demonstrated and, as will be shown in the thesis, this characteristic is maintained also in presence of significative ncertainties. The thesis is organized as follows, in section 2 the models of the UniBO RUAS sed in this thesis will be analyzed, in particlar a flly non-linear model of the helicopter bilt into the FlightLab environment will be presented together with the continos and discrete time models sed for the control synthesis. Also the choice of model ncertainties se for robstness assessment will be shown in this section. Section 3 will give a short introdction to the model predictive theory eplaining the wor principle of this class of controllers together with the stability and feasibility problems. Section 4 shows different model predictive architectres giving the mathematical details of each formlation and analysing, with the help of simlations, the advantages and disadvantages of each of them. As reslt of this analysis the best control formlation has been chosen and adopted to bild three different flight control systems presented in the following sections. In section 5 a Stability and Control Agmentation System SCAS) whose aim is to control the body frame speeds and the heading of the helicopter is presented. A comparison with a baseline LQR controller is shown by means of simlation, moreover the inflence of model ncertainties is shown sing a set of Monte Carlo simlations. The same has been done for a trajectory tracing and a path following system respectively presented in sections 6 and 7. Finally, section 8 will draw the conclsions and offer frther research recommendations. 1

12 2 Helicopter models For the prposes of this wor three different ind of models of the UniBo RUAS have been developed. Section 2.1 describes the assmptions made to bild a flly non-linear model of the helicopter in the FlightLab environment; starting from this a series of linear models sed for both simlation and control prposes have been derived by means of an identification procedre described in section 2.2. Finally, ncertainties have been added to the principal physical parameters of the model in order to evalate the robstness characteristics of control system, this ncertain model is described in section Non-linear model In order to better validate the control architectres developed in this thesis a non-linear model of the helicopter has been bilt into the FlightLab environment a tool specifically intended to bild rotorcraft models and provide analysis and simlation instrments). The rotorcraft has been conceptally divided into its main physical components, namely: Main rotor Bell-Hiller bar Tail Rotor Control Chain actators, rods, etc.) 11

13 Fselage Engine In the following each component model is described into the details jstifying the choices made dring the modelling phase Main rotor This component has been modelled as a 2-blade teetering rotor with a constant speed of 11 rpm. The blades have been considered rigid with a niform density distribtion. The blade section profile has been ept constant along the entire span and has been approimated by the TsAgi 14% shape; the aerodynamic data has been estimated by sing the CFD software pacage JavaFoil. The Peters-He three state dynamic inflow model has been adopted to calclate the inflow field and the aerodynamic interference between the main rotor wae and the fselage has been considered. figre 2.1: Main rotor and Hiller bar hb Hiller bar The stabilizer bar figre 2.2) has the fnction to provide a feedbac action that improves the flying qalities of the rotorcraft, moreover the system provides a portion of the torqe needed to control the main rotor blades pitch redcing the power needed by the actators. This components models only the rotor dynamics while the connection between the Bell- Hiller stabilizer, the control servos and the main rotor will be described in the net sections. The stabilizer is located directly above the main rotor and has been modelled as a second teetering rotor. As the main rotor it has been considered rigid with the same constant 12

14 speed of 11 rpm. The inertia distribtion has been calclated smming the contribtions of the plastic paddles, the spporting rod and the calibration brass masses. Paddles Calibration mass Spporting rod figre 2.2: Hiller bar The paddles are bilt sing a NACA15 profile and no aerodynamic interference with the remaining components has been considered since the stabilizer paddles are considerably smaller than the main rotor blades Tail rotor The tail rotor has been modelled as a simple Bailey 2 rotor since only the collective pitch control is applied. The blades have been considered rigid and niform, a NACA 1 airfoil has been sed to model the aerodynamic characteristics Control chain In the real helicopter the servo actators are connected, via a comple mechanical gear system, to a cople of swashplates. One is located on the tail rotor and controls the corresponding blades, the second controls the pitch of the stabilizer paddles and, via a mechanical mier, the pitch of the main rotor blades. This mier sms the contribte given from the swashplate to another which is proportional to the stabilizer flapping angle. The servos are controlled sing a Plse Width Modlated PWM) electric signal where the displacement of the actator is proportional to the dty cycle of the inpt. This comple system can be modelled by the following relationships: θ ψ ) = K X K X K β )cosψ K X K β ) sinψ c c 1lat a bh 2s 1long b bh 2c 2 A Baley rotor is a rotor in which only the coning angle is considered while the longitdinal and lateral flapping are not considered; this is the typical configration of tail rotors. 13

15 θ 2 2lat a 2long b ψ ) = K X cosψ K X sinψ 2.2 θ ) = K X 3 ψ 2.3 ped p Where θ 1, θ 2, θ3 are the blade pitch of the main rotor, the stabilizer and the tail rotor respectively, X, a, X b, X c X p are the servo control inpts and s 2c β,β 2 are the lateral and longitdinal flapping angles of the Bell-Hiller rotor. The following observations can be done: the main rotor cyclic pitches are a linear fnction of the cyclic controls and the Bell-Hiller stabilizer flapping angles; the stabilizer paddles do not have any collective pitch; the tail rotor cyclic pitches are forced to be according to the Bailey rotor model sed All the gains K present in the eqations have been obtained measring the pitch of each blade varying manally the control inpts on fied azimth positions and 9 degrees), figre 2.3 shows the reslt of the eperiment. figre 2.3: PWM to pitch data The eqations 2.1, 2.2, 2.3 have been implemented sing the Control System Graphic Editor CSGE, a modle of FlightLab) and connected to each single rotor. 14

16 2.1.5 Fselage This component models all the inertial and gravitational actions de to the entire UAS and the aerodynamic effects introdced by the fselage body. In this model the fselage has been considered as a rigid body; the mass and inertia variations de to the fel consmption have been neglected. The centre of gravity position and the inertia tensor have been measred directly on the model trogh dedicated eperiments. Several eperimental and CFD methods have been considered in order to calclate the aerodynamic coefficients of the fselage. All the approaches revealed to be too comple and time demanding in relation with the accracy needed in this application. In the model, data from fll-scale helicopters given in [1] has been sed see figre 2.4) figre 2.4: fselage aerodynamic data Engine Flight test data show that the rotor speed variation dring typical manoevres is abot 5 rpm that has a negligible effect on the dynamic behavior. For this reason in this model no engine dynamics have been added, the rotor speed has been ept constant at a nominal vale of 11 rpm Validation In order to have some indication abot the qality of the model some comparison with the real data can be made. Table 2.1 resmes some comparisons made with real flight data 15

17 Model Flight Data Collective pitch [deg] Roll attitde [deg] 2.3 ~3 Power reqired [hp].97 Table 2.1: model and real data comparison Moreover the loc nmber of the main rotor and the Hiller bar have been calclated from the model linearization and reslted in γ and γ. 59 respectively, which corresponds to the literatre data available for the same class of machines [9], [1]. A detailed validation with flight data has not been possible since, after a major accident, the rotorcraft has been repaired and modified. No flight data has been recorded yet in the new configration. f s 2.2 Linearized models In the formlation of the flight control systems sed in this wor a linear model of the rotorcraft dynamics is necessary, for this reason the FlightLab model described in the previos section has been linearized arond several trim conditions. In the following the strctre of the model and the linearization procedre sed to obtain it will be described. Finally, the validity of sch models will be demonstrated by comparing the response of the linear and the non-linear models sbject to the same inpts Model strctre For the prpose of this wor the classic state space linear model has been sed & = A B 2.4 where and are respectively the state and inpt vectors of the model and A and B the stability and control matrices. In order to obtain a good linear approimation of the plant is important not only to get the correct vales for the stability and the control matrices bt also to inclde all the significative states in the relative vector. 16

