Nonlinear Adaptive Control Law For ALFLEX Using Dynamic . - ICCAS

ICCAS2005June 2-5, KINTEX, Gyeonggi-Do, KoreaNonlinear Adaptive Control Law for ALFLEXUsing Dynamic Inversion and Disturbance Accommodation Control ObserverDaisaku Higashi* , Yuzo Shimada**, and Kenji Uchiyama**** Graduate Student, **Professor, ***Assistant ProfessorDepartment of Aerospace Engineering, College of Science and Technology, Nihon University7-24-1 Narashinodai, Funabashi, Chiba 274-8501, JAPAN(Phone: 81-47-469-5390; Fax: 81-47-467-9569; E-mail: daisaku0308@h7.dion.ne.jp)Abstract: In this paper, We present a new nonlinear adaptive control law using a disturbance accommodating control (DAC)observer for a Japanese automatic landing flight experiment vehicle called ALFLEX. A future spaceplane must have ability to dealwith greater fluctuations in the stability and control derivatives of flight dynamics, because its flight region is much wider than thatof conventional aircraft. In our previous studies, digital adaptive flight control systems have been developed based on a linear-parameter-varying (LPV) model depending on dynamic pressure, and obtained good simulation results. However, under previous control laws, it is difficult to accommodate uncertainties represented by disturbance and nonlinearity, and to design a stableflight control system. Therefore, in this study, we attempted to design a nonlinear adaptive control law using the DAC Observerand inverse dynamic methods. A good tracking property of the obtained system was confirmed in numerical simulation.Keywords: Guidance and Control, Adaptive Control, Disturbance Accommodating Control Observer, Spaceplane, ALFLEX1. INTRODUCTIONIn this paper, a new nonlinear adaptive control law using adisturbance accommodation control observer is presented for aJapanese automatic landing flight experiment vehicle calledALFLEX. ALFLEX’s flight experiments have been carriedout to demonstrate the overall system technology and evaluateautomatic landing technology for future Japanese spaceplanes.In this study, we focus on the guidance and control law forALFLEX. In particular, the purpose of this paper is to makethe roll rate and yaw rate of the vehicle track their referencesin designing the guidance and control law.In general, a spaceplane encounters greater fluctuations inthe stability and control derivatives of flight dynamics becauseits flight region is much wider than that of conventionalaircraft. Therefore, a conventional linear-time-invariant (LTI)expression is not appropriate in dealing with such a spaceplane’s characteristics, so that the employment of some sort ofadaptive mechanism is required.Fig.1 Automatic Landing Flight Experiment.In our previous studies of ALFLEX’s longitudinal motioncontrol, we developed a digital adaptive flight control systembased on a linear-parameter-varying (LPV) model, in whichunknown parameters were studied as functions of dynamicpressure and the number of adjustment parameters was re-duced as much as possible [1,2]. As a result of 1000 repetitions of the Monte-Carlo evaluation under the initial dispersion conditions, it was observed that ALFLEX successfullytouched down on the runway in almost all cases. Thus, theusefulness of the previous digital adaptive control laws wasconfirmed. However, if we attempt to apply the previouscontrol laws to ALFLEX’s lateral motion, it is difficult to dealwith uncertain terms s u c h a s oupling, disturbance andnonlinearity terms. Therefore, for the control of ALFLEX’slateral motion as a multi-input-multi-output (MIMO) system,the system should possess an estimation mechanism for suchuncertain terms and should be robust against disturbance.Baba, of the National Self-Defense Academy, Japan, et al.developed a flight control method based on inverse dynamicstransformation and singular perturbation theories [3,4,5]. Theobtained guidance and control system has the ability to directly control nonlinear dynamics and to make a vehiclefollow a predefined trajectory. However, a robustness problemfor uncertainties such as disturbance still remains for thismethod.Recently, a theory called the “linear adaptive theory” hasbeen proposed by Johnson, of the University of Alabama,USA. Using this theory with a disturbance accommodatingcontrol (DAC) observer [6,7], a nonlinear controlled systemcan be linearized by constructing feedback loops consisting ofthe terms estimated by the DAC observer and can also beprovided with adaptability, because, the DAC observer estimates the disturbance, uncertain and coupling terms of thecontrolled system. Thus, a MIMO nonlinear system can betransformed into a decoupled linear system, and the asymptotic stability of the system is guaranteed using the subspacestabilization technique. This stabilization technique is forfinding feedback gains such that the state variables of theobtained closed-loop system asymptotically approach asubspace that satisfies prescribed boundedness and stabilityconditions.Therefore, in this study, we attempt to apply the linearadaptive theory and subspace stabilization control techniquewith the DAC observer to a first-time-scale subsystem obtained using inverse dynamics transformation and singularperturbation approaches.Finally, to examine the effectiveness of the proposed system,numerical simulation was performed with ALFLEX’s6-degree-of-freedom nonlinear simulation model.

