Missile Roll Control Part-II Notes
Contents
1 Missile Control
1
1.1
Roll Position Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Roll Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Roll Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.4
Control Techniques in Roll Autopilots . . . . . . . . . . . . . . . . . . . .
3
1.5
Garnell’s Roll Autopilot-A Study and Analysis . . . . . . . . . . . . . . .
4
1.6
Modeling Plant and Servo . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.6.1
5
Analysis of Dynamics of Roll AP Design - Zero Order Servo . . .
References
7
i
Chapter 1 Missile Control An autopilot [1] is a closed loop system and it is a minor loop inside the main guidance loop; not all missile systems require an autopilot. (a) Broadly speaking autopilots either control the motion in the pitch and yaw planes, in which they are called lateral autopilots, or they control the motion about the fore and aft axis in which case they are called roll autopilots. (b) In aircraft autopilots, those designed to control the motion in the pitch plane are called longitudinal autopilots and only those to control the motion in yaw are called lateral autopilots. (c) For a symmetrical cruciform missile however pitch and yaw autopilots are often identical; one injects a g bias in the vertical plane to offset the effect of gravity but this does not affect the design of the autopilot.
1.1
Roll Position Autopilot
A simple block diagram of roll position autopilot is as shown in Fig.1.1.
1
Figure 1.1: General Block Diagram of Roll Position Autopilot[1]
(a) The roll position demand (φd ), in the case of Twist and Steer control, is compared with the actual roll position (φ), sensed by the roll gyro. (b) The error is amplified and fed to the servos, which in turn move the ailerons. (c) The movement of the ailerons, results in the change in the roll orientation of the missile airframe. (d) The changes in the airframe orientation due to external disturbances, biases etc are also shown in the achieved roll position. (e) The controlling action (feed back) continues till the demanded roll orientation is achieved.
1.2
Roll Derivatives
Aerodynamic derivatives enable control engineers to obtain transfer functions defining the response of a missile to aileron, elevator or rudder inputs. These derivatives are calculated from the total force from the wings, body and control surfaces on the assumption that control surfaces are in the central position. Assuming that the missile 2
is symmetrical in both planes i.e. in XY and XZ planes and that the missile is roll stabilized i.e., p ≈ 0,the airframe equations of motion given above can be further simplified
and used for analysis. As roll control is the intended application, let us consider a roll equation given by Eq. 1.1, Ap˙ = L = Lξ ξ + Lp p
(1.1)
Where, Lξ is rolling moment as a function of aileron angle. Bearing in mind that in most applications ξ is unlikely to exceed a few degrees, we regard Lξ as a constant. Lp is the damping derivative in roll and has dimensions of torque/unit roll rate. Since the torque will always oppose the roll motion its algebraic sign is invariably -ve. This derivative is often regarded as a constant for a given Mach number and operating height.
1.3
Roll Transfer Function
The roll transfer function (Roll rate/aileron deflection)
p(s) is obtained by rewriting ξ(s)
the Eqn. (1.1) as, p˙ − lp p = lξ ξ or in the transfer function form as, p(s) lξ −lξ /lp = = ξ(s) s − lp Ta s + 1
(1.2)
−lξ 1 can be regarded as a steady state gain and Ta = can be regarded as lp −lp aerodynamic time constant. Where
1.4
Control Techniques in Roll Autopilots
(a) Traditional or Conventional Design of Roll Autopilot as given in [1]. (b) Design of Roll Autopilot using Optimisation Technique. (Linear Quadratic Regulator) 3
(c) Design of Roll Autopilot using Sliding Mode Control. (d) Design of Roll Autopilot using Inertial Delay Control. (e) Design of Roll Autopilot using Disturbance Observer.
