Stable LPV realization of parametric transfer functions and its - PowerPoint PPT Presentation

Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design Franco Blanchini 1 , Daniele Casagrande 2 Stefano Miani 2 and Umberto Viaro 2 1 Dipartimento di Matematica e Informatica Universit´ a degli Studi di Udine 2 Dipartimento di Ingegneria Elettrica Gestionale e Meccanica Universit´ a degli Studi di Udine August 30, 2011 logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Transfer function versus state–space Plant: x ( t ) = Ax ( t )+ Bu ( t ) ˙ y ( t ) = Cx ( t ) Control: z ( t ) = Fz ( t )+ Gy ( t ) ˙ u ( t ) = Hz ( t )+ Ky ( t ) logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Transfer function versus state–space Plant: y ( s ) = P ( s ) u ( s ) Control: u ( s ) = K ( s ) y ( s ) Under suitable “no zero–pole cancellation” assumptions ∀ s ∈ C + det ( I − K ( s ) P ( s )) � = 0 , ⇔ � A + BKC � BH is such that Re ( λ i ) < 0 GC F logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Transfer function versus state–space Closed–loop stability does not depend on the chosen realization; Optimality does not depend on the chosen realization; (if you do not consider numerical problems) logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Parametric design Plant: x ( t ) = A ( w ) x ( t )+ B ( w ) u ( t ) ˙ y ( t ) = C ( w ) x ( t ) w ∈ W is a constant parameter Control: z ( t ) = F ( w ) z ( t )+ G ( w ) y ( t ) ˙ u ( t ) = H ( w ) z ( t )+ K ( w ) y ( t ) � A ( w )+ B ( w ) K ( w ) C ( w ) � B ( w ) H ( w ) G ( w ) C ( w ) F ( w ) must be Hurwitz for all w ∈ W . logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV design A(w) B(w) C(w) u w y F(w) G(w) H(w) K(w) Figure: Gain–scheduling control logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV design Plant: x ( t ) = A ( w ( t )) x ( t )+ B ( w ( t )) u ( t ) ˙ y ( t ) = C ( w ( t )) x ( t ) w ( t ) is an arbitrary time–varying parameter w ( t ) ∈ W Control: z ( t ) = F ( w ( t )) z ( t )+ G ( w ( t )) y ( t ) ˙ u ( t ) = H ( w ( t )) z ( t )+ K ( w ( t )) y ( t ) logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis STABILITY w(t) OPTIMALITY Pointwise optimality LPV stability1 logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis STABILITY w(t) OPTIMALITY Pointwise optimality LPV stability logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis p u h 1 y w − ≤ w ≤ w + y ( t ) = − α y ( t )+ w ( t ) u ( t ) , α > 0 , ˙ logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis p u h 1 y w − ≤ w ≤ w + y ( t ) = − α y ( t )+ w ( t ) u ( t ) , α > 0 , ˙ logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis p u h 1 y w − ≤ w ≤ w + y ( t ) = − α y ( t )+ w ( t ) u ( t ) , α > 0 , ˙ logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis Compensator: κ ( w ) κ ( w ) = − κ 0 s + β , β > 0 , w , κ 0 > 0 , logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis Compensator: κ ( w ) κ ( w ) = − κ 0 s + β , β > 0 , w , κ 0 > 0 , Closed loop polynomial: d ( s ) = s 2 +( β + α ) s + αβ + κ 0 . logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis Compensator: κ ( w ) κ ( w ) = − κ 0 s + β , β > 0 , w , κ 0 > 0 , Closed loop polynomial: d ( s ) = s 2 +( β + α ) s + αβ + κ 0 . Realizations � - β � � � - β 1 - κ 0 / w Σ 1 ( w ) = , Σ 2 ( w ) = − κ 0 / w 0 1 0 logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis Compensator: κ ( w ) κ ( w ) = − κ 0 s + β , β > 0 , w , κ 0 > 0 , Closed loop polynomial: d ( s ) = s 2 +( β + α ) s + αβ + κ 0 . Realizations � - β � � � - β 1 - κ 0 / w Σ 1 ( w ) = , Σ 2 ( w ) = − κ 0 / w 0 1 0 Closed–loop systems � − α � � � − κ 0 − α w A 1 ( w ) = , A 2 ( w ) = 1 − β − κ 0 / w − β � �� � � �� � LPV stable LPV unstable logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Motivation: parametric synthesis Compensator: κ ( w ) κ ( w ) = − κ 0 s + β , β > 0 , w , κ 0 > 0 , Closed loop polynomial: d ( s ) = s 2 +( β + α ) s + αβ + κ 0 . Realizations � - β � � � - β 1 - κ 0 / w Σ 1 ( w ) = , Σ 2 ( w ) = − κ 0 / w 0 1 0 Closed–loop systems � − α � � � − κ 0 − α w A 1 ( w ) = , A 2 ( w ) = 1 − β − κ 0 / w − β � �� � � �� � LPV stable LPV unstable Stability depends on the compensator realization: Rugh and Shamma (2000). logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stabilizability: separation principle Theorem P = ( A ( w ) , B ( w ) , C ( w )) LPV is stabilizable via linear LPV control iff it is possible to build an LPV state observer and an LPV stabilizing state feedback (dual problems). PLANT y u A(w) B(w) C(w) 0 x OBSERVER (w) ESTIMATED STATE FEEDBACK (w) logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stabilizability: duality Theorem P = ( A ( w ) , B ( w ) , C ( w )) LPV is stabilizable iff A ( w ) X + B ( w ) U ( w ) = XP ( w ) (state feedback eq. ) , RA ( w )+ L ( w ) C ( w ) = Q ( w ) R (state observer eq. ) . X full row rank, R full column rank, µ 1 ( P ( w )) < 0 , µ ∞ ( Q ( w )) < 0 . logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stabilizability: duality Theorem P = ( A ( w ) , B ( w ) , C ( w )) LPV is stabilizable iff A ( w ) X + B ( w ) U ( w ) = XP ( w ) (state feedback eq. ) , RA ( w )+ L ( w ) C ( w ) = Q ( w ) R (state observer eq. ) . X full row rank, R full column rank, µ 1 ( P ( w )) < 0 , µ ∞ ( Q ( w )) < 0 . Theorem P = ( A ( w ) , B ( w ) , C ( w )) is quadratically stabilizable iff PA ( w ) T + A ( w ) P + B ( w ) U ( w )+ U ( w ) T B ( w ) T < 0 , A ( w ) T Q + QA ( w )+ Y ( w ) C ( w )+ C T ( w ) Y ( w ) T < 0 , logo with P > 0 and Q > 0 . Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stable synthesis Assumption ( A ( w ) , B ( w ) , C ( w )) is (quadratically) LPV stabilizable. logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stable synthesis Assumption ( A ( w ) , B ( w ) , C ( w )) is (quadratically) LPV stabilizable. Question: Given a plant ( A ( w ) , B ( w ) , C ( w )) and a parametric compensator transfer function R ( s , w ) such that the closed–loop system is stable for constant values w ∈ W , is it possible to realize this compensator in such a way that the closed loop is LPV stable? logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stable synthesis Assumption ( A ( w ) , B ( w ) , C ( w )) is (quadratically) LPV stabilizable. Question: Given a plant ( A ( w ) , B ( w ) , C ( w )) and a parametric compensator transfer function R ( s , w ) such that the closed–loop system is stable for constant values w ∈ W , is it possible to realize this compensator in such a way that the closed loop is LPV stable? Theorem Yes, it is. logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

