Robustness analysis techniques for clearance of flight control laws

Transcription

1 Università di Siena Robustness analysis techniques for clearance of flight control laws Andrea Garulli, Alfio Masi, Simone Paoletti, Ercüment Türko glu Dipartimento di Ingegneria dell Informazione, Università di Siena Robust Control Workshop Udine, August 24-26, 2011

2 Università di Siena 1 COFCLUO project Clearance Of Flight Control Laws Using Optimization Funded by EC, Partners: - Linköping University (Svezia) - AIRBUS (Francia) - DLR (Germania) - ONERA (Francia) - FOI (Svezia) - Università di Siena (Italy)

3 Università 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

4 Università 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)

5 Università 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)

6 Università 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 (Linköping)

7 Università di Siena 6 Robust stability of LFR uncertainty models Consider the LFR system Σ : ẋ(t) = A(θ)x(t) = [ A+B (θ)(i D (θ)) 1 C ] x(t) where (θ) = diag(θ 1 I s1,...,θ nθ I snθ ) An equivalent representation of Σ is given by: ẋ = 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 nθ,θ nθ ] with 2 n θ vertices Ver[Θ] θ(t) = 0 (time-invariant uncertainty)

8 Università 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

9 Università 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 I 0 A B 0 I C D T 0 P(θ) 0 P(θ) W(θ) I 0 A B 0 I C D < 0, where S W(θ) = 1 +S1 T S 0 S 1 (θ) S0 T (θ)s1 T S0 T (θ)+ (θ)s 0 then the system Σ is robustly stable.,

10 Università di Siena 9 Remarks: multiaffine LF V(x;θ) = x T (P 0 + n θ j=1 θ j P j + n θ i=1 n θ j=i+1 θ i θ j P ij + ) x 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 2d 2 +2 n θ n(n+1) 2 free variables

11 Università di Siena 10 [Fu & Dasgupta 01] Let T i = blockdiag(0 s1,...,0 si 1,I si,0 si+1,...,0 snθ ), C i = T i C, D i = T i D for i = 1,...,n θ, and D 0 = I. If there exist 2n θ +2 matrices C µ,i R d n, D µ,i R d d, i = 0,...,n θ, s.t. [ ] [ ] Ci T [ ] C T [ ] µ,i Cµ,i Dµ,i + Ci Di 0, i = 1,...,n θ D T i D T µ,i and a symmetric P(θ) R n n, multiaffine in θ, such that θ Ver[Θ] P(θ) > 0 AT (θ)p(θ)+p(θ)a(θ) Π(θ) Π T (θ) ( D µ (θ)d 1 (θ)+d T (θ)dµ(θ) T ) where Π(θ) = ( P(θ)BD 1 (θ) C T µ(θ)+c T (θ)d T (θ)d T µ(θ) ) C(θ) = (θ)c, D(θ) = (θ)d I, C µ (θ) = C µ,0 + n θ i=1 θ ic µ,i, D µ (θ) = D µ,0 + n θ i=1 θ id µ,i then the system Σ is robustly stable. < 0,

12 Università 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) 2 free variables

13 Università di Siena 12 [Wang & Balakrishnan 02] If there exist: n θ +1 symmetric matrices Q 0,...,Q nθ 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 jq j > 0 AQ(θ)+Q(θ)AT +B (θ)n (θ)b T ( Q(θ)C T +B (θ)n (θ)d T) ( Q(θ)C T +B (θ)n (θ)d T) T N +D (θ)n (θ)d T < 0 then the system Σ is robustly stable.

14 Università di Siena 13 Remarks: candidate LF V(x) = x T (Q 0 + n θ j=1 θ j Q j ) 1x 2 n θ LMIs of dimension (n+d), 2 n θ LMIs of dimension n, 1 LMI of dimension d d(d+1) 2 +(n θ +1) n(n+1) 2 free variables easily extended to slowly time-varying parameters generalized to polynomial Lyapunov functions [Chesi et al., 04]

15 Università 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

16 Università di Siena 15 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 C µ,0, D µ,0 FD-cdµ 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

17 Università 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.

18 Università 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

19 Università 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)

20 Università di Siena 19 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

21 Università di Siena 20

22 Università di Siena 21 Example of GUI output 1 (Ma,Vc) Vc Ma Green tiles: cleared Red tiles: unstable (contain unstable models found by gridding) White tiles: unknown (not cleared and no unstable models)

23 Università di Siena 22 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).

24 Università di Siena 23 New version available A new version of the GUI is available at: garulli/lfr rai/... try it! (any feedback is welcome)

25 Università di Siena 24 Closed-loop integral longitudinal models Uncertain parameters and trim flight variables Symbol Description Nominal value C central tank 0.5 O outer tank 0 P payload 0.5 X center of gravity 0 M Mach number 0.86 V calibrated air speed 310 kt

