Control of High-Speed Underwater Vehicles

Control of High-Speed Underwater Vehicles

Gary J. Balas1, Jozsef Bokor2, Balint Vanek1, and Roger E.A. Arndt3

1 Aerospace Engineering and Mechanics, 110 Union St. SE, University ofMinnesota, Minneapolis, MN 55455 [email protected], [email protected]

2 MTA SZTAKI, Hungarian Academy of Science, Budapest XI Kende u. 13-17,H-1518 Budapest, POB 63, Hungary [email protected]

3 Civil Engineering and St. Anthony Falls Laboratory, 2 Third Ave. SE,University of Minnesota, Minneapolis, MN 55414 [email protected]

I (GB) was fortunate that early in my career I had the opportunity towork with Keith Glover. Keith had just begun his obsession with golf. Asimple game where a ball stands still and the player advances it from differentparts of the landscape into a hole. During a golf outing, Keith vigorouslystruck a ball off the fairway. It drove through a water hazard and miraculouslyended up a few feet off the green. I was amazed by the image I saw as theball moved through the water. Due to its velocity, a cavitation bubble wasgenerated behind it. I had no inkling that my golf outing with Keith wouldlead many years later to an interest in the control of supercavitating vehicles.

1 Introduction

An object traveling at very high speeds generates cavitation bubbles on thecorner of sharp contours due to pressure gradients that result in flow separa-tion. This is often the case if a propeller spins fast enough, the surroundingliquid is vaporized due to the decrease in pressure generating cavitation bub-bles [1]. As pressure surrounding the liquid decreases, the bubbles collapse.

B.A. Francis et al. (Eds.): Control of Uncertain Systems, LNCIS 329, pp. 25–44, 2006.© Springer-Verlag Berlin Heidelberg 2006

26 G.J. Balas, J.Bokor, B.Vanek, and R.E.A. Arndt

Cavitation bubbles contain significant energy that can lead to pitting on theblades and reduced efficiency. Minimization or reduction of cavitation is the fo-cus of significant engineering effort. Alternatively there is a significant benefitin some circumstances to increased cavitation or supercavitation. Supercavi-tation can be used by vehicles to dramatically reduce drag, enabling vehiclesable to maintain high high speed flight under water.

Supercavitation results in a single, stable bubble that can envelope anentire vehicle. Hence only the front tip of the vehicle, upon which is oftenmounted a cavitator, is in contact with the fluid. The cavitator is designedto create a low pressure cavity around the vehicle when traveling through thefluid at high speeds. The cavity may be ventilated with gas to increase itssize and reduce the operating speed needed to generate a cavity. Drag on asupercavitating vehicle is dramatically reduced due to the vehicle flying in agas rather than a fluid. The Russian military has exploited the benefits ofsupercavitation with the development of the Shkval torpedo which is claimedto reach speeds of 100 m/s [2]. This is roughly a factor of 5 faster than thefastest traditional underwater vehicles. A picture of Shkval vehicle is shownin Figure 1.

Fig. 1. Shkval vehicle

Steering and stabilization within the cavity area are significant technicalchallenges facing high speed supercavitating vehicles (HSSVs) [3]. The stabil-ity of the vehicle is a function of the cavitator size, shape and the size andimmersion of the control surfaces. The cavity generated by the cavitator sur-rounds the vehicle though they may interact at the vehicle trailing edge andat the points where the control surfaces impinge on the cavity wall. The lowbuoyancy acting on the body requires the vehicle weight to be supported bythe cavitator and control surfaces, if present. Body interaction with the cavitywalls which leads to high restoring forces known as planing. Planing can beused as a support force to improved maneuverability and eliminate the finactuators, there by reducing cost and complexity. Planing though can lead tolimit cycle oscillations and increases drag if not controlled. Hence, there aremany configurations and tradeoffs associated with control surface selection [5].

Control of High-Speed Underwater Vehicles 27

Even if the body is stable inside the cavity, the vehicle may not be stablewhen in contact with the cavity. Nonlinear interaction of the control surfacesand the body with the cavity wall is very important when calculating thefin and planing forces acting on the vehicle. Note that the distance betweenthe body and the cavity wall is on the order of 4-10 cm. The cavity wallexerts a large restoring force over the short period of time. The nature of thisinstability forces the vehicle back into the cavity, often resulting in limit cyclebehavior [3]. Cavity-vehicle interaction also exhibits strong memory effectsand cavity shape is a function of the history of the vehicle motion. Suppressionof limit cycles oscillations and disturbance attenuation require high bandwidthactuators and high sample rate real-time control algorithms.