ICCAS2005June 2-5, KINTEX, Gyeonggi-Do, Korea(d) Equations for vehicle’s positionx& E V cos γ cos χy& E V cos γ sin χz& E V sin γ .NotationsL, T , D, Y : lift, thrust, drag, and side force along eachwind axisl , M , N : moments of roll, pitch and yaw in body-axissystem: mass of vehiclemg: acceleration of gravityx E , y E , z E : c.g. position in Earth-fixed frameIi, j: moment of inertia: total velocity of vehicleVα, βφ , θ ,ψp , q, r: angle of attack and sideslip angle: Euler angles: roll, pitch, and yaw rates in body-axissystem: flight path angle and ground track angle: small parameter in singular perturbationtheoryγ,χε(13)(14)(15)2.2 Dynamic Inversion and Singular PerturbationThe inverse dynamics transformation theory is a techniquethat sequentially cancels a plant’s characteristic by sequentially forming an inversion system (dynamics), and realizes anideal transient response determined arbitrarily by a systemdesigner. On the other hand, the singular perturbation theory isan approach in which a control system is separated intofast-time-scale (FTS), middle-time-scale (MTS) and slowtime-scale (STS) subsystems according to the difference inresponse time between the state variables, and each feedbackloop is constructed according to each time scale. Hence, for aslower subsystem, the state variables of a faster subsystem canbe assumed to become steady values. Thus, in the proposedmethod, the order of the system can be reduced by employingtime scale separation, so that numerous state variables can becontrolled.2. CONTROL LAW FOR ALFLEX2.1 ALFLEX DynamicsOn the assumption of the flat-Earth model, a 6-degreeof-freedom of ALFLEX’s dynamics can be given by thefollowing equations [8].(a) Translational equations of motionV& (T cos α cos β D mAg ) mα& ( L T sin α mBg ) mV cos β(1) q tan β ( p cos α r sin α )&(β Y T cos α cos β mCg ) mV p sin α r cos α(2)(3)Here,A cos α cos β sin θ sin β sin φ cos θ ,(4) sin α cos β cos φ cos θB cos α cos φ cos θ sin α sin θ ,C cos α sin β sin θ cos β sin φ cos θ . sin α sin β cos φ cos θFig.2 Dynamic Inversion and Singular Perturbation(5)(6)(b) Rotational equations of motionε p& 1I xx I zz I xz2ε q& ε r& { I zz l I zx I zz pq I zz (I yy I zz )qr 2 I zx N I zx qr I zx (I xx I yy ) pq()1M I zx p 2 r 2 (I xx I zz ) qrI yy1I xx I zz I xz2} I zx l I pq I zx (I yy I zz )qr I xx I zx qr I xx (I xx I yy ) pq 2zx (7)x [V[Slow-time-scale variables]αβ φ θ ψxEyEzE ] , u T[Fast-time-scale variables]z [ p q r ], w [ δ a δ e δ r ](8)Next, we divide eqs. (1-15) into two groups as shown in(9)(c) Relationship between rates of Euler angles and body-axisangles(10)φ& p (q sin φ r cos φ ) tan θθ& p cos φ r sin φψ& (q sin φ r cos φ ) sec θIn this study, to apply singular perturbation control totrajectory control, we divide state variables in Eqs. (1-15) intoslow- and fast-time-scale variables.(11)(12)x& A1 ( x ) B1 ( x ) z C1 ( x ) u ,ε z& A2 ( x , z ) B2 (z ) C 2 ( x ) w .(16)(17)At this point, to describe STS dynamics, the parameter ε isset to zero. z x& A1 ( x ) [ B1 ( x ) C1 ( x ) ] u 0 A2 ( x , z ) B2 (z ) C 2 ( x ) w .(18)(19)

