Model simplification for AUV pitch-axis control design∗

Jan Petrich Daniel J. Stilwell†

The Bradley Department of Electrical and Computer Engineering,

Virginia Polytechnic Institute & State University, Blacksburg, VA 24061

{jpetrich,stilwell}@vt.edu

Abstract

Although the use of low-order equivalent models is common and extensively studied for control of

aircraft systems, similar analysis has not been performed for submersible systems. Toward an

improved understanding of the utility of low-order equivalent models for submersible systems, we

examine control design for pitch-axis motion of an autonomous underwater vehicle (AUV). Derived

from first principles, the pitch-axis motion of a streamlined AUV is described by third-order dynamics.

However, second-order approximate models are common for system identification and control design.

In this work, we provide theoretical justification for both the use of and limitations of a second-order

model, and we verify our results in practice via a series of case studies. We conclude that a

second-order pitch-axis model should often be sufficient for system identification and control design.

Keywords: autonomous underwater vehicles, attitude control, linear equivalent model, stability

1 Introduction

Complete nonlinear AUV models of autonomous underwater vehicles (AUVs) are introduced in Fossen

[1994], Gertler and Hagen [1967] and Arafat et al. [2006]. In general, the parameters of those models are

obtained through tow-tank experiments as described in Hwang [2003], Barros et al. [2008], Prestero

[2001] and Williams et al. [2006], or by employing computational fluid dynamics (CFD) tools as

described in Humphreys [2001], Jenkins et al. [2003], Sahin et al. [1997] and Geisbert [2007]. However,

conducting time-consuming and expensive tow-tank experiments or CFD simulations is prohibitive for∗The authors gratefully acknowledge the support of the National Science Foundation via Grant IIS-0238092, and the Office

of Naval Research via Grant N00014-03-1-0444.†Corresponding author

1

AUVs whose payload configuration and hydrodynamic characteristics change frequently depending on

mission requirements. Examples of such modular AUVs that easily accommodate payload sensors are the

Ocean Explorer Smith et al. [1995], the Mares AUV Cruz and Matos [2008], and the Remus

AUV Moline and Blackwell [2005]. In our work, we justify simplifying assumptions regarding the highly

nonlinear AUV dynamics in theory and practice, and we investigate the utility of a low-order pitch-axis

model for in-flight system identification as proposed in Petrich et al. [2007] and subsequent model-based

attitude control design.

In steady-state flight at zero angle of attack, the linearized pitch-axis motion of a streamlined AUV, as

derived from first principles, reduces to third order. Linear third-order AUV pitch-axis models can be

found in Fossen [1994], Silvestre and Pascoal [1997], Vuilmet [2005] and Prestero [2001]. Nevertheless,

system identification and control design for streamlined autonomous underwater vehicles often assume a

second-order model for the pitch-axis motion, see for example Fossen [1994], Prestero [2001],

Cristi and Healey [1989], Li et al. [2004], Kim et al. [2001] and Cristi et al. [1990]. However, a rigorous

justification for second-order approximate models has not been previously addressed. For this reason, this

work attempts to establish theoretical justification for using a second-order pitch-axis model for

parameter identification and attitude control design of streamlined AUVs. In order to analyze why a

second-order approximation seems to work well in practice, we introduce an error model that defines a

useful relationship between the third-order model and a second-order approximation. We use the error

model to quantify neglected system dynamics and discuss stability and performance of the closed-loop

third-order model when using a controller that is designed based on a second-order approximation. Our

analysis and field trials show that a second-order pitch-axis model often suffices in practice for control

design.

The use of second-order models is also common in the aircraft community, where reduced-order

equivalent models have been extensively studied for evaluating flight performance, see for

example Mitchell and Hoh [1982] and Hodgkinson [1982]. No equivalent work has been done to address

the utility of low-order equivalent models for AUVs.

Limited sensing ability poses fundamental challenges for obtaining reliable model parameters from flight

data. In particular, smaller AUVs are often not instrumented to measure linear velocities, and thus the

angle of attack α is unknown. With the limited instrumentation in mind, we address identifiability of the

underlying physical parameters of AUV systems, such as lift and drag coefficients, within the scope of

linear dynamic model identification. We prove that physical AUV parameters are not uniquely

identifiable using in-flight system identification techniques, as presented in Petrich et al. [2007] for linear

models and in Smallwood and Whitcomb [2001] for nonlinear models, and therefore the parameters of the

identified model do not determine the physical parameters of the actual system. Thus, for the purpose of

identifying physical AUV parameters, a third-order model does not provide any advantages over a

second-order approximation. Without model parameters, a state observer that estimates α and thus

enables state feedback cannot be directly implemented. We deduce output-feedback gains from an

identifiable, second-order approximation of the plant, and we show that those feedback gains stabilize the

underlying third-order plant if the error model is sufficiently small. We formally quantify the relationship

between size of the error model and stability margins of the control system.

When designing AUV controllers, model parameters are commonly assumed to be known. For example,

linear third-order model parameters, determined experimentally through tow-tank experiments and CFD

simulations, are presented for the Remus AUV in Prestero [2001] or the MARIUS vehicle in Fryxell et al.

[1994]. In contrast, we identify linear pitch models in situ utilizing only instrumented variables onboard

the AUV. Our work is motivated by vehicle systems that change their external shape and/or mass

distribution (e.g., new external payload) frequently, and for which a revised control system is required. In

this case, control design is expedited if a dynamic model can be identified rapidly from AUV flight data.

Thus we address control design in the context of system identification and corresponding parameter

estimation errors. Similar work has been presented in Rentschler et al. [2006], in which the authors

identify the parameters of a canonical third-order AUV pitch-axis model using numerical minimization

tools under the assumption that the model is stable. While we do not impose a stability assumption, our

focus is on the relationship between the third-order model and a second-order approximation. We

presume that a stabilizing, although perhaps poorly performing, control systems is available for which in

situ flight data can be acquired and a model can be identified.

