Conference PaperPDF Available

Design of Dynamic Inversion and Explicit Model Following Control Laws for Quadrotor Inner and Outer Loops

Authors:
  • University of Maryland
  • U.S. Army Combat Capabilities Development Command Aviation & Missile Center

Abstract and Figures

A quadrotor was assembled with commercial off-the-shelf (COTS) components readily available on the market as a platform for future research at Penn State. As a first step in this research, a model of the quadrotor is identified from flight data. Given the largely decoupled dynamics at low speed, frequency sweeps in different channels are performed separately on the roll, pitch, yaw and heave axes. A frequency-domain approach is used to perform system identification. First, frequency responses of the aircraft output are extracted from frequency-sweep flight data. Next, state-space models are fit to the frequency response data. Overall the identified model matched flight data well in both the frequency and time domain. Dynamic Inversion (DI) and Explicit Model Following (EMF) with LQR disturbance rejection control laws are developed for both an inner attitude loop and outer velocity loop. The control laws were developed to meet similar requirements, and have similar performance and robustness.
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Design of Dynamic Inversion and Explicit Model
Following Control Laws for Quadrotor Inner and
Outer Loops
Umberto Saetti
PhD Candidate
Joseph F. Horn
Professor
Department of Aerospace Engineering
The Pennsylvania State University
University Park, PA
Sagar Lakhmani
Graduate Student
Constantino Lagoa
Professor
Department of Electrical Engineering
The Pennsylvania State University
University Park, PA
Tom Berger
Aerospace Engineer
Aviation Development Directorate
U.S. Army AMRDEC
Moffett Field, CA
Abstract—A quadrotor was assembled with commercial off-
the-shelf (COTS) components readily available on the market as
a platform for future research at Penn State. As a first step in this
research, a model of the quadrotor is identified from flight data.
Given the largely decoupled dynamics at low speed, frequency
sweeps in different channels are performed separately on the roll,
pitch, yaw and heave axes. A frequency-domain approach is used
to perform system identification. First, frequency responses of the
aircraft output are extracted from frequency-sweep flight data.
Next, state-space models are fit to the frequency response data.
Overall the identified model matched flight data well in both the
frequency and time domain. Dynamic Inversion (DI) and Explicit
Model Following (EMF) with LQR disturbance rejection control
laws are developed for both an inner attitude loop and outer
velocity loop. The control laws were developed to meet similar
requirements, and have similar performance and robustness.
I. INTRODUCTION
Penn State has a long standing tradition of developing
flight control systems for rotorcraft. In particular, Dynamic
Inversion (DI) and Explicit Model Following (EMF) control
laws have been major areas of research throughout the past
two decades and have been implemented in various helicopter
related simulation studies [1]–[7]. However, few studies ac-
tually have implemented DI on small scale quadrotors [8].
The U.S. Army Aviation Development Directorate (ADD) has
applied system identification techniques, and both DI and EMF
to Micro Aerial Vehicles (MAVs) [9]–[12]. The purpose of
this paper is to merge the expertise on the rotorcraft system
identification and flight control design matured at Penn State
with the techniques developed at ADD, and to extend them
towards the application of the small scale quadrotors. For this
purpose, a quadrotor is assembled with commercial off-the-
shelf (COTS) components readily available on the market. The
quadrotor is meant to become a platform for future research
at Penn State and the base aircraft model of the Penn State’s
team for the AHS Micro-Aerial Vehicle Student Challenge.
This paper presents an overview of the hardware used on
the quadrotor, as well as the results of system identification
0Distribution Statement A: Approved for public release; distribution is
unlimited.
performed on the quadrotor. Next, the synthesis of Dynamic
Inversion (DI) and Explicit Model Following (EMF) control
laws for both an inner attitude loop and outer velocity loop
is presented. The performance of the DI and EMF controllers
is compared in the time and frequency domain. In addition,
a robustness analysis of the controllers is performed using
an unscented transformation of the identified model parameter
uncertainties. Finally, conclusions are presented.
II. HA RDWARE OVERVI EW
The in-house quadrotor, shown in Fig. 1 is comprised of
four main components: on-board computer, flight controller,
sensors, and electric motors. The Raspberry Pi is selected as an
on-board computer in this application. The on-board computer
acts as the brain of the Unmanned Aerial System (UAS) and is
responsible for processing the sensor data and communicating
with the flight controller. The flight controller used is the
PixHawk. It is responsible for running the built-in control
algorithm and operates on the MavLink message protocol.
The control algorithm allows to switch between manual flight
using the remote controller or an automated flight whenever
the pilot needs to. The flight controller has a built-in inertial
measurement unit (IMU), a gyroscope, a GPS receiver and a
barometer used to determine the changes in altitude. A ground
station (GS) and transmitter are also incorporated into the
control architecture of the UAS. The GS is primarily used
for monitoring the telemetry data during flight. A schematic
of the quadrotor is shown in Fig. 2.
