Non-Conservatism of LMI Conditions for Graziano Chesi Time-Varying - PowerPoint PPT Presentation

Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 1 \ 22 Non-Conservatism of LMI Conditions for Graziano Chesi Time-Varying Uncertain Systems www.eee.hku.hk/˜chesi Graziano Chesi Preliminaries Department of Electrical and Electronic Engineering SOS-Based Condition University of Hong Kong Non-Conservatism Conclusion IFAC Workshop on Uncertain Dynamical Systems Udine, Italy, August 23-26, 2011

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ LMIs are a standard tool for uncertain systems Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear, Preliminaries SOS-Based Condition polytopic/non-polytopic, time-invariant/time-varying Non-Conservatism uncertainty Conclusion

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear, Preliminaries SOS-Based Condition polytopic/non-polytopic, time-invariant/time-varying Non-Conservatism uncertainty Conclusion ◮ ... obtains conservative bounds on worst-case performances

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 2 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ LMIs are a standard tool for uncertain systems ◮ Mainly because one works with convex optimization ◮ ... can consider linear/nonlinear, Preliminaries SOS-Based Condition polytopic/non-polytopic, time-invariant/time-varying Non-Conservatism uncertainty Conclusion ◮ ... obtains conservative bounds on worst-case performances ◮ ... can (possibly) reduce the bounds conservatism

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3 \ 22 Graziano Chesi Basic problem: establishing robust stability www.eee.hku.hk/˜chesi Preliminaries SOS-Based Condition Non-Conservatism Conclusion

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3 \ 22 Graziano Chesi Basic problem: establishing robust stability www.eee.hku.hk/˜chesi ◮ Time-invariant (TI): there exist non-conservative LMI conditions, e.g. Preliminaries SOS-Based Condition [Bliman SICON 2004], [Scherer EJC 2006], [Chesi, Non-Conservatism Garulli, Tesi, Vicino SPRINGER 2009] Conclusion (in general asymptotically, possibly dependent on the uncertainty set)

LMIs in Uncertain Systems Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 3 \ 22 Graziano Chesi Basic problem: establishing robust stability www.eee.hku.hk/˜chesi ◮ Time-invariant (TI): there exist non-conservative LMI conditions, e.g. Preliminaries SOS-Based Condition [Bliman SICON 2004], [Scherer EJC 2006], [Chesi, Non-Conservatism Garulli, Tesi, Vicino SPRINGER 2009] Conclusion (in general asymptotically, possibly dependent on the uncertainty set) ◮ Time-varying (TV): what do we know?

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4 \ 22 Polytopic system with linear dependence: Graziano Chesi www.eee.hku.hk/˜chesi x ( t ) = A ( u ( t )) x ( t ) , ˙ u ( t ) ∈ U U = conv { u (1) , u (2) , . . . } A ( u ) = A 0 + � i u i A i , Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4 \ 22 Polytopic system with linear dependence: Graziano Chesi www.eee.hku.hk/˜chesi x ( t ) = A ( u ( t )) x ( t ) , ˙ u ( t ) ∈ U U = conv { u (1) , u (2) , . . . } A ( u ) = A 0 + � i u i A i , Preliminaries ◮ TI case: stable iff SOS-Based Condition Non-Conservatism A ( u ) is Hurwitz ∀ u ∈ U Conclusion

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 4 \ 22 Polytopic system with linear dependence: Graziano Chesi www.eee.hku.hk/˜chesi x ( t ) = A ( u ( t )) x ( t ) , ˙ u ( t ) ∈ U U = conv { u (1) , u (2) , . . . } A ( u ) = A 0 + � i u i A i , Preliminaries ◮ TI case: stable iff SOS-Based Condition Non-Conservatism A ( u ) is Hurwitz ∀ u ∈ U Conclusion ◮ TV case: stable iff ∀ ε > 0 ∃ δ > 0 : � x (0) � < δ ⇓ � x ( t ) � < ε ∀ t ≥ 0 and lim t →∞ x ( t ) = 0 ∀ u ( · ) ∈ U

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi Basic stability condition: Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi Basic stability condition: ◮ Time-invariant (TI) case: stable if Preliminaries ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) SOS-Based Condition Non-Conservatism Conclusion

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi Basic stability condition: ◮ Time-invariant (TI) case: stable if Preliminaries ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) SOS-Based Condition Non-Conservatism ◮ Time-varying (TV) case: stable if Conclusion ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U )

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi Basic stability condition: ◮ Time-invariant (TI) case: stable if Preliminaries ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) SOS-Based Condition Non-Conservatism ◮ Time-varying (TV) case: stable if Conclusion ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) ◮ In both cases the (same) condition is only sufficient

Basic Problem Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 5 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi Basic stability condition: ◮ Time-invariant (TI) case: stable if Preliminaries ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) SOS-Based Condition Non-Conservatism ◮ Time-varying (TV) case: stable if Conclusion ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) ◮ In both cases the (same) condition is only sufficient ◮ This condition is based on a common quadratic LF

Two Simple Examples Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems ◮ TI case: consider A ( u ) = A 0 + u 1 A 1 with 6 \ 22 � − 2 � 2 Graziano Chesi � � 1 0 www.eee.hku.hk/˜chesi A 0 = , A 1 = − 3 1 0 − 2 Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems ◮ TI case: consider A ( u ) = A 0 + u 1 A 1 with 6 \ 22 � − 2 � 2 Graziano Chesi � � 1 0 www.eee.hku.hk/˜chesi A 0 = , A 1 = − 3 1 0 − 2 ◮ A ( u ) is Hurwitz for all u ∈ U = [0 , 1], however � ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) Preliminaries SOS-Based Condition Non-Conservatism Conclusion

Two Simple Examples Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems ◮ TI case: consider A ( u ) = A 0 + u 1 A 1 with 6 \ 22 � − 2 � 2 Graziano Chesi � � 1 0 www.eee.hku.hk/˜chesi A 0 = , A 1 = − 3 1 0 − 2 ◮ A ( u ) is Hurwitz for all u ∈ U = [0 , 1], however � ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) Preliminaries SOS-Based Condition Non-Conservatism ◮ TV case: consider A ( u ( t )) = A 0 + u 1 ( t ) A 1 with Conclusion � � � � 0 1 0 0 A 0 = , A 1 = − 2 − 1 − 1 0

Two Simple Examples Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems ◮ TI case: consider A ( u ) = A 0 + u 1 A 1 with 6 \ 22 � − 2 � 2 Graziano Chesi � � 1 0 www.eee.hku.hk/˜chesi A 0 = , A 1 = − 3 1 0 − 2 ◮ A ( u ) is Hurwitz for all u ∈ U = [0 , 1], however � ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U ) Preliminaries SOS-Based Condition Non-Conservatism ◮ TV case: consider A ( u ( t )) = A 0 + u 1 ( t ) A 1 with Conclusion � � � � 0 1 0 0 A 0 = , A 1 = − 2 − 1 − 1 0 ◮ the origin is asymptotically stable for all u ( t ) ∈ U = [0 , 4], however � ∃ P > 0 : PA ( u ) + A ( u ) ′ P < 0 ∀ u ∈ ver ( U )

Why Conservative? Non-Conservatism of LMI Conditions for Time-Varying Uncertain Systems 7 \ 22 Graziano Chesi www.eee.hku.hk/˜chesi ◮ TI case: stable iff [Chesi et al. 2003] ∃ P ( u ) > 0 : P ( u ) A ( u ) + A ( u ) ′ P ( u ) < 0 ∀ u ∈ U deg P ( u ) ≤ b Preliminaries SOS-Based Condition Non-Conservatism Conclusion