PID controller parameters.
Abstract
In this chapter, we propose several Nyquistlike stability criteria for linear dynamical systems that are described by fractional commensurate order linear timeinvariant (FCOLTI) statespace equations (thus endowed with fractionalorder transfer functions) by means of the argument principle for complex analysis. Based on the standard Cauchy integral contour or its shifting ones, the stability conditions are necessary and sufficient, independent of any intermediate poles computation, domain transformation, and distribution investigation, which can be implemented graphically with locus plotting or numerically without any locus plotting. The proposed criteria apply to both single and multiple fractional cases as well and can be exploited in regularorder systems without any modification. Case study is included.
Keywords
 fractionalorder
 commensurate
 stability
 meromorphic/holomorphic
 argument principle
 Cauchy integral contour
1. Introduction
Fractionalorder calculus possesses a long history in pure mathematics. In recent decades, its involvements in systems, control, and engineering have attracted great attention; in the latest years, its significant extensions in various aspects of systems and control are frequently encountered [1, 2, 3, 4, 5, 6, 7, 8]. It turns out that phenomena modeled with fractionalorder calculus much more widely exist than those based on regularorder ones. It has been shown that fractionalorder calculus describes realworld dynamics and behaviors more accurately than the regularorder counterparts and embraces many more analytical features and numerical properties of the observed things; indeed, many practical plants and objects are essentially fractionalorder. Without exhausting the literature, typical examples include the socalled nonintegerorder system of the voltage–current relation of semiinfinite lossy transmission line [9] and diffusion of the heat through a semiinfinite solid, where heat flow is equal to the halfderivative of the temperature [10].
One of the major difficulties for us to exploit the fractionalorder models is the absence of solution formulas for fractionalorder differential equations. Lately, lots of numerical methods for approximate solution of fractionalorder derivative and integral are suggested such that fractionalorder calculus can be solved numerically. As far as fractionalorder systems and their control are concerned, there are mainly three schools related to fractionalorder calculus in terms of system configuration: (i) integerorder plant with fractionalorder controller, (ii) fractionalorder plant with integerorder controller, and (iii) fractionalorder plant with fractionalorder controller. The principal reason for us to bother with fractionalorder controllers is that fractionalorder controllers can outperform the integerorder counterparts in many aspects. For example, it has been confirmed that fractionalorder PID can provide better performances and equip designers with more parametrization freedoms (due to its distributed parameter features [4, 11, 12, 13]).
An important and unavoidable problem about fractionalorder systems is stability [13, 14, 15]. As is well known, stability in integerorder LTI systems is determined by the eigenvalues distribution; namely, whether or not there are eigenvalues on the close righthalf complex plane. The situation changes greatly in fractionalorder LTI systems, due to its specific eigenvalue distribution patterns. More precisely, on the one hand, eigenvalues of fractionalorder LTI systems cannot generally be computed in analytical and closed formulas; on the other hand, stability of the fractionalorder LTI systems is reflected by the eigenvalue distribution in some casesensitive complex sectors [13, 15], rather than simply the close righthalf complex plane for regularorder LTI systems. In this paper, we revisit stability analysis in fractional commensurate order LTI (FCOLTI) systems by exploiting the complex scaling methodology, together with the wellknown argument principle for complex analysis [16]. This work is inspired by the study for structural and spectral characteristics of LTI systems that is also developed by means of the argument principle [17, 18, 19]. The complex scaling technique is a powerful tool in stability analysis and stabilization for classes of linear and/or nonlinear systems; the relevant results by the author and his colleagues can be found in [20, 21, 22, 23, 24, 25]. Also around fractionalorder systems, the main results of this chapter are several Nyquistlike criteria for stability with necessary and sufficient conditions [26], which can be interpreted and implemented either graphically with loci plotting or numerically without loci plotting, independent of any prior pole distribution and complex/frequencydomain facts.
Outline of the paper. Section 2 reviews basic concepts and propositions about stability in FCOLTI systems that are depicted by fractional commensurate order differential equations or statespace equations. The main results of the study are explicated in Section 3. Numerical examples are sketched in Section 4, whereas conclusions are given in Section 5.