Experimental verification of our analysis is conducted using the Virginia Tech 475 AUV, shown in

Figure 1. The Virginia Tech 475 AUV is a conventional, streamlined vehicle that is nearly neutrally

buoyant, propelled at the stern, and controlled by movable control surfaces at the tail. It

weighs 8.5 kilograms and has a total length of 1 meter. The hull diameter is 0.12 meters (4.75 inches).

The Virginia Tech 475 AUV is capable of carrying various payload sensors needed for biological and

acoustic surveys, each of which changes the exterior shape and mass of the AUV. For this reason,

hydrodynamic properties such as lift and drag coefficients and added mass terms vary depending on the

payload. Thus, the ability to conduct in-flight system identification becomes particularly important. An

onboard attitude and heading reference system measures the vehicle’s pitch angle θ and pitch rate q. The

commanded elevator fin deflection used for attitude control is denoted δ.

Using three drastically different configurations of the Virginia Tech 475 AUV, we identify both

second-order and third-order linear pitch-axis models. We compare the resulting models and draw

Figure 1: Virginia Tech 475 AUV, named after its hull diameter of 4.75 inches.

conclusions about the performance and robustness of a control system that is designed based on a

second-order approximation. Comparing the full third-order dynamics to the second-order

approximation, we find a modest decrease in phase margin. We conclude that a linear second-order

pitch-axis model suffices for attitude control design of a slender and streamlined AUV.

This paper is organized as follows. In Section 2, we define linear third-order and second-order AUV

pitch-axis models. In Section 3, we introduce a model for the approximation error due to using a

second-order approximation for the third-order model. In Section 4, we investigate identifiability of the

underlying physical parameters (e.g., lift and drag) in the context of system identification. In Section 5,

we discuss inherent stability of the pitch-axis motion of a streamlined AUV, and in Section VI we

investigate control design. In Sections 7 and Section 8, we apply our results to the problems of system

identification and control design, respectively, for three configurations of the Virginia Tech 475 AUV.

2 Problem Statement

In this work we address the dynamics of a neutrally buoyant vehicle in level flight at zero angle of attack.

We assume a small separation between center of gravity and center of buoyancy, and a vehicle body with

three planes of symmetry. Under these assumptions, the linearized pitch dynamics are described by a

third-order model, see Fossen [1994], Prestero [2001] and Petrich et al. [2007]

α

q

θ

=

a22 a23 0

a32 a33 a34

0 1 0

α

q

θ

+

b21

b31

0

δ

y1

y2

=

0 1 0

0 0 1

α

q

θ

(1)

The states are angle of attack α, pitch rate q and pitch angle θ. The input is the elevator fin deflection δ.

As illustrated in Figure 2, the angle of attack α is defined between the body’s x-axis and the velocity

vector V , and the pitch angle is defined between the body’s x-axis and the XY -plane of the inertial

frame. Only pitch rate q and pitch angle θ are assumed to be instrumented, and thus the output signals

are y1 = q and y2 = θ.

α

V

q

θ

δ

xz

Y

X

Figure 2: A linear pitch-axis model describes the AUV motion in the body’s xz plane. Assuming constant

velocity V , the states are angle of attack α, pitch rate q, and pitch angle θ. The input is the elevator fin

deflection δ.

The model (1) is derived by restricting the motion of a rigid body in inviscid fluid to the xz-body plane

or dive plane, where x is the longitudinal axis and z is the vertical axis, and by linearizing the remaining

equations of motion around the steady-state flight condition. For the steady-state flight at zero angle of

attack, one can show that the vehicle’s speed V decouples completely from the remaining pitch motion.

Thus, we assume constant speed flight with V = V0. The coefficients of (1) are derived in Petrich et al.

[2007]

a32 =1Jy

V 20

{(mz −mx) +

12ρAbLCmα

}

a22 =1

2mzρV0Ab (CD0 + CLα)

a33 =1Jy

{12ρV 2

0 AbLCmq −mV0xcg

}

a23 =mx

mz

a34 = − 1Jy

zcgFw

b21 =1

2mzρV0AfCLδ

b31 =1

2JyρV 2

0 AfxfCLδ

(2)

The parameter Jy denotes the moment of inertia including added inertia along the pitch-axis. The AUV

parameters mx and mz represent the mass including added mass along x and z-axis, respectively. The

dry mass is m. The reference length is L and the location of the fins with respect to the center of

buoyancy is xf . The center of gravity is located at rcg =[xcg 0 zcg

]T. Hydrodynamic properties of

body and fins are characterized by the lift coefficients CLα and CLδand the corresponding reference

areas Ab and Af . The hydrodynamic coefficients Cmα < 0 and Cmq < 0 account for the body’s restoring

moment and viscous damping, respectively.

The transfer function corresponding to (1) isq(s)

θ(s)

=

s

1

β1s + β0

s3 + α2s2 + α1s + α0

δ(s)

=

s

1

G3(s) δ(s)

(3)

with the coefficientsβ1 = b31

β0 = a32b21 − b31a22

α2 = −a22 − a33

α1 = a22a33 − a34 − a23a32

α0 = a22a34

(4)

Although not fully justified, second-order pitch-models are found adequate for AUV control design in

practice, see for example Fossen [1994], Prestero [2001], Cristi and Healey [1989], Li et al. [2004],

Kim et al. [2001] and Cristi et al. [1990]. A typical second-order approximation isq

θ

=

A33 A34

1 0

q

θ

+

B31

0

δ (5)

with the corresponding transfer functionq(s)

θ(s)

=

s

1

B31

s2 −A33s−A34δ(s)

=

s

1

G0(s) δ(s)

(6)

In Fossen [1994], Prestero [2001] and Cristi et al. [1990], such a second-order model is derived by deleting

the state α in the third-order model (1), while in Cristi and Healey [1989], Li et al. [2004] and Kim et al.

