Contributions to control theory with applications to aerospace systems. Just beyond linear control...
Jean-Marc Biannic – HDR defense March 19th , 2010.
Contributions to control theory with applications to aerospace systems N
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Outline
Contributions to control theory with applications to aerospace systems N
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A quick overview of my scientific activities after PhD
Research : selected papers (1/2) LPV control, Gain-Scheduling and Robustness analysis J-M. Biannic, P. Apkarian. Missile autopilot design via a modified LPV synthesis technique. Aerospace Science and Technology, 1999. G. Ferreres, J-M. Biannic. Reliable computation of the robustness margin for a flexible aircraft. Control Engineering Practice, 2001. G. Ferreres, J-F. Magni, J-M. Biannic. Robustness analysis of flexible systems: Practical algorithms. International Journal of Robust and Nonlinear Control, 2003. J-M. Biannic, C. Roos, A. Knauf. Design and analysis of fighter aircraft flight control laws. European Journal of Control, 2006.
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A quick overview of my scientific activities after PhD
Research : selected papers (2/2) Saturations in control systems G. Ferreres, J-M. Biannic. Convex design of a robust anti-windup controller for an LFT model. IEEE Transactions on Automatic Control, 2007. C. Roos, J-M. Biannic. A convex characterization of dynamically constrained anti-windup controllers. Automatica, 2008. J-M. Biannic, S. Tarbouriech. Optimization and implementation of dynamic anti-windup compensators in aircraft control systems with multiple saturations. Control Engineering Practice, 2009. C. Roos, J-M. Biannic, S. Tarbouriech, C. Prieur, M. Jeanneau. On-ground aircraft control design using a parameter-varying anti-windup approach. Aerospace Science and Technology, 2010. Contributions to control theory with applications to aerospace systems N
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A quick overview of my scientific activities after PhD
Research : Toolboxes Freely online available from the author’s homepage : http://www.cert.fr/dcsd/idco/perso/Biannic/mypage.html Robustness analysis G. Ferreres, J-M. Biannic. Skew-µ Toolbox (SMT). µ-based MatlabToolbox for robustness analysis. Version 3, 2009.
LFT modelling J-M. Biannic, C. Doll and J-F. Magni. LFRT-SLK: A SIMULINK Interface to the LFR Toolbox. Version 1.1. July 2006.
Anti-Windup design J-M. Biannic, C. Roos. AWAST: The Anti-Windup Analysis and Synthesis Toolbox. Version 1.0. September 2008. Contributions to control theory with applications to aerospace systems N
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A quick overview of my scientific activities after PhD
Supervised PhD theses Vers une approche non linéaire de la commande d’attitude de satellites par jets de gaz. (Sontsada Valentin, 2004.) Contribution à la commande des systèmes saturés en présence d’incertitudes et de variations paramétriques. (Clément Roos, 2007) Aerospace Valley Award in 2008, GDR-MACS best PhD thesis award in 2009.
Inversion dynamique robuste & Applications aux systèmes aérospatiaux (Started in 2009 by Mario Hernandez) Analyse de robustesse en performance pour les systèmes de dimension élevée. (Started in 2010 by Laure Lafourcade)
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A quick overview of my scientific activities after PhD
Teaching (60 hours/year) 1993-1996 : Moniteur de l’enseignement supérieur (60h/an) Encadrement TD Mathématiques (UPS) Encadrement TD Automatique (SUP’AÉRO)
1996-1998 : Maître de conférence vacataire à SUP’AÉRO (50h/an) Encadrement TD Automatique Mise au point et encadrement de BE
From 1998 : Professeur vacataire à SUP’AÉRO (60h/an) Responsable du module Commande non linéaire Responsable du module Analyse de robustesse
From 1999 : Contribution à l’évolution de l’enseignement Mise en place d’un mini-projet : Pilotage-Guidage d’un avion civil en phase d’approche Membre du groupe scientifique ONERA/SUP’AÉRO Contributions to control theory with applications to aerospace systems N
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A quick overview of my scientific activities after PhD
European collaborations REAL (1998-2001) : Robust and Efficient Auto-Land Systems Design. European Project (ONERA, DLR, NLR, AIRBUS, Delft Univ.) Fruitful exchanges with S. Bennani, R. Luckner, H.D. Joos, P. Fabre and P. Menard.
