Documentation

# pidstd2

Create 2-DOF PID controller in standard form, convert to standard-form 2-DOF PID controller

## Syntax

```C2 = pidstd2(Kp,Ti,Td,N,b,c) C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts) C2 = pidstd2(sys) C2 = pid2(___,Name,Value) ```

## Description

`pid2` controller objects represent two-degree-of-freedom (2-DOF) PID controllers in parallel form. Use `pid2` either to create a `pid2` controller object from known coefficients or to convert a dynamic system model to a `pid2` object.

Two-degree-of-freedom (2-DOF) PID controllers include setpoint weighting on the proportional and derivative terms. A 2-DOF PID controller is capable of fast disturbance rejection without significant increase of overshoot in setpoint tracking. 2-DOF PID controllers are also useful to mitigate the influence of changes in the reference signal on the control signal. The following illustration shows a typical control architecture using a 2-DOF PID controller.

`C2 = pidstd2(Kp,Ti,Td,N,b,c)` creates a continuous-time 2-DOF PID controller with proportional gain `Kp`, integrator and derivative time constants `Ti`, and `Td`, and derivative filter divisor `N`. The controller also has setpoint weighting `b` on the proportional term, and setpoint weighting `c` on the derivative term. The relationship between the 2-DOF controller’s output (u) and its two inputs (r and y) is given by:

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

This representation is in standard form. If all of the coefficients are real-valued, then the resulting `C2` is a `pidstd2` controller object. If one or more of these coefficients is tunable (`realp` or `genmat`), then `C2` is a tunable generalized state-space (`genss`) model object.

`C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts)` creates a discrete-time 2-DOF PID controller with sample time `Ts`. The relationship between the controller’s output and inputs is given by:

`$u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$`

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` properties. (See Properties for more information). 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.

`C2 = pidstd2(sys)` converts the dynamic system `sys` to a standard form `pidstd2` controller object.

`C2 = pid2(___,Name,Value)` specifies additional properties as comma-separated pairs of `Name,Value` arguments.

## 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`. When `Ti` = `Inf`, the controller has no integral action. 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` `b` Setpoint weighting on proportional term. `b` can be: A real, nonnegative, and finite value.An array of real, nonnegative, finite values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `b` = 0, changes in setpoint do not feed directly into the proportional term. Default: 1 `c` Setpoint weighting on derivative term. `c` can be: A real, nonnegative, and finite value.An array of real, nonnegative, finite values.A tunable parameter (`realp`) or generalized matrix (`genmat`).A tunable surface for gain-scheduled tuning, created using `tunableSurface`. When `c` = 0, changes in setpoint do not feed directly into the derivative term. Default: 1 `Ts` Sample time. To create a discrete-time `pidstd2` controller, provide a positive real value (`Ts > 0`).`pidstd2` does not support discrete-time controller with undetermined sample time (`Ts = -1`). `Ts` must be a scalar value. In an array of `pidstd2` controllers, each controller must have the same `Ts`. Default: 0 (continuous time) `sys` SISO dynamic system to convert to standard `pidstd2` form. `sys` be a two-input, one-output system. `sys` must represent a valid 2-DOF 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 `pidstd2` controller, or to set other object properties such as `InputName` and `OutputName`. For information about available properties of `pidstd2` controller objects, see Properties.

## Output Arguments

 `C2` 2-DOF PID controller, returned as a `pidstd2` controller object, an array of `pidstd2` controller objects, a `genss` object, or a `genss` array. If all the coefficients have scalar numeric values, then `C2` is a `pidstd2` controller object. If one or more coefficients is a numeric array, `C2` is an array of `pidstd2` controller objects. The controller type (such as PI, PID, or PDF) depends upon the values of the gains. For example, when `Td` = 0, but `Kp` and `Ti` are nonzero and finite, `C2` is a PI controller. If one or more coefficients is a tunable parameter (`realp`), generalized matrix (`genmat`), or tunable gain surface (`tunableSurface`), then `C2` is a generalized state-space model (`genss`).

## Properties

 `b, c` Setpoint weights on the proportional and derivative terms, respectively. `b` and `c` values are real, finite, and positive. When you create a 2-DOF PID controller using the `pidstd2` command, the initial values of these properties are set by the `b`, and `c` input arguments, respectively. `Kp` Proportional gain. The value of `Kp` is real and finite. When you create a 2-DOF PID controller using the `pidstd2` command, the initial value of this property is set by the `Kp` input argument. `Ti` Integrator time. `Ti` is real and positive. When you create a 2-DOF PID controller using the `pidstd2` command, the initial value of this property is set by the `Ti` input argument. When `Ti` = `Inf`, the controller has no integral action. `Td` Derivative time. `Td` is real, finite, and nonnegative. When you create a 2-DOF PID controller using the `pidstd2` command, the initial value of this property is set by the `Td` input argument. When `Td` = 0, the controller has no derivative action. `N` Derivative filter divisor. `N` must be real and positive. When you create a 2-DOF PID controller using the `pidstd2` command, the initial value of this property is set by the `N` input argument. `IFormula` Discrete integrator formula IF(z) for the integrator of the discrete-time `pidstd2` controller `C2`. The relationship between the inputs and output of `C2` is given by: `$u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$` `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 `C2` is a continuous-time controller, `IFormula` is `''`. Default: `'ForwardEuler'` `DFormula` Discrete integrator formula DF(z) for the derivative filter of the discrete-time `pidstd2` controller `C2`. The relationship between the inputs and output of `C2` is given by: `$u={K}_{p}\left[\left(br-y\right)+\frac{1}{{T}_{i}}IF\left(z\right)\left(r-y\right)+\frac{{T}_{d}}{\frac{{T}_{d}}{N}+DF\left(z\right)}\left(cr-y\right)\right].$` `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 `pidstd2` controller with no derivative filter (`N = Inf`). When `C2` is a continuous-time controller, `DFormula` is `''`. Default: `'ForwardEuler'` `InputDelay` Time delay on the system input. `InputDelay` is always 0 for a `pidstd2` controller object. `OutputDelay` Time delay on the system Output. `OutputDelay` is always 0 for a `pidstd2` 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 or a 2-by-1 cell array of character vectors. Use this property to name the input channels of the controller model. For example, assign the names `setpoint` and `measurement` to the inputs of a 2-DOF PID controller model `C` as follows. `C.InputName = {'setpoint';'measurement'};` Alternatively, use automatic vector expansion to assign both input names. For example: `C.InputName = 'C-input';` The input names automatically expand to `{'C-input(1)';'C-input(2)'}`. 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: `{'';''}` `InputUnit` Input channel units, specified as a 2-by-1 cell array of character vectors. Use this property to track input signal units. For example, assign the units `Volts` to the reference input and the concentration units `mol/m^3` to the measurement input of a 2-DOF PID controller model `C` as follows. `C.InputUnit = {'Volts';'mol/m^3'};` `InputUnit` has no effect on system behavior. Default: `{'';''}` `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 2-DOF PDF controller in standard form. To do so, set the integral time constant to `Inf`. Set the other gains and the filter divisor to the desired values.

