Theoretical aspects of high-speed supercavitation vehicle control, B Vanek, J Bokor, G Balas

Tags: feedback linearization, relative degree, control design, tracking, Fin Force, rad/sec, system, control inputs, decomposition, controllability, zero dynamics, ys, controller design, supercavitation, control system, underwater vehicle, Gary Balas, feedback controller, delay systems, continuous time systems, coordinate system, invariant subspace, HSSV, cavity wall, system equations
Content: Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006
FrB11.2
Theoretical aspects of High-Speed Supercavitation Vehicle Control Baґlint Vanek, Joґzsef Bokor and Gary Balas
Abstract-- A Control System based on feedback linearization is developed for a high-speed supercavitating underwater vehicle. The supercavitation bubble surrounding the body leads to reduced drag but is also responsible for the undesired switched, nonlinear and delay dependent behavior caused by the phenomena known as planing. The theoretical contributions of the switched Control Design are discussed in connection with the mathematical description of the system. Special attention is made to understand and handle the complex and novel dynamics of the vehicle.
Fig. 1. Water tunnel experiment on supercavitation
I. PROBLEM DESCRIPTION
Based on the recent advancements [8], [6], [7] in simulation and control of a High-Speed Supercavitating Underwater Vehicle (HSSV), a new mathematical model was developed [1] to capture more details of the vehicle dynamics. The nonlinear interaction of the body with the cavity wall, showing memory effect, is very important, hence the way the cavity surface is described (cavity shape is a function of the history of the vehicle motion and cavitator area) plays an important role in the vehicle dynamics (Fig. 1). The theoretical aspects, i.e. controller design, controllability and tracking, raised by the novel system type are discussed in the following sections. The layout of the paper is as follows: a brief description of the generalized vehicle model developed in [1] is presented in Sec.II followed by the basic overview of the proposed control methodology (Sec.III). Section IV describes the theoretical design and controllability properties, including a solution for the Reference signal tracking. Simulation results are presented in Section V. The future direction of this research and conclusions are presented in Sec.VI.
II. MODEL DESCRIPTION
The system equations for the longitudinal motion of the HSSV are written in a local tangent reference frame attached to the nose of the vehicle (Fig. 2). The states in the statespace equations are: z(t)[m] nose vertical position; (t)[rad] body pitch angle; w(t)[m/s] vertical speed; and q(t)[rad/s] body pitch rate. The two control inputs are cav and fin the cavitator and fins deflection. In addition to the gravity force (Fg) another force (Fp) caused by the contact of the vehicle with the fluid surface can be present. It depends on the relative immersion depth (h ) and immersion angle () of the transom. Due to the lack of space the details of the system equations for the HSSV are omitted but the reader is referred to [1] for further details. The overall system equations can be written as:
x (t) =
Ax(t) + Bu(t) + Fg
if cT ()x(t) 0,
Ax(t) + Fp(t, x, ) + Bu(t) + Fg if cT ()x(t) 0,
(1)
This work was supported by the ONR, award number N000140110229, Dr. Kam Ng Program Manager B.Vanek, J.Bokor and G.Balas are with University of Minnesota, J.Bokor is on sabbatical leave from Hungarian Academy of Sciences Corresponding author: B. Vanek {[email protected]}
Fig. 2. The underwater vehicle with all the variables in the longitudinal plane during immersion
where x(t) X R4, x0 = x(t0), u(t) R2, xT (t) = [z^(t), (t), w(t), q(t)], z^(t) = z(t) + R - Rc, and cT () = [1 - , 0, L, 0] with denoting the delay operator: x(t) = x(t - ). Using this notation, Fp(t, x, ) can be written as:
Fp(x, )
=
P (1 -
h
R (x, )
+R
)2
(
1 + h (x, ) 1 + 2h (x, )
)(x,
),
(2)
h (x, ) =
R-1cT ()x(t) 0
if cT ()x(t) 0, if cT ()x(t) 0,
(3)
(x, ) =
cT ()x(t) - V -1R c 0
if cT ()x(t) 0, if cT ()x(t) 0,
(4)
where cT = [0, 1, /v, 0]. Eqs.1-4 describe the system as a bimodal, switched sys- tem. In the first (linear) mode the vehicle is flying inside the cavity and in the second mode it is planing e.g. on the bottom (or top) of the cavity. Note the characteristics of the switched system: (i) the switching hyperplane depends on the delayed state variable x(t - ), (ii) in the first mode the system dynamics is linear and in the second mode it is nonlinear input affine, i.e. the control inputs effect the dynamics linearly in both modes, and (iii) the switching condition does not depend on the control inputs. The reachability (controllability) properties of this system are important for control design.
