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.
Citation: Matušů R, Şenol B, Pekař L (2017) Robust stability of fractional order polynomials with complicated uncertainty structure. PLoS ONE 12(6): e0180274. https://doi.org/10.1371/journal.pone.0180274
Editor: Xiaosong Hu, Chongqing University, CHINA
Received: February 2, 2017; Accepted: June 13, 2017; Published: June 29, 2017
Copyright: © 2017 Matušů et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: This work was supported by the European Regional Development Fund under the project CEBIA‐Tech Instrumentation No. CZ.1.05/2.1.00/19.0376 and by the Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme project No. LO1303 (MSMT‐7778/2014). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Fractional order control represents promising and attractive research topic, which has been widely studied recently. In fact, the field of fractional order calculus itself is not new –, but the true boom of scientific works has exploded in various application areas over the last few years , . The applications of fractional order calculus can be found, among others, in physics , , bioengineering –, viscoelastic materials ,  and also , chaotic systems –, electronic circuits and fractance devices , , ultracapacitors , robotics –, signal processing , , and many other areas. Certainly, the field of automatic control is no exception to this trend, quite the opposite –. On the other hand, the robustness of control systems can already be seen as one of the classical (and fundamental) problems in control engineering theory – and practice . Naturally, the combination of robust and fractional order control is nowadays really appealing research discipline both for linear – and nonlinear – systems.
In control theory, a common way for the incorporation of uncertainty into mathematical model consists in the utilization of the parametric uncertainty. The systems with parametric uncertainty are supposed to have known and fixed structure (i.e. order), but some of their usually real parameters can vary (“slowly” in time) within assumed intervals. The typical problem related to the systems under parametric uncertainty is to investigate if such systems are stable (or such plants are stabilized) for all possible combinations of uncertain parameters, that is if the systems are robustly stable (or the plants are robustly stabilized). An array of methods was developed for robust stability analysis of integer order systems with parametric uncertainty , . The selection of suitable tool depends mainly on the uncertainty structure (i.e. on relations among uncertain coefficients and complexity of used functions). Generally, the more complex uncertainty structures require more complex analysis methods. However, the value set concept combined with the zero exclusion condition  represents a universal graphical tool which is applicable also for the most complicated uncertainty structures –.
As mentioned above, the robust and fractional order control has been widely combined by many researchers nowadays and thus a number of works on the issue of robust stability analysis of fractional order system have appeared lately. The robust stability test procedure for fractional order linear time-invariant (FO-LTI) systems of commensurate orders with interval uncertainty was firstly proposed in . The extension to the case of systems with also interval fractional orders was discussed in . Then, the robust stability problem for the general type of interval FO-LTI systems of noncommensurate orders was opened in . The state-space form of the interval FO-LTI systems was considered and their robust stability tested for the first time in  by means of the matrix perturbation theory. The alternative approach based on the Lyapunov inequality was subsequently presented in . The deficiency of the last two above-mentioned results (and also of many other works that followed) can be seen their conservativeness as the conditions are only sufficient ones. The necessary and sufficient condition for the interval FO-LTI systems was derived e.g. in  by using a complex Lyapunov inequality or in  in terms of linear matrix inequalities. However, both these works considered only the case of fractional order α ∈ [1, 2) and thus some further papers were focused on the α ∈ (0, 1) case–see e.g. . Then, the robust stability of FO-LTI interval systems with linear coupling relationships among the fractional order and other model parameters were studied for the cases of α ∈ [1, 2) and α ∈ (0, 1) in  and , respectively. The robust stability and stabilization of FO-LTI systems with polytopic uncertainty was considered e.g. in . However, the systems with more complicated uncertainty structures suffer from the lack of, especially analytical, tools. An exceptionally universal method is represented by the combination of the value set concept and the zero exclusion condition. Its classical integer order version  was extended to the fractional order cases e.g. in –.