III. SYS TE M IDE NT IFI CATI ON
System identification is done in two steps using the
CIFER R
[13] software tool. The identification procedure is
based on the use of frequency responses. First, frequency
responses of the aircraft output are extracted from frequency-
sweep flight data. Next, state-space models are fit to the fre-
quency response data. A frequency-domain modeling approach
is used, given that the dynamics of the quadrotor are unstable.
This would lead to divergence of the integration of the time-
domain equations and the consequent undermining of a time-
domain approach to system identification. Frequency sweeps in
TABLE 1: Lateral dynamics identified parameters.
Parameter Value CR Bound [%] Insensitivity [%]
Yv-0.3022 [1/s] 6.107 1.825
Yp0 [ft/(rad s2)] - -
Lv-0.8287 [rad/ft] 5.943 1.580
Lp0 [1/s] - -
Yδlat 0.0565 [ft/(s2%)] 4.170 2.077
Lδlat 33.5146 [rad/(s2%)] 3.297 1.144
different channels are performed on each axis, separately. The
axes in consideration are roll, pitch, yaw and heave. Piloted
frequency sweeps are used rather than automated inputs due
to their excellent broadband excitation. Notice that the built-in
controller is activated during the system identification process
to enhance the quadrotor stability and therefore to ease the
task of the pilot in performing the sweeps. A schematic of
the quadrotor control system is shown in Fig. 3. Given the
largely decoupled dynamics around each axis at low speed due
to the symmetry of the configuration, system identification in
performed separately around each axis.
A. Lateral Dynamics
The linear system describing the lateral dynamics is:
˙v
˙p
˙
φ
="YvYpg
LvLp0
0 1 0#"v
p
φ#+"Yδlat
Lδlat
0#δlat (1)
The parameters to be identified are the stability derivatives
Yv,Yp,Lv,Lp, and the control derivatives Yδlat ,Lδlat . The
outputs chosen to perform system identification, given their
good coherence factor, are p,˙v, and ayreconstructed in the
time domain [13]. Figure 4 shows input and output time history
used for lateral dynamics system identification. Figure 5 shows
the comparison between the approximated and real frequency
responses p
δlat ,˙v
δlat and ay
δlat . Figure 6 shows the comparison
between identified model and flight test data time responses.
Table 1 shows the value of identified parameters, the Cramer-
Rao bounds, and the insensitivities.
B. Longitudinal Dynamics
The linear system describing the longitudinal dynamics is:
˙u
˙q
˙
θ
="XuXqg
MuMq0
0 1 0 #"u
q
θ#+
Xδlong
Mδlong
0
δlong (2)
The parameters to be identified are the stability derivatives Xu,
Xq,Mu,Mq, and the control derivatives Xδlong ,Mδlong . The
outputs chosen for to perform system identification, given their
good coherence factor, are q,˙u, and axreconstructed in the
time domain. Figure 7 shows input and output time history
used for longitudinal dynamics system identification. Figure
8 shows the comparison between the approximated and real
frequency responses q
δlong ,˙u
δlong and ax
δlong . Figure 9 shows the
comparison between identified model and flight test data time
responses. Table 2 shows the value of identified parameters,
the CR bounds, and the insensitivities.
TABLE 2: Longitudinal dynamics identified parameters.
Parameter Value CR Bound [%] Insensitivity [%]
Xu-0.2568 [1/s] 5.302 1.814
Xq0 [ft/(rad s2)] - -
Mu1.1257 [rad/ft] 5.618 1.306
Mq0 [1/s] - -
Xδlong 0.0355 [ft/(s2%)] 10.73 5.321
Mδlong 27.9188 [rad/(s2%)] 4.155 1.109
TABLE 3: Directional dynamics identified parameters.
Parameter Value CR Bound [%] Insensitivity [%]
Nr-0.5617 [1/s] 25.19 9.713
Nδped 6.0308 [rad/(s2%)] 3.877 1.848
C. Directional Dynamics
The linear system describing the directional dynamics is:
˙r
˙
ψ=Nr0
1 0r
ψ+Nδped
0δped (3)
The parameters to be identified are the stability derivative
Nrand the control derivative Nδped . The output chosen for
to perform system identification is r. Figure 10 shows input
and output time history used for directional dynamics system
identification. Fig. 11 shows the comparison between the
approximated and real frequency responses r
δped . Figure 12
shows the comparison between identified model and flight test
data time responses. Table 3 shows the value of the identified
parameters, the CR bounds, and the insensitivities.
D. Vertical Dynamics
The linear system describing the vertical dynamics is:
˙w=Zww+Zδcoll δcoll (4)
The parameters to be identified are the stability derivative Zw
and the control derivative Zδcoll . The output chosen for to
perform system identification is wand azreconstructed in the
time domain. Fig. 13 shows input and output time history used
for vertical dynamics system identification. Fig. 14 shows the
comparison between the approximated and real frequency re-
sponses w
δcoll and az
δcoll . Fig. 15 shows the comparison between
identified model and flight test data time responses. Table 4
shows the value of the identified parameters, the CR bounds,
and the insensitivities.
