PI(D) tuning for Flight Control Systems via Incremental Nonlinear Dynamic Inversion

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by  Paul B. Acquatella

PI(D) tuning for Flight Control Systems via
Incremental Nonlinear Dynamic Inversion
Paul Acquatella B.∗,1 Wim van Ekeren∗,∗∗,2 Qi Ping Chu∗∗,3
∗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
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time-delay control (TDC) and proportional-integral-derivative control (PID). In this paper, we
1
0 show that incremental nonlinear dynamic inversion (INDI) — more familiar in the aerospace
2 community—areinfactequivalenttoTDC.Thisleadstoameaningfulandsystematicmethod
for PI(D)-control tuning of robust nonlinear flight control systems via INDI. We considered a
r
p reformulationof the plant dynamics inversionwhich removeseffector blending models fromthe
A resulting control law, resulting in robust model-free control laws like PI(D)-control.

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Keywords: aerospace, tracking, application of nonlinear analysis and design

]
Y
1. INTRODUCTION and ‘simplified NDI’, but the designation ‘incremental
S NDI’, introduced in (Chen and Zhang (2008)), is con-
. Ensuringstabilityandperformanceinbetweenoperational sidered to describe the methodology and nature of these
s
c pointsofwidely-usedgain-scheduledlinearPIDcontrollers type of control laws better (Chen and Zhang (2008); Chu
[ motivatestheuseofnonlineardynamicinversion(NDI)for (2010);Sieberlingetal.(2010)).INDIhasbeenelaborated

flight control systems. NDI cancels out nonlinearities in and applied theoretically in the past decade for advanced
3
v the model via state feedback, and then linear control can flight control and space applications (Sieberling et al.
1 be subsequentlydesignedtoclosethe systems’outer-loop, (2010); Smith (1998); Bacon and Ostroff (2000); Bacon
8 hence eliminating the need of linearizing and designing et al. (2000, 2001); Acquatella B. et al. (2012); Simpl´ıcio
9 different controllers for several operational points as in et al. (2013)). More recently, this technique has been ap-
8 gain-scheduling. plied also in practice for quadrotors and adaptive control
0 (Smeur et al. (2016a,b)).
. Inthispaperweconsidernonlinearflightcontrolstrategies
1 basedonincrementalnonlineardynamicinversion(INDI). In this paper, we present three main contributions in the
0
Using sensor and actuator measurements for feedback al- context of nonlinear flight control system design.
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lows the design of an incremental controlaction which, in
1 1)We revisitthe NDI/INDI controllaws andwe establish
combination with nonlinear dynamic inversion, stabilizes
: the equivalence between INDI and time-delay control
v thepartly-linearizednonlinearsystemincrementally.With
(TDC).
i
X this result, dependency on exact knowledge of the system
dynamics is greatly reduced, overcoming this major ro- 2)Basedonpreviousresultsreportedinthe roboticsliter-
r
a bustnessissuefromconventionalnonlineardynamicinver- ature showing the relationship between discrete formula-
sion. INDI has been considered a sensor-based approach tions of TDC and proportional-integral-derivative control
because sensor measurements were meant to replace a (PID), we show that an equivalent PI(D) controller with
large part of the vehicle model. gains<K,Ti,(Td)>tunedviaINDI/TDCismoremean-
ingful and systematic than heuristic methods, since one
Theoretical development of increments of nonlinear con-
considers desired error dynamics given by Hurwitz gains
trol action date back from the late nineties and started
<k ,(k )>.Subsequently,tuningtheremainingeffector
P D
with activities concerning ‘implicit dynamic inversion’ for
blending gain is much less cumbersome than designing a
inversion-based flight control (Smith (1998); Bacon and
whole set of gains iteratively.
Ostroff(2000)),where the architecturesconsideredinthis
paper were firstly described. Other designations for these 3)We also considerareformulationofthe plantdynamics
developments found in the literature are ‘modified NDI’ inversion as it is done in TDC which removes the effector
blending model (control derivatives) from the resulting
1 Research Engineer, Space Systems Dynamics Department. control law. This has not been the case so far in the
[e-mail]. reported INDI controllers, causing robustness problems
2 Graduate Student, Control & Operations Department.
because of their uncertainties. Moreover, this allows to
[e-mail].
consider the introduced term as a scheduling variable
3 Associate Professor, Control & Operations Department.
whichis only directly relatedto the proportionalgainK.
