[PPT] - Robustness analysis techniques for clearance of flight control laws PowerPoint Presentation

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Universit` a di Siena

Robustness analysis techniques for clearance of flight control laws

Andrea Garulli, Alfio Masi, Simone Paoletti, Erc¨ ument T¨ urko˜ glu

Dipartimento di Ingegneria dell’Informazione, Universit` a di Siena garulli@dii.unisi.it Robust Control Workshop – Udine, August 24-26, 2011

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Universit` a di Siena 1

COFCLUO project

Clearance Of Flight Control Laws Using Optimization Funded by EC, 2007-2010 Partners:

Link¨
ping University (Svezia)
AIRBUS (Francia)
DLR (Germania)
ONERA (Francia)
FOI (Svezia)
Universit`

a di Siena (Italy)

Robust Control Workshop – Udine, January 24-26, 2011

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COFCLUO project

Motivations: Validation of FCLs is a key issue in terms of time and costs Baseline solution in industry mainly relies on brute force simulation in a huge number of flight points Objectives: Detect worst-cases faster and/or more reliably than using Monte Carlo based techniques Give guarantees for whole regions of flight envelope to be cleared

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Models

Two types of models used within COFCLUO project: Complex Simulink models of closed-loop aircraft dynamics used for worst-case detection (provided by AIRBUS) Linear Fractional Representation (LFR) models, including rigid and flexible modes, for clearing whole regions LFR models derived from AIRBUS physical models (done by DLR and ONERA)

Robust Control Workshop – Udine, January 24-26, 2011

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UNISI contribution

Techniques for clearing entire regions of the flight envelope: robust aeroelastic stability (this talk) comfort criterion (this talk) robust stability of systems with saturations (Talk FrB06.1 in Milan)

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Robust aeroelastic stability

The largest real part of the closed-loop eigenvalues has to be negative, for all possible values taken by the uncertain parameters (aircraft mass configuration) and the trimmed flight variables (Mach number and calibrated air speed) Techniques adopted: Lyapunov-based analysis (UNISI) µ analysis (ONERA) IQCs (Link¨

ping)

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Robust stability of LFR uncertainty models

Consider the LFR system Σ : ˙ x(t) = A(θ)x(t) =

A + B∆(θ)(I − D∆(θ))−1C
x(t)

where ∆(θ) = diag(θ1Is1, . . . , θnθIsnθ ) An equivalent representation of Σ is given by: Σ :        ˙ x = Ax + Bq p = Cx + Dq q = ∆(θ)p , where x ∈ Rn, q, p ∈ Rd and d = nθ

i=1 si.

Assumptions: θ ∈ Θ = [θ1, θ1] × · · · × [θnθ, θnθ] with 2nθ vertices Ver[Θ] ˙ θ(t) = 0 (time-invariant uncertainty)

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Three methods based on parameter-dependent LFs

Dettori & Scherer (2000) Fu & Dasgupta (2001) Wang & Balakrishnan (2002) Methods based on polynomial LFs not suitable due to models size

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[Dettori & Scherer ’00] If there exist: a symmetric Lyapunov matrix P(θ) ∈ Rn×n, multiaffine in θ two matrices S0, S1 ∈ Rd×d such that ∀θ ∈ Ver[Θ]                    P(θ) > 0        I A B I C D       

T

    P(θ) P(θ) W(θ)            I A B I C D        < 0, where W(θ) =   S1 + ST

1

−S0 − S1∆(θ) −ST

0 − ∆(θ)ST 1

ST

0 ∆(θ) + ∆(θ)S0

  , then the system Σ is robustly stable.

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Remarks:

multiaffine LF

V (x; θ) = xT

P0 +

nθ

j=1

θjPj +

nθ

i=1

nθ

j=i+1

θiθjPij + · · ·

x
parameter-dependent multiplier W(θ), parameterized by S0, S1
2nθ LMIs of dimension (n + d), 2nθ LMIs of dimension n
2d2 + 2nθ n(n + 1)

2 free variables

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[Fu & Dasgupta ’01] Let Ti = blockdiag(0s1, . . . , 0si−1, Isi, 0si+1, . . . , 0snθ ), Ci = TiC, Di = TiD for i = 1, . . . , nθ, and D0 = −I. If there exist 2nθ + 2 matrices Cµ,i ∈ Rd×n, Dµ,i ∈ Rd×d, i = 0, . . . , nθ, s.t.

