Question Solved1 Answer Problem 1. Consider a third order system in the controllable canonical form dx(t) dt Ar(t) + Bu(t) x(t) + 0 1 -2 -2 u(t); y(t) = Cx(t) Y( = The following three are known about the transfer function G(s) VE), (i) The unit step response is asymptotic to 4, i.e. Yss = limt y(t) 4, and (ii) The unit step response satisfies y(0) = 0 and die 2–0 = 1, and (iii) Zeros of G(s) are repeated, i.e. z1 = 22. Obtain C in the output equation.

QEIYRU The Asker · Advanced Mathematics

Transcribed Image Text: Problem 1. Consider a third order system in the controllable canonical form dx(t) dt Ar(t) + Bu(t) x(t) + 0 1 -2 -2 u(t); y(t) = Cx(t) Y( = The following three are known about the transfer function G(s) VE), (i) The unit step response is asymptotic to 4, i.e. Yss = limt y(t) 4, and (ii) The unit step response satisfies y(0) = 0 and die 2–0 = 1, and (iii) Zeros of G(s) are repeated, i.e. z1 = 22. Obtain C in the output equation.
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Transcribed Image Text: Problem 1. Consider a third order system in the controllable canonical form dx(t) dt Ar(t) + Bu(t) x(t) + 0 1 -2 -2 u(t); y(t) = Cx(t) Y( = The following three are known about the transfer function G(s) VE), (i) The unit step response is asymptotic to 4, i.e. Yss = limt y(t) 4, and (ii) The unit step response satisfies y(0) = 0 and die 2–0 = 1, and (iii) Zeros of G(s) are repeated, i.e. z1 = 22. Obtain C in the output equation.
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=> standard form of controllable canonical form of order 3 is{:[A=[[0,1,0],[0,0,1],[-a_(3)-a_(2),-a_(1)]]quad B=[[0],[0],[1]]quad c=[[b_(3)-a_(3)b_(0),b_(2)-a_(2)b_(0),b_(1)-a_(1)b_(0)]]],[" where "G(s)=(b_(0)s^(3)+b_(1)s^(2)+b_(2)s+b_(3))/(s^(3)+a_(1)s^(2)+a_(2)s+a_(3))]:}from give A matrix, the values of{:[a_(1)","a_(2)","a_(3)" arl: "a_(3)=1","a_(2)=2","a_(1)=2],[=>G(s)=(b_(0)s^(3)+b_(1)s^(2)+b_(2)s+b_(3))/(s^(3)+2s^(2)+2s+1)]:}(i) unit step respouse i.e u(s)=(1)/(5){:[y(s)=G(s)*u(s)],[=(b_(0)s^(3)+b_(1)s^(2)+b_(2)s+b_(3))/(s^(3)+2s^(2)+2s+1)xx(1)/(s)],[y_(ss)=lim_(t rarr oo)y(t)=4]:}in s-domain{:[y_(ss)=lims_(s rarr0)y(s)=4],[=lim_(s rarr0)s[(b_(0)s^(3)+b_(1)s^(2)+b_(2)s+b_(3))/(s^(3)+2s^(2)+2s+1)xx(1) ... See the full answer