AIM-90-3371-CP Multivariable Control Law for Flat-Turn Strafing Maneuver by A Supermaneuverable Aircraft Chien Y. Huang Coprate Research Center Grumman Corporation, MS AO8-35 Bethpage, NY 11714 (PIIMF) control law that allows an advanced fighter aircraft to perform flat-turn sWing maneuver. Flat-turn strafing is used in terrain-following scenario to quickly acquire, track, and maybe destroy targets on the ground. This is a difficult task since it requires an airplane to yaw from left to right (and back) without gaining side velocity (i.e., with zero sideslip) while keeping the nose in level position. Side velocity (lateral acceleration) can be avoided by rolling the airplane such that the lift vector is tilted to cancel the side force, however, doing so leads to rapid banking maneuvers, which compromises pilot's ability to keep track of the target. An equally desirable companion maneuver is to be able to command side-velocity alone without yawing the airplane. Implicit model-following technique has been very successfully applied to the control of aircraft [7] and has been demonstrated to possess inherent reconfiguration properties [8]. It also has been shown to have good robustness against sensor noise and plant disturbance and reformulated for attitude pointing purpose [9]. Recently there is study to incorporate the model-following technique into the HOOrobust control framework [lo]. The organization of this paper is as follows. Section I1 formulates the control strategy and examines some of the properties. Additional robustness using loop transfer recovery is also described. Section I11 details the target supermaneuverable aircraft while Section IV presents the simulation results. Comments on additional control power involved and complexity of control algorithm are also made. Summary and conclusions are offered in Section V. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 ABSTRACT A control law that allows a supermaneuverable aircraft to perform flat-turn strafing maneuver is described. It is based on proportional-integral model-following concepts, which has inherent advantage of quickily obtaining a design that has satisfactory performance. The control law is applied to an advanced fighter aircraft and simulation results show that flat-turn strafing maneuvers are possible with the use of the model-following control law. I. Introduction Future aircraft are expected to meet higher performance and combat supeliority requirements, which has been characterized as supermaneuverability [ 11. Superagility or supermaneuverabilityindicates an aircraft's capability to perform maneuvers in post-stall or highangle-of-attack (or high-alpha) flight regimes or in situations where there is substantial cross-coupling between the modes. In the case of high-alpha maneuvers, the problems are loss of control effectiveness and nonlinear dynamics, which are being addressed by thrust vectoring and specialized control strategies. In the case of cross-coupled maneuvers, the problem is absence of dynamic modes separation, which requires control surfaces to be simultaneously deflected in an unusual combination and setting. Performing unconventional maneuvers in otherwise conventional flight regimes is a difficult task for a pilot, who is accustomed to aircraft that are controlled as single-input single-output (SISO)systems. Moreover, he may be hard-pressed to keep up with the associated fast-changing dynamics, resulting in a highworkload environment. All these factors point to the need of control laws that are truly multivariable in nature to carry out these multi-input multi-output. Yet there is additional handling quality requirement on the control law to achieve an apparent "feel" of a SISO system. Studies on controls for supermaneuverable aircraft have concentrated mostly in high-alpha vehicles using strategies such as nonlinear control [2], variable gain control [3], and robust control using loop-transfer recovery [41 and H=.optimization techniques [51. They are relatively few studies on highly dynamic maneuvers, one being a study of using nonlinear control to rapidly change attitudes [ 6 ] . The objective of this paper is to apply proportional-integral implicit model-following 11. Control Law Formulation The proportional-integral model-following (PIIMF) control law is reviewed here. For more details, refer