Radek Matušů

Graphical analysis of robust stability for systems with parametric uncertainty: an overview

Systems with parametric uncertainty play an important role in both the theory and practical applications of robust control. They are described by the mathematical model containing parameters that are not precisely known, but the values thereof lie within given intervals. This type of uncertainty can arise during the control of real processes, eg, as a consequence of modelling effort, imprecise measuring or the influence of certain external conditions. The number of robust stability analysis techniques has been developed during the previous years of research interest. However, many of them are highly specialized only for specific types of uncertainty structure. This paper offers an overview of illustrative examples aimed at usage of a universal graphical approach to robust stability analysis for systems with parametric uncertainty. The investigation method is based mainly on the combination of the value set concept and the zero exclusion condition, which is greatly advantageous especially for more complicated problems.

Single-Parameter Tuning of PI Controllers: From Theory to Practice

The paper deals with influence of a single scalar positive tuning parameter on performance properties of the closed control loop which contains algebraically designed PI controller while the response quality is evaluated by the size of first under- or overshoots. The controller coefficients are calculated from general solutions of diophantine equations in the ring of proper and Hurwitz stable rational functions. Subsequently, these controllers can be tuned by the only parameter. The contribution brings simple tuning rules and, moreover, it presents their possible practical application during control of real laboratory model assumed as system with parametric uncertainty.

DESIGN OF ROBUST PI CONTROLLERS AND THEIR APPLICATION TO A NONLINEAR ELECTRONIC SYSTEM

Design of Robust PI Controllers and their Application to a Nonlinear Electronic System The principal aim of the paper is to present a possible approach to the design of simple Proportional-Integral (PI) robust controllers and subsequently to demonstrate their applicability during control of a laboratory model with uncertain parameters through the Programmable Logic Controller (PLC) SIMATIC S7-300 by Siemens Company. The proposed and utilized synthesis consists of two steps. The former one is determination of controller parameters area, which ensures the robustly stable control loop and is based on computing/plotting the stability boundary locus while the latter one lies in the final choice of the controller itself relying on algebraic techniques. The basic theoretical parts are followed by laboratory experiments in which the 3rd order nonlinear electronic model has been successfully controlled in various working points.

Single-parameter tuning of PI controllers: Theory and application

Abstract The paper deals with the influence of a single tuning parameter on performance properties of the closed control loop which contains algebraically designed continuous-time PI controller while the response quality is evaluated by the size of first under- or overshoots. The controllers are obtained from general solutions of Diophantine equations in the ring of proper and stable rational functions and can be subsequently tuned via the only parameter. This article brings simple tuning rules, verifies their validity by means of simulation example and, moreover, it presents their possible practical application during control of real laboratory model assumed as system with parametric uncertainty.

Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach

The paper deals with a graphical approach to investigation of robust stability for a feedback control loop with an uncertain fractional order time-delay plant and integer order or fractional order controller. Robust stability analysis is based on plotting the value sets for a suitable range of frequencies and subsequent verification of the zero exclusion condition fulfillment. The computational examples present the typical shapes of the value sets of a family of closed-loop characteristic quasipolynomials for a fractional order plant with uncertain gain, time constant, or time-delay term, respectively, and also for combined cases. Moreover, the practically oriented example focused on robust stability analysis of main irrigation canal pool controlled by either classical integer order PID or fractional order PI controller is included as well.

The Kronecker summation method for robust stabilization applied to a chemical reactor

The paper focuses on robust stabilization where the suitable parameters of a simple continuous-time PI controller are determined through a combination of the Kronecker summation method, sixteen plant theorem, and an algebraic approach to control design in the ring of proper and stable rational functions. The initial theoretical background is followed by an illustrative experiment which includes computation of the controller and verification of control results for a continuous stirred tank reactor with exothermic reaction modelled as a fourth-order interval system.

Linear systems with unstructured multiplicative uncertainty: Modeling and robust stability analysis.

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.

Computation of robustly stabilizing PID controllers for interval systems

The paper is focused on the computation of all possible robustly stabilizing Proportional-Integral-Derivative (PID) controllers for plants with interval uncertainty. The main idea of the proposed method is based on Tan’s (et al.) technique for calculation of (nominally) stabilizing PI and PID controllers or robustly stabilizing PI controllers by means of plotting the stability boundary locus in either P-I plane or P-I-D space. Refinement of the existing method by consideration of 16 segment plants instead of 16 Kharitonov plants provides an elegant and efficient tool for finding all robustly stabilizing PID controllers for an interval system. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example in which the uncertain mathematical model of an experimental oblique wing aircraft is robustly stabilized.

Description and Analysis of Systems with Unstructured Additive Uncertainty

This contribution is focused on systems with unstructured additive uncertainty, their description and robust stability analysis. The work presents particularly the example of the additive uncertainty model creation on the basis of a third order integrating plant with parametric uncertainty by means of the selection of a nominal system and a suitable weight function. Moreover, it compares the results of robust stability border investigation for parametric, multiplicative and additive uncertainty model cases.

Robust stability of fractional order polynomials with complicated uncertainty structure

The main aim of this article is to present a graphical approach to robust stability analysis for families of fractional order (quasi-)polynomials with complicated uncertainty structure. More specifically, the work emphasizes the multilinear, polynomial and general structures of uncertainty and, moreover, the retarded quasi-polynomials with parametric uncertainty are studied. Since the families with these complex uncertainty structures suffer from the lack of analytical tools, their robust stability is investigated by numerical calculation and depiction of the value sets and subsequent application of the zero exclusion condition.