[e-mail].
2. FLIGHT VEHICLE MODELING 3. FLIGHT CONTROL LAW DESIGN
3.1 Nonlinear Dynamic Inversion
We areinterestedinEuler’sequationofmotionrepresent-
ing flight vehicles’ angular velocity dynamics
Let us define the control parameter to be the angular
Iω˙ +ω×Iω =MB (1) velocities, hence the output is simply y = ω. We then
where M ∈ R3 is the external moment vector in body consider an error vector defined as e = yd −y where yd
B
axes, ω ∈R3 is the angular velocity vector,and I ∈R3×3 denotes the smooth desired output vector (at least one
time differentiable).
the inertia matrix of the rigid body assuming symmetry
about the plane x−z of the body. Nonlineardynamicinversion(NDI)isdesignedtolinearize
and decouple the rotational dynamics in order to obtain
Furthermore, we will be interested in the time history of
an explicit desired closed loop dynamics to be followed.
the angular velocity vector, hence the dynamics of the
rotational motion of a vehicle (1) can be rewritten as the Introducing the virtual control input ν = ω˙des, if the
matrixG(ω)isnon-singular(i.e.,invertible)inthedomain
following set of differential equations
of interest for all ω, the nonlinear dynamic inversion
−1
ω˙ =I MB−ω×Iω (2) control consists in the following input transformation
(Slotine and Li (1990); Chu (2010))
where (cid:0) (cid:1)
−1
δ =G(ω) ν−f(ω) (6)
p L bCl
ω = q , MB = M =SQ cCm , which cancels all the nonline(cid:2)arities, an(cid:3)d a simple input-
"r # "N # "bCn # output linear relationship between the output y and the
new input ν is obtained as
I 0 I
xx xz
I = 0 I 0 , y˙ =ν (7)
yy
" I 0 I # Apart from being linear, an interesting result from this
xz zz
relationship is that it is also decoupled since the input ν
with p,q,r, the body roll, pitch, and yaw rates, respec- i
onlyaffects the outputy .Fromthis fact,the input trans-
tively; L,M,N, the roll, pitch, and yaw moments, respec- i
formation (6) is called a decoupling control law, and the
tively; S the wing surface area, Q the dynamic pressure,
resulting linear system (7) is called the single-integrator
b the wing span, c the mean aerodynamic chord, and
form. This single-integrator form (7) can be rendered ex-
Cl,Cm,Cn themomentcoefficientsforroll,pitch,andyaw,
ponentially stable with
respectively.Furthermore,letM be the sumofmoments
B
ν =y˙ +k e (8)
partially generated by the aerodynamics of the airframe d P
Ma and moments generatedby control surface deflections where y˙d is the feedforward term for tracking tasks, and
Mc,andwedescribeMB linearlyinthedeflectionanglesδ kP ∈ R3×3 a constant diagonal matrix, whose i−th
assuming the control derivatives to be linear as in Sieber- diagonal elements kPi are chosen so that the polynomials
ling et al. (2010) with (Mc)δ = ∂∂δMc; therefore s+kPi (i=p,q,r) (9)
MB =Ma+Mc =Ma+(Mc)δδ (3) may become Hurwitz, i.e., kPi < 0. This results in the
exponentiallystableanddecoupleddesired errordynamics
where
e˙+k e=0 (10)
P
La Lc δa which implies that e(t) → 0. From this typical tracking
Ma = Ma , Mc = Mc , δ = δe problem it can be seen that the entire control system
"Na # "Nc # "δr# will have two control loops (Chu (2010); Sieberling et al.
and δ corresponding to the control inputs: aileron, eleva- (2010)): the inner linearization loop (6), and the outer
tor, and rudder deflection angles, respectively. Hence the controlloop(8).ThisresultingNDIcontrollawdependson
dynamics (2) can be rewritten as accurateknowledgeoftheaerodynamicmoments,henceit
issusceptibletomodeluncertaintiescontainedinbothM
a
ω˙ =f(ω)+G(ω)δ (4)
and M .
c
with
In NDI control design, we consider outputs with relative
f(ω)=I−1 M −ω×Iω , G(ω)=I−1(M ) . degrees of one (rates), meaning a first-order system to
a c δ
be controlled, see Fig. 1. Extensions of input-output lin-
For practical imp(cid:0)lementations,(cid:1)we consider first-order ac- earizationforsystemsinvolvinghigherrelativedegreesare
tuator dynamics represented by the following transfer doneviafeedbacklinearization (SlotineandLi(1990);Chu
function (2010)).
