Main Content

# pid

PID controller in parallel form

## Description

Use `pid` to create parallel-form proportional-integral-derivative (PID) controller model objects, or to convert dynamic system models to parallel PID controller form.

The `pid` controller model object can represent parallel-form PID controllers in continuous time or discrete time.

• Continuous time — $C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}$

• Discrete time — $C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}$

Here:

• Kp is the proportional gain.

• Ki is the integral gain.

• Kd is the derivative gain.

• Tf is the first-order derivative filter time constant.

• IF(z) is the integrator method for computing the integral in discrete-time controller.

• DF(z) is the integrator method for computing the derivative filter in discrete-time controller.

You can then combine this object with other components of a control architecture, such as the plant, actuators, and sensors to represent your control system. For more information, see Control System Modeling with Model Objects.

You can create a PID controller model object by either specifying the controller parameters directly, or by converting a model of another type (such as a transfer function model `tf`) to PID controller form.

You can also use `pid` to create generalized state-space (`genss`) models or uncertain state-space (`uss` (Robust Control Toolbox)) models.

## Creation

You can obtain `pid` controller models in one of the following ways.

• Create a model using the `pid` function.

• Use the `pidtune` function to tune PID controllers for a plant model. Specify a 1-DOF PID controller type in the `type` argument of the `pidtune` function to obtain a parallel-form PID controller. For example:

```sys = zpk([],[-1 -1 -1],1); C = pidtune(sys,'PID');```
• Interactively tune the PID controller for a plant model using:

### Syntax

``C = pid(Kp,Ki,Kd,Tf)``
``C = pid(Kp,Ki,Kd,Tf,Ts)``
``C = pid(Kp)``
``C = pid(Kp,Ki)``
``C = pid(Kp,Ki,Kd)``
``C = pid(___,Name,Value)``
``C = pid``
``C = pid(sys)``

### Description

example

````C = pid(Kp,Ki,Kd,Tf)` creates a continuous-time parallel-form PID controller model and sets the properties `Kp`, `Ki`, `Kd`, and `Tf`. The remaining properties have default values.```

example

````C = pid(Kp,Ki,Kd,Tf,Ts)` creates a discrete-time PID controller model with sample time `Ts`.```

example

````C = pid(Kp)` creates a continuous-time proportional (P) controller.```

example

````C = pid(Kp,Ki)` creates a proportional and integral (PI) controller.```

example

````C = pid(Kp,Ki,Kd)` creates a proportional, integral, and derivative (PID) controller.```

example

````C = pid(___,Name,Value)` sets properties of the `pid` controller object specified using one or more `Name,Value` arguments for any of the previous input-argument combinations.```

example

````C = pid` creates a controller object with default property values. To modify the property of the controller model, use dot notation.```

