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# ltiblock.pid2

Tunable two-degree-of-freedom PID controller

## Syntax

blk = ltiblock.pid2(name,type)
blk = ltiblock.pid2(name,type,Ts)
blk = ltiblock.pid2(name,sys)

## Description

Model object for creating tunable two-degree-of-freedom PID controllers. ltiblock.pid2 lets you parametrize a tunable SISO two-degree-of-freedom PID controller. You can use this parametrized controller for parameter studies or for automatic tuning with Robust Control Toolbox™ tuning commands such as systune, looptune, or hinfstruct.

ltiblock.pid2 is part of the family of parametric Control Design Blocks. Other parametric Control Design Blocks include ltiblock.gain, ltiblock.ss, and ltiblock.tf.

## Construction

blk = ltiblock.pid2(name,type) creates the two-degree-of-freedom continuous-time PID controller described by the equation:

$u={K}_{p}\left(br-y\right)+\frac{{K}_{i}}{s}\left(r-y\right)+\frac{{K}_{d}s}{1+{T}_{f}s}\left(cr-y\right).$

r is the setpoint command, y is the measured response to that setpoint, and u is the control signal, as shown in the following illustration.

The tunable parameters of the block are:

• Scalar gains Kp, Ki, and Kd

• Filter time constant Tf

• Scalar weights b and c

The string type sets the controller type by fixing some of these values to zero (see Input Arguments).

blk = ltiblock.pid2(name,type,Ts) creates a discrete-time PID controller with sampling time Ts. The equation describing this controller is:

$u={K}_{p}\left(br-y\right)+{K}_{i}IF\left(z\right)\left(r-y\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}\left(cr-y\right).$

IF(z) and DF(z) are the discrete integrator formulas for the integral and derivative terms, respectively. The values of the IFormula and DFormula properties set the discrete integrator formulas (see Properties).

blk = ltiblock.pid2(name,sys) uses the dynamic system model, sys, to set the sampling time, Ts, and the initial values of all the tunable parameters. The model sys must be compatible with the equation of a two-degree-of-freedom PID controller.

### Input Arguments

name

PID controller Name, specified as a string. (See Properties.)

type

Controller type, specified as a string. Specifying a controller type fixes up to three of the PID controller parameters. type can take the following values:

StringController TypeEffect on PID Parameters
'P'Proportional onlyKi and Kd are fixed to zero; Tf is fixed to 1; Kp is free
'PI'Proportional-integralKd is fixed to zero; Tf is fixed to 1; Kp and Ki are free
'PD'Proportional-derivative with first-order filter on derivative actionKi is fixed to zero; Kp, Kd, and Tf are free
'PID'Proportional-integral-derivative with first-order filter on derivative actionKp, Ki, Kd, and Tf are free

Ts

Sampling time, specified as a scalar.

sys

Dynamic system model representing a two-degree-of-freedom PID controller.

## Properties

Kp,Ki,Kd,Tf,b,c

Parametrization of the PID gains Kp, Ki, Kd, the filter time constant, Tf, and the scalar gains, b and c.

The following fields of blk.Kp, blk.Ki, blk.Kd, blk.Tf, blk.b, and blk.c are used when you tune blk using a tuning command such as systune:

FieldDescription
ValueCurrent value of the parameter. blk.b.Value, and blk.c.Value are always nonnegative.
FreeLogical value determining whether the parameter is fixed or tunable. For example,
• If blk.Kp.Free = 1, then blk.Kp.Value is tunable.

• If blk.Kp.Free = 0, then blk.Kp.Value is fixed.

MinimumMinimum value of the parameter. This property places a lower bound on the tuned value of the parameter. For example, setting blk.Kp.Minimum = 0 ensures that Kp remains positive.
blk.Tf.Minimum must always be positive.
MaximumMaximum value of the parameter. This property places an upper bound on the tuned value of the parameter. For example, setting blk.c.Maximum = 1 ensures that c does not exceed unity.

blk.Kp, blk.Ki, blk.Kd, blk.Tf, blk.b, and blk.c are param.Continuous objects. For more information about the properties of these param.Continuous objects, see the param.Continuous object reference page.

IFormula, DFormula

Strings setting the discrete integrator formulas IF(z) and DF(z) for the integral and derivative terms, respectively. IFormula and DFormula can have the following values:

StringIF(z) or DF(z) Formula
'ForwardEuler'

$\frac{{T}_{s}}{z-1}$

'BackwardEuler'

$\frac{{T}_{s}z}{z-1}$

'Trapezoidal'

$\frac{{T}_{s}}{2}\frac{z+1}{z-1}$

Default: 'ForwardEuler'

Ts

Sampling time. For continuous-time models, Ts = 0. For discrete-time models, Ts is a positive scalar representing the sampling period. This value is expressed in the unit specified by the TimeUnit property of the model. To denote a discrete-time model with unspecified sampling time, set Ts = -1.