δ K
=G (s)= a , (5) 3.2 Incremental Nonlinear Dynamic Inversion
a
δ τ s+1
c a
and furthermore, we do not consider these actuator dy- The concept of incremental nonlinear dynamic inversion
namics in the control design process as it is usually the (INDI) amounts to the application of NDI to a system
case for dynamic inversion-basedcontrol.For that reason, expressed in an incremental form. This improves the ro-
we assume that these actuators are sufficiently fast in the bustness of the closed-loop system as compared with con-
control-bandwidthsense,meaningthat1/τ ishigherthan ventional NDI since dependency on the accurate knowl-
a
the control system closed-loop bandwidth. edge of the plant dynamics is reduced. Unlike NDI, this
X,Y,Z V,ψ,γ µ,α,β p,q,r
Reference Position FlightPath Attitude
Angleand RateControl
Trajectory Control Control
AirspeedControl
Fig.1.Fourloopnonlinearflightcontroldesign.We arefocusedonnonlineardynamic inversionofthe ratecontrolloop
(grey box) in the following. Image credits: Sonneveldt (2010).
control design technique is implicit in the sense that de- design is more dependent on accurate measurements or
sired closed-loop dynamics do not reside in some explicit accurateestimates ofω˙0,the angularacceleration,andδ0,
model to be followed but result when the feedback loops the deflection angles, respectively.
areclosed(BaconandOstroff(2000);Baconetal.(2000)).
Remark 1: By using the measured ω˙(t−λ) and δ(t−λ)
To obtain an incremental form of system dynamics, we incrementally we practically obtain a robust, model-free
consider a first-order Taylor series expansion of ω˙ (Smith controller with the self-scheduling properties of NDI.
(1998); Bacon and Ostroff (2000); Bacon et al. (2000,
Notice, however, that typical INDI control laws are nev-
2001); Sieberling et al. (2010); Acquatella B. et al. (2012,
ertheless also depending on effector blending models re-
2013)), not in the geometric sense, but with respect to a
suffiently small time-delay λ as flected in G0, which makes this implicit controller suscep-
tible to uncertainties in these terms. Instead, consider the
∂
following transformation as in (Chang and Jung (2009))
ω˙ = ω˙0+ f(ω)+G(ω)δ (ω−ω0)
∂ω(cid:2) (cid:3)(cid:12)(cid:12)ωδ==δω00 ω˙ =H +g¯·δ (16)
∂ (cid:12) with
+ ∂δ G(ω)δ ω=ω0(δ−(cid:12)δ0)+O(∆ω2,∆δ2) H(t)=f(ω)+(G(ω)−g¯)δ,
∼= ω˙0+f(cid:2)0(ω−(cid:3)ω(cid:12)(cid:12)(cid:12)0δ)=+δ0G0(δ−δ0) a(CndhawngithantdhJeufnogllo(w20in09g))(,bwuthenreotnl=im3iteind)ouorptciaosnes for g¯
(cid:12)
with 1 0 ··· 0 k 0 ··· 0
G1
ω˙0 ≡f(ω0)+G(ω0)δ0 =ω˙(t−λ) (11a) 0 1 0 kG2
where ω0 = ω(t − λ) and δ0 = δ(t − λ) are the time- g¯1 =kG·In =kG ... ... , g¯2 = ... ... .
delayed signals of the current state ω and control δ, re- 0 1  0 k 
spectively.Thismeansanapproximatelinearizationabout    Gn 
   
the λ−delayed signals is performed incrementally. Applying nonlinear dynamic inversion (NDI) to (16) re-
sults in an expression for the control input of the vehicle
For such sufficiently small time-delay λ so that f(ω) does
as
not vary significantly during λ, we assume the following
−1
δ(t)=g¯ ν(t)−H(t) . (17)
approximation to hold
ǫINDI(t)≡f(ω(t−λ))−f(ω(t))∼=0 (12) Considering H0 =ω˙0−g¯·δ(cid:2)0, the increm(cid:3)ental counterpart
which leads to of (17) results in a control law that is neither depending
∆ω˙ ∼=G0·∆δ (13) on the airframe model nor the effector blending moments
Here, ∆ω˙ = ω˙ − ω˙0 = ω˙ − ω˙(t − λ) represents the δ(t)=δ(t−λ)+g¯−1 ν−ω˙(t−λ) . (18)
incremental acceleration, and ∆δ = δ − δ0 represents
h i
the so-called incremental control input. For the obtained Remark 2:Theself-schedulingpropertiesofINDIin(15)
approximation ω˙ ∼= ω˙0 + G0(δ − δ0), NDI is applied to due to the term G0 are now lost, suggestingthat g¯should
obtain a relation between the incremental control input
be an scheduling variable.
