A manifold structure on the set of functional observers Jochen Trumpf University of W¨ urzburg Math. Institute Germany A manifold structure on the set of functional observers – p.1/12
Contents motivating problem A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends manifold and vector bundle structure A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends manifold and vector bundle structure application A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends manifold and vector bundle structure application L 2 -sensitivity of OAF-compensators A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends manifold and vector bundle structure application L 2 -sensitivity of OAF-compensators outlook A manifold structure on the set of functional observers – p.2/12
Contents motivating problem tracking observers definition and characterization manifold structure conditioned invariant subspaces and their friends manifold and vector bundle structure application L 2 -sensitivity of OAF-compensators outlook joint work with U. Helmke A manifold structure on the set of functional observers – p.2/12
Motivating problem Definition. Let ( C, A ) ∈ R p × n × R n × n . A linear subspace V ⊂ R n is called ( C, A ) -invariant if there exists an output injection matrix J such that ( A − JC ) V ⊂ V holds. Such a J is called a friend of V . A manifold structure on the set of functional observers – p.3/12
Motivating problem Definition. Let ( C, A ) ∈ R p × n × R n × n . A linear subspace V ⊂ R n is called ( C, A ) -invariant if there exists an output injection matrix J such that ( A − JC ) V ⊂ V holds. Such a J is called a friend of V . Problem: How much do perturbations in J affect V ? A manifold structure on the set of functional observers – p.3/12
Motivating problem Definition. Let ( C, A ) ∈ R p × n × R n × n . A linear subspace V ⊂ R n is called ( C, A ) -invariant if there exists an output injection matrix J such that ( A − JC ) V ⊂ V holds. Such a J is called a friend of V . Problem: How much do perturbations in J affect V ? cf. related work on stable subspaces by L. Rodman (various articles) or F. Velasco (LAA 301, pp. 15–49, 1999) A manifold structure on the set of functional observers – p.3/12
Motivating problem Let P ∈ R n × n be the orthogonal projector on V . Then ( A − JC ) V ⊂ V ⇐ ⇒ f ( P, J ) := ( I n − P )( A − JC ) P = 0 A manifold structure on the set of functional observers – p.4/12
Motivating problem Let P ∈ R n × n be the orthogonal projector on V . Then ( A − JC ) V ⊂ V ⇐ ⇒ f ( P, J ) := ( I n − P )( A − JC ) P = 0 Let ( P 0 , J 0 ) be such that f ( P 0 , J 0 ) = 0 . Consider ∂f ∂P | ( P 0 ,J 0 ) ( ˙ P ) = − ˙ PA 0 P 0 + ( I n − P 0 ) A 0 ˙ A 0 := A − J 0 C P, A manifold structure on the set of functional observers – p.4/12
Motivating problem Let P ∈ R n × n be the orthogonal projector on V . Then ( A − JC ) V ⊂ V ⇐ ⇒ f ( P, J ) := ( I n − P )( A − JC ) P = 0 Let ( P 0 , J 0 ) be such that f ( P 0 , J 0 ) = 0 . Consider ∂f ∂P | ( P 0 ,J 0 ) ( ˙ P ) = − ˙ PA 0 P 0 + ( I n − P 0 ) A 0 ˙ A 0 := A − J 0 C P, in the basis where � � � � � � 0 0 I k A 1 A 2 X and ˙ P 0 = , A 0 = P = [ P 0 , Ω] = 0 0 0 0 A 4 X ( Ω is skew-symmetric, here.) A manifold structure on the set of functional observers – p.4/12
Motivating problem We get � � ∂f 0 0 ∂P | ( P 0 ,J 0 ) ( ˙ P ) = A 4 X − XA 1 0 A manifold structure on the set of functional observers – p.5/12
Motivating problem We get � � ∂f 0 0 ∂P | ( P 0 ,J 0 ) ( ˙ P ) = A 4 X − XA 1 0 If A 1 and A 4 have disjoint spectra then the linear map X �→ A 4 X − XA 1 is injective, i.e. in this case the differential is injective. A manifold structure on the set of functional observers – p.5/12