to [8]. Let the plant be described by the usual (A, B, C, D) matrices and the model to be described by system matrix Am, namely Y = Ax+Bu = CX+DU Xm = Amxm X Presented at 1990 AIAA GNC Conference, Copyright Q 199Oby Grumman Corp. Published by AIAA with permission 499 (3) Implicit model-following incorporates the model in the quadratic cost function by weighting the difference between the actual and model state rates, that is, The state variable is augmented to include 6, JT Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 = 1/2 [X T T Q x + ~ xM U + U Ruldt (3 0 T T where Q = (A-Am) Qi(A-Am), M = (A-Am) QiB, and the cost function becomes T R = B QiB, and Qi is a weighting matrix. It can be shown that if B is square and invertible, then perfect model matching can always be achieved. If fact, the necessary feedback gain K is a simple algebraic relation given by K = B CT J = 1/2 T T [AXa Z A X ~ + ~ ASAU+AU X~ RAuIdt (14) 0 - 1 (A-Am) weighting on 6. The (linear quadratic) control obtained by optimizing Eq. 11 is with this gain, the closed-loop system becomes Ac = A - B K = Am 0 Typically B is not square; in this case the feedback giin wili depend on the structure of the control matrix and the weighting matrix Qi, Expandingand reatrangingEq. 15, we get T -1 T where BwL = (B QiB) B Q; and is the weighted left pseudo-inverse of B, while P is the solution of the Riccati equation associated with optimization of Eq. 5. Although the PIIMF control law possesses inherent sensitivity to plant parameter variations, we can improve its low-frequency disturbance-rejection characteristicsand provide zero steady-state error command response by adding proportional-integra1 compensation. Let Yd be the desired equilibfiumoutputs; then the perturbations around the POint (XO, UO) are: The second term on the right side provides proportional feedback of the error, while the third term provides the integral effect being sought. The dimension of the commanded input yd is chosen to equal the dimension of u, reflecting the fact that the number of controllable outputs is equal to the number of the controls available [12]. Eq. 16 is cast into incremental form in actual implementation to eliminate trim requirements and measurement bias. PIIMF control law as it stands if robust against parameter variations; however, its robustness against unmodeled dynamics is not well-known. Recently, the implicit model-following control law has been adapted to take incorporate loop-transfer recovery (LTR)[lo]. The objective of LTR is to force the loop-transfer function G K , where G(s) = C(sI-A)-lB and K is the feedback gain (in the forward loop), to a desirable shape. The steps involved in deriving K(s) are the design of a Kalman-Bucy filter, which in essence is the desired loop shape, and then Define output error integral 4 to be 500 CRCA is a generic advanced fighter aircraft intended for, among other things, studying the effects of damages and benefits of control reconfiguration. The aircraft is statically unstable (-12% mean aerodynamic chord) to achieve good transonic performance and to reduce supersonic trim drag. Thrust vectoring is available but not used here. Nine control surfaces exist. There are two canards with 30Qdihedral angle. The dihedral creates directional instability, but it provides an additional yaw power which is useful for control reconfiguration; and for our case, it allows further performance enhancement in lateral-directional axis. CRCA also has six trailing-edge ("E) surfaces, which are independently controlled to achieve both roll and pitch. Finally, a single rudder provides further directional control. The linear model used is obtained by perturbation using nonlinear CRCA model in terrain-following terrain avoidance (TFI'A) mode (altitude = sea level, Mach number = 0.9). Only the lateral-directional model is needed here as we use only anti-symmetric control. The state vector x of the model consists of sideslip angle (p. rad), roll rate (p, radsec), yaw rate (r, rad/sec), roll an le Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 the design of a Linear Quadratic regulator, which "recovers" the loop shape. Model-followingLTR will assume that the model transfer function (Cm(S) = C(s1A m j l B ) , modified by any robustness considerations, is the desirable loop shape, which if achieved, will naturally lead