Changing this property does not discretize or resample the model. Use c2d and d2c to convert between continuous- and discrete-time representations. Use d2d to change the sampling time of a discrete-time system.

Default: 0 (continuous time)

TimeUnit

String representing the unit of the time variable. This property specifies the units for the time variable, the sampling time Ts, and any time delays in the model. Use any of the following values:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

Default: 'seconds'

InputName

Input channel names. Set InputName to a string for single-input model. For a multi-input model, set InputName to a cell array of strings.

Alternatively, use automatic vector expansion to assign input names for multi-input models. For example, if sys is a two-input model, enter:

`sys.InputName = 'controls';`

The input names automatically expand to {'controls(1)';'controls(2)'}.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Input channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

Default: Empty string '' for all input channels

InputUnit

Input channel units. Use InputUnit to keep track of input signal units. For a single-input model, set InputUnit to a string. For a multi-input model, set InputUnit to a cell array of strings. InputUnit has no effect on system behavior.

Default: Empty string '' for all input channels

InputGroup

Input channel groups. The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example:

```sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];```

creates input groups named controls and noise that include input channels 1, 2 and 3, 5, respectively. You can then extract the subsystem from the controls inputs to all outputs using:

`sys(:,'controls')`

Default: Struct with no fields

OutputName

Output channel names. Set OutputName to a string for single-output model. For a multi-output model, set OutputName to a cell array of strings.

Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, enter:

`sys.OutputName = 'measurements';`

The output names to automatically expand to {'measurements(1)';'measurements(2)'}.

You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Output channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

Default: Empty string '' for all input channels

OutputUnit

Output channel units. Use OutputUnit to keep track of output signal units. For a single-output model, set OutputUnit to a string. For a multi-output model, set OutputUnit to a cell array of strings. OutputUnit has no effect on system behavior.

Default: Empty string '' for all input channels

OutputGroup

Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. For example:

```sys.OutputGroup.temperature = [1];
sys.InputGroup.measurement = [3 5];```

creates output groups named temperature and measurement that include output channels 1, and 3, 5, respectively. You can then extract the subsystem from all inputs to the measurement outputs using:

`sys('measurement',:)`

Default: Struct with no fields

Name

System name. Set Name to a string to label the system.

Default: ''

Notes

Any text that you want to associate with the system. Set Notes to a string or a cell array of strings.

Default: {}

UserData

Any type of data you wish to associate with system. Set UserData to any MATLAB® data type.

Default: []

## Examples

Tunable Two-Degree-of-Freedom Controller with a Fixed Parameter

Create a tunable two-degree-of-freedom PD controller. Then, initialize the parameter values, and fix the filter time constant.

```blk = ltiblock.pid2('pdblock','PD');
blk.b.Value = 1;
blk.c.Value = 0.5;
blk.Tf.Value = 0.01;
blk.Tf.Free = false;
blk```
```blk =

Parametric continuous-time 2-DOF PID controller "pdblock" with equation:

s
u = Kp (b*r-y) + Kd -------- (c*r-y)
Tf*s+1

where r,y are the controller inputs and Kp, Kd, b, c are tunable gains.

Type "showBlockValue(blk)" to see the current value and "get(blk)" to see all properties.
```

Controller Initialized by Dynamic System Model

Create a tunable two-degree-of-freedom PI controller. Use a two-input, one-output tf model to initialize the parameters and other properties.

```s = tf('s');
Kp = 10;
Ki = 0.1;
b = 0.7;
sys = [(b*Kp + Ki/s), (-Kp - Ki/s)];
blk = ltiblock.pid2('PI2dof',sys)```
```blk =

Parametric continuous-time 2-DOF PID controller "PI2dof" with equation:

1
u = Kp (b*r-y) + Ki --- (r-y)
s

where r,y are the controller inputs and Kp, Ki, b are tunable gains.

Type "showBlockValue(blk)" to see the current value and "get(blk)" to see all properties.
```

blk takes initial parameter values from sys.

If sys is a discrete-time system, blk takes the value of properties, such as Ts and IFormula, from sys.

Controller with Named Inputs and Output

Create a tunable PID controller, and assign names to the inputs and output.

```blk = ltiblock.pid2('pidblock','pid');
blk.InputName = {'reference','measurement'};
blk.OutputName = {'control'};```

blk.InputName is a cell array containing two strings, because a two-degree-of-freedom PID controller has two inputs.