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EMPIPT-11.7 (NASA -CR- 164321) APPLICATION O? VARIABLE N81-23093 STRUCTURE SYSTEM THEORY TO AIRCi;APT FLIGHT CONTROL Interim Report (Drexel 1 1Y , iv.) 42 p EC A03/MF AO1 CSCL O1C Unclas G3/08 42390 Application of Variable Structure System Theory to Aircraft Flight Control Interim Report May, 1981 Research Supported by N.A . S.A. Ames Research Center, NASA Grant No. NAG 2-8' Principal Investigator: Dr. Anthony J. Calise Dr. Isaac Kadushin Investigator: Research Assistant: Mr. Fred Kramer Mechanical Engineering & Mechanics Department Drexel University Philadelphia, PA 19104
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I The AV-8A portion of this research is partially supported b y the Naval Air Svstems Command.
CONTENTS Page NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Glide Slope Control For the Augmentor Wing Jet 2.
.01. Research Aircraft . . . . . . . . . . . . . . . . . . .
2.1 The System . . . . . . . . . . 2 . . . . . . . . . . . .
Control System Design . . . . . 2 2.2 . . . . . . . . . . . .
Motion. Results 2.3 Siidiug . . .
. . . . . . . . . . . . . 5 Design of Controls Required for 2.4 Reaching the Sliding Surface 12 , , , , , , , , , , , , , VSS Design For The AV-8A .
3. . . . . . . . . . . . 21 . . . . . .
3.1 VSS Design of the Attitude Loop, . , . , , . . , , . , 22 Comparison to Conventional Design. , . . , , , . . , 26 3.2 .
3.3 Numerical Results . . . . . . . . . . . 27 . . . . . . . .
4. Future Research . . . . 39 . . . . . . . . . . . . . .
. . . .
REFERENCES . . . . . 40 . . . . . . . . . . . . . . . . .
. . .
t + -i- NOMENCLATURE A = plant matrix B = control matrix d = localizer beam error, m N (RN) - engine RPM, q = pitch rate increment, rad/s s = distance from the sliding surface, s t = time, s m/s U - surge velocity in body frame, m/s v = inertial velocity increment, :a/s w - heave velocity in body frame, X - state vector.
= angle of attack increment, rad = flight path angle increment, rad Y stick) increment, deg (in) 6 - elevator angle (longitudinal e 6T = throttle increment, deg C = damping ratio = nozzle angle increment, rad ^(v) g - pitch angle increment, rad -ii- 1. Introduction This report summarizes the current status of our research on the application of Variable Structure System (VSS) theory tc design aircraft flight control systems. Two aircraft types are currently being investi- gated: the Augmentor Wing Jet. STOL Research Aircraft ( AWJSRA), and AV-8A AWJSRA design considers automatic control of longi- Harrier. The tudinal dynamics during the landing phase. The main task for the AWJSRA is to design an automatic landing system that captures and tracks a localizer beam. The control task for the AV-8A is to track velocity commands in a hovering flight configuration. Much of the effort since our last report [1] has been devoted to developing computer programs that are needed to carry out VSS design in a multivariable frame work, and in becoming familiar with the dynamics and control problems associated with the aircraft types under investigation. Numerous VSS design AWJSRA. The approaches pre- schemes were explored, particularly for the sented here are the ones that appear to be the best sLited for these aircraft types. Examples are given of the numerical results currently being generated. A brief summary of VSS theory was presented in [1].
-1- 2. Glide Slope Control For The Augmentor Wing Jet STOL Research Aircraft (AWJSRA) 2.1 The System The AWJSRA is a research aircraft modified from the De Havilland C-8-A turboprop by modifying the wing to include an augmentor flap system, boundary layer control and other lift augmentation systems, and by replacing the turboprop engine by a split flow jet engine. The system has been described in [2, 3, 4]. The purpose of the present work was to design a precise glide slope control system invariant to changes in some of the aircraft parameters.