III. CONTROLLER DESIGN CONSIDERATIONS The longitudinal axis HSSV model has linear and nonlinear time delay dynamics with three distinct modes: free
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flying, planing at the bottom and at the top of the cavity. The planing conditions are linear and define a switching hyper plane that separates the dynamic modes. Controllability results for bimodal switching systems are available for linear case under very specific conditions only [4]. To make use of linear controllability results, the approach taken in this paper is to apply different feedback laws in each modes to transform the system to linear time invariant, with the assumption of full state measurement. The control law design is synthesized in a new multivariable canonic coordinate frame. Extending the controllability results on bimodal switched LTI systems to time delayed switching conditions, requires the analysis of time delayed zero dynamics. The tracking problem is solved using a multivariable pole placement [1] extension of dynamic inversion. Given that the input variables enter linearly in both modes, and all the states are measurable (assumption): it is possible to select two outputs such that the relative degrees are identical in both modes. This allows the system to be transformed into LTI canonical form using linear state feedback in the central mode (1) and nonlinear state-delayed feedback in the planing mode (2,3). Hence the state equations in both modes can be linearized using a similar structure. After feedback linearization, the system has identical dynamics in both modes. This implies that the dynamics are continuous on the switching surface c()x = 0. To analyze controllability, the zero dynamics of (c(), Ac, Bc) have to be computed. The state space is time dependent due to the delay dependent switching condition. Using both inputs, the zero dynamics will be controllable. The tracking problem can be solved using multivariable pole placement extension of the switching dynamic inversion controllers. IV. THEORETICAL BACKGROUND OF THE CONTROLLER DESIGN Our approach relies on the assumption that the delay in the equations of motion can be eliminated by applying suitable feedback. Then the controllability analysis and the control design can be performed for the bimodal LTI system. Since the concept of relative degree plays a central role in this approach, its definition for nonlinear, time delay and LTI systems is given. A. Feedback linearization Consider a nonlinear input affine system:
m
x = f (x) + gi(x)ui, x X , u U (5)
i=1
yj = hj(x), yj Y, j = 1, . . . , p,
(6)
Definition 1: The system has a vector relative degree r = [r1, . . . , rp], ri 0, i, if at a point x0
(i) Lgj Lkf hi(x) = 0, . . . j = 1, . . . , m, k < ri-1,
(ii) The matrix
Lg1 Lrf1-1h1(x), . . . , Lg1 Lfr1-1h1(x)
AIA =
...,
(7)
Lgm Lfrp-1hp(x), . . . , Lgm Lfrp-1hp(x)
has rank p at x0. For linear time invariant (LTI) systems given by (A, B, C), we have that Lgj Lrfi-1hi(x) = ciAri-1bj and if p = m then the vector relative degree is defined if rankALT I = m. 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 time systems by introducing the discrete time shift operator as xt = xt- with denoting the given time delay. The time delay system is given now by (A(), B(), C()), i.e. the matrices depend on the delay operator. This implies that the coefficients are elements of the polynomial ring R[]. The relative degree is defined similarly to the LTI case as follows.
Definition 2: Given the single input - single output linear time delay system (A(), b(), c()). It has relative degree r > 0 if cAkb = 0, k = 0, . . . , r - 1 and cArb = 0. It has pure relative degree r if in addition cArb is an invertible element of R[].
