Let an uncertain LTI system : x = A ( ) x where is a notation that - PowerPoint PPT Presentation

General polynomial parameter-dependent Lyapunov functions for polytopic uncertain systems Dimitri PEAUCELLE † & Yoshio EBIHARA ‡ & Denis ARZELIER † & Tomomichi HAGIWARA ‡ † LAAS-CNRS - Toulouse, FRANCE ‡ Dpt. Electrical Engineering - Kyoto Univ., JAPAN

Introduction Robust stability in the Lyapunov context Let an uncertain LTI system : ˙ x = A ( ζ ) x where ζ is a notation that gathers all constant unknown bounded parameters. Stability is equivalent to the existence of a parameter-dependent Lyapunov function (PDLF) : V ζ ( x ) = x T P ( ζ ) x such that for all admissible uncertainties the LMIs hold P ( ζ ) > 0 , A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ ) < 0 (1) Considered case N N � � ➞ Affine polytopic systems A ( ζ ) = : ζ i ≥ 0 , ζ i = 1 ζ i A i i =1 i =1 � α j ( ζ ) P j : α j ( ζ ) = ζ j 1 1 ζ j 2 2 . . . ζ j N ➞ Polynomial PDLF (PPDLF) P ( ζ ) = N ➞ (1) is then a PPD-LMI. 1 24-28 July 2006, Kyoto

Outline ① Overview of existing techniques from the literature ”Sum-Of-Squares” / ”Positive coefficients” / ”Small-gain theorem” ➞ Large LMI problems ➞ Few results on convergence to exact robustness analysis tests ➞ Complex mathematical formulations ② Proposed approach : ”dilated LMIs” ➞ Same drawbacks ➞ Interpretations in terms of ”redundant system modeling” ③ Numerical example - robust H 2 guaranteed cost computation 2 24-28 July 2006, Kyoto

① Overview of techniques from the literature Solving PPD-LMIs such as A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ ) < 0 ? ”Sum-Of-Squares” approach [Chesi et al] - [Lasserre], [Parrilo] Express the PPD-LMI as a quadratic form of nomomials − ( A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ )) = ( α ( ζ ) ⊗ 1 ) T Q ( P )( α ( ζ ) ⊗ 1 ) is positive if SOS which is LMI problem : Q ( P ) + U ( ˜ P ) > 0 ➘ No proof of necessity ➘ Numerical construction of Q ( P ) and U ( ˜ P ) is complex ➘ Large LMIs with large number of variables ➚ Restrict to homogeneous forms to reduce the dimensions 3 24-28 July 2006, Kyoto

① Overview of techniques from the literature Solving PPD-LMIs such as A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ ) < 0 ? ”Positive coefficients” approach [Scherer], [Peres et al] - [P´ olya] As all parameters are positive ζ i ≥ 0 , N � ζ i ) d ( A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ )) = � ( α j ( ζ ) T j ( P ) i =1 is negative if all coefficient matrices are negative : T j ( P ) < 0 ➚ Proof of necessity for d large enough ( P ( ζ ) of fixed degree) ➘ Numerical construction of T j ( P ) is complex ➚ Large LMIs but no additional variables 4 24-28 July 2006, Kyoto

① Overview of techniques from the literature Solving PPD-LMIs such as A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ ) < 0 ? ”Small-gain theorem” approach [Bliman] - [Scherer], [Iwasaki] k =1 z k ˜ � m ➘ Assuming the sub-case A ( ζ ) = A 0 + A k : | z k | ≤ 1 A T ( ζ ) P ( ζ ) + P ( ζ ) A ( ζ ) = ( z { r } ⊗ 1 ) T R 1 ( P, z 2 ,...,m )( z { r } ⊗ 1 ) 1 1 it is negative for all | z 1 | ≤ 1 if there exists Q 1 ( z 2 ,...,m ) > 0 such that    Q 1 ( z 2 ,...,m ) 0  N 1 M T 1 R 1 ( P, z 2 ,...,m ) M 1 < N T 1 − Q 1 ( z 2 ,...,m ) 0 Choose Q 1 ( z 2 ,...,m ) polynomial and go on recursively with z 2 , . . . , z m . ➘ Numerical construction of the LMIs is complex ➘ Large LMIs and very large number of additional variables ➚ Proof of convergence to exact robustness test as degree of polynomials grow ➚ Extends to LFT modelling 5 24-28 July 2006, Kyoto