```Kp = 1; Ti = Inf; % No integrator Td = 3; N = 6; b = 0.5; % setpoint weight on proportional term c = 0.5; % setpoint weight on derivative term C2 = pidstd2(Kp,Ti,Td,N,b,c)```
```C2 = s u = Kp * [(b*r-y) + Td * ------------ * (c*r-y)] (Td/N)*s+1 with Kp = 1, Td = 3, N = 6, b = 0.5, c = 0.5 Continuous-time 2-DOF PDF controller in standard form ```

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

Create a discrete-time 2-DOF PI controller in standard form, using the trapezoidal discretization formula. Specify the formula using `Name,Value` syntax.

```Kp = 1; Ti = 2.4; Td = 0; N = Inf; b = 0.5; c = 0; Ts = 0.1; C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts,'IFormula','Trapezoidal')```
```C2 = 1 Ts*(z+1) u = Kp * [(b*r-y) + ---- * -------- * (r-y)] Ti 2*(z-1) with Kp = 1, Ti = 2.4, b = 0.5, Ts = 0.1 Sample time: 0.1 seconds Discrete-time 2-DOF PI controller in standard form ```

Setting `Td` = 0 specifies a PI controller with no derivative term. As the display shows, the values of `N` and `c` are not used in this controller. The display also shows that the trapezoidal formula is used for the integrator.

Create a 2-DOF PID controller in standard form, and set the dynamic system properties `InputName` and `OutputName`. Naming the inputs and the output is useful, for example, when you interconnect the PID controller with other dynamic system models using the `connect` command.

`C2 = pidstd2(1,2,3,10,1,1,'InputName',{'r','y'},'OutputName','u')`
```C2 = 1 1 s u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)] Ti s (Td/N)*s+1 with Kp = 1, Ti = 2, Td = 3, N = 10, b = 1, c = 1 Continuous-time 2-DOF PIDF controller in standard form ```

A 2-DOF PID controller has two inputs and one output. Therefore, the `'InputName'` property is an array containing two names, one for each input. The model display does not show the input and output names for the PID controller, but you can examine the property values to see them. For instance, verify the input name of the controller.

