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Application of Value Set Concept to Ellipsoidal Polynomial Families with Multilinear Uncertainty Structure
(Springer International Publishing Ag, 2019) Matusu, Radek; Senol, Bilal
The contribution intends to present the application of the value set concept to the ellipsoidal polynomial families with multilinear uncertainty structure. It is a follow-up to the previously published work, where the ellipsoidal polynomial families with affine linear uncertainty structure were studied. In the first parts of this paper, the basic terms related to the robustness under parametric uncertainty (e.g., uncertainty structure, uncertainty bounding set, family, and value set) are briefly recalled, with the accent on the ellipsoidal polynomial families. Subsequently, the non-convex value sets of the illustrative ellipsoidal polynomial family with multilinear uncertainty structure are plotted and analyzed. It is shown that the boundaries of the value set need not to mapped only from the boundaries in the parameter space but possibly also from the internal points.
Calculation of robustly relatively stabilizing PID controllers for linear time-invariant systems with unstructured uncertainty
(Elsevier Science Inc, 2022) Matusu, Radek; Senol, Bilal; Pekar, Libor
This article deals with the calculation of all robustly relatively stabilizing (or robustly stabilizing as a special case) Proportional-Integral-Derivative (PID) controllers for Linear Time-Invariant (LTI) systems with unstructured uncertainty. The presented method is based on plotting the envelope that corresponds to the trios of P-I-D parameters marginally complying with given robust stability or robust relative stability condition formulated by means of the H infinity norm. Thus, this approach enables obtaining the region of robustly stabilizing or robustly relatively stabilizing controllers in a P-I-D space. The applicability of the technique is demonstrated in the illustrative examples, in which the regions of robustly stabilizing and robustly relatively stabilizing PID controllers are obtained for a controlled plant model with unstructured multiplicative uncertainty and unstructured additive uncertainty. Moreover, the method is also verified on the real laboratory model of a hot-air tunnel, for which two representative controllers from the robust relative stability region are selected and implemented.
Description and Analysis of Systems with Unstructured Additive Uncertainty
(Springer International Publishing Ag, 2018) Matusu, Radek; Senol, Bilal
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.
Design of Robust PI Controllers for Interval Plants With Worst-Case Gain and Phase Margin Specifications in Presence of Multiple Crossover Frequencies
(Ieee-Inst Electrical Electronics Engineers Inc, 2022) Matusu, Radek; Senol, Bilal; Alagoz, Baris Baykant; Pekar, Libor
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.
Disturbance rejection FOPID controller design in v-domain
(Elsevier, 2020) Tufenkci, Sevilay; Senol, Bilal; Alagoz, Baris Baykant; Matusu, Radek
Due to the adverse effects of unpredictable environmental disturbances on real control systems, robustness of control performance becomes a substantial asset for control system design. This study introduces a v-domain optimal design scheme for Fractional Order Proportional-Integral-Derivative (FOPID) controllers with adoption of Genetic Algorithm (GA) optimization. The proposed design scheme performs placement of system pole with minimum angle to the first Riemann sheet in order to obtain improved disturbance rejection control performance. In this manner, optimal placement of the minimum angle system pole is conducted by fulfilling a predefined reference to disturbance rate (RDR) design specification. For a computer-aided solution of this optimal design problem, a multi-objective controller design strategy is presented by adopting GA. Illustrative design examples are demonstrated to evaluate performance of designed FOPID controllers. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University.
Fractional order proportional derivative control for time delay plant of the second order: The frequency frame
(Pergamon-Elsevier Science Ltd, 2020) Senol, Bilal; Demiroglu, Ugur; Matusu, Radek
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. (c) 2020 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Fractional Order Stability of Systems
(Ieee, 2017) Senol, Bilal; Matusu, Radek; Gul, Emine
This paper investigates and offers some stability analysis methods for systems of non-integer orders. Well known analysis methods such as Hurwitz, interlacing property, monotonic phase increment property are reconsidered in a fractional order way of thinking. A method to find the roots of the so-called fractional order polynomials is proposed and Hurwitz-like stability of the pseudo-polynomials is investigated. Effectiveness of the interlacing property and outcomes of the monotonic phase increment property for fractional order case is shown. Results are comparatively proved and illustrated clearly.