[2001] it is proposed without justification.

In the sequel, we assume the existence of lumped parameters A33, A34, and B31 defining a second-order

model (5) that captures the dominating modes and approximates the input-output behavior of (1). In

Section 7, we verify this assumption in practice.

For aircraft, such as in Etkin [1959] or Stevens and Lewis [2003], pitch motion is characterized by four

states: speed V , angle of attack α, pitch rate q and pitch angle θ. Detailed analysis shows that even for

aircraft steady-state and level flight, the speed V does not decouple from the remaining states of the

linear system. Aircraft weight forces are not compensated for by buoyancy forces, as in the case of a

neutrally buoyant AUV, and a negative (positive) pitch angle accelerates (decelerates) the aircraft.

However, it is widely accepted that the fourth-order pitch dynamics are governed by two modes, namely

short period and phugoid mode. Due to the time scale separation of both modes, it suffices to examine

the second-order short period mode for attitude control, see Blakelock [1991] and McRuer et al. [1973].

3 Error Model

In this section, we address the approximation of the third-order plant (1) using a second-order model (5).

We define an error model between (1) and (5) which is used for stability analysis.

Applying standard system identification techniques, e.g. Ljung [1987] and Eykhoff [1974], the parameters

of G0(s) in (6) are determined by matching the input-output behavior of model and plant. Due to the

model approximation and truncation of system modes, we generally find A33 6= a33, A34 6= a34

and B31 6= b31, see Antoulas [2005]. Thus, the physical significance of the parameters A33, A34, and B31 is

not apparent.

We investigate the relationship between second-order approximation and third-order representation of the

plant through an error model. To derive an error model, we decompose the third-order transfer

function (3) as q(s)

θ(s)

=

s

1

G2(s) Ge(s) δ(s) (7)

The transfer function G2(s) contains the second-order approximation (6) augmented with the system

identification errors ∆A33, ∆A34 and ∆B31

G2(s) =B31 + ∆B31

s2 − (A33 + ∆A33) s− (A34 + ∆A34)

and Ge(s) represents an error model due to the truncation of a zero and a pole located at ze and pe,

respectively. That is

Ge(s) =s− ze

s− pe

The coefficients ∆A33, ∆A34 and ∆B31 can be interpreted as identification errors for natural frequency,

damping ratio and zero frequency gain of the second order system.

A qualitative pole-zero-plot of a third-order plant (black) and a second-order approximation (gray) is

shown in Figure 3.

The parameters A33, A34 and B31 from the second-order model (6) relate to the third-order model (7)

through the selection of a specific state space realization of (7). Such a state space realization contains

the states and parameters of the second-order approximation (6) as well as a generic, augmented third

state xα

xα

q

θ

=

pα aq aθ

cα A33 A34

0 1 0

xα

q

θ

+

bα

B31 + bq

0

δ

y1

y2

=

0 1 0

0 0 1

xα

q

θ

(8)

For any cα 6= 0, the coefficients in (8) and (7) satisfy the equalities

pα = pe + ∆A33

bq = ∆B31

cα bα = (B31 + ∆B31) (pe − ze + ∆A33)

cα aq = (A33 − pe) ∆A33 + ∆A34

cα aθ = ∆A33 A34 − pe ∆A34

(9)

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.5

Pole−Zero Map

Real Axis

Imag

inar

y A

xis

Figure 3: System Identification process: A linear second-order model (gray) is identified for an underlying

third-order plant (black).

Using the realization (8), the coefficients of the third-order transfer function G3(s) from (3) can be

expressed

β1 = B31 + bq

β0 = cαbα − pα (B31 + bq)

α2 = −A33 − pα

α1 = pαA33 −A34 − cαaq

α0 = pαA34 − cαaθ

(10)

Figure 4 contains a signal diagram of (8) showing the second-order approximation and peripheral,

structured uncertainties that define the error system.

Both state space models (1) and (8) are a realization of G3(s) in (3) and describe the same input-output

behavior. They are therefore equivalent under a state space transformation, but contain different

[A33 A341 0

]

[B310

][q

θ

]

1

s

δ

Second order

approximation

bα[aq aθ

]

[bq0

]

[cα0

]1

s− pα

Figure 4: Signal diagram of the second-order approximation and the peripheral error model.

parameter sets. In general, we find xα 6= α. Thus, we conclude that the parameters of the third-order

state space models (1) and (8) cannot be determined uniquely from the input-output behavior of the

plant. A rigorous analysis addressing the identifiability of the physical AUV parameters (2) is presented

in the next section.

4 Identifiability of Model Parameters

In this section, we investigate identifiability of the physical AUV parameters (2) given the available input

and output signals. Although, a canonical third-order model can always be identified, it cannot be used

to infer physical parameters of the underlying plant. Thus, we show that the state space

representation (1) is not uniquely identifiable from input-output data.

For the identifiability analysis, we rewrite (1)

x = A(p) x + B(p) δ

y = C x + D δ(11)

where the state vector is x =[α q θ

]T, the output vector is y =

[q θ

]Tand p is

p =[a22 a23 a32 a33 a34 b21 b31

]T∈ R7 (12)

a vector of parameters that contains the coefficients of (1). The parameterization of A(p) and B(p)

follows immediately from (1) with the matrix components defined in (2).

Identifiability is discussed in Grewal and Glover [1976] for a variety of systems. For our purposes, we

recall that the input-output behavior of a linear, time-invariant system is uniquely determined by the

Markov parameters gi. For the state space representation (11), the Markov parameters are given by

g0 = D

gi = C Ai−1(p) B(p) i = 1, 2 . . .(13)

and the Markov parameter matrix defined in Grewal and Glover [1976] is

G(p) =

D

C B(p)

C A(p) B(p)

C A2(p) B(p)...