to robust model-following. This procedure is briefly described below. The first step is to design a Kalman-Bucy filter (i.e., find the filter gain Kf), so that its transfer function, -1 Gkf = C(sI-Am) K f , is Gm. The standard LTR approach calls for finding fictitious noise and disturbance matrices to make the equality true. Although this process can be carried out, it is by no means straight forward. Instead, the easiest way to obtain a filter gain Kf, which must be equal to B, is to simply assume that it is given. That is, if augment the plant = [;: A",] 5. (Q,rad), and yaw angle (w,rad); that is, x = [P p r Q w] The control surfaces (Fig. 1) are left and right canards (Cl, Cr, deg), left and right trailing-edges-#l (Tell, Telr, deg), left and right trailing-edge-#2 (Te21, Te2r. deg), and left and right elevators (El, Er, deg), and rudder (R, deg). These surfaces are further geared into three groups: canards, trailing-edge surfaces, and rudder. That is 6 = [Cn T Te R] . With these vectors defined, the system and control matrices are found to be, and set the KaIman-Bucy filter gain to be Kf = [-I Then -0.2538 -66.9342 8.2821 0 0 Gkf = Ca(sI-Ad"Kf C(SI-Am)-'B The second step is to "recover" the loop shape obtained by the formulation of Kalman-Bucy filter via a linear quadratic regulator. This is done by first setting T state weighting Q (e.g., to C aCa> and control weighting R (typically to 91) and decreasing the conrrol weighting. As p -+ 0, GK approaches Gm. The approach outlined above can also be extended to Hco formulations. Details can be found in [lo]. Although both LTR and Hoo model-following design methods are viable, their properties are not wellunderstood. They will be further examined in a future paper. We will confine here to use PIIMF to achieve the flat-turn maneuvers. HI. B = [ 0.0155 -5.4612 -0.0299 1.OOOO 0 -0.9990 0.0320 0 1.3049 0 -1.2709 0 0.0155 0 0 1.0000 0 0 0.0024 0.0020 0.0036 0.2336 2.0074 0.3737 0.lr 0 . r -0.y] (23) Note that very little power is available to impart side-force 0)despite canted canards and a large rudder. The Supermaneuverable Aircraft Model The proportional-integra1 implicit modelfollowing control law is applied to ControlReconfigurable Combat Aircraft (CRCA) [I31 (Fig. 1). 50 1 IV. Simulation Results requires simultaneous control surface movements, which can only be achieved with a truly multivariable control law. Proportional-integral model-following approach was used due to its formulation, which permits satisfactory performance to be quickly obtained. Simulation results show that flat-turn strafing maneuvers are possible with the use of the model-following control law. Several comments are in order. Simulations shown that flat-turn strafing maneuvers require considerable amount of control power. This is true for our aircraft since both canards and rudder produce sideslip as by product of commanding yaw and vice versa. Therefore, more surface deflections are needed to achieve necessary cancellations. With a different aircraft layout, it may be possible to reduce considerably the required control effort (for example, if a pure side-force control or thrust vectoring is available). This may be a design requirement for supermaneuverable aircraft. No attempt has been made to optimize further the design. One area for improvement is the model itself, which was not derived with modem aircraft in mind. One may accept less stability for performance gains. Finally, the control law appears to work rather well for linear models. It remains to be seen if this is true when it is tested in a nonlinear simulation. The model to be followed is given by Am =[ -0.7430 -1.oooO 0.0586 0 0 5.8700 0.0400 -0.5070 0 0 1.oooo 0 0 0 0 0 1.Oooo 0 0 0 -1O.oooO -4.oooO 0.8650 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 Because there are three inputs, up to three controls can be specified. The natural choice in this case is yd = [p 4 yIT, which means the observation matrix is To achieve an acceptable response, following weighting matrices are used: Qi = O.l*I5 Qg = diag(200,50,10) (26) Output responses and control efforts for a T command of 0.2 rad (1 1.5 deg) yaw (Le., yd = [O 0 0.21 are shown in Figs. 2 and 3. As can be seen, the control power required is large. In fact, rudder deflection is limited to +3@ (while the canards are limited to +60Qand -30*). To obtain