example

````C = pid(sys)` converts the dynamic system model `sys` to a parallel-form `pid` controller object.```

### Input Arguments

expand all

Proportional gain, specified as a real and finite value or a tunable object.

• To create a `pid` controller object, use a real and finite scalar value.

• To create an array of `pid` controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (`realp`) or generalized matrix (`genmat`).

• To create a tunable gain-scheduled controller model, use a tunable surface created using `tunableSurface`.

Integral gain, specified as a real and finite value or a tunable object.

• To create a `pid` controller object, use a real and finite scalar value.

• To create an array of `pid` controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (`realp`) or generalized matrix (`genmat`).

• To create a tunable gain-scheduled controller model, use a tunable surface created using `tunableSurface`.

Derivative gain, specified as a real and finite value or a tunable object.

• To create a `pid` controller object, use a real and finite scalar value.

• To create an array of `pid` controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (`realp`) or generalized matrix (`genmat`).

• To create a tunable gain-scheduled controller model, use a tunable surface created using `tunableSurface`.

Time constant of the first-order derivative filter, specified as a real and finite value or a tunable object.

• To create a `pid` controller object, use a real and finite scalar value.

• To create an array of `pid` controller objects, use an array of real and finite values.

• To create a tunable controller model, use a tunable parameter (`realp`) or generalized matrix (`genmat`).

• To create a tunable gain-scheduled controller model, use a tunable surface created using `tunableSurface`.

Sample time, specified as:

• `0` for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. Specify `Ts` in the time unit specified by the `TimeUnit` property.

In an array of `pid` controllers, the same `Ts` applies to all controllers.

PID controller models do not support unspecified sample time (`Ts = -1`).

Note

Changing `Ts` does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use `c2d` and `d2c`. To change the sample time of a discrete-time system, use `d2d`.

The discrete integrator formulas of the discretized controller depend upon the `c2d` discretization method you use, as shown in this table.

`c2d` Discretization Method`IFormula``DFormula`
`'zoh'``ForwardEuler``ForwardEuler`
`'foh'``Trapezoidal``Trapezoidal`
`'tustin'``Trapezoidal``Trapezoidal`
`'impulse'``ForwardEuler``ForwardEuler`
`'matched'``ForwardEuler``ForwardEuler`

For more information about `c2d` discretization methods, see `c2d`.

If you require different discrete integrator formulas, you can discretize the controller by directly setting `Ts`, `IFormula`, and `DFormula` to the desired values. However, this method does not compute new gain and filter-constant values for the discretized controller. Therefore, this method might yield a poorer match between the continuous-time and discrete-time PID controllers than using `c2d`.

Dynamic system, specified as a SISO dynamic system model or array of SISO dynamic system models. Dynamic systems that you can use include:

### Output Arguments

expand all

PID controller model, returned as:

• A parallel-form PID controller (`pid`) model object, when all the gains have numeric values. When the gains are numeric arrays, `C` is an array of `pid` controller objects.

• A generalized state-space model (`genss`) object, when the `numerator` or `denominator` input arguments includes tunable parameters, such as `realp` parameters or generalized matrices (`genmat`).

• An uncertain state-space model (`uss`) object, when the `numerator` or `denominator` input arguments includes uncertain parameters. Using uncertain models requires Robust Control Toolbox™ software.

## Properties

expand all

PID controller coefficients, specified as scalars. When creating a `pid` controller object or array of objects, specify these coefficients in the `Kp`, `Ki`, `Kd`, and `Tf` input arguments.

Discrete integrator formula IF(z) for the integrator of the discrete-time `pid` controller:

`$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$`

Specify `IFormula` as one of the following:

• `'ForwardEuler'`IF(z) = $\frac{{T}_{s}}{z-1}.$

This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the `ForwardEuler` formula can result in instability, even when discretizing a system that is stable in continuous time.

• `'BackwardEuler'`IF(z) = $\frac{{T}_{s}z}{z-1}.$

An advantage of the `BackwardEuler` formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.

• `'Trapezoidal'`IF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$

An advantage of the `Trapezoidal` formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the `Trapezoidal` formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.

When `C` is a continuous-time controller, `IFormula` is `''`.

Discrete integrator formula DF(z) for the derivative filter of the discrete-time `pid` controller:

`$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$`

Specify `DFormula` as one of the following:

• `'ForwardEuler'`DF(z) = $\frac{{T}_{s}}{z-1}.$

This formula is best for small sample time, where the Nyquist limit is large compared to the bandwidth of the controller. For larger sample time, the `ForwardEuler` formula can result in instability, even when discretizing a system that is stable in continuous time.

• `'BackwardEuler'`DF(z) = $\frac{{T}_{s}z}{z-1}.$

An advantage of the `BackwardEuler` formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result.

• `'Trapezoidal'`DF(z) = $\frac{{T}_{s}}{2}\frac{z+1}{z-1}.$

An advantage of the `Trapezoidal` formula is that discretizing a stable continuous-time system using this formula always yields a stable discrete-time result. Of all available integration formulas, the `Trapezoidal` formula yields the closest match between frequency-domain properties of the discretized system and the corresponding continuous-time system.

The `Trapezoidal` value for `DFormula` is not available for a `pid` controller with no derivative filter (`Tf = 0`).

When `C` is a continuous-time controller, `DFormula` is `''`.

This property is read-only.

Time delay on the system input. `InputDelay` is always 0 for a `pid` controller object.

This property is read-only.

Time delay on the system output. `OutputDelay` is always 0 for a `pid` controller object.

Sample time, specified as:

• `0` for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. `Ts` in specified in the time unit specified by the `TimeUnit` property.

If `pid` is an array of PID controllers, the same `Ts` applies to all controllers.

Time variable units, specified as one of the following:

• `'nanoseconds'`

• `'microseconds'`

• `'milliseconds'`

• `'seconds'`

• `'minutes'`

• `'hours'`

• `'days'`

• `'weeks'`

• `'months'`

• `'years'`

Changing `TimeUnit` has no effect on other properties, but changes the overall system behavior. Use `chgTimeUnit` to convert between time units without modifying system behavior.

Input channel name, specified as one of the following:

• Character vector

• `''` (no name specified)

Alternatively, assign the name `error` to the input of a controller model `C` as follows.

`C.InputName = 'error';`

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

Use `InputName` to:

• Identify channels on model display and plots.

• Specify connection points when interconnecting models.

Input channel units, specified as one of the following:

• A character vector.

• `''`, no units specified.

Use `InputUnit` to specify input signal units. `InputUnit` has no effect on system behavior.

For example, assign the concentration units `'mol/m^3'` to the input of a controller model `C` as follows.

`C.InputUnit = 'mol/m^3';`

Input channel groups. This property is not needed for PID controller models.

By default, `InputGroup` is a structure with no fields.

Output channel name, specified as one of the following:

• A character vector.

• `''`, no name specified.

For example, assign the name `'control'` to the output of a controller model `C` as follows.

`C.OutputName = 'control';`

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

Use `OutputName` to:

• Identify channels on model display and plots.

• Specify connection points when interconnecting models.

Output channel units, specified as one of the following:

• A character vector.

• `''`, no units specified.

Use `OutputUnit` to specify output signal units. `OutputUnit` has no effect on system behavior.

For example, assign the unit `'volts'` to the output of a controller model `C` as follows.

`C.OutputUnit = 'volts';`

Output channel groups. This property is not needed for PID controller models.

By default, `OutputGroup` is a structure with no fields.

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, `'System is MIMO'`.

User-specified data that you want to associate with the system, specified as any MATLAB data type.

System name, specified as a character vector. For example, `'system_1'`.

Sampling grid for model arrays, specified as a structure array.

Use `SamplingGrid` to track the variable values associated with each model in a model array, including identified linear time-invariant (IDLTI) model arrays.

Set the field names of the structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 11-by-1 array of linear models, `sysarr`, by taking snapshots of a linear time-varying system at times `t = 0:10`. The following code stores the time samples with the linear models.

` sysarr.SamplingGrid = struct('time',0:10)`

Similarly, you can create a 6-by-9 model array, `M`, by independently sampling two variables, `zeta` and `w`. The following code maps the `(zeta,w)` values to `M`.