C A2n−1(p) B(p)

∈ R2(2n+1) (14)

where n is the order of the system. For (11), n = 3. For the single-input system (11), the Markov

parameter matrix G(p) is a vector. The following theorem from Grewal and Glover [1976] states a

sufficient condition for local identifiability.

Theorem 4.1 The parameter vector p ∈ Rq is said to be locally identifiable at p ∈ Rq if

rank[

∂

∂pG(p)

]= q at p = p (15)

In other words, the Markov parameter matrix G(p) provides a local one-to-one map between system

parameters p and Markov parameters gi at p. Specifying Theorem 4.1 to the system at hand, we deduce

the following corollaries.

Corollary 4.2 Given the output signals y =[q θ

]Tfor system (1), the parameter vector p from (12) is

not identifiable.

Corollary 4.3 Suppose an augmented output signal y =[α q θ

]T, that is the output matrix of

system (1) satisfies C = I. Then, the parameter vector p from (12) is locally identifiable at every p, for

which poles and zeros of (1) do not coincide.

To prove Corollaries 4.2 and 4.3, we construct the Markov parameter matrix G(p) from (14) for each

case of C, and check the rank condition of the Jacobian matrix (15).

Corollaries 4.2 and 4.3 imply that identifying the parameter vector p requires that α is instrumented.

Therefore, deducing the physical AUV parameters (2) uniquely from the input-output behavior (1) is not

possible given the limited instrumentation of typical small AUV systems.

As compared to the seven parameters of (1), one can show that the five coefficients of transfer

function G3(s) in (3)

pc =[α0 α1 α2 β0 β1

]T∈ R5 (16)

are identifiable if the numerator and denominator polynomials in (3) do not share roots. In Section 7, we

estimate the coefficients pc and obtain a canonical third-order model G3(s) for the Virginia Tech 475

AUV.

5 Inherent Stability Assessment

In practice, linear second-order pitch-axis models are widely used for system identification and attitude

control design for AUVs. A common way to derive such a second-order model is to simply truncate α

in (1), see Fossen [1994], Prestero [2001] or Cristi et al. [1990]. For conventional vehicles, for which the

center of gravity is located below the center of buoyancy (zcg is positive), truncating α seems poorly

justified, because it always yields a stable system. In this section we show that fundamental system

properties such as stability need to be addressed more formally in the context of system simplification. In

particular, we examine the role of vehicle parameters in assessing stability of the third-order model (1),

and we demonstrate that stability assumptions about the second-order approximation do not imply

stability of the underlying third-order dynamics. For this reason, assuming a stable linear second-order

AUV pitch-axis model may not be feasible for system identification and control design.

We assess internal stability of the AUV’s pitch-axis model (1) by analyzing the coupling between α and

the remaining states q and θ. To expose these coupling terms, we propose a different representation of

the plant. We set δ = 0 and decompose the system (1) into two coupled subsystems

α(s) = Gα(s) q(s) and q(s) = Gq(s) a32 α(s)

with

Gα(s) =a23

s− a22and Gq(s) =

s

s2 − a33s− a34

(17)

Recalling q(s) = s θ(s), the transfer functions Gα(s) and Gq(s) originate from the first and second row

in (1), respectively. The overall internal system dynamics (1) are represented by the feedback system

shown in Figure 5.

Gq(s) Gα(s)αq

a32

+

Figure 5: Coupling between Gq(s) and Gα(s) through the parameter a32.

For a small horizontal separation xcg between center of gravity and center of buoyancy, we verify that the

poles of Gq(s) and Gα(s) remain in the open left half plane indicating stability of both subsystems. In

fact, the system parameters a33 and a34 in (2) suggest that Gq(s) resembles the motion of a physical

pendulum around the center of buoyancy. This motion is stable assuming both the existence of viscous

damping, modeled by a33 through the vehicle parameter Cmq < 0, and the location of the center of

gravity satisfying zcg > 0. Note that simply truncating α in (1) yields this stable pendular motion

q(s) = Gq(s) b31 δ(s)

The pole location of Gα(s) at a22 < 0 is characterized by the drag coefficient CD0 < 0 and the slope

CLα < 0 of the body’s lift coefficient, see (2).

We evaluate internal stability of (1) using the root locus of the series connection Gα(s) Gq(s) with respect

to the system parameter a32. Figure 6 shows a comparable numerical example of a root locus as function

of a32.

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−1.5

−1

−0.5

0

0.5

1

1.50.420.60.740.84

0.92

0.965

0.99

0.220.420.60.740.840.92

0.965

0.99

0.511.522.53

0.22

Root Locus

Real Axis

Imag

inar

y A

xis

a32 < 0

a32 > 0

a22 −

√

−a34

Figure 6: Root Locus representation of system poles as function of a32.

For a32 = 0, one recognizes the two complex poles of Gq(s) close to the imaginary axis and the single pole

of Gα(s) at a22. The parameter a32 is given in (2) and contains a destabilizing centripetal or munk

moment (mz −mx) V 20 > 0 and a restoring moment 1

2ρV 20 AbLCmα < 0 provided by the vehicle’s body.

Increasing a32 destabilizes the overall system by pushing two poles to the right half plane, while the pole

originally located at a22 moves further left. As illustrated in Figure 5, the parameter a32 acts as positive

feedback gain for the stable series connection Gα(s) Gq(s). Due to the proportionality between a32 and

V 20 , higher speeds are expected to increase the coupling.