18 In fll-scale helicopters the rotor flapping dynamics is significantly faster than the body motion and the related states can be neglected sing a qasi-static flapping approimation. In small-scale model helicopters instead, the main rotor dynamics slowed by the introdction of the stabilizer bar) and the body dynamics very fast de to the small scale) have comparable time constants and the rotor flapping dynamics have to be inclded in the linear model. The state and inpt vectors become then respectively: = [ a, b, c, d, p, q, r,, v, w, ϕ, ϑ, ψ ] = [,,, ] a b c p T T 2.5 where: a, b, c, d are the longitdinal and lateral flapping angles of respectively the main and stabilizer rotors; p, q, r, the body frame anglar rates;, v, w the body frame linear speeds; ϕ, ϑ, ψ the helicopter Eler angles;,,, the lateral and longitdinal cyclic pitch, the collective pitch and a b c p the tail rotor pitch respectively Model linearization procedre The standard linearization tool provided by FlightLab gives to the ser a list of states that can be selected and inclded in the state space and inpt vectors; the states available on this list depend on the particlar choices made by the ser when bilding the model. The linearization algorithm rns the non-linear model twice for each state inclded in the linearized vector; in each simlation a small distrbance is added or sbtracted from one of the states while the others are ept fied to the trim vale; moreover all the states not inclded in the vector are left free to evolve. The stability derivatives are obtained by calclating the time derivatives of the states in these conditions once the free states have reached a steady state condition. De to the strctre of the model, the rotor flapping states are not available in the standard tool list; on the other hand, the time scale of the fselage and the rotor dynamics are not 17

19 so mch separated as happens for fll scales rotorcrafts and the steady state assmption for the rotor dynamics can not be done. For this reason a different approach to obtain the linear models had to be taen. Since the model predictive control is fndamentally a time domain techniqe an identification procedre on the same domain has been adopted. The basic idea is to calclate the stability and control matrices by minimizing the difference between the response of the linear and the non-linear model sbject to the same inpts. For each trim condition a set of simlations with the nonlinear model have been carried ot recording the following data: 1) the system inpts ; 2) all the model states ; 3) the forces and moments introdced by each component The data sets have been obtained forcing each control and state ecept the flapping angles) to follow a or sine-sweep profiles while the remaining states were frozen in the trim condition. In this way it has been possible to isolate the contribtion of each control and state in the stability derivatives independently. Analysing the freqency content of the obtained data maes it possible to identify two main contribtions see figre 2.5): 1) a low-freqency contribtion de to the fselage and flapping dynamics from to ~ 5 Hz); 2) a seqence of higher harmonics corresponding to mltiples of the main rotor speed 36 and 73 Hz). Since the latter contribtions are not interesting for this application the data has been ct off above 1Hz. 18

20 figre 2.5: Typical power spectral density of the non linear model otpts Given the classic linear model formlation & = A B 2.6 The identification process has the objective to find the correct nmerical vales on the matrices A and B in order to eep the linear and the non-linear models response as similar as possible. In order to minimize the nmber of variables that have to been identified at the same time each row of the model 2.6 has been calclated separately. The identification algorithm calclates the terms in the i th row of the matrices by minimizing the following cost fnction: J T = 2 λ λ ) dt 2.7 nl l Where λ is a variable associated to the row considered and the sbscripts nl and l are referred to the non-linear and linear response respectively. The non-linear response is calclated directly sing the FlightLab model while the linear approimation is calclated by: λ = M 2.8 l nl nl 19

21 The rotor dynamics stability and control coefficients have been calclated sing as identification variable λ the time derivatives of the flapping angles a &, b&, c&, d& ). The rigid body dynamics has been identified indirectly, the forces and moments of each model component have been sed as identification variables. Once all the force and moments have been identified, the linear epression obtained have been sbstitted into the 6 d.o.f. rigid body eqations 2.9 allowing to obtain the complete linear model by analytically linearizing the remaining parts. This choice has been made becase facilitated the choice of the initial conditions for the identification algorithm and redced again the nmber of variables to be identified. F & = vr wq g sinθ m Fy v& = wp r g cosθ sinφ m Fz w& = q vp g cosθ cosφ m I p& = I I ) qr I r& pq) M I I yy zz q& = I r& = I zz yy I I zz yy ) rp I ) pq I z z z r 2 p& qr) M 2 p ) M z y Model validation In order to validate the reslts obtained both the linear and the non-linear model have been ecited by a common inpt seqence different from the ones sed in the identification process) and the responses obtained have been compared. figre 2.6 shows an eample of the comparisons, in this case a inpt has been imposed in the collective control. As can be seen there is a good correspondence between the linear and the nonlinear model, especially to the direct contribtions, the crosscopling effects are captred with lower accracy in this eample the lateral dynamics). The typical freqencies of both the flapping and the body dynamics are well captred giving an adeqate correspondence between the linear and the flly nonlinear models, moreover those freqencies are in the same range of what can be epected of this class of vehicles and reported in the nown literatre [1] 2

22 2.3 Simlation models figre 2.6 Comparison between linear and nonlinear models The direct implementation of the control systems in the FlightLab environment is too complicated and time demanding to be done in the first phase of the control architectre evalation. For this reason the simlations needed to evalate the characteristics of the control architectres have been carried in Simlin and a simplified model has been implemented as depicted in figre 2.7 figre 2.7: Simlation model scheme The body frame dynamics has been represented by the linear state space model developed in this section where all the states are spposed to be measred and available. The position of the helicopter has been represented in the North-East-Down NED) 21

23 reference frame and has been calclated rotating the body speeds with the fll non-linear direction cosine matri i T b and then integrating the rotated speeds. cosϑ cosψ sinϕ sinϑ cosψ cosϕ sinψ cosϕ sinϑ cosψ sinϕ sinψ i T = b cosϑ sinψ sinϕ sinϑ sinψ cosϕ cosψ cosϕ sinϑ sinψ sinϕ cosψ 2.1 sinϑ cosϑ sinϕ cosϑ cosϕ 2.4 Prediction model Model predictive control maes se of a plant of the model to calclate the control action. As will be eplained later, in this wor the prediction model is represented by a discrete time linear model, which has been obtained from the continos time model as follows. Given a constant sampling time T s the time derivative can be approimated by: 1) ) & = 2.11 T s by sbstitting it into the continos time formlation 2.6 we obtain: 1) = T A 1) ) T B ) 2.12 s s In order to maintain a simple notation the same symbols have been sed for both discrete and the continos time characteristic matrices, no confsion shold be raised by this choice since the difference will be clear by the contet. 1) = A ) B ) Model ncertainties In order to evalate the sensitivity of the control systems to the possible mismatches between the prediction model and the controlled plant a series of ncertainties have been artificially added to the simlation model sing the robst control toolbo of Matlab. 22

