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
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