```[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>) M.SamplingGrid = struct('zeta',zeta,'w',w)```

When you display `M`, each entry in the array includes the corresponding `zeta` and `w` values.

`M`
```M(:,:,1,1) [zeta=0.3, w=5] = 25 -------------- s^2 + 3 s + 25 M(:,:,2,1) [zeta=0.35, w=5] = 25 ---------------- s^2 + 3.5 s + 25 ...```

For model arrays generated by linearizing a Simulink® model at multiple parameter values or operating points, the software populates `SamplingGrid` automatically with the variable values that correspond to each entry in the array. For instance, the Simulink Control Design™ commands `linearize` (Simulink Control Design) and `slLinearizer` (Simulink Control Design) populate `SamplingGrid` automatically.

By default, `SamplingGrid` is a structure with no fields.

## Object Functions

The following lists contain a representative subset of the functions you can use with `pid` models. In general, any function applicable to Dynamic System Models is applicable to a `pid` object.

expand all

 `step` Step response of dynamic system `impulse` Impulse response plot of dynamic system; impulse response data `lsim` Plot simulated time response of dynamic system to arbitrary inputs; simulated response data `bode` Bode plot of frequency response, or magnitude and phase data `nyquist` Nyquist plot of frequency response `nichols` Nichols chart of frequency response `bandwidth` Frequency response bandwidth
 `pole` Poles of dynamic system `zero` Zeros and gain of SISO dynamic system `pzplot` Pole-zero plot of dynamic system model with additional plot customization options `margin` Gain margin, phase margin, and crossover frequencies
 `zpk` Zero-pole-gain model `ss` State-space model `c2d` Convert model from continuous to discrete time `d2c` Convert model from discrete to continuous time `d2d` Resample discrete-time model
 `feedback` Feedback connection of multiple models `connect` Block diagram interconnections of dynamic systems `series` Series connection of two models `parallel` Parallel connection of two models
 `pidtune` PID tuning algorithm for linear plant model `rlocus` Root locus plot of dynamic system `piddata` Access coefficients of parallel-form PID controller `make2DOF` Convert 1-DOF PID controller to 2-DOF controller `pidTuner` Open PID Tuner for PID tuning `tunablePID` Tunable PID controller

## Examples

collapse all

Create a continuous-time controller with proportional and derivative gains and a filter on the derivative term. To do so, set the integral gain to zero. Set the other gains and the filter time constant to the desired values.