more reasonable control deflections, the response must be slowed down. This can be accomplished by changing the integral weighting matrix Qc. In fact, if Qg is set to be diag(200,50,0.05), then the control effort will be within the surface limits, as shown in Figs. 4 and 5. Alternatively, we can command pure sideslip maneuvers without acquiring yaw and roll. Output responses for a command of 0.2 rad (11.5 deg) sideslip T (i.e., yd = [0.2 0 01 are shown in Figs. 6. In this case, it is difficult to null out roll and yaw completely. However, the maximum roll and yaw angles are no larger than 0.5 degrees. The control effort (not shown) for this case is also excessive. However, this can be reduced by lowering the desired sideslip angle (0.2 rad of p for this flight envelope is roughly equivalent to 220 fvsec). A final simulation showing a series of flat-turn maneuvers is shown in Fig. 7. In this scenario, yaw angles of 0.2 rad is commanded for 5 sec before switching to -0.2 rad. As shown by the plots, the command following is quite good. V. REFERENCES 1. Herbst, W. B., "Future Fighter Technologies," J. of Aircraft, Vol. 17, No. 8, Aug. 1980. 2. Garrard, William, Enns, Dale F., Snell, S. Antony, "Nonlinear Longitudinal Control of a Supermaneuverable Aircraft," Proc. of ACC, Pittsburgh, PA, June, 1989. 3. Ostroff, Aaron, "A Variable-Gain Output Feedback for High-Alpha Control," AIAA Guidance, Navigation, and Control Conference, Paper 89-3576CP, Boston, MA, Aug., 1989. 4. Looze, Douglas, and Freudenberg, James, "Linear Quadratic Design for Robust Performance of a Highly Maneuverable Aircraft," AIAA Guidance,Navigation, and Control Conference, AIAA paper 89-3457-CP, Boston, MA, Aug.. 1989. 5. Voulgaris, Petros, and Valavani, Lena, "High Performance Linear Quadratic and Hoo Designs for a 'Supermaneuverable' Aircraft," AIAA Guidance, Navigation, and Control Conference)AIAA paper 893456-CP, Boston, MA, Aug., 1989. 6 . Snell, S. Antony, Enns. Dale F., Garrard, William, "Nonlinear Control of a SupermaneuverableAircraft," AIAA Guidance, Navigation, and Control Conference, AIAA paper 89-3486-CP, Boston, MA, Aug., 1989. 7. Kreindler, E., and Rothschild, D., "Model-Following in Linear Quadratic Optimization," AIM Journal, Vol. 14, no. 7, July 1976, pp. 835-842. 8. Huang, Chien.Y., and Stengel, Robert F., "Restructurable Control Using Proportional-Integral Summary and Conclusions A control law that allows a supermaneuverable aircraft to perform flat-turn strafing maneuver is described. The maneuver is difficult as it 502 Iinplicit Model-Following," J . of Guidance. Control, and Navigarion, Vo1.3, No. 2. March-April, 1990. 9. Iluang. Chien Y.,"A Methodology for KnowleclgeBased Restructurable Control to Accorriinodate System Failures," P h . D . Thesis. Princeton Univcrsity, 1989. 10. Iluang, Ctiien Y.,"Application of Robust MotlclFollowing Concepls to Aircraft Control," Proc. of ACC. San Diego, CA. May, 1990. 11. Asseo. S . . "Applicalion of Optinial Control lo Perfect Model-Following." J . o/Aircra/r. Vol. 7, No. - 313. 12. Stengel, R., Stochastic Optimal Conrrol: Theory and Application. John Wiley & Sons, Inc., New York. 1986. 13. Mcrcadaiite, et at., "Control Recnnfigurable Combal Aircraft Piloted Simulation Development," Proc. o/ NAECON, Dayton, 011, May, 1988. pp. 512 - 519. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 4, July-Ailg., 1970. pp. 308 Fig. 2 Conlrol Reconfigurable Combal Alrcrafl (CRCA) 503 response to yaw command 12 10 8 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 6 4 2 0 -2 ( 1 2 3 4 5 6 7 time, sec Figure 2 Output responses to 0.2 rad yaw command response to yaw command 6 time, sec Figure 3 surface response to 0.2 rad yaw command 504 7 response to yaw command 15 10 - M Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 4 +T2 .r( 5- d 8 0 I 0 1 2 3 4 5 6 7 time, sec Figure 4 output response to 0.2 rad yaw command with less control power time, sec Figure 5 surface response to 0.2 rad yaw command with less control power 505 - - Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1990-3371 - time, sec Figure 6 Output responses to 0.2 rad sideslip command response to flat-turn yaw strafing maneuver 15 1C 5 3 a $ 0 c, .r( d 8 -5 -10 -15 0 5 15 10 20 time, sec Figure 7 Output responses to 0.2 flat-turn strafing maneuver 506 25

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