24 The set of parameters to be modelled as ncertain has been chosen analysing the basic principles of the helicopter flight dynamics. As well nown in these inds of vehicles the main sorce of control forces and momentms is the main rotor which can be represented by a simple dis flapping arond the hb. The flapping dynamics can be represented by the simplified eqations: b b& = p τ mr a a& = q τ mr c c& = p τ sr d d& = q τ sr 1 τ mr 1 a τ mr µ ΩR 1 c v τ µ ΩR sr 1 τ sr b µ Ω v v d µ Ω v R R B τ C τ a sr a mr a µ d µ w w a a w R Ω w R Ω B d) d A τ D τ mr b sr b b b A c) c 2.14 and the following parameters have been chosen as ncertain: the main and stabilizer rotor time constants: τ mr, τ sr, the control gains: A, B, C, D b a a b the Bell-Hiller mier gains A, B c d With reference to figre 2.8 the roll and pitch dynamics can be epressed as: I I yy p& = K q& = K β β Th Th mr mr ) b ) a 2.15 and only the inertia parameters have been considered ncertain. 23

25 figre 2.8: rotor momentms acting on the fselage For what concerns the linear motion the longitdinal and lateral dynamics can be epressed as: m& = gϑ Ta mv& = gϕ Tb 2.16 again only the mass parameter is sfficient to represent ncertainty. With the same considerations, also the following parameters have been considered ncertain: z-ais principal momentm of inertia I zz the collective and tail control gains. A detailed analysis on the ncertainty levels goes beyond the scope of this thesis; in order to have an indication of the robstness of the control system an error p to 25% on each ncertain parameter has been sed. 24

26 3 Model predictive control Mayne at al. in [11] define Model Predictive Control MPC) or Receding Horizon Control RHC) as is called sometimes as a form of control in which the crrent control action is obtained by solving on-line, at each sampling instant, a finite horizon open-loop optimal control problem, sing the crrent state of the plant as the initial state; the optimization yields an optimal control seqence and the first control in this seqence is applied to the plant. In other words, the predictive approach obtains the inpt to be applied to the process by minimizing the difference between the ftre reference and the predicted otpts of the system, the latter are calclated by means of a proper model of the plant. MPC differs from other conventional approaches since it solves the optimal control problem on-line for the crrent state of the plant, rather than determining offline a feedbac policy that provides the optimal control for all states). 25

27 figre 3.1 MPC base strctre As will be eplained later the MPC formlation is very general and incldes a wide class of algorithms, each of them has its own particlar characteristics bt shares the same base strctre see figre 3.1) and the principal components, namely: the prediction model; the objective fnction; the optimization solver; the constraints epression. The varios MPC approaches owe their sccess, especially in the process indstry, de to the ability of: handling mltivariable processes natrally; taing into accont for inpt and state constraints eplicitly; handling non minimm phase and nstable plants; being easy to tne. On the other hand, the se of MPC has been limited: de to the time domain natre of the approach which leads to partially lose the freqency domain information; 26

28 to systems with relatively slow dynamics de to the heavy comptational brden associated with the soltion of the constrained optimal problem. The first applications of MPC can be fond in the petro-chemical and process indstry where economical considerations reqired operating points on the bondary of the admissible sets defined by the constraints. Eamples of sch early predictive algorithms are given by IDCOM identification and command) proposed by Richalet et al in 1978 which employed a finite horizon plse response linear model, a qadratic cost fnction, and inpt and otpt constraints. Or by program qadratic dynamic matri control QDMC; Garcia& Morshedi, 1986) where qadratic programming is employed to solve eactly the constrained open-loop optimal control problem that reslts when the system is linear, the cost qadratic, and the control and state constraints are defined by linear ineqalities. Similarly to other technical inventions MPC was implemented by indstries long before the development of a well established theory, nevertheless a great research effort has been made by the academic commnity since the mid eighties and nowadays a strong conceptal and practical framewor for both practitioners and theoreticians has been bilt. The basic concepts of both linear and non-linear MPC can be fond respectively in [11] and [14]. In the following, the mathematical fondations of the predictive approach will be given together with the more important theoretical reslts regarding stability and feasibility problems. 3.1 MPC basics figre 3.2 MPC optimization 27

29 In order to obtain a rigoros formlation of the model predictive approach we assme that the plant can be modelled, has sally happens in the MPC literatre, by the difference eqation: 1) = f ), )) y ) = h ), )) 3.1 where is the state vector with n elements, the inpt vector of dimension m, y the z- dimensional otpt vector, f and h generic fnctions sed to model the process. The control action is obtained by solving, at each sampling instant, an open loop optimal problem defined by a cost inde in the form 3.2 and the set of constraints 3.3. The reslt of the optimization is a seqence of ftre controls = 1),..., H )] where only the first inpt 1) is actally applied to the plant. [ c H p 1 J, ) = L i), i)) F H )) 3.2 i= 1 p U X H p ) X f 3.3 The cost inde J is defined over a ftre time window, sally called prediction horizon, which starts at the actal sampling instant and ends H p steps later. Each step is long one sampling interval. The cost is given by the sm of two different contribtes: the stage cost L which is fnction of the predicted states and inpts and the terminal cost F which is fnction only of the state at the end of the prediction horizon. The constraints are formlated by means of the sets U, X and m R and gives the inpt constraints; the second is a sbset of states while the terminal set sed for stability reasons as will be eplained later. X R X f. The first is a sbset of n R and limits the admissible n f involves only the last state of the prediction and is As depicted in figre 3.2 a control horizon is often introdced, in this case only the first H c control moves are allowed while the remaining H p H are obliged to assme a fied c 28

30 vale; this is done in order to redce the degrees of freedom of the optimization redcing its compleity; in the following when is not specified H = H. c p 3.2 Stability The most important theoretical reslt in the MPC theory is given by the possibility of garanteeing the stability of the closed loop system, by properly choosing the basic components of the optimization problem. In order to demonstrate this property a local control law = ) has to be introdced, this control action is applied only when the state is inside the terminal constraint set X f. On this premise is possible to demonstrate that the constrained closed loop system is stable if: f S1. X f X, X f is closed and X f the state constraints are satisfied in the terminal set) S2. f ) U, X f the inpt constraints are respected by the local control law) S3. f, f )) X f X f X f is positively invariant sbject to f ) S4. F f, f )) F ) l, f )), X f The detailed demonstration can be fond in [1]. 3.3 Feasibility The predictive control approach is based on the ability of finding a soltion of the open loop optimization problem given by eqations 3.2 and 3.3 for each possible condition of the plant. Even before trying to find the soltion is worth to nderstand if sch a soltion eists or not, in other words we need to as: is the open loop problem feasible? 29

31 figre 3.3 terminal set and feasibility Analyzing the constraints strctre, see figre 3.3 which shows a bi-dimensional eample, is possible to nderstand how the presence of the terminal constraint can be a sorce of nfeasibility. In fact, given an initial condition ) the terminal constraint reqires to find a ftre control seqence in U, and a corresponding state trajectory in X, which drives the last state of the prediction inside the terminal set X f in less than one prediction horizon. Sch a condition is qite restrictive and, in or nowledge, there are no general reslts giving conditions for the feasibility of the finite horizon problem. The soltion of the feasibility problem is normally addressed at the implementation level, in other words when the choices of the prediction model, the cost fnction and the constraints epressions are made. The soltions investigated for the flight control system will be discssed in the net section together with the other implementation isses. 3