Motivating problem We get � � ∂f 0 0 ∂P | ( P 0 ,J 0 ) ( ˙ P ) = A 4 X − XA 1 0 If A 1 and A 4 have disjoint spectra then the linear map X �→ A 4 X − XA 1 is injective, i.e. in this case the differential is injective. Result. Let f ( P 0 , J 0 ) = 0 and σ ( A 0 | Im P 0 ) ∩ σ ( A 0 | R n / Im P 0 ) = ∅ Then locally around J 0 there exists a Lipschitz continuous function J �→ P ( J ) such that f ( J, P ( J )) = 0 A manifold structure on the set of functional observers – p.5/12
Tracking observers Consider the linear, time-invariant, finite-dimensional control system in state space form x = Ax + Bu ˙ (sys) y = Cx A manifold structure on the set of functional observers – p.6/12
Tracking observers Consider the linear, time-invariant, finite-dimensional control system in state space form x = Ax + Bu ˙ (sys) y = Cx Definition. A tracking observer for V x is a dynamical system v = Kv + Ly + Mu ˙ (obs) which is driven by u and by y and has the tracking property : v (0) := V x (0) ⇒ v ( t ) = V x ( t ) for all t ∈ R where x (0) and u ( . ) are arbitrary. A manifold structure on the set of functional observers – p.6/12
Tracking observers Theorem. (Luenberger, 1964 ) System (obs) is a tracking observer for V x if and only if V A − KV = LC (syl) M = V B In this case the tracking error e ( t ) = v ( t ) − V x ( t ) is governed by the differential equation ˙ e = Ke . A manifold structure on the set of functional observers – p.7/12
Tracking observers Theorem. (Luenberger, 1964 ) System (obs) is a tracking observer for V x if and only if V A − KV = LC (syl) M = V B In this case the tracking error e ( t ) = v ( t ) − V x ( t ) is governed by the differential equation ˙ e = Ke . Theorem. (Willems et al., ≈ 1980 ) Let V be of full row rank. For every tracking observer for V x there exists a friend J of Ker V such that ( A − JC ) | R n / Ker V is similar to K . Conversely, for every friend J of Ker V there exists a unique tracking observer for V x such that K is similar to ( A − JC ) | R n / Ker V . Especially, there exists a tracking observer for V x if and only if Ker V is ( C, A ) -invariant. A manifold structure on the set of functional observers – p.7/12
The manifold of tracking observers Theorem. (T., 2002) Let ( C, A ) be observable and let k and p be the numbers of rows of V and C , respectively. Then the set Obs k,k := { ( K, L, M, V ) | V A − KV = LC, M = V B, rk V = k } of tracking observer parameters is a smooth (sub)manifold of dimension k 2 + kp . A manifold structure on the set of functional observers – p.8/12
The manifold of tracking observers Theorem. (T., 2002) Let ( C, A ) be observable and let k and p be the numbers of rows of V and C , respectively. Then the set Obs k,k := { ( K, L, M, V ) | V A − KV = LC, M = V B, rk V = k } of tracking observer parameters is a smooth (sub)manifold of dimension k 2 + kp . Proof. The value (0 , 0) is a regular value of the map f : ( K, L, M, V ) �→ ( V A − KV − LC, M − V B ) The requirement rk V = k yields an open subset. A manifold structure on the set of functional observers – p.8/12
The manifold of tracking observers Theorem. (T., 2002) Consider the similarity action σ : GL( k ) × Obs k,k − → Obs k,k , ( S, ( K, L, M, V )) �→ ( SKS − 1 , SL, SM, SV ) The σ -orbit space Obs σ k,k of similarity classes [ K, L, M, V ] σ = { ( SKS − 1 , SL, SM, SV ) | S ∈ GL( k ) } . is a smooth manifold of dimension kp . A manifold structure on the set of functional observers – p.9/12
The manifold of tracking observers Theorem. (T., 2002) Consider the similarity action σ : GL( k ) × Obs k,k − → Obs k,k , ( S, ( K, L, M, V )) �→ ( SKS − 1 , SL, SM, SV ) The σ -orbit space Obs σ k,k of similarity classes [ K, L, M, V ] σ = { ( SKS − 1 , SL, SM, SV ) | S ∈ GL( k ) } . is a smooth manifold of dimension kp . Proof. The equations V A − KV = LC and M = V B are invariant under σ . The similarity action is free and has a closed graph mapping. Furthermore, dim GL( k ) = k 2 . A manifold structure on the set of functional observers – p.9/12
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