PI-Controller for simple Plant

Task:

1. Determine Hurwitz stability region for a simple plant with a PI-controller
2. Tune PI-controller to meet given specifications

The transfer function of the plant is given as:

This plant is to be controlled by a PI-controller:

We would like to now all parameters Kp and Tn for which the system is stable.

Note: This example assumes that you are familiar with the steps shown in Example 1.

1. As a first step we create a Simulink model:

2. Start PARADISE and load the model into PARADISE:

Start PARADISE by typing paradise at the MATLAB prompt

and load the Simulink model into PARADISE.

Since this model now contains two controller parameters we will open the PARAMETER SPECIFICATION window from the INPUT menu:

This is the Parameter specification window:

Lets open the parameter menus, and look which parameters PARADISE has found:

PARADISE has found both parameters. All parameters which begin with a letter K are by default classified as controller parameters.

In this case we only need to classify T_n as a controller parameter. Select T_n and use Edit -> Classify parameter:

Here we go:

3. Now let's specify the closed loop eigenvalue specifications:

As a first step we are interested in Hurwitz stability, i.e. all eigenvalues have to lie to the left of the imaginary axis.

Use Real part limitation as a boundary for the desired eigenvalue region.

4. Parameter Space:

After specifying the model and the eigenvalue specifications we can determine the stable regions:

Open the Parameter space window:

Use the Command: Run -> Execute grid to determine all controller parameters which satisfy the given specifications.

Ok, first let's rescale the axes:

As we can see from the following picture there are two stable regions.

The first has both negative Kp and Tn values. This represents positive feedback, which might lead to poor system performance, although the closed-loop is stable.

More interesting is the stable region which is bounded by Kp=0 and lies to the left of a curve with asymptote Tn=4. This means if we use a Tn>4, we can use any Kp>0 and we get a stable system.

Having done the Hurwitz stability check, we know that there are stabilizing controllers.

5. Actual PI-controller tuning

Our main goal is not just to analyse a PI-controller, but to design a PI-controller which leads to satisfactory closed-loop performance.

Closed-loop specifications for this system are:

Don't make the system slower than it already is, and provide sufficient damping.

The PI-controller achieves these specifications, if the closed loop eigenvalues lie to the left of a hyperbola shown in the following Gamma-Editor window.

This hypebola ensures a minimum damping corresponding to the asuymptotes. Furthermore it guarantees a short settling time.

Use the Command: Run -> Execute grid from the Parameter space window to determine

all controller parameters which satisfy the given specifications.

PARADISE determines the boundaries my mapping the eigenvalue specifications. Use the Options->Check Stability command to

check each region. There exists a semi-circle like stable region. Selecting parameter values from this set,

a satisfactory controller is designed (within minutes).

For example using

      K_p = 1 and T_n = 17

we get the following step response: