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For unstable plants, the priority of control goes to the stability of synthesis, which means to find a stabilizer controller. In the case where the plant is subjected to structured and unstructured uncertainties, the stability problem becomes more crucial. The problem was solved by a conservative method based on generalized Kharitonov's theorem and Nevanlinna-Pick's interpolation (NPI) technique. This paper introduces a proposed straightforward numerical approach for loop shaping the unstructured additive or multiplicative maximum uncertainty magnitudes. The approach finds controllers, which are capable of stabilizing the interval system while the uncertainty box is enlarged to its maximum dimensions. To illustrate, we introduce some numerical examples.

Since most of the theories in control engineering are stated for linear plant modeling, one may wonder whether such modeling is perfectly free from uncertainty. In fact, uncertainty in linear plant models may have several origins. Linearization or order reduction is not the only the reasons but also the measurement errors and the deviation of the operating point. Moreover, at high frequencies, both parameters and structure may change dramatically; uncertainty may exceed 100% at some frequency. Despite the existences of all details, one usually works with a simple law order nominal model and quantities of uncertainty [

Uncertainty may be grouped as structured (parametric) and unstructured. The former is formulated by bounding each uncertain parameter

Additive and multiplicative perturbations are the two classes of unstructured uncertainty usually considered in control systems. In these classes, the

The necessary and sufficient conditions for robust stability in the

In this paper, a proposed numerical method for loop shaping the unstructured additive and multiplicative maximum uncertainty magnitudes is introduced. The considered class of systems is assumed to have given parameter uncertainties. The method is based on a rally use of both generalized Kharitonov’s theorem and the Nevanlinna-Pick interpolation technique. The proposed method determines a different structure controller that stabilizes the interval system for the specific box of parameter uncertainty. The algorithm continues finding these stabilizers while the uncertainty box is enlarged to its maximum dimensions. The algorithm is executed such that the increment in the number of zeros and poles will be in its minimum value.

The robustness conditions (

Additional interpolation conditions depend on the relative degree of

In this section, a proposed approach to design a robust stabilizer controller for a class of SISO plants under mixed uncertainty is introduced [

Assume the nominal unstable plant and the perturbed plant are defined, respectively as,

For each value of

In other words, since it is assumed that the number of unstable poles in the plant should remain unchanged, the maximum allowable value of the parameter change in the plant model is found. It is then required that

Once

Next, the boundary function

The multiplicative unstructured uncertainty can be defined as

Since it is required that the function

The main important problem is the choice of

To illustrate, let us first consider a system of one unstable pole

In this case, it is sufficient to let

As mentioned earlier, when it is required to tolerate larger value of parameter variation, the frequency response of the maximum unstructured uncertainty can be used to generate the

For specific value say

Definition of the norm distance

Hence, the inequality (

Since the maximum magnitude of the unstructured uncertainty is known over the whole frequency range, then for specific value of

A general proper rational structure of

Obviously, these two quadratic functions can be assigned exactly for specific orders

where the column vectors

The least mean square linear regression is then used to obtain the values of the coefficients

From mathematical point of view, the solution vector

To release the solution from such cases, the solution is parameterized by the norm distance

An iterative bisection algorithm can carry out the proposed approach of determining

Let us first illustrate the proposed approach for loop shaping, that is, finding the

(1) Assume that the

(2) Assume that the

The

Note that the coefficients,

Consider the unstable plant defined by a nominal transfer function [

The poles are

To start the proposed approach, it is required first to evaluate the maximum structured uncertainty,

For this example, the extremal segments joining both Kharitonov’s vertices and segments for numerator and denominator are

Therefore, in this example, we have to consider only 12 (out of the theoretical 32 plants) extremal plants defined as

The unstructured uncertainty,

The proposed approach of shaping the unstructured uncertainty is performed for assumed one zero-two poles

Figure

The stabilizer controller is

To verify that the controller

Up to the value,

A summary of the results for

Greater value of

The stabilizer controller becomes of fifth order:

The Kharitonov templates are shown in Figure

Confirmation of the stabilized closed-loop system is illustrated also in the step responses of the nominal and interval systems (of 32 plants) as in Figure

If it is still required to tolerate larger value of

Magnitude of

Kharitonov’s templates with

Kharitonov’s templates with

Step response of the stabilized interval system.

In this case, the only difference from what we did above is the computation of the unstructured uncertainty

Additive uncertainty case with

Kharitonov’s templates with

A robust stabilization synthesis for systems under mixed structured (parameters) and unstructured multiplicative or additive uncertainties is considered. The implementation of Kharitonov’s theory and the maximum perturbation of the unstructured uncertainty with respect to the range of frequency of interest propose an approach for manipulating these mixed uncertainties. The NPI theorem is used to parameterize all stabilized controllers. A numerical straightforward approach for bounding (loop shaping) the unstructured uncertainty by a proper stable function is proposed to enlarge the box of parametric uncertainty in tandem with either multiplicative or additive uncertainty. The results show that it is always possible to enlarge the box on the expense of increasing the controller order.