32 4 Controller implementation In order to sccessflly apply MPC to practical control problems the base concept described in section 3 has to be specialized to meet the specific needs of the particlar application nder eam. In this section the implementation isses addressed dring the development of the flight control system will be described and the soltions obtained analyzed into the details, in particlar: 1. a fast optimization algorithm based on the soltion of a qadratic programming problem has been implemented; 2. the ability of following a given reference rather than reglate the plant towards the origin has been obtained by a proper coordinates transformation; 3. the feasibility problem has been addressed sing two different techniqes: a virtal set-point and a infinite prediction horizon; 4. zero steady state error has been obtained by sing integral actions and state observer techniqes. 31

33 4.1 Open loop optimal problem soltion The MPC theory does not provide a way to calclate the soltion of the optimization problem and, therefore, a sitable optimization algorithm has to be fond. In the literatre different classes of solvers sch as interior point, fast gradient or eplicit methods have been proposed each one with different characteristics and comptational brdens. For the prposes of this wor an interior point method especially developed for real time applications have been sed. This optimizer is able to solve qadratic programming optimization problems in the form of eq. 4.1 T T J = ν Hν ν G s. t ν min Mν ν ma 4.1 Where ν is the vector of optimization variables and for MPC problems corresponds to the ftre control seqence. In order to transform the optimization problem given by eqations 3.2, 3.3 into a QP problem the stage and terminal cost fnctions have to be chosen in the qadratic form 4.2 and the model has to be in the linear time invariant form 4.3. J H p = i= 1 1 T T T i) Q i) i) R i) ) H ) F H ) p p 4.2 1) = A ) B )

34 33 the ftre state seqence [1),,H p )] can be obtained by recrsively applying eqation 4.3 to the previos state of the seqence as follows:... ) 2)... 1) ) )... 2) 1) ) ) 2) 2) 3) 1) ) ) 1) 1) 2) ) ) 1) s B B A B A A s B AB B A A B A B AB A B A B A s s s = = = = = = 4.4 or, in a more compact notation S S = 4.5 where the bold character denotes a seqence of vectors, e.g. =[1),,H p )], and )] ), [ =. On the other hand, the cost fnction can be written in the matri form = ) ) ) 1) ) ) ) 1) ) 1) 1) ) 1) 1) p c c T p c c p p T p p H H H R R H H H H H F Q Q H H J M M O O M M M O M 4.6 In short T T R Q J = 4.7 By sbstitting 4.5 in 4.7 we can obtain the following:

35 J T T T = T S QS R) S QS ) const 4.8 For what concerns the constraints sally are epressed in the polyhedral form 4.9 over the entire prediction horizon. min min ma ma 4.9 In order to transform this epression in the desired form is necessary to eplicitly write the constraints over the ftre states and inpt seqences: M M min min min min M M ma ma ma ma 4.1 By sbstitting 4.5 in the ftre states epression, is possible to obtain the desired form. M M min min min min S S M ma M ma ma ma S 4.11 Both the cost fnction epression in 4.8 and the constraints 4.11 are written in the desired QP form. It is important to notice that the linear term G and the constraints limits ν min and ν ma depend on the initial condition which gives the closed loop behavior to this control architectre. 34

36 Is also worth to point ot that the conversion from the model predictive standard formlation to the qadratic programming one cannot be done completely offline, in fact both the cost fnction and the constraint formlation depend on the initial condition which changes at every sampling instant and, hence, the passage from 4.8 and 4.11 to 4.1 has to be done at each iteration. 4.2 Reference tracing The open loop optimization problem described by eqations 3.2 and 3.3 is written in the reglator form; in other words, the objective of the control system is to drive the state towards the origin. In this wor, on the other hand, the objective is to drive the otpts of the system y) to follow a given reference vector r). In order to se the reslts obtained for the reglator formlation also for the tracing is sfficient to perform an ais translation given by: = = ss ss 4.12 where the steady state vales ss and ss can be calclated by imposing: f y ss ss,, ss ss ) = ) = r 4.13 In particlar sing a linear prediction model the new eqilibrim point can be easily calclated by: ss ss A = C B D 1 W r r = W 4.14 the cost fnction 4.2 can be then transformed in: 35

37 36 ) ) ) ) ) ) ) F p p H i R Q H W r H i W r i i W r i J p = = 4.15 where the notation 2 Q denotes the qadratic form Q T and has been introdced in order to simplify notation. In order to transform 4.15 in to the QP formlation a seqence of ftre references r and steady state vales ss, ss are needed. There are two possible cases: the ftre set-point is nown and, therefore, is possible to select the window corresponding to prediction horizon sed in the optimization; only the actal reference vector r) is nown and is ept constant over the entire horizon. The seqence of steady state vales can be calclated from the reference in the two different cases respectively sing eqations 4.16 and 4.17 = ) 1) ) 1) p p ss ss H r r W W H M O M ) ) 1) = r W W H p ss ss M M 4.17 In the following eqation 4.18 will be sed for both cases r r ss ss W W = = 4.18 Again, the cost fnction can be written in the qadratic programming form obtaining:

38 J T T T T = T S QS R) 2S QS 2S QW r 2RW ) const 4.19 r which, besides the already mentioned dependence from, depends from the ftre reference seqence allowing the system to tae into accont for the set-point. 4.3 Feasibility garantee In order to se the predictive approach for the flight control system the feasibility of the open loop optimal problem has to be garanteed in any case. The main case of nfeasibility is de to the presence of the terminal state combined with a finite prediction horizon. As eplained in section 3.3 given an initial condition ) can happen that is impossible to reach the terminal set within the prediction horizon. In order to overcome this problem three different approaches can be adopted: move the terminal set towards the initial condition shortening the path from the initial point to it; enlarge the terminal set obtaining the same effect as the previos method; increase the prediction horizon in order to give more time to the system to reach the terminal set Virtal reference A terminal control law = ) has to be defined in order to garantee the stability of the f closed loop system. Since we are maing se of linear state space models of the plant the more natral way to define it is by the classic linear feedbac law = K ) = K r 4.2 ss ss 37

39 where K is obtained by the well-nown linear qadratic reglator algorithm sing the same weights sed in the MPC cost fnction Q and R) and = W KW. The stability conditions S1 and S2 are satisfied by defining the terminal set as: X f { X K r } = 4.21 min ma which, among the rest, depends on the reference vector r. This method maes se of this characteristic to move the terminal set towards the initial state ) in order to garantee feasibility. To do so the real reference r is sbstitted by a virtal reference z which is calclated together with the other optimization variables in order to minimize the cost fnction: J H p 1 = i= i) W z i) W z ) H p ) W z z r Q R F T 4.22 is worth to notice that the first part is similar to the standard cost fnction 4.15 besides the fact that the system is driven to follow the virtal reference rather than the actal one; the additional term 2 z r has the fnction to eep the virtal and the real set-points as near T as possible in a qadratic sense. The terminal cost can be calclated directly by imposing the stability condition S4 with strict eqality: T T T T A BK) F A BK) F A BK) Q A BK) K B RBK = 4.23 As done before the optimization problem can be transformed into the qadratic programming formlation obtaining: J = z T S T QS R T 2 S QW RW ) T T T W QW W RW z z T 2 S T QS 2W T QS 2T r 4.24 A final consideration has to be done on the comptational brden associated to the soltion of the optimal problem. Indicating with m the dimension of the inpt and with v the dimension of r is possible to calclate the dimension of the optimization vector which 38