This article presents a graphical approach to the robust stability analysis of families of fractional order polynomials (which can be considered as characteristic polynomials of investigated fractional order systems) with a particular emphasis on families of polynomials with complicated uncertainty structure based on plotting the numerically obtained value sets and utilization of the zero exclusion condition. This work is intended to accompany the contribution  and to put a stress on complex uncertainty structures such as multilinear, polynomial, and general, or even on the uncertain quasi-polynomials arising from the application of time-delay models.
The article is organized as follows. In Section 2, the fractional order polynomials with parametric uncertainty are defined. The Section 3 describes various structures of uncertainty and outlines the typical tools for their robust stability analysis. The graphical approach to robust stability investigation based on the value set concept and the zero exclusion condition is presented in Section 4. Further, Section 5 shows the practical applicability of the method by means of four illustrative examples with various complicated uncertainty structures. And finally, Section 6 offers some conclusion remarks.
2. Fractional order polynomials with parametric uncertainty
General continuous integro-differential operator (differintegral) is defined as , , : (1) where α is the order of the differintegration (typically α ∈ ℝ) and a and t are the limits of the operation. The differintegral can be defined in various ways. The three most common are Riemann-Liouville, Grünwald-Letnikov and Caputo definitions.
The Laplace transform of the differintegral which is defined in the Riemann-Liouville way is given by , , : (2) where integer n lies within (n– 1 < α ≤ n). Under the assumption of zero initial conditions, the Laplace transform is simply : (3) which holds true for all three mentioned differintegral definitions.
The fractional order polynomial with parametric uncertainty has the form: (4) where q is the vector of uncertainty, αn > αn−1 > ⋯ > α1 > α0 are real numbers and ρi for i = 0,…,n are coefficient functions.
The family of fractional order polynomials is then : (5) where Q is the uncertainty bounding set (commonly considered as a multidimensional box, i.e. individual components of vector q are bounded by intervals).
3. Structures of uncertainty
A level of complicatedness of the relations among coefficients of the polynomial Eq (4) (in other words the complexity of the coefficient functions ρi and their interconnections) is a crucial factor for the decision on a suitable tool for robust stability analysis both for integer and fractional order systems with parametric uncertainty. According to this, one can distinguish among several kinds of uncertainty structures. Standard classification for integer order systems is , , :
- Independent uncertainty structure (called interval one for Q in the shape of a box)
- Affine linear uncertainty structure (called polytopic one for Q in the shape of a polytope)
- Multilinear uncertainty structure
- Polynomial (polynomic) uncertainty structure
- General uncertainty structure
On top of that, so-called single parameter uncertainty is a special case, which can be seen as the simplest one despite the structure itself can be formally affine linear or even more complicated.
In the interval uncertainty, each uncertain parameter may enter into the one and only coefficient (although theoretically more uncertain parameters can enter into the same coefficient). This results in the mutual independence of all coefficients and possible utilization of the famous Kharitonov theorem. However, this holds true only for integer order version of interval polynomials. The case of fractional order interval polynomials is a bit more complicated since the real and imaginary parts can be mutually dependent and thus, the classical Kharitonov theorem is not directly applicable anymore (see e.g. ).
The affine linear uncertainty structure means that more uncertain parameters can enter into the same coefficient, but these coefficients must have the form of affine linear functions, i.e.: (6) where k is the dimension of the uncertainty vector q and am are constants for m = 0,…,k. The affine linear uncertainty structure appears very commonly in robust control practice because a simple interval controlled plant in feedback connection with a fixed controller generally leads to a closed-loop characteristic polynomial with affine linear uncertainty structure. A number of tools for investigation or robust stability for this structure can be found in the integer order robust control literature (e.g. the edge theorem, the 32 edge theorem (or similar generalized Kharitonov theorem) and more specialized 16 plant theorem). Robust stability of fractional order systems with affine linear uncertainty structure has been studied e.g. in .
