PI(D) tuning for Flight Control Systems via
Incremental Nonlinear Dynamic Inversion
Paul Acquatella B. ∗,1 Wim van Ekeren ∗,∗∗,2 Qi Ping Chu ∗∗,3

arXiv:1701.08981v3 [] 5 Apr 2017

∗
DLR, German Aerospace Center
Institute of System Dynamics and Control
D-82234 Oberpfaffenhofen, Germany
∗∗
Delft University of Technology, Faculty of Aerospace Engineering
2629HS Delft, The Netherlands

Abstract: Previous results reported in the robotics literature show the relationship between
time-delay control (TDC) and proportional-integral-derivative control (PID). In this paper, we
show that incremental nonlinear dynamic inversion (INDI) — more familiar in the aerospace
community — are in fact equivalent to TDC. This leads to a meaningful and systematic method
for PI(D)-control tuning of robust nonlinear flight control systems via INDI. We considered a
reformulation of the plant dynamics inversion which removes effector blending models from the
resulting control law, resulting in robust model-free control laws like PI(D)-control.
Keywords: aerospace, tracking, application of nonlinear analysis and design
1. INTRODUCTION
Ensuring stability and performance in between operational
points of widely-used gain-scheduled linear PID controllers
motivates the use of nonlinear dynamic inversion (NDI) for
flight control systems. NDI cancels out nonlinearities in
the model via state feedback, and then linear control can
be subsequently designed to close the systems’ outer-loop,
hence eliminating the need of linearizing and designing
different controllers for several operational points as in
gain-scheduling.
In this paper we consider nonlinear flight control strategies
based on incremental nonlinear dynamic inversion (INDI).
Using sensor and actuator measurements for feedback allows the design of an incremental control action which, in
combination with nonlinear dynamic inversion, stabilizes
the partly-linearized nonlinear system incrementally. With
this result, dependency on exact knowledge of the system
dynamics is greatly reduced, overcoming this major robustness issue from conventional nonlinear dynamic inversion. INDI has been considered a sensor-based approach
because sensor measurements were meant to replace a
large part of the vehicle model.
Theoretical development of increments of nonlinear control action date back from the late nineties and started
with activities concerning ‘implicit dynamic inversion’ for
inversion-based flight control (Smith (1998); Bacon and
Ostroff (2000)), where the architectures considered in this
paper were firstly described. Other designations for these
developments found in the literature are ‘modified NDI’
1

Research Engineer, Space Systems Dynamics
paul.acquatella@dlr.de.
2 Graduate
Student, Control & Operations
w.vanekeren@student.tudelft.nl.
3 Associate
Professor, Control & Operations
q.p.chu@tudelft.nl.

Department.
Department.
Department.

and ‘simplified NDI’, but the designation ‘incremental
NDI’, introduced in (Chen and Zhang (2008)), is considered to describe the methodology and nature of these
type of control laws better (Chen and Zhang (2008); Chu
(2010); Sieberling et al. (2010)). INDI has been elaborated
and applied theoretically in the past decade for advanced
flight control and space applications (Sieberling et al.
(2010); Smith (1998); Bacon and Ostroff (2000); Bacon
et al. (2000, 2001); Acquatella B. et al. (2012); Simplı́cio
et al. (2013)). More recently, this technique has been applied also in practice for quadrotors and adaptive control
(Smeur et al. (2016a,b)).
In this paper, we present three main contributions in the
context of nonlinear flight control system design.
1) We revisit the NDI/INDI control laws and we establish
the equivalence between INDI and time-delay control
(TDC).
2) Based on previous results reported in the robotics literature showing the relationship between discrete formulations of TDC and proportional-integral-derivative control
(PID), we show that an equivalent PI(D) controller with
gains < K, Ti , (Td ) > tuned via INDI/TDC is more meaningful and systematic than heuristic methods, since one
considers desired error dynamics given by Hurwitz gains
< kP , (kD ) >. Subsequently, tuning the remaining effector
blending gain is much less cumbersome than designing a
whole set of gains iteratively.
3) We also consider a reformulation of the plant dynamics
inversion as it is done in TDC which removes the effector
blending model (control derivatives) from the resulting
control law. This has not been the case so far in the
reported INDI controllers, causing robustness problems
because of their uncertainties. Moreover, this allows to
consider the introduced term as a scheduling variable
which is only directly related to the proportional gain K.

