The problem of stabilizing a second-order delay system using classical proportional-integral-derivative (PID) controller is considered. An extension of the Hermite-Biehler theorem, which is applicable to quasipolynomials, is used to seek the set of complete stabilizing PID parameters. The range of admissible proportional gains is determined in closed form. For each proportional gain, the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid or a triangle.

Dead times are often encountered in various engineering systems and industry processes such as electrical and communication network, chemical process, turbojet engine, nuclear reactor, and hydraulic system. In fact, delays are caused by many phenomena like the time required to transport mass, energy or information, the time processing for sensors, the time needed for controllers to execute a complicated algorithm control, and the accumulation of time lags in a number of simple plants connected in series [

Today, proportional-integral (PI) and proportional-integral-derivative (PID) controller types are the most widely used control strategy. It is estimated that over 90% of process control applications employ PID control thanks to its essential functionality and structural simplicity [

Several problems in process control engineering are related to the presence of delays. These delays intervene in dynamic models whose characteristic equations are of the following form [

One can consider the quasi-polynomials

The stability of the system with the characteristic equation (

Let

Under conditions (

A crucial stage in the application of the precedent theorem is to make sure that

Let

A second order system with delay can be mathematically expressed by a transfer function having the following form:

Clearly, the parameters

Let’s put

The application of the second condition of Theorem

We pass to the verification of the interlacing condition of

In this case, we graph the curves of

Representation of the curves of

Figure

The plot in Figure

Representation of the curves of

Representation of the curves of

Theorem

The

After determination of the roots of the imaginary part

For

We can use the interlacing property and do it as

From the previous equations we get the following inequalities:

Consider the plant given by relation (

We set the controller parameter

In this case, the stability region is defined by only two boundaries:

Region boundaries of Example

Consider the plant (

We now set the controller parameter

In this case, the stability region is defined by only two boundaries:

Region boundaries of Example

Plots of

As pointed out in Examples

We now state an important technical lemma that allows us to develop an algorithm for solving the PID stabilization problem. This lemma shows the behavior of the parameter

If

From (

As we can see from Figure

As we can see from Figure

The change of

As far as the odd roots of

We are ready to state the main results of our work.

Under the above assumptions on

For

the cross-section of the stabilizing region in the

the cross-section of the stabilizing region in the

The parameters

In view of Theorem

Choose

Find the roots

Compute the parameter

Determine the stability region in the plane

Go to step 1.

The stabilizing region of

In this work, we have proposed an extension of Hermit-Biehler theorem to compute the stability region for second-order delay system controlled by PID controller. The procedure is based first on determining the range of proportional gain value