The next and more complex level of relations among polynomial coefficients is represented by the multilinear uncertainty structure. It means that if all but one uncertain parameters are fixed, then ρi is affine linear in the remaining (non-fixed) parameter. Practically speaking, the coefficients can contain the product of uncertain parameters. The robust stability analysis for this uncertainty structure can already be quite a complicated task because the value sets are non-convex and tools based on extreme points or edges are not valid anymore. Well-known result for integer order polynomials with multilinear uncertainty structure is the mapping theorem, which works with the convex hull of the original family. Consequently, the analysis is easier but the cost is the sufficiency of the obtained results. A possible technique for investigation of fractional order polynomials with multilinear uncertainty structure can be found e.g. in .
The family with polynomial (polynomic) uncertainty structure contains the coefficient functions ρi with multivariable polynomials in uncertain parameters. The robust stability analysis is even more complicated because the value sets are not only non-convex but moreover, they can protrude from the convex hulls of the extremes. The polynomial uncertainty structure can be formally transformed into the multilinear one with different uncertainty bounding set but it is not very useful from the analysis point of view.
Finally, in general uncertainty structures the coefficients ρi can be arbitrary multivariable functions of components of q provided that ρi are continuous functions on assumed intervals. Practically no analytical tools are available for robust stability investigation in this general case.
Besides all the mentioned uncertainty structures the special type of retarded quasi-polynomial is a frequent object of interest from the control theory viewpoint. Assume (integer or fractional order) controlled time-delay plant: (7) where not only numerator b(s,q) and denominator a(s,q) polynomials are uncertain, but also the time-delay term Θ(q) can vary within supposed bounds (formally it could be included in the same vector of uncertainty q). Then suppose the plant Eq (7) is in the classical feedback connection with (integer or fractional order) controller: (8) The corresponding uncertain (integer or fractional order) closed-loop characteristic retarded quasi-polynomial is: (9) Simple graphical analysis of robust stability for this kind of fractional order quasi-polynomial is shown in .
4. Value sets and zero exclusion condition
As mentioned above, the complicated structures of uncertainty suffer from the lack of suitable techniques for robust stability analysis. However, a graphical method based on the combination of the value set concept and the zero exclusion condition  represents a universal tool, which can be applied to a wide range of uncertainty structures, including the most complex ones. Besides this, it can be used also for various regions of stability (robust D-stability). More details on parametric uncertainty, related robust stability analysis and several examples of the typical value sets for the integer order systems can be found in  and subsequently e.g. in , . The works – extended the concept of the value set to fractional order uncertain polynomials (or quasi-polynomials ).
The value set for the family of polynomials Eq (5) at the frequency ω ∈ ℝ is defined as : (10) which means that p(jω, Q) is the image of Q under p(jω,·). In practice, the value sets can be constructed by substituting s for jω, fixing ω and letting the vector of uncertain parameters q range over the set Q.
The zero exclusion condition for (Hurwitz) stability of the family of continuous-time polynomials Eq (5) can be formulated : Consider the invariant degree of polynomials in the family, pathwise connected uncertainty bounding set Q, continuous coefficient functions ρk(q) for k = 0, 1, 2,…, n and at least one stable member p(s, q0). Then the family P is robustly stable if and only if the origin of the complex plane (zero point) is excluded from the value set p(jω, Q) at all frequencies ω ≥ 0, i.e. P is robustly stable if and only if: (11)
In this work, the value sets for the fractional order families with complicated uncertainty structures are plotted by using a suitable sampling (gridding) of the uncertain parameters and direct calculation of related partial points of the value sets for a supposed frequency range. It means that the practical plotting of the Figs 1–4 from the next Section 5 and evaluation of the robust stability tests were performed as follows. A suitable set of non-negative frequencies was pre-selected and then the value set for each individual frequency was depicted. All those individual value sets are composed of the points corresponding to the images of all variations of the appropriately sampled uncertain parameters. When the suitable value sets were obtained, their position in relation to the origin of the complex plane had to be checked. As defined above, the family of (quasi-)polynomials is robustly stable if and only if the complex plane origin (zero point) is excluded from the value sets and all other required assumptions are fulfilled, especially the existence of at least one stable member of the family. This existence could be actually verified before the graphical analysis and if a chosen member is found unstable, the graphical test itself can be skipped because the whole family is robustly unstable. This technique is relatively easy-to-use, it leads to the robust stability results with the necessary and sufficient condition, and it is applicable even for the systems with very complicated uncertainty structures, which represents its main advantage. On the other hand, a long computational time for a high number of uncertain parameters is the weakness.
