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# make1DOF

Convert 2-DOF PID controller to 1-DOF controller

## Description

example

C1 = make1DOF(C2) converts the two-degree-of-freedom PID controller C2 to one degree of freedom by removing the terms that depend on coefficients b and c.

## Examples

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Design a 2-DOF PID controller for a plant.

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.

Convert the controller to one degree of freedom.

C1 = make1DOF(C2)
C1 =

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.

The new controller has the same PID gains and filter constant. However, make1DOF removes the terms involving the setpoint weights b and c. Therefore, in a closed loop with the plant G, the 2-DOF controller C2 yields a different closed-loop response from C1.

CM = tf(C2);
T2 = CM(1)*feedback(G,-CM(2));
T1 = feedback(G*C1,1);
stepplot(T2,T1,'r--')

## Input Arguments

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2-DOF PID controller, specified as a pid2 object or a pidstd2 object.

## Output Arguments

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1-DOF PID controller, returned as a pid or pidstd object. C1 is in parallel form if C2 is in parallel form, and standard form if C2 is in standard form.

For example, suppose C2 is a continuous-time, parallel-form 2-DOF pid2 controller. The relationship between the inputs, r and y, and the output u of C2 is given by:

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

Then C1 is a parallel-form 1-DOF pid controller of the form:

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

The PID gains Kp, Ki, and Kd, and the filter time constant Tf are unchanged. make1DOF removes the terms that depend on the setpoint weights b and c. For more information about 2-DOF PID controllers, see Two-Degree-of-Freedom PID Controllers.

The conversion also preserves the values of the properties Ts, TimeUnit, Sampling Grid, IFormula, and DFormula.