② Proposed ”dilated LMIs” approach Some characteristics ➘ Numerical construction of the LMIs is complex ➘ Large LMIs and large number of additional variables ➘ No proof of convergence to exact robustness test ➚ Alternative method ➚ Interpretation in terms of ”redundant system modeling” 6 24-28 July 2006, Kyoto

② Proposed ”dilated LMIs” approach Central tool : ”Finsler lemma” [Geromel 1998], [Peaucelle 2000] Stability of ˙ x = A ( ζ ) x is proved if T         P ( ζ )  x  x  x 0 ˙ � �  < 0 :  = 0 V ( x ) = A ( ζ ) − 1    ˙ P ( ζ ) ˙ ˙ x x x 0 A sufficient condition for that is the existence of G such that   � T G T < 0 P ( ζ ) 0 � � �  + G + A ( ζ ) − 1 A ( ζ ) − 1  P ( ζ ) 0 ➞ If P ( ζ ) is affine (order 1 PPDLF) it suffices to test on vertices : ζ i = 1 , ζ j � = i = 0 7 24-28 July 2006, Kyoto

② Proposed ”dilated LMIs” approach Redundant modeling - CDC’05 Consider the system with 2 equations x = A 2 ( ζ ) x x = A ( ζ ) x , A ( ζ ) ˙ ˙ Applying the same methodology leads to :   A ( ζ ) − 1 0 0   Π( ζ ) 0  + [ ∗ ] T < 0    + G A ( ζ ) − 1 0 0      Π( ζ ) 0  A ( ζ ) − 1 0 0 and one can prove that it corresponds to taking for ˙ x = A ( ζ ) x a PDLF T     1 1 P ( ζ ) = Π( ζ )         A ( ζ ) A ( ζ ) Special case of order 3 PPDLF. 8 24-28 July 2006, Kyoto

② Proposed ”dilated LMIs” approach Redundant modeling - ROCOND’06 Assume a given affine M ( ζ ) and the redundant equations x = A ( ζ ) x , M ( ζ ) ˙ ˙ x = M ( ζ ) A ( ζ ) x Applying the same methodology leads to :   A ( ζ ) − 1 0 0   Π( ζ ) 0  + [ ∗ ] T < 0    + G M ( ζ ) − 1 0 0      Π( ζ ) 0  M ( ζ ) − 1 0 0 and one can prove that it corresponds to taking for ˙ x = A ( ζ ) x a PDLF T     1 1 P ( ζ ) = Π( ζ )         M ( ζ ) M ( ζ ) Appropriate choices of M ( ζ ) improve the results. 9 24-28 July 2006, Kyoto

② Proposed ”dilated LMIs” approach Redundant modeling - MTNS’06 For a chosen set of monomial α j ( ζ ) = ζ j 1 1 ζ j 2 2 . . . ζ j N N , j ∈ { k 1 , . . . , k p } take the redundant equations α j ( ζ ) ˙ x = α j ( ζ ) A ( ζ ) x Applying the same methodology leads to LMIs for the robust analysis of ˙ x = A ( ζ ) x with a PPDLF T     1 1      α k 1 ( ζ ) 1   α k 1 ( ζ ) 1      P ( ζ ) = Π( ζ )     . .     . . . .             α k p ( ζ ) 1 α k p ( ζ ) 1 where Π( ζ ) is affine with respect to ζ . 10 24-28 July 2006, Kyoto

③ Numerical example Results are extended to H 2 guaranteed cost ➞ Allows on the numerical example to test the conservatism : The smaller the guaranteed H 2 norm is, the smaller is the conservatism. ➞ Academic example of order 3 ( x ∈ R 3 ) with 3 vertices ( N = 3 ). ➞ For P ( ζ ) = P (”quadratic stability”) : γ 2 = 18 . 15 (6 vars in LMIs) ➞ For P ( ζ ) of order 1 : γ 2 = 8 . 31 (52 vars in LMIs) ➞ For j ∈ { (100) } : γ 2 = 4 . 83 (217 vars in LMIs) ➞ For j ∈ { (100) , (200) , (010) , } : γ 2 = 3 . 73 (499 vars in LMIs) � j i ≤ 2 } : γ 2 = 2 . 67 (2101 vars in LMIs) ➞ For j ∈ { ( j 1 j 2 j 3 ) : ➞ Optimal value (expected by gridding) : γ 2 = 1 . 32 ✪ There is still work to be done : Reduce computation burden, Reduce conservatism... Compare numerically & theoretically the existing results. 11 24-28 July 2006, Kyoto