Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation Dimitri PEAUCELLE & Didier HENRION ‡ & Denis ARZELIER all with LAAS-CNRS - Toulouse, FRANCE ‡ also with Czech Technical University in Prague
Problem statement Well-posedness of interconnected system + A , E possibly non square z w ∇ ∈ ∇ ∇ uncertain z + w Outline ① Well-posedness and topological separation ② Robust stability of descriptor systems ③ Main result ④ Some results for robust analysis ⑤ Numerical example 1 GT MOSAR 9-10 juin 2005
Topological separation General framework w G (z, w)=0 w Well-Posedness: z w Bounded ( ¯ w, ¯ z ) ⇒ unique bounded ( w, z ) F (w, z)=0 z z ∃ θ topological separator: F (¯ z ) = { ( w, z ) : F ¯ z ( w, z ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) > − φ 1 ( || ¯ z || ) } G I ( ¯ w ) = { ( w, z ) : G ¯ w ( z, w ) = 0 } ⊂ { ( w, z ) : θ ( w, z ) ≤ φ 2 ( || ¯ w || ) } Related results : ➞ Stability ( θ Lyapunov certificate), Passivity ( θ storage function), IQC ... ➞ Robust analysis of Linear uncertain systems (Iwasaki) 2 GT MOSAR 9-10 juin 2005
Well-posedness in the considered case + Well-posedness: z w Null ( E − A∇ ) is empty ∀∇ ∈ ∇ ∇ z + w ✪ Includes classical µ theory framework: I − A∇ non-singular for all (structured) norm-bounded ∇ ✪ Results of the paper extend to block diagonal uncertainties: ∇ = diag ( δ R 1 I r 1 , . . . , δ R N R I r N R , δ C 1 I c 1 , . . . , δ C N C I c N C , ∆ C 1 , . . . , ∆ C N f ) 3 GT MOSAR 9-10 juin 2005
Well-posedness and descriptor systems ⇔ Es − A full rank for all s ∈ C + Stability of E ˙ x = Ax w = ∇ z for all ∇ = s − 1 I n ∈ C + ⇔ W.P . of Ez = Aw + x x + 4 GT MOSAR 9-10 juin 2005
Well-posedness and pole location Let a region defined as a half plane or a disk 2 s ∗ + d 3 ss ∗ ≤ 0 } D = { s ∈ C : d 1 + d 2 s + d ∗ D -Stability of E ˙ x = Ax ⇔ Es − A full rank for all s ∈ D w = ∇ z ⇔ W.P for all ∇ = ∇ ∇ . of Ez = Aw ∇ = { s − 1 I n : d 1 s − 1 s −∗ + d 2 s −∗ + d ∗ 2 s − 1 + d 3 ≥ 0 } . where ∇ 5 GT MOSAR 9-10 juin 2005
Well-posedness and polynomial descriptor systems Stability of A d x ( d ) + A d − 1 x ( d − 1) + · · · + A 1 ˙ x + A 0 x = 0 w = ∇ z for all ∇ = s − 1 I d n ∈ C + ⇔ W.P . of E z = A w · · · · · · A d O O A d − 1 A 1 A 0 − I O O I O O where E = A = − . . ... ... . . . . O O − I O I O 6 GT MOSAR 9-10 juin 2005
Robust stability of descriptor systems Robust stability ∀ ∆ ∈ ∆ ∆ of E (∆) ˙ x = A (∆) x with E (∆) = E A + ( B ∆ − E B )( E D − D ∆) − 1 E C A (∆) = A + ( B ∆ − E B )( E D − D ∆) − 1 C s − 1 ∈ C + s − 1 I n w = ∇ z O ⇔ W.P for all ∇ = : . of E z = A w O ∆ ∆ ∈ ∆ ∆ E A E B A B where E = A = E C E D C D 7 GT MOSAR 9-10 juin 2005
Main result: Quadratic separation w = ∇ z for all ∇ = ∇ ∇ W.P . of E z = A w � � I E ◦∗ ∇ ∗ ≤ O , ∀∇ ∈ ∇ ∇ Θ I ∇E ◦ ⇔ ∃ Θ : � � ⊥∗ � � ⊥ EE ◦ EE ◦ > O . Θ −A −A where the columns of E ◦ form an orthogonal basis of E ∗ � � ⊥ � � EE ◦ EE ◦ −A −A and the columns of span the null-space of . 8 GT MOSAR 9-10 juin 2005
