In control theory, a proper transfer function is a transfer function in which the degree of the numerator does not exceed the degree of the denominator. A strictly proper transfer function is a transfer function where the degree of the numerator is less than the degree of the denominator. The difference between the degree of the denominator (number of poles) and degree of the numerator (number of zeros) is the relative degree of the transfer function.
Example The following transfer function:
G ( s ) = N ( s ) D ( s ) = s 4 + n 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 {\displaystyle {\textbf {G}}(s)={\frac {{\textbf {N}}(s)}{{\textbf {D}}(s)}}={\frac {s^{4}+n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}}}
is proper, because
deg ( N ( s ) ) = 4 ≤ deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4\leq \deg({\textbf {D}}(s))=4} . is biproper, because
deg ( N ( s ) ) = 4 = deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4=\deg({\textbf {D}}(s))=4} . but is not strictly proper, because
deg ( N ( s ) ) = 4 ≮ deg ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=4\nless \deg({\textbf {D}}(s))=4} . The following transfer function is not proper (or strictly proper)
G ( s ) = N ( s ) D ( s ) = s 4 + n 1 s 3 + n 2 s 2 + n 3 s + n 4 d 1 s 3 + d 2 s 2 + d 3 s + d 4 {\displaystyle {\textbf {G}}(s)={\frac {{\textbf {N}}(s)}{{\textbf {D}}(s)}}={\frac {s^{4}+n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{d_{1}s^{3}+d_{2}s^{2}+d_{3}s+d_{4}}}}
because
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