Chien Huang, James Tylock, Paul Martorella, and Gareth Knowles Grumman Corporation, MS A08-35

Bethpage, NY 117 14

Preliminary control designs for the AIAA Control Design Challenge are described. Two methods, one based on classical and one based on modem control concepts, were applied. The designs were easy to carry out, which allowed fast turn-arounds. The designs were evaluated using a altitude-command, velocity-hold command for a low dynamic pressure flight condition. The results show that the goals can be met only if the low rate limit of actuators is raised. Comments about the design aircraft and control requirements are also included.

I. Introduction

Aircraft control design presents a number of challenges. Not only is an aircraft a nonlinear multi- input multi-output (MIMO) system, but it also has a number of constraints that precludes certain control structures and gains. Despite these difficulties, control design for aircraft in service has been quite satisfactory. This fact is even more surprising (especially for non- aircraft control engineers) considering that most (if not all) of the designs are based on single-input single-output (SISO) classical control techniques. One drawback of this approach is that extensive hand-tuning of conwol gains in the final stages is necessary in order to meet all design specifications. This method of aircraft control design is an art rather than a technique.

The relative success, to date, of the aircraft control design using SISO approaches may be attributed to the inherent separation of aircraft dynamics into longitudinal and lateral-directional axes. Thus, a loop can be designed one at time without much interaction from other loops. Future aircraft are expected to meet higher performance and combat superiority requirements characterized as supermaneuverability [ 11. Supermaneuverability refers to an aircraft's capability to perform maneuvers in post-stall flight regimes or in situations where there is substantial cross-coupling between the axes (or modes). Therefore, there is a need for aircraft control laws that are truly multivariable in nature to carry out these MIMO coupled maneuvers and at the same time achieve an apparent "feel" of a SISO system. The objective of this paper is to describe an aircraft control design approach that utilizes both SISO and MIMO techniques, where the final choice will be determined by the performance, stability, and complexity of the control law.

The methodology adopted is a combination of both linear and nonlinear design techniques. A well-understood

classical control based method [2,3] is first used to achieve a reasonable performance for the aircraft in question. The design uses simplified assumptions about the aircraft dynamics to place dominant poles in order to achieve basic flying qualities. The resulting control structure has proportional-integral loops to ensure zero steady-state errors. This exercise allows us to gain insight into the dynamics of the vehicle as well as its control power limitations 141. It also establishes a baseline performance for comparison of other methods.

Two modern control methods are then applied to a quasi-linear model of the aircraft. One is based on the proportional-integral implicit model-following approach. The model-following concept is an attractive methodology for control design as its objective is to force a plant to emulate the behavior of a model. By incorporating the desirable dynamic characteristics into the model, the weighting matrices and the gains are easily derived, and performance and handling quality requirements are naturally met. Experimenting with model-following techniques has had a long history at Grumman [5,6,7,8]. It has been demonstrated to possess inherent control reconfiguration properties [9] and to have good robustness against sensor noise and plant disturbance [lo]. Recently, study has been initiated to incorporate the model- following technique into the HA robust control framework 1111.

The other MIMO approach is based on nonlinear control techniques. It is expected that by explicitly taking into account the nonlinearities in the design, nonlinear control can better utilize the existing dynamics and control power to achieve the required performance. Research in nonlinear aircraft control has been carried out 112,131, though its use in actual applications is rare. In the second phase of this study, we will use lessons learned from our experimentations with nonlinear control [ 141 to modify existing algorithms to arrive at a suitable controller.

The results from the two MIMO designs are valid for selected flight regimes and will be optimized with respect to the nonlinear model. We intend to use PROTO-OPT [ 151, an interactive general optimization tool within our computer-aided control system design facilities, for this purpose. PROTO-OPT allows the control system designer to visually and interactively tune the controller. It also allows specification of hard and soft constraints, as well as include a check for stability of the final controller. The results of optimization can be displayed against original cost functionals for comparison, allowing the engineer to explore the trade-offs. This optimization

Presented at 1991 AIAA GNC Conference, Copyright 0 1991 by Grumman Corp. Published by AIAA with permission

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can check performance as well as stability and robustness. It may even be programmed to yield answers that require a minimum number of gain schedulings. Since the optimization uses a full set of nonlinear simulation equations, its results can be readily incorporated into the target aircraft; thus eliminating most of the ad-hoc control gain tuning.