2. Preliminaries and properties in FCOLTI systems
2.1 Preliminaries to fractionalorder calculus
Based on [13, 15], fractionalorder calculus can be viewed as a generalization of the regular (integerorder) calculus, including integration and differentiation. The basic idea of fractionalorder calculus is as old as the regular one and can be traced back to 1695 when Leibniz and L’Hôpital discussed what they termed the halforder derivative. The exact definition formula for the socalled
where
Basic facts about fractionalorder calculus are given as follows [13]:
• If
• If
• If
• Fractionalorder differentiation and integration are linear operations. Thus
• Under some additional assumptions about
• If
The fractionalorder calculus (1) and its properties are essentially claimed in the time domain. Therefore, it is generally difficult to handle these relations directly and explicitly. To surmount such difficulties, the Laplace transform of (1) is frequently used, which is given by
where
2.2 Definition and features of FCOLTI statespace equations
A scalar fractionalorder linear timeinvariant system can be described with a fractionalorder statespace equation in the form of
where
For simplicity, we employ
which is the fractionalorder transfer function defined from
In the following, the fractionalorder polynomial
is called the characteristic polynomial of the statespace equation (3).
We note by complex analysis ([16], p. 100) that
where
Bearing (6) in mind, our questions are (i) under what conditions
To address (i), let us return to (6) and observe for any
where we have used
To see under what conditions
where
Obviously,
Under the assumption that (7) and suppose that
Based on (6) and (9), the
By (10), it is not hard to see that
In the sequel, when the assumption (7) is true and
2.3 Closedloop configuration with FCOLTI systems
Consider the feedback system illustrated in Figure 1, in which we denote by
where
Fractionalorder transfer functions for
where
Now we construct the statespace equations for the open and closedloop systems of Figure 1. The openloop system can be expressed by the fractionalorder statespace equation:
In the closedloop system, we can write the closedloop statespace equation as
where
By definition, the characteristic polynomial for the closedloop system
where
with
Let us return to (15) and continue to observe that
In deriving (17), the determinant equivalence
which is nothing but the return difference relationship for the fractionalorder feedback system
3. Main results
3.1 Nyquist contours in the
z
/
s
domains
As another preparation for stability analysis in fractionalorder systems by means of the argument principle for meromorphic functions, we need to choose appropriate Nyquist contours.
Firstly, the simply closed curve defined on the
Secondly, the simply closed curve defined on the
where
Remarks about the contours
• In both cases, the origin of the complex plane is excluded from the contours themselves and their interiors. The reason for these specific contours is that
• One might suggest that in order to detour the origin, the small arc in
3.2 Stability conditions related to FCOLTI systems
Stability conditions in terms of the zeros distribution of
where
3.3 Stability criterion in FCOLTI systems
In what follows, a fractionalorder polynomial
where
vanishes nowhere over
In the above, the clockwise/counterclockwise orientation of
Under the given assumption about the concerned characteristic polynomial and the fact that
Bearing these facts in mind, let us apply the argument principle to (22) counterclockwisely with
More precisely, since
where
Note that all the roots of
The above equation says that
Several remarks about Theorem 1.
• Theorem 1 is independent of the contour and locus orientations; or the locus orientations can be alternatively defined after the locus is already drawn. The fractionally commensurate Hurwitz polynomial
• When the stability locus with respect to the infinite portion of
Since
• Each and all the conditions in Theorem 1 can be implemented only by numerically integrating
• The clockwise/counterclockwise orientation of
Next, a regularorder polynomial
where
vanishes nowhere over
3.4 Stability criteria for closedloop FCOLTI systems
Based on the return difference equation (31) claimed in the feedback configuration of Figure 1, together with the argument principle, the following
satisfies: (i)
To complete the proof, it remains to only show why we must work with the contour
which says clearly that if there are imaginary zeros of
On the contrary, if there exist no imaginary openloop poles, it is not hard to see that working with
As a
satisfies: (i)
In the above,
Several remarks about Theorems 3 and 4:
• The shifted contour
• Clearly, the detouring treatments in Theorems 3 and 4 do not exist in Theorems 1 and 2, since the stability conditions in the latter ones are claimed directly on the fractionalorder characteristic polynomials, in which transfer functions are not involved.