This paper investigates the control challenges associated with supercavi-tating vehicles using a low order, longitudinal axis vehicle model. The first partof the paper presents the simplified pitch-plane model of a HSSV vehicle. Thismodel draws heavily on the benchmark HSSV model proposed by Dzielski andKurdila [3]. That model is extended to include delay dependent interactionof the vehicle and cavity and a more refined planing model. The second partof the paper describes the control approach applied to the supercavitatingvehicle model. The plant model is described as a switched, bimodal systemwhose switching surface, the cavity boundary, is delay dependent [9]. Con-trollability of delay dependent, bimodal systems is discussed and a systematiccontrol design strategy based on dynamic inversion is proposed. A controlleris synthesized to track pitch angle, angular rate, vertical position and verticalspeed for the HSSV vehicle model using the proposed approach. Simulationsof the closed-loop vehicle are performed and analyzed in the third section ofthe paper. Challenges facing the control designer are highlighted with respectto actuator and sensor requirements, modeling issues, robustness and perfor-mance. The paper concludes with a summary of the results and directions forfuture study.

2 Modeling of Supercavitating Vehicles

Several mathematical descriptions of high speed supercavitation vehicles(HSSVs) are available in the literature [3, 6, 8]. A brief synopsis of these modelsfollows.

• A high fidelity, nonlinear six degrees-of-freedom (DOF) mathematicalmodel was proposed in [8]. Lookup tables are used to calculate the forcesgenerated by control surfaces as a function of immersion depth. The cen-terline of the cavity bubble is a function of the time history of the path ofthe vehicle nose affected by buoyancy.

• A two DOF longitudinal axis model for control design purposes was pro-posed in [3]. The simplified model is linear within the cavity and nonlinearwhen the vehicle is in contact with the cavity (planing). The planing con-


dition is defined as a function of vertical velocity and no time historydependency of the cavity is included.

• A one DOF longitudinal axis model was proposed to study the switchingbehavior of the supercavitating vehicle in [6]. The model contains onlylongitudinal translational motion and its interaction with the cavity. Theshape of the cavity boundary is described as a function of the time delayedposition of the vehicle nose.

This paper extends the two DOF longitudinal axis model developed in [3]to include the time dependent behavior of the cavity and redefines the planingcondition to be a function of the vehicle vertical position, pitch angle and timehistory of the cavity. The only characteristic of the longitudinal axis, nonlinearsix DOF model not included in the extended model is the nonlinear controlsurface force and moments dependence on the surface immersion depth andangle-of-attack.

2.1 Two Degree of Freedom Model with Delay

A natural way to simplify the complex behavior of the 6-DOF model is torestrict the motion of the vehicle to the longitudinal plane. Consider the twodegree of freedom vehicle shown in Figure 2.

Fig. 2. Schematic diagram of two degree of freedom model

The geometry of the vehicle is intended to approximate the geometry ofthe actual vehicle. The body consist of a conical and a cylindrical section,where the cylindrical section is twice as long as the front conical section. Thereference coordinate system is placed into the cavitator’s center, the x axisis the horizontal axis (perpendicular to z) in direction of the cavitator. θ isthe angular rotation of the x-axis. The z axis is oriented towards the centerof Earth. The body length is denoted by L and its radius is R. The bodyhas uniform density ρb, from which the mass and inertia are calculated. Thereference point is at the cavitator, hence the inertia matrix is computed at


that point. If the full vehicle body is in the cavity, hydrodynamic forces willonly act on the cavitator and fins. The cavitator drag coefficient is modeledas Cx = Cx0(1+σ) where σ is the cavitation number and Cx0 = 0.82 [4]. Theresulting lift on the cavitator is approximately:

Fcav =1

2πρR2

nV 2Cxαc = Clαc (1)

where Rn is the cavitator radius, ρ water density, V vehicle velocity, δc cav-itator deflection and αc = w

V + δc is the cavitator angle of attack. The forceacting on the fins is modeled as

Ffin = −nClαf (2)

where n denotes the relative effectiveness of the fins to the cavitator, δf the

fin deflection, q pitch rate and αf = w+qLV + δf is the fins angle of attack. All

angles are assumed to be small, hence small angle approximations apply tocalculate the forces. The dynamic equations of motion, after dividing throughby (πρmR2L), are 7

917L36

17L36

1160R2 + 133

405L2

w

q

= CV

1−nmL

−nm

−nm

−nmL

w

q

(3)

