International Journal of Control, Automation, and Systems Vol., No. 3, September 003 35 Airraft CAS Design with Input Saturation sing Dynami Model Inversion Sangsoo Lim and Byoung Soo Kim Abstrat: This paper presents a ontrol augmentation system (CAS) based on the dynami model inversion (DMI) arhiteture for a highly maneuverable airraft. In the appliation of DMI not treating atuator dynamis, signifiant instabilities arise due to limitations on the airraft inputs, suh as atuator time delay based on dynamis and atuator displaement limit. Atuator input saturation usually ours during high angles of attak maneuvering in low dynami pressure onditions. The pseudo-ontrol hedging (PCH) algorithm is applied to prevent or delay the instability of the CAS due to a slow atuator or ourrene of atuator saturation. The performane of the proposed CAS with PCH arhiteture is demonstrated through a nonlinear flight simulation. Keywords: Stability and ontrol, handling qualities, guidane and ontrol.. INTRODCTION Flight ontrol system design for fighter airraft ontinues to be one of the most demanding problems in the world of automati ontrol. The magnitude of the problem is driven by the nonlinear and unertain nature of airraft dynamis. Linear models of these systems are only valid for small regions in trim onditions. The onventional flight ontrol designs for this problem were to perform point designs for a large set of trim onditions and then onstrut a gain shedule by interpolating gains with respet to flight onditions. This proedure is time onsuming and expensive, but is well aepted and has yielded satisfatory results for many airraft. A more mathematial response to the issue of gain sheduling had appeared to overome these disadvantages. In referene, the systemati approah for automating gain shedule alulations has been desribed. This approah guarantees both stability and performane. An alternative to gain sheduling is to use ontrol design methods that diretly onsider the nonlinear nature of the problem. One suh alternative method is dynami model inversion (DMI). DMI has been applied suessfully to a number of flight ontrol problems [-4]. In this paper, a CAS based on DMI for high per- Manusript reeived September 6, 00; aepted Marh, 003. This work was partially supported by the Brain Korea Projet. Sang-Soo Lim is with the Shool of Mehanial and Aerospae Engineering, Gyeong-sang National niversity, Korea (e-mail: ssslim@intizen.om). Byoung Soo Kim is with the Shool of Mehanial and Aerospae Engineering, Gyeong-sang National niversity, Korea, Researher of ReCAPT (e-mail: bskim@gsnu.a.kr). formane airrafts is designed to enable the pilot to diretly ommand the angle of attak (or normal aeleration) and roll rate while ensuring the stability of the airframe [5,6]. In the appliation of DMI, signifiant problems of instability arise due to limitations on the airraft inputs, suh as atuator time delay based on slow dynamis and input saturation based on atuator displaement limit. These unstable AOA responses of the CAS whih remain untreated in the DMI method have been generated and reently developed by pseudo ontrol hedging (PCH) methodology in referenes [7,8], and is employed to protet the system from unstable response in the presene of slow atuation, atuator saturation and failures. This paper is organized as follows. Setion presents a design approah for α-cas based on DMI. Setion 3 outlines the appliation of the PCH algorithm to the proposed CAS. The evaluation of the CAS with PCH through the simulation is summarized in Setion 4. Conlusions are given in Setion 5.. DMI BASED α-cas Fig. depits the arhiteture of a ommand augmentation system (CAS) design based on DMI for a high performane airraft. The CAS is also required to stabilize the airframe while following normal aeleration (a n ) ommand and roll rate (P) ommand from the pilot. The CAS is responsible for maintaining zero angle of sideslip (or side aeleration) during maneuvers. The CAS onsists of two subsystems, the attitude orientation system and ommand augmentation logi. The former is an inner stabilization loop responsible for stabilizing the airframe while following the ommanded body rates. The latter ats as an outer-loop for