This definition has an obvious extension to the multivariable case. It requires the matrix A() to be invertible over RpЧp[]. To perform the analysis and design a controller, new state variables for equation 1 are chosen to be as xЇ = Tcx,:
xЇ1(t)
z(t)

xxЇЇ23((tt))
=
-V
(t) + (t)
w(t)
(8)
xЇ4(t)
q(t)
The matrix used for this coordinate transformation is:
c1 1 0 0 0
Tc = cc12A = 00
-V 1
1 0
00
(9)
c2 A
0 0 01
where the first two states were considered as output variables. (C = [c1 c2])
The state space equations in the new coordinate system
are:
xЇ =
AcxЇ(t) + Bcu(t) + FЇg AcxЇ(t) + FЇp(t, x, ) + Bcu(t) + FЇg
if cЇT ()xЇ(t) 0, if cЇT ()xЇ(t) 0,
(10)
where
0
1
0
0
0
Ac =
-110 0
-111 0
-120 0
-121 1
Bc =
c1
AB 0
-210 -211 -220 -221,
c2 AB (11)
The difference between Fgrav and FЇgrav is that Fgrav =
(Tc · Fgrav) + C1 where C1 is a constant associated with
the shift in the origin of the coordinate system. Similarly
FЇplane = (Tc · Fplane).
Note that the inputs enter linearly in the state equations
in both modes. In addition, it is assumed that all states can
be measured. This allows us to select two outputs defined as
y1 = x1 and y2 = x3 constituting an output matrix Cc, such that there exists pure vector relative degree in both modes,
and in addition, these are identical. The relative degree for
the modes are:
r11 = 2, r21 = 2, r11 + r21 = n = 4 Mode 1 (12) r12 = 2, r22 = 2, r12 + r22 = n = 4 Mode 2 (13)
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The consequence of this property is that one can apply state
feedback in both modes such that it eliminates time delay
in Mode 1 and nonlinearity (exact feedback linearization) in
Mode 2. This feedback is given by:
iMf c1-T1((y)1x3((tt))
-
Fx(t) 0,
-
FЇg
+
vI (t)
uf lc
=
iMf c1-T1((y)1x3((tt))
-
Fx(t) 0,
-
FЇg
-
FЇp(x,
)
+
vII (t)
(14)
where M1 = (CAB), y13 = [y1, y3]T , and the feed-
back gain F is defined by the controllability invariants ijk, i = 1, 2, j = 1, 2, k = 0, 1 of the linear part
A, B of the system (Eq.11).
The feedback linearized closed loop has the following
form in both mod0es:1 0 0
0 0
x c = 00
0 0
0 0
01 xc + 10
00
v1 v2
(15)
0000
01
and the switching condition is given by the sign of ys = c()xc.
B. Controllability analysis of the bimodal system The controllability of the linearized bimodal dynamics has to be analyzed and a tracking controller designed. Results on controllability of single input single output LTI systems with single switching surface and relative degrees r = r1 = r2 = 1 has been published by Heemels et al. [4]. In their approach the problem is reduced to analyzing the dynamics of the system on the switching surface. This is given by the zero dynamics derived with respect to the "switching output" ys. It was shown that the zero dynamics have to be controllable when using positive ys in one mode (negative ys in the second mode, respectively). Heemels et al. assumed that the system is both left and right invertible and the dynamics are continuous on the switching surface, i.e. A1x + b1u = A2x + b2u. The same results can be obtained using the following simple reasoning. Since the relative degree r = 1, under the above assumptions, the zero dynamics can be written as
(t) = H +
g1ys(t) g2ys(t)
if ys(t) 0, if ys(t) 0,
(16)
with Rn-1.
To compute the reachability subspace of the zero dynam-
ics, H
the Lie-algebra + g2ys- need to
of be
the vector spaces H + defined. ys+, ys- denotes
g1ys+ and a positive
(respectively negative) control to (H, g1) and (H, g2). It is
simple to show that this is the Lie - algebra generated by
Hx+[g1, g2]ys, ysT = the vectors {[g1, g2],
[ys+, ys-] and H[g1, g2], . .
this .,H
is given (at n-2[g1, g2]}.
