Fall 2001 16.318-5 Modal form One particular useful canonical form is called the Modal Form It is a diagonal representation of the state-space model Assume for now that the transfer function has distinct real poles pi (but this easily extends to the case with complex poles (s)(s-p1)(s-p2)…(s-p P2 Now define a collection of first order systems, each with state i →1=P11+71 S- P1 X 2 p2℃2+T2 U(s P2 X n Cntrl . Which can be written as )=A(t)+Bu( (t)= car(t)+ du( th 4 B Good representation to use for numerical robustness reasonsFall 2001 16.31 8–5 Modal Form • One particular useful canonical form is called the Modal Form – It is a diagonal representation of the state-space model. • Assume for now that the transfer function has distinct real poles pi (but this easily extends to the case with complex poles) G(s) = N(s) D(s) = N(s) (s − p1)(s − p2)···(s − pn) = r1 s − p1 + r2 s − p2 + ··· + rn s − pn • Now define a collection of first order systems, each with state xi X1 U(s) = r1 s − p1 ⇒ x˙ 1 = p1x1 + r1u X2 U(s) = r2 s − p2 ⇒ x˙ 2 = p2x2 + r2u . . . Xn U(s) = rn s − pn ⇒ x˙ n = pnxn + rnu • Which can be written as x˙(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) with A =   p1 ... pn   B =   r1 . . . rn   C =   1 . . . 1   T • Good representation to use for numerical robustness reasons