`C2.InputName`
```ans = 2x1 cell array {'r'} {'y'} ```

Create a 2-by-3 grid of 2-DOF PI controllers in standard form. The proportional gain ranges from 1–2 across the array rows, and the integrator time constant ranges 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]; Ti = [5:2:9;5:2:9];```

When you pass these arrays to the `pidstd2` command, the command returns the array of controllers.

```pi_array = pidstd2(Kp,Ti,0,Inf,0.5,0,'Ts',0.1,'IFormula','BackwardEuler'); size(pi_array)```
```2x3 array of 2-DOF PID controller. Each PID has 1 output and 2 inputs. ```

If you provide scalar values for some coefficients, `pidstd2` automatically expands them and assigns the same value to all entries in the array. For instance, in this example, `Td` = 0, so that all entries in the array are PI controllers. Also, all entries in the array have `b` = 0.5.

Access entries in the array using array indexing. For dynamic system arrays, the first two dimensions are the I/O dimensions of the model, and the remaining dimensions are the array dimensions. Therefore, the following command extracts the (2,3) entry in the array.

`pi23 = pi_array(:,:,2,3)`
```pi23 = 1 Ts*z u = Kp * [(b*r-y) + ---- * ------ * (r-y)] Ti z-1 with Kp = 2, Ti = 9, b = 0.5, Ts = 0.1 Sample time: 0.1 seconds Discrete-time 2-DOF PI controller in standard form ```

You can also build an array of PID controllers using the `stack` command.

```C2 = pidstd2(1,5,0.1,Inf,0.5,0.5); % PID controller C2f = pidstd2(1,5,0.1,0.5,0.5,0.5); % PID controller with filter pid_array = stack(2,C2,C2f); % stack along 2nd array dimension```

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

`size(pid_array)`
```1x2 array of 2-DOF PID controller. Each PID has 1 output and 2 inputs. ```

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 parallel-form `pid2` controller to standard form.

Parallel PID form expresses the controller actions in terms of proportional, integral, and derivative gains `Kp`, `Ki`, and `Kd`, and filter time constant `Tf`. You can convert a parallel-form `pid2` controller to standard form using the `pidstd2` command, provided that both of the following are true:

• The `pid2` controller can be expressed in valid standard form.

• The gains `Kp`, `Ki`, and `Kd` of the `pid2` controller all have the same sign.

For example, consider the following parallel-form controller.

```Kp = 2; Ki = 3; Kd = 4; Tf = 2; b = 0.1; c = 0.5; C2_par = pid2(Kp,Ki,Kd,Tf,b,c)```
```C2_par = 1 s u = Kp (b*r-y) + Ki --- (r-y) + Kd -------- (c*r-y) s Tf*s+1 with Kp = 2, Ki = 3, Kd = 4, Tf = 2, b = 0.1, c = 0.5 Continuous-time 2-DOF PIDF controller in parallel form. ```

Convert this controller to parallel form using `pidstd2`.

`C2_std = pidstd2(C2_par)`
```C2_std = 1 1 s u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)] Ti s (Td/N)*s+1 with Kp = 2, Ti = 0.667, Td = 2, N = 1, b = 0.1, c = 0.5 Continuous-time 2-DOF PIDF controller in standard form ```

The display confirms the new standard form. A response plot confirms that the two forms are equivalent.

```bodeplot(C2_par,'b-',C2_std,'r--') legend('Parallel','Standard','Location','Southeast')```

Convert a two-input, one-output continuous-time dynamic system that represents a 2-DOF PID controller to a standard-form `pidstd2` controller.

The following state-space matrices represent a 2-DOF PID controller.

```A = [0,0;0,-8.181]; B = [1,-1;-0.1109,8.181]; C = [0.2301,10.66]; D = [0.8905,-11.79]; sys = ss(A,B,C,D);```

Rewrite `sys` in terms of the standard-form PID parameters `Kp`, `Ti`, `Td`, and `N`, and the setpoint weights `b` and `c`.

`C2 = pidstd2(sys)`
```C2 = 1 1 s u = Kp * [(b*r-y) + ---- * --- * (r-y) + Td * ------------ * (c*r-y)] Ti s (Td/N)*s+1 with Kp = 1.13, Ti = 4.91, Td = 1.15, N = 9.43, b = 0.66, c = 0.0136 Continuous-time 2-DOF PIDF controller in standard form ```

Convert a discrete-time dynamic system that represents a 2-DOF PID controller with derivative filter to standard `pidstd2` form.