3. FLIGHT CONTROL LAW DESIGN

2. FLIGHT VEHICLE MODELING
We are interested in Euler’s equation of motion representing flight vehicles’ angular velocity dynamics
I ω̇ + ω × I ω = MB

(1)

where MB ∈ R3 is the external moment vector in body
axes, ω ∈ R3 is the angular velocity vector, and I ∈ R3×3
the inertia matrix of the rigid body assuming symmetry
about the plane x − z of the body.
Furthermore, we will be interested in the time history of
the angular velocity vector, hence the dynamics of the
rotational motion of a vehicle (1) can be rewritten as the
following set of differential equations

(2)
ω̇ = I −1 MB − ω × I ω
where

"
" #
" #
#
bCl
L
p
ω = q , MB = M = SQ cCm ,
N
r
bCn
"
#
Ixx
0 Ixz
0
Iyy
0
I=
,
Ixz
0
Izz
with p, q, r , the body roll, pitch, and yaw rates, respectively; L, M , N , the roll, pitch, and yaw moments, respectively; S the wing surface area, Q the dynamic pressure,
b the wing span, c the mean aerodynamic chord, and
Cl , Cm , Cn the moment coefficients for roll, pitch, and yaw,
respectively. Furthermore, let MB be the sum of moments
partially generated by the aerodynamics of the airframe
Ma and moments generated by control surface deflections
Mc , and we describe MB linearly in the deflection angles δ
assuming the control derivatives to be linear as in Sieber∂
ling et al. (2010) with (Mc )δ = ∂δ
Mc ; therefore
MB = Ma + Mc = Ma + (Mc )δ δ

(3)

where
Ma =

"

La
Ma
Na

#

, Mc =

"

Lc
Mc
Nc

#

, δ=

"

δa
δe
δr

#

and δ corresponding to the control inputs: aileron, elevator, and rudder deflection angles, respectively. Hence the
dynamics (2) can be rewritten as
ω̇ = f (ω) + G(ω)δ

(4)

with

f (ω) = I −1 Ma − ω × I ω , G(ω) = I −1 (Mc )δ .

For practical implementations, we consider first-order actuator dynamics represented by the following transfer
function
δ
Ka
= Ga (s) =
,
(5)
δc
τa s + 1
and furthermore, we do not consider these actuator dynamics in the control design process as it is usually the
case for dynamic inversion-based control. For that reason,
we assume that these actuators are sufficiently fast in the
control-bandwidth sense, meaning that 1/τa is higher than
the control system closed-loop bandwidth.

3.1 Nonlinear Dynamic Inversion
Let us define the control parameter to be the angular
velocities, hence the output is simply y = ω. We then
consider an error vector defined as e = yd − y where yd
denotes the smooth desired output vector (at least one
time differentiable).
Nonlinear dynamic inversion (NDI) is designed to linearize
and decouple the rotational dynamics in order to obtain
an explicit desired closed loop dynamics to be followed.
Introducing the virtual control input ν = ω̇des , if the
matrix G(ω) is non-singular (i.e., invertible) in the domain
of interest for all ω, the nonlinear dynamic inversion
control consists in the following input transformation
(Slotine and Li (1990); Chu (2010))