CT

i

DT

i

Cµ,i Dµ,i

+
CT

µ,i

DT

µ,i

Ci Di

≤ 0, i = 1, . . . , nθ

and a symmetric P(θ) ∈ Rn×n, multiaffine in θ, such that ∀θ ∈ Ver[Θ]       

P(θ) > 0   AT (θ)P(θ) + P(θ)A(θ) Π(θ) ΠT (θ) −

Dµ(θ)D−1(θ) + D−T (θ)DT

µ (θ)



 < 0, where Π(θ) =

P(θ)BD−1(θ) − CT

µ (θ) + CT (θ)D−T (θ)DT µ (θ)

C(θ) = ∆(θ)C,

D(θ) = ∆(θ)D − I, Cµ(θ) = Cµ,0 + nθ

i=1 θiCµ,i ,

Dµ(θ) = Dµ,0 + nθ

i=1 θiDµ,i

then the system Σ is robustly stable.

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Remarks:

multiaffine LF
parameter-dependent multipliers Cµ(θ) and Dµ(θ), parameterized by

Cµ,i, Dµ,i, i = 0, . . . , nθ

nθ + 2nθ LMIs of dimension (n + d), 2nθ LMIs of dimension n
(nθ + 1)(nd + d2) + 2nθ n(n + 1)

2 free variables

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[Wang & Balakrishnan ’02] If there exist: nθ + 1 symmetric matrices Q0, . . . , Qnθ ∈ Rn×n a symmetric scaling matrix N ∈ Rd×d such that, ∀θ ∈ Ver[Θ],                N > 0 Q(θ) = Q0 + nθ

j=1 θjQj > 0

  AQ(θ) + Q(θ)AT + B∆(θ)N∆(θ)BT

Q(θ)CT + B∆(θ)N∆(θ)DT
Q(θ)CT + B∆(θ)N∆(θ)DT T

−N + D∆(θ)N∆(θ)DT   < 0 then the system Σ is robustly stable.

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Remarks:

candidate LF

V (x) = xT

Q0 +

nθ

j=1

θjQj −1 x

2nθ LMIs of dimension (n + d), 2nθ LMIs of dimension n,

1 LMI of dimension d

d(d + 1)

2 + (nθ + 1)n(n + 1) 2 free variables

easily extended to slowly time-varying parameters
generalized to polynomial Lyapunov functions [Chesi et al., ’04]

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Dealing with complexity

The considered methods are generally computationally unfeasible for the clearance problems at hand ⇒ Find appropriate relaxations (trade off conservatism and computational burden) ⇒ Divide into simpler problems (partition the uncertainty domain) Strong requirement: easy-to-use tools

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Relaxations

Lyapunov function: multiaffine (mapdlf), affine (apdlf), constant (clf)
Multipliers: affine, constant, diagonal
Scalings: constant, diagonal

Relaxation Characteristics FD-cµ FD method with constant multipliers Cµ,0, Dµ,0 FD-cdµ FD method with constant diagonal multipliers Cµ,0, Dµ,0 DS-dS DS method with diagonal multipliers S0, S1 WB-dN WB method with diagonal scaling matrix N

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Progressive tiling

LTI uncertainty: partition the uncertainty domain into rectangular tiles, then test robustness in each tile Algorithm: 1) start with an hyperbox containing the entire uncertainty domain Θ if cleared then: done! if not 2) reduce the size of the uncleared tiles (by bisecting each side) 3) repeat until every tile is cleared or minimum tile size is reached Remark: for each tile the LFR is re-parameterized.