+V

0 79

0 17L36

w

q

+ CV 2

−nmL

1mL

−nm 0

δf

δc

+ Fgrav + Fplane(t, τ)

where C = 12Cx

Rn

R

2. The time delay between the fluid impinging on the

cavity and the aft end of the vehicle is δ(t) = LV . The time dependence of the

states and controls is suppressed in the paper unless it is needed for clarity.The force and moments due to planing, Fplane [7], and gravity, Fgrav, are

Fplane = −V 2

1mL

1m

(1− R

h + R)2(

1 + h

1 + 2h)αp, Fgrav =

79

17L36

g, (4)

h is the normalized immersion depth corresponding to the distance the bodypenetrates the water and αp is the immersion angle corresponding to the anglethe body surface makes with the cavity. R is the normalized median distanceRc−R

R between the transom and the cavity. The parameter values used in thenumerical calculations are given in Table 1.

In the original two degree-of-freedom model proposed in [3], the interac-tion of the vehicle and cavity was not a function of the cavitator position.Hence it was not delay dependent. Vertical velocity, w(t), was used to pre-dict the occurrence of planing assuming zero pitch angle and steady state


Table 1. System parameters for simulation model [3]

Parameter Description Value and Units

g Gravitational acceleration 9.81 ms2

m Density ratio, ρm

ρ2

n Fin effectiveness 0.5Rn Cavitator radius 0.0191mR Vehicle radius 0.0508mL Length 1.80mV Velocity 75m

s

σ Cavitation number 0.03Cx0 Lift coefficient 0.82

horizontal speeds. For short delay times, this is a valid assumption. In theextended model, the planing condition is a function of the cavity boundaryand a delayed function of the cavitator position. This requires an extensionof the vehicle state space description by two additional states:

• z(t) vertical position of vehicle at the nose,dz

dt= w − V θ.

• θ(t) vehicle pitch angle,dθ

dt= q.

The centerline of the cavity is determined by the path of the nose throughthe fluid, see Figure 2. It is assumed that the diameter of the cavity at anygiven location depends only on the length between its centerline position andthe actual position of the nose. The resulting equations of motion are

z

θwq

= A

zθwq

+ Bδe

δc+ Fgrav + Fplane(t, τ) (5)

The matrices A and B can be derived from equations (3, 2.1) and thedefinitions of the added states z and θ. The vehicle motion as stated above isa nonlinear, switched delayed-state dependent system. Contact with the gasfilled cavity bubble is responsible for the nonlinear effects. Three situationsare possible during the longitudinal motion: (i) the entire vehicle is insidethe cavity no forces are generated by planing, (ii) the transom (aft body) isimmersed in the bottom of the cavity (+ direction), or (iii) the planing occursas the transom hits the top of the cavity. In both situations the resultingplaning force acts in the opposite direction of the immersion, providing a largeenough force to bounce the body back into the cavity, leading to oscillatorymotion like a fast boat bouncing on the top of water.

The planing conditions and immersion angle are given by the followingequations:


h =

1R [z(t) + θL + R− z(t− τ) −Rc] if z(t− τ) + Rc < z(t) + θL + R

0 (inside cavity)1R [z(t− τ)−Rc − z(t)− θL + R] if z(t− τ)−Rc > z(t) + θL + R

(6)

αp =

θ − w(t− τ) + Rc

Vbottom contact

0 inside cavity

θ − w(t− τ)− Rc

Vtop contact

(7)

It can be seen by the geometric relationships how the two different planingconditions are stated. The position of the vehicle transom is a function of thevehicle nose position, rotation and vehicle radius at the transom. The cavity’sboundary is located at Rc distance from centerline position of the cavity τtime before (z(t − τ)), the immersion is the difference between the two. Theimmersion angle is a function of three different variables: The nominal cavityshape is due to a nonconstant cavity radius through the length of the body(Rc), the shape of the cavity distorted by the past vertical velocity of thevehicle (w(t− τ)), and the vehicle rotation during boundary penetration, (θ).All these expressions use small angle approximations around straight levelflight, assuming also that the horizontal speed of the vehicle (V ) is constantand the surrounding environment including the water temperature, pressuredo not change. To simplify the nonlinear equations of the planing forces, twonumerical constants are defined [7]:

κ1 =L

Rn(1.92

σ− 3)−1 − 1, κ2 = (1− (1− 4.5σ

1 + σ)κ

40171 )

12 (8)

Using constants κ1 and κ2, the relationship for the cavity radius at the tran-som is:

Rc = Rn(0.821 + σ

σ)

12 κ2 (9)

The expression for the contraction rate of the cavity (Rc) is:

Rc = −20

17(0.82

1 + σ

σ)

12 V

(1− 4.5σ1+σ κ

23171 )

κ2(1.92

σ − 3)(10)

The equations for the cavity shape are valid if the following condition is sat-isfied:

L > Rn(1.92

σ− 3). (11)

The state space equations describing the system in equation (3) represent abimodal, switched system. There are some specific characteristics of this modelthat we will use later. These are (i) in the first mode the system dynamics is


linear (inside cavity) and in the second mode it is nonlinear (planing) inputaffine, i.e. the control inputs effect the dynamics linearly in both modes, (ii) theswitching condition does not depend on the control inputs, (iii) the switchinghyperplane depends on the delayed state variable z(t− τ).

