Introduction to Feedback Systems 1999 nder YKSEL 3
Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3. 5 Observability
OBSERVABILITY Introduction to Feedback Systems / © 1999 Önder YÜKSEL • The linear, time-invariant system is said to be observable if the initial condition x(0) can be determined from the knowledge of the output y(t) & the input u(t) , in the interval 0<t<t. F , with t. F finite, for arbitrary x(0). 3/9/2021 3. 5 Observability 2
Remarks Introduction to Feedback Systems / © 1999 Önder YÜKSEL This is a simplified definition valid for LTI systems. In essence, any two Each state variable different states can must affect the output be distinguished, in an independent manner from the O/P 3/9/2021 3. 5 Observability 3
Lemma Introduction to Feedback Systems / © 1999 Önder YÜKSEL is observable if & only if is. i. e. observability involves A&C only 3/9/2021 3. 5 Observability 4
Proof of lemma 3/9/2021 3. 5 Observability Known Involves A&C only Known Introduction to Feedback Systems / © 1999 Önder YÜKSEL 5
Theorem Introduction to Feedback Systems / © 1999 Önder YÜKSEL is observable if & only if nm n 3/9/2021 has rank n. (Independent columns) 3. 5 Observability 6
Proof of theorem: Necessity Introduction to Feedback Systems / © 1999 Önder YÜKSEL Suppose rank Q 0<n Then there exists a vector 0 such that Q 0 =0 i. e. CAk =0 for k=0, 1, …n-1 Then C (t) =0 C (t)(x(0)+ )= C (t)x(0) Two different initial states, but same output Not observable 3/9/2021 3. 5 Observability 7
Proof of theorem: sufficiency Introduction to Feedback Systems / © 1999 Önder YÜKSEL • Suppose not observable & there exists 0 such that C (t) =0 for all t • By repeated differentiation of C (t) =0 ; for k=0, 1, …. n-1 3/9/2021 3. 5 Observability 8
Proof of theorem: sufficiency (Cont’d) Introduction to Feedback Systems / © 1999 Önder YÜKSEL For t=0 rank Q 0<n Not observable 3/9/2021 3. 5 Observability 9
Theorem: observability & coordinate tr. Introduction to Feedback Systems / © 1999 Önder YÜKSEL Observability is preserved under coordinate transformation Proof: 3/9/2021 3. 5 Observability 10
Theorem: controllability & feedback Introduction to Feedback Systems / © 1999 Önder YÜKSEL Controllability is preserved under state feedback + B + A K 3/9/2021 3. 5 Observability 11
Proof Introduction to Feedback Systems / © 1999 Önder YÜKSEL v + u B + x A K C u(t) for desired x(t. F) can be found x(t) for 0<t<t. F can be calculated Kx(t) can be calculated v(t) can be calculated 3/9/2021 3. 5 Observability 12
Note: Introduction to Feedback Systems / © 1999 Önder YÜKSEL Not true for observability ! 3/9/2021 3. 5 Observability 13
Theorem: Cancellation in TF Introduction to Feedback Systems / © 1999 Önder YÜKSEL If the input/output transfer function has a pole-zero cancellation, the system is either not controllable or not observable, or both. (i. e. no cancellation if & only if both controllable & observable) 3/9/2021 3. 5 Observability 14
What is cancellation? Introduction to Feedback Systems / © 1999 Önder YÜKSEL State space representation is n-dimensional. (state vector is n-dimensional) Characteristic polynomial |s 1 n-A| is nth order TF function is expected to have an nth order denominator 3/9/2021 3. 5 Observability 15
TF function has a cancellation if: Introduction to Feedback Systems / © 1999 Önder YÜKSEL Numerator & denominator have a common root i. e. N(s 0)=D(s 0)=0 for some s 0 Residue at one of the eigenvalues is zero Numerator & denominator have a common factor Degree of denominator polynomial is less than n 3/9/2021 3. 5 Observability 16
Justification Introduction to Feedback Systems / © 1999 Önder YÜKSEL Not controllable Not observable + u(t) + y(t) + 3/9/2021 3. 5 Observability 17
Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 18
Rotational mechanical system example Introduction to Feedback Systems / © 1999 Önder YÜKSEL Rot’l inertia: J 2 K Viscous friction: B 2 3/9/2021 Rot’l inertia: J 1 Viscous friction: B 1 3. 5 Observability 19
Circuit diagram Introduction to Feedback Systems / © 1999 Önder YÜKSEL K Node 1 J 1 B 1 Node 2 J 2 B 2 3/9/2021 3. 5 Observability 20
Introduction to Feedback Systems / © 1999 Önder YÜKSEL For simplicity take all parameter values as unity 3/9/2021 3. 5 Observability 21
Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 22
controllability Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 23
controllability Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 24
observability Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 25
Transfer function Introduction to Feedback Systems / © 1999 Önder YÜKSEL Cancellation due to uncontrollable mode 3/9/2021 3. 5 Observability 26
Minimal representation Introduction to Feedback Systems / © 1999 Önder YÜKSEL 3/9/2021 3. 5 Observability 27
End of Chapter Introduction to Feedback Systems / © 1999 Önder YÜKSEL Restart section Next chapter i 3/9/2021 Restart chapter General index End show 3. 5 Observability The End 28
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