40 gives a measre of the compleity of the optimal problem. Considering the case where the reference is nown over the entire prediction horizon and the virtal reference has to be calclated accordingly the dimension of the optimization vector becomes: h = m v) 4.25 H p while considering the reference constant over the entire horizon the dimension redces to: h = mh p v 4.26 posing m=4 and v=4, how is normal in the helicopter control problem, the comptational brden in the second case is 45 to 5% lower than the first case depending on the prediction horizon chosen and, hence, only the second case has been implemented. In order to evalate the performances of this architectre the speed control of the helicopter has been implemented. To do so is sfficient to introdce the model of the helicopter body dynamics as the plant model and select the body frame speeds, v, w and the heading angle ψ as otpts of the system. figre 4.1: performances as fnction of prediction horizon 39

41 figre 4.2: virtal set-point as fnction of prediction horizon Besides the obvios variations de to the calibration of the weight matrices Q and R is interesting to notice the inflence of the prediction time on the performances, figre 4.1 shows the reslts obtained by changing the prediction horizon from 1 to 1 steps. As can be seen going from 1 to 5 steps the performances progressively improve decreasing the settling and rise times. This changing can be eplained by looing at the virtal reference plots reported in figre 4.2, with longer prediction horizons the system can reach states frther away from the initial condition and this allows to choose a virtal reference nearer the actal one. Conversely a short prediction horizon forces the optimizer to choose a reference near the initial condition rather than near the reference obtaining poor performances. By frther increasing the prediction horizon, in the figre going from 5 to 1 steps, there is no gain in performances since the dynamics of the system are dominated by the constraints, in particlar the constraint imposed on the attitde angles ± 25 degrees) limits the acceleration to a maimm which can not be violated even choosing higher vales of reference. It is clear that sch behavior is fnction not only of the prediction horizon bt of the reference itself, moreover the best performances are obtained for longer horizons and, hence, with higher comptation brdens. Sch behavior is an heavy drawbac for this method since relates the comptational capabilities of the FCC with the performances of the control algorithm Infinite horizon In order to garantee feasibility of the open loop optimal problem a second method is to enlarge the prediction horizon; taing this concept to the limit this control architectre is 4

42 implemented considering an infinite horizon, is trivial to nderstand that with this choice is always possible to reach the terminal set however it is defined from every possible state. The stability conditions can be met by sing: f ) = X f = { } F = 4.27 on the other hand the standard formlation does not allow the system to be solved since an infinite nmber of nnown variables have to be fond. In order to overcome a problem the control horizon is introdced, the first H c control moves are allowed to change and are calclated by the optimization algorithm while the other inpts are obliged to be The cost fnction then becomes: = ss. J = H 1 c i= i) ss i) ss ) Q R i) 4.28 i= H c ss 2 Q the second term is still impossible to be calclated nmerically bt, if the system is stable, converges to: i= Hc 2 2 i) ss = H c ) Q ss P 4.29 where P is the soltion of the discrete Lyapnov matri eqation: A T PA P Q= 4.3 Since the helicopter is nstable relation 4.29 cannot be applied directly bt the system has to be stabilized first. To do so an inner control loop has been added as depicted in figre

43 r MPC CONTROL - HELICOPTER K figre 4.3: inner stabilizer loop again, a linear feedbac law has been sed bt in this architectre the weight matrices sed are not the same of the MPC cost fnction bt are calibrated in order to satisfy this linearity condition: min K, ] 4.31 ma [ min ma In this way is possible to garantee that the plant seen by the MPC system is stable for every possible state of the system. At this point the model sed to calclate the steady state condition and to predict the ftre response becomes: 1) = A BK) ) B ) 4.32 where the inpts are the reference for the stabilizer rather than the helicopter controls. The inpt constraints have to be modified as follows: K 4.33 min ma 42

44 figre 4.4: infinite horizon performances This architectre has been tested in the same framewor sed in the previos one and the reslts are shown in figre 4.4. As can be seen the prediction horizon has little inflence on the performances, all the signals are similar going from 1 to 5 prediction steps. Moreover is possible to notice that the response is similar to the best performances obtained with the previos architectre. Is worth to notice that the same architectre can be obtained enlarging the terminal set. If we choose X f = X, which obviosly garantees feasibility for every X, and a linear terminal control law f = K the stability condition S2 corresponds to 4.31 and the terminal cost calclated by 4.3 or by 4.23 are eqivalent Soft constraints The inpt constraints correspond to the physical limitation of the actators and cannot be eceeded in any case. On the other hand the state constraints are imposed by the designer for several different reasons bt do not correspond to any physical limitation of the helicopter. An eample of this is the attitde angle, in the previos eamples has been limited to be less than 25 degrees, bt the helicopter can reach also higher vales. 43

45 Modelling errors or distrbances, sch as wind gsts, can drive the system otside the limits imposed by the state constraints, when this happens the optimization problem cold become nfeasible. This happens when, given all the constraints the first states in the predicted seqence cannot be driven bac into the admissible state sbspace X. Since is very liely that this condition will occr dring real flights a way to restore feasibility is needed, two approaches can be sed in this case: a. detect the sitation and se a different controller, b. formlate the MPC architectre in order to be robst to sch a sitation. The first approach is more complicated since reqires a way to detect nfeasibility and needs a second control system capable of dealing with the sitation, conversely the second approach is more attractive even thogh implies a modification of the optimal control problem. The soltion adopted distinct two different classes of constraints: hard constraints which can not be eceeded nder any circmstances lie the inpt constraints) soft constraints that can be eceeded only if feasibility can not be obtained otherwise. The first class has been applied to the inpt constraints since those have to be respected anyway while the state constraints have been softened in order to enable the system to restore feasibility. To do so a set of slac variables ε has been introdced in the state constraints formlation: ε 4.34 min ma by means of these variables the admissible state sbspace can be moved in every direction as necessary to restore feasibility. In order to se sch possibility only when necessary the vector of slac variables is weighted in the cost fnction transforming it into: 44

46 J H p 1 = i= ) H p ) ε Q R F S 4.35 figre 4.5 shows the comparison between two different controllers, both have been implemented sing the infinite horizon architectre bt one ses all hard constraints while the second implements the soft constraint architectre described above. figre 4.5: comparison between soft and hard state constraints In the first case the problem becomes nfeasible after 11.2 seconds from the begging of the simlation, the inpt is set to the steady state vale and the system is controlled only by the inner stabilization loop. As can be seen the nfeasibility starts as soon as the attitde eceeds the limit vale and no control is given by the MPC algorithm ntil the inner stabilizer drives bac the system into the feasible region; is worth to point ot that once the MPC stops woring the stability of the closed loop is not garanteed anymore especially when controlling the non linear plant. Conversely when the state constraints are softened the nfeasibility does not occr even if the attitde reaches vales higher than the 25 degrees limit. At this point the MPC eeps controlling the system and all the stability garantees are valid. 45

