Editing Closed-loop controller (section)

==Closed-loop transfer function==
{{Main|Closed-loop transfer function}}
The output of the system ''y''(''t'') is fed back through a sensor measurement ''F'' to a comparison with the reference value ''r''(''t''). The controller ''C'' then takes the error ''e'' (difference) between the reference and the output to change the inputs ''u'' to the system under control ''P''. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (''SISO'') control system; ''MIMO'' (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through [[coordinate vector|vectors]] instead of simple [[scalar (mathematics)|scalar]] values. For some [[distributed parameter systems]] the vectors may be infinite-[[Dimension (vector space)|dimensional]] (typically functions).

[[File:simple feedback control loop2.svg|center|A simple feedback control loop]]

If we assume the controller ''C'', the plant ''P'', and the sensor ''F'' are [[linear]] and [[time-invariant]] (i.e., elements of their [[transfer function]] ''C''(''s''), ''P''(''s''), and ''F''(''s'') do not depend on time), the systems above can be analysed using the [[Laplace transform]] on the variables. This gives the following relations:

: <math>Y(s) = P(s) U(s)</math>
: <math>U(s) = C(s) E(s)</math>
: <math>E(s) = R(s) - F(s)Y(s).</math>

Solving for ''Y''(''s'') in terms of ''R''(''s'') gives

: <math>Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s).</math>

The expression <math>H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}</math> is referred to as the ''closed-loop transfer function'' of the system. The numerator is the forward (open-loop) gain from ''r'' to ''y'', and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If <math>|P(s)C(s)| \gg 1</math>, i.e., it has a large [[norm (mathematics)|norm]] with each value of ''s'', and if <math>|F(s)| \approx 1</math>, then ''Y''(''s'') is approximately equal to ''R''(''s'') and the output closely tracks the reference input.