E. Complete State Space Model
The complete state-space representation of the quadrotor
rigid body dynamics can be obtained by assembling the linear
systems representing the dynamics around each axis. The
resulting 6degree-of-freedom (DoF) state space system has
the following input and output vectors, and system and control
matrices:
xT= [v p φ u q θ r ψ w](5)
TABLE 4: Vertical dynamics identified parameters.
Parameter Value CR Bound [%] Insensitivity [%]
Zw-0.1734 [1/s] 39.72 16.06
Zδcoll -49.0651 [ft/(s2%)] 2.647 1.312
uT= [δlat δlong δped δcoll](6)
A=
YvYpg0 0 0 0 0 0
LvLp0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 0 XuXqg0 0 0
0 0 0 MuMq0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 Nr0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 Zw
(7)
B=
Yδlat 0 0 0
Lδlat 0 0 0
0 0 0 0
0Xδlong 0 0
0Mδlong 0 0
0 0 0 0
0 0 Nδped 0
0 0 0 0
0 0 0 Zδcoll
(8)
The eigenvalues of the system are calculated and presented
in Fig. 16. The system appears to be stable around the yaw
and heave axes. Both the pitch and roll axes have a negative
eigenvalue on the real axis and a pair of eigenvalues in the right
half plane, representing the classical hovering cubic dynamics
present in all hovering vehicles [14]. This indicates an unstable
mode in both pitch and roll. The similarity of the eigenvalues
in pitch and roll are due to the symmetry of the configuration
and is to be expected. The slight differences may be due to
the differences in the distribution of the electronics components
and sensors along the axes. Overall, these results agree with
the literature (e.g. [9], [12]).
IV. DYNA MI C INVERSION CONTRO LL ER
A. Inner Loop
A Dynamic Inversion (DI) control law is designed to
achieve stability, disturbance rejection, Attitude Command /
Attitude Hold (ACAH) response around the roll and pitch
axes, and Rate Command / Attitude Hold response around
the yaw and heave axes. A similar controller was previously
implemented on a B-430, as discussed in [7]. A general DI
scheme for a SISO system is shown in Fig. 17.
To model the DI controller, a modified state vector is
defined:
ˆ
xT= [p φ q θ r w](9)
as well as modified state and control matrices:
ˆ
A=
Lp0 0 0 0 0
1 0 0 0 0 0
0 0 Mq0 0 0
0 0 1 0 0 0
0 0 0 0 Nr0
0 0 0 0 0 Zw
(10)
TABLE 5: DI and EMF inner loop command filters
properties.
Command ωn[rad/s]ζ
Roll Attitude 10 0.7
Pitch Attitude 10 0.7
Vertical Speed 1 -
Yaw Rate 2 -
ˆ
B=
Lδlat 0 0 0
0 0 0 0
0Mδlong 0 0
0 0 0 0
0 0 Nδped 0
0 0 0 Zδcoll
(11)
Given a commanded reference trajectory ycmd(t), the interest
lays in controlling the output y(t)so that it follows the
command. In this particular application the reference trajectory
and the output are given respectively by Eq. 12 and Eq. 13:
yT
cmd = [φcmd θcmd rcmd Vzcmd](12)
yT= [φ θ r Vz](13)
where φ,θ, and r,Vzare the roll attitude, pitch attitude, roll
rate, and vertical speed (positive up) respectively. The output
matrix C that identifies the controlled states is given by:
C=C1
C2(14)
where
C1=010000
000100(15a)
C2=0001 0
00001(15b)
C1corresponds to the roll and pitch attitudes whereas C2is
related to the yaw rate and vertical speed. This partitioning is
due to the fact that the output equation has to be differentiated
two times to see the controls in the output equation while the
same procedure has to be done just once for rand Vzas given
in Eq. 16.
¨
φ
¨
θ
˙r
˙
Vz
=C1ˆ
A2x+C1ˆ
Aˆ
Bu
C2ˆ
Ax +C2ˆ
Bu (16)
Second order command filters are used for roll and pitch
attitudes, whereas first-order filters are utilized for vertical
speed and yaw rate. Table 5 shows the values used for the
command filters.
PID/PI compensation is used to reject external disturbances
and to compensate for differences between the inversion model
described in the next section and the bare-airframe dynamics.
The dynamic inversion control law is thus given by:
u=C1ˆ
Aˆ
B
C2ˆ
B1νC1ˆ
A2
C2ˆ
Ax(17)
TABLE 6: Outer loop command filter properties.
τ[s]
Vxcmd 1
Vycmd 1
where νis the pseudo-command vector and eis the error as
given respectively in Eq. 18 and Eq. 19.
νφ
νθ
νr
νVz
=
¨
φcmd
¨
θcmd
˙rcmd
˙
Vzcmd
+
eφ
eθ
er
eVz
KP+
˙eφ
˙eθ
0
0
KD+
Reφdt
Reθdt
Rerdt
ReVzdt
KI
(18)
e=ycmd y;(19)
KP,KD, and KIare 4-by-4 diagonal matrices identifying
respectively the proportional, derivative and integral gain ma-
trices. The gain values are discussed in more detail in Section
IV-C. A DI inner loop block diagram is shown in Fig. 18.
