Fall 2001 163110-7 Zeros in State Space models Roots of the transfer function numerator are called the system zeros Need to develop a similar way of defining/computing them using a state space model Zero: is a generalized frequency so for which the system can have a non-zero input u(t)=uoe5ot but exactly zero output y(t)=0Vt Note that there is a specific initial condition associated with this response 0, so the state response is of the form a(t)=coe5o →y(t)≡0 Given i= Az+ Bu, substitute the above to get tosoesot atoe sot Buoesot =[soI-A-BI Also have that y= C +Du=0 which gives 0 C D So we must solve for the so that solves: or I-A-B This is a generalized eigenvalue problem that can be solved in MATLAB using eig.m or tzero.m MaTLAB is a trademark of the mathworks IncFall 2001 16.31 10–7 Zeros in State Space Models • Roots of the transfer function numerator are called the system zeros. – Need to develop a similar way of defining/computing them using a state space model. • Zero: is a generalized frequency s0 for which the system can have a non-zero input u(t) = u0es0t , but exactly zero output y(t) ≡ 0 ∀t – Note that there is a specific initial condition associated with this response x0, so the state response is of the form x(t) = x0es0t u(t) = u0es0t ⇒ x(t) = x0es0t ⇒ y(t) ≡ 0 • Given ˙x = Ax + Bu, substitute the above to get: x0s0es0t = Ax0es0t + Bu0es0t ⇒  s0I − A −B  x0 u0 = 0 • Also have that y = Cx + Du = 0 which gives: Cx0es0t + Du0es0t = 0 →  C D  x0 u0 = 0 • So we must solve for the s0 that solves: or s0I − A −B C D x0 u0 = 0 – This is a generalized eigenvalue problem that can be solved in MATLABr using eig.m or tzero.m 2 2MATLAB
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