A study of advanced guidance laws for maneuvering target interception

1
A study of advanced guidance laws for maneuvering
target interception
Control Robotics Laboratory

• Student Felix Vilensky
• Supervisor Mark Moulin

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Introduction

• This project deals with missile-target
  interception. This is a highly non-linear and non
  stable control problem.
• We work with a simplified 2D model.
• We will discuss the following guidance laws
• PN (Proportional Navigation).
• Saturated PN.
• TDLQR(Time dependant LQR).
• OGL (Optimal guidance law).

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Plant - Interception problem
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PN controller
Target Acceleration Model
Plant
PN Controller
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PN controller - references

• Dhar,A.,and Ghose,D.(1993)
• Capture region for a realistic TPN guidance law.
• Chakravarthy,A.,and Ghose,D.(1996)
• Capturability of realistic Generalized True
  Proportional Navigation.
• Moulin,M.,Kreindler,E.,and ,Guelman,M(1996).
• Ballistic missile interception with
  bearings-only measurements.

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PN controller-Command acceleration and relative
distance vs. time
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PN controller Capturability limits
Initial parameters Capturability limits
lt86977
lt-1683
lt -0.82
gt0.04
gt0.174
(-0.0068,0.0232)
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PN controller -conclusions

• The performance of the PN controller is quite
  good. It enables intercepting a target in a wide
  range of initial conditions.
• Yet, the command acceleration is growing with
  time. And a real physical system cannot maintain
  an acceleration that is growing towards non
  physical values.
• We seek to find a controller that will work under
  the constraint of limited (saturated) command
  acceleration.

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Saturated PN

• The first and naive approach is to retain the PN
  controller and just to add at its output a
  lowpass filter and a saturation to get the
  command acceleration.

LP filter

• We use a Butterworth LPF of order 30 with cutoff
  frequency of 8 rad/sec. This filter will ensure
  that the command acceleration wont change too
  rapidly for the missile to follow.

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Saturated PN controller - Command acceleration
and relative distance vs. time
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Linear controllers

• Linear or partially linear controllers are easier
  to design than nonlinear ones.
• Linear control can be more easily optimized than
  nonlinear one.
• Recent papers used linear control design methods
• Hexner,G.,and Shima,T.(2007)
• Stochastic optimal control guidance law with
  bounded acceleration.
• Hexner,G.,Shima,T.,and Weiss,H.(2008)
• LQG guidance law with bounded acceleration
  command.

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Time dependant LQR

• Calculate every fixed interval of time (T) a new
  infinite horizon LQR.
• Use the following state variables
• Each time linearize the plant around
• i.e., around the relative speed and the
  distance at the time of calculation.
• Using this LQR controller till the next
  calculation.
• LQR recalculation period100ms.
• Plant sampling period about 50 ms.

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Time dependant LQR

• This linear system we use in each calculation

• The following J parameter is being minimized

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Time dependant LQR
Target Acceleration Module
State vector
Plant
controller
clock
LQR calculator
LQR controller
LPF and saturation
Gain vector
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Time dependant LQR Command acceleration behavior
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Pure LQR vs. TDLQR

• The TDLQR is designed using methods and
  intuition of optimal linear control.
• TDLQR is linear only in each time slice between
  calculations.
• There is a well known LQR guidance law, which is
  linear through all the engagement time. It is
  called OGL Optimal Guidance Law.
• While TDLQR is based on infinite horizon LQR,
  the OGL is a finite horizon LQR, which means that
  its control gain varies with time.

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Optimal Guidance Law
Thangavelu,R.,and Pardeep,S.(2007) A differential
evolution tuned Optimal Guidance Law.

• The OGL is obtained using the following
  linearization of the plant

• Where

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Optimal Guidance Law

• The OGL minimizes

• The optimal control is given by

• The OGL output is then passed through LPF an
  saturation, as explained earlier to get the
  command acceleration.

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Performance Analysis

• Miss distance vs. initial relative speed for PN
  (left) and saturated PN (right) controllers.

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Performance Analysis

• Miss distance vs. initial relative velocity for
  Time Dependant LQR, saturated PN and OGL
  controllers.

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Performance Analysis

• Miss distance vs. maximal command acceleration
  for Time Dependant LQR and saturated PN (left)
  and for OGL (right).

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Performance Analysis

• Miss distance vs. initial distance for Time
  Dependant LQR, saturated PN and OGL controllers.

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Performance Analysis

• Miss distances vs. initial distances for Time
  Dependant LQR, saturated PN and OGL controllers
  (for relatively small initial distances).

Initial distance Saturated PN (miss distance) Time dependant LQR (miss distance) OGL(miss distance)
10000 304.6893 301.162 250.6541
20000 231.6058 130.8969 18.3121
40000 26.603 182.5942 1811.2
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Performance Analysis-Results

• The effect of the lowpass filter and saturation
  block provides an optimal initial relative
  velocity for interception.
• The capturability is improved when the maximal
  command acceleration is increased for TDLQR and
  saturated PN, and gets worse for OGL.
• For large initial distances the miss distance
  grows monotonically with the initial distance.
• For small initial distances an optimal initial
  distance results in a minimal miss distance.

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Performance Analysis-Conclusions

• TDLQR has a clear advantage in the performance
  evaluation over both saturated PN and OGL .
• The miss distance for OGL is growing with the
  maximal command acceleration. OGL doesnt update
  linearization and thus applies non optimal
  command acceleration.

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Performance Analysis-Conclusions

• The optimum of initial relative velocity is
  obtained, since for high velocities the command
  acceleration cannot be high enough to complete
  the maneuver needed to get the missile into
  collision course with the target.
• The optimum of initial distance is obtained,
  since for small enough initial distances the
  missile covers too much distance (outruns the
  target) before the maneuver needed to get it into
  collision course with the target is completed.