Continuous-time PID controller

PID_ct.sq

PID controllers, or proportional-integral-derivative controllers, are probably the most popular kind of linear single-input single-output controllers. This is justified by their simplicity and their effectiveness for a large class of systems. Taking as input the difference between the desired set-point and the measured system output ("error" e(t)), they have three terms with easy-to-understand effects which are added up, and three parameters to adjust their weights:

Weights can be specified either separately for the three terms, or as a global gain kp and two time values Ti and Td which do not depend on the gain of the system. PID_ct.sq uses the latter parameterization. The control signal u(t) is

u(t)=kp(e(t)+int(e(t))/Ti+Td.de/dt)

The transfer function of the controller K(s) = U(s)/E(s), where U(s) and E(s) are the Laplace transforms of y(t) and e(t), respectively, is

Y(s)/E(s)=kp(1+1/Ti.s+Td.s)

Translating the conceptual simplicity of the PID into an effective design is not always straightforward. PID_ct.sq displays the graphics where common specifications can be checked; you can manipulate the PID parameters, the controller gain (kp) in the Bode, Nyquist, or root locus diagram, or the time values of the integrator and the derivator in the Bode, root locus, or open-loop poles diagram.

Figures

The figures are the same as those defined for RST_ct.sq, except for the Open-Loop Zeros and Poles and the Closed-Loop Poles which are not defined.

Settings

The System, Sample Time, method for converting to digital controller, and Damping Specification have the same effect as the corresponding menu entries defined in RST_ct.sq. Two new entries are defined.

PID Coefficients

The three parameters of the PID (kp, Ti and ) can be edited in a dialog box. For P, PI, or PD controllers, set the parameter of the missing component to the empty matrix [].

No Derivator On Reference

When the input of the PID controller is the error between the set-point and the measured output, discontinuities of the set-point are derived by the derivator component of the PID and yield infinite values for the control signal. To avoid that, the set-point is usually not derived. The control signal is

u(t)=kp(e(t)+int(e(t))/Ti)-kp.Td.dy/dt)

When No Derivator On Reference is checked, the set-point is not derived.


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