Observer design by pole placement

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Duality of the observer PP and controller PP problems Observer PP problem: Choose L, so that det ( zI −(A+LC) ) = p des (z) Controller PP problem: Choose K , so that det ( zI −(A+BK) ) = p des (z) Observer PP is not a new problem: det ( zI −(A+LC) ) ( (zI ) ) ⊤ = det −(A+LC) ( ) = det zI −(A ⊤ + C ⊤ L ⊤ ) ( ) = det zI −(Ã+ ˜B ˜K) =⇒ observer PP is controller PP for the dual system. ELEC 3035 (Part I, Lecture 7) Observer design 8 / 12
The results for state feedback PP can be restated for observer PP: Theorem: The eigenvalues of A+LC can be assigned choosing L to any locations in C if and only if A,C is observable. Observer canonical form ↔ Controller canonical form Lemma: • Let A,c and A ′ ,c ′ be two observable pairs and • assume that A and A ′ have the same char. polynomials. Then there is a unique similarity transformation given by the matrix T := ( O(A ′ ,c ′ ) ) −1 O(A,c) such that T −1 AT = A ′ and cT = c ′ . ELEC 3035 (Part I, Lecture 7) Observer design 9 / 12
Page 1 and 2: ELEC 3035, Lecture 7: Observer desi
Page 3 and 4: Output feedback control separation
Page 5 and 6: Error dynamics Our goal is to choos
Page 7: Observer design by pole placement T
Page 11 and 12: Feedback controller: σ̂x = (A+LC)

Observer design by pole placement

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