(6)
δ = G(ω)−1 ν − f (ω)
which cancels all the nonlinearities, and a simple inputoutput linear relationship between the output y and the
new input ν is obtained as
ẏ = ν
(7)
Apart from being linear, an interesting result from this
relationship is that it is also decoupled since the input νi
only affects the output yi . From this fact, the input transformation (6) is called a decoupling control law, and the
resulting linear system (7) is called the single-integrator
form. This single-integrator form (7) can be rendered exponentially stable with
ν = ẏd + kP e
(8)
where ẏd is the feedforward term for tracking tasks, and
kP ∈ R3×3 a constant diagonal matrix, whose i−th
diagonal elements kPi are chosen so that the polynomials
(i = p, q, r)
(9)
s + kPi
may become Hurwitz, i.e., kPi < 0. This results in the
exponentially stable and decoupled desired error dynamics
ė + kP e = 0
(10)
which implies that e(t) → 0. From this typical tracking
problem it can be seen that the entire control system
will have two control loops (Chu (2010); Sieberling et al.
(2010)): the inner linearization loop (6), and the outer
control loop (8). This resulting NDI control law depends on
accurate knowledge of the aerodynamic moments, hence it
is susceptible to model uncertainties contained in both Ma
and Mc .
In NDI control design, we consider outputs with relative
degrees of one (rates), meaning a first-order system to
be controlled, see Fig. 1. Extensions of input-output linearization for systems involving higher relative degrees are
done via feedback linearization (Slotine and Li (1990); Chu
(2010)).
3.2 Incremental Nonlinear Dynamic Inversion
The concept of incremental nonlinear dynamic inversion
(INDI) amounts to the application of NDI to a system
expressed in an incremental form. This improves the robustness of the closed-loop system as compared with conventional NDI since dependency on the accurate knowledge of the plant dynamics is reduced. Unlike NDI, this

X, Y, Z
Reference
Trajectory

V, ψ, γ
Flight Path
Angle and
Airspeed Control

Position
Control

µ, α, β
Attitude
Control

p, q, r
Rate Control

Fig. 1. Four loop nonlinear flight control design. We are focused on nonlinear dynamic inversion of the rate control loop
(grey box) in the following. Image credits: Sonneveldt (2010).
control design technique is implicit in the sense that desired closed-loop dynamics do not reside in some explicit
model to be followed but result when the feedback loops
are closed (Bacon and Ostroff (2000); Bacon et al. (2000)).
To obtain an incremental form of system dynamics, we
consider a first-order Taylor series expansion of ω̇ (Smith
(1998); Bacon and Ostroff (2000); Bacon et al. (2000,
2001); Sieberling et al. (2010); Acquatella B. et al. (2012,
2013)), not in the geometric sense, but with respect to a
suffiently small time-delay λ as

∂ 
ω̇ = ω̇0 +
(ω − ω0 )
f (ω) + G(ω)δ
ω=ω0
∂ω
δ=δ0


∂ 
+
G(ω)δ
∂δ

ω=ω0
δ=δ0

(δ − δ0 ) + O(∆ω 2 , ∆δ 2 )

∼
= ω̇0 + f0 (ω − ω0 ) + G0 (δ − δ0 )
with
ω̇0 ≡ f (ω0 ) + G(ω0 )δ0 = ω̇(t − λ)
(11a)
where ω0 = ω(t − λ) and δ0 = δ(t − λ) are the timedelayed signals of the current state ω and control δ, respectively. This means an approximate linearization about
the λ−delayed signals is performed incrementally.
For such sufficiently small time-delay λ so that f (ω) does
not vary significantly during λ, we assume the following
approximation to hold
(12)
ǫIN DI (t) ≡ f (ω(t − λ)) − f (ω(t)) ∼
=0
which leads to
∆ω̇ ∼
(13)
= G0 · ∆δ
Here, ∆ω̇ = ω̇ − ω̇0 = ω̇ − ω̇(t − λ) represents the
incremental acceleration, and ∆δ = δ − δ0 represents
the so-called incremental control input. For the obtained
approximation ω̇ ∼
= ω̇0 + G0 (δ − δ0 ), NDI is applied to
obtain a relation between the incremental control input
and the output of the system


