Question

b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) = 0 -1 ] y(t) = (1 out) u(t) , 0, \t <1 (1, t > 1 a) Find the state transition matrix and the impulse response matrix of the system. 2D) Determine whether the system is (i) completely state controllable, (ii) differentially control lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. (c) Repeat part (b) for to = 1. gd) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta- neously observable, (iv) detectable at time to = 0. timoto

Answer 1

Answers Given that - consider the systm *) = 67 to 1 m2 + ( Gos] uct) so, thel where bet? Ci, 14121 Yet) = {0, 13 act) find the the state tramiston matrix.. Answer. for to 261)=(0724) + (]uce) 8(4) - Co 1] 2ct) Impulse Response let =>(S_A)*8.0 +020 let (SIA) () = { $ :) -(0) (29. ] = {0 0]

feran] "- concora { " ) 07 OS SCS+ [sn]" = ( 5 ) (55_] BCEO 1 * ] (3) [01] =0 there we have found the Impulse response. o controlla libity check xu)= CO *}320+ [:744) 90.) = (01) 248) Contvbitum for controlability D={AB AB --- AMAS] HO 16 (3:13) 02 008 :')=16) o (common Hatton = 3or, so it un bombe)

o to=1 on the (A) CET = ($2006) then O-JC =($) (4][B]* - So Bit is controllable (d) checking observability o[a ACT ACT? ...] 80 condition for obserability ACC)(3) (matrie A matrix = (-9 D=li in - Ac matrine Ac matrie - > Mahine =o (so unobserable when t-o ºof!49003-178] {-4340 Cso If is obserable)