1.5
Garnell’s Roll Autopilot-A Study and Analysis
Consider an air to air homing missile whose roll moment of inertia is A = 0.96Kgm2 and is assumed to fly at a constant height of 1500m. The table 1.1 shows that the roll derivatives, aerodynamic gains and time constants vary largely due to the variability in the launch speeds in the range of M = 1.4 to M = 2.8. Assuming that other than the roll angle (output) the other states of the combined missile and servo dynamics and disturbances are not accessible, the missile is now required to be roll position stabilized.
Quantity −Lξ
−Lp Ta = Lξ Lp
−A Lp
M = 1.4 M = 1.6 M = 1.8 M = 2.0 M = 2.4 M = 2.8 7050
8140
9100
10200
11700
13500
22.3
24.9
27.5
30.3
34.5
37.3
0.043
0.0385
0.0349
0.0316
0.0278
0.0257
316
327
331
336
340
362
Table 1.1: Roll Derivatives, Gains and Time Constants
1.6
Modeling Plant and Servo
The combined dynamics of the airframe-servo combination can be deduced as Eq.(1.3) given below:φ(s) φ(s) ξ(s) = ∗ ξc (s) ξ(s) ξc (s) 2 lξ (ks ωns ) = ∗ 2 2 ) s(s − lp ) (s + 2µs ωns s + ωns 4
(1.3)
Where, ks is the servo gain, µs the damping factor and ωns the natural undamped frequency in rad/sec of the second order servo actuator. Considering the missile servo to be of zero order with gain ks , the block diagram of the roll position control loop with demanded roll position equal to zero using the aerodynamic derivatives is as shown in Fig.1.2.
φ=0
φ˙
ξ
δc
1/A s−lp
ksLξ
φ 1 s
Airframe
Actuator kg
Basic Roll AP Figure 1.2: Block Diagram of Roll Autopilot
1.6.1
Analysis of Dynamics of Roll AP Design - Zero Order Servo
In order to design the roll loop one must know the maximum anticipated induced rolling moment and the desired roll position accuracy. The combined dynamics of the airframeservo combination can be deduced as Eq.(1.4) given below:φ(s) φ(s) ξ(s) = ∗ ξc (s) ξ(s) ξc (s) lξ = ∗ (ks ) s(s − lp )
(1.4)
Where, ks is the servo gain. 5
(a) The aerodynamicist estimates that the largest rolling moments will occur at M = 2.8 due to unequal incidence in pitch and yaw and will have a maximum value of 1000 Nm. (b) If the maximum missile roll angle permissible is 1/20 rad then the stiffness of the loop must be not less than 1000 ∗ 20=20, 000 Nm/rad. (c) This means that in order to balance this disturbing moment we have to use 1000/13, 500 rad aileron, and this is approximately 4.2 deg. (d) The actual servo steady state gain -ks has to be negative in order to ensure a negative feedback system. Since the steady state roll angle φOSS for a constant disturbing torque L is given by φ0SS /L = 0.05/1000 = 0.05/(kg *ks *Lξ ), it follows that ks *kg must be not less than 20000/13500 = 1.48. (e) If kg is set at unity then ks must be 1.48. The open loop gain is now fixed at 1.48*Lξ /Lp = 535.
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References [1] Garnell, P., Guided Weapon Control Systems, Brassey’s Defence Publishers, London, 1980. [2] Nesline, F. W. and Zarchan, P., “Why Modern Controllers can go Unstable in Practice,” Journal of Guidance, Vol. 7, No. 4, 1984, pp. 495–500. [3] Blakelock, J. H., Automatic Control of Aircraft and Missiles,Second Edition, John Wiley and Sons,Inc, New York, 1990. [4] Siouris, G. M., Missile Guidance and Control Systems, Springer, New York, 2003. [5] Gurfil, P., “Zero-Miss Distance Guidance Law Based on Line of Sight Rate Measurement only,” Control Engineering Practice, Vol. 11, 2003, pp. 819–832. [6] Horton, M. P., “Autopilots for Tactical Missiles : An Overview,” Journal of Systems and Control Engineering, Vol. 209, 1995, pp. 127–138.
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