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Identiﬁcation of High-Order Linear Time-Invariant Models From Periodic

Nonlinear System Responses

Mahmoud A. Hayajnh

Graduate Assistant

School of Aerospace

Engineering

Georgia Institute of

Technology

Atlanta, GA 30332, USA

Umberto Saetti

Assistant Professor

Department of Aerospace

Engineering

Auburn University

Auburn, AL 36849, USA

J.V.R. Prasad

Professor

School of Aerospace

Engineering

Georgia Institute of

Technology

Atlanta, GA 30332, USA

ABSTRACT

This paper presents a ﬁrst step in the extension of subspace identiﬁcation toward the direct identiﬁcation of harmonic

decomposition linear time-invariant (LTI) models from nonlinear time-periodic (NLTP) system responses. The pro-

posed methodology is demonstrated through examples involving the NLTP dynamics of a ﬂapping-wing micro aerial

vehicle (FWMAV). These examples focus on the identiﬁcation of the vertical dynamics from various type of input-

output data, including LTI, LTP, and NLTP input-output data. The use of a harmonic analyzer to decompose the LTP

and NLTP responses into harmonic components is shown to introduce spurious dynamics in the identiﬁcation, which

make the identiﬁed model order selection challenging. A similar effect is introduced by measurement noise. The use

of model-order reduction and model-matching methods in the identiﬁcation process is studied to recover the harmonic

decomposition structure of the known system. The identiﬁed models are validated both in the frequency and time

domains.

INTRODUCTION

Systems with periodic dynamics exist across multiple disci-

plines of engineering, with examples including, but not lim-

ited to, bio-inspired robots (e.g., insects, birds, ﬁsh), space-

craft, rotorcraft, wind turbines, jet engines, and communica-

tion systems. The dynamics of these systems is generally rep-

resented by nonlinear time-periodic systems (NLTP). While

a signiﬁcant body of methods and tools is already available

for the identiﬁcation of linear time-periodic (LTP) systems

and their linear time-invariant (LTI) reformulations from lin-

ear systems’ responses, a number of signiﬁcant issues remain

open for the identiﬁcation of LTP systems and their LTI re-

formulations from nonlinear responses. This is especially true

for challenging applications in which time-periodicity is asso-

ciated with multivariable, nonlinear, and high-order dynam-

ics. Because aerospace vehicles with time-periodic dynam-

ics such as rotorcraft and ﬂapping-wing ﬂyers/micro aerial

vehicles (MAVs) are indeed characterized by multivariable,

nonlinear, and high-order dynamics, the development of new

methodologies is key for assessing their dynamic stability and

for performing ﬂight control design. A considerable amount

on literature exists on the extraction of LTP systems and

their LTI reformulations from NLTP systems via numerical

schemes. In rotorcraft applications, LTP systems and their

Presented at the Vertical Flight Society’s 9th Annual Electric

VTOL Symposium, San Jose, CA, USA, Jan 25–27, 2022. Copy-

right © 2022 by the Vertical Flight Society. All rights reserved.

linear time-invariant (LTI) reformulations are relevant to the

prediction of vibratory/rotor loads and to the analysis and de-

sign of active rotor control systems. A comprehensive sur-

vey of active rotor control system approaches is found in Ref.

(Ref. 1). Further, LTI reformulations of LTP systems enabled

to study the interference effects between the higher harmonic

control (HHC) and the aircraft ﬂight control system (AFCS) in

maneuvering ﬂight (Refs. 2–5). Recently, LTI reformulations

of LTP systems were employed in the design of load allevia-

tion control (LAC) laws (Refs. 6,7). A comprehensive survey

of the methods used to extract harmonically-decomposed LTI

models of rotorcraft is found in Ref. 8. In ﬂapping-wing ap-

plications, LTI reformulations of LTP systems are important

for the study of dynamic stability. Methods span averaging

methods (Refs. 9–11), Floquet theory (Refs. 12–15), and har-

monic decomposition (Refs. 16,17).

All these methods, however, focused on the extraction of LTI

reformulations of LTP systems from physics-based models via

numerical schemes, rather than from system identiﬁcation. In

fact, only a limited amount of studies sought the identiﬁcation

of the LTP dynamics of rotorcraft or ﬂapping-wing vehicles.

For rotorcraft LTP system identiﬁcation, see Ref. 8and the

references therein. For ﬂapping-wing, see Refs. 18–20. Sur-

prisingly, to the best knowledge of the authors, very little pub-

lished work exists which focused on the direct identiﬁcation

of harmonic decomposition LTI models for any of these vehi-

cles. It is important to point out that the application of time-

1

invariant reformulations to convert a LTP model identiﬁcation

problem into a LTI one is far from straightforward because of

the need of deﬁning an approach to return from the LTI identi-

ﬁed model to its LTP counterpart. To handle this difﬁculty, as

well as to simplify the application of model identiﬁcation to

MIMO LTP systems, periodic extensions of subspace model

identiﬁcation methods have been developed over the last few

years. Early works in this direction include the LTP exten-

sions of the intersection algorithm 21 and of the MOESP al-

gorithm 22,23 and the approach of Ref. 24. In Ref. 25 an ap-

proach to LPV model identiﬁcation subject to the requirement

of a periodic scheduling sequence has been proposed. In re-

cent years novel approaches to subspace identiﬁcation of LTP

systems have been proposed, both in the time-domain and in

the frequency-domain using harmonic transfer functions (see,

e.g., Refs. 26–31).

