Value-set-based approach to robust stability analysis for ellipsoidal families of fractional-order polynomials with complicated uncertainty structure

This paper presents the application of a value-set-based graphical approach to robust stability analysis for the ellipsoidal families of fractional-order polynomials with a complex structure of parametric uncertainty. More specifically, the article focuses on the families of fractional-order linear time-invariant polynomials with affine linear, multilinear, polynomic, and general uncertainty structure, combined with the uncertainty bounding set in the shape of an ellipsoid. The robust stability of these families is investigated using the zero exclusion condition, supported by the numerical computation and visualization of the value sets. Four illustrative examples are elaborated, including the comparison with the families of fractional-order polynomials having the standard box-shaped uncertainty bounding set, in order to demonstrate the applicability of this method. © 2019 by the authors.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [LO1303 (MSMT-7778/2014)

Zaman gecikmesi içeren üçüncü derece sistemler için kesir dereceli PD denetleyici tasarımı

Due to the lack of integral operator, proportional derivative controllers have difficulties in providing stability and robustness. This difficulty is especially felt in higher order systems. In this publication, analytical design method of fractional proportional derivative controllers is presented to ensure the stability of third order systems with time delay. In this method, it is aimed to achieve the frequency characteristics of a standard control system to ensure stability. It is aimed to provide the desired gain crossover frequency, phase crossover frequency and phase margin properties of the system. In this way, the stability and robustness of the system can be obtained by choosing the appropriate values. The reason for choosing a fractional order controller is that the controller parameters to provide these features can be tuned more accurately. In order for the obtained stability to be robust to unexpected external effects, it is aimed to flatten the system phase. In the literature, phase flattening is performed by setting the phase derivative to zero at a specified frequency value. This can lead to mathematical complexity. In this publication, the phase flattening process is provided graphically by correctly selecting the frequency characteristics given above. Thus, an accurate and reliable controller design method is presented, avoiding mathematical complexity. The effectiveness of the proposed method has been demonstrated on three different models selected from the literature. The positive contribution of the method to the system robustness has been proven by changing the system gain at certain rates.İntegral operatörünün eksikliğinden dolayı, oransal türev denetleyiciler kararlılık ve dayanıklılığı sağlama konularında zorlanabilmektedir. Bu zorluk, özellikle yüksek dereceli sistemlerde kendini daha çok hissettirmektedir. Bu yayında, zaman gecikmesi içeren üçüncü derece sistemlerin kararlılığının sağlanması için kesir dereceli oransal türev denetleyicilerin analitik tasarım yöntemi sunulmuştur. Bu yöntemde kararlılığın sağlanması için standart bir kontrol sisteminin sahip olduğu frekans özelliklerine ulaşılması hedeflenmiştir. Sistemin istenen kazanç kesim frekansı, faz kesim frekansı ve faz payı özelliklerini sağlaması hedeflenmiştir. Bu şekilde uygun değerler seçilerek sistemin kararlılığı ve dayanıklılığı elde edilebilecektir. Kesir dereceli bir denetleyicinin seçilme sebebi de bu özelikleri sağlayacak denetleyici parametrelerinin daha doğru şekilde ayarlanabilmesidir. Elde edilen kararlılığın beklenmeyen dış etkilere karşı dayanıklı olması için de sistem fazının düzleştirilmesi hedeflenmiştir. Literatürde faz düzleştirme işlemi, faz türevinin belirlenen bir frekans değerinde sıfırlanması ile gerçekleştirilmektedir. Bu da matematiksel karmaşıklığa yol açabilmektedir. Bu yayında ise faz düzleştirme işlemi yukarıda verilen frekans özelliklerinin doğru şekilde seçilmesi ile grafiksel olarak sağlanmaktadır. Böylece matematiksel karmaşıklıktan kaçınılarak, doğru ve güvenilir bir denetleyici tasarım yöntemi sunulmuştur. Önerilen yöntemin etkinliği literatürden seçilmiş üç farklı model üzerinde gösterilmiştir. Yöntemin sistem dayanıklılığına pozitif katkısı ise sisteme kazancının belli oranlarda değiştirilmesi ile ispatlanmıştır

Fractional order proportional derivative control for time delay plant of the second order: The frequency frame