The procedure described is applied to a high- performance aircraft as provided by the AIAA Control Design Challenge. This paper describes the first phase of the study, whose goals are to learn about thedynamics of the aircraft and the control power limitations. Linear models have been generated and a preliminary evaluation done using the techniques outlined. The final objective is to perform a longitudinal maneuver as specified in the problem statement.

The organization of the paper is as follows. Section II details our effort in characterizing the aerodynamics and dynamics of the design aircraft. Section I11 describes a classical control design for the aircraft, while Section IV relates a parallel design using Linear Quadratic model- following concepts. Section V presents preliminary linear and nonlinear simulation results for a prescribed maneuver in the longitudinal axis. Finally, Section VI offers a summary of the work and some thoughts on the design thus far.

11. Dynamics of the Design Aircraft

The first step in our design process is to study the aerodynamics associated with the aircraft. This procedure allows us to ascertain the degree of nonlinearity and, later, the validity range of the linear models and designs (which will help to determine the gain schedule). From the simulation program, we find that the equations of motion can be succinctly summarized as follows:

Force Equations:

CL = 0.95*ACL(M,a,6dh) + CLnz(M) * nz (1) CY = ACY(M,a,P)+CYg,(M,a)*& + CYM~(M,~) *

* a t - ACYgr(M,a,6r) (2) CD = 1.02 * ACD(M,CL) + ACD(h) (3)

Moment Equations:

Cl = ACl(M,a,P) + Clga(M,a) * 6a + CIMt(M,a) * 6dt - AClgr(M,a,6r) +

2v * [ CLp(M,a) * p + Cl,(M,a) * r 1 (4)

Cm = ACm(M,a,Mh) + CmnZ(M) * nz t - C - * [ Cmq(M,a) * q + Cmb(M,a) * & 3 (9 2v

Cn = ACn(M,a,P) + Cng,(M,a) * 6a + CnMt(M,a) * 6dt + Cngr(M,a,P) * 6r + b - 2v * [ Cnp(M,a) * p + Cq(M,a) * r 1

Where CD = drag Coefficient CL = lift coefficient CY = sideforce coefficient Cl = rolling moment'coefficient Cm = pitching moment coefficient Cn = yawing moment coefficient b = span length - c = reference chord length h = altitude M = Mach number nz = vertical acceleration p = roll rate q = pitch rate r = yaw rate a = angle of attack P = sideslip angle 6a = aileron deflection 6dh = symmetric deflection of the horizontal tails 6r = rudder deflection 6dt = differential deflection of the horizontal tails

All of the aerodynamic terms are obtained via table lookup. The dependencies of each term are indicated by the variables enclosed in parenthesis. By examining the table, we were able to extract the actual data points (which probably correspond to wind-tunnel measurements) and fit them with one- and two-dimensional cubic splines. Some of these plots for Mach = 0.6 are shown in Fig. 1 to 5.

The longitudinal terms, ACL(M,a,Gdh) and ACl(M,a,P) are shown in Fig. 1 and 2, respectively. C b and Cma. two important coefficients, are simply the slopes of these curve (for a given 6dh). From the plots, it can be seen that both coefficients are piecewise linear over a large range of angle of attack, which implies that linear designs may be adequate in the longitudinal axis. Three lateral-directional terms, ACn(M,a,P), ACYg,(M,a,6r), AClg,(M,a,6r), are shown in Fig. 3, 4, and 5 , respectively. The weathercock stability Cnp. which is the slope of ACn(M,a,P) for a given a, shows some mild variations across P. CY&,, the rudder sideforce effective is shown to drop off as the angle of attack increases, but remain relatively linear for most of rudder deflection. AClgr, the rolling moment due to rudder is shown to be significant and varies a great deal with angle of attack, Despite the nonlinearities we find that, overall, the airplane is aerodynamically very "clean."

