Documentation

# pidstd

Create a PID controller in standard form, convert to standard-form PID controller

## Syntax

```C = pidstd(Kp,Ti,Td,N) C = pidstd(Kp,Ti,Td,N,Ts) C = pidstd(sys) C = pidstd(Kp) C = pidstd(Kp,Ti) C = pidstd(Kp,Ti,Td) C = pidstd(...,Name,Value) C = pidstd ```

## Description

`C = pidstd(Kp,Ti,Td,N)` creates a continuous-time PIDF (PID with first-order derivative filter) controller object in standard form. The controller has proportional gain `Kp`, integral and derivative times `Ti` and `Td`, and first-order derivative filter divisor `N`:

`$C={K}_{p}\left(1+\frac{1}{{T}_{i}}\frac{1}{s}+\frac{{T}_{d}s}{\frac{{T}_{d}}{N}s+1}\right).$`

`C = pidstd(Kp,Ti,Td,N,Ts)` creates a discrete-time controller with sample time `Ts`. The discrete-time controller is:

`$C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$`

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` inputs. (See Properties for more information about `IFormula` and `DFormula`). If `DFormula` = `'ForwardEuler'` (the default value) and `N` ≠ `Inf`, then `Ts`, `Td`, and `N` must satisfy `Td/N > Ts/2`. This requirement ensures a stable derivative filter pole.

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

`C = pidstd(Kp)` creates a continuous-time proportional (P) controller with `Ti` = `Inf`, `Td` = 0, and `N` = `Inf`.

`C = pidstd(Kp,Ti)` creates a proportional and integral (PI) controller with `Td` = 0 and `N` = `Inf`.

`C = pidstd(Kp,Ti,Td)` creates a proportional, integral, and derivative (PID) controller with `N` = `Inf`.

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

`C = pidstd` 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`. Default: 1 `Ti` Integrator time. `Ti` can be: A real and positive value.An array of real and positive values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. Default: `Inf` `Td` Derivative time. `Td` 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 `Td` = 0, the controller has no derivative action. Default: 0 `N` Derivative filter divisor. `N` can be: A real and positive value.An array of real and positive values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `N` = `Inf`, the controller has no filter on the derivative action. Default: `Inf` `Ts` Sample time. To create a discrete-time `pidstd` controller, provide a positive real value (`Ts > 0`).`pidstd` does not support discrete-time controller with undetermined sample time (`Ts = -1`). `Ts` must be a scalar value. In an array of `pidstd` controllers, each controller must have the same `Ts`. Default: 0 (continuous time) `sys` SISO dynamic system to convert to standard `pidstd` form. `sys` must represent a valid controller that can be written in standard form with Ti > 0, Td ≥ 0, and N > 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 `pidstd` controller, or to set other object properties such as `InputName` and `OutputName`. For information about available properties of `pidstd` controller objects, see Properties.

## Output Arguments

 `C` `pidstd` object representing a single-input, single-output PID controller in standard form. The controller type (P, PI, PD, PDF, PID, PIDF) depends upon the values of `Kp`, `Ti`, `Td`, and `N`. For example, when `Td` = `Inf` and `Kp` and `Ti` are finite and nonzero, `C` is a PI controller. Enter `getType(C)` to obtain the controller type. When the inputs `Kp`,`Ti`, `Td`, and `N` or the input `sys` are arrays, `C` is an array of `pidstd` objects.

## Properties

 `Kp` Proportional gain. `Kp` must be real and finite. `Ti` Integral time. `Ti` must be real, finite, and greater than or equal to zero. `Td` Derivative time. `Td` must be real, finite, and greater than or equal to zero. `N` Derivative filter divisor. `N` must be real, and greater than or equal to zero. `IFormula` Discrete integrator formula IF(z) for the integrator of the discrete-time `pidstd` controller `C`: `$C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$` `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 `pidstd` controller `C`: `$C={K}_{p}\left(1+\frac{1}{{T}_{i}}IF\left(z\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\right).$` `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 `pidstd` controller with no derivative filter (`N = Inf`). When `C` is a continuous-time controller, `DFormula` is `''`. Default: `'ForwardEuler'` `InputDelay` Time delay on the system input. `InputDelay` is always 0 for a `pidstd` controller object. `OutputDelay` Time delay on the system Output. `OutputDelay` is always 0 for a `pidstd` 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

### Create Continuous-Time Standard-Form PDF Controller

Create a continuous-time standard-form PDF controller with proportional gain 1, derivative time 3, and a filter divisor of 6.

`C = pidstd(1,Inf,3,6);`
```C = s Kp * (1 + Td * ------------) (Td/N)*s+1 with Kp = 1, Td = 3, N = 6 Continuous-time PDF controller in standard form ```

The display shows the controller type, formula, and coefficient values.

