Sysquake Pro – Table of Contents

Sysquake – Table of Contents

Sysquake for LaTeX – Table of Contents

PID_ct.sq

Continuous-time PID controller

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 r(t) and the measured system output y(t) ("error" e(t)=r(t)-y(t)), they have three terms with easy-to-understand effects which are added up, and three parameters to adjust their weights:

• a proportional term (the larger the error, the larger the control signal to reduce it);
• an integral term (if a nonzero control signal is required to cancel out the error, the control signal is increased until the error vanishes);
• a derivative term (the evolution of the error is anticipated to increase damping).

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

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

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.

For set-point tracking, filtering the same way the measured output and the set-point by considering only the error e(t)=y(t)-r(t) does not give a good transient behavior when the set-point is discontinuous. The set-point is usually not differentiated. In addition, the proportional term of the controller kp applied to the set-point can be reduced by a factor b smaller than 1. A third common improvement is to filter the derivative term to limit the amplification of noise at high frequencies (this is actually required to have a causal controller); the filter is parameterized with a number N, typically between 10 and 20, which is the bandwidth of the effect of the derivator term. In the Laplace domain, the control signal is

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, Sampling Period, 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 differentiated by the derivator component of the PID and yield infinite values for the control signal (see above).

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

Display Frequency Line

When selected, moving the mouse above a frequency response (Bode or sensitivity) will display a corresponding line in other frequency responses, Nyquist diagrams, and zero/pole diagrams.