4. Numerical illustrations
4.1 Example description for Theorems 1 and 2
Consider a single fractionalorder commensurate system [15] with the characteristic polynomial
where the commensurate order
In what follows, the
4.2 Numerical results for Theorems 1 and 2
The following cases are considered in terms of
•
The same conclusions can be drawn by examining the
•
The same conclusions can be drawn by examining the
•
The instability conclusions in each
•
The instability conclusions in each
•
The instability conclusions in each
•
Stability in each case of
Based on the numerical results, the stability/instability conclusions based on the
4.3 Example description for Theorem 3
Consider the feedback configuration of Figure 1 used for automatic voltage regulator (AVR) in generators, which is formed by the subsystems [6]:
where
In the following, we focus merely on verifying the closedloop stability based on Theorem 3, based on the parametrization results therein. To this end, the
The socalled optimal controller parameters are listed in Table 1.







Case 1  1.2623  0.5531  0.2382  1.2555  1.1827 
Case 2  1.2623  0.5526  0.2381  1.2559  1.1832 
4.4 Numerical results for Theorem 3
Based on Table 1, the stability loci in the two cases are plotted in Figure 10. The stability loci for the two cases cannot be distinguished from each other graphically. By counting the outmost circle as one clockwise encirclement around the origin, then one can count another counterclockwise encirclement after zooming into the local region around the origin; it follows that the net encirclements number is zero. Indeed, our numerical phase increment computations in either case yields that
5. Conclusions
Stability is one of the imperative and thorny issues in analysis and synthesis of various types of fractionalorder systems. By the literature [28, 29, 30], the frequently adopted approaches are through single/multiple complex transformation such that fractionalorder characteristic polynomials are transformed into standard regularorder polynomials, and then stability testing of the concerned fractionalorder systems is completed by the root distribution of the corresponding regularorder polynomials. In view of the root computation feature, such existing approaches are direct in testing methodology.
In this paper, we claimed and proved an indirect approach that is meant also in the
Acknowledgments
The study is completed under the support of the National Natural Science Foundation of China under Grant No. 61573001.
References
 1.
Fedele G, Ferrise A. Periodic disturbance rejection with unknown frequency and unknown plant structure. Journal of the Franklin Institute. 2014; 351 :10741092  2.
Fedele G, Ferrise A. Periodic disturbance rejection for fractionalorder dynamical systems. Fractional Calculus and Applied Analysis. 2015; 18 (3):603620  3.
Li MD, Li DH, Wang J, Zhao CZ. Active disturbance rejection control for fractionalorder system. ISA Transactions (The Journal of Automation). 2013; 52 :365374  4.
Padula F, Visioli A. Tunning rules for optimal PID and fractionalorder PID controllers. Journal of Process Control. 2011; 21 :6981  5.
Podlubny I, Petráš I, Vinagre BM, O’Leary P, Dorčák L. Analogue realizations of fractionalorder controllers. Nonlinear Dynamics. 2002; 29 :281296  6.
Ramezanian H, Balochian S, Zare A. Design of optimal fractionalorder PID controllers using particle swarm optimization for automatic voltage regulator (AVR) system. Journal of Control, Automation and Electrical Systems. 2013; 24 :601611  7.
Raynaud HF, Zergainoh A. Statespace representation for fractional order controllers. Automatica. 2000; 36 :10171021  8.
Tavazoei MS. A note on fractionalorder derivatives of periodic functions. Automatica. 2010; 48 :945948  9.
Wang JC. Realizations of generalized warburg impedance with RC ladder networks and transmission lines. Journal of the Electrochemical Society. 1987; 134 (8):19151920  10.
Podlubny I. FractionalOrder Differential Equations. San Diego: Academic Press; 1999  11.
