Online Identification of parameters in time dependent differential - PowerPoint PPT Presentation

Online Identification of parameters in time dependent differential equations (using partial observation) Johann Baumeister Goethe University, Frankfurt, Germany IMPA – Rio de Janeiro / October 30th to November 3rd 2017 Supported by the Alexander von Humboldt Foundation 1 / 28

Outline Model reference adaptive systems–Introduction Model reference adaptive systems–Complete observation Addendum–Noisy data Model reference adaptive systems–Observation by a projection Concluding remarks October 28, 2017 2 / 28

Introduction Process equation ( ∗ ) D t w + A 0 ( w ) + A ( q , w ) = f ( t ) in V ∗ , t ∈ (0 , T ); w (0) = ζ = f ( t ) , t ∈ (0 , ∞ ); w (0) = ζ ( V ∗ = H − 1 (Ω)) D t w + cw −∇ ( q ∇ w ) ���� � �� � A 0 ( w ) A ( q , w ) or = f ( t ) , t ∈ (0 , ∞ ); w (0) = ζ ( V ∗ = H − 1 (Ω)) D t w − ∆ w + qw � �� � ���� A 0 ( w ) A ( q , w ) or . . . . . .  V , H , P separable Hilbert spaces   → V ∗ Gelfand triple ,  V ֒ → H ֒    → V ∗ linear  A 0 ( · ) : V −     w ( t ) ∈ V state at time , t ∈ [0 , T ) (Model data) q ∈ Q ad ⊂ P parameter (to be identified)   → V ∗ bilinear (has to be specified)  A ( · , · ) : Q × V −     ζ ∈ H initial state     → V ∗ f : (0 , ∞ ) − 3 / 28

Introduction: Weak solution and assumptions Weak solution w : [0 , ∞ ) − → H with w ( t ) ∈ V , t > 0 , � D t w ( t ) , v � + � A 0 ( w ( t )) , v � + � A ( q , w ( t )) , v � = � f ( t ) , v � for all t > 0 , v ∈ V w (0) = ζ Assumption (A0) (1) (Continuity of A 0 ) |� A 0 ( u ) , v �| ≤ c 0 � u � V � v � V , u , v ∈ V , with a constant c 0 ≥ 0 . (2) (Continuity of A ( · , · )) � A ( q , u ) , v � ≤ c 0 ( q ) � u � V � v � V for all q ∈ Q ad , u , v ∈ V , where c 0 ( q ) ≥ 0 is a constant which may depend on q . (3) (Coercivity of A 0 ( · ) + A ( q , · )) � A 0 ( u ) , u � + � A ( q , u ) , u � + β ( q ) � u � 2 H ≥ c 1 ( q ) � u � 2 V for all q ∈ Q ad , u ∈ V where β ( q ) , c 1 ( q ) are constants which may depend on q with β ( q ) ≥ 0 , c 1 ( q ) > 0 . (4) f ∈ L 2 (0 , T ; V ∗ ) for all T > 0 , ζ ∈ H . 4 / 28

Introduction: Parameter to solution map ( ∗ ) D t w + A 0 ( w ) + A ( q , w ) = f ( t ) in V ∗ , t ∈ (0 , T ); w (0) = ζ → V : D t w ( t ) ∈ V ∗ , t ∈ (0 , T ] } . W ((0 , T ); V ) := { w : (0 , T ] − W ((0 , T ); V ) is a Hilbert space and each function in w ∈ W ((0 , T ); V ) may be considered as w ∈ C ([0 , T ]; H ) . Fact Let the assumption (A0) hold. Then there exists for each q ∈ Q ad a uniquely determined weak solution w of ( ∗ ) which satisfies: w ∈ W ((0 , T ); V ) , w ∈ C ([0 , T ]; H ) for all T > 0 . (1) In other words: The parameter to solution map (PtS) from Q ad into V ⊂ H is well defined. 5 / 28

Introduction: Plant Model equation ( ∗ ) D t z + A 0 ( z ) + A ( p , z ) = f ( t ) in V ∗ , t ∈ (0 , T ); z (0) = ζ ( p , z ) is a plant iff z is a weak solution of ( ∗ ) |� A ( q , z ( t )) , v �| ≤ c 2 � q � P � v � V , t ≥ 0 , q ∈ P , v ∈ V , with a constant c 2 ≥ 0 Identication problem with complete observation Given a plant ( p , z ), find the parameter p from an observation of the state z on [0 , ∞ ) . 6 / 28

(MRAS): Model reference adaptive systems Idea Given a plant ( p , z ) consider a reference system with state t �− → u ( t ) driven by a time dependent parameter t �− → q ( t ) which is adapted by matching the state z of the modelled system and the state u of the reference system. Historically Used by engineers to control online a finite dimensional system whose parameters are not known in advance. Goal Find a reference system and an adaption rule such that t →∞ ( u ( t ) − z ( t )) = θ , lim lim t →∞ ( q ( t ) − p ) = θ . Analysis Stability arguments for linear nonautonomous systems. Realization Here, partial differential equations. Applicability to each type of a differential equations which contains parameters, even to their stationary equations (artificial dynamics). Difficulties In infinite dimensional systems the observation of the state z is crucial. 7 / 28