NASTAC (2004-2007) : Nonlinear Analysis and Synthesis Techniques for Aircraft Control. Garteur Project (AIRBUS, SAAB, FOI, ONERA, DLR, NLR, LAAS, Univ. of Leicester, de Monfort and Bristol), Fruitful exchanges with D. Bates, A. Marcos, M. Hagstrom, J. Robinson, F. Villaume, M. Jeanneau...
NICE (2010-2012) : Nonlinear Innovative Concepts for Design and Evaluation. EDA project involving many european partners, Promising expected collaboration between ONERA, LAAS, Univ Tor Vergata (Lucca Zaccarian) and DASSAULT (L. Goerig). Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Some thoughts on LPV control Introduction For aerospace applications, the LPV representation (
x˙ y
= A(θ)x + B(θ)u = C (θ)x + D(θ)u
(1)
is very useful since it captures the behavior of many plants on a large operating range. Nonlinear effects may also be described by a quasi-LPV version of (??) in which: θ = θ(x )
(2)
Unfortunately, this last relation cannot be directly handled by LPV design techniques. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Some thoughts on LPV control For technical reasons, LPV systems are often rewritten into a more suitable format... Polytopic model Σ(θ) =
A(θ) B(θ) C (θ) D(θ)
∈ Co {Σ1 , . . . , ΣN }
(3)
LFT model Σ(θ) =
A(θ) B(θ) C (θ) D(θ)
Σ3
Σ(θ) Σ5 Σ1
N
= Fu (M, Θ)
(4)
Θ
Σ4 Σ2
. x u
M
x y
Contributions to control theory with applications to aerospace systems
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LPV control versus Gain-Scheduling
Some thoughts on LPV control Design techniques Σ3
Θ
Σ4 Σ2
Σ(θ) Σ5
. x
Σ1 Σ K3 Σ K4 Σ K2
ΣK (θ)
Σ K5
Σ K1
xK
x
M y
u K
. xK
Θ
LPV design techniques have emerged in the mid-90’s along with the development of LMI optimization tools. Convexity follows from the particular structure of the LPV controller copying that of the plant... Many extensions have been proposed recently: to introduce bounds on θ˙ to handle perturbations on θ or unavailable parameters
Convexity is often lost and the computational burden increases... Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Some thoughts on LPV control Polytopic or LFT model ? Consider for example the second-order LPV plant: "
0 1 −θ2 −θ
x˙ = S4
#
(
x ⇔
x˙ 1 = x2 x˙ 2 = −θ(θx1 + x2 )
S3
.x
+
2
1/s
(5)
.
x2
x1
1/s
x1
−
unavoidable conservatism
+ +
θ
M
2
θ
w1
I 2 /s
S5
S1
θ
z1
θ
w2
z2
θ(t)
θ .I 2
θ(t)
M
M(s)
S2
A(θ) ∈ Co {A(S1 ), A(S3 ), A(S5 )}
A(θ) = Fu (M, θI2 )
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LPV control versus Gain-Scheduling
Some thoughts on LPV control Polytopic or LFT design ? As seen above, polytopic models are often more conservative than LFT models, except for affine parametric dependence, LFT-based LPV design techniques are generally more conservative than their polytopic-based counterparts, Conservatism of LFT design increases with the number of parameters and their repetition, Numerical complexity of the LFT design grows rapidly with the repetition of each parameter (scaling variables), Numerical complexity of the polytopic approach grows with the number of vertices of the polytope (number of LMI constraints), Implementation of LFT controllers is easier...