B. Outer Loop
The outer loop controller tracks forward and lateral ground
velocities in the heading frame. The heading frame is a vehicle
carried frame where the x-axis is aligned with the current
aircraft heading, the z-axis is positive up in the inertial frame,
and the y-axis points to the right, forming a left-handed
orthogonal coordinate system. Eq. 20 shows the rotation from
body to the heading frame
Th/b ="cos θsin φsin θcos φsin θ
0 cos φsin φ
sin θsin φcos θcos φcos θ#(20)
so that the velocities in the heading frame are given by:
"Vx
Vy
Vz#=Th/b "u
v
w#(21)
The command filter for both lateral and forward velocities, Vx
and Vy, is first order. The time constants are given in Table 6.
The filtered velocities are subtracted from their measurements
to find the error, which goes through a PI controller. The feed-
forward signal is subsequently added, leading to the desired
commands:
νVx
νVy=˙
Vxcmd
˙
Vycmd +eVx
eVyKP+ReVxdt
ReVydtKI(22)
Lateral velocity is controlled by acting on the roll angle.
Consider the lateral velocity equation of motion
˙
Vy=YvVy+cmd (23)
DI is applied, leading to the following inverse law:
φcmd =1
g(νVyYvVy)(24)
Similarly, longitudinal velocity is controlled by acting on the
pitch angle. Consider the longitudinal velocity equation of
motion ˙
Vx=XuVxcmd (25)
DI is applied, leading to the following inverse law:
θcmd =1
g(νVxXuVx)(26)
A DI outer loop block diagram is shown in Fig. 19.
C. Error Dynamics
Feedback compensation is needed both in the inner and
the outer loop for two main reasons. Firstly, we use approx-
imations and the inversion is not exact. Secondly, there are
external disturbances to the system. Similarly as in [15] it can
be demonstrated that for a Dynamic Inversion controller
e(n)=νy(n)
cmd (27)
where nis the number of times the output equation has to
be derived in order for the controls to appear explicitly in the
output equation. For the states for which this has to be done
twice, a PID control strategy applied to the pseudo-command
vector is given by:
ν= ¨ycmd(t) + KD˙e(t) + KPe(t) + KIZt
0
e(τ)(28)
Substituting 28 into 27, we obtain the closed-loop error dy-
namics
¨e(t) + KD˙e(t) + KPe(t) + KIZt
0
e(τ)= 0 (29)
The gains can be chosen so that the frequencies of the error
dynamics are of the same order as the command filters,
ensuring that the bandwidth of the response to disturbances is
comparable to the one of an input given by a pilot. By taking
the Laplace transform and therefore switching to frequency
domain the error dynamics becomes:
e(s)s2+KDs+sKP+1
sKI= 0 (30)
or equivalently
e(s)s3+KDs2+KP+KI= 0 (31)
In order to obtain gains that would guarantee a desired re-
sponse, the error dynamics can be set equal to the third-order
system given in Eq. 32:
(s2+ 2ζωns+ωn2)(s+p)=0 (32)
Developing the product between the polynomials leads to
s3+ (p+ 2ζωn)s2+ (2ζ ωnp+ωn2)s+ωn2p= 0 (33)
Setting the coefficients of the polynomial equal to the the gains
of Eq. 31, results in:
KD= 2ζωn+p(34a)
KP= 2ζωnp+ωn2(34b)
KI=ωn2p(34c)
Specifically, this approach is used for φand θin the inner
loop.
Similarly, for those states for which PI compensation is
applied, the pseudo-command vector is given by:
ν= ˙ycmd (t) + KPe(t) + KIZt
0
e(τ)(35)
TABLE 7: DI inner loop disturbance rejection frequencies,
damping ratios, and integrator poles.
ωn[rad/s]ζp
φcmd 10 0.7 2
θcmd 10 0.7 2
rcmd 1 0.7 -
Vzcmd 1 0.7 -
TABLE 8: DI outer loop disturbance rejection frequencies
and damping ratios.
ωn[rad/s]ζ
Vxcmd 1 0.7
Vycmd 1 0.7
which leads to the following closed-loop error dynamics:
˙e(t) + KPe(t) + KIZt
0
e(τ)= 0 (36)
and, therefore, to:
s+KP+1
sKI= 0 (37)
The resulting gains are:
KP= 2ζωn(38a)
KI=ωn2(38b)
This type of compensation is applied to Vxand Vyin the outer
loop, and to Vzand rin the inner loop. Table 7 and Table 8
show the natural frequencies, damping ratios, time constants,
and the integrator pole values, respectively, for the inner and
the outer loop. Note that the integrator pole pis usually chosen
to be one-fifth of the natural frequency, corresponding to about
one-fifth of the loop crossover frequency [16]. Further, the
outer loop error dynamics needs to be at a significantly lower
frequency than the equivalent inner loop dynamics (e.g. for
the longitudinal velocity ωn= 1 rad/s, which is one order
of magnitude less than ωn= 10 rad/s of the pitch attitude
error dynamics). The numerical value of the resulting gains
are shown in Table 9 and Table 10.