LPV stable realization Definition Given T ( s , w ) = N ( s , w ) d ( s , w ) , w ∈ W , the systems z ( t ) = F ( w ) z ( t )+ G ( w ) ω ( t ) , ˙ ξ ( t ) = H ( w ) z ( t )+ K ( w ) ω ( t ) , is a parametric realization of T ( s , w ) if T ( s , w ) = H ( w )( sI − F ( w )) − 1 G ( w )+ K ( w ) , ∀ w ∈ W . logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Basic result Definition LPV stable realization . Assuming that d ( s , w ) is a Hurwitz polynomial for all w ∈ W , the realization � F ( w ) � G ( w ) Σ ( w ) = , H ( w ) K ( w ) is LPV stable if z ( t ) = F ( w ( t )) z ( t ) (1) ˙ is asymptotically stable for any function w ( t ) ∈ W . logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Basic result Definition LPV stable realization . Assuming that d ( s , w ) is a Hurwitz polynomial for all w ∈ W , the realization � F ( w ) � G ( w ) Σ ( w ) = , H ( w ) K ( w ) is LPV stable if z ( t ) = F ( w ( t )) z ( t ) (1) ˙ is asymptotically stable for any function w ( t ) ∈ W . Theorem Each parametric transfer function T ( s , w ) , with d ( s , w ) Hurwitz, admits an LPV stable realizations. logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Procedure Procedure logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design

Procedure Procedure 1 Take any realization � ˜ � ˜ F ( w ) G ( w ) ˜ ˜ H ( w ) K ( w ) logo Blanchini Casagrande Miani Viaro Stable LPV realization for LPV design