As such, the objective of this paper is to extend the use

of subspace identiﬁcation of higher-order LTI systems in

harmonic decomposition form from nonlinear time-periodic

(NLTP) system responses. The paper begins with a detailed

description of the proposed methodology. The methodology is

demonstrated through examples involving the NLTP dynam-

ics of a ﬂapping-wing micro aerial vehicle (FWMAV). Ex-

amples focus on the identiﬁcation of the FWMAV dynamics

from various type of input-output data, including LTI, LTP,

and NLTP input-output data. The effect of a harmonic an-

alyzer to decompose the LTP and NLTP responses into har-

monics is assessed on the identiﬁcation process, along with

the effect of measurement noise. The identiﬁed models are

validated both in the frequency and time domains.

METHODOLOGY

Mathematical Background

Consider a nonlinear time-periodic (NLTP) system in ﬁrst-

order form:

˙

x

x

x=f

f

f(x

x

x,u

u

u,t)(1a)

y

y

y=g

g

g(x

x

x,u

u

u,t)(1b)

where x

x

x∈Rnis the state vector, u

u

u∈Rmis the control in-

put vector, y

y

y∈Rlis the output vector, and tis the dimen-

sional time in seconds. The nonlinear functions f

f

fand g

g

gare

T-periodic in time such that:

f

f

f(x

x

x,u

u

u,t) = f

f

f(x

x

x,u

u

u,t+T)(2a)

g

g

g(x

x

x,u

u

u,t) = g

g

g(x

x

x,u

u

u,t+T)(2b)

Note that the fundamental period of the system is T=2π

ωsec-

onds, where ωis the frequency of excitation in rad/s. Let x

x

x∗(t)

and u

u

u∗(t)represent a periodic solution of the system such that

x

x

x∗(t) = x

x

x∗(t+T)and u

u

u∗(t) = u

u

u∗(t+T).

Let x

x

x∗(t)and u

u

u∗(t)represent a periodic solution of the system

such that x

x

x∗(t) = x

x

x∗(t+T)and u

u

u∗(t) = u

u

u∗(t+T). Then, the

NLTP system can be linearized about the periodic solution.

Consider the case of small disturbances:

x

x

x=x

x

x∗+∆x

∆x

∆x(3a)

u

u

u=u

u

u∗+∆u

∆u

∆u(3b)

where ∆x

∆x

∆xand ∆u

∆u

∆uare the state and control perturbation vectors

from the candidate periodic solution. A Taylor series expan-

sion is performed on the state derivative and output vectors.

Neglecting terms higher than ﬁrst order results in the follow-

ing equations:

f

f

f(x

x

x∗+∆x

∆x

∆x,u

u

u∗+∆u

∆u

∆u,t) = f

f

f(x

x

x∗,u

u

u∗,t)F

F

F(t)∆x

∆x

∆x+G

G

G(t)∆u

∆u

∆u(4a)

g

g

g(x

x

x∗+∆x

∆x

∆x,u

u

u∗+∆u

∆u

∆u,t) = g

g

g(x

x

x∗,u

u

u∗,t)P

P

P(t)∆x

∆x

∆x+Q

Q

Q(t)∆u

∆u

∆u(4b)

where:

F(t) = ∂f(x,u)

∂xx∗,u∗,G(t) = ∂f(x,u)

∂ux∗,u∗(5a-b)

P(t) = ∂g(x,u)

∂xx∗,u∗,Q(t) = ∂g(x,u)

∂ux∗,u∗(5c-d)

Note that the state-space matrices in Eq. (5) have T-periodic

coefﬁcients. Equations (4a) and (4b) yield a linear time-

periodic (LTP) approximation of the NLTP system of Eq. (1)

as follows:

∆

∆

∆˙

x

x

x=F

F

F(t)∆x

∆x

∆x+G

G

G(t)∆u

∆u

∆u(6a)

∆

∆

∆y

y

y=P

P

P(t)∆x

∆x

∆x+Q

Q

Q(t)∆u

∆u

∆u(6b)

Hereafter, the notation is simpliﬁed by dropping the ∆in front

of the linearized perturbation state and control vectors while

keeping in mind that these vectors represent perturbations

from a periodic equilibrium. Next, the state, input, and out-

put vectors of the LTP systems are decomposed into a ﬁnite

number of harmonics of the fundamental period via Fourier

analysis:

x

x

x=x

x

x0+

N

∑

i=1

x

x

xic cosiψ+x

x

xis siniψ(7a)

u

u

u=u

u

u0+

M

∑

j=1

u

u

ujc cos jψ+u

u

ujs sin jψ(7b)

y

y

y=y

y

y0+

L

∑

k=1

y

y

ykc cos kψ+y

y

yks sin kψ(7c)

As shown in Ref. 16, the harmonic decomposition method-

ology can be used to transform the LTP model into an ap-

proximate higher-order linear time-invariant (LTI) model in

ﬁrst-order form:

˙

X

X

X=A

A

AX

X

X+B

B

BU

U

U(8a)

Y

Y

Y=C

C

CX

X

X+D

D

DU

U

U(8b)

where the augmented state, control, and output vectors are:

X

X

XT=x

x

xT

0x

x

xT

1cx

x

xT

1s... x

x

xT

Nc x

x

xT

Ns (9a)

U

U

UT=u

u

uT

0u

u

uT

1cu

u

uT

1s... u

u

uT

Mc u

u

uT

Ms(9b)

Y

Y

YT=y

y

yT

0y

y

yT

1cy

y

yT

1s... y

y

yT

Lc y

y

yT

Ls(9c)

with A

A

A∈Rn(2N+1)×n(2N+1),B

B

B∈Rn(2N+1)×m(2M+1),C

C

C∈

Rl(2L+1)×n(2N+1), and D

D

D∈Rl(2L+1)×m(2M+1).

