Meccanica e scienze avanzate dell ingegneria

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

Table of contents List of figres...3 List of tables...5 Abstract...6 1 Introdction...7 2 Helicopter models...11 2.1 Non-linear model...11 2.1.1 Main rotor...12 2.1.2 Hiller bar...12 2.1.3 Tail rotor...13 2.1.4 Control chain...13 2.1.5 Fselage...15 2.1.6 Engine...15 2.1.7 Validation...15 2.2 Linearized models...16 2.2.1 Model strctre...16 2.2.2 Model linearization procedre...17 2.2.3 Model validation...2 2.3 Simlation models...21 2.4 Prediction model...22 2.5 Model ncertainties...22 3 Model predictive control...25 3.1 MPC basics...27 3.2 Stability...29 3.3 Feasibility...29 4 Controller implementation...31 4.1 Open loop optimal problem soltion...32 4.2 Reference tracing...35 4.3 Feasibility garantee...37 4.3.1 Virtal reference...37 4.3.2 Infinite horizon...4 1

4.3.3 Soft constraints...43 4.4 Offset-free control...46 4.4.1 Integral action...46 4.4.2 State observer...5 5 Stability and Control Agmentation System...53 5.1 SCAS control strctre...53 5.2 Simlation reslts...55 6 Trajectory Tracing System...59 6.1 System architectre...6 6.2 Simlation reslts...61 7 Path Following System...68 8 Conclsion and otloo...71 Bibliography...73 2

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...47 3

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

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...55 5

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

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

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

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

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

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 2.5. 2.1 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

Fselage Engine In the following each component model is described into the details jstifying the choices made dring the modelling phase 2.1.1 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 2.1.2 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

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. 2.1.3 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. 2.1.4 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 2.1 1 1c 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

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

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 2.1.6 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. 2.1.7 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

Model Flight Data Collective pitch [deg] 7 6.5-7 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 3. 44 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. 2.2.1 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

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. 2.2.2 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

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 3-2-1-1 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

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

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 2.9 2.2.3 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 3-2-1-1 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

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

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 ) 2.13 2.5 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

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

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

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

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