To ascertain the stability of the airplane, we performed small perturbations off selected aim points to generate the linear models. For the longitudinal mode,

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the state vector consists of angle of attack (a, rad), pitch rate (9, rawsec), pitch angle, (0, radsec), and forward speed (V, fdsec); that is, x = [a q 8 VIT. The controls are horizontal tails (Mh, deg), and throttle angle (GPLA, deg); that is, u = [ 6dh 6PLAIT. For M = 0.5 and h = 9800 ft, we find that:

1 A = [ 0 1.oOOO 0 0

-0.7795 1.OOOO 0.0000 -0.0002 -5.4746 -2.5047 O.OOO0 0.0004

-3.5957 0 -32.1426 -0.0145

r -0.0016 -o.oooo 1 I = I -0.r91 0.0000

0 -0.0713 0.2094 1

Based on this model, the short-period mode has a damping Cs9 = 0.6 and a natural frequency w = 2.72 rad/sec, while the phugoid mode has a damping $, = 0108 and a natural frequency o = 0.081 rad/sec. Therefore, longitudinally the aircraft in flight condition is very stable with a long phugoid time constant at this flight condition.

For the lateral-directional mode, the state vector consists of sideslip angle (p, rad), roll rate (p, rad/sec), yaw rate (r, radsec), roll angle (@,rad), yaw angle (v, rad); that is, x = [p p r @ yIT. The controls are horizontal tails ( a t , deg), aileron (6a, deg), and rudder.(&, deg). For the same flight condition, we find that:

SP

P

-0.1925 0.0798 -0.9968 0.0594 0 -27.0720 -2.2162 1.3970 0 0 4.6830 -0.0745 -0.5745 0 0 0 1.OOOO 0.0800

0 1.0032 0 0

-0.0003 -0.ooOO -0.OOO6

0.1707 0.1695 -0.0249 0.0210 0.0022 -0.0486

0 0

0 0

0 0

0 0 0

Based on this model, the Dutch-roll mode is lightly damped with damping ratio cdr = 0.157 and natural frequency odr = 2.62 ra/sec. Therefore, compensation is necessary to further improve the aircraft lateral-directional stability and performance.

Preliminary control designs are carried out in this flight regime, which was selected because it is at the comer of the flight envelope specified. Due to dynamic pressure involved, this flight condition is expected to pose

challenges to the control design. To check performance, we perform a longitudinal maneuver specified in the Challenge problem statement. The maneuver is to follow an altitude ramp command of 50 ft/m within 50 ft, while holding the speed constant (within 0.01 Mach). The design procedures and simulations are given in the next sections.

111. Control Design Method 1:

Our classical control design is derived from the design requirements and aircraft dynamics. Since the design calls for command following, proportional-integral (PI) loops are used. Additional damping is added through rate feedback. Because the baseline aircraft is longitudinally very stable, we used minimal inner loop compensation, concentrating mostly on the outer-loop compensation to achieve guidance. The resultant control design is shown in Fig. 6.

As can be seen from the diagram, the design includes a velocity-command loop as well as an altitude loop. For the altitude-command loop, the h command is subtracted with the measured h and passed through a standard PI control block. Vertical speed, , which is obtained as the integral of nz and acts as damping term, is added later. The net result is an overall pitch command, which is subtracted with measured 0 and fed into a simple proportional theta-loop with damping. Since positive pitch requires negative elevator, the resultant surface command is multiplied by -1. For the velocity-command loop, we feedback forward speed as well as the horizontal acceleration, which acts as the damping term. The net result adjusts the throttle angle to control the thrust. Both loops are designed separately, with the underlying assumption that there is very little effect of coupling between the loops.

We used root-locus method to check that the outer- loop did not compromise the stability and then use time- history to check for performance. For example, KhP was set to 0.19 because that is the value that allowed us to stay within the actuator rate limits. The final gains of the design, after repeated root-locus with other gains and time- history plots, are shown in Table 1.

IV. Control Design Method 2:

The model-following approach is based on modem control techniques. It uses linear quadratic regulator concepts, but adapts the structure to achieve command following. An advantage of the design is that it provides both inner-loop and outer-loop control in the same formulation. The proportional-integra1 model-following (PIIh4F) control law is reviewed here. For more details, refer to [7]. Let the plant be described by the usual (A, B, C , D) matrices and the model to be described by system matrix A,, namely

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. y = c x

. X, = A,x,

Implicit model-following incorporates the model in the quadratic cost function by weighting the difference between the actual and model state rates, that is,

O o .

J= 1/21 (x - im)T 0 00

= 1/21 [xTQx+2xT 0

T T where Q = (A-A,) Qi(A-A,), =@-A,) QiB,

We can improve the design's low-frequency disturbance-rejection characteristics and provide zero steady-state error command response by adding proportional-integral compensation. Let yd be the desired equilibrium outputs; then the perturbations around the corresponding set point (xo, u0) are:

TQiB, and Qi is a weighting matrix.