26 Università di Siena 25 LFR models for aeroelastic stability Models representative of closed-loop aeroelastic longitudinal dynamics in frequency range [0,15] rad/sec. Model n d θ 1, s 1 θ 2, s 2 θ 3, s 3 θ 4, s 4 C C, 16 CX C, 14 X, 4 OC C, 26 O, 24 OCX C, 24 O, 22 X, 4 POC C, 42 O, 24 P, 13 MV M, 26 V, 28 Parameters not appearing in block are set to the nominal values

27 Università di Siena 26 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

28 Università di Siena 27 Aeroelastic stability results: C and CX models C model: progressive tiling CX model: progressive tiling Method (lf) Cleared NOPs t(sec) Method (lf) Cleared NOPs t(sec) DS (clf) DS (apdlf) DS-dS (clf) DS-dS (apdlf) FD-cµ (clf) FD-cµ (apdlf) FD-cdµ (clf) FD-cdµ (apdlf) WBQ-dM DS (clf) DS (apdlf) DS-dS (clf) DS-dS (apdlf) FD-cµ (clf) FD-cµ (apdlf) FD-cdµ (clf) FD-cdµ (apdlf) WBQ-dM Cleared: rate of uncertainty domain cleared NOPs: number of LMI tests performed

29 Università di Siena 28 Aeroelastic stability results: OC and OCX models OC model: progressive tiling Method (lf) Cleared NOPs t(h:m:s) DS-dS (clf) : 26 : 32 DS-dS (apdlf) : 31 : 51 FD-cµ (clf) : 55 : 30 FD-cµ (apdlf) : 06 : 11 FD-cdµ (clf) : 22 : 00 FD-cdµ (apdlf) : 31 : 50 WBQ-dM : 14 : 48 OCX model: progressive tiling Method (lf) Cleared NOPs t(h:m:s) DS (clf) : 00 : 28 DS (apdlf) : 07 : 22 DS-dS (clf) : 10 : 09 DS-dS (apdlf) : 53 : 36 FD-cµ (clf) : 31 : 34 FD-cµ (apdlf) : 17 : 47 FD-cdµ (clf) : 52 : 25 FD-cdµ (apdlf) : 34 : 34 WBQ-dM : 34 : 12

30 Università di Siena 29 Aeroelastic stability results: POC model POC model: progressive tiling Method Cleared OPS t (h:m:s) DS-dS (clf) : 42 : 48 FD-cµ (clf) : 38 : 10

31 Università di Siena 30 Aeroelastic stability results: MV model MV model: progressive tiling Method Rate NOPs Time Time/OP (h:m:s) (h:m:s) DS-dS (apdlf) : 35 : 25 0 : 01 : 25 DS-dS (clf) : 32 : 13 0 : 00 : 35 FD-cµ (apdlf) : 09 : 45 0 : 10 : 24 FD-cµ (clf) : 00 : 29 0 : 09 : 21 FD-cdµ (apdlf) : 59 : 54 0 : 01 : 02 FD-cdµ (clf) : 28 : 02 0 : 00 : 22 Rate: clearance rate (cleared / clearable)

32 Università di Siena 31 1 (Ma,Vc) 1 (Ma,Vc) Vc 0 Vc Ma Ma DS-ds with clf DS-ds with apdlf

33 Università di Siena 32 1 (Ma,Vc) 1 (Ma,Vc) Vc 0 Vc Ma Ma FD-cµ with clf FD-cµ with apdlf

34 Università di Siena 33 1 (Ma,Vc) 1 (Ma,Vc) Vc 0 Vc Ma Ma FD-cdµ with clf FD-cdµ with apdlf

35 Università di Siena 34 Adaptive tiling MV model: adaptive tiling Method clf apdlf Rate NOPs Time (h:m:s) DS-dS : 35 : : 27 : : 20 : : 10 : 37 FD-cµ : 09 : : 19 : : 21 : : 05 : 11 FD-cdµ : 59 : : 13 : : 18 : : 39 : 18

36 Università di Siena 35 Comments apdlf conditions need to solve fewer optimization problems than clf ones 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)

37 Università di Siena 36 Comfort analysis Joint work with Anders Hansson and Ragnar Wallin (Linköping University) Comfort index formulated as H 2 performance from wind velocity to acceleration at specific points along the aircraft fuselage Need robust H 2 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

38 Università di Siena 37 Comfort index J c = 1 π ω ω W(jω) 2 T(jω;δ) 2 Φ v (ω)dω Φ v (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 H 2 analysis (upper bound to worst-case comfort performance)

39 Università di Siena 38 Robust H 2 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 H 2 norm

40 Università di Siena 39 Finite-frequency H 2 norm Given the system ẋ(t) = Ax(t)+Bw(t) z(t) = Cx(t) with transfer function G(s), the finite-frequency H 2 norm is defined as where G(jω) 2 2, ω = 1 2π W o ( ω) = 1 2π ω ω ω Tr{G(jν) G(jν)}dν = Tr ω is the finite-frequency observability Gramian { } B T W o (ω)b (jνi A) C T C(jνI A) 1 dν.