2

Subspace Identiﬁcation

Consider now a discrete-time representation of the harmonic

decomposition system in Eq. (8) with unknown coefﬁcient

matrices:

X

X

X[k+1] = A

A

AdX

X

X[k] + B

B

BdU

U

U[k](10a)

Y

Y

Y[k] = C

C

CX

X

X[k] + D

D

DU

U

U[k](10b)

where the subscript dstrands for discrete. This system rep-

resents the approximate LTI dynamics to be identiﬁed. To do

so, subspace identiﬁcation is used. The choice of subspace

identiﬁcation is justiﬁed by its single-step approach to solv-

ing for the unknown system coefﬁcients, as opposed to an

iterative process. Here, a discrete-time approach to identi-

ﬁcation is adopted as the Hankel matrices constructed from

input-output data with continuous-time methods may become

ill-conditioned for high -order systems because of their block-

Vandermonde structure (Ref. 32). Because the systems con-

sidered in this study may be of high order, depending on the

number of harmonics of interest, discrete-time subspace iden-

tiﬁcation offers increased numerical stability. Based on this

framework, the identiﬁcation problem is stated as follows:

given smeasurements of input U

U

U[k]and output Y

Y

Y[k]generated

by the unknown system in Eq. (1) and then decomposed in

its harmonics using a harmonic analyzer, determine the order

of the unknown system (i.e.,n(2N+1)) and the coefﬁcient

matrices A

A

Ad,B

B

Bd,C

C

C, and D

D

Dup to within a similarity transfor-

mation. The general procedure of the subspace identiﬁcation

algorithm is summarized from Refs. 23,33,34 and is articu-

lated in ﬁve major steps.

The ﬁrst step involves the construction of the block Hankel

matrices from the given input-output data. The input block

Hankel matrices are deﬁned as:

U

U

Ublock =

U

U

U1U

U

U2U

U

U3... U

U

Uj

U

U

U2U

U

U3U

U

U4... U

U

Uj+1

.

.

..

.

..

.

.....

.

.

U

U

UiU

U

Ui+1U

U

Ui+2... U

U

Ui+j−1

U

U

Ui+1U

U

Ui+2U

U

Ui+3... U

U

Ui+j

U

U

Ui+2U

U

Ui+3U

U

Ui+4... U

U

Ui+j+1

.

.

..

.

..

.

.....

.

.

U

U

U2iU

U

U2i+1U

U

U2i+2... U

U

U2i+j−1

(11)

=U

U

Up

U

U

Uf(12)

where the subscripts pand findicate past and future data, re-

spectively, iis the number of block rows, and jis the number

of block columns. The number of block rows iis arbitrarily

chosen such that it is larger than the order of the system and

j=s−2i+1. In Ref. 23 iis recommended to be equal to

twice the ratio between the maximum order and the number

of outputs. The output block Hankel matrices Y

Y

Ypand Y

Y

Yfare

found in a similar way. Note that in Eq. (11) past data corre-

sponds to rows up to the ith, whereas future data correspond

to rows after the ith. By applying recursive substitution to Eq.

(10), one obtains:

Y

Y

Yp=Γ

Γ

ΓiX

X

Xp+H

H

HiU

U

Up(13a)

Y

Y

Yf=Γ

Γ

ΓiX

X

Xf+H

H

HiU

U

Uf(13b)

where the observability matrix is deﬁned as:

Γ

Γ

Γi=C

C

C C

C

CA

A

AdC

C

CA

A

A2

d... C

C

CA

A

Ai−1

dT(14)

The matrix H

H

Hiis a block Toeplitz matrix of the following

form:

H

H

Hi=

D

D

D0

0

0... ... 0

0

0

C

C

CB

B

BdD

D

D0

0

0... 0

0

0

C

C

CA

A

AdB

B

BdC

C

CB

B

BdD

D

D... 0

0

0

.

.

..

.

..

.

.....

.

.

C

C

CA

A

Ai−2

dB

B

BdC

C

CA

A

Ai−3

dB

B

Bd... C

C

CB

B

BdD

D

D

(15)

Additionally, the past and future states are stacked states de-

ﬁned as:

X

X

Xp=X

X

X0X

X

X1... X

X

Xi−1(16a)

X

X

Xf=X

X

XiX

X

Xi+1... X

X

Xi+j−1(16b)

The second step involves the computation of the oblique pro-

jection by means of QR decomposition. In this step, the pro-

jection of the future output space along the future input space

into the joint space of the past input and output, Y

Y

Yf/U

U

UfU

U

Up

Y

Y

Yp,

is found. This projection can be thought of as the problem

of predicting the future outputs Y

Y

Yfusing the information ob-

tained from the past data U

U

Up

Y

Y

Ypand the knowledge of the fu-

ture inputs U

U

Uf. Then, the observability matrix Γ

Γ

Γiis extracted

from this projection by means of singular value decomposi-

tion (Ref. 23).

