Question

Linear time-invariant sys tems. Consider a SISO, LTI system: t(t) Az(t) + bu(t) y(t) cr (t) with IC x(0) 3o. If the nominal input is a non-zero constant, i.e. if u(t)= u, under what conditions does there exist a constant nominal solu- tion (t) nominal output zero? Under what conditions do there exist constant nom- inal solutions that satisfy g = u for all u? To for some axo? Under what conditions is the corresponding

Answer 1

Dear students,

This question is about solving the SISO,LTI system.
Answer number 2 Given here, A SI LO LT I system is given as follows silt) Axe (++bult) -- 0 y (ta cults . (;) & alo)= no ŭ (t) z u ( A constant) Jaking the Laplace's transform. on both sides of eduation (i), we get, sx (d) - Xlo). Ax (s)+ b() Ljilt) + 3x18-xlo)) sxls)-80= Axls) + bu E a X!!! [S-A)= Hot by sex x(s): Hot boy S o SA

elm xls)Anot ben Xlsl not bu. - - (ni). s no (s not bu) (1) -A) SIA sloo- A) sls-A) S-A Slo-A) bu Ala-Al 6 1 ( Breaking it into s s - A I the particolo bu: Als-A) + Bijs (Multiplying both sides day slo-A)) Or bu= (A + B) 3 - ARA Comparing both sicks u get, Ast Byro & A, A = bu & Bio - As Putting these interes in the ediation (W), ne

shot by not bu - bu slo-A) (A-A) ALA-A) AS Orxlal: no bubu DAT Als-A) AS Xlo) - (notbul 1 - bu Jaking inverse Laplou's trensform on both of the sides ,we get 1* Xlo!= 1% [ not link to the high For alta no me get, no = (not be eat – bu ults Cultl=u (A constant) wy

Comparing both sides ,we get, nothu but no or bu bulo on [u: 1 Simularly, not be co ch noc buna Nominal output is given by yltia culti se ( mot bange at bin u Lt) .: According to the duestion, yltico och enot terug enten by con Ob (exo they (30 04hp (anty-o) - bu On (cnother bu?) + +[Acnot bea]

:: Comporing both sidas ,we get, актера на со + bu? (0» схо. Бас со : " (u-c) O c (rot by a bu? в “ мо+ bx НА НА o А4 bu АААА, har bu2 o A+ Lч & Acnot beuro Hмо thи го (), Ол |---- Рио

so output will be equal to zero for | 4 x + b ) Again, y lt= a ( for all conditions) or utby? - (not they eat-o Bu ut home - Senorben) (37 684) f )eo de Cuthy? . cno-beu) - At (enot bela) Color comparing both sides of the education, we utbu - Chorbeu o - (%) (tbcu.

butting from edication (Vi) to eduation ( we get, ut bu?e on st bis zo to the output ylt) is u. for b = -
Thanks and regards.