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Adaptive tiling

Idea: combine progressive tiling approach with an adaptive choice of the relaxation (different Lyapunov function or multiplier) Rationale: use conservative but fast methods first, then switch to more powerful and computationally demanding ones only for the uncleared tiles Tested on clearance problems: adaptation on the Lyapunov function constant → affine → multiaffine

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Gridding

Before attempting to clear a tile, the tile is gridded and stability of models on the grid is checked If at least one model on the grid is unstable, the tile is skipped and temporarily marked as unstable (portions of the tile can be later cleared, as partitioning proceeds) Three types of tiles when max number of partitions is reached: Cleared Unstable (contain unstable models found by gridding) Unknown (not cleared and no unstable models)

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Software and GUI

All techniques implemented in MATLAB using: LFR toolbox YALMIP SDPT3 A Graphical User Interface (GUI) developed to set up clearance problems and display results

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Example of GUI output

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc

Green tiles: cleared Red tiles: unstable (contain unstable models found by gridding) White tiles: unknown (not cleared and no unstable models)

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Main GUI features

Inputs: model selection choice of methods, relaxations and tiling options restriction to polytopic flight envelopes shifted stability (for slowly divergent modes) Outputs: 2D plots of cleared, unstable and unknown regions number of optimization problems solved and elapsed time rates of cleared, unstable and unknown domain clearance rate: ratio between cleared region and “clearable” domain (tiles that do not contain unstable models found by gridding).

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New version available

A new version of the GUI is available at:

www.dii.unisi.it/∼garulli/lfr rai/

... try it! (any feedback is welcome)

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Closed-loop integral longitudinal models

Uncertain parameters and trim flight variables Symbol Description Nominal value C central tank 0.5 O

uter tank

P payload 0.5 X center of gravity M Mach number 0.86 V calibrated air speed 310 kt

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LFR models for aeroelastic stability

Models representative of closed-loop aeroelastic longitudinal dynamics in frequency range [0,15] rad/sec. Model n d θ1, s1 θ2, s2 θ3, s3 θ4, s4 C 20 16 C, 16 − − − CX 20 18 C, 14 − − X, 4 OC 20 50 C, 26 O, 24 − − OCX 20 50 C, 24 O, 22 − X, 4 POC 20 79 C, 42 O, 24 P, 13 − MV 20 54 M, 26 V , 28 − − Parameters not appearing in ∆ block are set to the nominal values

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Uncertainty set

MV model Flight variables M and V are bounded in a polytope, representing the considered flight envelope Robustness analysis has been carried out on the smallest rectangle including the polytope Other models Fuel loads (C and O), payload (P) and position of center of gravity (X) take normalized values between 0 and 1 Corresponding uncertainty domains are hyper-boxes in the appropriate dimensions

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Aeroelastic stability results: C and CX models

C model: progressive tiling

Method (lf) Cleared NOPs t (sec) DS (clf) 1 1 9.78 DS (apdlf) 1 1 10.97 DS-dS (clf) 1 3 10.15 DS-dS (apdlf) 1 1 4.15 FD-cµ (clf) 1 1 5.81 FD-cµ (apdlf) 1 1 8.77 FD-cdµ (clf) 1 3 8.43 FD-cdµ (apdlf) 1 1 4.93 WBQ-dM 1 3 4.38

CX model: progressive tiling

Method (lf) Cleared NOPs t (sec) DS (clf) 1 1 30.88 DS (apdlf) 1 1 44.47 DS-dS (clf) 1 5 48.59 DS-dS (apdlf) 1 1 16.33 FD-cµ (clf) 1 1 18.60 FD-cµ (apdlf) 1 1 32.30 FD-cdµ (clf) 1 5 32.37 FD-cdµ (apdlf) 1 1 18.91 WBQ-dM 1 5 9.16 Cleared: rate of uncertainty domain cleared NOPs: number of LMI tests performed

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Aeroelastic stability results: OC and OCX models

OC model: progressive tiling

Method (lf) Cleared NOPs t (h:m:s) DS-dS (clf) 1 73 0 : 26 : 32 DS-dS (apdlf) 1 41 0 : 31 : 51 FD-cµ (clf) 1 33 2 : 55 : 30 FD-cµ (apdlf) 1 1 0 : 06 : 11 FD-cdµ (clf) 1 85 0 : 22 : 00 FD-cdµ (apdlf) 1 49 0 : 31 : 50 WBQ-dM 1 169 0 : 14 : 48

OCX model: progressive tiling

Method (lf) Cleared NOPs t (h:m:s) DS (clf) 1 185 327 : 00 : 28 DS (apdlf) 1 1 3 : 07 : 22 DS-dS (clf) 1 745 11 : 10 : 09 DS-dS (apdlf) 1 265 10 : 53 : 36 FD-cµ (clf) 1 185 41 : 31 : 34 FD-cµ (apdlf) 1 1 0 : 17 : 47 FD-cdµ (clf) 1 841 8 : 52 : 25 FD-cdµ (apdlf) 1 385 13 : 34 : 34 WBQ-dM 1 2129 4 : 34 : 12