In the third step, the singular value decomposition of the

weighted oblique projection is computed. In this step, the

order of the system is determined as the number of the non-

zero singular values. In practice, the order of the system is

found by comparing the singular values with a small thresh-

old greater than zero. In step four, the shift property of the

observability matrix is used to obtain the identiﬁed system

and output matrices A

A

Adand C

C

C. In step ﬁve, the control and

feedthrough matrices B

B

Bdand D

D

Dare calculated using the least

square method. Lastly, the identiﬁed system is transformed

back to continuous-time form.

Model-Order Reduction

Because system identiﬁcation if performed for a harmonic de-

composition model, input-output data from the NLTP system

needs to be decomposed into harmonics of the fundamental

frequency of the system. To do so, the input-output data is

processed with a harmonic analyzer to extract the harmonic

coefﬁcients of the signal. This has the adverse effect of in-

troducing spurious dynamics in the identiﬁcation. To remove

3

these spurious dynamics from the identiﬁed system, model-

order reduction is employed. Those spurious dynamics in-

troduced that are relatively slow compared to the known fun-

damental frequency of the system are ﬁrst removed via trun-

cation. The states corresponding to these dynamics are typi-

cally identiﬁed by spectral analysis of the identiﬁed system.

However, some prior knowledge on the system is required

to understand which dynamics are indeed spurious. Next,

those spurious dynamics that are faster than the fundamen-

tal frequency are removed using singular perturbation the-

ory (Ref. 35). More speciﬁcally, under the assumption that

the identiﬁed dynamics are stable, residualization is used to

further reduce the order of the model (Ref. 35).

The state vector of identiﬁed dynamics is partitioned into fast

and slow components:

X

X

XT= [X

X

XT

sX

X

XT

f](17)

Then, the identiﬁed dynamics can be re-written as:

˙

X

X

Xs

˙

X

X

Xf=A

A

AsA

A

Asf

A

A

Afs A

A

Af X

X

Xs

X

X

Xf+B

B

Bs

B

B

BfU

U

U(18)

By neglecting the dynamics of the fast states (i.e.,˙

X

X

Xf=0

and performing a few algebraic manipulations, the equations

for a reduced-order system with the state vector composed of

the slow states is:

˙

X

X

Xs=ˆ

A

A

AX

X

Xs+ˆ

B

B

BU

U

U(19)

where:

ˆ

A

A

A=A

A

As−A

A

AsfA

A

A−1

fA

A

Afs (20a)

ˆ

B

B

B=B

B

Bs−A

A

AsfA

A

A−1

fB

B

Bf(20b)

Model Matching

Because subspace identiﬁcation yields an unstructured sys-

tem, the states of the identiﬁed system do not generally have

a physical meaning. In addition, the identiﬁed system will in

general not be in harmonic decomposition form. However,

when the system dynamics are know a priori, the physical

meaning of the states can be recovered as the identiﬁed system

matrices are up to within a similarity transformation matrix of

the harmonic decomposition model (Ref. 36). Consider the

identiﬁed unstructured dynamics in continuous time:

Gc:˙

X

X

X=A

A

AX

X

X+B

B

Bu

u

u

Y

Y

Y=C

C

CX

X

X+D

D

DU

U

U(21)

Additionally, consider a structured model with unknown co-

efﬁcients θ

θ

θ.

Gs(θ

θ

θ):˙x

˙x

˙x=A

A

A(θ

θ

θ)x

x

x+B

B

B(θ

θ

θ)u

u

u

y

y

y=C

C

C(θ

θ

θ)x

x

x+D

D

D(θ

θ

θ)u

u

u(22)

The model matching problem consists of ﬁnding those un-

known coefﬁcients that minimize the H∞norm of the differ-

ence between Gcand Gs. Formally,

θ

θ

θ∗=arg min||Gc(s)−Gs(s;θ)||∞(23)

This minimization problem is a non-convex non-smooth

optimization problem. This problem can be reformulated as

a structured control problem for which robust computational

techniques are available (Ref. 37).

Summary

In summary, the proposed methodology for the direct identi-

ﬁcation of LTI harmonic decomposition models is articulated

in ﬁve major steps:

1. Generation of the input-output data from the NLTP sys-

tem.

2. Processing the input-output data with a harmonic ana-

lyzer to extract the harmonics of the fundamental fre-

quency of the system.

3. Application of subspace identiﬁcation to identify the

higher-order LTI dynamics.

4. Removal of spurious higher-order dynamics introduced

by the harmonic analyzed via model-order reduction.

5. Application of model-matching methods to recover the

harmonic decomposition form of the identiﬁed LTI ap-

proximation to the NLTP system.