= 0) by Denote
this subspace by R(H, [g1, g2]). Thus a necessary condition
for controllability is that R(H, [g1, g2]) = Rn-1, i.e., the
pair (H, [g1, g2]) has to be controllable. This is a Kalman
- like rank condition, since in a given mode one can use
only positive (respectively negative) control in the zero dy-
namics, imposing an additional condition on H. A sufficient
condition is that if H has an even number of eigenvalues
with zero real parts, then the zero dynamics are controllable
with nonnegative inputs. More results on controllability with nonnegative inputs can be found in [3], [9]. This result is extended for our application as follows. Consider the MISO system with B RnЧm and ys = Cx. Also consider the case, when there is a direction p Im{B} such that the system is left and right invertible corresponding to the direction p. Using the notation B = [ p BЇ], one has the system:
x = Ax + pup + BЇuЇ, ys = Cx.
(17)
Let us denote by V the largest (A, p) - invariant subspace in C = ker{C} and by W the smallest (C, A) invariant subspace over Im{p}. It follows that system has the following decomposition induced by a choice of basis in V and W:
= A11 + v
(18)
up
=
1
(-A12
-
BЇ21uЇ +
v)
(19)
= A22 + BЇ22uЇ + Gys,
(20)
Since r = 1, = ys, equation (20) describes the dynamics of the system on C. Rewriting the zero dynamics equation
as
= P + QuЇ + Rys.
(21)
assuming that Q is monic. Proposition 1: If the pair (P, Q) is controllable, then is controllable "without" using ys, e.g. by applying uЇ = Q#(-Rys + w). If the pair (P, Q) is not controllable, then the conditions of controllability with unconstrained uЇ but nonnegative ys is 1) The pair (P, [Q R]) has to be controllable. 2) Consider the decomposition induced by the reachability subspace R(P, Q),
1 = P111 + P122 + QuЇ + R1ys
(22)
2 =
P222 + R2ys,
(23)
where R2 = 0. Then the imaginary part of the eigenvalues of P22 cannot be zero. Remark 1: The first condition is a Kalman-rank condition. The second one can be given in some alternative forms using e.g. results from [3], [9]. For the high speed supercavitating vehicle model, this result has to be applied to a time delay system. The following approach is taken. Since only one delay time is present in the switching condition, it is possible to discretize the system with an extended state space by including the delayed state variable. Since feedback linearization has been already applied, it is possible to use a backward difference scheme defined for LTI systems that preserves the geometry needed to analyze the zero dynamics. The discrete time state equations are:
x(t + 1) = Adx(t) + Bdv(t), ys = Cdx(t) (24)
where
1 T 0 0 0
0
0
Ad = 000
1 0 0
0 1 0
0 T 1
000 ,
Bd
=
24011
T T
22T 0 42T

,
10000
0
0
Cd = [1, 0, L, 0, -1] (25)
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with T denoting the sample time.
The next step is to find the relative degrees by selecting
one of the inputs, v1 for example. They are identically r = 2
for both modes since the feedback linearization and state
transform resulted in the same linear canonic form in both
modes.
To obtain the zero dynamics one has to construct a state
transform matrix Tcd from the row vectors spanning the orthogonal complement of V and Im{Bd1} where Im{Bd1} is the first column of Bd. It can be shown that V = span{cs, csAd} and the remaining 3 rows of Tcd is selected from imB1 resulting in the transform:


cs
Tcd = 10
0 0
csA 0 1
0 0
00
(26)
0 41T 0 -21T 0
Using this state transform [T (t), T (t)]T = Tcdx(t) and that V is (Ad, Bd1) invariant, leads to the following decomposition:
(t + 1) =
0 0
a12 a22
(t) +
0 0
0 e22
0 e23
(t)+
+
0 b21
v1(t) +
0 f22
v2(t)
(27)
ys = [1 0] (t) switching condition
(28)
(t + 1) = P (t) + R(t) + Qv2(t),
(29)
where
p11 p12 p13
0 r12
0
P = 0 p22 p23 , R = 0 r22 , Q = 0 .