```Kp = 1; Ki = 0; % No integrator Kd = 3; Tf = 0.5; C = pid(Kp,Ki,Kd,Tf)```
```C = s Kp + Kd * -------- Tf*s+1 with Kp = 1, Kd = 3, Tf = 0.5 Continuous-time PDF controller in parallel form. ```

The display shows the controller type, formula, and parameter values, and verifies that the controller has no integrator term.

Create a discrete-time PI controller with trapezoidal discretization formula.

To create a discrete-time PI controller, set the value of `Ts` and the discretization formula using `Name,Value` syntax.

`C1 = pid(5,2.4,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s`
```C1 = Ts*(z+1) Kp + Ki * -------- 2*(z-1) with Kp = 5, Ki = 2.4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PI controller in parallel form. ```

Alternatively, you can create the same discrete-time controller by supplying `Ts` as the fifth input argument after all four PID parameters, `Kp`, `Ki`, `Kd`, and `Tf`. Since you only want a PI controller, set `Kd` and `Tf` to zero.

`C2 = pid(5,2.4,0,0,0.1,'IFormula','Trapezoidal')`
```C2 = Ts*(z+1) Kp + Ki * -------- 2*(z-1) with Kp = 5, Ki = 2.4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PI controller in parallel form. ```

The display shows that `C1` and `C2` are the same.

When you create a PID controller, set the dynamic system properties `InputName` and `OutputName`. This is useful, for example, when you interconnect the PID controller with other dynamic system models using the `connect` command.

`C = pid(1,2,3,'InputName','e','OutputName','u')`
```C = 1 Kp + Ki * --- + Kd * s s with Kp = 1, Ki = 2, Kd = 3 Continuous-time PID controller in parallel form. ```

The display does not show the input and output names for the PID controller, but you can examine the property values. For instance, verify the input name of the controller.

`C.InputName`
```ans = 1x1 cell array {'e'} ```

Create a 2-by-3 grid of PI controllers with proportional gain ranging from 1–2 across the array rows and integral gain ranging from 5–9 across columns.

To build the array of PID controllers, start with arrays representing the gains.

```Kp = [1 1 1;2 2 2]; Ki = [5:2:9;5:2:9];```

When you pass these arrays to the `pid` command, the command returns the array.

```pi_array = pid(Kp,Ki,'Ts',0.1,'IFormula','BackwardEuler'); size(pi_array)```
```2x3 array of PID controller. Each PID has 1 output and 1 input. ```

Alternatively, use the `stack` command to build an array of PID controllers.

`C = pid(1,5,0.1) % PID controller`
```C = 1 Kp + Ki * --- + Kd * s s with Kp = 1, Ki = 5, Kd = 0.1 Continuous-time PID controller in parallel form. ```
`Cf = pid(1,5,0.1,0.5) % PID controller with filter`
```Cf = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 1, Ki = 5, Kd = 0.1, Tf = 0.5 Continuous-time PIDF controller in parallel form. ```
`pid_array = stack(2,C,Cf); % stack along 2nd array dimension`

These commands return a 1-by-2 array of controllers.

`size(pid_array)`
```1x2 array of PID controller. Each PID has 1 output and 1 input. ```

All PID controllers in an array must have the same sample time, discrete integrator formulas, and dynamic system properties such as `InputName` and `OutputName`.

Convert a standard form `pidstd` controller to parallel form.

Standard PID form expresses the controller actions in terms of an overall proportional gain `Kp`, integral and derivative time constants `Ti` and `Td`, and filter divisor `N`. You can convert any standard-form controller to parallel form using the `pid` command. For example, consider the following standard-form controller.