2.2 Control System Design The control system was designed to use the existing controls: elevator angie and the engine thrust - an independent system controlled the throttle. The jet nozzle was set to a nominal value of 90°.
b y The equilibrium trajectory was chosen to be a 7.5° glide at 30.9 m/sec (b0 knots) starting at an altitude of 396.5 m (1300 ft). Other system parameters are given in (3,4]. The design was based on the linearized model in [4] modified to a wind axis coordinate system: x Ax + Bu where Nh ] .KT [d, 100 9, 100 a, v, 100 q, (2.1-2) T - u [100 5v, 100 S e , 5 T ] (2.1-3) 0 0 0^ 0 -.309 .309 i 0 0 0 0 1 (2.1-4) A ' 0 .042 -.52 -.94 1.03 -.36 .0007 0 0 -.097 .043 -.052 .004 -1.36 0 0 .0174 -.0816 0 0 0 0 -1 0 0 0 0 0 B ' 0 0 0 (2.1-5) -.015 0 1.2 0 0 .72 -2- At the equilibrium trajectory, the nozzle is perpendicular to the aircraft's longitudinal axis. Thus nozzle angle variation is a rather poor velocity control, as can be seen from the small value of the control derivative.
It was, therefore, decided to keep the nozzle angle constant.
The variable structure control requires controls capable of almost instantaneous changes. The only fast control available in the AWJSRA is the elevator angle. the engine thrust control has a time constant of about 1 sec. The control system consir.ts of two loops: the first loop is an internal loop with state variables [9, a', v, q] controlled by the elevator angle 6 e and designed as a variable structure control system [5, 6]. This is mainly an attitude control sysLdm. Speed control is achieved through changes in angle of attack. The control system parameters were determined by placement of the eigenvalues at the desired position when the system is in sliding mode along the surface s - C 1 (100 8) + C 2 (100 a) + C 3 v + (100 q) - 0 (2.2) The eigenvalues were placed so that the resulting inner loop system will have natural frequency of 1.5 rad/sec, damping ratio C - 0.7 and a real eigeuvalue of 0.1 sec 1 . The resulting sliding surface is s - + 3.82 (100 8) - 2.22 (100 a) - 0.934 v + (100 q) - 0 (2.3) and the resulting c.losad loop system, in sliding mode is: 100 -3.82 2.22 100 8 .934 0 100 . n - -3.88 1.77 .67 100 a (2.4) + 0 6Nh v
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v -.36 with q determined from (2.3) ; 100q - -3.82 (100 8) + 2.22 (100 a) + 0.934 v (2.5) The velocity control through attitude can be seen by considering the changes in 8, a, and q required to counter a change in v, so that s-0.
As can be seen from (2. 3), a positive v can be obtained by a decrease in a, and vice versa.
The use of a variable structure control system in sliding mode for the attitude control makes the aircraft control invariant to changes in the coefficients of the state matrix governing the pitch rate, q.
-3- depend on the position of the airplane c.g.. Thus, These coefficients the AWJSRA control will be invariant ► o changes in the c.g positioc.
Such a feature may be important for future applications of the variable structure control to airplanes with large c.g,varia:ion such as transport and military flying vehicles.
The outer loop control system consists of the "beam error" control effected through thrust variation. As the thrust direction in steady state is practically perpendicular to the flight path, changes in thrust cause changes in the vertical acceleration and, thus induce changes in angle of attack. As a result, a change in flight path angle occurs.
Since beam error is proportional to y , this error is eliminated after motion. Changes in a also cause changes in aircraft attitude a transient and velocity. These are controlled by the inner loop, which is in sliding mode along the surface s. The engine rpm, which controls the thrust is in turn controlled by the throttle and is unaffected by other state variables.
The outer loop was designed under the assumption that the inner loop q was eliminated Thus, the variable mode.
is already in the sliding new state vector is thus using (2.5) . The (2.6) xl T . [d, 100 9, 100 a, v, 6%] those of (2.4).
for 100 9, 100 a, v are motion and the equations of The outer contl-ol loop was designed by minimizing the quadratic performance index (2.7-1) + 6e ] dt J - 1/2 I Ix Q x l T 1 s with 0 0 0 0 0 0 0 0 1 (2.7-2) 0 0 Q 0 C 1 0 1 0 0 0 0 0 0 0 and is the time at which the sliding mode begins.