We conclude that stability of both subsystems Gα(s) and Gq(s) does not imply stability of (1), and

simply truncating α may not preserve stability properties of the linear pitch-axis model. Thus,

second-order approximations, which are derived by deleting α, may not be reliable for system

identification and control design.

6 Control Design

In this section, we combine the results from previous sections to illustrate a control design approach that

utilizes the second-order approximation (6). We derive conditions under which output-feedback gains

derived from the second-order approximation ensure that the underlying third-order dynamics are

stabilized.

Referring to the state space representation (1), we distinguish between the AUV’s pitch motion defined

by the outputs q and θ and the AUV’s heave motion defined by the output α. It is well known that AUV

heave motion is non-minimum phase, see Narasimhan and Singh [2006] and Narasimhan et al. [2006], and

thus the corresponding transfer function contains a pole and/or a zero in the right half plane. However, a

similar statement about the pitch motion is not true in general.

Our control design strategy focuses on the case of stabilizing an unstable plant (7). The stability analysis

in Section 5, in particular the root locus approach in Figure 6, concludes that an unstable plant (7)

satisfies a32 > 0. With this assumption, it follows from (4) that

ze = −β0

β1= a22 − a32

b21

b31< 0

for a conventional vehicle with b31 < 0, b21 < 0 and a22 < 0. Thus, the zero ze of the transfer

functions (3) and (7) is located in the open left half plane.

For the control design process, we address output-feedback for the AUV pitch-axis models (6) and (7)

and select output-feedback gains Kp and Kd similar as in Prestero [2001]. However, for stability analysis,

we find it useful to suppose a single output θ(s). Then, the feedback gains Kp and Kd define a

PD-controller transfer function

Gc(s) = Kp + sKd (18)

recalling q(s) = s θ(s). We choose the feedback law (18) over a more general PID-framework that is

commonly used for AUV attitude control, see Fossen [1994], Petrich et al. [2007] or Rentschler et al.

[2006], because PD-control suffices to stabilize the second-order system. Integral control may be added to

meet performance criteria such as zero steady-state error, overshoot etc.

When selecting feedback gains for the second-order model G0(s) from (6), one commonly specifies the

natural frequency ω0 > 0 and the damping coefficient D > 0 for the closed-loop system

H2(s) =Gc(s) G0(s)

1 + Gc(s) G0(s)=

n2(s)d2(s)

Then, the denominator polynomial satisfies

d2(s) = s2 + (B31Kd −A33) s + B31Kp −A34

= s2 + 2Dω0s + ω20

and the nominal feedback gains are

Kp =1

B31

(A34 + ω2

0

)

Kd =1

B31(A33 + 2Dω0)

(19)

Since Kp and Kd are chosen exclusively based on a second-order approximation, nothing can be said

about whether or not the feedback law δ(s) = −Gc(s)θ(s) suffices to stabilize the third-order dynamics.

In the following, we augment the nominal feedback gains (19) such that closed-loop stability is

guaranteed for the underlying third-order plant (7). The third-order closed-loop system is

H3(s) =Gc(s) G2(s) Ge(s)

1 + Gc(s) G2(s) Ge(s)=

n3(s)d3(s)

(20)

The following proposition addresses the selection of such feedback gains.

Proposition 6.1 Suppose the second-order approximation satisfies |∆B31| < |B31| and assume that the

truncated mode is stable implying pe < 0. For the controller (18), let the feedback gains be

Kp =1

B31

(A34 + ω2

0 + γp

)

Kd =1

B31(A33 + 2Dω0 + γd)

(21)

with γp > 0, γd > 0 and suppose ω0 and D are selected to satisfy

D2 >1

4Ke

(1 +

2γp

ω20

)(22)

where

Ke = 1 +∆B31

B31∈ (0, 2)

Then, the closed-loop system (20) consisting of the third-order plant (7) and the controller (18) is

asymptotically stable if

−zeKeγp > |pe + 2ze| |A34|+ |pe + ze| |∆A34|

−zeKeγd > |pe + 2ze| |A33|+ |pe + ze| |∆A33|(23)

In other words, we increase robustness against parameter estimation errors and truncated pitch dynamics

by augmenting the nominal feedback gains (19) with sufficiently large, positive constants γp and γd.

Imposing the assumption |∆B31| < |B31|, we suppose that the sign of the input parameter B31 is

identified correctly. It is easy to verify that |∆B31| < |B31| implies

sign (B31) = sign (B31 + ∆B31)

Proof of Proposition 6.1: Using the assumptions ze < 0 and pe < 0, we divide the inequalities (23)

by −ze > 0 and note that

Ke γp > 2 |A34|+ |∆A34|

Ke γd > 2 |A33|+ |∆A33|

Using∣∣∣∣∆B31

B31

∣∣∣∣ < 1, we conclude

Ke γp >

∣∣∣∣∆B31

B31

∣∣∣∣ |A34|+ |∆A34|

Ke γd >

∣∣∣∣∆B31

B31

∣∣∣∣ |A33|+ |∆A33|

(24)

Similarly, we invoke 0 < Ke < 2 in (23), and find that

−zeKeγp > |pe + zeKe| |A34|+ |pe| |∆A34|

> |pe − zeKe| |A34|+ |pe| |∆A34|

−zeKeγd > |pe + zeKe| |A33|+ |pe| |∆A33|

> |pe − zeKe| |A33|+ |pe| |∆A33|

(25)

The denominator polynomial of the closed-loop transfer function H3(s) from (20) is

d3(s) = s3 + κ2s2 + κ1s + κ0

The coefficients κi, i = 0, 1, 2 are

κ2 = −pe − (A33 + ∆A33) + (B31 + ∆B31) Kd

κ1 = pe (A33 + ∆A33)− (A34 + ∆A34) + (B31 + ∆B31)Kp − ze (B31 + ∆B31) Kd

κ0 = pe (A34 + ∆A34)− ze (B31 + ∆B31) Kp

(26)

Referring to Routh’s Stability Criterion Routh [1930], all roots of the polynomial g(s) have negative real

part if and only if

κi > 0, i = 0, 1, 2 and κ2κ1 > κ0

Thus, we substitute Kp and Kd from (21) into (26) and check the signs of each coefficient κi individually.