İkinci derece zaman gecikmeli modeller için kesir dereceli oransal-integral denetleyici tasarımında analitik yaklaşım
(2022) Şenol, Bilal; Demiroğlu, Uğur; Matusu, Radek
Bu yayın, ikinci derece zaman gecikmeli modellerin kararlılık ve dayanıklı performansı için kesir dereceli oransal-integral denetleyicinin adım adım tasarımına odaklanmaktadır. Analitik olarak elde edilmiş denklemler genelleştirilmiştir ve söz konusu modeller için kullanılabilir. Yöntemin ana hedefi, Bode çizimindeki kazanç ve faz kesim frekansları arasında kalan faz eğrisini düzleştirmektir. Bu şekilde, kazanç değişimlerine karşı dayanıklılık sağlanacaktır. Bunun yanısıra, tüm sistemin kararlılığı temin edilecektir. Tasarım aşamasında, literatürde var olan çalışmaların aksine sadece kazanç kesim frekansı değil, kazanç ve faz kesim frekanslarının her ikisi de ele alınmıştır. Ayrıca, faz düzleştirme işlemi faz türevinin sıfıra eşitlenmesi ile sağlanmamıştır. Bu yayın, probleme farklı bir bakış açısı getirmektedir. İki farklı denetleyici hesaplanmıştır. İlk denetleyici, istenen kazanç kesim frekansı ve faz payı özelliklerini sağlamaktadır. İkinci ise faz kesim frekansı ve kazanç payını temin etmektedir. Daha sonra bu denetleyiciler bağlanmıştır ve her iki durumu da sağlayan tek bir denetleyici elde edilmiştir. Önerilen denklemler, literatürden iki farklı model üzerine uygulanmış ve sonuçlar grafiksel olarak verilmiştir.
Linear systems with unstructured multiplicative uncertainty: Modeling and robust stability analysis
(Public Library Science, 2017) Matusu, Radek; Senol, Bilal; Yeroglu, Celaleddin
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.
On the Effects of the Frequency Frame on DC Motor Example: Fractional Order PI-PD Case
(Ieee, 2019) Demiroglu, Ugur; Senol, Bilal; Matusu, Radek
This is a comparative study investigating the effects of the controller design method, frequency frame on a DC motor example. Both fractional order proportional integral and proportional derivative controllers are designed for mentioned example which is a first order plus time delay model. The two controllers are in different structures in that they can show different behaviors. The frequency frame approach is a frequency domain method which intends to provide the objective system improved stability and robustness by shaping the Bode curves. In contrary to existing similar methods, shaping is done by a graphical point of view, not by complicated mathematical operations. The reason why a fractional order controller is selected comes from the necessity of a common constant. The study in the paper aims to show the effectiveness of the frequency frame by providing desired frequency specifications with different controllers. Both controllers are calculated for a DC motor transfer function and the comparative results are given.
Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems
(Mdpi, 2021) Tufenkci, Sevilay; Senol, Bilal; Matusu, Radek; Alagoz, Baris Baykant
Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems.
An overview of FOPID controller design in v-domain: design methodologies and robust controller performance
(Taylor & Francis Ltd, 2023) Tufenkci, Sevilay; Alagoz, Baris Baykant; Senol, Bilal; Matusu, Radek
The complex v-plane is an emerging design domain for fractional order control system design. Recently, several works demonstrated the advantages of tuning FOPID controllers in v-plane. These approaches essentially perform the minimum angle pole placement to a target angle within the stability region of the v-plane and facilitate fractional order control system design tasks because of inherently guaranteed stabilisation of fractional order transfer functions. Accordingly, the optimal FOPID controller tuning problem can be simplified to placement of minimum angle system pole to a target point within the stability region of the v-plane. After reviewing previous v-domain design works, authors investigate prominent target points that can result in improved FOPID control performance for the v-domain design task. The consideration of target points in polar coordinates can provide two design parameters (angle and magnitude), which can convey essential system knowledge associated with the stability and control performance of FOPID control systems. In this perspective, effects of minimum angle pole positions on control performance indices are investigated in detail, and some prominent target points to manage FOPID design in v-domain have been reported. The v-domain design examples are illustrated to reveal the effects of the sampled pole positions on the robust control performance.
