STABILIZATIO ON OF LONGITUDINAL AIRCRAFT MOTION USING MODEL PREDICTIVE CONTROL AND EXACT LINEARIZATION

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1 8 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES STABILIZATIO ON OF LONGITUDINAL AIRCRAFT MOTION USING MODEL PREDICTIVE CONTROL AND EXACT LINEARIZATION Čeliovsý S.*, Hospodář P.** *CTU Prage, Faclty of Electrical Engineering, Department of Control Engineering, **Aerospace Research and Test Establishment, Prage, Czech Repblic Abstract This wor deals with constrained control of nonlinear model of an aircraft. A methodology applying for this control is based on an eact linearization (EL in connectionn with linear model predictive control (MPC techniqes. Introdction The prpose of this control methodology is the flight recovery problem. Primary tas of an atomatic recovery system is to regain control over an aircraft in case of nintended manevers or ncontrolled sitations (e.g.. pilot loses orientation. In sch sitation nonlinear movement is assmed and therefore classical control methods cannot be sed. Those reqire model with small deviations and control arond the eqilibrim. Whole procedre has several limiting factors. Above all there are stall angle of attac and sideslip angle (separation of the flow aerodynamic limitation, pitch, roll and yaw rate (mechanical limitations of the plane, aerodynamic loads (physiological pilot limitations and a control limit (amplitde and rate of control srfaces deflection and thrst of engine. Based on these facts combination of inpt constrained MPC with EL were chosen. An important advantage is that EL model can be controlled with a linear MPC with fied model. On the other hand it is not easy to define constraints of linearized model. As a controlled platform for a plane behavior a model of flying laboratory Vilí (Fig. was created. In this case it was modeled as a nonlinear form of longitdinal behavior with two inpts. Problem formlation Pre mathematical eqation of motion for aircraft model is in non-linear form: & = f (, ( where is state vector, is inpt vector and f is differentiable field on. Now we can define projection of or non-linear space to new linear space and change inpts: = Φ(, = α(,v ( where is state of new linear model and v is new virtal inpt. Most of nonlinear dynamic models (aircraft model too can be described as control affine system: & = f ( + g ( ( By the help of Lie derivative it is possible to create static feedbac that transforms nonlinear to linear model. & = f ( + g( & = A ( v = D( + α( Matrices A and B are depended on transformation form. They are typically composed from series of integrators. Matrices D and α are determined by Lie derivatives and will be described below. MPC is a type of control which ses optimal state-feedbac and predictive strategy for optimal design seqence of control action with reference to ftre states and otpts of the system. MPC is based on a linear, discrete time state-space model of the model. Qadratic programming (QP was typically sed for

2 ČELIKOVSKÝ S., HOSPODÁŘ P. soltion of MPC. A main advantage of QP is a simple implementation of inpts/otpts constraints. Qadratic objective fnction for linearized system is: J ( v = q ( w + r v ( = = where w is reference vector and q and r are state respectively inpt weights. Mathematical model of an aircraft A formlation of aircraft flight dynamics is derived from Newton s second law of motion. This can be investigated from the viewpoint of stability or flying qalities and can be sed for modeling of aircraft motion [, ]. There are following simplifying assmptions: the aircraft is rigid body with fied mass, gravitational acceleration is constant and the aircraft is withot strctral deformation. Following eqation and variables are relative to the body aes system (Fig,.. Fig. Body ais components; definition of moments L, M, N; forces X, Y, Z; velocities, v, w; anglar rates p, q, r; angle of attac α and sideslip angle β. Eqation of motion The general forces and moments eqations can be described by Newton s second law of motion in translational and rotational form []. We consider rotational ais system, where the derivate operator applied to vectors has two parts: one for the rate of change of the vector, and one for ais system rotation. v F = + t m + Ω v = Fw + Fa Ft (6 ( I Ω M = + Ω I Ω = t M a (7 Where F represents the sm of all eternally applied forces, m is the mass of aircraft, v is the velocity vector, Ω is the total anglar velocity of the aircraft, M represents the sm of all applied torqes and I is moment of inertia matri. Thereby we get si component eqations of motion, where on the left side there are time derivatives of anglar rates and velocities and on the right side there are description of motion and eternal inflences. That is where the eternal forces depend on the weight vector F w, the aerodynamic force vector F a and the thrst vector F t respectively. We assme the thrst prodced by the engine acts only parallel to the aircraft s longitdinal ais. Eternal moments are those de to aerodynamics.. Aerodynamic model The aircraft aerodynamic model mainly depends on the following factors: the airspeed V (Mach nmber or Reynolds nmber respectively and density of the airflow ρ; the geometry of the aircraft (wing area S, wing span b and mean aerodynamic chord c MAC ; the orientation of the aircraft relative to the airflow [] (angle of attac α and side slip angle β; the control srface deflections δ and the anglar rates p, q, r. There are other variables sch as the time derivatives of the aerodynamic angles that also play a role, bt these effects are less prominent. The volme of the aerodynamic forces and moments is determined by the amont of air variables in different directions. In the standard way of modeling the aerodynamic forces and moments are given by the following eqations: F a = q S ci ( α, β, δ, p, q, r,... i = L, D, Y (8 M a = q S ci ( α, β, δ, p, q, r,... i (9 i = l, m, n = b, c, b i MAC where q ρv is dynamic pressre. Each dimensionless aerodynamic coefficient from (8 and (9 can be described as a fnction of other flight parameters: cl = cl( α, β, δ, p, q, r, M,Re,... (