V. EXPLICIT MO DE L FOLLOWING CONTRO LL ER
A. Inner Loop
An Explicit Model Following (EMF) control law is de-
signed to achieve stability, disturbance rejection, ACAH re-
TABLE 9: DI inner loop compensation gains.
KDKPKI
φcmd 16 128 200
θcmd 16 128 200
rcmd - 1.4 1
Vzcmd - 0.3 0.5
TABLE 10: Outer loop compensation gains.
KPKI
Vxcmd 1.4 1
Vycmd 1.4 1
sponse around the roll and pitch axes, and RCAH response
around the yaw and heave axes. A general EMF scheme for
a SISO system is shown in Fig. 20. The present study does
not assume perfect inverse plant dynamics, in fact the inverse
plant is based on a set of decoupled first- and second-order
linear models, where the stability and control derivatives are
taken from the identified model. Specifically, the inverse plant
on the roll, pitch, and yaw axes is approximated respectively
by the following inverse transfer functions:
δlat
φ=s(sLp)
Lδlat
(39a)
δlong
θ=s(sMq)
Mδlong
(39b)
δped
r=sNr
Nδped
(39c)
δcoll
Vz
=sZw
Zδcoll
(39d)
The equivalent time delay τf, used to delay the ideal response
such that it can be physically followed by the controlled
system, is estimated to be τf= 0.056 s, representative of
the brushless motors time constant plus some additional delay
given by the sensors. LQR is used for stability and disturbance
rejection. The states and dynamics of the system used to obtain
the LQR gains are:
xT=p φ Rφ q θ Rθ r Rr Vzz(40)
A=
LvLp0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 MuMq0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 Nr000
0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 Zw0
0 0 0 0 0 0 0 0 1 0
(41)
B=
Lδlat 0 0 0
0 0 0 0
0 0 0 0
0Mδlong 0 0
0 0 0 0
0 0 0 0
0 0 Nδped 0
0 0 0 0
0 0 0 Zδcoll
0 0 0 0
(42)
An EMF inner loop block diagram is shown in Fig. 21.
B. Outer Loop
The EMF outer loop controller tracks forward and lateral
velocities the heading frame. Command filter are first order and
share the same time constants as the DI command filters. The
present study does not assume perfect inverse plant dynamics,
in fact the inverse plant is based on a set of decoupled 1st order
linear models, where the stability and control derivatives are
pulled from the identified model. Specifically, the inverse plant
for forward and lateral speed is approximated respectively by
the following inverse transfer functions:
θcmd
Vx
=sXu
g(43a)
φcmd
Vy
=sYv
g(43b)
LQR is used independently on each axis for disturbance
rejection. The system used to synthesize the gains for the
forward speed is:
˙
Vx
˙x=Xu0
1 0Vx
x+g
0θcmd (44)
where xis the longitudinal position in the heading frame. The
system used to synthesize the gains for the lateral speed is:
˙
Vy
˙y=Yv0
1 0Vy
y+g
0φcmd (45)
where yis the lateral position in the heading frame. An EMF
outer loop block diagram is shown in Fig. 22.
C. Disturbance Rejection
Since the loop on the right hand side of Fig. 20 is
independent from the the inverse plant, the LQR gains for
both inner and outer loop can be determined from the models
discussed in Section V-A and V-B. The cost function being
minimized is:
J=Zt
0xTQx +uTRu(46)
The weight matrices are designed according to [17] and are of
the form:
Q=diag hα2
1
(x1)2
max ·· · α2
n
(xl)2
max i(47)
R=ρ diag hβ2
1
(u1)2
max ·· · β2
m
(um)2
max i(48)
where (xi)2
max and (uj)2
max are the largest desired response
and input for that particular component of the output/input,
Pl
i=1 α2
i= 1 and Pm
j=1 β2
i= 1 are used to add an
additional relative weighting on the various components of the
output/control input, and ρis used as the relative weighting
between the control and state penalties. Table 11 and Table 12
show the penalties on the states and controls used in the inner
loop LQR design, where ρ= 2. Table 13 and Table 14 show
the penalties on the states and controls used in the outer loop
LQR design, where ρ= 0.25 for both lateral and longitudinal
axes.
VI. RE SU LTS
A. Time Domain
Batch simulations are run to test and compare the EMF
and DI controllers. The gains are adjusted such that the two
control strategies give similar performances. The results shown
in this section refer to a lateral speed command doublet.
Figure 23 shows the heading frame velocities, whereas the
Fig. 24 shows the body frame velocities. Figure 25 and
Fig. 26 show respectively the angular rates and the attitude
TABLE 11: EMF inner loop LQR state penalties.
(xi)max α2
i
p π/180 1/10
φ π/180 1/10
Rφ0.05π/180 1/10
q π/180 1/10
θ π/180 1/10
Rθ0.05π/180 1/10
r π/180 1/10
Rr π/180 1/10
Vz1 1/10
z1 1/10
TABLE 12: EMF inner loop LQR control penalties.