### Create Discrete-Time PI Controller with Trapezoidal Discretization Formula

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

`C = pidstd(1,0.5,'Ts',0.1,'IFormula','Trapezoidal') % Ts = 0.1s`

This command produces the result:

```Discrete-time PI controller in standard form: 1 Ts*(z+1) Kp * (1 + ---- * --------) Ti 2*(z-1) with Kp = 1, Ti = 0.5, 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`, `Ti`, `Td`, and `N`.

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

### Create PID Controller and Set System Properties

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

`C = pidstd(1,0.5,3,'InputName','e','OutputName','u');`

### Create Grid of Standard-Form PID Controllers

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

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

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

These commands produce a 2-by-3 array of discrete-time `pidstd` objects. All `pidstd` 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 the `stack` command to build arrays of `pidstd` objects.

```C = pidstd(1,5,0.1) % PID controller Cf = pidstd(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 Parallel-Form `pid` Controller to Standard Form

Parallel PID form expresses the controller actions in terms of an proportional, integral, and derivative gains Kp, Ki, and Kd, and a filter time constant Tf. You can convert a parallel form controller `parsys` to standard form using `pidstd`, provided that:

• `parsys` is not a pure integrator (I) controller.

• The gains `Kp`, `Ki`, and `Kd` of `parsys` all have the same sign.

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

These commands produce a parallel-form controller:

```Continuous-time PIDF controller in standard form: 1 1 s Kp * (1 + ---- * --- + Td * ------------) Ti s (Td/N)*s+1 with Kp = 2, Ti = 0.66667, Td = 2, N = 0.4```

### Create `pidstd` Controller from Continuous-Time Dynamic System

The dynamic system

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

represents a PID controller. Use `pidstd` to obtain H(s) to in terms of the standard-form PID parameters Kp, Ti, and Td.

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

These commands produce the result:

```Continuous-time PID controller in standard form: 1 1 Kp * (1 + ---- * --- + Td * s) Ti s with Kp = 9, Ti = 1.5, Td = 0.33333```

### Create `pidstd` Controller from Discrete-Time Dynamic System

You can convert a discrete-time dynamic system that represents a PID controller with derivative filter to standard `pidstd` form.

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

The resulting `pidstd` 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 = pidstd(sys)`

you obtain the result:

```Discrete-time PIDF controller in standard form: 1 Ts 1 Kp * (1 + ---- * ------ + Td * ---------------) Ti z-1 (Td/N)+Ts/(z-1) with Kp = 2.75, Ti = 0.045833, Td = 0.0075758, N = 0.090909, Ts = 0.1```

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

Similarly, you cannot write `sys` in standard form using the `Trapezoidal` formula for both integrators. Doing so would result in negative `Ti` and `Td`, which also returns an error.

### Discretize Continuous-Time `pidstd` Controller

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

```Cc = pidstd(1,2,3,4); % continuous-time pidf controller Cd1 = c2d(Cc,0.1,'zoh')```
```Discrete-time PIDF controller in standard form: 1 Ts 1 Kp * (1 + ---- * ------ + Td * ---------------) Ti z-1 (Td/N)+Ts/(z-1) with Kp = 1, Ti = 2, Td = 3.2044, N = 4, 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 standard form: 1 Ts*z 1 Kp * (1 + ---- * ------ + Td * -----------------) Ti z-1 (Td/N)+Ts*z/(z-1) with Kp = 1, Ti = 2, Td = 3, N = 4, Ts = 0.1```

## Tips

• Use `pidstd` either to create a `pidstd` controller object from known PID gain, integral and derivative times, and filter divisor, or to convert a dynamic system model to a `pidstd` object.

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

• Create arrays of `pidstd` controllers by:

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

• To create or convert to a parallel-form controller, use `pid`. Parallel form expresses the controller actions in terms of proportional, integral, and derivative gains Kp, Ki and Kd, and a filter time constant Tf:

`$C={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1}.$`
• There are two ways to discretize a continuous-time `pidstd` 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. (For more information, see Discretize Continuous-Time pidstd 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-time and discrete-time `pidstd` controllers than using `c2d`.