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The authors have declared that no competing interests exist.

This article deals with continuous-time Linear Time-Invariant (LTI) Single-Input Single-Output (SISO) systems affected by unstructured multiplicative uncertainty. More specifically, its aim is to present an approach to the construction of uncertain models based on the appropriate selection of a nominal system and a weight function and to apply the fundamentals of robust stability investigation for considered sort of systems. The initial theoretical parts are followed by three extensive illustrative examples in which the first order time-delay, second order and third order plants with parametric uncertainty are modeled as systems with unstructured multiplicative uncertainty and subsequently, the robust stability of selected feedback loops containing constructed models and chosen controllers is analyzed and obtained results are discussed.

The robustness of control systems represents an attractive research topic whose necessity is boosted by everyday control engineering practice [

Basically, uncertainty in LTI SISO systems can be taken into consideration in two main ways. Either one can use a model with parametric uncertainty [_{∞}) [

The powerful tool for (not only) uncertainty description in LTI Multiple-Input Multiple-Output (MIMO) systems is represented by the Linear Fractional Transformations (LFTs) [

Be reminded that all variations in studied sort of systems are supposed to occur only “slowly” as the LTI systems are considered. Some results on the robust stability of time-variant systems can be found e.g. in [

This article deals with one kind of unstructured uncertainty, known as the multiplicative uncertainty. Its main aim is to present an approach to construct a multiplicative uncertainty model from an LTI SISO system with real parametric uncertainty and also to depict a technique for robust stability analysis. Within the scope of the presented illustrative examples, the first order time-delay, second order and third order plants with parametric uncertainty are modeled as systems with unstructured multiplicative uncertainty (by means of the suitable choice of a nominal system and a weight function), and sequentially, the robust stability of selected feedback loops with obtained plant models and several controllers is investigated. The primary technical contribution of this paper lies in its survey of several specific techniques for the construction of unstructured multiplicative uncertainty models, with the emphasis on the frequently used transfer functions of the controlled plants. Moreover, the work discusses the conservatism in the robust stability tests of the feedback control loops where the “original” parametric uncertainty plant is replaced by an unstructured multiplicative uncertainty model. Some preliminary results related to this paper and a comparison of parametric and unstructured approaches to uncertainty modeling can be found in [

The article is organized as follows. In Section 2, uncertainty modeling techniques with a special accent on unstructured multiplicative uncertainty are briefly described. Section 3 then provides the fundamentals of robust stability analysis for the class of systems being studied. Next, the set of three comprehensive examples (for first order time-delay, second order, and third order plant) focused on modeling and robust stability analysis is presented in the extensive Section 4. And finally, Section 5 offers some concluding remarks.

A preliminary version of this article was presented at the 19th International Conference on Systems, Zakynthos, Greece, 2015 [

The first and fundamental step in robust control is to respect the difference between the true behavior of a control loop and its mathematical description by means of exploiting the uncertain model. Roughly speaking, one fixed “nominal” model is replaced by a whole family of models represented by some neighborhood of the nominal one. This neighborhood can be quantified essentially by means of two main approaches.

The first technique, using parametric uncertainty, supposes known structure of the system (known order), but the imprecisely known real physical parameters. In practice, the parametric uncertainty is given through intervals which bound the uncertain parameters. For details, see e.g. [

On the other hand, the second, unstructured uncertainty approach does not even need any knowledge of the model structure and its description grounds in restriction of the frequency characteristics spread [

The parametric uncertainty approach is natural and advantageous from the viewpoint of its relative simplicity, while the unstructured uncertainty approach is favorable especially for unmodeled dynamics or nonlinearities and, furthermore, preferential for H_{∞} control design methods. Furthermore, various mixtures of uncertainties were investigated e.g. in [

One can distinguish among several types of unstructured uncertainty models, i.e. multiplicative and additive models and their inverse versions, which allow describing also the unstable dynamics [_{0}(_{M}(_{M}(

The scheme of the multiplicative uncertainty (

The multiplicative uncertainty has formally two variants, i.e. the input and output one. The input version corresponds to

Obviously, both versions are equivalent for SISO case, but they have to be distinguished for MIMO systems. The analogical structure holds true also for the inverse multiplicative uncertainty.

The choice of a suitable weight function is the important part of the model creation as will be shown in the examples hereinafter. For the weight function, the following inequality, where the left-hand side represents normalized perturbation (relative error), must be fulfilled:

Many theoretical tools presume that all members of the family _{0}(_{M}(

Under the assumption of multiplicative uncertainty, the closed-loop system is robustly stable if and only if [_{0}(_{0}(_{M}(_{0}(_{0}(

Furthermore, the robust stability condition (

This key section presents examples of the possible construction of multiplicative uncertainty model for the commonly used forms of controlled plants, i.e. for the first order time-delay plant, second order plant and third order plant. Moreover, robust stability is analyzed for a closed loop with the second or third order plant model and selected feedback controllers.