Biswas K, Sen S, Dutta PK. Realization of a constant phase element and its performance study in a differentiabtor circuit. IEEE Transactions on Circuits and Systems  I. 2006; 53 (9):802807  12.
Chen YQ. Ubiquitous fractional order controls? In: Plenary talk in the Second IFAC Symposium on Fractional Derivatives and Applications (IFAC FDA2006); Port, Portugal; 2006  13.
Chen YQ, Petráš I, Xue D. Fractional order control—A tutorial. In: Proceedings of the 2009 American Control Conference; 2009. pp. 13971411  14.
Alagoz BB. A note on robust stability analysis of fractional order interval systems by minimum argument vertex and edge polynomials. IEEE/CAA Journal of Automatica Sinica. 2016; 3 (4):411417  15.
Radwan AG, Soliman AM, Elwakil AS, Sedeek A. On the stability of linear systems with fractionalorder elements. Chaos, Solitons & Fractals. 2009; 40 :23172328  16.
Stein EM, Shakarchi R. Complex Analysis. Princeton/Oxford: Princeton University Press; 2003. 379p  17.
Zhou J. Generalizing Nyquist criteria via conformal contours for internal stability analysis. Systems Science & Control Engineering. 2014; 2 :444456. DOI: 10.1080/21642583.2014.915204  18.
Zhou J, Qian HM. Pointwise frequency responses framework for stability analysis in periodically timevarying systems. International Journal of Systems Science. 2017; 48 (4):715728. DOI: 10.1080/00207721.2016.1212430  19.
Zhou J, Qian HM, Lu XB. An argumentprinciplebased framework for structural and spectral characteristics in linear dynamical systems. International Journal of Control. 2017; 90 (12):25982604. DOI: 10.1080/00207179.2016.1260163  20.
Zhou J, Qian HM. Stability analysis of sampleddata systems via open/closedloop characteristic polynomials contraposition. International Journal of Systems Science. 2017; 48 (9):19411953. DOI: 10.1080/00207721.2017.1290298  21.
Zhou J. Complex scaling circle criteria for Lur’e systems. International Journal of Control. 2019; 92 (5):975986. DOI: 10.1080/00207179.2017.1378439  22.
Zhou J. Interpreting Popov criteria in Lure systems with complex scaling stability analysis. Communications in Nonlinear Science and Numerical Simulation. 2018; 59 :306318. DOI: 10.1016/j.cnsns.2017.11.029  23.
Zhou J, Gao KT, Lu XB, et al. Mathematical Problems in Engineering. 2018; 2018 :8492735. DOI: 10.1155/2018/8492735. 14p  24.
Zhou J, Gao KT, Lu XB. Stability analysis for complicated sampleddata systems via descriptor remodeling. IMA Journal of Mathematical Control and Information. 2018. DOI: 10.1093/imamci/dny031  25.
Zhou J. Stability analysis and stabilization of linear continuoustime periodic systems by complex scaling. International Journal of Control. 2018. DOI: 10.1080/00207179.2018.1540888  26.
Zhou J. Complexdomain stability criteria for fractionalorder linear dynamical systems. IET Control Theory and Applications. 2017; 11 (16):27532760. DOI: 10.1049/ietcta.2016.1578  27.
Watanabe R, Miyazaki H, Ehto S. Complex Analysis. Tokyo: Pefukan Press; 1991  28.
Monje CA, Chen YQ, Vinagre BM, Xue DY, Feliu V. FractionalOrder Systems and Controls: Fundamentals and Applications. London/New York: Springer; 2010  29.
RajasMoreno A. An approach to design MIMO fractionalorder controllers for unstable nonlinear systems. IEEE/CAA Journal of Automatica Sinica. 2016; 3 (3):338344  30.
Soorki MN, Tavazoei MS. Constrained swarm stabilization of fractional order linear time invariant swarm systems. IEEE/CAA Journal of Automatica Sinica. 2016; 3 (3):320331
Notes
 The fractionalorder integral portion in the PID is approximated by K I / s λ + 0.0001 in order to avoid definition problem at the origin when it is in the form of K I / s λ .