36 International Journal of Control, Automation, and Systems Vol., No. 3, September 003 traking pilot ommands. Referenes 5 and 6 detail the design of the ommand augmentation logi, the ommand transformation from body rates to Euler angle rates, and the model inversion. In the CAS of Fig., the angle of attak (α) or pith rate (Q) ommand an be used instead of a n -ommand aording to the region of flight envelope or onfiguration hange. In this paper, the fous is on a high-α maneuver therefore, the α-cas following the α-ommand (rather than the a n -ommand) is desribed. The α ommand augmentation logi produing pith rate ommand, Q from a ommanded angle of attak, α of the pilot is based on the expression of (). W + VP Q = an g0 + g osφ os, () where a n is the aeleration along the normal diretion of the reverse Z-axis of the airraft body axes. By approximating that W α, α an be expressed from () as follows: α = ( Q VP ang0 + g osφ os ). () Designate the right hand sides of () with pseudoontrol variableσ, i.e., α = σ. (3) Proportional plus integral ontrol laws of the form will be used to follow the ommanded AOA as in the following (4). α α t p i 0 σ = K ( α α) + K ( α α) d τ. (4) The ontroller gains ( K pα and K iα ) are to be seleted on the basis of the speed of AOA response and traking error. The required pith rate to follow a speified α-ommand an be derived from () using (3) and (4). P n z a y Q = Kpα( α α) + Kiα ( α α) dτ + ( VP + an g g 0 os φ os ). Command Augmentation Logi & Turn Coordination Command Transformation S Attitude Control φ,, ψ P, Q, R, φ,, a, a,, V W y z, 3 Fig.. Command augmentation system. Inverse Model sing Neural Network φ,, ψ, φ,, ψ (5) Airraft α ( VP + a g 0 gosφ os ) n ( VP + a g 0 gosφ os ) Pith Attitude K Q Q α iα Dynamis and Kpα + + α s Orientation S System Fig.. Outer-loop of α-cas for stability analysis. It is usual to design the inner-loop dynamis of the attitude orientation system to be muh faster than that of the outer-loop ommand augmentation traking loop. Moreover it is assumed that the ommanded body angular rates are exatly followed. These assumptions within the bounds of an infinitely fast and stable inner-loop (i.e., a perfet inner-loop is assumed) makes outer-loop analysis possible. Fig. represents the outer-loop of the α-cas by assuming that the inner-loop is muh faster than the outer-loop and thus Q =Q. From Fig., we an obtain the losed-loop transfer funtion as α() s Kpαs+ Kiα = α () s s + Kpαs+ K. (6) iα In this study, a % settling time of the inner-loop ( t Qs ) is designed 3 times faster than that of the outerloop ( t αs ). The ontroller gains are seleted to satisfy the Handling Qualities Speifiation [9]. That is, t α s =.4 se and t Qs = 0.8 se orrespond to the short period natural frequeny and damping ratio of.7 (rad /se) and 0.707, respetively. As a result, ontroller gains are hosen as K pα =3.83 (se - ), iα K =7.35 (se - ). (7) The only pith axis is treated from following [6] beause this paper is onerned with AOA ommand augmentation system and AOA input-output. The attitude orientation system provides stabilization of the airframe and a traking funtion of the attitude rates. In order to enable the use of DMI, the vehile attitude rate ommands in pith axes must be derived as follows: = Q osφ R sinφ. (8) This is the Euler angle rate equation. The Q is the derived input of the attitude orientation system in (5) and it is transformed into using (8) to make the omposition of () possible (Fig. 4). The attitude orientation system will be designed to trak these ommands with as little error as possible. Fig. ontains the attitude orientation system (i.e., the inner stabilization loop of the ommand augmentation system). Note that integrating attitude ommands are n