(ui)max β2
i
δlat 0.01 1/4
δlong 0.01 1/4
δped 0.01 1/4
δcoll 0.01 1/4
response. Figure 27 shows the controls time history and Fig.
28 shows the actuator perturbations from trim. EMF and DI
show an outstanding agreement both in the on-axis and off-axis
response.
B. Frequency Domain
Figure 29 shows the inner attitude loop broken-loop re-
sponse (loop broken at the input to the mixer) for both the DI
and EMF controllers. Table 15 shows the inner attitude loop
crossover frequencies and gain and phase margins of both the
DI and EMF controllers. Both loops have similar crossover
frequencies ωcφby design. The inner loops were designed
with a reduced phase margin requirement of PM >35 deg. The
EMF design meets this requirement with PM = 40.3 deg, while
the DI design nearly meets the requirement with PM = 34.4
deg. Both designs meet the nominal gain margin requirement
of GM >6dB.
Figure 30 shows the attitude disturbance response for
both the DI and EMF controllers. Table 16 lists the attitude
disturbance rejection bandwidth (DRB) and peak (DRP) [18]
values for the two controllers. The DI controller has a higher
DRB and DRP than the EMF controller. This demonstrates
Bode’s Integral Theorem between the two controllers, where
a reduction in magnitude in one frequency range of the Sensi-
tivity function comes at the cost of an increase in magnitude
at another frequency range.
TABLE 13: EMF outer loop LQR state penalties.
(xi)max α2
i
Vx1 1/2
x1 1/2
Vy1 1/2
y1 1/2
TABLE 14: EMF outer loop LQR control penalties.
(ui)max β2
i
θcmd π/180 1
φcmd π/180 1
TABLE 15: Lateral axis broken loop gain and phase margins.
ωc[rad/sec] GM [dB] PM [deg]
BLφDI 17.8 9.22 34.3
BLφEM F 20.4 7.8 40.3
BLvDI 1.59 18.4 66.8
BLvEM F 1.6 21.5 67.4
TABLE 16: Lateral axis disturbance response gain and phase
margins.
DRB [rad/s] DRP [dB]
φ0dDI 5.12 4.45
φ0dEM F 3.69 2.83
v0/vdDI 1.77 1.89
v0/vdEM F 1.24 1.59
Figure 31 shows the closed-loop attitude response of the
two controllers as compared to their common command model.
Both designs have excellent model tracking performance
across a wide frequency range. In addition, the closed-loop
attitude response of both controllers is lower-order, validating
the simple inverse model designs used for the outer loop of
each design.
Figure 32 shows the outer velocity loop broken-loop re-
sponse (loop broken at the input to the inner loop) for both the
DI and EMF controllers, and Table 15 lists the outer velocity
loop crossover frequencies and gain and phase margins. Note
that by design, the outer-loop crossover frequencies are about
ωcvωcφ/10, to have good frequency separation between the
inner and outer loops. Furthermore, the outer loops of both DI
and EMF designs have sufficient gain and phase margins.
Figure 33 shows the velocity disturbance response for both
the DI and EMF controllers. Table 16 lists the velocity DRB
and DRP values for the two controllers. Both controllers have
similar values for velocity DRB and DRP. Finally, Fig. 34
shows the closed-loop velocity response to velocity command.
Both controllers have similar closed-loop responses and excel-
lent model following performance.
These results show that the DI and EMF controllers,
designed with different methods but to meet the same spec-
ifications, produce very similar results. Both designs will be
taken to flight to assess their performance.
C. Robustness Analysis
Stability and performance robustness analyses of the DI
and EMF controllers were performed by perturbing the iden-
tified stability and control derivatives of the bare-airframe and
evaluating stability margins, disturbance rejection bandwidth
(DRB), and disturbance rejection peak (DRP). Essentially, it
is desired to evaluate the statistics (mean and variance) of
the controller performance based on the known statistics of
the identified parameters. To do this robustness analysis in an
efficient manner, an unscented transformation [19] was used,
as explained for the roll axis below.
To perform the unscented transformation, the covariance
matrix Pfor the vector of identified parameters:
xT= [YvLvYδlat Lδlat ](49)
is needed. The covariance matrix was taken directly from the
CIFER R
identification results as:
P=H1(50)
where His the parameter identification Hessian matrix [13].
The next step in performing the unscented transformation is
determining the sigma points x(i), that will be used as the
perturbed parameters:
x(i)=¯
x+˜
x(i), i = 1, ..., 2n(51)
where:
¯
xis a vector of the nominal parameter values,
n= 4 is the number of identified parameters being
perturbed,
˜
x(i)= +(nP)T
ifor i= 1, ..., n
˜
x(i+n)=(nP)T
ifor i= 1, ..., n, and
(nP)T
icorresponds to the ith row of the matrix (nP).
