The function **has a zero ats**=−4 and poles ats=−2 ands=−3. Note that T (∞)=0. Because of this, some texts would say that T (s) **hasazeroats**=∞.

As an example, consider the function T (s)=4 s/4+1 s 2 /6+5s/6+1 =4 s/4+1 (s/2+1)(s/3+1) The function **has a zero ats**=−4 and poles ats=−2 ands=−3.No tet hat T (∞)=0. Because of this, some texts would say that T (s) **hasazeroats**=∞.

We say that G (s) **hasazeroats**=1. In fact, G (s) has=n m\zerosat1."In this waya proper transfer function has exactlyn-poles andn-zeros counting zeros atinnity.

Poles and zeros Zeros IfP (s) = y (s) u (s) **hasazeroats**=s 0, then for all inputs u (s 0) the output y (s 0) =0. This is equivalent to, 0=C (s 0 I A) 1 B+D Using a state-space Laplacedomain form (ats=s 0) we have, s 0 x (s 0) =Ax (s 0) +Bu (s 0) 0 =Cx (s 0) +Du (s 0) This can be rearranged into matrix form, * (s 0 I A) B C D ** x (s 0) u ...

... of the estimate ^ K ik is given by (12), and the vector of parameters can betaken as =[p r;:::;p 0;q n 1;:::;q 0] T: (15) From hydrodynamic properties of the model under study, it follows that the problem (13)-(14) must be considered subject to the following constraints (Perez andFossen, 2008b): ^ K ik (s) **hasazeroats**=0 ...

(If (f,s) hasapoleats=1, thenr>0 and if (f,s) **hasazeroats**=1 thenr<0, otherwiser=0. ) Exercise 6a) Letr 0 (f) be the order of pole or zero of (f,s) ats=0.