One way to design the control law of a mini-UAV
B. Lang* , P. Paquier†
National Engineering Institute in Mechanics and Microtechnologies (ENSMM), Besançon, France
and
J.-M. Friedt‡
Business or Academic Affiliation 2, Besançon, France
This paper deals with a method used to design the control law of the μDrone MAV.
This vehicle uses six propellers to fly and the dynamic model approximation for the
motion is a MIMO linear time-invariant system. As we want to design a linear regulator,
it is necessary to build a robust feedback control law. The LQ state feedback regulator
design is applied to a standard model, tacking into account some perturbations. This is
why the model is augmented with a perturbation vector and an observable subsystem is
extracted in order to build a state estimator whose gain is the solution of a LQ problem.
The subsystem is then decomposed into a controllable set and an uncontrollable one. The
use of an asymptotic rejection strategy of the influence of uncontrollable modes gives the
possibility to find a state feedback applied only to the controllable ones. Here again
feedback matrix is chosen as the solution of a LQ problem. To compute the weighting
matrices of quadratic criterions we use a “partial observability gramian”. The great
advantage of this method is due to the use of only three scalars to synthesize the control
law.
Nomenclature
x, y , z



ui
Fi
Mi
i
ki
= centre of gravity coordinates of the MAV
= roll angle
= pitch angle
= yaw angle
= actuator control input for solid
Si
= thrust produced by the ith actuator
= moment produced by the ith actuator
= time constant of the ith actuator
= DC gain of the ith actuator
u1
x
u2
u3 u 4
u
=
y
=
yr
= reference vector
y
z 

u5
 T
u6  input vector
T
output vector
18
= state vector of initial linear MAV model  R
x
A, B, C = matrices of initial state-space model
Associate Professor, ENSMM, blang@ens2m.fr.
Coordinator of the UAV project, ENSMM, Address/Mail Stop for second author.
‡
Research and Development Engineer, Department Name, Address/Mail Stop for fourth
author (etc).
*
†
1
xp
= perturbation vector
I.
Introduction
paper deals with a method used to design the control law of the μDrone. This MAV
THIS
uses six propellers to fly and it is not obvious to control the six brushless motors
simultaneously. In order to build a dynamical model of the MAV, we worked on the
assumption that it can be described with seven solids in interaction. After a linearization of
the equations in the vicinity of a horizontal trim, the model is a multi-input, multi-output
(MIMO) linear time-invariant dynamic system. This approximation can be used only if it is
possible to build a robust feedback control law.
The interest of the present paper is to provide a simple method to reach the needed
robustness, using the LQ state-feedback regulator design applied to a “standard model”,
tacking into account some perturbations. This method is detailed in Ref. 1 and very recently
in Ref. 2.
The paper is organized as follows. Section II gives some details on the MAV model
and the control design is explained in section III. Section IV gives complements about the
choice of adjustment parameters.
II.
Model of the μDrone
The μDrone is a “gyro dyne” with six propellers. We worked on the assumption that it can be described with
seven solids in interaction.
Solid S 0 is the body.
Solids S1 and S 2 are made up of a fix pitch propeller, the motor shaft and the blades holder. The two
propellers are counter-rotating and provide the lift of the MAV.
Solids S 3 and S 4 include a variable pitch propeller, the motor shaft and the variable pitch system. These
propellers provide the trim stabilization.
Solids S 5 and S 6 are like the two precedents and provide MAV propulsion.
The detailed description of this mechanical system modeling can be found in Ref. 3.
Let Fi denotes the thrust produced by the ith actuator and M i the associated moment. If
u i denotes the
control input of the actuator, we assume that
(1)
dFi
 Fi (t )  k i .u i (t )
dt
M i (t )  k i .Fi (t )
i
With this assumption and after linearization of the mechanical model, the MIMO linear time-invariant
system, denoted DRO, is an approximation of the MAV behavior 4,5. DRO is a system with 6 inputs, 6 outputs,
and state-vector x is of size 18.
(2)
dx
 A.x  B.u
dt
y  C .x
The eigenvalues of the A matrix are so defined: 10 are zero, 6 are real negative and 2 are purely imaginary.
The rank of controllability matrix is 2 and that of observability matrix is 4.
III.
Control design of the MAV
2
In order to build a “standard model” we introduce a perturbation vector x p and we work on the assumption
that each perturbation is constant. It is then possible to give a simple model for these perturbations
(3)
dxp
dt
 A p .x p
where A p is a square matrix of zeros. In our application
x p  R 2 and the perturbations are moments acting
on roll and pitch rates. The initial system DRO is augmented and becomes
(4)
d xe
 Ae .x e  Be .u
dt
y  C e .x e
 A A12 
 B
Ae  
, Be    , Ce  C O

