OPTIMAL STATE SPACE CONTROL SYSTEM DESIGN APPROACH FOR MISSILE ROLL AUTOPILOT Notes based on papers (1) Why Modern Controllers Can Go Unstable in Practice” by F.W.Nesline and P.Zarchan and (2) ”Combined Optimal/Classical Approach to Robust Missile Autopilot Design” by F.W.Nesline, B.H.Wells and P.Zarchan.
Contents
1 Missile Control
1
1.1
Control Techniques in Roll Autopilots . . . . . . . . . . . . . . . . . . . .
1.2
MODERN CONTROL APPLIED TO MISSILE ROLL AUTOPILOT[1],[2],[3]
1 2
1.2.1
Fundamentals of Optimal Control . . . . . . . . . . . . . . . . . .
3
1.2.2
Application to Missile Roll Autopilot . . . . . . . . . . . . . . . .
4
1.2.3
Solved Example . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
References
6
i
Chapter 1 Missile Control An autopilot [4] 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
Control Techniques in Roll Autopilots
(a) Traditional or Conventional Design of Roll Autopilot as given in [4] [1] [2]. (b) Design of Roll Autopilot using Optimisation Technique. (Linear Quadratic Regulator) 1
(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.
1.2
MODERN CONTROL APPLIED TO MISSILE ROLL AUTOPILOT[1],[2],[3]
The transfer function which forms the basis of roll autopilot design is φ˙ Kδ (s) = δ ωRR
1 s 1 + ωRR
(1.1)
where φ˙ is the roll rate, δc is the fin deflection command, Kδ is the fin effectiveness and ωRR is the roll rate bandwidth. This transfer function model can be transformed into a simplified model given by φ¨ = −ωRR .φ˙ + Kδ .δc ˙ φ(0) = φ˙ 0
(1.2) (1.3)
The controller requires roll angle φ as a state which cannot be directly measured. ˙ Thus the roll angle can be calculated The ideal rate gyro can measure the roll rate, φ. by integrating the output of the gyro, φ˙ m , to obtain the estimated roll angle, phim . Denoting p = φ˙ m , the augmented system using eqn.1.3 can be described by two first order differential equations given by φ˙ m = p
(1.4)
p˙ = −ωRR .p + Kδ .δc
(1.5)
The above set of equations can be expressed in state space model as x˙ = A x + B u
(1.6)
2
" where x =
#
p "
B=
1.2.1
φm
0 Kδ
" ,A=
0
1
0 − ωRR
# ,
# .
Fundamentals of Optimal Control
For the optimal control problem, given an initial state x(t0 ), we want to find a control vector u that drives the state x(t0 ) to the desired final state xd (tf ) in such a way that a selected performance index of the form
Z J=
g(x, u, t)dt
(1.7)
is minimised. The functional form of the performance index can be expressed in a variety of forms. One useful form which includes a quadratic index and a penalty for physical constraints such as expenditure of control energy is given as
Z J=
xT Q x + uT R udt
(1.8)
where Q and R are weighting matrices. If we apply the principle of calculus of variations to the minimisation of the performance index, we obtain the Riccati equation. The Riccati equation is a set of non-linear differential equations which must be solved for the Riccati gains S(t) and is given as = S(t)BR−1 B T S(t) − S(t)A − AT S(t) − Q
(1.9)
For the case in which final time tf approaches infinity, the Riccati gain matrix, S(t), becomes a constant matrix, S, and reduces to
SBR−1 B T S − SA − AT S − Q = 0
(1.10) 3
Using the quadratic performance index defined in eqn.1.8, it can be shown that for a linear feedback control, the optimal control law is u = R−1 B T S x
1.2.2
(1.11)
Application to Missile Roll Autopilot
For the given problem of eqn.1.6, the quadratic performance index that is to be minimised is
Z
inf
J=
[( 0
δCM X 2 2 δCM X 2 ˙ 2 ) φ +( ) φ + δc 2 ]dt φM X φ˙ M X
(1.12)
where φM X is the maximum desired value of roll angle φ, φ˙ M X is the maximum desired value of roll rate, φ˙ and δCM X is the maximum desired value of fin deflection command δc . Comparing eqn.1.8 and 1.12 gives # "δ 2 CM X 0 2 Q = φM X δCM X 2 0 φ˙ 2
(1.13)
MX
and R = 1. Another form can be
Z
inf
J=
[ 0
1 φM X
2 2φ
+
1 φ˙ 2M X
φ˙ 2 +
1 δCM X
2 δc
2
]dt
(1.14)
in which case
" Q=
and R =
# 0
1 φM X
0
2
(1.15)
1 φ˙ 2M X
1 . δCM X 2
4
Substituting the matrices A, B, Q and R into the Riccati eqn.1.10, with the Riccati gain matrix given by the symmetric matrix " # S11 S12 S12
S22
(1.16)
the matrices can be expanded and simplified to arrive at the following equations:δCM X 2 2 2 2 − S12 Kδ = 0 φM X
(1.17)
S11 − ωRR S12 − S12 S22 Kδ 2 = 0
(1.18)
,
and 2S12 − 2ωRR S22 +
1.2.3
δCM X 2 − S22 2 Kδ 2 = 0 2 ˙ φM X
(1.19)
Solved Example
Using the table given [1] repeated below,
Figure 1.1: Nominal Values of Inputs for Roll Autopilot [1]
the control law can be calculated as u = −3φ − 0.103φ˙
(1.20)
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References [1] Nesline, F. W. Wells, B. and Zarchan, P., “Combined Optimal/Classical Approach to Robust Missile Autopilot Design,” Journal of Guidance and Control , Vol. 4, No. 3, May-Jun 1981, pp. 316–322. [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] Robert C.Nelson, D., Flight Stability and Automatic Control , McGraw Hill Book Company, New York, 1989. [4] Garnell, P., Guided Weapon Control Systems, Brassey’s Defence Publishers, London, 1980.
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