5. Illustrative examples
In order to show the practical applicability of the graphical approach to robust stability analysis discussed hereinbefore, four illustrative examples with families of fractional order (quasi-)polynomials are presented in this Section. The first three examples deal with multilinear, polynomial and general uncertainty structure, successively, and the last one focuses on a family of retarded quasi-polynomials.
5.1 Multilinear uncertainty structure
The value sets depicted for the frequency range from 0 to 2.1 [rad/s] with the step 0.03 by means of sampling the uncertain parameters (with the step 0.02) are shown in Fig 1. In accordance with the process described in the previous Section, the family Eq (12) is robustly stable because it contains a stable member and the origin of the complex plane (zero point) is excluded from the plotted value sets.
5.2 Polynomial uncertainty structure
The value sets are now plotted for the frequency range 0:0.05:1.8 [rad/s] and both uncertain parameters are sampled with the step 0.01 –see Fig 2. Obviously, the family Eq (13) is not robustly stable because the complex plane origin is included in the value sets.
5.3 General uncertainty structure
The use of the frequency range 0:0.1:3 [rad/s] and sampling the uncertain parameters with the step 0.02 lead to the value sets which are shown in Fig 3. Similarly as in the previous case, the family Eq (14) is robustly unstable due to the inclusion of the zero point in the value sets.
5.4 Uncertain quasi-polynomials
The aim of the last example is to decide on the robust (in)stability of the family of retarded quasi-polynomials. In , the fractional order PI controller: (15) was designed for the first order plus time delay plant: (16) For the purpose of this article, a potential change of ±20% in the gain, time constant, and time-delay term is supposed, i.e.: (17) According to Eq (9), the corresponding family of fractional order closed-loop characteristic retarded quasi-polynomials is: (18) where K ∈ [0.8,1.2], T ∈ [0.8,1.2] and Θ ∈ [0.08,0.12].
The value sets are depicted for the frequency range ω = 0:0.1:7 [rad/s] and for the sampled gain K = 0.8:0.02:1.2, time constant T = 0.8:0.02:1.2, and time delay term Θ = 0.08:0.002:0.12. The result can be seen in Fig 4 which clearly demonstrates that the family Eq (18) is robustly stable (the zero point is excluded and the family contains a stable member).
This article was focused on a graphical approach to robust stability investigation for families of fractional order polynomials or even quasi-polynomials with complicated uncertainty structure. The four illustrative examples demonstrated the application of the values set concept and the zero exclusion condition for the families of fractional order polynomials with multilinear uncertainty structure, polynomial uncertainty structure, general uncertainty structure, and for the family of the fractional order retarded quasi-polynomials. The obtained results showed the effectivity of the method for robust stability analysis of fractional order polynomials with various complex uncertainty structures. The potential directions for future research can be seen in robust stability analysis of e.g. fractional order anisochronic systems with internal delays and uncertain parameters , fractional order systems with spherical uncertainty  or fractional order systems with complicated uncertainty structures combined with the uncertain fractional orders.
- 1. Oldham KB, Spanier J. Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. New York–London: Academic Press; 1974.
- 2. Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York, USA: John Wiley and Sons; 1993.
- 3. Podlubný I. Fractional Differential Equations. San Diego, CA, USA: Academic Press; 1999.
- 4. Machado JAT, Kiryakova V, Mainardi F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation. 2011; 16(3): 1140–1153.
- 5. Gutiérrez RE, Rosário JM, Machado JAT. Fractional Order Calculus: Basic Concepts and Engineering Applications. Mathematical Problems in Engineering. 2010; 19 p.
- 6. Dzieliński A, Sierociuk D, Sarwas G. Some applications of fractional order calculus. Bulletin of the Polish Academy of Sciences: Technical Sciences. 2011; 58(4): 583–592.