For κ2, we find

κ2 = −pe − (A33 + ∆A33) + Ke (A33 + 2Dω0 + γd)

= −pe −∆A33 +∆B31

B31A33 + Ke (2Dω0 + γd)

Recalling the lower bound for Keγd from (24), we conclude

κ2 > −pe + Ke2Dω0 > 0 (27)

For κ1, we find

κ1 = pe (A33 + ∆A33)− (A34 + ∆A34) + Ke

(A34 + ω2

0 + γp

)− zeKe (A33 + 2Dω0 + γd)

= (pe − zeKe) A33 + pe∆A33 − zeKe (2Dω0 + γd) +∆B31

B31A34 −∆A34 + Ke

(ω2

0 + γp

)

With the lower bound for Keγp from (24), we simplify

κ1 > (pe − zeKe) A33 + pe∆A33 − zeKe (2Dω0 + γd) + Keω20

Invoking the lower bound for −zeKeγd from (25), we find

κ1 > −zeKe2Dω0 + ω20 > 0 (28)

For κ0, we find

κ0 = pe (A34 + ∆A34)− zeKe

(A34 + ω2

0 + γp

)

= (pe − ze) A34 + pe∆A34 − zeKe

(ω2

0 + γp

) (29)

Again, invoking the lower bound for −zeKeγp from (25), we conclude

κ0 > −zeKeω20 > 0

Finally, it remains to show that κ2κ1 > κ0. A lower bound for κ2κ1 is found by multiplying (27) and

(28), that is

κ2κ1 > (−pe + Ke2Dω0)(−zeKe2Dω0 + ω2

0

)

> −zeK2e (2Dω0)

2

Using the control design requirement (22), we rewrite the previous inequality as

κ2κ1 > −zeKe

(ω2

0 + 2γp

)(30)

We find an upper bound of κ0 by applying (25) to (29), that is

κ0 = (pe − ze) A34 + pe∆A34 − zeω20 − zeγp

< −zeKeω20 − zeKe2γp

(31)

Combining (30) and (31), we conclude

κ2κ1 > −zeKe

(ω2

0 + 2γp

)> κ0

Thus, all stability criteria are met.

¤

For control design purposes, Proposition 6.1 can be used to derive stabilizing output-feedback gains based

a second-order approximation (5). The assumptions in Proposition 6.1 include bounds on modeling errors

parameterized as coefficient errors ∆A33, ∆A34 and ∆B31, as well as signs of real-valued, truncated

zero ze and pole pe. Using an experimentally identified linear pitch-axis model for the Virginia Tech 475

AUV, we provide insight into the selection of γp and γd in Section 8.

7 Experimental Results, System Identification

In this section, we estimate the parameters of a linear second-order and third-order AUV pitch-axis

model using real data collected during field trials. The detailed parameter estimation process is presented

for the nominal configuration of the Virginia Tech 475 AUV shown in Figure 1. Comparing the identified

pitch-axis models for the Virginia Tech 475 AUV, we find that the difference between third-order model

and second-order approximation is marginal. In addition, we identify and compare third-order and

second-order models for two dramatically different payload configurations: the Virginia Tech 475 AUV

equipped with a forward-looking sonar, shown in Figure 7, and the Virginia Tech 475 AUV equipped

with a towed-hydrophone array, shown in Figure 8. All field trials are conducted at a constant propeller

rate of 2400 rpm which corresponds to a speed of approximately 1 ms−1.

We presume the closed loop system illustrated in Figure 9 in order to identify the parameters of a generic

pitch model Σ. The controller transfer function K(s) and the commanded pitch angle θr(t) are known,

Figure 7: Virginia Tech 475 AUV with an attached Forard Looking Sonar (AUV/FLS configuration).

Figure 8: Virginia Tech 475 AUV with an attached towed sensor array (AUV/TSA configuration).

and the collected data set contains the AUV’s trajectory θ(t), q(t) and δ(t). We simulate the trajectories

for pitch angle θ(t), pitch rate q(t), and elevator fin deflection δ(t) using the closed loop system from

Figure 9. The notation θ, q, δ distinguishes the simulated trajectories from the real trajectories θ, q, δ.

Applying a Nelder-Mead Simplex Method Lagarias et al. [1998], we estimate the parameters of the linear

pitch modelΣ

− controllerK(s)

−

δ

q

θ

θr

Figure 9: Closed loop system consisting of pitch model Σ that is second or third order, a controller transfer

function K(s) and a known commanded pitch angle θr.

pitch model Σ (second or third order) by numerically minimizing the cost function

J = θrms + τ qrms + δrms (32)

The root-mean-square error qrms is multiplied by τ = 1 second to obtain compatible units. The individual

root-mean-square errors are

ηrms =

√1T

∫ T

0‖η(t)− η(t)‖2 dt , η = θ, q, δ (33)

For the data set in Figure 10, T = 180 seconds.

Case 1: Using a second-order model (5) for Σ. The numerical minimization yields J = 3.70 deg, and the

parameter estimates are

A34 = −0.27, A33 = −0.91 and B31 = −1.84 (34)

Thus, the second-order model (5) is

˙q˙θ

=

−0.91 −0.27

1 0

q

θ

+

−1.84

0

δ (35)

The associated transfer function G0(s) from (6) is

G0(s) =θ(s)

δ(s)=

−1.84s2 + 0.91s + 0.27

(36)

Using the second-order model (35), the real and simulated trajectories are plotted in Figure 10 in gray

and black, respectively. Figure 10 shows pitch angle θ(t), θ(t) in the top plot and elevator fin deflection

δ(t), δ(t) in the bottom plot.

