Some results related to modeling systems with uncertainty Neskolko - PowerPoint PPT Presentation

Some results related to modeling systems with uncertainty Neskolьko rezulьtatov ob modelirovanie sistem s neopredel¨ ennost�mi Dimitri PEAUCELLE - Dmitri i Жanoviq Poselь LAAS-CNRS - Université de Toulouse - FRANCE Sankt-Peterburg Dekabrь 2012

Introduction ■ Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 1

❶ Motivations for uncertain descriptor modeling ■ Models issued from physics are naturally in descriptor form E ˙ x = Ax + Bu ● Example: mechanical system M ¨ θ + C ˙ θ + Kθ = u M : inertia ; C : friction ; K : stiffness ; u external forces           ¨ ˙  − C − K  M O θ θ  I  =  +  u     ˙ O I θ I O θ O ● Example: robotic systems with Lagrange formulations [MG89] E may not be invertible ● Example: networks of systems with algebraic constraints describing links Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 2

❶ Motivations for uncertain descriptor modeling ■ Descriptor models can be converted to usual models ● Example: mechanical system θ + M − 1 C ˙ ¨ θ + M − 1 Kθ = M − 1 u ● Assumes that M − 1 is well-conditioned and known ● If some parameters are unknown: M (∆) , C (∆) , K (∆) θ + M − 1 (∆) C (∆) ˙ ¨ θ + M − 1 (∆) K (∆) θ = M − 1 (∆) u ● Increased complexity of the model ■ Descriptor models are preferable for describing systems with uncertainty Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 3

❶ Motivations for uncertain descriptor modeling ■ Example: DC motor I ˙ ω = bu regulated in speed u = − ω ● Parameters are assumed uncertain I = 1 + δ 1 , b = 1 + δ 2 ω = − 1 + δ 2 (1 + δ 1 ) ˙ ω = − (1 + δ 2 ) ω ⇒ ˙ ω 1 + δ 1 ● Model is rational w.r.t. uncertainties: exists an LFT representation ∆ w z ∆ ∆ Σ    A B ∆ − 1 + δ 2 = A + B ∆ ∆( I − D ∆ ∆) − 1 C ∆ =  ⋆ ∆ 1 + δ 1 C ∆ D ∆ ▲ Can be build with Robust Control toolbox of Matlab or LFRT [Mag05] Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 4

❶ Motivations for uncertain descriptor modeling ω = − 1+ δ 2 ● Building the LFT for ˙ 1+ δ 1 ω ▲ 1st step: descriptor form with no denominators ω + δ 1 ˙ ˙ ω = − ω − δ 2 ω ▲ 2nd step: all multiplications correspond to a feedback w 1 = δ 1 z 1 w 1 = δ 2 z 2 ω + w 1 = − ω − w 2 ˙ : , z 1 = ˙ z 2 = ω ω � � � � � � 0 δ 1 z 1 w 1 ▲ 3rd step: descriptor LFT: ∆ = , z ∆ = , w ∆ = , 0 δ 2 z 2 w 2     � � � � 1 0 0 − 1 − 1 − 1 ˙ ω ω = , w ∆ = ∆ z ∆  − 1 1 0   0 0 0  z ∆ w ∆ 0 0 1 1 0 0 ▲ last step: invert the left-hand side matrix Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 5

❶ Motivations for uncertain descriptor modeling ω = − 1+ δ 2 ● LFT for ˙ 1+ δ 1 ω − 1     1 0 0 − 1 − 1 − 1    δ 1 0 − 1+ δ 2     1+ δ 1 =  ⋆ − 1 1 0 0 0 0      0 δ 2    0 0 1 1 0 0   − 1 − 1 − 1   0  δ 1   =  ⋆ − 1 − 1 − 1    0 δ 2  1 0 0 − 1            δ 1 0  − 1 − 1  δ 1 0  − 1 � � = − 1 +  I − − 1 − 1      0 0 0 0 1 δ 2 δ 2 − 1      1 + δ 1  − 1 δ 2 � � = − 1 − δ 1 δ 2   0 1 1    − 1+ δ 2 � �  = − 1 + δ 1 − δ 2 1+ δ 1 = − 1+ δ 2 1+ δ 1 = − 1 − δ 1 δ 2 1+ δ 1 1 ■ The example shows the interest of descriptor models, even if only for technical manipulations Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 6

❶ Motivations for uncertain descriptor modeling ■ All fractional models have affine descriptor representations [MAS03] ▲ Proof: All fractional models can be converted to an LFT        ˙ x  A B ∆  x  =  , w ∆ = ∆ z ∆  z ∆ C ∆ D ∆ w ∆ ▲ the LFT gives the affine descriptor form:       − B ∆ ∆  ˙  I x  A  =  x  I − D ∆ ∆ O z ∆ C ∆ ● Can give representations of smaller dimensions ▲ Example       1 δ 1 δ 2 ω ˙ − 1  =  ω ⇔ (1 + δ 1 ) ˙ ω = − (1 + δ 2 ) ω  0 1 + δ 1    − 1 δ 2 z 1 0 0 1 1 z 2 Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 7

