Find a PI controller for a plant in state space form:

Or for the plant as a transfer function:

Specify the type of controller:

Proportional:

Proportional integral:

Proportional integral and derivative:

Filtered version of PID:

Evaluate the performance of the resulting controller:

Reference following:

Disturbance rejection:

Get the PID parameters in different standard forms suitable for different implementation technologies:

Ideal form:

Series form:

Parallel form:

Obtain the SystemsModelControllerData object and extract a property from it:

Get the property directly:

Get a list of properties:

Find the computed controllers, PID feedback controller, and feedforward filter:

The feedback controller transfer function and the PID controller:

By default, there is no feedforward filter:

Find closed loop transfer functions from reference, process disturbance, and sensor noise to output:

The reference to output transfer function measures the ability to follow reference changes:

The disturbance to output transfer function measures the ability to reject process disturbances:

The sensor noise to output transfer function measures the ability to reject measurement noise:

Find transfer functions from reference, process disturbance, and sensor noise to control output:

The reference to control transfer function measures the control effort to follow reference:

The disturbance to output transfer function measures the control effort to reject the disturbance:

The sensor noise to output transfer function measures the control effort to reject sensor noise:

By default, the controller architecture is a proportional integrating controller:

Specify the controller architecture:

Proportional ("P") controller:

Proportional integral ("PI") controller:

Proportional derivative ("PD") controller:

Proportional integral derivative ("PID") controller:

Filtered PD ("PFD") controller:

Filtered PID ("PIFD") controller:

Use integral action to eliminate steady-state reference following error:

Derivative action may result in a faster reference response:

A faster response is accompanied by higher control effort:

The tuning rule is automatically determined:

The property "TuningRule" gives the tuning rule that was used:

Obtain the automatically selected tuning rule for a given lsys and "carch":

The "CohenCoon" tuning rule allows for PD and PFD architectures:

Controllers designed with the "LoopShaping" rule result in a maximum sensitivity of about 1.4:

The Nyquist plot of the open-loop transfer function lies outside the sensitivity circle of radius 1/1.4:

The "TyreusLuyben" rule may give a stabilizing controller for an unstable system:

The "ZieglerNicholsFrequency" rule may also stabilize the system:

The default feedforward transfer function is unity:

This may result in a large overshoot to step reference inputs:

Use the PIDFeedforward option to improve the tracking performance:

Automatically compute the reference weights for the feedforward filter:

The weights:

The feedforward filter transfer function:

Specify the derivative filter parameter:

Automatically compute the derivative filter parameter and obtain its value:

Specify a parameter estimation method:

Different estimation methods lead to different controllers:

For a system with three cascaded water tanks, find a PID controller for a constant water level in tank 3:

The transfer function from the feed rate in tank 1 to the liquid level in tank 3:

Compute a liquid-level PID controller for the system:

The reference response from the input to the first tank to the water level in the third tank:

The PID controller also improves the gain and phase margins:

An isothermal continuously stirred tank reactor (CSTR):

The transfer function from the dilution rate to the product concentration:

Compute a PID controller that controls the product concentration:

The control effort for a set-point change of 0.1 gmol/liter in the product concentration:

The achieved product concentration:

Use a tuning rule, which gives a faster response:

A faster response is accompanied by an increased peak control effort:

A DC motor has the transfer function model below. Find a PID position controller and simulate its reference and disturbance responses:

Reference response:

Disturbance response, typically from varying loads:

Reference as well as disturbance:

Using a different tuning rule:

In a continuous stirred-tank reactor where the reaction occurs, the objective is to cause the temperature to track a desired trajectory by using the fluid temperature as the control input:»

The nonlinear model of the system:

The reference trajectory is modeled as the response of a first-order system:

The open-loop system does not track the desired trajectory:

Design a PID controller and compute the response of the nonlinear system with the controller:

The response of the linearized system with the controller:

Compare the responses: