An internal model principle for observers J. Trumpf J.C. Willems - PowerPoint PPT Presentation

Definition of an observer Observer properties An internal model principle The state space case Summary An internal model principle for observers J. Trumpf J.C. Willems July 2007 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is an observer? Plant w 2 w 1 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is an observer? Plant w 2 R 1 ( σ ) w 1 + R 2 ( σ ) w 2 = 0 w 1 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is an observer? Plant w 2 R 1 ( σ ) w 1 + R 2 ( σ ) w 2 = 0 w 1 r ❅ � ˆ w 2 ❅ R 1 ( σ ) w 1 + ˆ ˆ R 2 ( σ ) ˆ w 2 = 0 � Observer Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is an observer? Plant w 2 R 1 ( σ ) w 1 + R 2 ( σ ) w 2 = 0 ✓✏ w 1 e r − ✒✑ ❅ � ˆ w 2 ❅ R 1 ( σ ) w 1 + ˆ ˆ R 2 ( σ ) ˆ w 2 = 0 � Observer Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary The error system Given a plant B = { ( w 1 , w 2 ) | R 1 ( σ ) w 1 + R 2 ( σ ) w 2 = 0 } and an observer ˆ w 2 ) | ˆ R 1 ( σ ) w 1 + ˆ B = { ( w 1 , ˆ R 2 ( σ ) ˆ w 2 = 0 } for B , the error system is defined as w 2 ) ∈ ˆ B e = { e | ∃ ( w 1 , w 2 ) ∈ B , ( w 1 , ˆ B : e = ˆ w 2 − w 2 } The elimination theorem says that B e is an LTID system. Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is a good observer? We postulate that the fundamental property any reasonable observer should have is B e is autonomous. The classical cases are: B e stable ⇒ asymptotic observer (discrete time) B e nilpotent ⇒ dead-beat observer B e = 0 ⇒ exact observer Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is a good observer? We say that ˆ B contains an internal model of B if B ⊆ ˆ B This is equivalent to the existence of S such that � � ˆ ˆ � � R 1 R 2 = S R 1 R 2 Then the error system is B e = { e | SR 2 e = 0 } which is autonomous ( SR 2 = ˆ R 2 is nonsingular square since w 1 is full input to ˆ B ). Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is a good observer? B ⊆ ˆ B ⇒ ∃ S : B e = Ker SR 2 ( σ ) Recall: w 2 is observable from w 1 ⇔ ( w 1 , w 2 ) , ( w 1 , ˜ w 2 ) ∈ B implies w 2 = ˜ w 2 ⇔ R 2 ( λ ) has full column rank for all λ ∈ C Under this condition we get full pole placement : for any π there exists S such that det SR 2 = π , can even choose SR 2 = I . Note: same story for w 2 detectable or reconstructible from w 1 . Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Main result Theorem B e autonomous implies B contr . ⊆ ˆ B. Corollary Any asymptotic (dead-beat, exact) observer for a controllable system contains an internal model. Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Proof sketch � M 1 � � Γ 1 � ˆ B contr . = Im , B contr . = Im M 2 Γ 2 where Γ 1 is square and has full rank ( w 1 is full input to ˆ B ). Then, ( B contr . ) e is given by � 0 � � M 1 � � l � − Γ 1 = e M 2 − Γ 2 l ′ and is autonomous iff � M 1 � � Γ 1 � − Γ 1 � � = rk M 1 − Γ 1 = rk Γ 1 = rk rk M 2 − Γ 2 Γ 2 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Proof sketch Since Γ 1 has full column rank this implies the existence of a rational T such that � M 1 � � Γ 1 � = T M 2 Γ 2 i.e. B contr . ⊆ ˆ B contr . Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary What is an observer? x z σ x = Ax + Bu u K r (P) y = Cx y ✓✏ e Plant − ✒✑ G ˆ σξ = F ξ + Gy + Hu z (O) ˆ z = J ξ H Observer Gains Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Luenberger’s equations The existence of Z such that ZA − FZ = GC H = ZB σ x = Ax + Bu y = Cx K = JZ z = Kx implies ( d := ξ − Zx ) σξ = F ξ + Gy + Hu ˆ z = J ξ σ d = Fd e = Jd Z then maps x -trajectories to corresponding ξ -trajectories. Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Fuhrmann’s interpretation       x Zx ξ u u u → ˆ       γ : B f − B f ,  �→  =:       y y y     ˆ z z z is an injective behavior homomorphism, i.e. the observer contains an internal model! We know from the above theorem that this is true for every asymptotic observer if the system is controllable. Fuhrmann and Helmke proved this statement in the state space case in 2002. Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Outline Definition of an observer 1 Observer properties 2 An internal model principle 3 The state space case 4 Summary 5 Trumpf, Willems An internal model principle for observers

Definition of an observer Observer properties An internal model principle The state space case Summary Summary Everything is easy once we have an internal model. This is the case for reasonable observers. Outlook properness algorithms nD case Trumpf, Willems An internal model principle for observers