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Aeroelastic stability results: POC model

POC model: progressive tiling

Method Cleared OPS t (h:m:s) DS-dS (clf) 1 993 33 : 42 : 48 FD-cµ (clf) 1 105 142 : 38 : 10

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Aeroelastic stability results: MV model

MV model: progressive tiling

Method Rate NOPs Time (h:m:s) Time/OP (h:m:s) DS-dS (apdlf) 0.9931 1030 24 : 35 : 25 0 : 01 : 25 DS-dS (clf) 0.9875 1282 12 : 32 : 13 0 : 00 : 35 FD-cµ (apdlf) 1 174 30 : 09 : 45 0 : 10 : 24 FD-cµ (clf) 0.9993 218 34 : 00 : 29 0 : 09 : 21 FD-cdµ (apdlf) 0.9921 1202 20 : 59 : 54 0 : 01 : 02 FD-cdµ (clf) 0.9895 1346 8 : 28 : 02 0 : 00 : 22 Rate: clearance rate (cleared / clearable)

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−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc −1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc

DS-ds with clf DS-ds with apdlf

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−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc −1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc

FD-cµ with clf FD-cµ with apdlf

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−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc −1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 (Ma,Vc) Ma Vc

FD-cdµ with clf FD-cdµ with apdlf

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Adaptive tiling

MV model: adaptive tiling

Method clf apdlf Rate NOPs Time (h:m:s) DS-dS 7 0.9931 1030 24 : 35 : 25 2 5 0.9931 1042 27 : 27 : 54 4 3 0.9931 1110 25 : 20 : 00 6 1 0.9931 1236 20 : 10 : 37 FD-cµ 7 1 174 30 : 09 : 45 2 5 1 179 31 : 19 : 19 4 3 1 188 32 : 21 : 30 6 1 1 206 36 : 05 : 11 FD-cdµ 7 0.9921 1202 20 : 59 : 54 2 5 0.9921 1214 23 : 13 : 35 4 3 0.9921 1266 21 : 18 : 30 6 1 0.9921 1362 15 : 39 : 18

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Comments

apdlf conditions need to solve fewer optimization problems than clf

nes but this reduces the computational time only if number of problems

is significantly smaller (compare FD-cµ and FD-cdµ for OC and OCX) choosing structurally simpler multipliers can increase the time if the number of optimization problems grows too much (see FD with apdlf for OC and OCX) choosing the most powerful method is not always wise (or possible) difficult to pick “best” method a priori adaptation of LF structure has reduced times for DS-ds (15%) and FD-cdµ (25%), not for FD-cµ (again: difficult to predict this a priori)

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Comfort analysis

Joint work with Anders Hansson and Ragnar Wallin (Link¨

ping University)

Comfort index formulated as H2 performance from wind velocity to acceleration at specific points along the aircraft fuselage Need robust H2 analysis to account for uncertainty in the flight parameters Limited frequency range: – industrial practice computes comfort index on a limited freq interval – LFR models are valid only in specific freq range

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Comfort index

Jc =

1

π ω

ω

|W(jω)|2|T(jω; δ)|2Φv(ω) dω Φv(ω): power spectral density of wind velocity T(jω; δ): aircraft transfer function (with uncertain parameters δ) W(jω): comfort filter ≫ Current industrial practice: gridding of the parameter space + numerical integration in desired frequency range (lower bound to worst-case comfort performance) ≫ Contribution: robust finite-frequency H2 analysis (upper bound to worst-case comfort performance)

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Robust H2 analysis

Vaste literature (Paganini, Feron, Stoorvogel, Iwasaki, Sznaier,...) Both time-domain and frequency-domain techniques Frequency-domain techniques provide an upper bound to the system frequency gains, for all frequencies ω and admissible uncertainties δ → Need tools to compute finite-frequency H2 norm

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Finite-frequency H2 norm

Given the system    ˙ x(t) = Ax(t) + Bw(t) z(t) = Cx(t) with transfer function G(s), the finite-frequency H2 norm is defined as G(jω)2