δ = δ0 + G−1
ν − ω̇0
(14)
0
Note that the deflection angle δ0 that corresponds to ω̇0
is taken from the output of the actuators, and it has
been assumed that a commanded control is achieved sufficiently fast according to the assumptions of the actuator
dynamics in (5). The total control command along with
the obtained linearizing control ∆δ can be rewritten as
h
i
ν
−
ω̇(t
−
λ)
.
(15)
δ(t) = δ(t − λ) + G−1
0

The dependency of the closed-loop system on accurate
knowledge of the airframe model in f (ω) is largely decreased, improving robustness against model uncertainties contained therein. Therefore, this implicit control law

design is more dependent on accurate measurements or
accurate estimates of ω̇0 , the angular acceleration, and δ0 ,
the deflection angles, respectively.
Remark 1: By using the measured ω̇(t − λ) and δ(t − λ)
incrementally we practically obtain a robust, model-free
controller with the self-scheduling properties of NDI.
Notice, however, that typical INDI control laws are nevertheless also depending on effector blending models reflected in G0 , which makes this implicit controller susceptible to uncertainties in these terms. Instead, consider the
following transformation as in (Chang and Jung (2009))
ω̇ = H + ḡ · δ
(16)
with
H(t) = f (ω) + (G(ω) − ḡ)δ,
and with the following (but not limited) options for ḡ
(Chang and Jung (2009)), where n = 3 in our case




1 0 ··· 0
kG1 0 · · · 0
0 1

 0 kG2

 , ḡ2 =  .
.
ḡ1 = kG ·In = kG 
..
..
 ..



.
.
.
.
.
0
1
0
kGn
Applying nonlinear dynamic inversion (NDI) to (16) results in an expression for the control input of the vehicle
as


(17)
δ(t) = ḡ −1 ν(t) − H(t) .

Considering H0 = ω̇0 − ḡ · δ0 , the incremental counterpart
of (17) results in a control law that is neither depending
on the airframe model nor the effector blending moments
h
i
(18)
δ(t) = δ(t − λ) + ḡ −1 ν − ω̇(t − λ) .

Remark 2: The self-scheduling properties of INDI in (15)
due to the term G0 are now lost, suggesting that ḡ should
be an scheduling variable.
3.3 Time Delay Control and Proportional Integral control

Time delay control (TDC) (Chang and Jung (2009)) departs from the usual dynamic inversion input transformation of (16)


(19)
δ(t) = ḡ −1 ν(t) − H̄(t)
where H̄ denotes an estimation of H, being the nominal
case when H̄ = H which results in perfect inversion.
Instead of having an estimate, the TDC takes the following
assumption (Chang and Jung (2009)) analogous to (12)
ǫT DC (t) ≡ H(t − λ) − H(t) ∼
(20)
= 0.
This relationship is used together with (16) to obtain what
is called time-delay estimation (TDE) as the following

H̄ = H(t − λ) = ω̇(t − λ) − ḡ · δ(t − λ)

(21)

In addition, ǫ(t) is called TDE error at time t. Combining
the equations we obtain the following TDC law


(22)
δ(t) = δ(t − λ) + ḡ −1 ν − ω̇(t − λ)
which is in fact equivalent to the INDI control law obtained
in (18). Appropriate selection of ḡ must ensure stability
according to (Chang and Jung (2009)), and ideally, this
term should be tuned according to the best estimate of the
true effector blending moment ĝ(ω̃) for measured angular
velocities ω̃.
So far we have considered derivations in continuous-time.
For practical implementations of these controllers and
for the matters of upcoming discussions, sampled-time
formulations involving continuous and discrete quantities
as in (Chang and Jung (2009)) are more convenient and
restated here. For that, considering that the smallest λ one
can consider is the equivalent of the sampling period ts of
the on-board computer. The sampled formulation of (22)
may be expressed as


