CONTROL STRATEGIES FOR A CLASS OF

1
CONTROL STRATEGIES FOR A CLASS OF
SUPERCAVITATING VEHICLES ANUKUL
GOEL 8/23/2002 Masters Thesis Defense
Advisors Dr. Andrew J. Kurdila
Dr. Rick Lind Committee Dr. Norman Fitz-Coy
2
OUTLINE

• Introduction and Motivation
• Equations of Motion
• Control Objectives
• LQR Control Synthesis
• Nonlinear Simulation
• Conclusion

3
INTRODUCTION

• Motivation

• Navy needs next generation torpedoes and high
  speed underwater
• vehicles such as SUPERCAVITATING VEHICLES
• Limited research available in public domain for
  modeling and
• control of supercavitating vehicles.
• This project was to model and control and
  prototypical
• supercavitating torpedo.

• My contributions to the project

• Equation of motion (Anand and Anukul)
• Linear equations of motion (Anand and Anukul)
• Control Synthesis (Anukul)
• Nonlinear Simulations (Anukul)

4
CAVITATION

• At High Velocity

• Fluid Velocity Increases
• Fluid Pressure Drops below
• Vapor Pressure
• Fluid Vaporizes

Tip Vortex Cavitation
Supercavitation
5
ISSUES WITH SUPERCAVITATION
Advantages
Disadvantages

• High Frequency Motions
• Prediction of Cavity
• Control and Maneuvering
• Maintaining the Cavity

• Low Skin Friction Drag
• High Velocities

6
VEHICLE MODEL CONTROL SURFACES
Lc
Le1
De1
Dc
W
Cavitator

• 2 Elevators
• 2 Rudders
• 1 Cavitator

1 DOF Fin
1 DOF Cavitator
7
CAVITATOR/FIN MODEL
Cl for Cavitator
Half angle 150
Cl for Fin
Angle of attack (deg)
Sweepback angle0 deg
Angle of attack (deg)
8
EQUATIONS OF MOTION NONLINEAR

• Equations of motion are identical for torpedo
  and aircraft.

• Consider force equation

• Additional 9 equations for moment, orientation
  and position.

• Components of forces and moments are unique for
  torpedo.

• Consider force in body fixed X-axis direction.

• Similar decomposition for all forces and moments.

9
EQUATIONS OF MOTION LINEAR
10
OPEN-LOOP DYNAMICS

• Consider root locus of open-loop poles for
  linearizing airspeed

• Open-loop system is always unstable
• No Phugoid mode (Similar to F-16 aircraft)

11
CONTROL OBJECTIVE

• Track a pitch or roll rate command up to 30
  deg/s
• and

• maintain an overshoot less than 15.
• have rise time less 0.5s.
• have a steady-state error less than 5.

• Avoid actuator saturation

Control Surface Limits
12
ASSUMPTIONS FOR CONTROL SYNTHESIS

• The cavity model is fixed cavity. Thus there is
  no variation of immersion
• and the torpedo is symmetrically situated in
  the cavity.
• Longitudinal controls are cavitator and
  horizontal fins only
• Lateral controls are 2 vertical fins (rudders).
• It is assumed that 9 states (velocities, angular
  rates and orientation ) are
• available for feedback for tracking
  controller.
• It is assumed that control surface deflection of
  small order (0.1 deg for
• longitudinal and 0.01 for lateral) are
  achievable.
• Propulsive force is assumed to be constant during
  flight, the constant value
• being obtained during trim optimization.

13
LQR SYNTHESIS

• Find Controller K such that u-Kx minimizes

• x is the state
• u is the control from K
• Q and R are weighting matrices

• In case of longitudinal feedback 4 states and
  tracking error

• In case of lateral feedback 5 states and
  tracking error

14
LQR LONGITUDINAL

• More weighting on cavitator to ensure more use
  of elevator
• Weighting of zero on all states to allow for
  their variation

Cavitator Deflection
Linear Response
Cavitator (deg)
q (deg/s)
Time (s)
Time (s)
15
LQR LATERAL
Linear Response
Rudder Deflection
Rudder (deg)
Roll rate (deg/s)
Time (s)
Time (s)
16
NONLINEAR SIMULATION
de1 (deg)

• Command 15 deg/s doublet
• of pitch rate

de1 rate (deg/s)

• Rise Time 0.17s
• Overshoot 11.53
• Minimal lateral response
• Unstable for commands gt 30 deg/s

Time (s)
17
NONLINEAR SIMULATION
dr1 (deg)
p (deg/s)
Time (s)

• Command 15 deg/s doublet
• of roll rate

q (deg/s)

• Rise Time 0.5s
• Overshoot 0
• Coupled motion
• Unstable for commandsgt50 deg/s

Time (s)
18
ROBUSTNESS GAIN/PHASE MARGINS
x
19
ROBUSTNESS GAIN/PHASE MARGINS
Longitudinal (Inner and Outer loop) Gain Margin
inf Phase Margin 57deg
Magnitude (dB)
Phase(deg)
Frequency (rad/sec)
Magnitude (dB)
Lateral (Inner and Outer loop) Gain Margin
50.36 dB Phase Margin inf
Phase(deg)
Frequency (rad/sec)
20
ROBUSTNESSPERTURBED RESPONSE

• 20 Perturbation given in values of cl, cd,
  immersion.
• Small variation in A, B matrices.
• Stable eigenvalues remain stable for variation
  with velocity.
• Closed-loop response was acceptable with nominal
  controller.

• Tracking objectives are met
• Small variations in airspeed

21

• 10 uncertainty in coefficients of lift and drag
  of cavitator
• introduced into system during synthesis.
• Good Lateral performance obtained.
• Good Longitudinal controller not obtained
  possibly as the
• controller tries to stabilize the uncertain
  poles.

22

• Higher overshoot and
• rise time.
• Saturation for commands
• higher than 20 deg/s.

p (deg/s)
Time (s)

• Rise time 0.025s
• Overshoot 0
• Saturation for commands
• higher than 40deg/s

p (deg/s)
Time (s)
23
CONCLUSIONS

• A model for supercavitating torpedo has been
  achieved
• A LQR controller has been obtained than gives
  good
• performance for pitch and roll rate
  tracking.
• A controller for lateral dynamics has been
  obtained
• that shows good performance.
• The controller for longitudinal is yet to be
  obtained.
• The model for cavity is assumed to be fixed.
  Dynamic
• model for cavity is yet to be obtained.
• The Pitch and Roll rate controllers are for
  inner-loop
• model. An outer-loop controller can be
  obtained for
• guidance and navigation

FUTURE WORK