0 0 p33
00
q31
(30)
The zero dynamics are described by the last equation. (The
same approach can be repeated when selecting the second
column of Bd.) Using Proposition 1, it can be seen that due to their special structure, the (P, Q) pair is controllable. This
implies that the dynamic inversion controller with switching
and an pole placement for tracking error stability can be
applied to control the bimodal system.
C. Multivariable Pole Placement for Tracking
The performance objective of the control design is to track desired state commands. The inversion based controller has the following form:
u1(t) u2(t)
= (CAB)-1(
x1(t) x2(t)
ref
- [u]
x1(t) x2(t)
-
- [l]
x3(t) x4(t)
- [Gc] - [Pc(t, )] -
v1(t) v2(t)
)
(31)
The reference tracking part of the controller:
v1(t) v2(t)
= [Їu]
x1(t) - x1,ref (t) x2(t) - x2,ref (t)
+[Їl]
x3(t) - x3,ref (t) x4(t) - x4,ref (t)
(32)
[u] =
-110 -210
-111 -211
[l] =
-120 -220
-121 -221
(33)
[Їu] =
-Ї110 0
-Ї111 0
[Їl] =
0 -Ї220
0 -Ї221
(34)
The feedback linearized closed-loop has the following form in all modes:
x c =
0 -110 0 -210
1 -111 0 -211
0 -120 0 -220
0 -121 1 -221
xc-
- BcB-1
-110 -210
-111 -211
-120 -220
-121 -221
xc +
v1 v2 (35)
Acl
=
Ac
-
BcFin0v
+
BcFctr 1
=
0
0
-Ї0110
-Ї111 0
0 0
0 1

(36)
0
0 -Ї220 -Ї221
The closed-loop system is stable for a given set of Ї
coefficients. With feedback linearization, the system behaves
the same regardless of the interior switching state, hence
one linear outer loop controller can guarantee stability and
appropriate tracking properties for the complex system. A
variety of linear Design Approaches can be used for that
purpose [5], [2], but the focus of this paper is on the feed-
back linearization controller. Hence a simple pole-placement
controller is synthesized as the outer-loop tracking controller.
The inner loop dynamics after feedback linearization, using
the new canonical coordinates are:
x 1
0 1 0 0 x1
xx 23
=
00
0 0
0 0
01 xx23
+
Bc
f c
(37)
x 4
0 0 0 0 x4
where f,c denotes the additional deflection of the fins and cavitator commanded by the pole-placement controller. The objective is to place the closed-loop poles to obtain the desired tracking response. This system is nilpotent, because all eigenvalues of A are zero. The first two states relative to vertical position and speed are controlled by the fins (f ) and the other two states relative to vehicle angle and angle rate are controlled by the cavitator due to the lack of cross coupling. The feedback linearization controller is denoted by Finv and the tracking controller by f and c respectively. The resulting reference tracking controller has the following form:
f c
= (CAB)-1
-Ї110 -Ї111 x1(t) - x1,ref (t)
0
0
x2(t) - x2,ref (t)
+
0 -Ї220
0 -Ї221
x3(t) - x3,ref (t) x4(t) - x4,ref (t)
+
x 2,ref x 4,ref
}
+ (38)
The closed-loop has the following dynamics:
Acl = Ac - BcFinv + Bc
f c
(39)
The tracking part of the controller is responsible for the location of the poles. The eigenvalues of the system are:
1,2 = -0.5Ї221 ± 0.5 (-Ї221)2 - 4Ї220
(40)
3,4 = -0.5Ї121 ± 0.5 (-Ї121)2 - 4Ї120
which are stable based on the selection of the Ї coefficients.
