Theoretical aspects of high-speed supercavitation vehicle control

Theoreticalaspects of

HSSV Control

MotivationProblem Description

Vehicle Configuration

ControlDesignControl Approach

ControllabilityAnalysis

Simulation

SummaryReferences

Theoretical aspects of High-SpeedSupercavitation Vehicle Control

Bálint Vanek1 József Bokor1,2 Gary J. Balas1

1Department of Aerospace Engineering and MechanicsUniversity of Minnesota

2on sabbatical leave from Hungarian Academy of SciencesThis work was supported by ONR, Dr. Kam Ng program officer

American Control Conference, Minneapolis 2006


HSSV Control





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Outline

1 MotivationProblem DescriptionVehicle Configuration

2 Control DesignControl ApproachControllability AnalysisSimulation

3 SummaryReferences

(Image:PSU/ARL)


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Supercavitation

• Pressure of fluid dropsdue to high speed,leading to vaporization

• Reduced skin frictiondrag

• Planing force can beused to sustain the tail

• Transition tosupercavitation needs effort

• Control surfaces immersionchanges

• Large, nonlinear planingforces


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Longitudinal Vehicle States

Cavity envelopeCavity centerline

Torpedo’s symmetry line

imm.

imm.

fin

cav

p

fin

g

cav

• The switching hyperplane depends on the delayedstate variable x(t − τ),

• First mode the system dynamics is linear, second modeis nonlinear input affine

• Switching condition does not depend on the controlinputs


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System Equations

x(t) = Ax(t) + Bu(t) + Fg + Fp(t , x , δ) (1)

Bimodal, Switched System:

Fp(x , δ) = P(1− R′

h′(x , δ) + R′ )2(

1 + h′(x , δ)

1 + 2h′(x , δ))α(x , δ), (2)

h′(x , δ) =

{R−1c(δ)x(t) if c(δ)x(t) ≤ 0,

0 if c(δ)x(t) ≥ 0,(3)

α(x , δ) =

{cα(δ)x(t)− V−1Rc if c(δ)x(t) ≤ 0,

0 if c(δ)x(t) ≥ 0,(4)


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Control Approach

• Feedback laws for each mode transforms system to LTI• Control design synthesized in a new multivariable

canonic coordinate frame• Extending controllability results on bimodal switched

LTI systems to time delayed, switched, with the analysisof time delayed zero dynamics

• Tracking problem using multivariable pole placement


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Control Architecture

Vehicle

fincavitator

Feedback Linearizing Controller

cB.Actv

gravF 0x

∫ sC

cA

planeF

δ

(Bimodal)

sy

cx

)( τ−txc

(switching)


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Feedback LinearizationInput variables enter linearly in both modes. Coordinatetransformation for canonical coordinates:

Tc =

1 0 0 00 −V 1 00 1 0 00 0 0 1

(5)

Linear dynamics in new coordinates:

Ac =

0 1 0 0

−α110 −α111 −α120 −α1210 0 0 1

−α210 −α211 −α220 −α221,

Bc =

0

c1AB0

c2AB

(6)

Two outputs: y1 = x1 and y2 = x3, vector relative degree inboth modes are identically (2,2).


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State Feedback

Applying a switched nonlinear feedback:

uflc =

{(CAB)−1(y13(t)− Fαx(t)− Fg + vI(t)(CAB)−1(y13(t)− Fαx(t)− Fg − Fp(x , δ) + vII(t)

(7)the switching condition is given by the sign of ys = c(δ)xc

• Nilpotent system with identical linear dynamics in bothmodes

• The system is continuous on the switching hypersurface• The relative degrees are equal in both modes


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System Decomposition

Selecting a direction p ∈ Im{B} such that the system is leftand right invertible:

x = Ax + pup + Bu, ys = Cx . (8)

the largest (A, p) - invariant subspace in ker{C} and thesmallest (C, A) invariant subspace over Im{p} induce thefollowing decomposition:

ξ = A11ξ + γv (9)

up =1γ

(−A12η − B21u + v) (10)

η = Pη + Qu + Rys, (11)

Last equation denoting the zero dynamics on the switchinghyperplane assuming Q is monic.


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Controllability ConditionsIf the pair (P, Q) is controllable, then η is controllablewithout using ys,If the pair (P, Q) is not controllable, then usingunconstrained u and nonnegative ys

1 The pair (P, [Q R]) has to be controllable.2 Consider the decomposition induced by the reachability

subspace R(P, Q),

η1 = P11η1 + P12η2 + Qu + R1ys (12)η2 = P22η2 + R2ys, (13)

where R2 6= 0. Then the imaginary part of the eigenvaluesof P22 cannot be zero.