```Kp = 2; Ti = 3; Td = 4; N = 50; C_std = pidstd(Kp,Ti,Td,N)```
```C_std = 1 1 s Kp * (1 + ---- * --- + Td * ------------) Ti s (Td/N)*s+1 with Kp = 2, Ti = 3, Td = 4, N = 50 Continuous-time PIDF controller in standard form ```

Convert this controller to parallel form using `pid`.

`C_par = pid(C_std)`
```C_par = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 2, Ki = 0.667, Kd = 8, Tf = 0.08 Continuous-time PIDF controller in parallel form. ```

Convert a continuous-time dynamic system that represents a PID controller to parallel `pid` form.

The following dynamic system, with an integrator and two zeros, is equivalent to a PID controller.

`$H\left(s\right)=\frac{3\left(s+1\right)\left(s+2\right)}{s}.$`

Create a `zpk` model of H. Then use the `pid` command to obtain H in terms of the PID gains `Kp`, `Ki`, and `Kd`.

```H = zpk([-1,-2],0,3); C = pid(H)```
```C = 1 Kp + Ki * --- + Kd * s s with Kp = 9, Ki = 6, Kd = 3 Continuous-time PID controller in parallel form. ```

Convert a discrete-time dynamic system that represents a PID controller with derivative filter to parallel `pid` form.

Create a discrete-time zpk model that represents a PIDF controller (two zeros and two poles, including the integrator pole at `z` = 1).

`sys = zpk([-0.5,-0.6],[1 -0.2],3,'Ts',0.1);`

When you convert `sys` to PID form, the result depends on which discrete integrator formulas you specify for the conversion. For instance, use the default, `ForwardEuler`, for both the integrator and the derivative.

`Cfe = pid(sys)`
```Cfe = Ts 1 Kp + Ki * ------ + Kd * ----------- z-1 Tf+Ts/(z-1) with Kp = 2.75, Ki = 60, Kd = 0.0208, Tf = 0.0833, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in parallel form. ```

Now convert using the `Trapezoidal` formula.

`Ctrap = pid(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')`
```Ctrap = Ts*(z+1) 1 Kp + Ki * -------- + Kd * ------------------- 2*(z-1) Tf+Ts/2*(z+1)/(z-1) with Kp = -0.25, Ki = 60, Kd = 0.0208, Tf = 0.0333, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in parallel form. ```

The displays show the difference in resulting coefficient values and functional form.

For this particular dynamic system, you cannot write `sys` in parallel PID form using the `BackwardEuler` formula for the derivative filter. Doing so would result in `Tf < 0`, which is not permitted. In that case, `pid` returns an error.

Discretize a continuous-time PID controller and set integral and derivative filter formulas.

Create a continuous-time controller and discretize it using the zero-order-hold method of the `c2d` command.

```Ccon = pid(1,2,3,4); % continuous-time PIDF controller Cdis1 = c2d(Ccon,0.1,'zoh')```
```Cdis1 = Ts 1 Kp + Ki * ------ + Kd * ----------- z-1 Tf+Ts/(z-1) with Kp = 1, Ki = 2, Kd = 3.04, Tf = 4.05, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in parallel form. ```

The display shows that `c2d` computes new PID gains for the discrete-time controller.

The discrete integrator formulas of the discretized controller depend on the `c2d` discretization method. For the `zoh` method, both `IFormula` and `DFormula` are `ForwardEuler`.

`Cdis1.IFormula`
```ans = 'ForwardEuler' ```
`Cdis1.DFormula`
```ans = 'ForwardEuler' ```

If you want to use different formulas from the ones returned by `c2d`, then you can directly set the `Ts`, `IFormula`, and `DFormula` properties of the controller to the desired values.

```Cdis2 = Ccon; Cdis2.Ts = 0.1; Cdis2.IFormula = 'BackwardEuler'; Cdis2.DFormula = 'BackwardEuler';```

However, these commands do not compute new PID gains for the discretized controller. To see this, examine `Cdis2` and compare the coefficients to `Ccon` and `Cdis1`.

`Cdis2`
```Cdis2 = Ts*z 1 Kp + Ki * ------ + Kd * ------------- z-1 Tf+Ts*z/(z-1) with Kp = 1, Ki = 2, Kd = 3, Tf = 4, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PIDF controller in parallel form. ```

## Version History

Introduced in R2010b