2,¯ ω = 1

2π ¯

ω −¯ ω

Tr {G(jν)∗G(jν)} dν = Tr

BT Wo(ω)B
where

Wo(¯ ω) = 1 2π ¯

ω −¯ ω

(jνI − A)−∗CT C(jνI − A)−1dν. is the finite-frequency observability Gramian

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Finite-frequency observability Gramian

The finite-frequency observability Gramian can be computed as Wo(¯ ω) = L(¯ ω)∗Wo + WoL(¯ ω), where Wo is the standard observability Gramian and L(¯ ω) = j 2π ln

(A + j¯

ωI)(A − j¯ ωI)−1 . Main idea: use the above result together with standard robust H2 theory to provide an upper bound to the robust finite frequency H2 norm of a system with parametric uncertainty

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Problem formulation

LFR system:              ˙ x(t) = Ax(t) + Bqq(t) + Bww(t) p(t) = Cpx(t) + Dpqq(t) z(t) = Czx(t) + Dzqq(t) q(t) = ∆(δ)p(t) ∆(δ) = diag(δ1Is1, . . . , δnδIsnδ ) ∈ ∆ := {∆(δ) : −1 ≤ δi ≤ 1} M =   M11 M12 M21 M22   =     A Bq Bw Cp Dpq Cz Dzq     S(M; ∆) = M22 + M21∆(I − M11∆)−1M12 The robust finite-frequency H2 norm of the system is defined as sup

∆∈∆

S(M; ∆)2

2,¯ ω = 1

2π sup

∆∈∆

¯

ω −¯ ω

Tr {S(M; ∆)∗S(M; ∆)} dν

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Robust H2 theory

The system S(M, ∆) has robust H2 norm less than γ2 if there exist Hermitian matrices Wo, P+, P− > 0, X > 0 (X commuting with ∆) , satisfying      AP− + P−AT + BqXBT

q

P−CT + BqXDT CP− + DXBT

q

DXDT −

X

I



    < 0     AP+ + P+AT + BqXBT

q

P+CT + BqXDT CP+ + DXBT

q

DXDT −

X

I



   < 0   Wo I I P+ − P−   > 0 Tr

BT

wWoBw

< γ2

(SDP)

[Paganini, ACC’97]

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Main result

Let Wo, P+, P−, X, be the solution of (SDP). Then, an upper bound to the robust finite-frequency H2 norm is given by sup

∆∈∆

S(M; ∆)2

2,¯ ω ≤ Tr

BT

w(WoL(¯

ω) + L(¯ ω)∗Wo)Bw

where

L(¯ ω) = j 2π ln

(A + j¯

ωI)(A − j¯ ωI)−1 with A = A + (P−CT + BqXDT )R−1C and R =   X I   − DXDT

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Skecth of the proof

From standard robust H2, (SDP) guarantees that there exists Y (jω)

such that S(M, ∆)(jω)∗S(M, ∆)(jω) ≤ Y (jω), ∀ω, ∀∆ ∈ ∆

Y (jω) admits a spectral factorization Y = (N −1M2)∗(N −1M2), whose

state space realization is given by N −1M2 =   A + (P−CT + BqXDT )R−1C Bw R− 1

2 C

  and Wo is its (standard) observability Gramian

The finite-frequency observability Gramian of the spectral factor N −1M2

is given by WoL(¯ ω) + L(¯ ω)∗Wo

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Extension to dynamic scaling

Choosing constant scaling matrices X in (SDP) leads in general to conservative upper bounds Standard robust H2 theory exploits dynamic scaling matrices of the form X(s) =

Cψ(sI − Aψ)−1 I
X
Cψ(sI − Aψ)−1 I

∗ An upper bound to the robust finite-frequency H2 norm is obtained by solving a slightly more involved SDP (details in CDC 2010 paper)

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Numerical example

A =          −2.5 0.5 −50 −1 0.5 −0.5 −5 100 −100          Bq =          0.25 −0.5          Bw =          5 5            Cp Cz   =     1 1       Dpq Dzq   =     1     ∆(δ) = δI2, − 1 ≤ δ ≤ 1

Exact robust H2 norm: γ2 = 1.5311, attained for δ = 0.25 Exercise: compute the robust finite-frequency H2 norm for ω = 50 rad/s (true value: 0.8919)