(MRAS): References In the next part we follow mainly W. Scondo Ein Modellabgleichungs Verfahren zur adaptiven . . . Dissertation, Goethe Universit¨ at Frankfurt/Main, 1987 J. Baumeister, W. Scondo, M.A. Demetriou and I.G. Rosen On-line parameter estimation for infinitedimensional dynamical systems SIAM J. Control Optim. 35 (1997), 678-713 In the third part we are inspired by R. Boiger and B. Kaltenbacher A online parameter identification method for time dependent partial differential equations Inverse Problems, 32 (2016), 28 pp. See also P. K¨ ugler Online parameter identification without Ricatti-type equations in a class of time-dependent partial differential equations: an extended state approach with potential to partial observations Inverse Problems, 26 (2010) 8 / 28

(MRAS): complete observation Model equation and plant ( ∗ ) D t z + A 0 ( z ) + A ( p , z ) = f ( t ) in V ∗ , t ∈ (0 , T ); z (0) = ζ (MRAS)  in V ∗ , t > 0 , D t u + C ( u − z ( t )) + A 0 ( z ( t )) + A ( q , z ( t )) = f ( t )    u (0) = u 0 in P ∗ , t > 0 , D t q − b ( z ( t ) , u − z ( t )) = θ    q (0) = q 0 What is b in the adaptation rule? It satisfies � A ( q , z ( t )) , v � = � q | b ( z ( t ) , v ) � for all parameters q ∈ P and all t ≥ 0 . Such a mapping b exists since � A ( · , z ( t ) , v � ∈ P ∗ ≡ P . due to the fact that ( p , z ) is a plant. 9 / 28

(MRAS): well-posedness of the system e := u − z , r :== q − p . These error quantities are solutions of D t e + Ce + A ( r , z ( t )) = θ , t > 0 , e (0) = e 0 = u 0 − ζ , D t r − b ( z ( t ) , e ) = θ , t > 0 , r (0) = r 0 = q 0 − p . (A priori estimate) � t � e ( t ) � 2 H + � r ( t ) � 2 � e ( s ) � 2 V ds ≤ � e ( t 0 ) � 2 H + � r ( t 0 ) � 2 P + 2 c 5 P , t > t 0 . t 0 (Consequences) → L ( e , r )( t ) := � e ( t ) � 2 H + � r ( t ) � 2 t �− P is decreasing (L is a Ljapunov-type function). � ∞ sup t ∈ [0 , ∞ ) � e ( t ) � 2 H + � r ( t ) � 2 t 0 � e ( s ) � 2 P + 2 c 5 V ds < c := � e (0) � 2 H + � r (0) � 2 P . The error system has a uniquely determined solution ( e , r ) in W ((0 , T ) , V × P ) for all T > 0 . (MRAS) has a uniquely determined weak solution ( u , q ) in W ((0 , T ) , V × P ) for all T > 0 . 10 / 28

(MRAS): identifiabilty Fact (Output-identifiability) t →∞ � u ( t ) − z ( t ) � H = 0 lim Fact (Parameter-identifiability) t →∞ � q ( t ) − p � P = 0 lim if the plant ( p , z ) is asymptotically persistently excited. Definition Let ( p , z ) be a plant. The state z is asymptotically persistently excited if there exist numbers l > 0 , µ > 0 and a sequence ( t n ) n ∈ I N in (0 , ∞ ) with lim n t n = ∞ such that the following condition holds: ∀ h ∈ P ∀ n ∈ I N ∃ t n , 1 , t n , 2 ∈ [ t n , t n + l ] ∃ v ∈ V\{ θ } �� � � t n , 2 � � � � A ( h , z ( t )) , v � ds � ≥ µ � h � P � v � V � � � t n , 1 11 / 28

(MRAS): Richness Persistently exciteness is the key property to ensure that adaptation can be achieved in a sequence of intervalls [ t n , t n + l ] with lim n t n = ∞ . Notice � D t q | q � = � q | b ( z ( t ) , e ) � = � A ( q , z ( t )) , e � Conditions like persistently exciteness are also called richness conditions . Here is stronger richness condition which plays a fundamental role in the next part. Definition Let ( p , z ) be a plant. The state z is uniformly persistently excited if there exist numbers l > 0 , µ > 0 such that ∀ h ∈ P ∀ t 0 ∈ (0 , ∞ ) ∃ t 1 , t 2 ∈ [ t 0 , t 0 + l ] ∃ v ∈ V\{ θ } � t 2 �� � � � � � A ( h , z ( t )) , v � ds � ≥ µ � h � P � v � V � � � t 1 12 / 28

(MRAS): Richness/References Observation The term � t 2 � A ( h , z ( t )) , v � ds t 1 is the variational right hand side of the Fr´ echet-derivative of the parameter to solution mapping in the parameter p . We should note that the excitation conditions are, in practice, difficult to verify. In M.A. Demetriou and I.G. Rosen On the persistence of excitation in the adaptive estimation of distributed parameter systems IEEE Trans. on Autom. Control 39, 1994 a careful study and analysis of the excitation conditions for a parabolic equation with a constant parameter can be found. In the context of adaptive control of finite dimensional systems richness conditions are analyzed for instance in K.S. Narendra and A.M. Annaswammy Persistent excitation in adaptive systems Int. J. Control 45, 1987 13 / 28