ICCAS2005June 2-5, KINTEX, Gyeonggi-Do, KoreaIn eq.(18), we considereq.(18) can be written as[zu ] to be inputs such thatTx& K1 ( x x ) K 2 ( x x ) dt.tobserver. To do so, we consider a quadratic spline functionwith respect to time.z p1 f [ x(t )] c1 c2 t c3 t 2(20)(30)0From eqs.(18) and (20),[zThus, eqs.(29) and (30) can be rewritten into the followingforms according to reference [6].u ] is generated usingTt z 1 u [ B1 ( x ) C1 ( x ) ] A1 ( x ) ( x x ) K 2 0 ( x x ) dt . (21)e& p z p1 Lδ a δ a(31a)z& p Γz p(31b)Here,Substituting z from eq.(21) into eq.(19), w can be obtained. However, since this w is obtained by assuming that[z p z p1z p2]z p3Tε 0 , there exists a difference between the actual variablez and w and the FTS variable z and w . Therefore, todecrease both differences, we consider the next system calledFTS.Here, z& A3 ( x , z ) B3 ( z ) C 3 ( x ) w(22) z z z ,(23) w w w .In this system, the FTS control input w can be designed forEq.(22) to maintain z close to zero.3. LINEAR ADAPTIVE CONTROL LAW3.1 Linearization of system using DAC observerAs mentioned above, ALFLEX’s dynamics is treated as anonlinear system including disturbance. As controlled variables, we consider roll rate and yaw rate in the FTS subsystem.Therefore, eqs. (7) and (9) are expressed by the followingnonlinear differential equations.p& mL ( p, r , w) Lδ a δ ar& m N ( p , r , w ) N δ r δ r(24)(25)Here, the first term represents the coupling, uncertain anddisturbance terms and the second term represents the controlinput term. Since eqs.(24) and (25) have similar forms, henceforth we will consider to design an observer for the firstequation eq. (24).First, we define the error in roll rate ase p p p .(26)Differentiating this, the differential equation with respect toroll rate error is obtained.(27)e& p p& p& p& mL ( p, r , w) Lδ a δ aHere, p represents the predefined reference roll rate. Denoting the first and second uncertain terms as(28)z p1 p& mL ( p, r , w) ,eq. (27) can be rewritten ase& p z p1 Lδ a δ a .(29)Here, to estimate the unknown term z p1 , we employ the DAC 0 1 0 , Γ 0 0 1 . 0 0 0 (31c)As an observer for eqs.(31a) and (31b), we introduceeˆ& p zˆ p1 Lδ a δ a k1 ( e p eˆ p ),zˆ& p Γzˆ p k 2 4 ( e p eˆ p ).(32a)(32b)Subtracting eqs.(32a) and (32b) from eqs.(31a) and (31b),respectively, and combining the results, the following DACobserver is obtained. e& p eˆ& p k1 1 0 0 e p eˆ p z& zˆ& k2 4Γ z p zˆ p pp (33) k1 1 0 0 k 0 1 0 e p eˆ p 2 k 0 0 1 z p zˆ p 3 k 0 o 0 4 If observer gain k [ k1 k2 k3 k4 ]T is selected so that theabove equation becomes stable, [ eˆ&zˆ& ] approachesp[ e& ppz& p ] with time.Next, to design the tracking system for the roll rate, as astate variable, we consider the integral of tracking error,tracking error and its derivative.x [ x1x2x3 ] e p dτ 0tTTepe& p , (34)As an actuator of the vehicle, we assume the followingfirst-order lag element with the time constant T [s].1T1Tδ&a δ a u p(35)Differentiating eq.(34) and using eq.(35), the state differentialequation becomes 0 x1 0 1 0 x1 0 d 0 up.0x2 0 0 1 x2 z p1 L dt 1 x3 0 0 x3 z p 2 δ a T T T (36)To simplify the problem, we divide the total input into twoparts.(37)u p u p, h u p ,eHere, u p ,h is a control so that the second term in eq.(36) iscancelled out.