(23)
δ(k) = δ(k − 1) + ḡ −1 ν(k − 1) − ω̇(k − 1)
where it has been necessary to consider ν at sample k−1 for
causality reasons. Replacing the sampled virtual control ν
according to (8) we have


(24)
δ(k) = δ(k − 1) + ḡ −1 ė(k − 1) + kP e(k − 1)
and we can consider the following finite difference approximation of the error derivatives as angular accelerations
are not directly measured
ė(k) = [e(k) − e(k − 1)]/ts .
(25)
Consider now the standard proportional-integral (PI) control
Z t

δ(t) = K e(t) + TI−1
e(σ)dσ + δDC ,
(26)
0

where K ∈ R3×3 denotes a diagonal proportional gain
matrix, TI ∈ R3×3 a constant diagonal matrix representing a reset or integral time, and δDC ∈ R3 denotes a
constant vector representing a trim-bias, which acts as
a trim setting and is computed by evaluating the initial
conditions. The discrete form of the PI is given by
k−1
X

ts e(i) + δDC
(27)
δ(k) = K e(k − 1) + TI−1
i=0

When substracting two consecutive terms of this discrete
formulation, we can remove the integral sum and achieve
the so-called PI controller in incremental form

δ(k) = δ(k − 1) + K · ts ė(k − 1) + TI−1 · e(k − 1) (28)
Following the same steps, and for completeness, we also
present the PID extension by simply considering the extra
derivative term ë

δ(k) = δ(k−1)+K ·ts TD ë(k−1)+ ė(k−1)+TI−1 ·e(k−1) ,
where TD ∈ R3×3 denotes a constant diagonal matrix
representing derivative time.
3.4 Equivalence of INDI/TDC/PI(D)
Having in mind the found the equivalence between INDI
and TDC, and comparing terms from (24) with (28),

we have the following relationships as originally found
in (Chang and Jung (2009)) which are the relationship
between the discrete formulations of TDC and PI in
incremental form
K = (ḡ · ts )−1 , TI = kP−1
(29)
Whenever the system under consideration is of secondorder controller canonical form, we will have error dynamics of the form ë + kD ė + kP e = 0, and considering the
newly introduced derivative gain kD related to ë we have
−1
K = kD · (ḡ · ts )−1 , TI = kD · kP−1 , TD = kD
(30)
This suggests not only that an equivalent discrete PI(D)
controller with gains < K, Ti , (Td ) > can be obtained
via INDI/TDC, but doing so is more meaningful and
systematic than heuristic methods. This is because we
begin the design from desired error dynamics given by
Hurwitz gains < kP , (kD ) > and what follows is finding
the remaining effector blending gain ḡ either analytically
whenever G is well known, with a proper estimate Ĝ, or by
tuning according to closed-loop requirements. As already
mentioned, details on a sufficient condition for closed-loop
stability under discrete TDC, and therefore applicable to
its equivalent INDI, can be found in (Chang and Jung
(2009)) and the references therein.
In essence, this procedure is more efficient and much less
cumbersome than designing a whole set of gains iteratively.
Moreover, for flight control systems, the self-scheduling
properties of inversion-based controllers have suggested superior advantages with respect to PID controls since these
must be gain-scheduled according to the flight envelope
variations. The relationships here outlined suggests that
PID-scheduling shall be done at the proportional gain K
via the effector blending gain ḡ, and not over the whole
set of gains < K, Ti , (Td ) >.
4. LONGITUDINAL FLIGHT CONTROL
SIMULATION
In this section, robust PI tuning via INDI is demonstrated
with a simple yet significant example consisting of the
tracking control design for a longitudinal launcher vehicle
model. The second-order nonlinear model is obtained from
(Sonneveldt (2010); Kim et al. (2004)), and it consists on
longitudinal dynamic equations representative of a vehicle
traveling at an altitude of approximately 6000 meters,
with aerodynamic coefficients represented as third order
polynomials in angle of attack α and Mach number M .
The nonlinear equations of motion in the pitch plane are
given by