International Journal of Control, Automation, and Systems Vol., No. 3, September 003 37 onsistent with the Euler angle rate ommand. Next, the Euler pith angle rate equations are differentiated with respet to time yielding. = ( Q Rφ )os φ ( Qφ + R )sinφ. (9) The moment equation [0] an next be substituted in these expressions when the vehile pith rate on the right-hand side is eliminated. If this step is arried out, the seond-order nonlinear differential equation for the Euler pith angle an be obtained as = F + G + G + G. (0) 4 a 5 e 6 r Development of the attitude orientation system is based on the seond-order nonlinear differential equation (0). These equations an be transformed to a linear, time-invariant form by designating the righthand side of this equation by pseudo-ontrol variable ν. Proportional plus derivative ontrol laws an next be designed for eah of these systems as ν = k ( ) + ( ) + p k d. () is negleted during modeling beause it is a negligible value. (0) is transformed into (), a set of three moment oeffiient nonlinear equations [6], using the relationship in Fig. 3. These transformations enable us to apply the DMI proess. In this paper only pith axes are desribed but () is used as a set of three equations in order to make inversion proess desription possible. CL = F+ G a _ md + G e _ md + G3 r _ md, CM = F + G4 a _ md + G5 e _ md + G6 r _ md, () C = F + G + G + G. N 3 7 a _ md 8 e _ md 9 r _ md The variables F, F, F 3, G, G,,G 9 denote nonlinear state dependent funtions. Based on the above relationship, the nonlinear DMI beomes the relation of moment oeffiients ( CL, CM, C N ) to ontrol surfae defletions ( a_ md, e _ md, r _ md ) as follows: _ md G G G3 CL F a _ md G4 G5 G 6 CM F e = G _ md 7 G8 G9 CN F3. (3) r It is diffiult to prove mathematially that the G ij s are invertible beause every airraft has a different Gij value respetively. However, we an figure out that there is a : orrespondene relationship that exists between ontrol surfae ( e, a, r ) and atual CL, CM, C N. The e, a, r whih have dominant effets along pith, roll and yaw axes relating to CL, CM, C N, bring G into existene. The speifi proess for DMI used in this paper is shown in Fig. 3. The seond-order nonlinear equations for the Euler angles are transformed to a linear, time invariant form of the three pseudo-ontrol variables ( ν, ν, ν 3 ). Then, kinemati relations beome moment equations (L, M, N). Next, moment equations are transformed to moment oeffiient equations. The last step in this proess is the ommanded ontrol input derivation proess, whih is the major newly designed logi in this paper. A detailed blok diagram of the pith pseudoontrol loop that ontrols stability and a part of the attitude orientation system is given in Fig. 4. This blok diagram an be redued to yield the pith pseudo-ontrol loop transfer funtion whih is a standard form as () s k s+ k d p = () s s + kds+ kp. (4) It an trak a step and a ramp pith rate ommands with zero steady state error. The feedbak gains k p, k d an be related to the natural frequeny ω n and the damping ratio ς as k = ω n, k =ςω n. (5) p 3. PSEDO-CONTROL HEDGING ARCHITECTRE Input saturation presents a signifiant problem for ontrol. It also implies ontrollability and invertibility issues during saturation. This temporary loss of ontrol effetiveness violates the neessary onditions for φ = ν = ν ψ = ν 3 Pseudo Control input to Kinemati Relations P Q R Fig. 3. DMI proess. Q Kineti Relations to Moments L M N d Moments to Moment Coeffiients K d C C C L M N Final Goal of the Model Inversion + K p + + Fig. 4. Pith orientation pseudo ontrol loop. ν a _ md e _ md r _ md
