# pid

Create PID controller in parallel form, convert to parallel-form PID controller

## Syntax

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

## Description

`C = pid(Kp,Ki,Kd,Tf)` creates a continuous-time PID controller with proportional, integral, and derivative gains `Kp`, `Ki`, and `Kd` and first-order derivative filter time constant `Tf`:

`$C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}.$`

This representation is in parallel form. If all of `Kp`, `Ki`, `Kd`, and `Tf` are real, then the resulting `C` is a `pid` controller object. If one or more of these coefficients is tunable (`realp` or `genmat`), then `C` is a tunable generalized state-space (`genss`) model object.

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

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

IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter. By default,

`$IF\left(z\right)=DF\left(z\right)=\frac{{T}_{s}}{z-1}.$`

To choose different discrete integrator formulas, use the `IFormula` and `DFormula` properties. (See Properties for more information about `IFormula` and `DFormula`). If `DFormula` = `'ForwardEuler'` (the default value) and `Tf` ≠ 0, then `Ts` and `Tf` must satisfy `Tf > Ts/2`. This requirement ensures a stable derivative filter pole.

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

`C = pid(Kp)` creates a continuous-time proportional (P) controller with `Ki` = 0, `Kd` = 0, and `Tf` = 0.

`C = pid(Kp,Ki)` creates a proportional and integral (PI) controller with `Kd` = 0 and `Tf` = 0.

`C = pid(Kp,Ki,Kd)` creates a proportional, integral, and derivative (PID) controller with `Tf` = 0.

`C = pid(...,Name,Value)` creates a controller or converts a dynamic system to a `pid` controller object with additional options specified by one or more `Name,Value` pair arguments.

`C = pid` creates a P controller with `Kp` = 1.

## Input Arguments

 `Kp` Proportional gain. `Kp` can be: A real and finite value.An array of real and finite values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `Kp` = 0, the controller has no proportional action. Default: 1 `Ki` Integral gain. `Ki` can be: A real and finite value.An array of real and finite values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `Ki` = 0, the controller has no integral action. Default: 0 `Kd` Derivative gain. `Kd` can be: A real and finite value.An array of real and finite values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `Kd` = 0, the controller has no derivative action. Default: 0 `Tf` Time constant of the first-order derivative filter. `Tf` can be: A real, finite, and nonnegative value.An array of real, finite, and nonnegative values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `Tf` = 0, the controller has no filter on the derivative action. Default: 0 `Ts` Sample time. To create a discrete-time `pid` controller, provide a positive real value (`Ts > 0`). `pid` does not support discrete-time controller with unspecified sample time (`Ts = -1`). `Ts` must be a scalar value. In an array of `pid` controllers, each controller must have the same `Ts`. Default: 0 (continuous time) `sys` SISO dynamic system to convert to parallel `pid` form. `sys` must represent a valid PID controller that can be written in parallel form with `Tf` ≥ 0. `sys` can also be an array of SISO dynamic systems.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Use `Name,Value` syntax to set the numerical integration formulas `IFormula` and `DFormula` of a discrete-time `pid` controller, or to set other object properties such as `InputName` and `OutputName`. For information about available properties of `pid` controller objects, see Properties.

## Output Arguments

 `C` PID controller, represented as a `pid` controller object, an array of `pid` controller objects, a `genss` object, or a `genss` array. If all the gains `Kp`, `Ki`, `Kd`, and `Tf` have numeric values, then `C` is a `pid` controller object. When the gains are numeric arrays, `C` is an array of `pid` controller objects. The controller type (P, I, PI, PD, PDF, PID, PIDF) depends upon the values of the gains. For example, when `Kd` = 0, but `Kp` and `Ki` are nonzero, `C` is a PI controller. If one or more gains is a tunable parameter (`realp`), generalized matrix (`genmat`), or tunable gain surface (`tunableSurface`), then `C` is a generalized state-space model (`genss`).