Application to descriptor systems w = ∇ z for all ∇ = s − 1 I n ∈ C + . Stability of E ˙ x = Ax ⇔ W.P . of Ez = Aw Choice of separator: Θ � �� � − E ◦∗ P � � O I = − 2 Re ( s − 1 ) E ◦∗ PE ◦ E ◦∗ s −∗ I E ◦ s − 1 − PE ◦ O E ◦∗ PE ◦ > O LMI result ❶ : E ◦∗ P � � ⊥∗ O � � ⊥ EE ◦ EE ◦ < O − A − A PE ◦ O E ∗ X ∗ = XE ≥ O , A ∗ X ∗ + XA < O Equivalent to ❷ : 9 GT MOSAR 9-10 juin 2005
Example of descriptor system 1 1 ˙ x = x Scalar ’switch’ system: 0 α ● x ( t ) = 0 the system is stable If α � = 0 � � ⊥ ❶ EE ◦ = [ ] → LMI p > 0 − A � � ❷ X = → LMI x 1 ≥ 0 , 2 x 1 + 2 x 2 α < 0 x 1 x 2 ● ˙ If α = 0 x = x the system is unstable 1 � � ⊥ ❶ = → LMI p > 0 , 2 p < 0 EE ◦ − A 1 � � ❷ X = → LMI x 1 ≥ 0 , 2 x 1 < 0 x 1 x 2 10 GT MOSAR 9-10 juin 2005
Application to uncertain descriptor systems : | δ | ≤ ¯ Robust Stability of E (∆) ˙ x = A (∆) x , ∆ = δ I δ s − 1 ∈ C + s − 1 I n w = ∇ z O ⇔ for all ∇ = : W.P . of . | δ | ≤ ¯ E z = A w O δ I m δ Choice of separator: E ◦∗ 2 Θ 1 E ◦ −E ◦∗ E ◦∗ 1 P 2 Θ 2 2 s − 1 E ◦ : ∇E ◦ = 1 Θ = − P E ◦ O O 1 ∆ E ◦ 2 Θ 2 ∗ E ◦ O Θ 3 2 E ◦∗ 1 P E ◦ E ◦∗ 2 Θ 3 E ◦ 1 ≥ O , 2 ≥ O δ Θ 2 ∗ + ¯ 2 ( Θ 1 + ¯ δ Θ 2 + ¯ E ◦∗ δ 2 Θ 3 ) E ◦ 2 ≤ O constrained by the LMIs δ Θ 2 ∗ + ¯ 2 ( Θ 1 − ¯ δ Θ 2 − ¯ δ 2 Θ 3 ) E ◦ E ◦∗ 2 ≤ O 11 GT MOSAR 9-10 juin 2005
Reducing conservatism ◆ Original system s − 1 I n ˙ E A E B x A B x O = , ∇ = E C E D z ∆ C D w ∆ O δ I m ❖ Augmented system O I O O I O O O ¨ ˙ x x O E A E B O O A B O ˙ x x = O E C E D O O C D O z ∆ w ∆ E A O O E B A O O B ˙ ˙ z ∆ w ∆ E C O O E D C O O D s − 1 I 2 n O , ∇ = O δ I 2 m 12 GT MOSAR 9-10 juin 2005
Reducing conservatism ◆ Quadratic separation on Original system ➞ Lyapunov stability with V ( η, ∆) = η ∗ Pη ❖ Quadratic separation on Augmented system ➞ Lyapunov stability with V ( η, ∆) = η ∗ P (∆) η where in the case E = I (not descriptor) ∗ A (∆) A (∆) P (∆) = P I I 13 GT MOSAR 9-10 juin 2005
Robust stability analysis example − 16 0 0 8 0 0 0 0 3 0 0 0 − 8 − 16 0 0 0 0 4 8 0 9 2 0 − 19 − 17 0 0 10 9 0 0 0 0 0 0 E = , A = − 20 10 0 0 0 0 0 0 0 1 4 0 − 10 0 3 0 0 0 5 0 1 0 2 0 − 6 − 6 3 1 0 0 3 0 3 0 7 0 ◆ Quadratic separation on Original system ➞ LMIs feasible up to ¯ δ = 0 . 22 - infeasible for ¯ δ = 0 . 23 ❖ Quadratic separation on Augmented system ➞ LMIs feasible up to ¯ δ = 0 . 45 - infeasible for ¯ δ = 0 . 46 14 GT MOSAR 9-10 juin 2005
Conclusions ➞ Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation is valuable for extensions of known results to de- scriptor systems. ➞ Applying existing methods to artificially augmented systems gives new less conservative results (more about this in Seville) 15 GT MOSAR 9-10 juin 2005
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