- 7. Hilfer R. Applications of fractional calculus in physics. Singapore: World Scientific; 2000.
- 8. Sabatier J, Agrawal OP, Machado JAT. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Dordrecht, Netherlands: Springer; 2007.
- 9. Magin RL. Fractional Calculus in Bioengineering. Connecticut, USA: Begell House; 2006.
- 10. Magin RL. Fractional Calculus in Bioengineering: A Tool to Model Complex Dynamics. In: Proceedings of the 13th International Carpathian Control Conference. High Tatras, Slovakia; 2012.
- 11. Magin RL, Ovadia M. Modeling the cardiac tissue electrode interface using fractional calculus. Journal of Vibration and Control. 2008; 14(9–10): 1431–1442.
- 12. Perdikaris P, Karniadakis GE. Fractional-Order Viscoelasticity in One-Dimensional Blood Flow Models. Annals of Biomedical Engineering. 2014; 42(5): 1012–1023. pmid:24414838
- 13. Adolfsson K, Enelund M, Olsson P. On the Fractional Order Model of Viscoelasticity. Mechanics of Time-Dependent Materials. 2005; 9(1): 15–34.
- 14. Heymans N. Dynamic measurements in long-memory materials: fractional calculus evaluation of approach to steady state. Journal of Vibration and Control. 2008; 14(9–10): 1587–1596.
- 15. Petráš I, Bednárová D. Fractional-order chaotic systems. In: Proceedings of the 14th IEEE International Conference on Emerging Technologies & Factory Automation. Palma de Mallorca, Spain; 2009: 1031–1038.
- 16. Radwan AG, Moaddy K, Salama KN, Momani S, Hashim I. Control and switching synchronization of fractional order chaotic systems using active control technique. Journal of Advanced Research. 2014; 5(1): 125–132. pmid:25685479
- 17. Li C, Chen G. Chaos in the fractional order Chen system and its control. Chaos, Solitons & Fractals. 2004; 22(3): 549–554.
- 18. Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons & Fractals. 2016; 89: 447–454.
- 19. Dorčák Ľ, Terpák J, Petráš I, Dorčáková F. Electronic realization of the fractional-order systems. Acta Montanistica Slovaca. 2007; 12(3): 231–237.
- 20. Krishna BT, Reddy KVVS. Active and passive realization of fractance device of order 1/2. Active and Passive Electronic Components. 2008; 5 p.
- 21. Zhang L, Hu X, Wang Z, Sun F, Dorrell DG. Fractional-order modeling and State-of-Charge estimation for ultracapacitors. Journal of Power Sources. 2016; 314: 28–34.
- 22. Silva MF, Machado JAT, Lopes AM. Fractional order control of a hexapod robot. Nonlinear Dynamics. 2004; 38(1–4): 417–433.
- 23. Lima MFM, Machado JAT, Crisóstomo M. Experimental signal analysis of robot impacts in a fractional calculus perspective. Journal of Advanced Computational Intelligence and Intelligent Informatics. 2007; 11(9): 1079–1085.
- 24. Ostalczyk P, Stolarski M. Fractional-Order PID Controllers in a Mobile Robot Control. IFAC Proceedings Volumes. 2009; 42(13): 268–271.
- 25. Delavari H, Lanusse P, Sabatier J. Fractional Order Controller Design for A Flexible Link Manipulator Robot. Asian Journal of Control. 2013; 15(3): 783–795.
- 26. Sheng H, Chen YQ, Qiu TS. Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. London, UK: Springer; 2012.
- 27. Das S, Pan I. Fractional Order Signal Processing: Introductory Concepts and Applications. Heidelberg, Germany: Springer; 2012.
- 28. Chen Y, Petráš I, Xue D. Fractional Order Control–A Tutorial. In: Proceedings of the 2009 American Control Conference. St. Louis, MO, USA; 2009.
- 29. Podlubný I. Fractional-Order Systems and PIλDμ-Controllers. IEEE Transactions on Automatic Control. 1999; 44(1): 208–214.