In the case of affine parametric dependence, polytopic design should be preferred. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
A modified LPV approach Enforcing affine parametric dependence in an LFT model Θ(t)
Θ(t)
~
I
M(s) u
y
u
~ M(s)
y
(a)
(b)
A M(s) = Cz Cy (
⇒
M(s)
1+ τs
Bw Dzw Dyw
Bu Dzu Dyu
˜ A ˜ ˜ M(s) = Cz ˜y C
˜w B 0 0
˜u B ˜ zu D ˜ Dyu
˜ +B ˜ w ΘC ˜z )x + (B ˜u + B ˜ w ΘD ˜ zu )u x˙ = (A ˜ ˜ y = Cy x + Dyu u
(6)
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LPV control versus Gain-Scheduling
A modified LPV approach Remarks ˙ is bounded, the approximation error between LFT As soon as Θ models (a) and (b) becomes arbitrarily small when τ → 0, ˙ and exploiting the properties of the In most cases, given a bound on Θ plant, an upper-bound on τ can be obtained, Given a polytope for Θ: Θ ∈ Co {Θ1 , . . . , ΘN } using (??), the "modified" LFT model is easily rewritten in a polytopic format: ˜ +B ˜ w Θi C ˜z )x + (B ˜u + B ˜ w Θi D ˜ zu )u x˙ = (A Mi (s) : (7) ˜y x + D ˜ yu u y =C Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
A modified LPV approach LPV design model zu
Wu. (s)
Weighting function on control inputs
Wref (s) . u
Reference model
+ ec
− z
u
K? y
A(s) Actuator
w Θ ur
F(s)
Θ
z Θ
G(s)
+
−
Wp (s)
e
Plant (LFT)
ze
Weighting on error
y + +
Regulated outputs
Wy
Wz
Measurement noises
wm
w=
ec
wm u
z=
P i (s)
zu
zε
y
Polytopic design model Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
A modified LPV approach Application to a pich-axis missile control problem 1.4
1.3
L2 induced performance level γ
1.2
1.1
1
0.9
0.8
0.7 −2 10
−1
0
10
10
1
10
filter time constant τ
L2 gain vs γ Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
A modified LPV approach Application to a pich-axis missile control problem 20
40
15
30
10
Angle of attack (deg)
normal acceleration az
20
10
0
5
0
−5
−10 −10
−20
−30
−15
0
0.5
1
1.5
2 time (sec)
2.5
3
3.5
−20
4
600
0
0.5
1
1.5
2 time (sec)
2.5
3
3.5
4
0
0.5
1
1.5
2 time (sec)
2.5
3
3.5
4
40
20 400 0
α
−20
0
Parameter M
tail−deflection rate (deg/s)
200
−200
−40
−60
−80
−100
−400
−120 −600 −140
−800
0
0.5
1
1.5
2 time (sec)
2.5
3
3.5
4
−160
J-M. Biannic, P. Apkarian. Missile autopilot design via a modified LPV synthesis technique. Aerospace Science and Technology, 1999. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Gain-Scheduling Principle Grid the operating domain, Trim and linearize the plant about each grid point, Perform local syntheses via well-established linear methods, Perform linear interpolation to get the global controller. H H5 Ki+1,j
H4
Ki+1,j+1 K(V,H)
H3 Ki,j
Ki,j+1
H2 V H1 V1
V2
V3
V4
V5
V6
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LPV control versus Gain-Scheduling
Gain-Scheduling Recent improvements Born in the 60s, gain scheduling techniques have been successfully applied on many aerospace applications. They offer two major advantages over LPV techniques, since they are: numerically much cheaper, less conservative.