❶ Motivations for uncertain descriptor modeling ■ General descriptor models E xx ˙ x + E xπ π = Ax + Bu y + E xy ˙ x + E yπ π = Cx + Du ● x : state ; u : inputs ● π : linearly constrained signals ● E xx and A may not be square   x ˙ E ˙ A square and ˙ ▲ Can be converted to ˆ x = ˆ x + ˆ Bu with ˆ E and ˆ ˆ A ˆ x = ˆ   π λ ▲ Not recommend: increased size of the model Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 8

Outline ❶ Motivations for uncertain descriptor modeling ❷ Affine polytopic models ❸ LFT models - frequency dependent models ❹ Augmented descriptor models and conservatism reduction Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 9

❷ Affine polytopic models ■ All fractional models have affine descriptor representations E xx (∆) ˙ x + E xπ (∆) π = A (∆) x + B (∆) u y + E xy (∆) ˙ x + E yπ (∆) π = C (∆) x + D (∆) u ● Models also used for polynomial non-linear models [CTF02] ■ General affine descriptor data    − E xx (∆) − E xπ (∆) A (∆) B (∆)  = M (∆) − E yx (∆) − E yπ (∆) C (∆) D (∆) ● Different classes of affine models ▲ intervals: M � M (∆) � M (element-wise m ij ≤ m ij (∆) ≤ m ij ) ▲ parallelotopes: M (∆) = M | 0 | + � ¯ p p =1 δ p M | p | : | δ p | ≤ 1 � v ] � M [ v =1 ... ¯ ▲ polytopes: M (∆) ∈ co ● intervals ⊂ parallelotopes ⊂ polytopes Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 10

❷ Affine polytopic models x2eq x1 ■ Example: two mass spring system k u M1 w2 M2 w1 f M 1 ¨ x 1 + f ( ˙ x 1 − ˙ x 2 ) + k ( x 1 − x 2 ) = u + w 1 M 2 ¨ x 2 + f ( ˙ x 2 − ˙ x 1 ) + k ( x 2 − x 1 ) = w 2 ● General affine data model   x 1 ¨ x 2 ¨     � �   u + w 1  M 1 0 f − f k − k ˙ x 1   =    ˙ x 2 0 − f − k M 2 f k w 2     x 1 � �� � = M (∆) x 2 Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 11

❷ Affine polytopic models ■ Example: two mass spring system - continued ● 4 uncertain parameters M i ≤ M i ≤ M i , f ≤ f ≤ f , k ≤ k ≤ k � � M 1 0 f − f k − k M (∆) = 0 − f − k M 2 f k ● Interval model generates conservatism (elements assumed independent)   0 − f − k  M 1 f k  0 M 2 − f f − k k   0 − f − k  M 1 f k �  0 − f − k M 2 f k    M 1 0 f − f k − k �  0 − f − k M 2 f k Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 12

❷ Affine polytopic models ■ Example: two mass spring system - continued � � 0 − f − k M 1 f k M (∆) = 0 M 2 − f f − k k ● Parallelotopic model ▲ Nominal model: M | 0 | (center of the intervals)    1 0 0 0 0 0 M (∆) = M | 0 | + δ M 1 1 2 ( M 1 − M 1 )  0 0 0 0 0 0 . . .    0 0 0 0 1 − 1 1 + δ k 2( k − k )  0 0 0 0 − 1 1 � �� � M | k | ▲ M | k | "axis" of variations along δ k . Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 13

❷ Affine polytopic models ■ Example: two mass spring system - continued � � 0 − f − k M 1 f k M (∆) = 0 − f − k M 2 f k ● Polytopic model: 2 4 = 16 vertices ( δ i = ± 1 in parallelotopic model) � � 0 − f − k M 1 f k M [1] = 0 − f − k M 2 f k � � M 1 0 f − f k − k M [2] = 0 − f − k M 2 f k . . . � � 0 − f − k M 1 f k M [16] = 0 − f − k M 2 f k Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 14

❷ Affine polytopic models ■ Some techniques to deal with affine uncertain models ● General type of analysis criteria η T Θ(∆) η < 0 , ∀ η � = 0 : M (∆) η = 0 ▲ Example: negative definite derivative of Lyapunov function V = x T P (∆) x � � � � � � O P (∆) x ˙ ˙ V = η T η < 0 , ∀ η = � = 0 : η = 0 − I A (∆) P (∆) O x ( M ⊥ (∆) matrix generating the null-space of M (∆) ) ● Usual LMI type result M ⊥ T (∆)Θ(∆) M ⊥ (∆) < O ▲ Problem: M ⊥ (∆) is a rational function of ∆ ▲ Solutions: SOS [HG05, Sch06], Polyá [CGTV09, OdOP08] Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 15

❷ Affine polytopic models ■ Some techniques to deal with affine uncertain models ● General type of analysis criteria η T Θ(∆) η < 0 , ∀ η � = 0 : M (∆) η = 0 ( M ⊥ (∆) matrix generating the null-space of M (∆) ) ● Usual LMI type result M ⊥ T (∆)Θ(∆) M ⊥ (∆) < O ● Slack variables results (variables issued from the "creation" Finsler lemma) Θ(∆) < FM (∆) + M T (∆) F T ∃ F : ▲ Conservative, but sufficient to test at the vertices ▲ Θ(∆) affine (parameter-dependent Lyapunov function) ▲ [OBG99, PABB00, EH04, PDSV09] Sankt-Peterburg, Dekabrь 2012 D. Peaucelle 16