and the output of the system
−1
δ =δ0+G0 ν−ω˙0 (14) 3.3 Time Delay Control and Proportional Integral control
Note that the deflection angle(cid:2)δ0 that(cid:3)corresponds to ω˙0 Time delay control (TDC) (Chang and Jung (2009)) de-
is taken from the output of the actuators, and it has parts from the usual dynamic inversioninput transforma-
been assumed that a commanded control is achieved suf- tion of (16)
ficiently fast accordingto the assumptionsofthe actuator δ(t)=g¯−1 ν(t)−H¯(t) (19)
dynamics in (5). The total control command along with
where H¯ denotes an estimation of H, being the nominal
the obtained linearizing control ∆δ can be rewritten as (cid:2) (cid:3)
case when H¯ = H which results in perfect inversion.
−1
δ(t)=δ(t−λ)+G0 ν−ω˙(t−λ) . (15) Insteadofhavinganestimate,theTDCtakesthefollowing
assumption (Chang and Jung (2009)) analogous to (12)
h i
The dependency of the closed-loop system on accurate ǫTDC(t)≡H(t−λ)−H(t)∼=0. (20)
knowledge of the airframe model in f(ω) is largely de-
creased, improving robustness against model uncertain- Thisrelationshipisusedtogetherwith(16)toobtainwhat
ties contained therein. Therefore,this implicit controllaw is called time-delay estimation (TDE) as the following
H¯ =H(t−λ)=ω˙(t−λ)−g¯·δ(t−λ) (21) we have the following relationships as originally found
in (Chang and Jung (2009)) which are the relationship
Inaddition, ǫ(t)iscalledTDE error attime t.Combining between the discrete formulations of TDC and PI in
the equations we obtain the following TDC law incremental form
δ(t)=δ(t−λ)+g¯−1 ν−ω˙(t−λ) (22) K =(g¯·t )−1, T =k−1 (29)
s I P
whichisinfactequivalent totheINDIcontrollawobtained
(cid:2) (cid:3) Whenever the system under consideration is of second-
in (18). Appropriate selection of g¯ must ensure stability
ordercontrollercanonicalform,wewillhaveerrordynam-
according to (Chang and Jung (2009)), and ideally, this
ics of the form e¨+k e˙ +k e = 0, and considering the
termshouldbetunedaccordingtothebestestimateofthe D P
newly introduced derivative gain k related to e¨we have
true effector blending moment gˆ(ω˜) for measured angular D
velocities ω˜. K =k ·(g¯·t )−1, T =k ·k−1, T =k−1 (30)
D s I D P D D
So far we have considered derivations in continuous-time.
This suggests not only that an equivalent discrete PI(D)
For practical implementations of these controllers and
controller with gains < K, T , (T ) > can be obtained
i d
for the matters of upcoming discussions, sampled-time
via INDI/TDC, but doing so is more meaningful and
formulations involving continuous and discrete quantities
systematic than heuristic methods. This is because we
as in (Chang and Jung (2009)) are more convenient and
begin the design from desired error dynamics given by
restatedhere.Forthat,consideringthatthesmallestλone
Hurwitz gains < k , (k ) > and what follows is finding
P D
can consider is the equivalent of the sampling period t of
s the remaining effector blending gain g¯ either analytically
the on-board computer. The sampled formulation of (22)
wheneverGiswellknown,withaproperestimateGˆ,orby
may be expressed as
tuning according to closed-loop requirements. As already
−1
δ(k)=δ(k−1)+g¯ ν(k−1)−ω˙(k−1) (23) mentioned,detailsonasufficientconditionforclosed-loop
whereithasbeennecessarytoconsiderνatsamplek−1for stability under discrete TDC, and therefore applicable to
(cid:2) (cid:3)
causalityreasons.Replacingthe sampledvirtualcontrolν its equivalent INDI, can be found in (Chang and Jung
according to (8) we have (2009)) and the references therein.
