# 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(z) = DF(z) = Tsz/(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`).A tunable generalized matrix (`genmat`), such as a gain surface for gain-scheduled tuning, created using `gainsurf` (requires Robust Control Toolbox™ software). 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`).A tunable generalized matrix (`genmat`), such as a gain surface for gain-scheduled tuning, created using `gainsurf` (requires Robust Control Toolbox software). 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`).A tunable generalized matrix (`genmat`), such as a gain surface for gain-scheduled tuning, created using `gainsurf` (requires Robust Control Toolbox software). 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`).A tunable generalized matrix (`genmat`), such as a gain surface for gain-scheduled tuning, created using `gainsurf` (requires Robust Control Toolbox software). 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 undetermined sample time (`Ts = -1`). `Ts` must be a scalar value. In an array of `pid` controllers, each controller must have the same `Ts`. `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 single 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`) or generalized matrix (`genmat`), 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` values are real and finite. `Tf` Derivative filter time constant. The `Tf` property stores the derivative filter time constant of the `pid` controller object. `Tf` are real, finite, and greater than or equal to zero. `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. To denote a discrete-time model with unspecified sample 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 sample 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 sample 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 plotsExtracting subsystems of MIMO systemsSpecifying 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 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 plotsExtracting subsystems of MIMO systemsSpecifying connection points when interconnecting models Default: Empty string `''` for all output 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 output 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: `[]` `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

PID Controller with Proportional and Derivative Gains, and Filter Time Constant (PDF Controller)

Create a continuous-time controller with proportional and derivative gains, and filter time constant (PDF controller).

```Kp=1; Ki=0; 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.

Discrete-Time PI Controller

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

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

`C = pid(5,2.4,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s`

This command produces the result:

```Discrete-time PI controller in parallel form: Ts*(z+1) Kp + Ki * -------- 2*(z-1) with Kp = 5, Ki = 2.4, Ts = 0.1```

Alternatively, you can create the same discrete-time controller by supplying `Ts` as the fifth argument after all four PID parameters `Kp`, `Ki`, `Kd`, and `Tf`.

`C = pid(5,2.4,0,0,0.1,'IFormula','Trapezoidal');`

PID Controller with Custom Input and Output Names

Create a PID controller, and set dynamic system properties `InputName` and `OutputName`.

`C = pid(1,2,3,'InputName','e','OutputName','u');`

Array of PID Controllers

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

Create a grid of PI controllers with proportional gain varying row to row and integral gain varying column to column. To do so, start with arrays representing the gains.

```Kp = [1 1 1;2 2 2]; Ki = [5:2:9;5:2:9]; pi_array = pid(Kp,Ki,'Ts',0.1,'IFormula','BackwardEuler');```

These commands produce a 2-by-3 array of discrete-time `pid` objects. All `pid` objects in an array must have the same sample time, discrete integrator formulas, and dynamic system properties (such as `InputName` and `OutputName`).

Alternatively, you can use `stack` to build arrays of `pid` objects.

```C = pid(1,5,0.1) % PID controller Cf = pid(1,5,0.1,0.5) % PID controller with filter pid_array = stack(2,C,Cf); % stack along 2nd array dimension```

These commands produce a 1-by-2 array of controllers. Enter the command:

`size(pid_array)`

to see the result

```1x2 array of PID controller. Each PID has 1 output and 1 input.```

Convert PID Controller from Standard to Parallel Form

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 times Ti and Td, and filter divisor N. You can convert any standard form controller to parallel form using `pid`.

```stdsys = pidstd(2,3,4,5); % Standard-form controller parsys = pid(stdsys) ```

These commands produce a parallel-form controller:

```Continuous-time PIDF controller in parallel form: 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 2, Ki = 0.66667, Kd = 8, Tf = 0.8```

Convert Dynamic System to Parallel-Form PID Controller

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

The dynamic system

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

represents a PID controller. Use `pid` to obtain H(s) to in terms of the PID gains Kp, Ki, and Kd.

```H = zpk([-1,-2],0,3); C = pid(H)```

These commands produce the result:

```Continuous-time PID controller in parallel form: 1 Kp + Ki * --- + Kd * s s with Kp = 9, Ki = 6, Kd = 3```

Convert Discrete-Time Zero-Pole-Gain Model to Parallel-Form PID Controller

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

```% PIDF controller expressed in zpk form sys = zpk([-0.5,-0.6],[1 -0.2],3,'Ts',0.1) ```

The resulting `pid` object depends upon the discrete integrator formula you specify for `IFormula` and `DFormula`. For example, if you use the default `ForwardEuler` for both formulas:

`C = pid(sys)`

returns the result

```Discrete-time PIDF controller in parallel form: Ts 1 Kp + Ki * ------ + Kd * ----------- z-1 Tf+Ts/(z-1) with Kp = 2.75, Ki = 60, Kd = 0.020833, Tf = 0.083333, Ts = 0.1```

Converting using the `Trapezoidal` formula returns different parameter values:

`C = pid(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')`

This command returns the result:

```Discrete-time PIDF controller in parallel form: Ts*(z+1) 1 Kp + Ki * -------- + Kd * ------------------- 2*(z-1) Tf+Ts/2*(z+1)/(z-1) with Kp = -0.25, Ki = 60, Kd = 0.020833, Tf = 0.033333, Ts = 0.1```

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

Discretize a Continuous-time PID Controller

First, discretize the controller using the `'zoh'` method of `c2d`.

```Cc = pid(1,2,3,4) % continuous-time pidf controller Cd1 = c2d(Cc,0.1,'zoh')```

`c2d` computes new parameters for the discrete-time controller:

```Discrete-time PIDF controller in parallel form: Ts 1 Kp + Ki * ------ + Kd * ----------- z-1 Tf+Ts/(z-1) with Kp = 1, Ki = 2, Kd = 3.0377, Tf = 4.0502, Ts = 0.1```

The resulting discrete-time controller uses `ForwardEuler` (Ts/(z–1)) for both `IFormula` and `DFormula`.

The discrete integrator formulas of the discretized controller depend upon the `c2d` discretization method, as described in Tips. To use a different `IFormula` and `DFormula`, directly set `Ts`, `IFormula`, and `DFormula` to the desired values:

```Cd2 = Cc; Cd2.Ts = 0.1; Cd2.IFormula = 'BackwardEuler'; Cd2.DFormula = 'BackwardEuler'; ```

These commands do not compute new parameter values for the discretized controller. To see this, enter:

`Cd2`

to obtain the result:

```Discrete-time PIDF controller in parallel form: 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```

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### Tips

• Use `pid` either to create a `pid` controller object from known PID gains and filter time constant, or to convert a dynamic system model to a `pid` object.

• To deisgn a PID controller for a particular plant, use `pidtune` or `pidTuner`.

• 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 this example.) 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`.