Proper transfer function

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

deg ⁡ ( N ( s ) ) = 4 ≰ deg ⁡ ( D ( s ) ) = 3 {\displaystyle \deg({\textbf {N}}(s))=4\nleq \deg({\textbf {D}}(s))=3} .

A not proper transfer function can be made proper by using the method of long division.

The following transfer function is strictly proper

G ( s ) = N ( s ) D ( s ) = 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 {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}}}}

because

deg ⁡ ( N ( s ) ) = 3 < deg ⁡ ( D ( s ) ) = 4 {\displaystyle \deg({\textbf {N}}(s))=3<\deg({\textbf {D}}(s))=4} .

Implications

A proper transfer function will never grow unbounded as the frequency approaches infinity:

| G ( ± j ∞ ) | < ∞ {\displaystyle |{\textbf {G}}(\pm j\infty )|<\infty }

A strictly proper transfer function will approach zero as the frequency approaches infinity (which is true for all physical processes):

G ( ± j ∞ ) = 0 {\displaystyle {\textbf {G}}(\pm j\infty )=0}

Also, the integral of the real part of a strictly proper transfer function is zero.

References

  • Transfer functions - ECE 486: Control Systems Spring 2015, University of Illinois
  • ELEC ENG 4CL4: Control System Design Notes for Lecture #9, 2004, Dr. Ian C. Bruce, McMaster University