SIMULATION MODEL

The proposed methodology is demonstrated through exam-

ples involving the dynamics of a ﬂapping-wing micro aerial

vehicle (FWMAV). Consider the NLTP vertical dynamics of

a FWMAV from Ref. 10:

˙w

¨

φ=g−kd1|˙

φ|w−kL˙

φ2

−kd2|˙

φ|˙

φ−kd3w˙

φ+"0

1

Ifcosωt#U(24)

where wis the vertical speed, ˙

φis the wing ﬂapping speed,

and g is the gravitational acceleration. Additionally, kd1,kd2,

kd3, and kLare constant parameters, IFis the ﬂapping moment

of inertia, ωis the ﬂapping frequency, and Uis the amplitude

of the ﬂapping control input. The state vector is x

x

xT=wφ

and the control vector is u

u

u=U. A high-order LTI approxi-

mation to the NLTP dynamics at hover is found using the har-

monic balance algorithm described in Refs. 17,38. The state

and control input harmonics retained in this process are up to

the ﬁrst (i.e., N=1 and M=1). It follows that the higher-order

LTI system has the following state and control input vectors:

X

X

XT=w0˙

φ0w1c˙

φ1cw1s˙

φ1s(25a)

U

U

UT=U0U1cU1s(25b)

where the state and input vectors have dimensions of

n(2N+1) = 6 and m(2M+1) = 3, respectively. The

numerical values of the system parameters are taken

from Ref. 10, which result in the following model:

4

˙w0

¨

φ0

˙w1c

¨

φ1c

˙w1s

¨

φ1s

=

−4 0 0 −0.032 0 −0.105

0−75.51 −428.4 0 −1.408e+3 0

0−0.065 −2.892 0 −165.9 0

−856.9 0 0 −54.74 0 −179.4

0−0.211 164.5 0 −5.107 0

−2.817e+3 0 0 151.0 0 −96.28

w0

˙

φ0

w1c

˙

φ1c

w1s

˙

φ1s

+

0 0 0

0 3.628e+6 0

0 0 0

7.256e+600

0 0 0

0 0 0

U0

U1c

U1s

(26)

The state vector is re-arranged to show the existence of two uncoupled subsystems:

˙w0

¨

φ1c

¨

φ1s

¨

φ0

˙w1c

˙w1s

=

−4−0.032 −0.105 0 0 0

−856.9−54.747 −179.46 0 0 0

−2.817e+3 151.0−96.28 0 0 0

0 0 0 −75.51 −428.4−1.408e+3

0 0 0 −0.065 −2.892 −165.9

0 0 0 −0.211 164.5−5.107

w0

˙

φ1c

˙

φ1s

˙

φ0

w1c

w1s

+

0 0 0

7.256e+600

0 0 0

0 3.628e+6 0

0 0 0

0 0 0

U0

U1c

U1s

(27)

The zeroth-harmonic of the vertical speed, w0, is coupled with

the ﬁrst-harmonic states of the ﬂapping speed, ˙

φ1cand ˙

φ1s).

These states are decoupled from the remaining three states

(i.e.,˙

φ0,w1cand w1s), which are in turns coupled together.

The zeroth harmonic of the control input is shown to affect

the ﬁrst subsystem only, whereas the ﬁrst cosine harmonic of

the control input affects solely the second subsystem. To bet-

ter understand the dynamic properties of each subsystem, the

modal participation factors are computed (Ref. 39) and shown

in Fig. 1. Figure 1a shows that the vertical speed contributes

to the heave mode exclusively through its zeroth harmonic,

whereas its ﬁrst harmonic contributes to the ﬂap mode. On

the other hand, Fig. 1b shows the ﬂapping speed to contribute

to the heave mode with its ﬁrst harmonics , and to the ﬂap

mode solely through its zeroth harmonic. The heave mode

has its base eigenvalue at −3.53, whereas the ﬂap mode has

its base eigenvalue at −75.75. These results indicate signif-

icant frequency separation and modal participation between

the two modes. Based on these considerations, the subsystem

consisting of w0,˙

φ1cand ˙

φ1swill be identiﬁable by perturbing

the system through the zeroth harmonic of the control input,

and by measuring the response of the zeroth harmonic of the

vertical speed and the ﬁrst harmonic of the ﬂapping speed.

Conversely, the other subsystem which includes ˙

φ0,w1cand

w1swill be identiﬁable by perturbing the system through the

ﬁrst cosine harmonic of the control input and by measuring

the response of the ﬁrst harmonic of the vertical speed and the

zeroth harmonic of the ﬂapping speed.

RESULTS

The proposed methodology is demonstrated through exam-

ples involving the FWMAV dynamic model described above.

These examples focus the identiﬁcation of the subsystem cor-

responding to the heave dynamics from various type of input-

output data. First, identiﬁcation is performed directly from

input-output data collected from the harmonic decomposition

model (i.e., the high-order LTI model). Note that this data

is already decomposed into harmonics. Next, the perturba-

tion response corresponding to the LTP system states is re-

constructed from the LTI input-output data using Eq. (7c). A

harmonic analyzer is applied to re-extract the harmonic coefﬁ-

cients of the reconstructed input-output data. This data is then

used in the identiﬁcation process. This is done to assess the

effect of the harmonic analyzer on the identiﬁcation process.

Next the identiﬁcation is repeated by applying the harmonic

5

1s 0 1c

Harmonics [N/rev]

0

20

40

60

80

100

120

Modal Participation Facrors [%]

Flap Mode

Heave Mode

(a) Vertical speed.

1s 0 1c

Harmonics [N/rev]

0

20

40

60

80

100

120

Modal Participation Facrors [%]

Flap Mode

Heave Mode

(b) Flapping speed.

