Question

10. Consider the nonlinear state equation di(t) = z{(t) – u(t)zy(t) + W T ult) éz(t) = x2(t) + 2x1(t) - u(t)xi(t) For homogeneous initial conditions, compute the nominal solution correspond- ing to the nominal input ū(t) = t for all t > 0. (Hint: search for monomial solutions in time.) Then, linearize the dynamics about the nominal solution. Is the linear system time-varying?

Answer 1

Ta Consider the non-linear equation 2, (t) = xilt) - ult)u(t) * Xift) rislt) - azit) + 2x1[t) - uLt) xi(t) For homogenous initial conditions compute the nominal solution using input ult/= t Let xi(t) = g xi(t) = dy - y - ty + 1/4 ole yhty (+-t) Divide by yz on both sides dk tk (it-t) = Let -1 = ka Differentiate on I both sides. I + 4 d = dk It is the equation standard form of linear of ist order. y (1.F) e. SQC1-F) du + IF - Alt-t)dtent-ty kltēt%) Svetem tot + ca s e a da to -ēt lettera a. zt dt de tot de

$(te%) -% te y = - 4 oth where is a constant For 6-0, y = qe jo y = 11 € tin teres xift)t siz (t) = X(t) + 2 *, (t) - ult)xi(t) - xy(t) + (2 - t)u.lt) dyzet) - xylt), (2-t)xılt) 1F - JE!) d et equation ) 2168) (et) - 5 €* (2-t) x.lt).dt XLt) - t xult) et . Jēt (at)t dt - Set (it-4) ot Integration 22 Set. tot - Stetot By Parts = 2 (-ēkt - 5-est) - (-ett-56-€“Jztat) 3. ? lett -et) + ett"72 (-e't-e)) .. étta ne X2(t): ť+cēt . Yes the linear system is time warying