41 Università di Siena 40 Finite-frequency observability Gramian The finite-frequency observability Gramian can be computed as W o ( ω) = L( ω) W o +W o L( ω), where W o 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 H 2 theory to provide an upper bound to the robust finite frequency H 2 norm of a system with parametric uncertainty

42 Università di Siena 41 Problem formulation LFR system: ẋ(t) = Ax(t)+B q q(t)+b w w(t) p(t) = C p x(t)+d pq q(t) z(t) = C z x(t)+d zq q(t) q(t) = (δ)p(t) (δ) = diag(δ 1 I s1,...,δ nδ I snδ ) := { (δ) : 1 δ i 1} M = M A B q B w 11 M 12 = C p D pq 0 M 21 M 22 C z D zq 0 S(M; ) = M 22 +M 21 (I M 11 ) 1 M 12 The robust finite-frequency H 2 norm of the system is defined as sup S(M; ) 2 2, ω= 1 2π sup ω ω Tr{S(M; ) S(M; )}dν

43 Università di Siena 42 Robust H 2 theory The system S(M, ) has robust H 2 norm less than γ 2 if there exist Hermitian matrices W o, P +, P > 0, X > 0 (X commuting with ), satisfying AP +P A T +B q XBq T P C T +B q XD T [ ] CP +DXBq T DXD T X 0 < 0 0 I AP + +P + A T +B q XBq T P + C T +B q XD T [ ] CP + +DXBq T DXD T X 0 < 0 (SDP) 0 I W o I > 0 I P + P Tr { B T ww o B w } < γ 2 [Paganini, ACC 97]

44 Università di Siena 43 Main result Let W o, P +, P, X, be the solution of (SDP). Then, an upper bound to the robust finite-frequency H 2 norm is given by } sup S(M; ) 2 2, ω Tr {Bw(W T o L( ω)+l( ω) W o )B w where with L( ω) = j 2π ln[ (A+j ωi)(a j ωi) 1] A = A+(P C T +B q XD T )R 1 C and R = X 0 0 I DXD T

45 Università di Siena 44 Skecth of the proof From standard robust H 2, (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 1 M 2 ) (N 1 M 2 ), whose state space realization is given by N 1 M 2 = A+(P C T +B q XD T )R 1 C B w R 1 2 C 0 and W o is its (standard) observability Gramian The finite-frequency observability Gramian of the spectral factor N 1 M 2 is given by W o L( ω)+l( ω) W o

46 Università di Siena 45 Extension to dynamic scaling Choosing constant scaling matrices X in (SDP) leads in general to conservative upper bounds Standard robust H 2 theory exploits dynamic scaling matrices of the form X(s) = {[ C ψ (si A ψ ) 1 I ]} X {[ C ψ (si A ψ ) 1 I ]} An upper bound to the robust finite-frequency H 2 norm is obtained by solving a slightly more involved SDP (details in CDC 2010 paper)

47 Università di Siena 46 Numerical example A = B q = B w = C p = C z D pq D zq = (δ) = δi 2, 1 δ 1 Exact robust H 2 norm: γ 2 = , attained for δ = 0.25 Exercise: compute the robust finite-frequency H 2 norm for ω = 50 rad/s (true value: )

48 Università di Siena 47 1 G(jω) ω = 50 S(M, ) 2 2, ω ω [rad/s] δ ω [rad/s] Gain plots for different values of δ Finite-frequency H 2 norm for different values of ω and δ

49 Università di Siena 48 G(jω) 2 2, ω sup S(M, ) 2 2, ω n ψ = 0 n ψ = 1 n ψ = ω = ω [rad/s] Robust finite-frequency H 2 norm

50 Università di Siena 49 Comparison with low-pass filtering An approximation of the robust finite-frequency H 2 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 H 2 norm of the resulting system m n ψ = n ψ = n ψ = n ψ = Robust H 2 norm for the system with low-pass filter n ψ G 2 2, Robust finite-frequency H 2 norm

51 Università di Siena 50 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 H 2 norm with ω = 15 rad/s, computed in N p partitions of the uncertainty interval [ 1,1] (constant scaling)

52 Università di Siena ω = 15 G(jω) ω [rad/s] Gain plots for different values of δ

53 Università di Siena 52 S(M, ) 2 2, ω gridding + integration ω = 15 N p = 200 ω = 15 N p = 100 ω = 15 N p = 50 ω = 15 N p = δ Robust finite-frequency H 2 analysis (pos.#12)

54 Università di Siena robust, ω = + gridding, ω = + robust, ω = 15 gridding, ω = 15 S(M, ) 2 2, ω δ Robust finite-frequency H 2 analysis (N p = 20, pos.#4)

55 Università di Siena 54 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

56 Università di Siena 55 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

57 Università di Siena 56 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 thanks!