q̄S
α̇ = q +
Cz (α, M ) + bz (M )δ ,
(31a)
mVT


q̄Sd
Cm (α, M ) + bm (M )δ ,
(31b)
q̇ =
Iyy
where
Cz (α, M ) = ϕz1 (α) + ϕz2 (α)M,
Cm (α, M ) = ϕm1 (α) + ϕm2 (α)M,
bz (M ) = 1.6238M − 6.7240,
bm (M ) = 12.0393M − 48.2246,

with

and
3

ϕz1 (α) = −288.7α + 50.32α |α| − 23.89α,
ϕz2 (α) = −13.53α |α| + 4.185α,
ϕm1 (α) = 303.1α3 − 246.3α |α| − 37.56α,
ϕm2 (α) = 71.51α |α| + 10.01α.
These approximations are valid for the flight envelope of
−10◦ ≤ α ≤ 10◦ and 1.8 ≤ M ≤ 2.6. To facilitate
the control design, the nonlinear longitudinal model is
rewritten in the more general state-space form as
ẋ1 = x2 + f1 (x1 ) + g1 u
(32a)
ẋ2 = f2 (x1 ) + g2 u
(32b)
where:
x1 = α,
x2 = q
g1 = C1 bz ,
g2 = C2 bm
and


q̄S
f1 (x1 ) = C1 ϕz1 (x1 ) + ϕz2 (x1 )M ,
C1 =
,
mVT


q̄Sd
.
f2 (x1 ) = C2 ϕm1 (x1 ) + ϕm2 (x1 )M ,
C2 =
Iyy

The control objective considered here is to design a PI
autopilot via INDI that tracks a smooth command reference yr with the pitch rate x2 . It is assummed that the
aerodynamic force and moment functions are accurately
known and the Mach number M is treated as a parameter
available for measurement. Moreover, for this second-order
system in non-lower triangular form due to g1 u and f2 (x1 ),
pitch rate control using INDI is possible due to the timescale separation principle (Chu (2010); Sieberling et al.
(2010)). With respect to actuator dynamics modeled as in
(5), we consider Ka = 1, and τa = 1e−2 .
4.1 Pitch rate control design
First, introduce the rate-tracking error
z2 = x2 − x2ref
(33)
the z2 −dynamics satisfy the following error
ż2 = ẋ2 − ẋ2ref
(34)
for which we design the following exponentially stable
desired error dynamics
(35)
ż2 + kP2 z2 = 0, kP2 = 50 rad/s.
According to the results previously outlined, the incremental nonlinear dynamic inversion control law design follows
from considering the approximate dynamics around the
current reference state for the dynamic equation of the
pitch rate as in (13)
q̇ ∼
(36)
= q̇0 + ḡ · ∆δ
assuming that pitch acceleration is available for measurement and the scalar ḡ to be a factor of the accurately
known estimate of g2
ḡ = kG · ĝ2 , kG = 1.
This is rewritten in our formulation as
(37)
ẋ2 ∼
= ẋ20 + ḡ · ∆u
where recalling that ẋ20 is an incremental instance before
ẋ2 , and therefore the incremental nonlinear dynamic inversion law is hence obtained as

u = u0 + ḡ −1 ν − ẋ20 ,
(38)

ν = −kP2 z2 + ẋ2ref ,
or more compactly

u = u0 + ḡ −1 − kP2 z2 − ẋ20 + ẋ2ref

(39)
(40)

This results as desired, in the following z2 −dynamics
(41)
ż2 = ẋ20 + ḡ · ∆u − ẋ2ref .
Notice that we are replacing the accurate knowledge of f2
by a measurement (or an estimate) as f2 ∼
= ẋ20 , which will
result in a control law which is not entirely dependent on
a model, hence more robust.
We now consider these continuous-time formulations in
sampled-time. To that end, we replace the small λ with
the sampling period ts so that tk = k · ts is the k−th
sampling instant at time k, and therefore
u(k) = u(k − 1)+
 (42)