In return, some drawbacks exist and have justified recent works in order to improve the original approach: interpolation of dynamic controllers (not completely solved yet), stability-preserving interpolation methods, gain-scheduling for quasi-LPV plants (velocity-based linearization). Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Gain-Scheduling Illustrations of stability-preserving interpolation techniques P P (Calculation point for Ki(s)) i−1 i
Pi+1
Validity of compensator Ki(s)
θ min x2
θ max
A priori defined trajectory
Trajectory segments inside the intersection of two consecutive ellipsoids
Pi+1 Pi P i−1
Stability ellipsoid Polyedral validity domain of the polytopic model
x1
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LPV control versus Gain-Scheduling
Gain-Scheduling Interpolation of locally robust (polytopic) controllers
Σi,j
θ2
θ2i+1 (1+δ)θ2
i
θ2i (1−δ)θ2
i
θ1i (1−δ)θ1
θ1i+1
θ1
(1+δ)θ1
i
i
J-M. Biannic, C. Roos, A. Knauf. Design and analysis of fighter aircraft flight control laws. European Journal of Control, 2006. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Controller a posteriori Validation Gain scheduling by interpolation
Modified LPV design (based on a polytopic approach)
of locally robust controllers Linear interpolation
θ
Wref (s)
C
with
u
+
ec
−
1/s
y
x
x
^ A(s)
+ LFT model of the plant possibly including uncertainties
xr
K(θ )
u
^xa
Θ(t)
∆
Standard form for ...
M(s)
robustness analysis
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LPV control versus Gain-Scheduling
Robustness analysis tools Time-invariant (µ-based) analysis Scope: validation of gain-scheduled controllers on the continuum of LTI models on the whole operating domain Principle: worst case (∆=∆* ) (smallest norm destabilizing case)
δ2 UNSTABLE
kmax
Nominal case (∆=0)
det(I−M(jw) ∆ ) = 0 STABLE
δ1
Contributions: G. Ferreres, J-F. Magni, J-M. Biannic. Robustness analysis of flexible systems: Practical algorithms. IJRNC, 2003. G. Ferreres, J-M. Biannic. Skew-µ Toolbox (SMT). µ-based Matlab Toolbox for robustness analysis. Version 3, 2009. C. Roos , J-M. Biannic. Efficient computation of guaranteed stability domains for high-order parameter dependent plants. ACC, 2010. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Robustness analysis tools Time-varying analysis: beyond quadratic stability Scope: validation of gain-scheduled or LPV controllers including time-varying effects such as possible interactions between parameters and states evolutions. IQC-based analysis: A general and powerful framework which unfortunately usually leads to high order LMI problems (KYP lemma). Alternative approaches (frequency-domain) should be investigated (extension of µ analysis). Parameter-Dependent Lyapunov Functions (PDLF): A natural extension of the quadratic stability concept which enables to introduce bounds on the parameters variations. Numerical complexity is also an issue to which a solution is proposed in the following contribution: J-M. Biannic, C. Pittet. LPV analysis of switched controllers in satellite attitude control systems. AIAA GNC, 2010. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Robustness analysis tools On the optimization of PDLF In order to reduce the computational burden while controlling conservatism, polynomial functions can be used and optimized on a grid of the parameter space {δ1 , . . . , δN }: ∀j = 1, . . . , N, Pr i i=0 δj Pi > 0 P P r δji (Ac (δj )0 Pi + Pi Ac (δj )) + ρmin ri=1 iδji−1 Pi < 0 i=0 Pr Pr δ i (A (δ )0 P + P A (δ )) + ρ i−1 Pi < 0 c j i i c j max i=0 j i=1 iδj
(8)
By such an approach, the numerically expensive and possibly conservative KYP lemma is avoided. However, the validity of the PDLF between grid points should be checked a posteriori. After re-writting (??) as an LFT, this can be achieved by a µ analysis test which also can provide worst cases (if any) so as to update the grid... ... see also Laure Lafourcade’s PhD thesis started this year. Contributions to control theory with applications to aerospace systems N
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LPV control versus Gain-Scheduling
Preliminary conclusions Enhanced gain-scheduling approaches (including modified LPV design) offer many possibilities to the control engineer, especially in aerospace in which they fit very well: © low conservatism © low numerical complexity → possibility to handle many parameters § no guarantee a priori... but can be checked a posteriori!