−1
δ(k)=δ(k−1)+g¯ e˙(k−1)+kPe(k−1) (24) In essence, this procedure is more efficient and much less
andwecanconsiderthe followingfinite difference approx- cumbersomethandesigningawholesetofgainsiteratively.
(cid:2) (cid:3)
imation of the error derivatives as angular accelerations Moreover, for flight control systems, the self-scheduling
are not directly measured propertiesofinversion-basedcontrollershavesuggestedsu-
perioradvantageswithrespecttoPIDcontrolssincethese
e˙(k)=[e(k)−e(k−1)]/t . (25)
s must be gain-scheduled according to the flight envelope
variations. The relationships here outlined suggests that
Considernowthe standardproportional-integral (PI)con-
PID-scheduling shall be done at the proportional gain K
trol
t via the effector blending gain g¯, and not over the whole
−1
δ(t)=K e(t)+TI e(σ)dσ +δDC, (26) set of gains <K, Ti, (Td)>.
0
Z
where K ∈ R3×3(cid:0)denotes a diagonal p(cid:1)roportional gain
matrix, T ∈ R3×3 a constant diagonal matrix represent- 4. LONGITUDINAL FLIGHT CONTROL
I
ing a reset or integral time, and δ ∈ R3 denotes a SIMULATION
DC
constant vector representing a trim-bias, which acts as
a trim setting and is computed by evaluating the initial Inthissection,robustPItuningviaINDIisdemonstrated
conditions. The discrete form of the PI is given by with a simple yet significant example consisting of the
k−1 tracking controldesign for a longitudinal launcher vehicle
−1 model.Thesecond-ordernonlinearmodelisobtainedfrom
δ(k)=K e(k−1)+T t e(i) +δ (27)
I s DC
(Sonneveldt (2010); Kim et al. (2004)), and it consists on
i=0
(cid:0) X (cid:1) longitudinaldynamicequationsrepresentativeofavehicle
When substracting two consecutive terms of this discrete traveling at an altitude of approximately 6000 meters,
formulation, we can remove the integral sum and achieve with aerodynamic coefficients represented as third order
the so-called PI controller in incremental form polynomials in angle of attack α and Mach number M.
−1
δ(k)=δ(k−1)+K·ts e˙(k−1)+TI ·e(k−1) (28) The nonlinear equations of motion in the pitch plane are
given by
Following the same steps(cid:0), and for completeness, w(cid:1)e also
presentthePIDextensionbysimplyconsideringtheextra q¯S
α˙ =q+ C (α,M)+b (M)δ , (31a)
z z
derivative term e¨ mV
T(cid:20) (cid:21)
δ(k)=δ(k−1)+K·t T e¨(k−1)+e˙(k−1)+T−1·e(k−1) , q¯Sd
s D I q˙ = C (α,M)+b (M)δ , (31b)
m m
where TD ∈ R3×3 d(cid:0)enotes a constant diagonal matri(cid:1)x Iyy (cid:20) (cid:21)
representing derivative time. where
Cz(α,M)=ϕz1(α)+ϕz2(α)M,
3.4 Equivalence of INDI/TDC/PI(D)
Cm(α,M)=ϕm1(α)+ϕm2(α)M,
b (M)=1.6238M−6.7240,
Having in mind the found the equivalence between INDI z
b (M)=12.0393M−48.2246,
and TDC, and comparing terms from (24) with (28), m
and with
ϕz1(α)=−288.7α3+50.32α|α|−23.89α, ν =−kP2z2+x˙2ref, (39)
or more compactly
ϕz2(α)=−13.53α|α|+4.185α,
−1
ϕm1(α)=303.1α3−246.3α|α|−37.56α, u=u0+g¯ −kP2z2−x˙20 +x˙2ref (40)
ϕm2(α)=71.51α|α|+10.01α. This results as desired,(cid:0)in the following z2−dyn(cid:1)amics
These approximations are valid for the flight envelope of z˙2 =x˙20 +g¯·∆u−x˙2ref. (41)
−10◦ ≤ α ≤ 10◦ and 1.8 ≤ M ≤ 2.6. To facilitate
Notice that we are replacing the accurate knowledge off2
trhewercitotnetnroinl tdheesigmno,rethgeenneornallinsteaatre-lsopnagcietufodrinmalasmodel is byameasurement(oranestimate)asf2 ∼=x˙20,whichwill
result in a control law which is not entirely dependent on
x˙1 =x2+f1(x1)+g1u (32a) a model, hence more robust.