−10

0

10

θin

deg

0 20 40 60 80 100 120 140 160 180

−10

0

10

δin

deg

time in sec

Figure 10: Real (gray) versus simulated (black) AUV trajectories using a second-order model. Top: pitch

angle θ(t), θ(t), and bottom: elevator fin deflection δ(t), δ(t).

Case 2: Using a third-order model (8) for Σ. For a canonical third-order realization, the numerical

minimization yields J = 2.97 deg. The associated transfer function G3(s) from (3), containing the

coefficients pc, is

G3(s) =θ(s)

δ(s)=

−1.35s− 2.77s3 + 2.30s2 + 1.74s + 0.35

(37)

The parameters in model (8) are deduced by fixing the parameters A33, A34 and B31 from (34) and by

incorporating the canonical coefficients pc into the parameter identities (10). Setting cα = 1, the

parameter estimates are

pα = −1.39, aq = −0.21, aθ = 0.03, bα = −0.89 and bδ = 0.49 (38)

Thus, the state space model (8) is given by

˙xα

˙q˙θ

=

−1.39 −0.21 0.03

1 −0.91 −0.27

0 1 0

xα

q

θ

+

−0.89

−1.35

0

δ (39)

Using the third-order model (39), the real and simulated trajectories are plotted in Figure 11 in gray and

black, respectively. Figure 11 shows pitch angle θ(t), θ(t) in the top plot and elevator fin deflection

δ(t), δ(t) in the bottom plot.

−10

0

10

θin

deg

0 20 40 60 80 100 120 140 160 180

−10

0

10

δin

deg

time in sec

Figure 11: Real (gray) versus simulated (black) AUV trajectories using a third-order model. Top: pitch

angle θ(t), θ(t), and bottom: elevator fin deflection δ(t), δ(t).

Figure 10 and Figure 11 show that the simulated AUV trajectories are hardly distinguishable between

using the second-order model (35) and using the third-order model (39). For reasons of comparison, we

summarize the individual root-mean-square errors (33) in Table 1. We infer that the numerical fit in the

time domain barely improves when using a third-order model.

Table 1: System identification RMS errors (33) for θ, q and δ.

second-order model third-order model

θrms in deg 1.4172 0.9268

qrms in deg s−1 0.8977 0.8029

δrms in deg 1.3857 1.2404

J in deg 3.7006 2.9701

The bode plots and pole-zero plots for the second-order transfer function (36) and the third-order

transfer function (37) are shown in Figure 12 in gray and black, respectively.

In the frequency domain, we find in a qualitative sense that a second-order approximation captures the

dominating system dynamics with sufficient accuracy, see Figure 12. Although, the third-order model

contains an additional pole-zero pair (pe, ze), the additional mode does not have a significant impact on

the input-output behavior of the plant.

In order to quantify the difference between any two transfer functions P1(jω) and P2(jω), we adopt

an L2-error metric

∆ (P1, P2) = ‖P1 − P2‖L2

=1π

∫ ∞

0|P1(jω)− P2(jω)|2 dω

(40)

For the linear AUV pitch-axis models (36) and (37), we find ‖G0(s)‖L2 = 2.62, ‖G3(s)‖L2 = 2.68

and ∆(G0, G3) = 0.4244.

In addition to the nominal AUV configuration, we conduct system identification trials using the

AUV/FLS configuration from Figure 7 and the AUV/TSA configuration from Figure 8. For the

AUV/FLS configuration, the cost function J from (32) decreases from 4.90 deg when using a second-order

model to 4.83 deg when using a third-order model. For the AUV/TSA configuration, the cost function J

from (32) decreases from 2.50 deg when using a second-order model to 2.37 deg when using a third-order

model. Thus, for both AUV/payload configurations the improvement in predicting the AUV’s trajectory

using a third-order model instead of a second-order model is marginal in the time domain. To evaluate

the results in the frequency domain, we show the bode plots of estimated second-order and third-order

models including the associated L2 error metric ∆ in Figure 13 for the AUV/FLS configuration and in

Figure 14 for the AUV/TSA configuration.

−10

0

10

20

magnit

ude

indB

10−2

10−1

100

0

45

90

135

180

phase

indeg

Hz

−3 −2.5 −2 −1.5 −1 −0.5 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

imagin

ary

axis

real axis

Figure 12: Nominal vehicle configuration: Bode plots (upper plots) and pole/zero locations (lower plot) of

the second-order pitch model (35) in gray and the third-order pitch model (39) in black. ∆ = 0.42.

−20

0

20

40

magnit

ude

indB

10−2

10−1

100

0

45

90

135

180

phase

indeg

Hz

−3 −2.5 −2 −1.5 −1 −0.5 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

imagin

ary

axis

real axis

Figure 13: Forward Looking Sonar (FLS) vehicle configuration: Bode plots (upper plots) and pole/zero

locations (lower plot) of the second-order pitch model (35) in gray and the third-order pitch model (39) in

black. ∆ = 2.34.

8 Experimental Results, Control Design

In this section, we investigate whether the second-order model (35) suffices for designing a controller that

guarantees stability for the underlying third-order dynamics (39). For this reason, we first design a

−20

0

20

40

magnit

ude

indB

10−2

10−1

100

0

45

90

135

180

phase

indeg

Hz

−3 −2.5 −2 −1.5 −1 −0.5 0

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

imagin

ary

axis

real axis

Figure 14: Towed sensor array (AUV/TSA) vehicle configuration: Bode plots (upper plots) and pole/zero

locations (lower plot) of the second-order pitch model (35) in gray and the third-order pitch model (39) in

black. ∆ = 0.23.

nominal feedback controller (19) based on the second-order model. Using the parameter estimates

obtained in Section 7, we explicitly compute an error model between (8) and (5), and design a second

feedback controller (21) using Proposition 6.1. Tracking performance and overall robustness are evaluated

for both cases.