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Application for the HSSV

HSSV has time delay dependency Discretize the systemusing backward difference scheme

Cd = [1, 0, L, 0,−1] (14)

The relative degrees are identically r = 2 for both modes.Zero dynamics obtained:

Tcd =

cs

csA1 0 0 0 00 0 1 0 00 β41T 0 −β21T 0

(15)


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Switched Zero DynamicsDecomposing the system to: [ξT (t), ηT (t)]T = Tcdx(t)

ξ(t + 1) =

[0 a120 a22

]ξ(t) +

[0 0 00 e22 e23

]η(t)+

+

[0

b21

]v1(t) +

[0

f22

]v2(t) (16)

ys =[1 0

]ξ(t) switching condition (17)

η(t + 1) = Pη(t) + Rξ(t) + Qv2(t), (18)

where

P =

p11 p12 p130 p22 p230 0 p33

, R =

0 r120 r220 0

, Q =

00

q31

. (19)

The zero dynamics are described by the last equation.The HSSV is controllable.


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Tracking: Multivariable PolePlacement

Feedback linearized system

∫cT pplF

FLA

FLBrefx

FLxv 1−cT

xdtd

11−M

[u1(t)u2(t)

]= (CAB)−1(

[x1(t)x2(t)

]ref− [αu]

[x1(t)x2(t)

]−

− [αl ]

[x3(t)x4(t)

]− [Gc]− [Pc(t , τ)]−

[v1(t)v2(t)

]) (20)


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Reference Tracking Controller

[v1(t)v2(t)

]= [αu]

[x1(t)− x1,ref (t)x2(t)− x2,ref (t)

]+ [αl ]

[x3(t)− x3,ref (t)x4(t)− x4,ref (t)

](21)

• The system behaves the same regardless of theinterior switching state

• One linear outer loop controller can guarantee stabilityand appropriate tracking

• Position and angle command dynamics are decoupled


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Vehicle Parameters

Parameter Description Value and Units

g Gravitational acceleration 9.81 ms2

m Density ratio, ρmρ 2

Rn Cavitator radius 0.01905mR Vehicle radius 0.0508mRc Cavity radius at tail 0.0647mL Length 1.8mV Velocity 75m

sσ Cavitation number 0.035

C1 Cav. lift coefficient 2685 N/radC2 Fin lift coefficient 1343 N/rad


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Simulation Setup

• Obstacle avoidance maneuver, 75m/s, 17.5mtranslation within 4 s

• Pressure, temperature, viscosity etc. are constant• 10% disturbance on cavity wall• Model mismatch due to actuators (Gact = 1

200s+1 )• Trajectory reference commands and reduce limit cycle

oscillations.


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17.5 m Maneuver

0 1 2 3 4 5 6 7−10

0

10

20z

(m)

Basic

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

time (s)

h (

m)

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

pit

ch r

ate

(rad

/sec

)

Basic

0 1 2 3 4 5 6 7−4000

−2000

0

2000

4000

time (s)

Fin

Fo

rce

(N)

• High planing depth induces high actuator deflections• Good reference tracking• High pitch rate oscillations and accelerations due to the

cavity disturbance


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Role of Disturbance Model

0 1 2 3 4 5 6 7−1

−0.5

0

0.5

1

pit

ch r

ate

(rad

/sec

)

High Noise

0 1 2 3 4 5 6 7−5000

0

5000

time (s)

Fin

Fo

rce

(N)

0 1 2 3 4 5 6 7−0.5

0

0.5

pit

ch r

ate

(rad

/sec

)

Slow Noise

0 1 2 3 4 5 6 7−4000

−2000

0

2000

4000

time (s)

Fin

Fo

rce

(N)

• Higher noise ⇒ higher pitch rate and accelerations ontail but same immersion depth

• Control scheme is sensitive to bandwidth ofdisturbance model


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Different Amplitude Maneuvers

0 1 2 3 4 5 6 7−4000

−2000

0

2000

4000

Fin

Fo

rce

(N)

0 1 2 3 4 5 6 7−2000

−1000

0

1000

Cav

. Fo

rce

(N)

time (s)

BasicAMP 15AMP 20

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

pit

ch r

ate

(rad

/sec

)

0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

time (s)

pla

ne

dep

th (

m) Basic

AMP 15AMP 20

• Planing depth is influenced by maneuver amplitude• Fin deflection is not changed• Cavitator deflection is more sensitive


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Summary• Supercavitation is a promising way to increase the

speed of underwater vehicles• Control design challenges including delayed state

dependency, nonlinearities and switching with noisyswitching surface were analyzed

• The controllability problems related with the systemwere analyzed

• An inversion based control methodology was developed• A successful implementation of a two-loop control

strategy was demonstrated• Further development requires increased collaboration

between fluid and control researchers

Future goals: three dimensional trajectory tracking(asymmetric fin immersion and non-vertical planing forces),robust constraint fulfillment, control the vehicle using onesurface.


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References

G.J. Balas, J. Bokor, B. Vanek and R.E.A. ArndtControl of Uncertain Systems: Modelling,Approximation and Design.Control of High-Speed Underwater VehiclesSpringer-Verlag, 2006.

DARPA Advanced Technology OfficeUnderwater ExpressBAA06-13 Proposer Information Pamphlet (PIP), 2005.

G.J. Balas, Z. Szabó and J. BokorControllability of bimodal LTI systemsaccepted to: IEEE Transactions on Automatic Control,2006.

Theoretical aspects of high-speed supercavitation vehicle control

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