38 International Journal of Control, Automation, and Systems Vol., No. 3, September 003 DMI. Avoiding saturation is important for systems where the open-loop plants are highly unstable, but otherwise introdues onservatism. The method desribed in this setion is Pseudo- Control Hedging (PCH) for the CAS based on DMI. The oneptual desription of this method is that the PCH signal subtrats the over ontrolled atuator signal bakwards (hedged). This over ontrolled signal is originated based on input saturation or time-delay in the atuator. The purpose of the PCH is to prevent the airraft from entering unstable flight onditions while the input is saturated or delayed. Thus, the PCH algorithm is applied only to the attitude orientation system responsible for inner-loop stability of the CAS. The longitudinal α-cas with PCH is illustrated in Fig. 5. Where f denotes transformation of ommanded pseudo ontrol input (ν ) to Proportional plus Derivative ontrol logi. An approximate dynami inversion element is developed to determine atuator ommands of the form ˆ (,, ) md = f x x ν, (6) where ν is the pseudo-ontrol signal and represents a desired that is expeted to be approximately ahieved by md. This DMI blok is designed without onsideration of the atuator model. This ommand ( md ) will not neessarily equal the atual ontrol ( ) due to the atuator. The PCH signal (ν h ) is omposed of the differene between the ommanded pseudo-ontrol input (ν ) and the ahieved pseudo-ontrol input ( ν ˆ ). The ommanded pseudo ontrol input (ν ) of the output of the PD ontroller is expressed as follows: ν = k ( ) + k ( ). (7) p d To obtain the estimated atual pseudo ontrol input ( ν ˆ ), reverse alulations from atual elevator defletion angle input () to the pith angular aeleration ( ) are expressed as follows: νˆ = = fˆ ( x, x, ). (8) Therefore, the PCH signal (ν h = ν νˆ ) an be written as follows: ν = ν νˆ = f( x, x, ) fˆ ( x, x, ˆ ). (9) h md This PCH signal should be subtrated from the ommand signal of beause the pseudo ontrol signal orresponds to the angular aeleration. However, the attitude orientation system in Fig. 5 uses as the input instead of beause this value is negligible. Thus, after integrating this PCH signal (ν h ), it was subtrated (hedged) from. 4. SIMLATION RESLTS AND DISCSSION The performane of the α-cas using the DMI desribed in previous setions was analyzed through simulation. The F-6 model in [0] was hosen as a mathematial model in the simulation. The simulation fouses only on the longitudinal airraft motion. 4.. The influene of slow atuator dynamis The CAS using DMI is very sensitive to the variane of the atuator dynamis sine the atuator dynamis is not onsidered in the DMI method. To show this, the stabilizer atuator dynamis was modeled as a first order system and two time onstants in the atuator model were tested. One (0.0 se) orresponds to fast atuator dynamis, and the other (0. se) represents slow atuator dynamis. In this simulation an α-ommand (α = 5 deg) was given to the CAS from the level flight trim ondition of Mah 0.55 at an altitude of 0,000 ft. The angle of attak response of the airraft without the PCH arhiteture is given in Fig. 6. Fig. 6(a) and 6(b) represent the α-responses with fast and slow atuators, respetively. As shown in Fig. 6, the stability of the CAS using DMI beomes degraded as the atuator dynamis slows down. α Comand Augmentation and Command Transformation Logi ν h f s ν md DMI Atuator Airraft νˆ Q M α Fig. 5. DMI based α-cas with PCH ompensation. Fig. 6. Effets of slow or fast atuator dynamis.