−1
− kP2 z2 (k − 1) − ẋ2 (k − 1) + ẋ2ref (k − 1) ,
ḡ
where due to causality relationships we need to consider
the independent variables at the same sampling time k −1.
Referring back to the derived relationship between INDI
and PI control, the equivalent PI control in incremental
form is


u(k) = u(k − 1) + K · ts ż2 (k − 1) + TI−1 z2 (k − 1) ,
(43)
with
K = (ḡ · ts )−1 , TI = kP−1
(44)
2
The nature of the desired error dynamics (proportional)
gain kP2 is therefore of an integral control action, whereas
the effector blending gain ḡ act as proportional control.
Having designed for desired error dynamics, and for a
given sampling time ts , tuning a pitch rate controller is
only a matter of selecting a proper effector blending gain
ḡ according to performance requirements.
Remark 3: Notice at this point that having the PI control
in incremental form introduces a finite difference of the
error state, which is the equivalent counterpart of what
has been considered the acceleration or state derivative
ẋ20 in INDI controllers.
Remark 4: Notice also that designing the PI control
gains via INDI is highly beneficial, since only the effector
blending gain is the tuning variable. This strongly suggests
that robust adaptive control can be achieved by scheduling
this variable online during flight and not over the whole
set of gains.
Simulation results for the INDI/PI control are presented in
Figure 2, considering smooth rate doublets for a nominal
longitudinal dynamics model at Mach 2. For both controllers, the same zero-mean Gaussian white-noise with
standard deviation sdq = 1e−3 rad/s is added to the rates
to simulate noisy measurements. The designed INDI gains
of kP2 = 50 rad/s and kG = 1 are mapped to PI gains
resulting in K = 100 ĝ2−1 and TI = 0.02 s, both controllers
showing identical closed-loop response as expected.
With this example, it is demonstrated how a self-scheduled
PI can be tuned via INDI by departing from desired error
dynamics with the gain kP2 , and considering an accurate
effector blending model estimate ḡ = ĝ2 .

−6

4

4

−4

2

2

−2

0

0

−2

2

0

−2
−4

−4

qREF
qI N DI
qP I
qI N DI − qP I

−6
0

5

10

15

time (s)

αI N DI
αI N DI − αP I
δI N DI
δI N DI − δP I

−6
20

0

5

10

15

time (s)

δ (deg)

6

α (deg)

q (deg/s)

6

4
6
20

Fig. 2. INDI/PI nominal tracking control simulation of the
flight model (31) for kP2 = 50 rad/s and kG = 1
5. CONCLUSIONS
This paper presented a meaningful and systematic method
for PI(D) tuning of robust nonlinear flight control systems
based on results previously reported in the robotics literature (Chang and Jung (2009)) regarding the relationship between time-delay control (TDC) and proportionalintegral-derivative control (PID). The method was demonstrated in the context of an example for the pitch rate
tracking of a conventional longitudinal nonlinear flight
model, showing the same tracking performance under
nominal conditions.
Being incremental nonlinear dynamic inversion (INDI)
equivalent to TDC clearly suggests that imposing desired error dynamics, as usual for INDI control laws,
and then mapping these into an equivalent incremental
PI(D)-controller together with control derivatives leads to
a meaningful and systematic PI(D) gain tuning method,
which is very difficult to do heuristically.
We considered a reformulation of the plant dynamics inversion which reduces knowledge of the effector blending
model (control derivatives) from the resulting control law,
reducing feedback control dependency on accurate knowledge of both the aircraft and effector blending models,
hence resulting in robust and model-free control laws like
the PI(D) control. Since usual flight control systems involves gain scheduling over the flight envelope, another
key benefit of this result is that scheduling only the effector blending gain seems promising for adaptive control
systems.
ACKNOWLEDGEMENTS
M. Ruf, N. Tekles, and G. Looye are acknowledged for
discussions leading to improvements of this paper.
REFERENCES
Acquatella B., P., Falkena, W., van Kampen, E., and Chu,
Q.P. (2012). Robust Nonlinear Spacecraft Attitude Control using Incremental Nonlinear Dynamic Inversion. In