- 30. Petráš I. Stability of fractional-order systems with rational orders: A survey. Fractional Calculus & Applied Analysis. 2009; 12(3): 269–298.
- 31. Ahmed BS, Sahib MA, Gambardella LM, Afzal W, Zamli KZ. Optimum Design of PIλDμ Controller for an Automatic Voltage Regulator System Using Combinatorial Test Design. PLOS ONE. 2016; 11(11): e0166150. pmid:27829025
- 32. Doyle J, Francis B, Tannenbaum A. Feedback Control Theory. New York, USA: Macmillan; 1992.
- 33. Barmish BR. New Tools for Robustness of Linear Systems. New York, USA: Macmillan; 1994.
- 34. Bhattacharyya SP, Chapellat H, Keel LH. Robust control: The parametric approach. Englewood Cliffs, New Jersey, USA: Prentice Hall; 1995.
- 35. Skogestad S, Postlethwaite I. Multivariable Feedback Control: Analysis and Design. Chichester, UK: John Wiley and Sons; 2005.
- 36. Bhattacharyya SP, Datta A, Keel LH, Linear Control Theory: Structure, Robustness, and Optimization. USA: CRC Press, Taylor & Francis Group; 2009.
- 37. Kasnakoğlu C. Investigation of Multi-Input Multi-Output Robust Control Methods to Handle Parametric Uncertainties in Autopilot Design. PLOS ONE. 2016; 11(10): e0165017. pmid:27783706
- 38. Petráš I, Chen YQ, Vinagre BM. A robust stability test procedure for a class of uncertain LTI fractional order systems, In: Proceedings of the International Carpathian Control Conference. Malenovice, Czech Republic; 2002.
- 39. Petráš I, Chen YQ, Vinagre BM, Podlubný I. Stability of linear time invariant systems with interval fractional orders and interval coefficients. In: Proceedings of the International Conference on Computation Cybernetics. Vienna, Austria; 2005.
- 40. Petráš I, Chen YQ, Vinagre BM. Robust stability test for interval fractional order linear systems. In: Unsolved Problems in the Mathematics of Systems and Control. Blondel VD, Megretski A, Eds. Princeton, NJ: Princeton University Press. 2004; 208–211.
- 41. Chen YQ, Ahn H-S, Podlubný I. Robust Stability Check of Fractional Order Linear Time Invariant Systems With Interval Uncertainties. In: Proceedings of the IEEE International Conference on Mechatronics & Automation. Niagara Falls, Canada; 2005.
- 42. Ahn H-S, Chen YQ, Podlubný I. Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Applied Mathematics and Computation. 2007; 187(1): 27–34.
- 43. Ahn H-S, Chen YQ. Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica. 2008; 44(11): 2985–2988.
- 44. Lu J-G, Chen G. Robust Stability and Stabilization of Fractional-Order Interval Systems: An LMI Approach. IEEE Transactions on Automatic Control. 2009; 54(6): 1294–1299.
- 45. Lu J-G, Chen YQ. Robust Stability and Stabilization of Fractional-Order Interval Systems with the Fractional Order α: The 0 < α <1 Case. IEEE Transactions on Automatic Control. 2010; 55(1): 152–158.
- 46. Liao Z, Peng C, Li W, Wang Y. Robust stability analysis for a class of fractional order systems with uncertain parameters. Journal of The Franklin Institute. 2011; 348(6): 1101–1113.
- 47. Li C, Wang J. Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0 < α <1 case. Journal of The Franklin Institute. 2012; 349(7): 2406–2419.
- 48. Lu J-G, Chen YQ. Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fractional Calculus & Applied Analysis. 2013; 16(1): 142–157.
- 49. Tan N, Özgüven ÖF, Özyetkin MM. Robust stability analysis of fractional order interval polynomials. ISA Transactions. 2009; 48(2): 166–172. pmid:19193371
- 50. Şenol B, Yeroğlu C. Computation of the Value Set of Fractional Order Uncertain Polynomials: A 2q Convex Parpolygonal Approach. In: Proceedings of the 2012 IEEE International Conference on Control Applications. Dubrovnik, Croatia; 2012.
