# Dragon Notes

UNDER CONSTRUCTION
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# Control Systems:Routh Table

Routh Table

[Same-sign first column $$\ra$$ Stability]: # of sign changes in the first column $$\up =$$ # of poles lying in the right half-plane
[Same-sign first col. $$\up +$$ row of $$0$$'s $$\ra$$ Marginal stability]: no lhp/rhp poles imply $$j\omega$$ poles, hence oscillations

Epsilon Method

[First column zeros $$\up\ra\epsilon\hspace{3px}$$]: after completing the table, choose $$\epsilon$$ as positive or negative and observe sign changes for stability

Entire row of zeros

$$\bb{1}$$ Form polynomial using entries of row immediately above zeros' row as coefficients
$$\up\bb{2}$$ Start with power of $$s$$ in the label column, continue by skipping every other power of $$s$$
$$\up\bb{3}$$ Differentiate the polynomial w.r.t. $$s$$
$$\up\bb{4}$$ Replace the row of zeros with coefficients of obtained polynomial

[Purely odd/even polynomial factors $$\ra$$ row of zeros]: a row of zeros occurs when a purely even or purely odd polynomial factors the original polyn.
[Row of zeros $$\ra\up$$ root symmetry]: a purely even/odd polynomial's roots are $$(1)$$ symmetrical and real, $$(2)$$ symmetrical and imaginary, or $$(3)$$ quadrantal
[Root symmetry $$+\up$$ no rhp roots $$\ra$$ no lhp roots]: else symmetry's violated
[No row of zeros $$\ra\up$$ no $$j\omega$$ roots]: $$j\omega$$ roots are symmetric about origin (complex roots can still occur, non-quadrantal)
[No $$j\up\omega$$ roots above row of $$0$$'s]: polyn. formed by those rows lacks symmetry