This paper intends to tune fractional order proportional derivative controller for the performance, stability and robustness of second order plus time delay plant. The tuning method is based on the previously proposed “frequency frame” which is a rectangular frame enclosing gain and phase margins limited with gain and phase crossover frequencies in the Bode plot. Edges of the frame are tuned to achieve desired crossover frequencies and margins. By shaping the curves of the Bode plot, improvements are observed in the performance and robustness of the second order plus time delay system controlled by a fractional order proportional derivative controller. Generalized equations to obtain the parameters of the fractional order proportional derivative controller for second order plus time delay plant are given. In contrast to existing studies, this method reduces mathematical complexity when providing desired properties. Three examples are considered and effectiveness of the frequency frame is shown. © 2020 The Franklin Institut

Tuning of PI λ Controllers for FOPTD Plants via the Stability Boundary Locus

This study intends to present the systematic design procedure of stabilizing fractional order proportional integral controllers for first order plus time delay (FOPTD) plants via the stability boundary locus (SBL) method. Proposed method gives generalized computation equations for tuning related controllers within desired frequency ranges. Besides, equations to compute the real root boundary (RRB) of the SBL are presented. In spite of the classical design technique, this paper proposes to find the starting and ending points and the stability region of the stability boundary of the system by heuristic approach. This method ensures the stability by analytically obtained formulas instead of testing the regions of the SBL with arbitrary selected points. This proves the advantage and the contribution of the proposed procedure. Obtained equations are applied on examples and the results are illustratively given. Comparisons with the literature showed the effectiveness of the proposal.Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [LO1303 (MSMT-7778/2014)

An examination of complex fractional order physical phenomena in IOPD controller design

This research focuses on the fractional complex order plant (FCOP). The significant contribution is the role of complex plant models in system stability and robustness and associated physical phenomena. A general transfer function is studied in the paper. Other plant models may be built with this structure since the FCOP is a general mathematical form covering integer order plant (IOP) and fractional order plant (FOP). Using the equations produced with the proposed technique and the recommended integer order proportional derivative (IOPD controller, physical changes in integer, fractional and complex coefficients, and orders are observed within this paper. Analysis of the plant controlled with an IOPD controller is done by applying an integrator to reveal the differences. The effects of the parameters are discussed together with the visuals, supported by simulations. The aim is to tune the controller parameters to achieve the phase and specifications as the researcher desired. It is observed that the integrator greatly takes part in reducing the steady-state error. The IOP with the integrator showed the lowest steady-state error, and also, the settling and overshoot time were enhanced. Increase in the phase margin also caused an increase in the phase crossover frequency. It is also observed that the fractional order affected the phase crossover frequency comparing with the IOP, and the complex order modification also had an effect comparing to the fractional order version. The complex order of the system is considered with its conjugate components in the imaginary part thus, the results are found separately for each case. © 2023 John Wiley & Sons Ltd

Design of robust PI controllers for interval plants with worst-case gain and phase margin specifications in presence of multiple crossover frequencies

This article deals with the computation of robustly performing Proportional-Integral (PI) controllers for interval plants, where the performance measures are represented by the worst-case Gain Margin (GM) and Phase Margin (PM) specifications, in the event of multiple Phase Crossover Frequencies (PCFs) and/or Gain Crossover Frequencies (GCFs). The multiplicity of PCFs and GCFs poses a considerable complication in frequency-domain control design methods. The paper is a continuation of the authors' previous work that applied the robust PI controller design approach to a Continuous Stirred Tank Reactor (CSTR). This preceding application represented the system with a single PCF and a single GCF, but the current article focuses on a case of multiple PCFs and GCFs. The determination of a robust performance region in the P-I plane is based on the stability/performance boundary locus method and the sixteen plant theorem. In the illustrative example, a robust performance region is obtained for an experimental oblique wing aircraft that is mathematically modeled as the unstable interval plant. The direct application of the method results in the (pseudo-)GM and (pseudo-)PM regions that "illogically" protrude from the stability region. Consequently, a deeper analysis of the selected points in the P-I plane shows that the calculated GM and PM boundary loci are related to the numerically correct values, but that the results may be misleading, especially for the loci outside the stability region, due to the multiplicity of the PCFs and GCFs. Nevertheless, the example eventually shows that the important parts of the GM and PM regions, i.e., the parts that have an impact on the final robust performance region, are valid. Thus, the method is applicable even to unstable interval plants and to the control loops with multiple PCFs and GCFs

A system with multiplicative uncertainty.

<p>A system with multiplicative uncertainty.</p