# getComponents

Extract SISO control components from a 2-DOF PID controller

## Description

`[`

decomposes the 2-DOF PID controller `C`

,`X`

] =
getComponents(`C2`

,`looptype`

)`C2`

into two SISO control
components. One of the control components, `C`

, is a 1-DOF PID
controller. The other, `X`

, is a SISO dynamic system. When
`C`

and `X`

are connected in the loop
structure specified by `looptype`

, the resulting closed-loop
system is equivalent to the 2-DOF control loop.

For more information about 2-DOF PID control architectures, see Two-Degree-of-Freedom PID Controllers.

## Examples

### Extract SISO Components from 2-DOF PID Controller

Decompose a 2-DOF PID controller into SISO control components, using each of the feedforward, feedback, and filter configurations.

To start, obtain a 2-DOF PID controller. For this example, create a plant model and tune a 2-DOF PID controller for it.

```
G = tf(1,[1 0.5 0.1]);
C2 = pidtune(G,'pidf2',1.5)
```

C2 = 1 s u = Kp (b*r-y) + Ki --- (r-y) + Kd -------- (c*r-y) s Tf*s+1 with Kp = 1.12, Ki = 0.23, Kd = 1.3, Tf = 0.122, b = 0.664, c = 0.0136 Continuous-time 2-DOF PIDF controller in parallel form.

`C2`

is a `pid2`

controller object, with two inputs and one output. Decompose `C2`

into SISO control components using the feedforward configuration.

`[Cff,Xff] = getComponents(C2,'feedforward')`

Cff = 1 s Kp + Ki * --- + Kd * -------- s Tf*s+1 with Kp = 1.12, Ki = 0.23, Kd = 1.3, Tf = 0.122 Continuous-time PIDF controller in parallel form. Xff = -10.898 (s+0.2838) ------------------ (s+8.181) Continuous-time zero/pole/gain model.

As the display shows, this command returns the SISO PID controller `Cff`

as a `pid`

object. The feedforward compensator `X`

is returned as a `zpk`

object.

Decompose `C2`

using the feedback configuration. In this case as well, `Cfb`

is a `pid`

controller object, and the feedback compensator `X`

is a `zpk`

model.

`[Cfb,Xfb] = getComponents(C2,'feedback');`

Decompose `C2`

using the filter configuration. Again, the components are a SISO `pid`

controller and a `zpk`

model representing the prefilter.

`[Cfr,Xfr] = getComponents(C2,'filter');`

## Input Arguments

`C2`

— 2-DOF PID controller

`pid2`

object | `pidstd2`

object

2-DOF PID controller to decompose, specified as a
`pid2`

or `pidstd2`

controller
object.

`looptype`

— Loop structure

`'feedforward'`

(default) | `'feedback'`

| `'filter'`

Loop structure for decomposing the 2-DOF controller, specified as
`'feedforward'`

, `'feedback'`

, or
`'filter'`

. These correspond to the following control
decompositions and architectures:

`'feedforward'`

—`C`

is a conventional SISO PID controller that takes the error signal as its input.`X`

is a feedforward controller, as shown:If

`C2`

is a continuous-time, parallel-form controller, then the components are given by:$$\begin{array}{c}C\left(s\right)={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1},\\ X\left(s\right)=\left(b-1\right){K}_{p}+\frac{\left(c-1\right){K}_{d}s}{{T}_{f}s+1}.\end{array}$$

The following command constructs the closed-loop system from

*r*to*y*for the feedforward configuration.T = G*(C+X)*feedback(1,G*C);

`'feedback'`

—`C`

is a conventional SISO PID controller that takes the error signal as its input.`X`

is a feedback controller from*y*to*u*, as shown:If

`C2`

is a continuous-time, parallel-form controller, then the components are given by:$$\begin{array}{c}C\left(s\right)=b{K}_{p}+\frac{{K}_{i}}{s}+\frac{c{K}_{d}s}{{T}_{f}s+1},\\ X\left(s\right)=\left(1-b\right){K}_{p}+\frac{\left(1-c\right){K}_{d}s}{{T}_{f}s+1}.\end{array}$$

The following command constructs the closed-loop system from

*r*to*y*for the feedback configuration.T = G*C*feedback(1,G*(C+X));

`'filter'`

—`X`

is a prefilter on the reference signal.`C`

is a conventional SISO PID controller that takes as its input the difference between the filtered reference and the output, as shown:If

`C2`

is a continuous-time, parallel-form controller, then the components are given by:$$\begin{array}{c}C\left(s\right)={K}_{p}+\frac{{K}_{i}}{s}+\frac{{K}_{d}s}{{T}_{f}s+1},\\ X\left(s\right)=\frac{\left(b{K}_{p}{T}_{f}+c{K}_{d}\right){s}^{2}+\left(b{K}_{p}+{K}_{i}{T}_{f}\right)s+{K}_{i}}{\left({K}_{p}{T}_{f}+{K}_{d}\right){s}^{2}+\left({K}_{p}+{K}_{i}{T}_{f}\right)s+{K}_{i}}.\end{array}$$

The following command constructs the closed-loop system from

*r*to*y*for the filter configuration.T = X*feedback(G*C,1);

The formulas shown above pertain to continuous-time, parallel-form
controllers. Standard-form controllers and controllers in discrete time can
be decomposed into analogous configurations. The
`getComponents`

command works on all 2-DOF PID
controller objects.

## Output Arguments

`C`

— SISO PID controller

`pid`

object | `pidstd`

object

SISO PID controller, returned as a `pid`

or
`pidstd`

controller object. The form of
`C`

corresponds to the form of the input controller
`C2`

. For example, if `C2`

is a
standard-form `pidstd2`

controller, then
`C`

is a `pidstd`

object.

The precise functional form of `C`

depends on the loop
structure you specify with the `looptype`

argument, as
described in Input Arguments.

`X`

— SISO control component

`zpk`

model

SISO control component, specified as a zero-pole-gain
(`zpk`

) model. The precise functional form of
`X`

depends on the loop structure you specify with
the `looptype`

argument, as described in Input Arguments.

## Version History

**Introduced in R2015b**

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