Strictly proper

In control theory, a strictly proper transfer function is a transfer function where the degree of the numerator is less than the degree of the denominator.

Example

The following transfer function is not strictly 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:$deg(\; extbf\{N\}(s))\; =\; 4\; less\; deg(\; extbf\{D\}(s))\; =\; 4$.

The following transfer function however, is strictly proper:$extbf\{G\}(s)\; =\; frac\{\; extbf\{N\}(s)\}\{\; extbf\{D\}(s)\}\; =\; frac\{n\_\{1\}s^\{3\}\; +\; n\_\{2\}s^\{2\}\; +\; n\_\{3\}s\; +\; n\_\{4\{s^\{4\}\; +\; d\_\{1\}s^\{3\}\; +\; d\_\{2\}s^\{2\}\; +\; d\_\{3\}s\; +\; d\_\{4$because:.

Implications

A strictly proper transfer function will approach zero as the frequency approaches infinity.:$extbf\{G\}(pm\; jinfty)\; =\; 0$which is true for all physical processes.

See also

* Proper transfer function