Figure 1: Modal participation factors for the vertical dynam-

ics of the FWMAV in hover.

analyzer directly to the LTP input-output data. The process

is repeated with white noise applied to the identiﬁcation data

to simulate measurement noise. Lastly, the proposed identi-

ﬁcation method is applied to input-output data gathered from

the NLTP system. The identiﬁcation process uses the MOESP

algorithm (Ref. 23).

Identiﬁcation from LTI Input-Output Data

As a ﬁrst example, the proposed identiﬁcation method is ap-

plied to input-output data obtained using the harmonic decom-

position model in Eqs. (26) and (27). This case is chosen ﬁrst

as the input-data is already decomposed into harmonics of the

fundamental frequency of the system. Practically, the system

is perturbed in its zeroth-harmonic input U0with a doublet

starting at t=5 sec and the resulting response of w0,˙

φ1c, and

˙

φ1sis measured. The LTI system response is shown in Fig. 2

with a red dashed line.

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

-0.3

-0.2

-0.1

0

w [m/s]

NLTP LTP LTI

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7

Time [s]

-500

0

500

Figure 2: Response of the NLTP, LTP and high-order LTI ver-

tical dynamics of a FWMAV to a doublet input in the ﬂapping

torque.

The singular values resulting from the SVD of the oblique

projection are shown in Fig. 3. One can see a clear gap af-

ter the third singular value, showing that only three harmonic

states are controllable, thus identiﬁable, when using the zeroth

harmonic input. The eigenvalues of the identiﬁed system are

shown in Fig. 4. The identiﬁed eigenvalues match well with

those of the heave dynamics subsystem.

0 5 10 15 20 25 30 35 40 45 50

Order

10-15

10-10

10-5

100

log( )

Singular Values

X 3

Y 0.1494

Figure 3: Singular values when using LTI input-output data

directly (no noise).

It is worth noting that, in reality, the state/output measure-

ments are not readily decomposed into harmonics of the fun-

damental frequency of the system. It follows that a harmonic

analyzer must be used to recover the harmonic coefﬁcients

of the input-output data. In the special case just considered

where the harmonic coefﬁcients are readily available, the har-

monic analyzer was assumed to be perfect.

The identiﬁcation process is repeated for the case where white

6

-80 -70 -60 -50 -40 -30 -20 -10 0

-200

-150

-100

-50

0

50

100

150

200

LTI (exact)

LTI (identified)

Figure 4: Identiﬁed vs. true eigenvalues using LTI input-

output data directly (no noise).

noise is added to the LTI input-output data. The white noise

has a signal-to-noise ration of 20 when applied to the out-

put data. The singular values for this case are shown in Fig.

5. Like for the case without noise, a gap is still clearly seen

between the third and fourth singular values of the identiﬁed

dynamics. As such, the order of the identiﬁed model is chosen

as 3 and its eigenvalues are shown in Fig. 6.

0 5 10 15 20 25 30 35 40 45 50

Order

10-3

10-2

10-1

100

log( )

Singular Values

X 3

Y 0.1506

Figure 5: Singular values when using LTI input-output data

directly (with noise).

Harmonic Analyzer Effect on the Identiﬁcation

Because the input-output data from either the LTP or NLTP

dynamics is not readily available in its harmonic components,

a harmonic analyzer must be used to compute the harmonics

of the output signals. However, the harmonic analyzer may

introduce distorsions in the signal that may hinder the identi-

ﬁcation process. In this section, the effect of the harmonic an-

alyzer on the identiﬁcation process is assessed. To do so, the

-80 -70 -60 -50 -40 -30 -20 -10 0

-200

-150

-100

-50

0

50

100

150

200

LTI (exact)

LTI (identified)

Figure 6: Identiﬁed vs. true eigenvalues using LTI input-

output data directly (with noise).

perturbation response corresponding to the LTP system states

is ﬁrst reconstructed from the LTI input-output data using Eq.

(7c). Then, the harmonic coefﬁcients are extracted from the

perturbation response using the harmonic analyzer. When this

data is used in the identiﬁcation process, the jump in the sin-

gular values occurs at the eleventh singular value rather than

at the third, as shown in Fig. 7. This indicates that the har-

monic analyzer introduced spurious dynamics in the identiﬁed

model. These are high-frequency dynamics introduced by the

windowing effect of the harmonic analyzer. The eigenvalues

of the identiﬁed 11-state system are shown in Fig. 8. In spite

of the introduction of high-frequency dynamics and thus extra

eigenvalues, the three eigenvalues corresponding to the heave

dynamics subsystem are identiﬁed correctly. Model-order re-

duction is then used to retain only those three states associated

with the heave dynamics, which are shown against the eigen-

values of the known heave dynamics in Fig. 8.

0 5 10 15 20 25 30 35 40 45 50

Order

10-12

10-10

10-8

10-6

10-4

10-2

100

102

log( )

Singular Values

X 11

Y 2.287e-06

Figure 7: Harmonic analyzer effect on the singular values.

7

-80 -70 -60 -50 -40 -30 -20 -10 0

-500

-400

-300

-200

-100

0

100

200

300

400

500

LTI (exact)

LTI (identified)

Reduced-Order LTI (identified)

Figure 8: Identiﬁed vs. true eigenvalues when using the har-

monic analyzer.