To develop a control strategy for this system, the following considerationswill be applied. Properties (i) and (ii) allow for feedback linearization in bothmodes. This will be performed in a coordinate system that reflect the mostthe geometric structure of the problem. It will be shown that this designstep results in linear dynamics in both modes ensuring continuous dynamicson the switching hyper surface. Since the latter depends on delayed statevariables, controllability has to be analyzed and a controller has to be designedthat ensures stability and tracking performance. This is investigated in thefollowing section.

3 Theoretical Background of the Controller Design

Our approach relies on the assumption that the delay in the equations ofmotion can be eliminated by applying a suitable feedback. Then the control-lability analysis and the controller design can be performed for bimodal LTIsystems. Since the concept of relative degree plays a central role in this ap-proach, its definition for nonlinear, time delay and LTI systems will be givenfirst.

3.1 Feedback Linearization

Given a nonlinear input affine system:

x = f(x) +m

i=1

gi(x)ui, x ∈ X , u ∈ U (12)

yj = hj(x), yj ∈ Y, j = 1, . . . , p, (13)

has a vector relative degree r = [r1, . . . , rp], ri ≥ 0, ∀i, if at a point x0

(i) LgjLk

fhi(x) = 0, . . . j = 1, . . . , m, k < ri−1,

(ii) The matrix

AIA =

Lg1Lr1−1f h1(x), . . . , Lg1L

r1−1f h1(x)

...,

LgmL

rp−1f hp(x), . . . , Lgm

Lrp−1f hp(x)

(14)

has rank p at x0.


For linear time invariant (LTI) systems given by (A, B, C), we have thatLgjL

ri−1f hi(x) = ciA

ri−1bj and if p = m then the vector relative degree isdefined if rankALTI = n where n is the state dimension.

The concept of relative degree can be extended to time delay systems, too.Usually this is defined for a discrete time equivalent of the continuous timesystems by introducing the discrete time shift operator Δ as Δxt = xt−τ withτ denoting the given time delay.

The time delay system is given now by (A(Δ), B(Δ), C(Δ)), i.e. the ma-trices depend on the delay operator. This implies that the coefficients areelements of the polynomial ring R[Δ]. The relative degree is defined similarlyto the LTI case as follows.

Definition 1 Given the single input - single output linear time delay system(A(Δ), b(Δ), c(Δ)). It has relative degree r > 0 if cAkb = 0, k = 0, . . . , r− 1and cArb = 0. It has pure relative degree r if in addition cArb is an invertibleelement of R[Δ].

This definition has an obvious extension to the multivariable case. It re-quires the matrix ATD is invertible over Rp×p[Δ]. To perform the analysisand design a controller, new state variables for equation 3 are chosen as:

x1(t)x2(t)x3(t)x4(t)

=

z(t)

−V θ(t) + w(t)θ(t)q(t)

(15)

The matrix used for this coordinate transformation is:

Tc =

cT1

cT1 AcT2

cT2 A

=

1 0 0 00 −V 1 00 1 0 00 0 0 1

(16)

The state space equations in the new coordinate system are:

˙x =Acx(t) + Bcu(t) + Fg if cT (δ)x(t) ≤ 0,

Acx(t) + Fp(t, x, δ) + Bcu(t) + Fg if cT (δ)x(t) ≥ 0,(17)

where

Ac =

0 1 0 0

−α110 −α111 −α120 −α121

0 0 0 1−α210 −α211 −α220 −α221,

Bc =

0

cT1 AB0

cT2 AB

(18)

and Fgrav = TcFgrav + C1 where C1 is a constant associated with the shift inthe origin of the coordinate system. Similarly Fplane = TcFplane.


