Fall 2001 16.318-8 . Consider the two transfer functions GI s)=C(sI-A)B+D G2(s)=C(sI-A)B+D Does g1(s)≡G2(s)? GI(s=C(sI-A)B+D C(TT(SI-A)(TT-B+D (C(s-A)-7(T-B)+D A)T(B)+ C(sI-A)B+D=G2(s) So the transfer function is not changed by putting the state-space model through a similarity transformation . Note that in the transfer function 618+628+63 G + a2s+ we have 6 parameters to choose · But in the related state- space model, we have A-3×3,B-3×1, C-1x3 for a total of 15 parameters Is there a contradiction here because we have more degrees of free- dom in the state-space model? No. In choosing a representation of the model, we are effectively choosing a t. which is also 3 x3 and thus has the remaining 9 degrees of freedom in the state-spaFall 2001 16.31 8–8 • Consider the two transfer functions: G1(s) = C(sI − A) −1 B + D G2(s) = C¯(sI − A¯) −1 B¯ + D¯ Does G1(s) ≡ G2(s) ? G1(s) = C(sI − A) −1 B + D = C(T T −1 )(sI − A) −1 (T T −1 )B + D = (CT)  T −1 (sI − A) −1 T  (T −1 B) + D¯ = (C¯)  T −1 (sI − A)T −1 (B¯) + D¯ = C¯(sI − A¯) −1 B¯ + D¯ = G2(s) • So the transfer function is not changed by putting the state-space model through a similarity transformation. • Note that in the transfer function G(s) = b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 we have 6 parameters to choose • But in the related state-space model, we have A−3×3, B −3×1, C − 1 × 3 for a total of 15 parameters. • Is there a contradiction here because we have more degrees of free￾dom in the state-space model? – No. In choosing a representation of the model, we are effectively choosing a T, which is also 3 × 3, and thus has the remaining 9 degrees of freedom in the state-space model