The following state-space matrices represent a discrete-time 2-DOF PID controller with a sample time of 0.05 s.

```A = [1,0;0,0.6643]; B = [0.05,-0.05; -0.004553,0.3357]; C = [0.2301,10.66]; D = [0.8905,-11.79]; Ts = 0.05; sys = ss(A,B,C,D,Ts);```

When you convert `sys` to 2-DOF 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.

`C2fe = pidstd2(sys)`
```C2fe = 1 Ts 1 u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * --------------- * (c*r-y)] Ti z-1 (Td/N)+Ts/(z-1) with Kp = 1.13, Ti = 4.91, Td = 1.41, N = 9.43, b = 0.66, c = 0.0136, Ts = 0.05 Sample time: 0.05 seconds Discrete-time 2-DOF PIDF controller in standard form ```

Now convert using the `Trapezoidal` formula.

`C2trap = pidstd2(sys,'IFormula','Trapezoidal','DFormula','Trapezoidal')`
```C2trap = 1 Ts*(z+1) 1 u = Kp * [(b*r-y) + ---- * -------- * (r-y) + Td * ----------------------- * (c*r-y)] Ti 2*(z-1) (Td/N)+Ts/2*(z+1)/(z-1) with Kp = 1.12, Ti = 4.89, Td = 1.41, N = 11.4, b = 0.658, c = 0.0136, Ts = 0.05 Sample time: 0.05 seconds Discrete-time 2-DOF PIDF controller in standard form ```

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

For some dynamic systems, attempting to use the `Trapezoidal` or `BackwardEuler` integrator formulas yields invalid results, such as negative `Ti`, `Td`, or `N` values. In such cases, `pidstd2` returns an error.

Discretize a continuous-time standard-form 2-DOF PID controller and specify the integral and derivative filter formulas.

Create a continuous-time `pidstd2` controller and discretize it using the zero-order-hold method of the `c2d` command.

```C2con = pidstd2(10,5,3,0.5,1,1); % continuous-time 2-DOF PIDF controller C2dis1 = c2d(C2con,0.1,'zoh')```
```C2dis1 = 1 Ts 1 u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * --------------- * (c*r-y)] Ti z-1 (Td/N)+Ts/(z-1) with Kp = 10, Ti = 5, Td = 3.03, N = 0.5, b = 1, c = 1, Ts = 0.1 Sample time: 0.1 seconds Discrete-time 2-DOF PIDF controller in standard form ```

The display shows that `c2d` computes new PID coefficients 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`.

`C2dis1.IFormula`
```ans = 'ForwardEuler' ```
`C2dis1.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.

```C2dis2 = C2con; C2dis2.Ts = 0.1; C2dis2.IFormula = 'BackwardEuler'; C2dis2.DFormula = 'BackwardEuler';```

However, these commands do not compute new coefficients for the discretized controller. To see this, examine `C2dis2` and compare the coefficients to `C2con` and `C2dis1`.

`C2dis2`
```C2dis2 = 1 Ts*z 1 u = Kp * [(b*r-y) + ---- * ------ * (r-y) + Td * ----------------- * (c*r-y)] Ti z-1 (Td/N)+Ts*z/(z-1) with Kp = 10, Ti = 5, Td = 3, N = 0.5, b = 1, c = 1, Ts = 0.1 Sample time: 0.1 seconds Discrete-time 2-DOF PIDF controller in standard form ```

## Tips

• To design a PID controller for a particular plant, use `pidtune` or `pidTuner`. To create a tunable 2-DOF PID controller as a control design block, use `tunablePID2`.

• To break a 2-DOF controller into two SISO control components, such as a feedback controller and a feedforward controller, use `getComponents`.

• Create arrays of `pidstd2` controllers by:

In an array of `pidstd2` 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 `pid2`. Parallel form expresses the controller actions in terms of proportional, integral, and derivative gains Kp, Ki and Kd, and a filter time constant Tf. For example, the relationship between the inputs and output of a continuous-time parallel-form 2-DOF PID controller is given by:

`$u={K}_{p}\left(br-y\right)+\frac{{K}_{i}}{s}\left(r-y\right)+\frac{{K}_{d}s}{{T}_{f}s+1}\left(cr-y\right).$`
• There are two ways to discretize a continuous-time `pidstd2` 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 Standard-Form 2-DOF 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 `pidstd2` controllers than using `c2d`.