In our situation the inputs enter linearly in the state equations in bothmodes. In addition, it is assumed that all states can be measured. This allowsus to select two outputs defined as y1 = x1 and y2 = x3, such that there existspure vector relative degree in both modes, and in addition, these are identical,i.e. by defining:

Cc = 1 0 0 00 0 1 0

(19)

The relative degree for the modes are:

r21 = 2, r1

2 = 2, r21 + r1

2 = n = 4 Mode 1 (20)

r21 = 2, r2

2 = 2, r21 + r2

2 = n = 4 Mode 2 (21)

The consequence of this property is that one can apply state feedback inboth modes such that this will eliminate time delay in Mode 1 and nonlinearity(exact feedback linearization) in Mode 2.

This feedback is given by:

uflc =M−1

1 (y13(t) + Fαx(t) + Fg + vI(t)) if cT (δ)x(t) ≤ 0,

M−11 (y13(t) + Fαx(t) + Fg − Fp(x, δ) + vII(t)) if cT (δ)x(t) ≥ 0,

(22)where M1 = (CcAcBc)

−1, y13 = [y1, y3]T , and the feedback gain Fα is defined

by the controllability invariants αijk, i = 1, 2, j = 1, 2, k = 0, 1 of thelinear part A, B of the system.

Applying this control with identical external signals vI = vII = v, thefeedback linearized closed loop has the following form in both modes:

xFL = AFLxFL + BFLv, (23)

rwhere

AFL =

0 1 0 00 0 0 00 0 0 10 0 0 0

, BFL =

0 01 00 00 1

, (24)

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

3.2 Controllability Analysis of the Bimodal System

The controllability of the linearized, bimodal dynamics needs to be analyzedand a tracking controller designed. Results on controllability of single inputsingle LTI systems with single switching surface and relative degrees r = r1 =r2 = 1 has been published by [12].


In their approach the problem is reduced to analyzing the dynamics of thesystem on the switching surface. This is given by the zero dynamics derivedwith respect to the “switching output” ys. It was shown that the zero dynamicshave to be controllable when using positive ys in one mode (negative ys in thesecond mode, respectively). It was assumed that the system is both left andright invertible and the dynamics is continuous on the switching surface, i.e.A1x + b1u = A2x + b2u.

Since the relative degree r = 1, under the above assumptions, the zerodynamics can be written as:

η(t) = Hη +g1ys(t) if ys(t) ≤ 0,

g2ys(t) if ys(t) ≥ 0,(25)

with η ∈ Rn−1.It can be proved that the problem (25) is equivalent to the following (sign

constrained) switching problem with systems (H, g1) and (H,−g2) with a non-negative input u, see [19]. Then the same results can be obtained using thefollowing reasoning: in the unconstrained input case to compute the reachabil-ity set the following Lie-algebra of the vector spaces Hη + g1u and Hη− g2u,i.e., the Lie-algebra generated by Hη + [g1 − g2][u u]T , needs to be defined.Denote this set by R(H, [g1,−g2]). Thus a necessary condition of controlla-bility is that R(H, [g1, −g2]) = Rn, i.e., the pair (H, [g1,−g2]) has to becontrollable. This is a Kalman - like rank condition. Since one can use onlysign constrained inputs, this imposes an additional condition on H . A suffi-cient condition is that if H has an even number of eigenvalues with zero realparts, then the zero dynamics is controllable with nonnegative inputs. Moreresults on controllability with nonnegative inputs can be found in [11, 13].

This result is extended for our application as follows. Consider the MISOsystem with B ∈ Rn×m and ys = Cx. Also consider the case, when thereis a direction p ∈ ImB such that the system is left and right invertible cor-responding to the direction p. Using the notation B = [ p B], one has thesystem:

x = Ax + pup + Bu, ys = Cx. (26)

Let us denote by V∗ the largest (A, p) - invariant subspace in C = kerCand by W∗ the smallest (C, A) invariant subspace over Imp. It follows thatsystem has the following decomposition induced by a choice of basis in V∗ andW∗:

ξ = A11ξ + γv (27)

up =1

γ(−A12η − B21u + v) (28)

η = A22η + B22u + Gys, (29)


Since r = 1, ξ = ys, equation (29) describes the dynamics of the systemon C. Rewrite the equation of this zero dynamics as

η = Pη + Qu + Rys. (30)

assuming that Q is monic.

Proposition 1 If the pair (P, Q) is controllable, then η is controllable “with-out” using ys, e.g. by applying u = Q#(−Rys + w). If the pair (P, Q) is notcontrollable, then the conditions of controllability with unconstrained u butnonnegative ys is the following.

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 + Q1u + R1ys (31)

η2 = P22η2 + R2ys, (32)

where R2 = 0. Then P22 has no real eigenvalues.