For the nominal control design (19), we choose

ω0 = 2πf0, f0 = 0.5Hz and D =1√2

and obtain the nominal feedback gains

K(1)p = −5.22 and K

(1)d = −1.91 (41)

In addition to the nominal controller, we invoke Proposition 6.1 and compute additive feedback gains γp

and γd, which provide robustness against model uncertainties caused by truncated system dynamics and

system identification errors. For this reason, the third-order model (7) is populated using the estimated

parameters of (39) and the parameter equalities (9). The parameters of the error model (42) are

∆A34 = −0.88, ∆A33 = −1.09, ∆B31 = 0.49, pe = −0.30 and ze = −2.05 (42)

Invoking Proposition 6.1, we find γp = 2.17 and γd = 4.37, and the feedback gains (21) become

K(2)p = −6.40 and K

(2)d = −4.28 (43)

The superscripts (1) and (2) denote the nominal control design approach (19) and the robust control

design approach (21), respectively.

Recalling the controller’s transfer function Gc(s) from (18), we plot the open-loop transfer functions in

Figure 15. We use the second-order model G0(s) from (36) and show the bode plots of Gc(s)G0(s) in

gray, solid for the nominal feedback gains (41) and dashed for the robust feedback gains (43). Similarly,

we use the third-order model G3(s) from (37) and show the bode plots of Gc(s)G3(s) in black, solid for

the nominal feedback gains (41) and dashed for the robust feedback gains (43). For all four cases, phase

margins ϕm and required actuator bandwidth fa are summarized in Table 2. The gain margins are

infinite.

For the Virginia Tech 475 AUV, the parameter estimation in Section 7 reveals only a small difference

between the linear third-order pitch-axis model and the second-order approximation, see Figure 12. For

this reason, the respective open-loop transfer functions coincide well. In fact, the nominal feedback

gains (41), chosen based on a second-order model (35), already stabilize the underlying third-order

dynamics for the particular error model parameters (42). Only the phase margin reduces by 10 deg.

From Proposition 6.1 follows that the feedback gains (43) stabilize the underlying third-order system (7)

for all |∆A34| ≤ 0.88, |∆A33| ≤ 1.09 and |∆B31| ≤ 0.49. Thus, the feedback gains (43) increase robustness

−10

0

10

20

30

40m

agnitude

indB

10−2

10−1

100

101

−180

−135

−90

−45

0

f in Hz

phase

indeg

Figure 15: Open-loop transfer functions using second-order model (gray) and third order model (black).

The bode plots are shown for using nominal feedback gains (41) (solid) and robust feedback

gains (43) (dashed).

as compared to the nominal feedback gains (43). We find increased phase margins ϕm visible in Figure 15

and listed in Table 2. However, the required actuator bandwidth fa increases as a trade-off between

robustness and actuator effort is inevitable.

In order to evaluate the tracking performance, we plot the sensitivity transfer function for all four cases

in Figure 16. We show the bode plots of (1 + Gc(s)G0(s))−1 in gray, solid for the nominal feedback

gains (41) and dashed for the robust feedback gains (43). Similarly, we plot (1 + Gc(s)G3(s))−1 in black,

solid for the nominal feedback gains (41) and dashed for the robust feedback gains (43). Good tracking

performance is indicated by small sensitivity magnitude. In Figure 16, no significant difference is visible

between using a third-order model (black) and a second-order approximation (gray). However, we find

better tracking when using the robust controller (dashed) from (43) as compared to using the nominal

Table 2: Following Figure 15: phase margins ϕm and actuator bandwidths fa.

second-order model third-order model

nominal

gains (41)

(solid gray)

ϕm = 69.3 deg

fa = 0.66 Hz

(solid black)

ϕm = 58.9 deg

fa = 0.56 Hz

robust

gains (43)

(dashed gray)

ϕm = 85.9 deg

fa = 1.27 Hz

(dashed black)

ϕm = 79.2 deg

fa = 0.97 Hz

controller (solid) from (41). This is expected since we have increased actuator bandwidth in order to

attain increased robustness.

9 Conclusions

In this paper, we establish theoretical and practical justification for using linear second-order AUV

pitch-axis models for system identification and attitude control design. Our analysis rests on deriving an

error model which defines parameter correlations between the true third-order model and the

second-order approximation. We are motivated by severe size and weight constraints of miniature AUVs

that only allow for limited onboard instrumentation. In particular, we assume that the angle of attack α

is not known.

We prove that the physical parameters of a linear third-order model are not identifiable without

measuring α. Thus, we show that vehicle parameters such as hydrodynamic coefficients or added mass

terms cannot be obtained using online system identification.

We investigate stability of a streamlined AUV and show that truncating the state α always yields a

stable second-order model. However, rigorous stability analysis shows that the underlying third-order

dynamics may very well be unstable. Thus naive truncation of a state may yield incorrect results.

Field trials are an important component of our work. We use standard system identification techniques

to identify the parameters of a second-order pitch-axis model as well as the coefficients of a canonical

third-order realization for three different AUV configurations, and we show that a third-order model

yields negligible improvement over a second-order approximation in predicting pitch-axis motion.

−30

−20

−10

0

10 m

agnit

ude

indB

10−2

10−1

100

101

0

30

60

90

120

f in Hz

phase

indeg

Figure 16: Sensitivity transfer function using second-order model (gray) and third order model (black).

The bode plots are shown for using nominal feedback gains (41) (solid) and robust feedback gains (43)

(dashed).

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