47 4.4 Offset-free control One of the more common reqirements for a control system is to avoid errors at regime. The presence of mismatches between the real behavior and the predicted dynamics, de to modeling errors and eternal distrbances, is the case of sch errors in the predictive approach. Once again is possible to modify the control system architectre in order to avoid this behavior. One possible approach is qite classic and consists into adding the integral of the tracing error into the state vector and penalize it into the cost fnction. A less obvios approach maes se of a particlar formlation of the plant model where the inpts are moved into the state vector; an observer estimates them by sing only the information on the system otpts coming from the sensors Integral action The idea of compensate for steady state errors sing an integral action is based on the classic PID architectre. In order to implement this featre into the MPC control the first step is to obtain a prediction model where the integral of the tracing error e is calclated. To do so is sfficient to epand the linear model as follows: & A B r e = C e I & 4.36 the ftre state becomes fnction of two different inpts: the actal inpts which have to be optimized and the reference r that has to be treated as a parameter by the optimization algorithm. In order to obtain the ftre control seqence is possible to se the same procedre seen before and obtain: = S S S rr 4.37 which can be inclded into the cost fnction in order to obtain the QP formlation: J T T T T = T S QS R) 2S QS 2S QW r 2RW r 2S QS ) const 4.38 rr 46

48 The control scheme is then modified in order to inclde the feedbac of the integral error as depicted in figre 4.6 figre 4.6: Integral action architectre scheme In figre 4.7 the comparison between two different controllers is shown. The infinite horizon control developed in section and the same controller agmented with the integral action. figre 4.7: effects of the integral action A small distrbance has been introdced at the second 2, as can be seen the baseline controller, indicated in red, presents a constant tracing error. Conversely the introdction of the integral action allows the system to cancel this error. 47

49 On the other hand a significative overshoot can be noticed on the step response of the system when the integral action is introdced. This is cased by the interaction between the constraints and the integral called wind-p. As can be seen the lateral acceleration of the helicopter his limited by the attitde constraint, when this limit it reached the integral of the error eeps increasing withot any frther effect on the performances. When the setpoint is reached the contribte of the integral is higher than the necessary and the system eeps accelerating. At this point the error is inverted and the integral error decreases reaching the correct vale. In the control literatre several anti wind-p schemes can be fond, all rely on the same basic concept: when the satration is reached the integration action is freezed at the reached vale. The same concept has been applied here by sing the following relations: e& i e& i = r i = y i min, i otherwise i ma, i 4.39 where the sbscript i identifies the single component of the error vector and are reported in Table 4.1. i r, y i i i min, i 1 sp, ϑ ϑ min 2 v sp, v ϕ ϕ min ma, i ϑ ma ϕ ma 3 w sp, w c c min cma 4 ψ sp, ψ r r min r ma Table 4.1: anti wind-p limit variables The anti wind-p scheme adopted adds a non linearity on the controlled plant and, therefore, is not possible to inclde this featre on the prediction model. This leads to a mismatch between the predicted integral error and the one actally calclated bt this error has no conseqences on the practical application of this control architectre. 48

50 figre 4.8: effect of the anti wind-p system In figre 4.8 the effect of the anti wind-p system is clearly shown: the ability of canceling the steady state error has been maintained withot variations while the overshoot cased by the integral wind-p has been eliminated obtaining lot better tracing performances. Finally a robstness analysis has been performed by means of Monte Carlo simlations, the controlled plant has been sbstitted by the ncertain model described in section 2.5 and the ncertainties have been randomly chosen at each simlation. In figre 4.9 a representative set of the simlations has been reported, is worth to notice that all the constraints have been respected and the tracing error at regime has been cancelled by the presence of the integral action. 49

51 5 figre 4.9: Monte Carlo simlation State observer In order to obtain offset-free control with MPC a different approach has been proposed in [17], the prediction model is agmented by sing the delta- formlation reported in eqation 4.4 and an observer estimates both the states and inpts in order to minimize the mismatches between the predicted and the real otpts. ) ) ) 1) 1) I B I B A = 4.4 These estimates are calclated as follows: )) ) ˆ ) ) ˆ ) ˆ 1) ˆ 1) ˆ y C K I B I B A obs = 4.41

52 and are sed by the MPC algorithm as initial states of the optimization. The overall control scheme is reported in figre 4.1. figre 4.1: state observer control architectre In this scheme the offset free control is achieved since the mismatches between the predicted otpts and the actal ones are lmped into the estimates of the nmeasred variables and the inpts. In figre 4.11 the reslts of a Monte Carlo simlation performed with the state observer control architectre are shown. Zero steady state error has been obtained bt the attitde constraints has not been respected, this is cased by the fact that the real and the estimated vales are different and the actions taen by the MPC control to respect the constraints do not obtain the desired reslt on the real plant. 51

53 figre 4.11: Monte Carlo simlation 52

54 5 Stability and Control Agmentation System One of the more diffsed ways to control the motion of the UAS is trogh remote piloting; the operator directly controls the air vehicle from the grond station sing an Hman Machine Interface HMI) really similar to those sed in manned aircrafts. As already mentioned helicopters are nstable and heavily copled systems and are really difficlt to pilot, the SCAS system has the aim of stabilize and decople all the controlled ais of the helicopter in order to simplify the tas of the operator; moreover the same system cold be also sed as an inner control loop of more comple Gidance, Navigation and Control GNC) systems. In the following sections the control system architectre will be described in depth and a set of simlations will be shown in order to demonstrate its performances. 5.1 SCAS control strctre The aim of the SCAS is to control the body speeds and the heading of the helicopter, since these variables are inclded in the standard linear approimation of the plant see eq.2.5) this can be directly sed into the control system definition. Given the analysis presented in section 4 the following base architectres have been joined together: 53

55 54 the infinite prediction horizon control, the integral error action with the antiwind-p correction the soft constraints formlation. For clarity in the following the complete controller formlation is reported. The inner stabilizer loop has been obtained by the classic LQR control procedre imposing the weight matrices as: I R I Q 1, = = 5.1 and the prediction model has been set as follows: r I B e C BK A e = & & 5.2 with = C 5.3 The cost fnction to be minimized is ) = = ) ) ) H c i S P c a R a Q a r W H W r i W r i J ε 5.4 where the sbscript a indicates the state and inpts of the agmented model 5.2. The weight matrices Q and R are different from the ones sed to calibrate the inner stabilizer and have been tned in order to avoid response overshoot for low and medim aggressive manoevres while maimizing performances.