AIAA Guidance, Navigation, and Control Conference.
American Institute of Aeronautics and Astronautics,
Inc. (AIAA-2012-4623).
Acquatella B., P., van Kampen, E., and Chu, Q.P. (2013).
Incremental Backstepping for Robust Nonlinear Flight
Control. In EuroGNC 2013, 2nd CEAS Specialist Conference on Guidance, Navigation, and Control.
Bacon, B.J. and Ostroff, A.J. (2000). Reconfigurable
Flight Control using Nonlinear Dynamic Inversion with
a Special Accelerometer Implementation. In AIAA
Guidance, Navigation, and Control Conference and Exhibit. (AIAA-2000-4565).
Bacon, B.J., Ostroff, A.J., and Joshi, S.M. (2000). Nonlinear Dynamic Inversion Reconfigurable Controller utilizing a Fault-tolerant Accelerometer Approach. Technical
report, NASA Langley Research Center.
Bacon, B.J., Ostroff, A.J., and Joshi, S.M. (2001). Reconfigurable NDI Controller using Inertial Sensor Failure
Detection & Isolation. IEEE Transactions on Aerospace
and Electronic Systems, 37, 1373–1383.
Chang, P.H. and Jung, J.H. (2009). A Systematic Method
for Gain Selection of Robust PID Control for Nonlinear Plants of Second-Order Controller Canonical Form.
IEEE Transactions on Control Systems Technology,
17(2), 473–483.
Chen, H.B. and Zhang, S.G. (2008). Robust Dynamic
Inversion Flight Control Law Design. In ISSCAA 2008,
2nd International Symposium on Systems and Control
in Aerospace and Astronautics.
Chu, Q.P. (2010). Advanced Flight Control. Lecture notes,
Delft University of Technology, Faculty of Aerospace
Engineering.
Kim, S.H., Kim, Y.S., and Song, C. (2004). A Robust
Adaptive Nonlinear Control Approach to Missile Autopilot Design. Control Engineering Practice, 33(6),
1732–1742.
Sieberling, S., Chu, Q.P., and Mulder, J.A. (2010). Robust
Flight Control Using Incremental Nonlinear Dynamic
Inversion and Angular Acceleration Prediction. Journal
of Guidance, Control and Dynamics, 33(6), 1732–1742.
Simplı́cio, P., Pavel, M., van Kampen, E., and Chu, Q.P.
(2013). An Acceleration Measurements-based Approach
for Helicopter Nonlinear Flight Control using Incremental Nonlinear Dynamic Inversion. Control Engineering
Practice, 21(8), 1065–1077.
Slotine, J.J. and Li, W. (1990). Applied Nonlinear Control.
Prentice Hall Inc.
Smeur, E.J., Chu, Q.P., and de Croon, G.C. (2016a).
Adaptive Incremental Nonlinear Dynamic Inversion for
Attitude Control of Micro Air Vehicles. Journal of
Guidance, Control and Dynamics, 39(3), 450–461.
Smeur, E.J., de Croon, G.C., and Chu, Q.P. (2016b).
Gust Disturbance Alleviation with Incremental Nonlinear Dynamic Inversion. In IEEE/RSJ International
Conference on Intelligent Robots and Systems (IROS).
Smith, P.R. (1998). A Simplified Approach to Nonlinear Dynamic Inversion Based Flight Control. In
AIAA Atmospheric Flight Mechanics Conference, 762–
770. American Institute of Aeronautics and Astronautics, Inc. (AIAA-98-4461).
Sonneveldt, L. (2010). Adaptive Backstepping Flight Control for Modern Fighter Aircraft. PhD thesis, Delft University of Technology, Faculty of Aerospace Engineering.

