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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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