Remark 1 The first condition is a Kalman-rank condition. The second onecan be given in some alternative forms using e.g. results from [11, 13].

For the high speed supercavitating vehicle model, this result has to beapplied to a time delay system. The following approach is taken.

Since only one delay time is present in the switching condition, it is possibleto discretize the system with extended state space by including the delayedstate variable. Since feedback linearization has been already applied, it ispossible to use a backward difference scheme defined for LTI systems thatpreserves the geometry needed to analyze the zero dynamics. The resultingdiscrete time state equations are:

x(t + 1) = Adx(t) + Bdv(t), ys = Cdx(t) (33)

where

Ad =

1 T 0 0 00 1 0 0 00 0 1 T 00 0 0 1 01 0 0 0 0

, Bd =

0 0

β21T β22T0 0

β41T β42T0 0

, Cd = [1, 0, v, 0,−1] (34)

where T denoted the sample time.The next step is to find the relative degrees by selecting one of the inputs,

say v1 first. They are identically r = 2 for both modes since the feedbacklinearization and state transform resulted in the same linear canonic form inboth modes.


To obtain the zero dynamics one has to construct a state transform matrixTcd from the row vectors spanning the orthogonal complement of V∗ andimBd1 where imBd1 is the first column of Bd.

It can be shown that V∗⊥ = span{cs, csAd} and that the remaining threerows of Tcd is selected from ImB⊥1 resulting in the transform:

Tcd =

cs

csA1 0 0 0 00 0 1 0 00 β41T 0 −β21T 0

(35)

Using this state transform [ξT (t), ηT (t)]T = Tcdx(t) and that V∗ is (Ad, Bd1)invariant, the following decomposition is obtained:

ξ(t + 1) =0 a12

0 a22ξ(t) +

0b21

v1(t) +0

e22v2(t) (36)

ys = 1 0 ξ(t) switching condition (37)

η(t + 1) = Pη(t) + Rξ(t) + Qv2(t), (38)

where

P =

p11 p12 p13

0 p22 p23

0 0 p33

, R =

0 r12

0 r22

0 0

, Q =

00

q31

. (39)

The zero dynamics are described by the last equation. (The same approachcan be repeated when selecting the second column of Bd.) Using Proposition1, it can be seen that due to their special structure, the (P, Q), pair is con-trollable. This implies that the dynamic inversion controller with switchingand pole placement for tracking error stability can be applied to control thebimodal system.

4 Multivariable Pole Placement

With feedback linearization the system behaves the same regardless of theinterior switching state, hence a single outer loop controller can guaranteestability and appropriate tracking properties for the complex system. A varietyof linear design approaches can be used for that purpose [15–18], but the focusof this paper is on the feedback linearization controller. Hence a simple pole-placement controller is synthesized as the outer-loop tracking controller.

The inner loop dynamics after feedback linearization is given by

xFL = AFLxFL + BFLv, y = CsxFL, (40)


where using the structure of Cs one can see that the first and the third statevariables are measured. The system (Cc, AFL, BFL)has relative degrees forr1 = 2, r2 = 2 and the dynamic inversion controller designed for tracking canbe written as:

v = M−11 ( xref

c24 − Fppl(xFL − xrefc )), (41)

where xrefc is the reference state vector transformed to the canonic state coor-

dinates, xrefc24 is a 2-dimensional vector containing the 2nd and the 4th elements

of xrefc and the gain matrix designed for pole allocation has the structure:

Fppl =−α110 −α111 0 0

0 0 −α220 −α221(42)

Defining the tracking error as etr = xFL−xrefc , its dynamics can be written

as

etr = Acletr,

where Acl = AFL −BFLFppl, and

Acl =

0 1 0 0

−α110 −α111 0 00 0 0 10 0 −α220 −α221

. (43)

This shows that the coefficients of the tracking error characteristic equa-tion (thus the poles, too) are arbitrarily assigned by the choice of the feedbackgain Fppl. The closed-loop and the tracking error system are stable for a givenset of α coefficients.

The structure of the complete control-loop system is shown in Figures 3and 4 where the inner-loop is responsible to feedback linearize the system andthe outer-loop is responsible for reference tracking. As stated before the plantis switched and nonlinear, hence different controllers are used in the threedifferent modes, which are selected by a state dependent switching based onthe planing model and measurements. As noise enters the cavity wall, whichis the switching surface also these selections are not always perfect, hencethe outer-loop must be robust enough to handle that “noise.” After feedbacklinearization the reference tracking part needs only to be designed for thelinear model. It is possible to track position, velocity, angle and angle ratecommands with different weights if they are selected consistently. The poleplacement controller is synthesized in the canonic coordinates. The specialstructure of the feedback linearized system results in independent controlposition and angle tracking.