O
O Ap 
with
As we want to elaborate the control law with a state-feedback, it is necessary to build a state estimator. The
first step is to compute the observability staircase form of  Ae , Be , Ce  and extract an observable subsystem
(5)
d xo
 Aeo .x o  Beo .u
dt
y  C eo .x o
It is then possible to build a state estimator
d xˆ o
 Aeo .xˆ o  Beo .u  K o .( y  Ceo .xˆ o )
dt
The gain matrix K o is chosen as the solution of the LQ problem
(6)

find


v   KK o .x o that minimize J o   x o .Qo .x o  v .Ro .v .dt for
0
and define a positive scalar
(7)
T
T
d xo
 AeoT .x o  CeoT .v
dt
k o such as
K o  k o .KK oT
At this point, the problem is to make a good choice for the weighting matrices
Qo and Ro of the quadratic
criterion J o . In order to adjust easily Qo , we use a “partial observability gramian” defined as
(8)
where
Too
Go (Too )   exp( AeoT .t ).CeoT .Ceo . exp( Aeo .t ).dt
0
Too  k o .To with To positive, called the filtering horizon and k o the shape factor.
As the pair
 Aeo , Ceo  is observable, Go (Too ) is asymmetric positive matrix, and it is possible to choose
Qo  Too .Go (Too )
1
(9)
Ro  I
with I the identity matrix.
The state estimator is then
u 
K o . 
 y
Now let us compute the controllability staircase form of  Aeo , Beo , C eo  with the similarity transformation
(10)
matrix
d xˆ o
  Aeo  K o .C eo .xˆ o  Beo
dt
Toc
3
(11)
d  x nc   Aeonc

dt  x c   Aeo 21
x 
C eoc . nc 
 xc 
y  C eonc
As
O   x nc   O 
.

.u
Aeoc   x c   Beoc 
with
 x nc 
 x   Toc .x o
 c
Aeo 21 is not a matrix of zeros, uncontrollable modes x nc perturb controllable modes x c . In order to
reject the influence of uncontrollable modes it is possible to use a linear transformation
(12)
u (t )  v(t )  Ga .x nc (t )
~
x (t )  x (t )  T .x (t )
c
(13)
c
a
d~
xc
 Aeoc .~
x c  Aeo 21  Aeoc .Ta  Ta . Aeonc  Beoc .Ga .x nc  Beoc .v
dt
y  C eoc .~
x c  C eonc  C eoc .Ta .x nc
Now, it is often possible to find
(14)
Ga and Ta are constant matrices, leading to
where
nc
Ga and Ta such as
Aeo 21  Aeoc .Ta  Ta . Aeonc  Beoc .Ga   O
Ceonc  Ceoc .Ta   O
With these conditions equations (11) become
(15)
d~
xc
 Aeoc .~
x c  Beoc .v
dt
d x nc
 Aeonc .x nc
where the pair  Aeoc , Beoc  is controllable
dt
y  C eoc .~
xc
Once more, it is possible to compute the solution of the LQ problem
find



d~
xc
T
T
v   K c .~
x c which minimize J c   ~
x c .Qc .~
x c  v .Rc .v .dt for
 Aeoc .~
x c  Beoc .v
0
dt
In order to adjust weighting matrices
Qc and Rc of the quadratic criterion J c , define a positive scalar k c ,
Tc such as
Tc  To / k c
a positive control horizon
(16)
and the “partial observability gramian”
(17)
Tc
T
T
Goc (Tc )   exp( Aeoc
.t ).Ceoc
.Ceoc . exp( Aeoc .t ).dt
0
It is then possible to choose
(18)
T
Rc  Tc .Beoc
.Goc (Tc ).Beoc
T
Qc  Ceoc
.Ceoc
Now, it is possible to elaborate the feedback control
4
(20)
where
v  K c .~
x c  Kr .y r
K r is a constant matrix and y r the reference vector.
K r matrix we use equations (15) in which vector v is replaced by expression (20)
~
dxc
  Aeoc  Beoc .K c .~
x c  Beoc .K r . y r
dt
y  C eoc .~
xc
In order to determine
(21)
In permanent rate we want
(22)
y  y r . As  Aeoc  Beoc .K c  is a stability matrix, it is invertible and
y  C eoc . Aeoc  Beoc .K c  .Beoc .K r . y r
1
Then, the choice
(23)

