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.

Example

The following transfer function is proper:$extbf\{G\}(s)\; =\; frac\{\; extbf\{N\}(s)\}\{\; extbf\{D\}(s)\}\; =\; frac\{s^\{4\}\; +\; n\_\{1\}s^\{3\}\; +\; n\_\{2\}s^\{2\}\; +\; n\_\{3\}s\; +\; n\_\{4\{s^\{4\}\; +\; d\_\{1\}s^\{3\}\; +\; d\_\{2\}s^\{2\}\; +\; d\_\{3\}s\; +\; d\_\{4$because:$deg(\; extbf\{N\}(s))\; =\; 4\; leq\; deg(\; extbf\{D\}(s))\; =\; 4$.The following transfer function however, is not proper:$extbf\{G\}(s)\; =\; frac\{\; extbf\{N\}(s)\}\{\; extbf\{D\}(s)\}\; =\; frac\{s^\{4\}\; +\; n\_\{1\}s^\{3\}\; +\; n\_\{2\}s^\{2\}\; +\; n\_\{3\}s\; +\; n\_\{4\{d\_\{1\}s^\{3\}\; +\; d\_\{2\}s^\{2\}\; +\; d\_\{3\}s\; +\; d\_\{4$because:$deg(\; extbf\{N\}(s))\; =\; 4\; leq\; deg(\; extbf\{D\}(s))\; =\; 3$.

Implications

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

See also

* Strictly proper