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## Determination of PID controller parameters from step response speciﬁcations

June 30, 2015 page compiled on June 30, 2015 at 7:13pm
This note describes how to design a PID controller for a system deﬁned by second order diﬀerential equation based on requirements for a step response speciﬁed by the rise time and the settling time.

The goal is to determine the three PID parameters () from the plant transfer function and (rise time and settling time).

Consider the following mechanical system

is the mass of the car, is the damping coeﬃcient and is the spring constant. To illustrate, assuming standard SI units:

The ﬁrst step is to derive the mathematical model for the system. This means ﬁnding a diﬀerential equation that relates the output (the displacment ) to the input, which is the applied force .The fFriction force between the mass M and the ground is ignored in this example.

The ﬁrst step is to make a free body diagram

Applying Netwon laws gives

or

Taking Laplace transform and assuming zero initial conditions gives

The transfer function is deﬁned as the ratio of the output to the input in the Laplace domain. Here the input is , which is the external force, and the output is which is the displacement. Taking the Laplace transform of the above diﬀerential equation gives the transfer function

Using block diagram the transfer function is illustrated as

The PID controller is now added. The transfer function of the PID controller itself is

The controller is added to the system and the loop is closed. The following diagram represents the updated system with the controller in place

Let be the open loop transfer function

Hence the closed loop transfer function is

Therefore

 (1)

The closed loop transfer function (1) shows there are three poles.

Putting one pole at a distance of away from the imaginary axis, while the remaining two poles are the dominant poles results in the following diagram

The denominator of equation (1) can be rewritten as

Equating coeﬃcients gives

Solving for PID parameters results in

These are the PID parameters as a function of and .

and are determined in order to obtain the PID parameters.

The time response speciﬁcations are now introduced in order to determine these parameters. Assuming these are the time domain requirments

1.
The settling time
2.
The rise time

Using the following for criterion

 (3)

And the rise time is given by

But , hence

 (4)

From (3) and (4) are solved for

Solving numerically gives

Hence the solution is

and

Substituting the values for and in (2), and the values given for and , gives the PID parameters

Using Matlab, the step response is found