Ts -4- The resulting throttle control is: d th a - C 1 " 1 (2.8) with C 1 - (5.53, -6.0^; , 5.27, -5.91, -4.81 (2.9) 2.3 Sliding Motion Results The system dynamics described in (2.2) were simulated in the sliding mode, using a second order Runge-Kutta method. The system was required to decrease a 10m. initial beam error. The results are shown in Figures 2.1 to 2.5. The beam error decreased to 5% of its initial value in about 11 seconds ( see Fig. 2.1). The re was practically no overshoot. The motion towards the equilibrium glida path was accompanied by a slight nose down pitching (Fig. 2.2) and a very small increase in velocity ( Fig. 2.4). The main effect was a considerable increase in thrust as can be seen from the increase in r.p . m. (Fig. 2.5). This is the main path control and according to (2,31 is preferred by human pilots. The nose down tilt required to hold the speed approximately constant is also described in (3,41. Such a control technique, which couples speed, attitude, and path control may be a heavy burden on the human pilot and, therefore, degrade his rating (opinion) of the system. However, the automatic control system is fully capable of both path, attitude, and speed control regardless of their coupling.
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m N m m 'm m m G4 (a") 39 ` -11- 2.4 Design of Controls Required for Reaching the Sliding Surface The design procedure for reaching the sliding surface is described The requirement is that the norm of s should always decrease, in (1, 51.
i.e., (2.10) ss < 0 (1] The procedure described in was generally used, however, since the rpm factor N influences the motion of the inner loop variables, an additional component had to be added to the control, as shown in Chapter VIII
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sa - s( Ea i x i - 1.2 (2.11)
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iml where a l a 0 a4 . 2.14 a 2 = .0147 a5 = .173 (2.12) a 3 a 1.03 a6 = .8 To satisfy ( 2.10), the following control structure is appropriate: 6 ai sxi > o (2.13) 6 E i xi ^i { B i sxi < o whe re -a i > a 1 /1.2 , -6 1 < a 1 /1.2 (2.14) The following selection was made: a l - 0 6 1 .0
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62 ' a 3 = 1.6 63-0 a, 3.6 64 v a5 0 0.3 65 M 0 (2.15) s 1.3 66 = 0 a6 -12- The aircraft and control system dynamics both off and on the sliding surface were simulated and the results are shown in Figures 2.7 - 2.13.
The response is initially slower due to a throttle command limit that was imposed such that < 10.7°, which corresponds to N a 98.5 (normal take-off power setting). The general motion is similar to that of ideal sliding, with the exception that d e is no longer continuous (Fig. 2.13).
The beam error (Fig. 2.7) settles in 14 seconds and is accompanied by f small changes in a (Fig. 2.8), nose down pitching (Fig. 2.9) and minor velocity variations (Fig. 2.10). The engines RPM response is given in Fig. 2.11, and exhibits the effect of limiting 6 T . The value of s during the run is shown in Fig. 2.12. Note that sliding occurs almost immediately and is maintained throughout the maneuver.
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3. VSS Design For The AV-8A This portion of our research considers the VSS design of a velocity comrn&id control system for the AV-8A in hovering flight. Both longitudinal any lateral dynamics will he considered, however, this report will only address control of the longitudinal velocity components (surge and heave).
Reference gives the linearized model for the Harrier dynamics (6] for airspeeds between 0 and 120 knots. We have elected to use the values for 30 knots, which result in the following mode'_ for the s y stem dynamics in the bodv frame: (3.1) z - :\x + B whe re (3.2) x T - ) u , w , a , q , RN) (5 e (3. 3) RN cj ^T - , n , -.035 -,02 -9.8 0 .002 -.011 -.105 -1.66 0 -.309 A- 0 J 0 I 0 (3.4) .005b 0 0 -.13 .00 I 0 0 0 0 -4.8b 0 -9.3 .16 .28 0 B - 0 0 0 (3.5) .2 0 0 0 0 4.8b In the design nozzle angle is held fixed (-i-0), so that the onl y :x ans of achieving a u_ is b y pitching the aircraft. Vertical velocit y is c controlled b y R\ c . In the design of s y stems with a command input, it is customary to redefine the state and control perturbations about a commanded equilibrium state and control obtained b y setting u - u , w - w and c c solving for the remainiag states and controls by equating (3.1) to zero.