3 STABILIZATION OF LONGITUDINAL AIRCRAFT MOTION USING MODEL PREDICTIVE CONTROL AND EXACT LINEARIZATION This description is too detailed for mathematical modeling and too difficlt for measrement/comptation. Therefore it is described in a simplified way where the effect of each flight parameter is separated. For eample the aerodynamic lift coefficient is then given by: q c MAC c L = c L Lα α Lq Lδ δ e e ( V The aerodynamic coefficient eqals smmation of initial vale (vale on lift line at zero angle of attac, prodct of lift line derivatives and angle of attac, prodct of non-dimensional pitch rate derivatives and non-dimensional pitch rate and prodct of control derivatives and control deviation (elevator. The aerodynamic coefficients (more precisely aerodynamic derivatives are sally obtained from wind tnnel or by comptational methods []. In this case data from flight tests [6, 7] were sed (Fig., where the aerodynamic derivatives were estimated by maimm lielihood method [8]. & = c & & & = c 6 = c = c sin ( 7 sin ( ( cos( 8 9 cos + ( The longitdinal motion of aircraft is a dynamic system with for states/otpts y = = [V, α, q, θ] (consective airspeed, angle of attac, pitch rate and pitch angle and two inpts = [δ e, F t ] (elevator deflection and thrst of engine. All parts which are independent on state variables are sbstitted by constants (see Appendi.. Steady flight Comptation of eqilibrim is a classical way of how to determine initial vales of states and inpts of nonlinear model. In this case we have for eqations and si nnowns (for states and two inpts. However, we can se additional eqations for vertical speed (i.e. change of altitde that we assme zero: h & = sin( = = ( Frther more typically an attitde of airspeed and pitch rate ( is assmed zero in eqilibrim. Thereby we get three eqations with three nnowns: = c cos( = c6 7 9 c sin( ( = c Fig. model Vilí dring flight test. Longitdinal part of motion The aircraft eqations of motion in general form are a set of copled nonlinear differential eqations. However, in or case these eqations can be simplified. Assming zero lateral states (side slip angle, roll and yaw rates, roll angle, a separated longitdinal motion in nonlinear form can be obtained: where is chosen airspeed. Analytically soltion is de to goniometric fnctions relatively complicated. Therefore Nelder-Mead simple method is sed for solving of this static optimization problem. Eact linearization An assmption for this transformation is an accomplishment of an invertible matri D [9]. In or case it is necessary to se dynamic feedbac for accomplishment of this assmption. It means, that the elevator