56 Finally, the inpt and state constraints are reported in Table 5.1 Variables Symbol Limits Inpts tr 1 tr Flapping angles Anglar speeds Long and lateral speeds a, b, c, d 15 15deg p, q, r 2 2rad / s, v 1 1m / s Vertical speed w 5 5m / s Long and lat. attitde angles ϑ, φ 35 35deg Yaw angle ψ 36 36deg Table 5.1: SCAS constraints In order to point ot the advantages given by the se of the model predictive approach, in particlar the ability of enforcing constraints and se of the information abot ftre set points, an LQR control system has been developed and calibrated for comparison prposes. This control approach has been chosen since maintains the same basic strctre of the MPC, in particlar handles mltiple inpt mltiple otpt systems and maes se of a cost fnction similar to the one sed in the predictive controllers. Moreover in the SCAS also the integral action has been inclded by sing the following agmented system as plant model & A = e & C B e 5.5 The baseline controller has been calibrated in order to have similar performance with respect to the MPC system for low and medim aggressive manoevres. 5.2 Simlation reslts Two different sets of simlations have been carried on in order to evalate the controller characteristics. In the first set the SCAS has been applied to a linear model of the helicopter identical to the one sed in the prediction model, in the second set model 55

57 ncertainties have been applied to simlation model and a Monte Carlo simlation have been carried on to evalate the robstness characteristics of the system. Figres 5.1 and 5.2 show the reslts obtained in two separate simlations where a series of acceleration manoevres where performed on respectively the longitdinal and lateral ais. The speed set-point has been designed in order to obtain manoevres with increasing aggressiveness in order to evalate the response of both the systems. figre 5.1: longitdinal accelerations 56

58 figre 5.2: lateral accelerations As epected dring the low and medim aggressive manevers the response of inpt, attitde and speed variables is qite similar comparing the MPC and the LQR controllers. A significative difference can be noticed when the system is ased to perform really aggressive manevering, in this case the MPC controller has the ability of stabilizing the system and perform the manevers with a smooth behavior. Moreover both the inpt and attitde constraints are respected. On the contrary the baseline SCAS does not have the information abot inpt constraints and, when the calclated inpt is significatively larger than the satrations, the system loses the ability of stabilizing and controlling the helicopter. In order to evalate the robstness characteristics of controller a Monte Carlo simlation has been performed sing the ncertain model shown in section 2.5. In figre 5.3 a redced nmber of simlations have been reported, as can be seen both the inpt and attitde constraints are still respected by the control system while offset free tracing is achieved. 57

59 figre 5.3: Monte Carlo Simlation 58

60 6 Trajectory Tracing System In order to increase the level of atonomy of the system and redce the wor load of the hman operator, the position control of the rotorcraft can be atomatically eected by the FCS. The motion control of air vehicles can be roghly divided in two different classes: trajectory tracing, where the UAS is reqired to trac a time parameterized reference, and path following, where the vehicle is reqired to converge and follow a given path withot any time specification. In this section the first approach is presented, the trajectory is constitted by a seqence of three-dimensional positions, epressed in the North-East-Down NED) reference frame, and a target heading angle to be reached at a specific time. Differently from the classic approach, which divides the vehicle gidance and the control problem into an inner and an oter loop, the system presented here has been bilt sing an integrated approach an both the navigation and control problems are solved simltaneosly. As done for the SCAS first the system architectre will be presented and eplained into the details, afterwards a set of simlations will be shown to demonstrate the system capabilities. 59

61 6.1 System architectre Normally the position of the helicopter is epressed into an earth fied reference frame, typically the NED frame; on the other hand the helicopter dynamics is epressed into the body frame. In order to calclate the actal position of the helicopter is necessary to rotate the body speed from one frame to the other sing the actal Eler angles; since this operation is non-linear it can not be directly inclded in the prediction model for the MPC architectre sed here. A linear approimation of the helicopter inematics can be obtained by epressing the position into the body frame ais at the beginning of the horizon and se it trogh all the prediction horizon. If the change of attitde over the horizon is small the position dynamics can be approimated as: s= & [, v, w] 6.1 where s is the position of the helicopter epressed in the new frame. The fll linear model sed to predict the ftre state of the helicopter can be epressed as & b A = s & L b B s b 6.2 where L is a matri, which selects the body speeds from the state vector b. At this point is sfficient to rotate both the actal position and the ftre reference from the NED to the actal body frame to close the loop. 6

62 figre 6.1: Trajectory tracing system architectre Finally, regarding the MPC architectres proposed in section 4 an infinite prediction horizon has been sed together with a soft state constraints formlation, in this case the integral action is not needed as simlations will demonstrate. The baseline controller has been bilt arond the same linear approimation of the helicopter inematics sed for the MPC, the system 6.2 has been sed as a plant and the following control law has been implemented. = r K 6.3 where = W KW Simlation reslts As done for the SCAS the first set of simlations presented shows the reslts obtained by the system when the helicopter dynamics and the model sed in the prediction are identical. Is worth to notice that, in this case, a difference between the prediction model and the controlled plant is still present de to the linear approimation of the helicopter inematics shown in the previos section. Figres 6.2 and 6.3 show again two different simlation where a reposition manoevre has been performed on the longitdinal and lateral ais respectively. As for the SCAS, the 61

63 performances of the MPC and the baseline controller are the same for low and medim aggressive manoevres. Is important to notice that for the medim and high aggressive manoevres the LQR controller does not respect the attitde constraints, in particlar for the aggressive repositions the attitde angles eceed the 9 degrees which clearly nfeasible on the real helicopter. figre 6.2: longitdinal reposition 62

64 figre 6.3: Lateral reposition manevers Another advantage given by the predictive control formlation is the ability of taing into accont information abot the ftre set-point when this is available. The control of the position of an UAS is the typical case where this happens since the desired flight plan is normally delivered before starting the mission. In order to evalate the change of performances given by sch ability the previos control formlation has been modified in order to se a ftre seqence reference. Figres 6.4, 6.5, 6.6,6.7 show respectively the position, speed, attitde and controls of the helicopter performing the lateral reposition manevers nder the control of the standard and modified MPC systems. Observing the lateral ais variables, namely the East position, the lateral speed and the roll angle is possible to see how the information abot the ftre set-point allows the system to anticipate the manever redcing significantly the tracing error. Moreover the modified system starts the manevers in a smoother way which is reflected on the other ais response where the error is significantly redced. 63

65 figre 6.4: Constant vs. nown reference, NED position figre 6.5: Constant vs. nown reference, body speeds 64

66 figre 6.6: constant vs. nown reference, attitde angles figre 6.7: constant vs. nown reference, controls displacement The robstness test has been carried on the predictive architectre, as observed for the SCAS controller the predictive architectre is able of maintaining stability and the ability of 65

67 enforcing all the constraints. Moreover, as anticipated, zero steady state error is achieved withot the need of an integral action. figre 6.8: Monte Carlo simlation Finally a measre of the comptational brden associated to the control has been evalated measring the time needed to solve the online part of the control algorithm, in order to obtain real time performances this time as to be less than one sample interval, in the specific case 5ms. The tests have been carried on a laptop compter eqipped with an Intel Centrino Do processor whose comptational capability has been measred sing a benchmaring software and obtaining a capacity of 15 GFLOPS Giga Floating point Operations per second). A standard embedded controller for aerospace applications has a more limited capability of abot 2 GFLOPS [23], in order to nderstand if sch a controller is capable of real time performances the times obtained in simlation have been increased proportionally to the comptational capabilities and the reslts reported in 66

68 figre 6.9: Time to obtain the control action As can be seen the control algorithm is fast enogh to be implemented on a standard control system. 67