Identiﬁcation from LTP Input-Output Data Once the ef-

fect of the harmonic analyzer is understood, it is applied to

identify the high-order LTI dynamics from LTP data. The LTP

data is generated by feeding the same doublet input used for

the LTI dynamics into the LTP system, and by measuring the

perturbation response in the vertical speed and ﬂapping angle.

The LTP system response is shown in Fig. 2with a blue line.

Then, the harmonic analyzer is applied to the output data to

extract the signals w0,˙

φ1c, and ˙

φ1s. The singular values of the

identiﬁed dynamics are shown in Fig. 9. A gap is observed

between the seventh and eighth singular values. As such, the

order of the system is selected as seven. The eigenvalues of

the identiﬁed 7-state system are shown in Fig. 10. Again,

while high-frequency eigenvalues are introduced by the har-

monic analyzer, the three eigenvalues corresponding to the

heave dynamics subsystem appear to be identiﬁed correctly.

The spurious high-frequency dynamics is truncated and resid-

ualized and the 3-state model eigenvalues are shown in Fig.

10.

To validate the identiﬁed 3-state dynamics in the time domain,

responses of the identiﬁed dynamics and LTP dynamics are

compared for a doublet input different than that used in the

identiﬁcation process. The responses are shown in Fig. 11,

where the output of the identiﬁed system nearly overlaps that

of the LTP system. Note that Eq. (7c) was used to reconstruct

the perturbation response from the LTI system output. These

results suggest the suitability of the proposed approach for

the identiﬁcation of the high-order LTI dynamics from LTP

system responses, when applied to simple FWMAV models.

The process is repeated for the case where white noise is ap-

plied to the input-output data prior to the use of the harmonic

analyzer. Signal-to-noise ratio is chosen again as 20. The sin-

gular values plot from the identiﬁcation is shown in Fig. 12.

When compared to Fig. 9, a clearer jump is noted after the

seventh singular value. Nonetheless, the harmonic analyzer

introduces four extra eigenvalues in the identiﬁcation. Once

again, in spite of the four extra eigenvalues, the three eigenval-

0 5 10 15 20 25 30 35 40 45 50

Order

10-10

10-8

10-6

10-4

10-2

100

102

log( )

Singular Values

X 7

Y 0.0005974

Figure 9: Singular values when using LTP input-output data

(no noise).

-80 -70 -60 -50 -40 -30 -20 -10 0 10

-800

-600

-400

-200

0

200

400

600

800

LTI (exact)

LTI (identified)

Reduced-Order LTI (identified)

Figure 10: Identiﬁed vs. true eigenvalues using LTP input-

output data (no noise).

ues corresponding to the heave dynamics are correctly iden-

tiﬁed. The eigenvalues after the application of model-order

reduction to remove the spurious dynamics are shown in Fig.

13 and correspond to those of the original heave dynamics

subsystem.

Identiﬁcation from NLTP Input-Output Data As a last ex-

ample, the identiﬁcation process is performed based on input-

output data obtained from the NLTP dynamics. After obtain-

ing the NLTP system response using the same control input

doublet used in the previous examples, the periodic trim solu-

tion is subtracted from the input-output data to ﬁnd the control

input and state perturbations. The resulting signals are pro-

cessed with the harmonic analyzer to decompose the signal

into harmonics of the fundamental frequency of the system.

The Fourier coefﬁcients of the input-output data are then used

in the identiﬁcation process. The singular values resulting

from the subspace identiﬁcation are shown in Fig. 14. In this

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

-0.1

0

0.1

0.2

LTP

LTI

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [s]

-500

0

500

Figure 11: Identiﬁed LTI vs. original LTP responses follow-

ing a control input doublet.

0 5 10 15 20 25 30 35 40 45 50

Order

10-5

10-4

10-3

10-2

10-1

100

101

log( )

Singular Values

X 7

Y 0.00216

Figure 12: Singular values when using LTP input-output data

(with noise)

ﬁgure, a clear jump is seen at the ninth singular value, indicat-

ing that once again the harmonic analyzer introduces spurious

dynamics. The eigenvalues of the identiﬁed 9-state system

as well as the reduced-order model eigenvalues are shown in

Fig. 15. The eigenvalues of the reduced-order model are very

similar to those of the known system. To validate the iden-

tiﬁed 3-state dynamics in the time domain, responses of the

identiﬁed dynamics and NLTP dynamics are compared for a

doublet input different than that used in the identiﬁcation pro-

cess. The responses are shown in Fig. 16, where the output of

the identiﬁed system nearly overlaps that of the NLTP system.

These results suggest the suitability of the proposed approach

also for the identiﬁcation of the high-order LTI approximate

dynamics from NLTP system responses, when applied to sim-

ple FWMAV models.

-80 -70 -60 -50 -40 -30 -20 -10 0 10

-400

-300

-200

-100

0

100

200

300

400

LTI (exact)

LTI (identified)

Reduced-Order LTI (identified)

Figure 13: Comparison between the identiﬁed and original

systems’ eigenvalues when using the harmonic analyzer on

the LTP signals with measurement noise.

0 5 10 15 20 25 30 35 40 45 50

Order

10-10

10-8

10-6

10-4

10-2

100

102

log( )

Singular Values

X 9

Y 0.0005539

Figure 14: Singular values when using NLTP input-output

data (no noise).