## Properties

 `Kp, Ki, Kd` PID controller gains. The `Kp`, `Ki`, and `Kd` properties store the proportional, integral, and derivative gains, respectively. `Kp`, `Ki`, and `Kd` are real and finite. `Tf` Derivative filter time constant. The `Tf` property stores the derivative filter time constant of the `pid` controller object. `Tf` is real, finite, and nonnegative. `IFormula` Discrete integrator formula IF(z) for the integrator of the discrete-time `pid` controller `C`: `$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$` `IFormula` can take the following values: `'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 `''`. Default: `'ForwardEuler'` `DFormula` Discrete integrator formula DF(z) for the derivative filter of the discrete-time `pid` controller `C`: `$C={K}_{p}+{K}_{i}IF\left(z\right)+\frac{{K}_{d}}{{T}_{f}+DF\left(z\right)}.$` `DFormula` can take the following values: `'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 `''`. Default: `'ForwardEuler'` `InputDelay` Time delay on the system input. `InputDelay` is always 0 for a `pid` controller object. `OutputDelay` Time delay on the system Output. `OutputDelay` is always 0 for a `pid` controller object. `Ts` Sample 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. PID controller models do not support unspecified sample time (```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 sample time of a discrete-time system. Default: `0` (continuous time) `TimeUnit` Units for the time variable, the sample time `Ts`, and any time delays in the model, specified as one 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 name, specified as a character vector. Use this property to name the input channel of the controller model. For example, 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`. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: Empty character vector, `''` `InputUnit` Input channel units, specified as a character vector. Use this property to track input signal units. For example, assign the concentration units `mol/m^3` to the input of a controller model `C` as follows. `C.InputUnit = 'mol/m^3';` `InputUnit` has no effect on system behavior. Default: Empty character vector, `''` `InputGroup` Input channel groups. This property is not needed for PID controller models. Default: `struct` with no fields `OutputName` Output channel name, specified as a character vector. Use this property to name the output channel of the controller model. For example, assign the name `control` to the output of a controller model `C` as follows. `C.OutputName = 'control';` You can use the shorthand notation `y` to refer to the `OutputName` property. For example, `C.y` is equivalent to `C.OutputName`. Input channel names have several uses, including: Identifying channels on model display and plotsSpecifying connection points when interconnecting models Default: Empty character vector, `''` `OutputUnit` Output channel units, specified as a character vector. Use this property to track output signal units. For example, assign the unit `Volts` to the output of a controller model `C` as follows. `C.OutputUnit = 'Volts';` `OutputUnit` has no effect on system behavior. Default: Empty character vector, `''` `OutputGroup` Output channel groups. This property is not needed for PID controller models. Default: `struct` with no fields `Name` System name, specified as a character vector. For example, `'system_1'`. Default: `''` `Notes` Any text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if `sys1` and `sys2` are dynamic system models, you can set their `Notes` properties as follows: ```sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes``` ```ans = "sys1 has a string." ans = 'sys2 has a character vector.' ``` Default: `[0×1 string]` `UserData` Any type of data you want to associate with system, specified as any MATLAB® data type. Default: `[]` `SamplingGrid` Sampling grid for model arrays, specified as a data structure. For model arrays that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model in the array. This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data 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 should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. For example, suppose you create a 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, suppose you create a 6-by-9 model array, `M`, by independently sampling two variables, `zeta` and `w`. The following code attaches 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 example, the Simulink Control Design™ commands `linearize` and `slLinearizer` populate `SamplingGrid` in this way. Default: `[]`

## 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, as described in Tips. 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. ```

## Tips

• Use `pid` to:

• Create a `pid` controller object from known PID gains and filter time constant.

• Convert a `pidstd` controller object to a standard-form `pid` controller object.

• Convert other types of dynamic system models to a `pid` controller object.

• To design a PID controller for a particular plant, use `pidtune` or `pidTuner`. To create a tunable PID controller as a control design block, use `tunablePID`.

• Create arrays of `pid` controller objects by:

In an array of `pid` controllers, each controller must have the same sample time `Ts` and discrete integrator formulas `IFormula` and `DFormula`.

• To create or convert to a standard-form controller, use `pidstd`. Standard form expresses the controller actions in terms of an overall proportional gain Kp, integral and derivative times Ti and Td, and filter divisor N:

`$C={K}_{p}\left(1+\frac{1}{{T}_{i}}\frac{1}{s}+\frac{{T}_{d}s}{\frac{{T}_{d}}{N}s+1}\right).$`
• There are two ways to discretize a continuous-time `pid` controller:

• Use the `c2d` command. `c2d` computes new parameter values for the discretized controller. The discrete integrator formulas of the discretized controller depend upon the `c2d` discretization method you use, as shown in the following 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 the `c2d` reference page. For more information about `IFormula` and `DFormula`, see Properties .

• If you require different discrete integrator formulas, you can discretize the controller by directly setting `Ts`, `IFormula`, and `DFormula` to the desired values. (See Discretize a Continuous-Time PID Controller.) 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- and discrete-time `pid` controllers than using `c2d`.