69 7 Path Following System In this section the path following approach is sed for controlling the helicopter s position. As eplained previosly the aim of sch system is to converge on and follow a given path with no time specification. In other words the control action has to be fnction only on the actal position and not of the time at which the position is reached. Conversely, the predictive approach is a time domain techniqe and cannot be directly applied to solve the path following problem bt can be sed transforming the latter in a seqence of trajectory tracing problems. At each sampling instant the system calclates a ftre reference trajectory as fnction of the desired path and the crrent position then, by means of the system developed in the section 6, the control actions needed to follow the trajectory are calclated maintaining the MPC advantages. To calclate the crrent reference trajectory the system simply finds the point on the path nearer the helicopter, this point becomes the starting point of the ftre trajectory. Then the desired position is propagated in the ftre by a simple linear interpolation between the waypoints. 68

70 figre 7.1: Path following principle As sal the performances of the path following system will be compared with the ones of the trajectory tracing system with nown reference while eecting a series of manoevres. The first test shown reports the reslts following a path with different angles trn that has to be performed eeping the heading fied to the north. figre 7.2: path following vs. trajectory tracing In figre 7.2 the trajectory followed by the two systems is shown, as can be seen the performances are qite similar. The difference can be noticed eamining the time histories of the speeds, figre 7.3, and attitde, figre 7.4. While the trajectory tracing system needs to compensate for the non instantaneos accelerations with a speed overshoot the 69

71 path following system reaches the target speed withot this problem, this is also reflected in a smoother manevering and a lower demand in attitde angles. figre 7.3: speed time plot figre 7.4: attitde time plot 7

72 8 Conclsion and otloo In this thesis the latest reslts achieved at the Bologna University regarding the UniBo RUAS project have been presented. In particlar a series of flight control systems based on the model predictive control theory have been illstrated, its performances have been assessed and compared with a baseline linear qadratic reglator. First, a set of base architectres has been shown and compared each other in order to gain a better nderstanding of the advantages and disadvantages introdced by each one. As reslt of this analysis the most promising architectres have been joined together and sed to solve the problem of flight control of a small scale helicopter. The MPC theory has been applied to three different control systems: a SCAS, a trajectory tracing and a path following system. A comparison with a baseline LQR controller has been carried on and, for all the three configrations, the predictive control showed several advantages: constraints on both the inpt and states of the controlled plant can be eplicitly introdced in the flight control system formlation and have been sed to tae into accont for control satrations and flight envelope protection, the presence of constraints ensres stability of the system also performing high aggressive manoevres where the baseline controller fails, 71

73 the predictive approach has the ability of sing all the information abot the ftre reference, when this is available, and ses it to enhance the control performances. On the other hand the comptational brden associated with the MPC approach is a lot heavier than the classic control techniqes, nevertheless than to the capabilities of modern processors and new optimization algorithms applied in this thesis real time performances can be achieved by state of the art microcontrollers. Frther development of this wor will inclde the implementation of the proposed architectres in the actal flight control compter and validation trogh hardware in the loop simlation and actal fight testing. Moreover, a frther investigation on the copling between model predictive control and state observers has to be done in order to better asses the effects on control and constraints introdced by state estimation errors. 72

74 Bibliography [1] R. Pretolani, Progetto e realizzazione del sistema di navigazione, gida e controllo per n elicottero con capacità di volo atonomo, II School of Engineering, University of Bologna, PhD Thesis, 27 [2] B. Teodorani Progetto e realizzazione del sistema di gestione atonoma del volo e controllo in remoto per n velivolo UAV ad ala rotante. II School of Engineering, University of Bologna, PhD Thesis, 27 [3] F. Zanetti Design, implementation and tests of Real-time Feed-forward controller and navigation system for a small scale UAV helicopter II School of Engineering, University of Bologna, PhD Thesis, 29 [4] F.Zanetti, S.Colatti, G.M. Saggiani, V.Rossi "Rotary wing UAV: HIL tests for a modelbased FeedForward controller" 34th ERF meeting proceedings, Liverpool, UK, Sept 28. [5] A. Bdiyono, S. Wibowo Optimal Tracing Controller Design for a Small Scale Helicopter, Jornal of Bionic Engineering, Vol.4, No. 4, pp , 27 73

75 [6] L. Marconi, R. Naldi, Robst fll degree of freedom tracing control of a helicopter, Atomatica, Vol. 43, No. 11, pp , 27 [7] K. Peng, G. Cai, B. M. Chen, M Dongb, K. Y. Lma, T. H. Lee Design and implementation of an atonomos flight control law for a UAV helicopter, Atomatica, Vol. 45, No. 1, pp , 29 [8] Gareth D. Padfield. Helicopter Flight Dynamics: The Theory and Application of Flying Qalities and Simlation Modelling. Second Edition, Bramwell Pblishing 27 [9] R. Cnha, C.Silvestre, SimModHeli: A Dynamics Simlator for Model-Scale Helicopters The 11 th Mediterranean conference on control and atomation proceedings, 23 [1] B.Mettler, C.Dever and E.Feron Scaling effects and Dynamic Characteristics of Miniatre Rotorcraft Jornal of Gidance, Control and Dynamics Vol.27,No. 3, 24 [11] D.Q. Mayne, J. B. Rawlings, C. V. Rao, P. O. M. Scoaert, Constrained model predictive control: Stability and optimality, Atomatica, Vol 36, pp , 2 [12] T.Zo, S.Y.Li, B.C.Ding, A dal mode nonliear MPC with enlarged terminal constraint sets, Acta Atomatica Sinica, Vol 32, 26 [13] L. Imsland, J. A. Rossiter, B.Plymers, J. Syens, Robst Triple Mode MPC American Control Conference Proceedings, 26 [14] Y.I. Lee, B. Kovaritaisb, M. Cannon Constrained receding horizon predictive control for nonlinear systems Atomatica Vol 38, pp , 22 [15] D.Limon, I.Alvarardo,T.Alamo, E.F. Camacho MPC for tracing constant references for constrained linear systems Atomatica Vol 44, pp , 28 74

76 [16] V.Eadatylos, C.J. Taylor, A. Chotai Non-minimal model predictive control with an integral-of-error state variable UKACC International Conference, Glasgow, Agst 26 [17] U. Maeder, F. Borrelli, M. Morari Linear offset-free Model Predictive Control Atomatica Vol 45, pp , 29 [18] G. V. Raffo, M. G. Ortega, F. R. Rbio An integral predictive/nonlinear Hinf control strctre for a qadrotor helicopter Atomatica Vol 46, pp , 21 [19] S. Olar, D. Dmr Compact eplicit MPC with garantee of feasibility for tracing Proceedings of the 44th IEEE Conference on Decision and Control,Seville 25 [2] E.Frazzoli, M.A.Dahleh, E.Feron Trajectory tracing control design for aotnomos helicopters sing a bacstepping algorithm Procedings of the America Control Conference, Chicago 2 [21] Fabio A. Almeida, Waypoint Navigation Using Constrained Infinite Horizon Model Predictive Control AIAA Gidance, Navigation and Control Conference and Ehibit, 28 [22] Handling qalities reqirements for military rotorcrafts. Techincal Report ADS- 33DPRF, United States Army, St-Lois Missori, 1996 [23] 75