Model Matching

Model matching is performed by leveraging the robust control

toolbox in MATLAB®. It is assumed that the outputs corre-

spond to the states in harmonic decomposition form and that

the feedthrough matrix is equal to zero (i.e.,D

D

D=0

0

0). The un-

structured matrices identiﬁed with subspace identiﬁcation are

converted to continuous-time form using the d2c function.

Next, the continuous-time state-space model thus obtained

is compared to the parametrized state-space model using the

function hinfstruct. This command solves for those un-

known parameters θ

θ

θof the structured system that minimize

the H∞norm of the difference between the identiﬁed dynam-

ics and the structured system. In the minimization process, the

H∞norm tolerance is set to 1e−20, and the target gain is set

to 1e−3. The minimization problem is run for all of the ex-

amples presented above. The system matrix A

A

Afor each case is

9

-80 -70 -60 -50 -40 -30 -20 -10 0 10

-800

-600

-400

-200

0

200

400

600

800

LTI (exact)

LTI (identified)

Reduced-Order LTI (identified)

Two complex

eigenvalues

Figure 15: Identiﬁed vs. true eigenvalues using NLTP input-

output data (no noise).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.2

-0.1

0

0.1

0.2

w [m/s]

NLTP LTI

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [s]

-500

0

500

Figure 16: Identiﬁed LTI vs. original NLTP responses fol-

lowing a control input doublet.

found and compared to the original upper left 3x3 matrix from

Eq. (27). The Frobenius norm is used to ﬁnd the percentage

error between the identiﬁed and known structured system ma-

trices. Table 1shows the percentage error of the difference

between the exact and the identiﬁed matrices for all of these

cases above, where the error is deﬁned as:

e=|A

A

A(θ

θ

θ∗)−A

A

Aexact|F

|A

A

Aexact|F

(28)

The identiﬁed dynamics from LTI input-output data gives the

best match with the exact dynamics. In fact, the match is al-

most perfect. This does not come as a surprise as it is the case

where the harmonic analyzer is assumed to be perfect. The

remaining cases show that the use of the harmonic analyzer

reduces the accuracy of the identiﬁed dynamics, and that the

identiﬁcation from NLTP dynamics is less precise than that

from LTP data.

Table 1: Percentage error of the difference between the iden-

tiﬁed structured system matrix and the exact dynamics in har-

monic decomposition form.

Input-Output Data Type Error, e[%]

LTI 9.0843e-07

LTI + Noise 0.0767

LTI + Harmonic Analyzer 0.0026

LTP 0.328

LTP + Noise 0.4514

NLTP data 4.1337

CONCLUSION

In this work, the use of subspace identiﬁcation was extended

toward the direct identiﬁcation of higher-order LTI systems in

harmonic decomposition form from nonlinear time-periodic

system (NLTP) responses. The methodology was demon-

strated through examples involving the NLTP dynamics of a

ﬂapping-wing micro aerial vehicle (FWMAV). Examples fo-

cused on the identiﬁcation of the heave dynamics from vari-

ous type of input-output data, including LTI, LTP, and NLTP

input-output data. The effect of a harmonic analyzer to de-

compose the LTP and NLTP responses into harmonics was as-

sessed on the identiﬁcation process. The effect of white noise

on the identiﬁcation process was studied as well. Based on

this work, the following conclusions can be reached.

1. The application of harmonic analyzers to decompose

input-output data into harmonics of the fundamental fre-

quency of the system introduces spurious dynamics in

the identiﬁed system. These spurious dynamics make it

challenging to determine the correct order of the system.

When the order of the system is known, these spurious

dynamics can be removed using model-order reduction

methods such as truncation and residualization. How-

ever, some prior knowledge of the system is necessary

to remove the spurious dynamics introduced by the har-

monic analyzer.

2. The mismatch between the identiﬁed and exact systems

when the identiﬁcation is performed from LTI input-

output data (i.e., for the case where the harmonic ana-

lyzer is perfect) is very small. The mismatch grows, but

is still acceptable, if the identiﬁcation is performed from

harmonically-decomposed LTP and NLTP input-output

data.

3. Noise is shown to have a negative effect on the accuracy

of the identiﬁcation. Additionally, noise makes it harder

to determine the true order of the system.

4. Model matching allowed to recover the harmonic decom-

position structure in the identiﬁed model. However, pre-

vious knowledge of the system to be identiﬁed is neces-

sary for this step.

Future work will concentrate on the process of determining

the spurious dynamics when there is limited or no knowledge

10

of the system to be identiﬁed. Furthermore, future work will

focus on extending the methodology to more complex and

higher-order systems such as the longitudinal ﬂight dynamics

of FWMAVs, to helicopter rotors, and to rotorcraft in general.

ACKNOWLEDGMENTS

This research was partially funded by the Government under

the Vertical Lift research Center of Excellence (VLRCOE)

program at Georgia Tech under Agreement No. W911W6-17-

2-0002. The U.S. Government is authorized to reproduce and

distribute reprints for Government purposes notwithstanding

any copyright notation thereon. The views and conclusions

contained in this document are those of the authors and should

not be interpreted as representing the ofﬁcial policies, either

expressed or implied, of the US Army Technology Develop-

ment Directorate, CCDC AvMC or the U.S. Government.

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