Fall 2001 16.3112-6 Example: Loss of observability Typical scenario: consider system G(s)of the form a s+a y so that s+a 1 S +1s+ Clearly a pole-zero cancelation in this system(pole s The state space model for the system is ac+ +(a-1 y=x1+2 A B C=11,D=0 a The Observability/ Controllability tests are(a= 2) 「11 rank rank CA rank[B AB rank /I System controllable, but unobservable. Consistent with the picture Both states can be influenced by But e -at mode dynamics canceled out of the output by the zeroFall 2001 16.31 12—6 Example: Loss of Observability • Typical scenario: consider system G(s) of the form 1 s + a x1 s + a s + 1 x2 u → → → y so that s + a s + 1 · 1 s + a G(s) = • Clearly a pole-zero cancelation in this system (pole s = −a) • The state space model for the system is: x˙ 1 = −ax1 + u x˙ 2 = −x2 + (a − 1)x2 y = x1 + x2    −a 0 a − 1 −1  , B = ∙ 1 0 ¸ , C = £ 1 1 ¤ ⇒ A = , D = 0 • The Observability/Controllability tests are (a = 2): ∙ ∙ C CA ¸ = rank 1 −1 1 ¸ = 1 < n = 2 −1 rank ∙ 1 −2 1 ¸ = 2 0 £ B AB ¤ rank = rank • System controllable, but unobservable. Consistent with the picture: — Both states can be influenced by u — But e−at mode dynamics canceled out of the output by the zero