4 ČELIKOVSKÝ S., HOSPODÁŘ P. deflection (inpt is added in integrator thereby elevator rate is the new inpt and elevator deflection is changed into state. We define the elevator inpt as a new state: = ( Now we determine nonlinear transformation Φ:. The system in new coordinates can be epressed as: & = & = v & = & = & = v where the first new state is chosen as: = sin( & = & sin( + cos( & (6a (6b (6c (6d (6e (7 by sbstitting derivatives of state from ( to (7 and proper arrangement we get: = & = sin( ( c +... cos( ( c6 7 9 cos( +... (8 cos( ( + c 8 The reslt of the last term is too complicated to be solved by analytical method. Therefore Matlab symbolic toolbo for soltion (6b was sed. Second part of decopled system is determined by last three eqations (6c, (6d, (6e: = = (9 = c Last three state derivatives are also solved by symbolic toolbo. From (6 it is easy to form state and inpt matrices (: A = B = ( Last part of completion of the eact linearization is to provide second eqation from (. Using affine system assmption we can easily determine matrices D and α as a part which is dependent on inpt (matri D and the rest (matri α. The feedbac linearizing control law is given by: = D( ( v α( ( Model Predictive control Conventionally MPC is formlated in discrete time[]. Assming controlled aircraft model ( the description by the linear discrete time difference eqation is: + = A ( Following states on prediction horizon T p can be described as follows: M = A = A = A + A = A = A Bv + A + ABv Bv A Bv + ABv ( Last eqation can be rewritten to matri form: = A ( where each variable with bar is described as follows: + = + M + B AB B = M A B A A A = M A A B B O L B v v + v = M v + ( Now we can define final form of objective fnction for MPC with eact linearized system: J T T ( v = ( A w Q( A w + v Rv min v Sbject to: min ( + α( v D( α( D + min ma ma (6 (7a (7b where Q and R are weight matrices of states and inpts respectively. The bo constraints (7a of original model were transformed into state dependent constraints (7b. Figre illstrates the feasible area of and. It has for corners

5 STABILIZATION OF LONGITUDINAL AIRCRAFT MOTION USING MODEL PREDICTIVE CONTROL AND EXACT LINEARIZATION a,b,c,d. The coordinates of corners are ( min, ma, ( ma, ma, ( ma, min, ( min, min, respectively. procedre. Let s have two points from the previos problem. Fig. constrained transformation Fig. Feasible area of the control inpts original coordinates Corners of original model were step by step transformed in to new coordinates (7b. Figre illstrates the feasible area of virtal inpts v and v after each point in the feasible area of and is mltiplied by matri D and translated by α. Constrained point c, b are transformed into c and b. c = D( c + α( (9 b = D( b + α( Line between new points can be described: v ( c lv ( c = + ( v ( b = + lv ( b where line parameters and l are soltion of two eqations with two nnowns: v ( c v ( b l = v ( c v ( b ( v ( b v ( c v ( b v ( b = v ( c v ( b The constrained line is now defined as: v = + ( lv Final relationship between original and virtal constraints is formlated as follows: ma ( v + lv Fig. Feasible area of the virtal control inpts The new coordinates have the following relationship with the old ones: a = D( a + α( b = D( b + α( c D( c α( (8 = + d = D( d + α( Each point pairs (c, b, (b, a, (a, d, (d, c are conseqently sed for comptation of virtal inpt constraints ( which create complete trapeze constraints for virtal inpts. The comptational process is shown on figre 6. This procedre creates enclosed area for virtal constraints. The feasible area in Figre of the transformed inpts is not sitable for se with the predictive control algorithm. However, the bo constraints of original inpt can be rewritten by following

6 ČELIKOVSKÝ S., HOSPODÁŘ P. Fig. 6 constraints comptational process Case stdy In this section, we present algorithm and simlation reslts sing MATLAB and SIMULINK. We sppose iterative process that is described below. At initialization step we start from steady flight condition ( that satisfies inpt constraints and constant ( for nonlinear simlation ( is compted. MPC process starts with nonlinear state comptation of the prediction horizon. This is open loop soltion of aircraft eqation ( and feedbac linearizing control ( with actal state as initial condition and virtal inpts v on the horizon prediction. Obtained otpt is sed to constrct the constraints of the virtal inpt v at each step of the prediction horizon. Net we se the calclated inpt constrained seqence to generate new virtal inpts (6 sbject to (7b. Constrained conditions are controlled and if they are not satisfactory the MPC process is repeated. After decision of satisfied condition is net simlation step compted. Following algorithm smmarizes the steps of the proposed methodology. Algorithm Initialization Model preparation ( Steady state flight condition ( Simlation Solve open loop aircraft eqation ( with EL( and virtal inpts v MPC Open loop simlation on prediction horizon (( Constraints comptation ( v = MPC soltion (6(7b while constrained condition 6 Open loop comptation on prediction horizon (( 7 Constraints comptation ( 8 v = MPC soltion (6(7b end while end MPC retrn to Simlation ntil t sim < T final end Simlation The performance of this scheme has been verified via simlation. The aircraft is controlled from steady regime and conseqently its flight is distrbed by a wind gst (addition to angle of attac. Character of wind distrbance is shown in Fig. 7 (dash line. air speed [m.s - ] pitch rate [deg.s - ] time [s] pitch angle [deg] Fig. 7 States of aircraft longitdinal motion Wind gst amplitde is two degree and we can see from Fig. 7 that real angle of attac error is smaller than one degree. The error means difference between actal state and reference which is obtained from steady flight soltion. angle of attac [deg] measred otpt wind distrbance. time [s] 6