x˙2 =f2(x1)+g2u (32b) We now consider these continuous-time formulations in
where: sampled-time. To that end, we replace the small λ with
x1 =α, x2 =q the sampling period ts so that tk = k · ts is the k−th
g1 =C1bz, g2 =C2bm sampling instant at time k, and therefore
and u(k)=u(k−1)+
(42)
f1(x1)=C1 ϕz1(x1)+ϕz2(x1)M , C1 = q¯S , g¯−1 −kP2z2(k−1)−x˙2(k−1)+x˙2ref(k−1) ,
mVT where due to causality relationships we need to consider
(cid:2) (cid:3)
(cid:2) (cid:3) q¯Sd theindependentvariablesatthesamesamplingtimek−1.
f2(x1)=C2 ϕm1(x1)+ϕm2(x1)M , C2 = .
I
yy Referring back to the derived relationship between INDI
(cid:2) (cid:3)
and PI control, the equivalent PI control in incremental
The control objective considered here is to design a PI
form is
autopilot via INDI that tracks a smooth command refer-
−1
ence yr with the pitch rate x2. It is assummed that the u(k)=u(k−1)+K·ts z˙2(k−1)+TI z2(k−1) ,
aerodynamic force and moment functions are accurately (43)
(cid:2) (cid:3)
knownandthe Machnumber M istreatedasaparameter with
availableformeasurement.Moreover,forthissecond-order −1 −1
K =(g¯·t ) , T =k (44)
systeminnon-lowertriangularformduetog1uandf2(x1), s I P2
pitch rate control using INDI is possible due to the time- The nature of the desired error dynamics (proportional)
scale separation principle (Chu (2010); Sieberling et al. gaink is thereforeofanintegralcontrolaction,whereas
P2
(2010)).Withrespecttoactuatordynamicsmodeledasin the effector blending gain g¯ act as proportional control.
(5), we consider Ka =1, and τa =1e−2. Having designed for desired error dynamics, and for a
given sampling time t , tuning a pitch rate controller is
s
4.1 Pitch rate control design only a matter of selecting a proper effector blending gain
g¯ according to performance requirements.
First, introduce the rate-tracking error
Remark3:NoticeatthispointthathavingthePIcontrol
z2 =x2−x2 (33)
ref in incremental form introduces a finite difference of the
the z2−dynamics satisfy the following error error state, which is the equivalent counterpart of what
z˙2 =x˙2−x˙2 (34) has been considered the acceleration or state derivative
ref
for which we design the following exponentially stable x˙20 in INDI controllers.
desired error dynamics Remark 4: Notice also that designing the PI control
z˙2+kP2z2 =0, kP2 =50 rad/s. (35) gains via INDI is highly beneficial, since only the effector
Accordingtotheresultspreviouslyoutlined,theincremen- blendinggainisthetuningvariable.Thisstronglysuggests
tal nonlinear dynamic inversioncontrollaw design follows thatrobustadaptivecontrolcanbeachievedbyscheduling
from considering the approximate dynamics around the this variable online during flight and not over the whole
current reference state for the dynamic equation of the set of gains.
pitch rate as in (13)
SimulationresultsfortheINDI/PIcontrolarepresentedin
q˙∼=q˙0+g¯·∆δ (36) Figure 2, considering smooth rate doublets for a nominal
assuming that pitch acceleration is available for measure- longitudinal dynamics model at Mach 2. For both con-
ment and the scalar g¯ to be a factor of the accurately trollers, the same zero-mean Gaussian white-noise with
known estimate of g2 standard deviation sdq =1e−3 rad/s is added to the rates
g¯=kG·gˆ2, kG =1. tosimulatenoisymeasurements.The designedINDI gains
of k = 50 rad/s and k = 1 are mapped to PI gains
P2 G
This is rewritten in our formulation as resultinginK =100gˆ−1 andT =0.02s,bothcontrollers
2 I
x˙2 ∼=x˙20 +g¯·∆u (37) showing identical closed-loop response as expected.
where recalling that x˙20 is an incremental instance before Withthisexample,itisdemonstratedhowaself-scheduled
x˙2, and therefore the incremental nonlinear dynamic in- PI can be tuned via INDI by departing from desirederror
version law is hence obtained as dynamics with the gain k , and considering an accurate
−1 P2
u=u0+g¯ ν−x˙20 , (38) effector blending model estimate g¯=gˆ2.
(cid:0) (cid:1)
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