Define output error integral

The state variable is augmented to include

and the cost function becomes

T Axa+2Axa SAu+AuT

At this point, we discretize both the system equations and the cost functions and carry out the design in digital domain. This approach ensures that the gains computed are optimized with respect to the sampling time. It has the additional advantage of taking into account any higher frequency dynamics that may cause problem when the actual onboard implementation is carried out. Doing so, we get the following equations:

Eq.9: X k + l =

Eq. 11: x ~ ~ + ~

Eq. 13: Axk = X k - X o

where At is the sampling time. r -I

Optimization of this quadratic cost functions leads to discrete-time Riccati equation, from which a steady-state solution P can be calculated. The feedback gain then is given by

Auk

Recasting Eq. 29 in incremental form, we get

For this design, the state-vector is x = [a q 8 V hIT. Although this is a state feedback case, all of the required outputs are available. However, a is said to be noisy. Therefore, in the final design, dependency on angle of attack will be removed. Two sets of gains were computed. The first set uses higher control energy to ensure that the design goal (altitude following within 50 ft while holding velocity within 0.01 M) is met. The model used is a faster response than the plant and the weighting matrices are set to be

0

339

6 = diag(l.0,0.7) (3 1)

The resultant gains are shown in Table 2. A second set of gains were also computed using

significantly less throttle, but it does not meet the altitude following specification (but velocity hold is achieved). For that case, the model used is exactly the same as the plant (which means the design becomes only a guidance control) and the weighting matrices are set to be

i = I5 Q6 = diag(l.0,0.05) (32)

onlinear Simulation Results

Both control designs are implemented into the nonlinear simulation program furnished by AIAA. We inserted dynamics into all surface actuators (first-order with time constant of 0.05 sec) and a 24 deg/sec rate limit, since they are not present in the simulation but specified in the problem statement. We were careful to make sure that we did not exceed the 24 deg/sec actuator rate limit, which we found to be a very low value that constrained our design.

The time responses of altitude, horizontal tail, forward speed, and throttle angle of all three designs are shown in Fig. 8 to 10. The nonlinear simulations of h and 6dh followed closely that of linear responses, while V and GPLA show discrepancies for large throttle angles. The classical design has more altitude overshoot than that of model-following design, but uses less throttle to keep velocity in check. None of the designs met the goal of altitude following due to the rate-limit on the horizontal tail and the low dynamic pressure flight condition chosen.

VI. Summary and Conclusions

Preliminary control designs for the AIAA Control Design Challenge are described. The approach was to first ascertain the dynamics of the aircraft and then construct linear models for an initial check on the aircraft stability and design. Two methods, one based on classical and the other on modem control concepts were applied in this phase. They were evaluated for an altitude-command and velocity-hold maneuver. Both designs proved to be simple to use, thus allowing fast turn-around.

The initial conclusions about the challenge problem are that the aircraft used is aerodynamically very "clean," thus lending to relatively easy control designs. The surface rate limit of 24 deg/sec seems unreasonable given the current state-of-art, which typically allows for 100 deglsec. The number of independent surfaces (i.e., the control redundancy) is small, which implies that MIMO design techniques may not be needed. It remains to be seen whether some unconventional maneuver will be performed that requires MIMO designs.

We intend to continue the control designs for the next phase of challenge problem. We will apply the nonlinear

control method and optimization procedures as well as derive the lateral-directional design. Finally, more complete simulations will be run to fully check the control laws.

EFERENCES

1.

2.

3

4.

5.

4.

7.

8.

9.

10,

11.

12.

13.

14.

Herbst, W. B., "Future Fighter Technologies," J. of Aircraft, Vol. 17, No. 8, Aug. 1980. Martorella, P., et al., "Control Definition Study For Advanced Vehicle," NASA Contractor Report 3738, November 1983. Klein, R., Lapins, M., Martorella, R.P. and Sturm, M., "Control Law Development For a a Close- Coupled Canard, Relaxed Static Stability Fighter," presented at AIAA 20th Aerospace Science Meeting, January 1982, Orlando, Florida. Beaufere, Henry, et. al., "Control Power Requirements for Statically Unstable Aircraft, Vol. I and 11," AFWAL-TR-87-3018, June 1987. Kreindler, E., and Rothschild, D., "Model-Following in Linear Quadratic Optimization," AIAA Journal, Vol. 14, no. 7, July, 1976, pp. 835-842. Gran, R., Berman, H., and Rossi, M., "Optimal Digital Flight Control for Advanced Fighter Aircraft," J. of Aircraft, Vol. 14, No. 1, Jan, 1977,