7 STABILIZATION OF LONGITUDINAL AIRCRAFT MOTION USING MODEL PREDICTIVE CONTROL AND EXACT LINEARIZATION elevator rate deflection [deg.s - ] time [s] v - thrst of engine [N] time [s] Fig. 8 inpts dring wind distrbances v Fig. 9 constrained virtal inpts Previos figre shows constrained inpts. De to dynamic eact linearization is new inpt defined (. Therefore elevator rate deflection as original inpt sed instead of elevator deflection. Both inpts satisfy amplitde limits (elevator rate deflection - [ /s], thrst of engine [N]. Virtal inpts sed for feedbac linearizing control are shown in Fig. 9. There are for lines which correspond to constrained transformation ( and define Simlation time:. [s] feasible control area (filled srface. The cross represents vale of MPC soltion. Figre shows the same problem with the difference that the feasible areas of linearized model on the prediction horizon are shown. On the feasible area of each prediction step the virtal inpt vale is mared by a circle. All vales are inside the feasible region thereby is graphically demonstrated flfillment of constrained condition. inpts v v Fig. feasible areas of linearized model on the prediction horizon horizon prediction 7

8 ČELIKOVSKÝ S., HOSPODÁŘ P. Appendi The longitdinal part of eqations of motion derived from (6, (7 are described in aerodynamic ais system: S F V& T = q ( cd D αα Dδ eδe osα m m + g sin( α θ S q cmac & α = q cl Lα α Lq Lδ δe e Vm V Ft g + q sinα os( α θ mv V ScMAC q cmac q& = q cm m α α mq m δ δ e e I y V & θ = q ( Each constants of nonlinear eqation ( are defined as follows: ρs ρs c = cd c = cd α m m ρs c = cdδe c = m m ρs c = g c6 = cl m ρs ρs c7 = clα c8 = cmacclq m m ( ρs ρs c9 = clδe c = cmaccm m I y ρs ρs c = cmaccmα c = cmaccmq I y I y ρs c = cmaccmδe I Conclsion y [] B. N. Pamadi, Performance, stability dynamics, and control of airplanes, (nd edition, AIAA, [] Etin, B., Dynamics of atmospheric flight (Wiley and Sons, Inc., N.Y. 97 [] Pačes P, Čensý T, Hanzal V, Draler K, Vašo O. A combined angle of attac and angle of sideslip smart probe with twin differential sensor modles and dobled otpt signal. IEEE Sensors Concil,, p ISBN [] J.D. Jr. Anderson, Comptational flid dynamics the basics with applications (McGraw-Hill international edition, 99 [6] V. Klain, E. A. Morelli: Aircraft system identification, AIAA 6 [7] R. V. Jategaonar, Flight vehicle system identification A time domain methodology, AIAA 6 [8] P. Hospodář: Estimation of aerodynamic characteristic sing otpt error method, Czech Aerospace proceedings /. [9] A. Isidori, Nonlinear control system. Commnications and Control Engineering Serie, Springer Verlag, 989 [] J. Miciejowsi, Predictive control with constraints, Prentice Hall, London, Copyright Statement The athors confirm that they, and/or their company or organization, hold copyright on all of the original material inclded in this paper. The athors also confirm that they have obtained permission, from the copyright holder of any third party material inclded in this paper, to pblish it as part of their paper. The athors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the pblication and distribtion of this paper as part of the ICAS proceedings or as individal off-prints from the proceedings. In this wor, a new method was provided in order to apply standard MPC techniqes for eact linearizable system with inpts constraints. An iterative process was designed and at every step the optimization problem involved only linear dynamics and constrints. Ftre wor will investigate the etension of restrictions on the stats and speed inpts. References [] J. R. Raol, J. Singh, Flight mechanics modeling and analysis, CRC Press, 9 8