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
Universit` a di Siena 1 COFCLUO project Clearance Of Flight Control Laws Using Optimization Funded by EC, 2007-2010 Partners: - Link¨ oping University (Svezia) - AIRBUS (Francia) - DLR (Germania) - ONERA (Francia) - FOI (Svezia) - Universit` a di Siena (Italy) Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 2 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 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 3 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
Universit` a di Siena 4 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) Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 5 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¨ oping) Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 6 Robust stability of LFR uncertainty models Consider the LFR system A + B ∆( θ )( I − D ∆( θ )) − 1 C � � Σ : x ( t ) = A ( θ ) x ( t ) = ˙ x ( t ) where ∆( θ ) = diag ( θ 1 I s 1 , . . . , θ n θ I s nθ ) An equivalent representation of Σ is given by: x = Ax + Bq ˙ Σ : , p = Cx + Dq q = ∆( θ ) p where x ∈ R n , q, p ∈ R d and d = � n θ i =1 s i . Assumptions: θ ∈ Θ = [ θ 1 , θ 1 ] × · · · × [ θ n θ , θ n θ ] with 2 n θ vertices Ver [Θ] ˙ θ ( t ) = 0 (time-invariant uncertainty) Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 7 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 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 8 [Dettori & Scherer ’00] If there exist: a symmetric Lyapunov matrix P ( θ ) ∈ R n × n , multiaffine in θ two matrices S 0 , S 1 ∈ R d × d such that ∀ θ ∈ Ver [Θ] P ( θ ) > 0 T I 0 I 0 0 P ( θ ) 0 A B A B < 0 , P ( θ ) 0 0 0 I 0 I 0 0 W ( θ ) C D C D where S 1 + S T − S 0 − S 1 ∆( θ ) 1 , W ( θ ) = − S T 0 − ∆( θ ) S T S T 0 ∆( θ ) + ∆( θ ) S 0 1 then the system Σ is robustly stable. Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 9 Remarks: • multiaffine LF n θ n θ n θ � � � � � V ( x ; θ ) = x T P 0 + θ j P j + θ i θ j P ij + · · · x j =1 i =1 j = i +1 • parameter-dependent multiplier W ( θ ) , parameterized by S 0 , S 1 • 2 n θ LMIs of dimension ( n + d ) , 2 n θ LMIs of dimension n • 2 d 2 + 2 n θ n ( n + 1) free variables 2 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 10 [Fu & Dasgupta ’01] Let T i = blockdiag (0 s 1 , . . . , 0 s i − 1 , I s i , 0 s i +1 , . . . , 0 s nθ ) , C i = T i C , D i = T i D for i = 1 , . . . , n θ , and D 0 = − I . If there exist 2 n θ + 2 matrices C µ,i ∈ R d × n , D µ,i ∈ R d × d , i = 0 , . . . , n θ , s.t. � � � � � � C T C T � � i µ,i + ≤ 0 , i = 1 , . . . , n θ C µ,i D µ,i C i D i D T D T i µ,i and a symmetric P ( θ ) ∈ R n × n , multiaffine in θ , such that ∀ θ ∈ Ver [Θ] P ( θ ) > 0 A T ( θ ) P ( θ ) + P ( θ ) A ( θ ) Π( θ ) < 0 , Π T ( θ ) D µ ( θ ) D − 1 ( θ ) + D − T ( θ ) D T � � − µ ( θ ) µ ( θ ) + C T ( θ ) D − T ( θ ) D T P ( θ ) BD − 1 ( θ ) − C T � � where Π( θ ) = µ ( θ ) C ( θ ) = ∆( θ ) C , D ( θ ) = ∆( θ ) D − I , C µ ( θ ) = C µ, 0 + � n θ D µ ( θ ) = D µ, 0 + � n θ i =1 θ i C µ,i , i =1 θ i D µ,i then the system Σ is robustly stable. Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 11 Remarks: • multiaffine LF • parameter-dependent multipliers C µ ( θ ) and D µ ( θ ) , parameterized by C µ,i , D µ,i , i = 0 , . . . , n θ • n θ + 2 n θ LMIs of dimension ( n + d ) , 2 n θ LMIs of dimension n • ( n θ + 1)( nd + d 2 ) + 2 n θ n ( n + 1) free variables 2 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 12 [Wang & Balakrishnan ’02] If there exist: n θ + 1 symmetric matrices Q 0 , . . . , Q n θ ∈ R n × n a symmetric scaling matrix N ∈ R d × d such that, ∀ θ ∈ Ver [Θ] , N > 0 Q ( θ ) = Q 0 + � n θ j =1 θ j Q j > 0 AQ ( θ ) + Q ( θ ) A T + B ∆( θ ) N ∆( θ ) B T Q ( θ ) C T + B ∆( θ ) N ∆( θ ) D T � � < 0 Q ( θ ) C T + B ∆( θ ) N ∆( θ ) D T � T � − N + D ∆( θ ) N ∆( θ ) D T then the system Σ is robustly stable. Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 13 Remarks: • candidate LF � − 1 n θ � � V ( x ) = x T Q 0 + θ j Q j x j =1 • 2 n θ LMIs of dimension ( n + d ) , 2 n θ LMIs of dimension n , 1 LMI of dimension d • d ( d + 1) + ( n θ + 1) n ( n + 1) free variables 2 2 • easily extended to slowly time-varying parameters • generalized to polynomial Lyapunov functions [Chesi et al., ’04] Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 14 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 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 15 Relaxations • Lyapunov function: multiaffine (mapdlf), affine (apdlf), constant (clf) • Multipliers: affine, constant, diagonal • Scalings: constant, diagonal Relaxation Characteristics FD-c µ FD method with constant multipliers C µ, 0 , D µ, 0 FD-cd µ FD method with constant diagonal multipliers C µ, 0 , D µ, 0 DS-dS DS method with diagonal multipliers S 0 , S 1 WB-dN WB method with diagonal scaling matrix N Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 16 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. Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 17 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 Robust Control Workshop – Udine, January 24-26, 2011
Universit` a di Siena 18 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) Robust Control Workshop – Udine, January 24-26, 2011
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