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- Strix – Saab Bofors (BAE)
- Precision Guided Mortar Munition (PGMM) – ATK
- Projectile Equations of Motion
- Controllability
- Bang-Bang Control
- Impulsive dynamical systems

- “Naïve” control strategy
- Simulations
- Future directions

STRIX Main Parts

Package

Programming

unit

Launch unit

Projectile

Sustainer

Fuze System

Impact Sensor

Electronics & Power Supply

Control Rockets Assembly

Warhead

Fin Assembly

Target Seeker

3. Ballistic phase

4. Guidance phase

Forward observer

2. Launch

1. Preparations

Guidance Phase

Hit

Find

Sustainer

separation

Electric

arming

Guidance with

control rockets

Target seeker

activation

Target

acquisition

and

selection

Proportional

navigation

STRIX Target Impact

- KILL
- Initiation of warhead from impact sensor
- Penetration of ERA and main armour
- Behind armour effect (pressure etc.)

Projectile Model

Equations of Motion

Forces:

Equations for Rotational rates

Moments

Definition: Given the system

Controllability: The pair (A,B) is said to be controllable iff at the initial time t0 there exist a control function u(t) which will transfer the system from its initial state x(t0) to the origin in some finite time. If this statement is true for all time, then the system is "Completely Controllable".

- Is the “full-information” nonlinear model of the projectile, with no wind, controllable such that it will land within a terminal set T, for a given number of discrete, fixed magnitude impulses?
- Note that the control impulses have additional constraints which include:
- each control impulse can only be fired once
- presences of a dwell-time between firings
- finite burn time

- Three approaches to the nonlinear controllability problem with finite, discrete impulses are investigated:
- Bang-Bang control
- Impulsive dynamical systems
- Naïve control design

- The problem is to find a feasible bang-bang control action that takes the system from a given initial point to a given terminal point with time being a free parameter.
- Minimum fuel optimal control problem
- Unfortunately, the theory of minimum-fuel systems is not as well developed as the theory of minimum-time systems. Also the design of fuel-optimal controllers is more complex that time-optimal controllers. In fact there may not exist a fuel-optimal control, with a finite number of discrete thrusters, that drive the projectile from any initial state to the origin (controllable).

- Many systems exhibit both continuous- and discrete-time behaviors which are often denoted as hybrid systems. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist of three elements:
- a continuous-time differential equation, which governs the motion of the dynamical system between impulsive or resetting events;
- a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs;
- and a criterion for determining when the states of the system are to be reset.

The projectile control problem can be viewed as an impulsive dynamical system, whose analysis can be quite involved. In the general situation, such systems can exhibit infinitely many switches, beating, etc.

Controllability of hybrid systems is a hot topic currently, and despite the numerous papers on the topic efficient numerical algorithms that provide control algorithms is still lacking.

- Projectile control algorithms are often synthesized in an ad hoc manner. These solutions are logic based and involve testing a performance criteria at each time step.
- Consider the following control strategy to drive a projectile from a given state to a target set:
- If current state in the target set, STOP
- Given current point (after apex). If an impulse is not active then compute the corresponding impact state and the miss distances.
- Numerically integrate EOM to determine impact location

- If miss distance is within tolerance, NO ACTION taken.
- If miss distance is less than target set, FIRE in positive direction
- If miss distance is more than target set, FIRE in negative direction
- Under some conditions, NCS results in optimal solution.

- Ideal assumptions include:
- Restricting the problem to two dimensions (x,y)
- No wind, target location, projectile position/velocity known
- Each impulse can be fired more than once
- Infinitely many impulses
- Point mass model
- Physical parameters:
- weight, 33 lbs
- muzzle velocity and angle: 235 m/s, 50 degs
- impulse duration and magnitude: 0.015 +/- .0002s, 5.0 +/-0.3 g
- sample time,0.005s
- Impact error tolerance: 0.1m
- Unaided projectile path: 2772 m

Projectile Path

Undisturbed

+ 300 Meters

- 300 Meters

Point mass

model

Impact point computed exactly, 39 shots required

- Due to numerical errors, two extra shots were needed
- Chattering caused by numerical integration errors, which is typical of NCS algorithm

Rigid body

model

Impact point within 0.1m, 9 shots required

- Due to numerical errors, series of extra shots were needed

Impact Distribution for +200m Target

Impulse Distribution

Positive impulse

Negative impulse

8 shots

Tradeoff Accuracy

- Total impulse force constant, #shots x impulse = 24*5g
- More shots: Increased accuracy, more complex
- Less shots: Less accurate, cheaper

Initial Findings

- It is easier to hit targets beyond the initial trajectory
- Function of the limited flight time of the projectile and computation delay
- If the target is overshot, the projectile may not be able to react fast enough to bring it down in time.
- Current configurations allow for no more than a 225 meter overshoot and 310 meter undershoot

Summary

- Interesting class of control systems for which there has been a limited amount of theoretical results
- For the short term, focus on better understanding the naïve control strategy
- Rigid body equations of motion
- Atmospheric disturbances
- Trajectory tracking versus end point control

- Over the long term, develop a mathematical framework for control of nonlinear systems with a finite number of discrete, finite duration, fixed magnitude impulses.

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- B. Friedrich, ATK, Private Communication.
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