K r   C eoc .( Aeoc  Beoc .K c ) 1 .Beoc

1
leads to the solution. In our application, this last matrix is not invertible; it is the reason why we use the
expression of the pseudo inverse and it appears that it is not possible to choose reference  r except for the zero
value.
Now the use of expression (12) leads to the control vector
u  v  Ga .x nc  K c .~
x c  K r . y r  Ga .x nc
which gives
(24)
u  K c .Ta  Ga
x 
K c . nc   K r . y r
 xc 
Toc in (11) it is possible to express u as a feedback control
u  K c .Ta  Ga K c .Toc .x o  K r . y r
but with the transformation matrix
(25)
According to separation principle and using state estimator expression (10), it follows easily the regulator
equations
(26)
with
y 
d xo
 Areg .x o  Breg . r 
dt
y
y 
u  C reg .x o  Dreg . r 
y
Areg  Aeo  K o .C eo  Beo .K c .Ta  Ga
Breg  Beo .K r
Ko 
C reg  K c .Ta  Ga
Dreg  K r
IV.
O
K c .Toc
K c .Toc
Choice of the adjustment parameters
Among the main objectives, the stability of state matrix estimator Areg in (26) is essential. It is clear that
To , k o and k c will influence stability, acting on K o and K c matrices. The first step is to assign a
fixed value to k o and k c parameters; often 5 is a good value to start. Now the filtering horizon To is adjusted
parameters
5
so that Areg is a stability matrix; it is often effective to start with the slowest time constant value of the initial
model.
In order to quantify the robustness stability of the feedback, the regulator, without reference, can be
converted to a transfer function matrix. Starting with
(27)
we obtain
(28)
where
d xo
 Areg .x o  K o . y
dt
u  C reg .x o
u ( s )   K ( s ). y ( s )
K ( s)  KK c .sI  Aeo  Beo .KK c  K o .Ceo  .K o
1
(29)
KK c  K c .Ta  Ga
K c .Toc
(It is interesting to notice the symmetry of expression (29) in relation to matrix gains
KK c and K o ).
Similarly, the initial model of the MAV is
(30)
Now define
(31)
(32)
(33)
(34)
y ( s )  DRO ( s ).u ( s )
Lu ( s)  K ( s).DRO ( s) the input loop transfer
L y ( s)  DRO ( s ).K ( s ) the output loop transfer


1
1


M st  min 
,
 the static margin
max Lu ( j ) max L y ( j ) 





1
1


M dyn  min 
,
 the dynamic margin
max .Lu ( j )
max

.
L
(
j

)


y


Then, when Areg is a stability matrix, the stability of the global feedback is robust in relation to any relative
error such as
(35)
 DRO ( j )
 M st
DRO ( j )
it is why
M st is expressed in %
Similarly, the stability of the feedback is robust in relation to any relative error such as
(36)
 DRO ( j )
 .M dyn
DRO ( j )
Notice that: M dyn  M Td where
M Td is the time delay margin.
Now it is not difficult to compute static and dynamic margin for different values of
To , k o and k c . It then may
be possible to find a triplet for which the stability margins and the dynamic performances are fulfilled.
The reason why this strategy is often efficient is due to the natural robustness of a LQ problem solution and to
the notion of loop transfer recovery (LTR).
In our application the following results are obtained
M st  42.2 %, M dyn  32.7 ms.
6
References
Books
1De Larminat P., Automatique, commande des systèmes linéaires, deuxième édition, Hermès, 2000, Chaps.16, 19-20.
2De Larminat P., Automatique appliquée, Hermès, 2007.
Reports, Theses, and Individual Papers
3Verdot, T., and Passave, J., “Lois de comportement et système de navigation inertielle,” PFE, ENSMM, Besançon,
France 2005.
4Paquier, P., Brocard, G., Lang, B., and Friedt, J.-M., “Projet μDrone – ENSMM (Besançon) Dossier de synthèse,”
Concours international universitaire de drones miniatures, ENSMM, Besançon, Sept. 2005.
5Bonneuil, A., Mairiniac, L., Poudevigne, E., and Poupon, E., “Vers une plateforme avionique autonome,” PFE,
ENSMM, Besançon, France, 2007.
μDrone project : <URL : http://www.ens2m.fr/drone>
7