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Fgrav
x0
vehicle
v - h- Bc - h? -
- h? - Cs
-ys
6
6BMB B
Ac xЇ(t)
-
6xЇ( )
Fplane (Bimodal)
Feedback Linearizing Controller (switching)
nose pos. (m)
20 Basic 10
0
-10
0
1
2
3
4
5
6
7
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
time (s)
plane depth (m)
Fig. 5. Simulated trajectory of high speed supercavitating vehicle
Fig. 3. Inner loop: Switching feedback linearizing controller
Feedback linearized system
xref- Tc-1 - f - Fppl v-p BF L- fx F-L
6
6
xF L -
AF L
pitch rate (rad/sec)
0.4 Basic 0.2
0
-0.2
-0.4
0
1
2
3
4
5
6
7
4000
Fin Force (N)
2000
0 Fig. 4. Outer loop: Inversion controller for tracking -2000
The structure of the feedback controller can be seen in Figure 3-4. The inner-loop feedback linearizes the system and the outer-loop provides reference tracking. Different controllers are used in the 3 switched modes. They are selected by a switching logic based on the planing model and measurements. The cavity wall is noisy and it is the switching surface. Hence the outer-loop must be robust to handle that "noise." After feedback linearization the reference tracking part needs only to be designed for a linear model. It is possible to track both position, velocity, angle and angle rate commands with different weights if they are selected consistently. The designed pole placement controller also operates on the transformed canonic coordinates. The special structure of the feedback linearized system enables control of position and angle without cross coupling.
V. CONTROL OF A SUPERCAVITATING VEHICLE MODEL
Simulations are performed in MATLAB/SIMULINK envi-
ronment and parameter dependencies are analyzed in com-
parison with a reference setup. The maneuver is an obstacle
avoidance maneuver: the horizontal speed is constant 75m/s
while the vehicle moves up 17.5m and returns to continue
its Straight Path as seen in Figure 6.
It is assumed that the water conditions (pressure, tem-
perature, viscosity etc.) are constant during the 4 second
maneuver. A noise component is added to the cavity wall
resulting in an uncertain switching surface. The cavity wall
varies between 90 - 110% of the nominal cavity gap. The
noise is modeled as a random White Noise process passing
through
a
filter
Gn
=
1 600s+1
.
Another
simplification
in
the
original problem is the lack of actuator model. It is included
in
the
simulation
setup
with
dynamics:
Gact
=
1 200s+1
.
The
performance specifications are to track trajectory reference
commands and reduce limit cycle oscillations. The reference
-4000
0
1
2
3
4
5
6
7
time (s)
Fig. 6. Control deflection and pitch rate with basic setup
tracking properties received the higher priority as compared
with the damping the oscillatory behavior. The following
controller gains were selected:-Ї110 = -40000; -Ї111 = -400; -Ї220 = -90000; -Ї221 = -600 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 still negligible comparing
with the inversion based contribution to compensate against
the effect of planing.
The
basic
actuator
model
is:
Gact
=
1 200s+1
which
has
bandwidth 30Hz, noticeably slower actuators are not able to
stabilize the system, while faster actuators has better perfor-
mance, with less oscillation. The knowledge about delay time
also plays an important role in the controller performance.
The vehicle tracks the reference signal when the uncertainty
is small, but oscillations grow due to imprecise knowledge
of the delay, immersion into the liquid is getting deeper and
lasts longer, which results in instability when the uncertainty
in delay time exceeds approximately 20 percent.
The noise model can be modified two different ways:
magnitude and filter. The maximum planing depth remains
the same but the actuator deflection is more radical if the
noise has larger magnitude, it has larger spikes and has longer
settling times, Figure 8.
Changing the noise filter leads to different characteristic -
if it is faster than the sampling time of the sensors than it has
no effect, if it is slow enough than the controller can hardly
deal with it, it leads to larger planing depths and oscillations.