- 51. Şenol B, Yeroğlu C. Robust Stability Analysis of Fractional Order Uncertain Polynomials. In: Proceedings of the 5th IFAC Workshop on Fractional Differentiation and its Applications. Nanjing, China; 2012.
- 52. Yeroğlu C, Şenol B. Investigation of robust stability of fractional order multilinear affine systems: 2q-convex parpolygon approach. Systems & Control Letters. 2013; 62(10): 845–855.
- 53. Şenol B, Ateş A, Alagoz BB, Yeroğlu C. A numerical investigation for robust stability of fractional-order uncertain systems. ISA Transactions. 2014; 53(2): 189–198. pmid:24103699
- 54. Moornani KA, Haeri M. Robust stability testing function and Kharitonov-like theorem for fractional order interval systems. IET Control Theory and Applications. 2010; 4(10): 2097–2108.
- 55. Şenol B, Yeroğlu C. Frequency boundary of fractional order systems with nonlinear uncertainties. Journal of The Franklin Institute. 2013; 350(7): 1908–1925.
- 56. Yeroğlu C, Tan N. Classical controller design techniques for fractional order case. ISA Transactions. 2011; 50(3): 461–472. pmid:21497807
- 57. Yeroğlu C, Özyetkin MM, Tan N. Frequency Response Computation of Fractional Order Interval Transfer Functions. International Journal of Control, Automation, and Systems. 2010; 8(5): 1009–1017.
- 58. Matušů R, Prokop R. Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach. Mathematical Problems in Engineering. 2015; 9 p.
- 59. Ma Y, Lu J-G, Chen W, Chen Y. Robust stability bounds of uncertain fractional-order systems. Fractional Calculus & Applied Analysis. 2014; 17(1): 136–153.
- 60. Lu J, Ma Y, Chen W. Maximal perturbation bounds for robust stabilizability of fractional-order systems with norm bounded perturbations. Journal of The Franklin Institute. 2013; 350(10): 3365–3383.
- 61. Jiao Z, Zhong Y. Robust stability for fractional-order systems with structured and unstructured uncertainties. Computers & Mathematics with Applications. 2012; 64(10): 3258–3266.
- 62. Liu S, Jiang W, Li X, Zhou X-F. Lyapunov stability analysis of fractional nonlinear systems, Applied Mathematics Letters, 2016, Vol. 51, pp. 13–19.
- 63. Yin C, Zhong S, Huang X, Cheng Y. Robust stability analysis of fractional-order uncertain singular nonlinear system with external disturbance. Applied Mathematics and Computation. 2015; 269: 351–362.
- 64. Chang X-H, Xiong J, Park JH. Fuzzy robust dynamic output feedback control of nonlinear systems with linear fractional parametric uncertainties. Applied Mathematics and Computation. 2016; 291: 213–225.
- 65. Matušů R, Prokop R. Graphical analysis of robust stability for systems with parametric uncertainty: an overview. Transactions of the Institute of Measurement and Control. 2011; 33(2): 274–290.
- 66. Matušů R, Prokop R. Robust Stability Analysis for Systems with Real Parametric Uncertainty: Implementation of Graphical Tests in Matlab. International Journal of Circuits, Systems and Signal Processing. 2013; 7(1): 26–33.
- 67. Matušů R, Pekař L. Robust Stability of Thermal Control Systems with Uncertain Parameters: The Graphical Analysis Examples. Manuscript submitted to Applied Thermal Engineering. 2017.
- 68. Gu K, Kharitonov VL, Chen J. Stability of Time-Delay Systems. Boston, USA: Birkhäuser; 2003.
- 69. Luo Y, Chen Y. Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica. 2012; 48(9): 2159–2167.
- 70. Matušů R. Spherical Families of Polynomials: A Graphical Approach to Robust Stability Analysis. International Journal of Circuits, Systems and Signal Processing. 2016; 10: 326–332.