A study of advanced guidance laws for maneuvering target interception Student: Felix Vilensky Supervisor: Mark Moulin Control & Robotics Laboratory

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).

Plant - Interception problem

PN controller Plant PN Controller PN Controller TargetAcceleration Model Model

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.

PN controller-Command acceleration and relative distance vs. time

PN controller – Capturability limits Initial parametersCapturability limits <86977 <-1683 <= >=0.04 >=0.174 ( ,0.0232)

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.

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 won’t change too rapidly for the missile to follow.

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

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.

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 period:100ms. Plant sampling period: about 50 ms. Time dependant LQR

This linear system we use in each calculation: The following J parameter is being minimized:

Time dependant LQR LQR calculator Plant TargetAcceleration Module Module LQR controller LPF and saturation saturation clock State vector Gain vector controller

Time dependant LQR – Command acceleration behavior

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.

Optimal Guidance Law The OGL is obtained using the following linearization of the plant: Where: Thangavelu,R.,and Pardeep,S.(2007) A differential evolution tuned Optimal Guidance Law.

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.

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

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

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

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

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)

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.

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 doesn’t update linearization and thus applies non optimal command acceleration.

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.