The perturbed values of the stability and control derivatives,
given by the sigma points x(i), are then used to evaluate
the stability margins and disturbance rejection characteristics
(DRB and DRP) of the controllers. This becomes the nonlinear
transformation of the sigma points x(i), with the outputs being
gain and phase margins, DRB, and DRP. Note that through
this unscented transformation, only 2nevaluations are done to
assess the robustness of the controller through the statistics of
the stability margins, DRB, and DRP. Once the 2nvalues of
gain and phase margins, DRB, and DRP are computed, they
are used to draw 1σ,2σ, and 3σconfidence ellipsoids on the
specifications, as shown in Figs. 35 through 38.
Figure 35 shows the inner attitude loop stability margin ro-
bustness for both the EMF and DI controllers. The confidence
ellipsoids are flat (appearing as lines), with gain and phase
margin varying together with perturbations to the stability and
control derivatives. This is because the inner loop crossover
frequency is around ωc= 20 rad/s, while the perturbed
stability and control derivatives affect the rigid body dynamics
of the quadrotor (namely the lateral hovering cubic) at a much
lower frequency. At high frequency (around ω= 20 rad/s),
these effects are seen as a pure gain shift. Furthermore, since
the DI controller inner attitude broken-loop response BLφ
phase curve is flat around crossover frequency (Fig. 29), a gain
shift in the broken-loop response has a larger effect on the gain
margin than the phase margin, as seen by the orientation of
the DI confidence ellipsoids in Fig. 35. The DI confidence
ellipsoids lie along the Level 1/Level 2 boundary, staying
within around 2 deg of the boundary. For the EMF controller,
a gains in the gain of the broken-loop response affects both
the gain and the phase margin. However, as seen in Fig. 35,
the confidence ellipsoids for the EMF controller are wholy
contained within the Level 1 region.
Figure 36 shows the outer velocity loop stability margin
robustness for both the EMF and DI controllers. For the
outer loop, with its lower crossover frequency, there is a
larger spread in the confidence ellipsoids as compared to the
inner loop. However, the 3σconfidence ellipsoids for both
DI and EMF controllers are within the Level 1 region of
the specification. In addition, the EMF controller confidence
ellipsoids are more compact than the DI controller confidence
ellipsoids, suggesting higher robustness to variations in the
quadrotor stability and control derivatives.
Finally, the disturbance rejection performance robustness
of the EMF and DI controllers is shown in Fig. 37 for the
attitude loop and Fig. 38 for the velocity loop. There is a
similar level of disturbance rejection performance robustness
in the inner and outer loops for the both controllers, as seen
by the similarly sized confidence ellipsoids.
VII. CONCLUSION
A quadrotor was assembled with commercial off-the-
shelf (COTS) components readily available on the market.
A frequency-domain approach was used to perform system
identification using two steps. First, frequency responses of
the aircraft output were extracted from frequency-sweep flight
data. Second, state-space models were fit to the frequency
response data. Next, DI and EMF controllers were designed
for both inner attitude loop and outer velocity loop for the
quadrotor. DI feedback gains were tuned based on desired error
dynamics whereas EMF feedback gains were obtained using
LQR. Based on this work, the following conclusions can be
reached:
1) Given the largely decoupled dynamics at low speed,
system identification was performed separately on
each axis. Overall the identified model matched flight
data closely in both the frequency and time domain.
This indicates that the assumptions of linearity and
decoupling of each axis are justified in hover condi-
tions for this type of configuration.
2) The identified model contained a classical hovering
cubic in both pitch and roll axes. The similarity of the
eigenvalues in pitch and roll are due to the symmetry
of the configuration and is to be expected.
3) DI and EMF performances were compared both in
the time and frequency domains. Both time and
frequency responses show an outstanding match be-
tween the two control laws, which were tuned to
meet the same requirements. Overall, the gain and
phase margins, disturbance rejection performance,
and model following performance for the inner and
outer loop of both DI and EMF are satisfactory and
match closely to each other.
4) An unscented transformation was used to evaluate
stability and performance robustness statistics from
the identified bare-airframe parameters uncertainty
statistics. The results showed similar outer-loop sta-
bility robustness and inner- and outer-loop perfor-
mance robustness for both the DI and EMF con-
trollers. Higher inner-loop stability robustness was
seen for the EMF controller.
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Fig. 1: Quadrotor UAS.
Fig. 2: Quadrotor schematic.
Fig. 3: Quadrotor control system schematic.
Fig. 4: Input and output time history for lateral dynamics system identification.
Fig. 5: Real and approximated roll axis frequency response.
Fig. 6: Comparison between nonlinear and linear roll axis time response.
Fig. 7: Input and output time history for longitudinal dynamics system identification.
Fig. 8: Real and approximated pitch axis frequency response.
Fig. 9: Comparison between nonlinear and linear pitch axis time response.
Fig. 10: Input and output time history for directional dynamics system identification.
Fig. 11: Real and approximated yaw axis frequency response.
Fig. 12: Comparison between nonlinear and linear yaw axis time response.
Fig. 13: Input and output time history for heave dynamics system identification.