The control signals change less rapidly and have smaller
spikes. The simulation (Fig.9) shows a large magnitude noise
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pitch rate (rad/sec)
Fin Force (N)
0.5 Slow Noise 0
-0.5
0
1
2
3
4
5
6
7
4000
2000
0
-2000
-4000
0
1
2
3
4
5
6
7
time (s)
Fig. 7. Trajectory tracking: Slow cavity noise
1 High Noise 0.5
0
-0.5
-1
0
1
2
3
4
5
6
7
5000
0
Cav. Force (N)
Fin Force (N)
pitch rate (rad/sec)
4000
2000
0
-2000
-4000
0
1
2
3
4
5
6
7
1000
0
-1000
Basic AMP 15
AMP 20
-2000
0
1
2
3
4
5
6
7
time (s)
0.4
0.2
0
-0.2
-0.4
0
1
2
3
4
5
6
7
0.4 Basic
0.3
AMP 15
AMP 20
0.2
0.1
0
0
1
2
3
4
5
6
7
time (s)
plane depth (m)
pitch rate (rad/sec)
Fin Force (N)
-5000
0
1
2
3
4
5
6
7
time (s)
Fig. 9. Trajectory tracking: 15, 18, 20m amplitude maneuvers
Fig. 8. Trajectory tracking: High cavity noise
with Rc ± 0.5(Rc - R) cavity radius and a slow noise case
when
the
noise
filter
is
Gn
=
1 60s+1
.
As expected the maximum planing depth increases as the
maneuver become more radical. It is interesting to note that
the control signals do not follow the same trend. The fin
deflection remain basically the same, while the cavitator
deflection has small contribution from the increased planing,
possibly because the planing angle is different in the different
cases.
VI. SUMMARY AND future research Supercavitation is a very promising way to increase the speed of underwater vehicles at the expense of a complicated vehicle architecture. Successful development of such a system will require increased collaboration between fluid and control researchers. As an intermediate step the control design challenges including delayed state dependency, nonlinearities and switching with noisy switching surface were analyzed and an inversion based control methodology was proposed on a recently developed 2-DOF mathematical model of the HSSV. The ultimate goal for the future research is implementation of a three dimensional trajectory tracking controller on the HSSV test vehicle. The 3-D motion will no longer have a symmetry plane, hence the asymmetric fin immersion and non-vertical planing forces leading to non input affine systems require special attention. Furthermore robust constraint fulfillment remains an open issue, which can be attempted to solve by RHC control based methods.
REFERENCES [1] Gary J. Balas, Joґzsef Bokor, Baґlint Vanek, and Roger E.A. Arndt. Control of Uncertain Systems: Modelling, Approximation, and Design, chapter Control of High-Speed Underwater Vehicles, pages 25­44. LNCIS. Springer-Verlag, 2006. [2] G.J. Balas, R. Chiang, A.K. Packard, and M. Safanov. Robust control toolbox. MUSYN Inc. and The MathWorks, Natick MA, 2005. [3] R.F. Brammer. Controllability in linear autonomous systems with positive controllers. SIAM J. Control, 10:329­353, 1972. [4] M.K. Cё amlibel, W.P.M.H. Heemels, and J.M. Schumacher. On the controllability of bimodal piecewise linear systems. In:Alur R, Pappas GJ (eds.) hybrid systems: Computationand Control LNCS 2993, Springer, Berlin, 250­264, 2004. [5] J.C. Doyle, K. Glover, P. Khargonekar, and B. Francis. State-space solutions to standard H2 and H control problems. IEEE Trans Auto Control 34:831­847, 1989. [6] J. Dzielski and A. Kurdila. A benchmark control problem for supercavitating vehicles and an initial investigation of solutions. Journal of Vibration and Control, 9(7):791­804, 2003. [7] I.N. Kirschner, D.C. Kring, A.W. Stokes, and J.S. Uhlman. control strategies for supercavitating vehicles. J Vibration and Control 8:219­ 242, 2002. [8] I.N. Kirschner, B. J. Rosenthal, and J.S. Uhlman. Simplified dynamical systems analysis of supercavitating high-speed bodies. In Fifth International Symposium on Cavitation (CAV2003), Osaka,Japan, 2003. [9] S.H. Saperstone and J.A. Yorke. Controllability of linear oscillatory systems using positive controls. SIAM J. Control, 9:253­262, 1971.
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