Robust Control Workshop – Udine, January 24-26, 2011

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10

−2

10 10

2

0.2 0.4 0.6 0.8 1

|G(jω)|2 ← ¯ ω = 50 ω [rad/s]

10

−2

10 10

2

10

4

−1 −0.5 0.5 1 0.5 1 1.5

S(M, ∆)2

2,¯ ω

δ ¯ ω [rad/s]

Gain plots for different values of δ Finite-frequency H2 norm for different values of ¯ ω and δ

Robust Control Workshop – Udine, January 24-26, 2011

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10

−2

10 10

2

0.5 1 1.5 2

G(jω)2

2,¯ ω

← ¯ ω = 50 ¯ ω [rad/s] sup∆∈∆ S(M, ∆)2

2,¯ ω

nψ = 0 nψ = 1 nψ = 2

Robust finite-frequency H2 norm

Robust Control Workshop – Udine, January 24-26, 2011

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Comparison with low-pass filtering

An approximation of the robust finite-frequency H2 norm can be obtained by cascading the system with a low-pass filter F(s) =

¯ pm (s+¯ p)m and then

computing the standard robust H2 norm of the resulting system

m 1 2 3 4 5 nψ = 0 2.0216 1.9660 1.9652 1.9649 1.9648 1.9648 nψ = 1 1.8836 1.8254 1.8117 1.8049 1.8093 1.8073 nψ = 2 1.8482 1.8002 1.7997 1.7994 1.7993 1.7994 nψ = 3 1.8421 1.7939 1.7929 1.7930 1.7930 1.7918 Robust H2 norm for the system with low-pass filter nψ 1 2 3 G2

2,50

1.2631 1.2233 1.1953 1.1931 Robust finite-frequency H2 norm

Robust Control Workshop – Udine, January 24-26, 2011

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CFCL application

Model of a civil aircraft including both rigid and flexible body dynamics δ: fuel tanks level, normalized in the range [−1, 1] Resulting uncertain system (including approximated Von Karman filter modeling the wind spectrum and output filters for the specified position): LFR with 21 states and ∆ block of size 14 Model derived in the frequency range between 0 and 15 rad/s (no physical meaning outside this range) Proposed solution: robust finite-frequency H2 norm with ¯ ω = 15 rad/s, computed in Np partitions of the uncertainty interval [−1, 1] (constant scaling)

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10

−2

10

−1

10 10

1

10

2

10

3

0.5 1 1.5 2 2.5 3 3.5

|G(jω)|2 ¯ ω = 15→ ω [rad/s]

Gain plots for different values of δ

Robust Control Workshop – Udine, January 24-26, 2011

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−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7

S(M, ∆)2

2,¯ ω

¯ ω = 15 Np = 200 ¯ ω = 15 Np = 100 ¯ ω = 15 Np = 50 ¯ ω = 15 Np = 20

gridding + integration

δ

Robust finite-frequency H2 analysis (pos. #12)

Robust Control Workshop – Udine, January 24-26, 2011

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−1 −0.5 0.5 1 1 2 3 4 5 6 7

S(M, ∆)2

2,¯ ω

robust, ¯ ω = +∞ gridding, ¯ ω = +∞ robust, ¯ ω = 15 gridding, ¯ ω = 15 δ

Robust finite-frequency H2 analysis (Np = 20, pos. #4)

Robust Control Workshop – Udine, January 24-26, 2011

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Final remarks

Proposed techniques and relaxations managed to solve most COFCLUO clearance problems Lyapunov techniques can handle mixed LTI/LTV uncertainties, but are time consuming Relaxations allow one to address the key trade off between performance and computational burden No a priori hierarchy among the relaxations Good matching with worst-case analysis

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Ongoing work

Combine techniques for LTI uncertainties with recent work on memoryless nonlinearities to cope with saturations and rate limiters in actuators (IFAC talk FrB06.1) Comfort: alternative approach based on µ-analysis and adaptive partitioning of frequency domain (preliminary results in IFAC talk TuA06.2) Feedback form industry

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Those who are interested can find much more in:

Optimization Based Clearance of Flight Control Laws

A. Varga, A. Hansson, G. Puyou (Eds.)

Springer, LNCIS vol. 416, to appear in September 2011 .................................................................................thanks!

Robust Control Workshop – Udine, January 24-26, 2011