Fig. 14: Real and approximated heave axis frequency response.
Fig. 15: Comparison between nonlinear and linear heave axis time response.
Fig. 16: Eigenvalues of the identified model.
Fig. 17: DI block diagram.
Fig. 18: DI inner loop.
Fig. 19: DI outer loop.
Fig. 20: EMF block diagram.
Fig. 21: EMF inner loop.
Fig. 22: EMF outer loop.
Fig. 23: Heading frame velocities time response.
Fig. 24: Body frame velocities time response.
Fig. 25: Angular rates time response.
Fig. 26: Euler angles time response.
Fig. 27: Controls time response.
Fig. 28: Actuators time response.
Fig. 29: Inner-loop broken-loop response (loop broke at input
to mixer).
Fig. 30: Attitude disturbance response.
Fig. 31: Attitude command closed-loop response.
Fig. 32: Outer-loop broken-loop response (loop broke at
input to inner-loop).
Fig. 33: Velocity disturbance response.
Fig. 34: Velocity command closed-loop response.
Fig. 35: Inner attitude loop stability robustness.
Fig. 36: Outer velocity loop stability robustness.
Fig. 37: Inner attitude loop performance robustness.
Fig. 38: Outer velocity loop performance robustness.
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List of Tables. List of Examples. Preface. 1 The Kinematics and Dynamics of Aircraft Motion. 1.1 Introduction. 1.2 Vector Kinematics. 1.3 Matrix Analysis of Kinematics. 1.4 Geodesy, Earth's Gravitation, Terrestrial Navigation. 1.5 Rigid-Body Dynamics. 1.6 Summary. 2 Modeling the Aircraft. 2.1 Introduction. 2.2 Basic Aerodynamics. 2.3 Aircraft Forces and Moments. 2.4 Static Analysis. 2.5 The Nonlinear Aircraft Model. 2.6 Linear Models and the Stability Derivatives. 2.7 Summary. 3 Modeling, Design, and Simulation Tools. 3.1 Introduction. 3.2 State-Space Models. 3.3 Transfer Function Models. 3.4 Numerical Solution of the State Equations. 3.5 Aircraft Models for Simulation. 3.6 Steady-State Flight. 3.7 Numerical Linearization. 3.8 Aircraft Dynamic Behavior. 3.9 Feedback Control. 3.10 Summary. 4 Aircraft Dynamics and Classical Control Design. 4.1 Introduction. 4.2 Aircraft Rigid-Body Modes. 4.3 The Handling-Qualities Requirements. 4.4 Stability Augmentation. 4.5 Control Augmentation Systems. 4.6 Autopilots. 4.7 Nonlinear Simulation. 4.8 Summary. 5 Modern Design Techniques. 5.1 Introduction. 5.2 Assignment of Closed-Loop Dynamics. 5.3 Linear Quadratic Regulator with Output Feedback. 5.4 Tracking a Command. 5.5 Modifying the Performance Index. 5.6 Model-Following Design. 5.7 Linear Quadratic Design with Full State Feedback. 5.8 Dynamic Inversion Design. 5.9 Summary. 6 Robustness and Multivariable Frequency-Domain Techniques. 6.1 Introduction. 6.2 Multivariable Frequency-Domain Analysis. 6.3 Robust Output-Feedback Design. 6.4 Observers and the Kalman Filter. 6.5 LQG/Loop-Transfer Recovery. 6.6 Summary. 7 Digital Control. 7.1 Introduction. 7.2 Simulation of Digital Controllers. 7.3 Discretization of Continuous Controllers. 7.4 Modified Continuous Design. 7.5 Implementation Considerations. 7.6 Summary. Appendix A F-16 Model. Appendix B Software. Index.
Flight Control Optimization on a Fully Compounded Helicopter with Redundant Control Effectors
  • B R Geiger
B.R. Geiger, Flight Control Optimization on a Fully Compounded Helicopter with Redundant Control Effectors, M.S. Thesis, The PEnnsylvania State University, May 2005.
Adaptive Model Inversion Control of a Helicopter with Structural Load Limiting
  • N Sahani
  • J F Horn
N. Sahani, and J.F. Horn, Adaptive Model Inversion Control of a Helicopter with Structural Load Limiting, AIAA Journal of Guidance, Control, and Dynamics, Vol. 29,(2), March-April 2006.
Multi-Input Multi-Output Model-Following Control Design Methods for Rotorcraft
  • J M Spires
  • J Horn
J.M. Spires, and J.F Horn, Multi-Input Multi-Output Model-Following Control Design Methods for Rotorcraft, AHS 71 st Annual Forum, Virginia Beach, VA, May 2015.
Use of Harmonic Decomposition Models in Rotorcraft Flight Control Design with Alleviation of Vibratory Loads
  • U Saetti
  • J F Horn
U. Saetti, and J.F. Horn, Use of Harmonic Decomposition Models in Rotorcraft Flight Control Design with Alleviation of Vibratory Loads, European Rotorcraft Forum, Milan, Italy, September 2017.