Robust Control of Air Flow Speed in Laboratory Model of Hot-Air Tunnel: Multiplicative Uncertainty-Based Approach
(Springer International Publishing Ag, 2023) Matusu, Radek; Senol, Bilal; Shaikh, Ibrahim
This contribution deals with the comparison of control simulations and real-world control experiments on a laboratory model of a hot-air tunnel using robustly relatively stabilizing Proportional-Integral(-Derivative) (PI(D)) controllers. The control synthesis takes advantage of the recently published authors'work that has presented amethod for calculation of robustly relatively stabilizing PID controllers for Linear Time-Invariant (LTI) systemswith unstructured uncertainty. Even though controller design uses the plant model with unstructured multiplicative uncertainty, the simulations are based on sampling the uncertain parameters in a corresponding model with parametric uncertainty.
Robust PI Control of Interval Plants With Gain and Phase Margin Specifications: Application to a Continuous Stirred Tank Reactor
(Ieee-Inst Electrical Electronics Engineers Inc, 2020) Matusu, Radek; Senol, Bilal; Pekar, Libor
The paper is focused on robust Proportional-Integral (PI) control of interval plants with gain and phase margin specifications and on the application of this approach to a Continuous Stirred Tank Reactor (CSTR). More specifically, the work aims at the determination of PI controller parameter regions, for which not only robust stability but also some level of robust performance of the closed-loop control system is guaranteed, and this robust performance is represented by the required gain and phase margin that has to be ensured for all potential members of the interval family of controlled plants, even for the worst case. The applied technique is based on the combination of the previously published generalization of stability boundary locus method (for specified gain and phase margin under the assumption of fixed-parameter plants) with the sixteen plant theorem. This extension enables the direct application of the method to design the robustly performing PI controllers for interval plants. The effectiveness of the improved method is demonstrated on a CSTR, modeled as the interval plant, for which the robust stability and robust performance regions are obtained.
Robust stability of fractional order polynomials with complicated uncertainty structure
(Public Library Science, 2017) Matusu, Radek; Senol, Bilal; Pekar, Libor
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.
Robust Stability of Fractional-Order Linear Time-Invariant Systems: Parametric versus Unstructured Uncertainty Models
(Wiley-Hindawi, 2018) Matusu, Radek; Senol, Bilal; Pekar, Libor
The main aim of this paper is to present and compare three approaches to uncertainty modeling and robust stability analysis for fractional-order (FO) linear time-invariant (LTI) single-input single-output (SISO) uncertain systems. The investigated objects are described either via FO models with parametric uncertainty, by means of FO unstructured multiplicative uncertainty models, or through FO unstructured additive uncertainty models, while the unstructured models are constructed on the basis of appropriate selection of a nominal plant and a weight function. Robust stability investigation for systems with parametric uncertainty uses the combination of plotting the value sets and application of the zero exclusion condition. For the case of systems with unstructured uncertainty, the graphical interpretation of the utilized robust stability test is based mainly on the envelopes of the Nyquist diagrams. The theoretical foundations are followed by two extensive, illustrative examples where the plant models are created; the robust stability of feedback control loops is analyzed, and obtained results are discussed.
Tuning of PI? Controllers for FOPTD Plants via the Stability Boundary Locus
(Ieee, 2018) Demiroglu, Ugur; Matusu, Radek; Senol, Bilal
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.
Two Approaches to Description and Robust Stability Analysis of Fractional Order Uncertain Systems
(Ieee, 2016) Matusu, Radek; Senol, Bilal
The principal goal of this contribution is to present two different approaches to description and robust stability analysis of continuous-time fractional order uncertain systems. The first approach uses fractional order models with parametric uncertainty and robust stability of corresponding closed-loop control systems is investigated by using the value set concept in combination with the zero exclusion condition. On the other hand, the second approach is based on fractional order models with unstructured multiplicative uncertainty created by choosing a nominal plant and a suitable weight function. In this case, the graphical robust stability test utilizes the envelopes of the Nyquist diagrams. Both methods are illustrated by means of the joint example.
Value-Set-Based Approach to Robust Stability Analysis for Ellipsoidal Families of Fractional-Order Polynomials with Complicated Uncertainty Structure
(Mdpi, 2019) Matusu, Radek; Senol, Bilal; Pekar, Libor
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.

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