Tnis detail is omitted here but the definitions are implied.
- ► 1- 3.1 VSS Design of the Attitude Loop The VSS design for attitude control is based on the controller structure shown in Figure 3.1. The sliding surface is defined bv: s - C 1 6e + q 69 - 0- e c (3.6) where e c - k 1 6u , 6u - u-u c (3. 7) ana a is regarded as a constant or slowly varying input. In sliding c mode (s-o) we have from (3.4) and (3.6) .
T 1 e - -e + e , 1 - 1/C 1 (3.8) C 1 which is stable for an y ' 0. The transient response iF dictated by C 1 and is invariant with respect to remaining state variables. The design of k 1 and C 1 is based on Fig. 3.2. The closed loop poles were chosen from Fig. 3.3 taken from [7]. Selecting w - 2 rad/s and a damping parameter of 3 sec -1 (which corresponds to - .75), the resultine valur.s for C 1 k and are: C 1 - 3.0 s -1 k1 - .136 s/m (3.9) The heaving motion is controlled using a conventional proportional control law M - .1378 w - .8 RN (3.10) c To guarantee reaching and existance of the sliding mode, it is sufficient that ss < 0 (3.11) Differentiating (3.6) and assuming - 1 - 0, we obtain c s( ss - a x i + .: 5 e ) (3.12) i-1 whe re u v m L L W 6J L u V L u cn L v L u G !'1 6!
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;.he re (3.15) i /.2 B < a 1 /.2 ai > a Allowing for possible variations in the parameters in (3.4) with the exception of A(1,3), the following selections were made .200 .222 19.6 (3.16) a = 20.4 3 13.7 15.0 .008 The last term in (3.14) basically controls the time required to reach the sliding surface, and k was chosen as 4.0 in-s.
3.2 Comparison to a Conventional Design A conventional design of the attitude loop for the control structure of Figure 3.2 follows the same lines except that the transfer function for e - e is second order c e(s) .2 k2 (3.17) e c s2 + (.13 + .2k 3 ) + .2 k2 (s) whe re (3.18) 6 e -k., ( 6 -8 )-k 3 q c -26- Placing the closed loop poles to match the response time of the variable structure s y stem we obtain k l - .0816 s/m k , - 50 in k 3 - 24.35 in-s 2 (3.19) Mote that the gains in (3,19) are considerably higher than the gains in (3.14) for the variable structure control in the vicinity of the sliding surface (s-o). This should aid in avoiding control saturation and instabilit y due to large command inputs.
3.3 Numerical Results The numerical results of this section compare the VSS control to a conventional control design for response stability under the presence of saturating control. The magnitude of was limited to 4 inches. Two 6e levels of responses are given, corresponding to initial velocit y errors of -3 m/s and -10 m/s. Figures 3.4 to 3.7 give the VSS response for an initial velocity error of -3 m/s. Note from Fig. 3.6 that sliding ini- tiates at 2.5 seconds, when the response is essentially complete. Figure 3.7 shows that there is little coupling with the heave d ynamics. Figures 3.3 to 3.10 give the velocity pitch attitude, and longitudinal stick responses for an initial velocit y error of -10 m/s. Note the similarity of response in velocit y with that of Fig. 3.4.
Figures 3.11 - 3.14 show the velocit y and longitudinal stick responses with proportional control for the same conditions. Note that for an initial velocity error of -10 m/s, the proportional control is on the verge of instabilit y , exhibiting 25°0 overshoot and prolonged periods of control saturation. Figures 3.13 and 3.8 and Fig. 3.10.
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t0 N N N 'I (D m 4. Future Research The research for the next reporting period will examine the behavior of the SWJSRA and the AV-8A subject to system parameter variations and external disturbances. In addition, transient responses will be generated using nonlinear models for these aircraft. In the case of the AV-8A, we propose to examine using nozzle angle as a control to achieve reaching of the sliding surface. Currently, reaching takes up most of the transient response, and increasing k in the controller design leads tc unstable behavior in the presence of large command inputs.
For next y ear, we propose to examine other aircraft types currently of interest to NASA Ames. In particular, a tail-sitter vehicle has been discussed with the technical project monitor.
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