Figs

Initially, assume a frequently used model of a first order time-delay system described by the transfer function:

Recall that, instead of working directly with the parametric system, the model with unstructured multiplicative uncertainty is going to be created and used in the ensuing considerations.

The creation of the multiplicative model (_{0}(_{M}(

Firstly, the nominal model is chosen by taking the mean values of the uncertain parameters i.e. this nominal model still contains the time-delay term:

In the next step, the appropriate weight function, considered as the uncertainty envelope, has to be found in order to fulfill inequality (

The first weight function is determined on the basis of the equation from [_{M1}(

The second weight function is constructed by using the recommendation from [_{0} is the relative uncertainty at steady-state (low frequencies), 1/_{∞} is the magnitude of the weight at high frequencies. The suitable values for this example are _{0} = 1/3, _{∞} = 3 and they result in:
_{M2}(

The good fit of the obtained weight _{M3}(

Nonetheless, not only the weight function but also nominal model can be modified in order to derive a different model with unstructured multiplicative uncertainty. For example, it could be advantageous (because of facilitated controller design) to use a time-delay free nominal model and to consider the time-delay term as the uncertainty. So, the alternative nominal model for the example being studied can be:

Naturally, the set of normalized perturbations will differ from the previously plotted one. For this case, just one weight function is chosen in an analogical way as the function (

A comparison of normalized perturbations and the weight (

In the next example, consider a second order system with two different time constants given by the transfer function:
_{1} ∈ ⟨9,11⟩ and _{2} ∈ ⟨0.9,1.1⟩.

The nominal model is chosen through the average values of the uncertain parameters, i.e.:

A first representative of possible weight functions can be found by using the recommendation for unmodeled dynamics (_{0} = 0.1, _{∞} = 0.36 and

The selection of the second appropriate weight function is based on a simple idea. The worst possible case of uncertainty (which has to be covered by this weight function) in the model (_{1} = 9 and _{2} = 0.9. This combination of parameters directly corresponds to the “uppermost” magnitude characteristics of normalized perturbation from Figs

Bode magnitude plots of both weight functions _{M5} (_{M6} (_{1} = 9:0.2:11 and _{2} = 0.9:0.02:1.1. The zoomed version of the same Bode magnitude plots is depicted in

Thus, if the second (exact) weight (

This (second order plant) example also includes the robust stability analysis. For this purpose, a trio of PI controllers is supposed:

For the first PI controller (_{M}(_{0}(_{0}(

The similar envelope of Nyquist diagrams for the second controller (

Finally, the robust stability condition is visualized for the last PI controller (_{M}(

Obviously, the utilization of a simpler first order weight function (

Alternatively, robust stability can be graphically tested and visualized by means of the condition in the form (_{M}(

For the final case, assume a third order system (inspired by Burns [_{1} ∈ ⟨3,5⟩ and _{2} ∈ ⟨1,3⟩.

The nominal system is considered again as the transfer function with the mean values of the uncertain parameters from (

The appropriate weight was initially chosen as:
^{−1} rad/s (under the assumption of only a constant in the numerator) and that the Bode plot has a peak near this frequency. The magnitude of this peak depends on the size of a damping ratio. The lower damping ratio, the higher peak. However, the requested “final” slope at the high frequencies is only -20 dB/decade and thus the influence of the second order denominator is compensated by the first order polynomial in the numerator. Moreover, this term contributes to the Bode plot rise from the frequency 1 rad/s, i.e. near the beginning of the peak. The final weight function, after slight manual adjustment, has the form:

The Bode magnitude plots of normalized perturbations for all combinations of parameters according to _{1} = 3:0.1:5, _{2} = 1:0.1:3 together with the Bode magnitude plot of the weight function (

So, the final model for the third example is:

The plant family (

This article focused on the modeling and robust stability analysis of continuous-time LTI SISO systems with unstructured multiplicative uncertainty. The examples presented herein have shown the techniques for the construction of multiplicative uncertainty models from systems with parametric uncertainty via the selection of suitable nominal models and weight functions. Moreover, the robust stability of the feedback control loops that contain multiplicative uncertainty plants was analyzed and their conservatism in comparison with the usage of “original” parametric uncertainty plants was discussed.

The authors are grateful to Andreas Chernel for his improvement of the English in one of the versions of this manuscript. Nevertheless, all potential imperfections in this final published version are of sole responsibility of the authors.

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