By a time-scale separation argument, they handle a significant class of quasi-LPV systems, thus covering some "soft" nonlinearities and providing a path at the boundary of linear systems... However, despite recent works, LPV or gain-scheduling approaches generally fail to take correctly into account saturation effects which unfortunately are ubiquitous in aerospace applications (and others). And yet, saturated linear systems admit a quasi-LPV description... Contributions to control theory with applications to aerospace systems N
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Saturations in Control Systems
Standard form for saturated systems normalized deadzones w
z
Φ
− + −1
Λ
L(s)
Λ
L(s)
M(s)
"
M(s) = C (sI − A)
ΣNL :
x˙
z
w
−1
B + D = Fl
0 −I I I
#
!
,Λ
−1
= Ax + Bw = Cx + Dw = Φ(z) = [φ(z1 ), . . . , φ(zm )]0
L(s)Λ
(9)
with: φ(zi ) = zi − sat1 (zi ) Contributions to control theory with applications to aerospace systems N
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Saturations in Control Systems
Weakness of the quasi-LPV description w
z
w
modelling
Φ
M(s)
Θ( z)
z
M(s)
design / analysis
φ(zi ) ∈ [0 1[ (10) zi The parameter Θ is then highly dependent on the states which is not explicitely "captured" by LPV techniques, for which any θi belongs to the interval [0 1] or [0 ρi ] if zi is bounded... Θ(z) = diag [θ1 (z1 ) . . . θm (zm )] ,
θi (zi ) =
φ(z) slope ρ = 1 − 1/ z
z
conservative area
z
Not only this description is rather conservative but its validity must be checked a posteriori (z 0 s.t. : " (?) +
#
A B v CJ Q 0 AJ −Z i h Cp 0 Q
"
Bw + Bv DJ BJ −S + DS Dp S
Q Zi 0 + Q[Ci 0]0 Zi + [Ci 0]Q 1
#
S
0 0 − γ2 I
0 , i = 1, . . . , m
(18)
⇒ Nonlinear system Fu (Σ(s), Φ) is stable on EP = x /x 0 Q −1 x ≤ 1 , R∞ ⇒ ∀x0 ∈ EP , 0 zp (t)0 zp (t) dt ≤ γ Contributions to control theory with applications to aerospace systems N
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Saturations in Control Systems
Application to anti-windup control Dynamic anti-windup synthesis : LMI formulations The above BMI problem becomes convex... in the full-order case (nJ = n) [a] in the static-case (nJ = 0) [a] in the full-order case, with constrained poles (Re(λi (AJ ) < −α) [b] in the reduced-order case, with fixed poles [a] b [a] J-M. Biannic, S. Tarbouriech. Optimization and implementation of dynamic anti-windup compensators in aircraft control systems with multiple saturations. Control Engineering Practice, 2009 b [b] C. Roos, J-M. Biannic. A convex characterization of dynamically constrained anti-windup controllers. Automatica, 2008. Contributions to control theory with applications to aerospace systems N
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Saturations in Control Systems
Application to anti-windup control More details about fixed-poles synthesis... J(s) = M0 +
n1 X M1i i=1
s + λi
+
n2 X i=1
−1
= DJ + CJ (sI − AJ )
s2
M2i + 2ηi ωi + ωi2
BJ
˜ J = BJ S and D ˜ J = DJ S, BMI (??) becomes an LMI With B "
(?) +
#
A B v CJ Q 0 AJ −Z i h Cp 0 Q
"
˜J Bw + Bv D ˜J B −S + DS Dp S
#
0 0 − γ2 I