✲v ✲ Bc✲ ❄

Fgrav

✲ ✲ ❄

x0

✲

✲ δ

✻x(t − τ )

Acx(t)✛

▼

Fplane

(Bimodal)

✛

✻

✛

FeedbackLinearizingController

(switching)

✛

✛

✛

✻Cs

✲ ys

vehicle

Fig. 3. Inner loop: Switching feedback linearizing controller

✲xrefT−1

c✲ ✲ Fppl

✲vpBF L

✲ ✲xF L ✲xF L

✛AF L

✻✻

Feedback linearized system

Fig. 4. Outer loop: Inversion controller for tracking

5 Control of a Supercavitating Vehicle Model

Simulations are performed in Matlab/Simulink environment and all pa-rameter dependencies are analyzed with respect to the reference setup. Themaneuver is an obstacle avoidance maneuver: the horizontal speed is constant75m/s while the vehicle moves up 17.5m and returns to continue its straightpath as seen in Figure 5.

It is assumed that pressure and cavitation number remain constant duringthe 4 second maneuver. In addition to the original problem, an additional noisecomponent is added to the cavity wall leading to uncertainty in the cavity wallposition uncertain - the controller switching surface. The cavity wall is not inRc distance from the cavity centerline, it ranges between Rc ± 0.1(Rc − R)with a filter Gn = 1

600s+1 . Recall the baseline control problem assumes a


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10

20

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)

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−5

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w (

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time (s)

BasicAMP 15AMP 20

Fig. 5. Simulated trajectory of high speed supercavtating vehicle

unity actuator model. Figure 7 shows the effect of including a 30 Hz, firstorder actuator model.

The performance specifications are to track θ reference commands andreduce limit cycle oscillations. The reference tracking properties received thehigher priority as compared with the damping the oscillatory behavior. Thefollowing controller gains were selected:

−α110 = −40000;−α111 = −400;−α220 = −90000;−α221 = −600 (44)

With which the resulting eigenvalues are −300;−300;−200;−200.The resulting contribution from the tracking part of the controller with

these high gains is negligible comparing with the inversion based part’s con-tribution to compensate against the effect of planing.

Actuator bandwidth has a significant impact on closed-loop performance.The time response of the closed-loop system with the addition of a 30 Hz firstorder actuator is shown in Figure 7. Increased oscillations of the vehicle inthe cavity are clearly evident after cavity strikes during the maneuver. Thebaseline response in Figure 6 has significantly less cavity wall strikes. All othersimulation results presented have unity as the actuator model, i.e. no actuator.

The value of the delay time also plays an important role in the controllerperformance. This is due to the feedback controller directly inverting the delay.To see this, a 5 ms reduction in the delay time is introduced into the simu-lation. The vehicle tracks the reference signal well though oscillations due tocavity wall strikes last longer and the maximum planing depth increases whenthe delay is not known precisely, see Figure 8.

The cavity wall is often not a clear glass boundary between the vaporand the fluid. Rather there may be a turbulent boundary associated withthat interface. The effect of a non-perfect cavity boundary is modeled asthe nominal boundary with the addition of a “noisy” fluid. The boundary ismodeled as Rc ± 0.1(Rc −R). In the simulation the maximum planing depthremains the same but the actuator deflection is more radical, see Figure 9.


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δ cav (

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Fig. 6. Trajectory tracking: No actuator

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Basic

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n (

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Basic

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−0.2

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δ cav (

rad

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Fig. 7. Trajectory tracking: First order 30 Hz actuator model

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/m)

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−0.2

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time (s)

δ cav (

rad

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Fig. 8. Trajectory tracking: Uncertain time delay


Increasing the noise magnitude results in larger actuator deflection spikes andhas longer settling times.

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−0.5

0

0.5

time (s)

δ cav (

rad

)

Fig. 9. Trajectory tracking: Cavity noise

A larger magnitude noise of size Rc ± 0.5(Rc − R) cavity radius and aslow noise case when the noise filter is Gn = 1

60s+1 , Figure 10. As expectedthe maximum planing depth increases as the maneuver become more radicalbut it is interesting to note that the control signals are not following thesame trend, they fin deflection remain basically the same, while the cavitatordeflection has small contribution from the increased planing, possibly becausethe planing angle is different in the different cases.