Chin, J., H. Berman and J. Ellinwood. X-29A Flight Control System Design Experiences. AIAA GNC Conference. Paper 82-1538, 1982. Huang, Chien Y., "Multivariable Control Law for Flat-Turn Strafing Maneuver by a Supermaneuverable Aircraft," Proc. of AIAA Guidance, Navigation, and Control Conference, Aug. 20-22, 1990, pp.499-506. Huang, Chien Y. and Stengel, Robert F., "Restructurable Control Using Proportional-Integral Implicit Model Following," J. Guidance, Control, and Dynamics, Vol. 13, No. 2, March-April 1990,

Huang, Chien Y., "A Methodology for Knowledge- Based Restructurable Control to Accommodate System Failures," Ph. D . Thesis, Princeton University, 1989. Huang, Chien Y., "Application of Robust Model- Following Concepts to Aircraft Control," American Control Conference, San Diego, CA, May, 1990. Also available as Grumman Corporate Research Center Technical Paper. Lane, Stephen H., and Stengel, Robert F., "Flight Control Design Using Non-linear Inverse Dynamics," Automatica, Vol. 24, No. 4, 1988, pp. 471-483. Huang, Chien Y. and Knowles, Gareth J., "Application of Nonlinear Control Strategies to Aircraft at High Angle of Attack," IEEE Control and Decision Conference, Dec. 4-9, 1990, Honolulu, Hawaii, pp. 188-193. Huang, Chien Y., et al., "Analysis and Simulation of Nonlinear Control Strategies to High Angle-of-

pp. 32-37.

pp. 303-309.

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Attack Maneuvers," submitted to AIM Guidance, Navigation, and Control Conference, New Orleans, LO, 1991.

16. Reilly, J., Eadan E., Huang, C., and Levine, W.S., "A Computer-Aided Optimization-Based Controller Design Tool," to appear in 1991 American Control Conference, Boston, MA.

Table 1. Gains for classical control

Table 2. PIIMF gains, set 1

Table 3. PIIMF gains, set 2

Fig. 1. ACL(M,a,Sdh) for

341

Fig. 2. ACm(M,a,Sdh) for M = 0.6

0

28 -92 Fig. 3. ACn(M,a,P) for M = 0.6

. 0

Fig. 4. ACY&Vl,a,Sr) for M = 0.6

342

60 0 Fig. 5. AClgr(M,a,Gr) for M = 0.6

~~~ ~

Fig. 6. Block diagram of the classical control design

Fig. 7. Block diagram of the model-following control design

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Time (sec)

ppp -___-

............................ ..... . i . . ....... i . . ...... .:. ........ 0

.................................................................. 2

0 1 2 3 4 5 6 7 8 9 10 - 10

Time (sec)

Time (sec)

Time (sec)

Fig. 8. Nonlinear simulation using classical control design

344

-___I- pppp-

............................ : ................... .................................................. 1 1 m m

,-.-.-. ,-,-I-

...................

550 ........ .;. . . . . . . . .;. . . . . . . . ,. ....... .:. ........ .:. A

......... i. ....... ..j. ........ j... . . . . . .;... ...... I . ....... .;. ........

. . ......................

10 Time (sec)

, . . . ,-.-VI

n

Time (sec)

Fig. 9. Nonlinear simulation using PIIMF, gain set 1

345

5

0

-5

.IO

Time (sec)

..................... ..............................

Ir ............................................................................

-

0 1 2 3 4 5 6 7 8 9 10 Time (sec)

_ _ _ _ - ~ ~

0 540 ............................ : ......... ; ......... :.. ....................... 2

>

A

......... I.. ...... .:. . ......... :. ........ .: ......... 3z ~~ ___-- 530

0 1 2 3 4 5 6 7 8 9 10 Time (sec)

80

60

40

20 0 1 2 3 4 5 6 7 8 9 10

Time (sec)

i I

I

J

Fig. 10. Nonlinear simulation using PIIMF, gain set 2

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