International Journal of Control, Automation, and Systems Vol., No. 3, September 003 39 Fig. 7 depits the α-response with the ompensation of the PCH result in Fig. 6(b). The ompensation of the PCH makes the CAS stable despite the slow atuator, as shown in Fig. 7. 4.. Influene of atuator saturation Atuator saturation or ontrol surfae defletion limit usually ours in a high α maneuver in a low dynami pressure region. In the simulation in Fig. 8, a high α-ommand (α = 0 deg) was given to the CAS from the low dynami pressure trim ondition of Mah 0.33 and altitude of 5,000 ft. As we expeted, the stabilizer defletion was saturated with this high α-ommand input by the pilot. Fig. 8(a) without PCH shows that the system is extremely unstable due to atuator saturation, while Fig. 8(b) with PCH illustrates that PCH works well even despite atuator saturation. Fig. 9(a) displays the stabilizer angle orresponding to Fig. 8(a). It is saturated to the displaement limit for a large time period and then demonstrates a bang-bang type response, indiating instability. However, Fig. 9(b) represents the stabilizer angle response with PCH, orresponding to Fig. 8(b). From the PCH signal of Fig. 9() and the stabilizer defletions in Fig. 8, it is known that some amount of exessive ommand input generated in the ontrol loops is subtrated by the PCH signal to prevent or delay the instability of the CAS due to a slow atuator or ourrene of atuator saturation. 6. CONCLSION Time (se) Fig. 7. Compensation of PCH for slow atuator. Time (se) Fig. 8. Effet of PCH in ase of atuator saturation. Fig. 9. Stabilizer defletion response with or without PCH. The CAS using DMI has advantages ompared to a linear system based CAS, for example, it does not need gain sheduling and is very robust on airraft onsidering gravity variation. However, this DMI based CAS makes the losed-loop system very unstable with slow atuator dynamis or in the instane of atuator saturation. In this paper the struture of the α- CAS using DMI is desribed and the arhiteture of the PCH is suggested to ompensate for the instability tendeny related to the atuator. Through the simulation it is demonstrated that the PCH applied to the CAS is very effetive to ompensate for the slow atuator dynamis or atuator saturation. REFERENCES [] R. Eberhardt and K. A. Wise, Automated Gain Shedules for Missile Autopilots sing Robustness Theory, Pro. 99 AIAA Aerospae Siene onferene. [] D. Enns, Robustness of Dynami Inversion vs. µ synthesis: Lateral-Diretional Flight Control Example, Pro. 990 AIAA Guidane, Navigation, Contr. Conf., August 990. [3] S. H. Lane and R. F. Stingel, Flight Control Design sing Nonlinear Inverse Dynamis, Automatia, vol. 4, pp. 47-483, 988. [4] C. Huang. et al., Analysis and Simulation of a Nonlinear Control Strategy for High Angle of Attak Maneuvers, Pro. 990 AIAA Guidane, Navigation, Control Conf., Portland, OR, August 990. [5] P. K. A. Menon, Nonlinear Command Augmentation System for a High Performane Airraft, Pro. of the AIAA Guidane, Navigation, and Control Conferene, Monteray, CA, vol. I, pp. 70-730, August 993. [6] B. S. Kim and A. J. Calise, Nonlinear Flight Control sing Neural Networks, Journal of
30 International Journal of Control, Automation, and Systems Vol., No. 3, September 003 Guidane, Control, and Dynamis, vol. 0, no., pp. 6-33, 997. [7] E, Johnson and A. J. Calise, Neural Network Adaptive Control of Systems with Input Saturation, Pro. of the Amerian Control Conferene, 00. [8] M. Idan, M. Johnson, and A. J. Calise, A Hierarhial Approah to Adaptive Control for Improved Flight Safety, Pro. of the AIAA Guidane, Navigation, and Control Conferene, 00. [9] D. MLean, Automati Flight Control Systems, Prentie-Hall, In., Englewood Cliffs, NJ, pp. 5-65, 988. [0] B. Stevens and F. Lewis, Airraft Control and Simulation, John Wiley & Sons, In., NY, pp. 80-, 99. Sang-Soo Lim is a Ph.D. andidate at the Shool of Mehanial and Aerospae Engineering of Gyeong-sang National niversity. He is a flight test pilot in the Korean Air Fore. His areas of interest inlude airraft flight ontrol law design, airraft testing and evaluation. Byoung Soo Kim reeived his Ph.D. degree from the Shool of Aerospae Engineering of Georgia Teh in.s.a. in 994. His researh area is flight ontrol system design for manned /unmanned airraft and algorithm development of neural network based adaptive ontrols.
International Journal of Control, Automation, and Systems Vol., No. 3, September 003 3