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−0.2

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rad

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hp

rim

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/m)

BasicAMP 15AMP 20

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−2

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δ fin (

rad

)

0 1 2 3 4 5 6 7−0.4

−0.2

0

0.2

0.4

δ cav (

rad

)

time (s)

Fig. 10. Trajectory tracking: 15, 18, 20m amplitude maneuvers

6 Summary and Future Research

This paper describes the control challenges associated with supercavitatingvehicles. The feedback control problem features a delayed switched system


with significant nonlinearities affecting the design. There are a number of openquestions to be resolved before such a vehicle can be successfully controlled.These include:

• Desired maneuvering requirements and performance metrics• Actuation methods, sensing requirements and accuracy needed to meet

performance requirements.• Robustness of feedback controller to modeling errors and poor physical

models

The performance and robustness requirements facing the design of a feedbackcontroller will significantly shape the design of future supercavitating vehicles.

Acknowledgments

This work was funded by the Office of Naval Research, award numberN000140110229, Dr. Kam Ng Program Manager. The first author would liketo thank Professor Keith Glover for many stimulating conversations on thedesign of feedback control algorithms and model reduction for engineeringsystems. I am a better control engineer because of the lessons he taught me.

References

1. Sturgeon, V (2001), “Racing through water: Supercavitation,” Caltech Under-graduate Research Journal vol. 1, no. 2, 16–21

2. Ashley S (2001) “Warp drive underwater,” Scientific American, 42–51.3. Dzielski J, Kurdila, A (2003), “A Benchmark control problem for supercavitat-

ing vehicles and an initial investigation of solutions,” J. Vibration and Control9:791–804

4. Logvinovich GV (1972), “Hydrodynamics of free-boundary flows,” translatedfrom the Russian (NASA-TT-F-658), US Department of Commerce, Washing-ton D.C.

5. Arndt REA, Balas GJ, Wosnik M (2005), “Control of cavitating flows: A per-spective,” accepted for publication, Japan Society of Mechanical Engineers In-ternational Journal, Japan

6. Kirschner IN, Rosenthal BJ, Uhlman JS (2003), “Simplified dynamical systemanalysis of supercavitating high-speed bodies,” Fifth International Symposiumon Cavitation, Osaka Japan

7. Paryshev EV, (2002), “The Plane problem of immersion of an expanding cylin-der through a cylindrical free surface of variable radius,” International SummerScientific School on High-Speed Hydrodynamics, Cheboksary, Russia

8. Kirschner IN, Kring DC, Stokes AW, Uhlman JS (2002), “Control strategiesfor supercavitating vehicles,” J Vibration and Control 8:219–242

9. Shao Y, Mesbahi M, Balas GJ (2003), “Planing, switching and supercavitatingflight control,” AIAA Guidance, Navigation and Control Conference, AIAA-2003-5724


10. Balas G, Bokor J, Szabo Z (2003), “Invariant Subspaces for LPV Systems andTheir Applications,” IEEE Trans Automat Contr, 48:2065–2068

11. Brammer RF (1972), “Controllability in linear autonomous systems with posi-tive controllers,” SIAM J. Control, 10:329–353

12. Camlibel MK, Heemels WPMH, Schumacher JM (2004), On the controllabil-ity of bimodal piecewise linear systems. In: Alur R, Pappas GJ (eds.) HybridSystems: Computationand Control, LNCS 2993, Springer, Berlin, 250–264

13. Saperstone SH, Yorke JA (1971), “Controllability of linear oscillatory systemsusing positive controls,” SIAM J. Control, 9:253–262

14. Balas G, Bokor J, Szabo (2004), “Tracking of continuous LPV systems usingdynamic inversion,” Conference on Decision and Control 3:2929–2933

15. Doyle JC, Glover K, Khargonekar P, Francis B (1989), “State-space solutions tostandard H2 and H∞ control problems,” IEEE Trans Auto Control 34:831–847

16. McFarlane DC, Glover K (1989), Robust Controller Design using NormalisedCoprime Factor Plant Descriptions, Springer-Verlag Lecture Notes in Controland Information Sciences, V138

17. McFarlane DC, Glover K (1992), “A Loop Shaping Design Procedure using H∞

Synthesis,” IEEE Trans Auto Control 37:759–76918. Balas GJ, Packard AK, Doyle JC, Glover K, Smith R (1993) µ-Analysis and

Synthesis Toolbox, MUSYN Inc. and The MathWorks, Natick MA19. G. Balas , Z. Szabo and J. Bokor, Controllability of bimodal LTI systems.

Submitted to the 2006 American Control Conference, Minneapolis, MN 2005.

Control of High-Speed Underwater Vehicles

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