Open access peer-reviewed chapter - ONLINE FIRST

Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems

By Jun Zhou

Submitted: April 22nd 2019Reviewed: June 18th 2019Published: August 14th 2019

DOI: 10.5772/intechopen.88119

Downloaded: 40

Abstract

In this chapter, we propose several Nyquist-like stability criteria for linear dynamical systems that are described by fractional commensurate order linear time-invariant (FCO-LTI) state-space equations (thus endowed with fractional-order 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 regular-order systems without any modification. Case study is included.

Keywords

  • fractional-order
  • commensurate
  • stability
  • meromorphic/holomorphic
  • argument principle
  • Cauchy integral contour

1. Introduction

Fractional-order 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 fractional-order calculus much more widely exist than those based on regular-order ones. It has been shown that fractional-order calculus describes real-world dynamics and behaviors more accurately than the regular-order counterparts and embraces many more analytical features and numerical properties of the observed things; indeed, many practical plants and objects are essentially fractional-order. Without exhausting the literature, typical examples include the so-called non-integer-order system of the voltage–current relation of semi-infinite lossy transmission line [9] and diffusion of the heat through a semi-infinite solid, where heat flow is equal to the half-derivative of the temperature [10].

One of the major difficulties for us to exploit the fractional-order models is the absence of solution formulas for fractional-order differential equations. Lately, lots of numerical methods for approximate solution of fractional-order derivative and integral are suggested such that fractional-order calculus can be solved numerically. As far as fractional-order systems and their control are concerned, there are mainly three schools related to fractional-order calculus in terms of system configuration: (i) integer-order plant with fractional-order controller, (ii) fractional-order plant with integer-order controller, and (iii) fractional-order plant with fractional-order controller. The principal reason for us to bother with fractional-order controllers is that fractional-order controllers can outperform the integer-order counterparts in many aspects. For example, it has been confirmed that fractional-order 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 fractional-order systems is stability [13, 14, 15]. As is well known, stability in integer-order LTI systems is determined by the eigenvalues distribution; namely, whether or not there are eigenvalues on the close right-half complex plane. The situation changes greatly in fractional-order LTI systems, due to its specific eigenvalue distribution patterns. More precisely, on the one hand, eigenvalues of fractional-order LTI systems cannot generally be computed in analytical and closed formulas; on the other hand, stability of the fractional-order LTI systems is reflected by the eigenvalue distribution in some case-sensitive complex sectors [13, 15], rather than simply the close right-half complex plane for regular-order LTI systems. In this paper, we revisit stability analysis in fractional commensurate order LTI (FCO-LTI) systems by exploiting the complex scaling methodology, together with the well-known 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 fractional-order systems, the main results of this chapter are several Nyquist-like 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/frequency-domain facts.

Outline of the paper. Section 2 reviews basic concepts and propositions about stability in FCO-LTI systems that are depicted by fractional commensurate order differential equations or state-space 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.

Notations and terminologies of the paper. Rand Cdenote the sets of all real and complex numbers, respectively. Ikdenotes the k×kidentity matrix, while C+is the open right-half complex plane, namely, C+=sC:Res>0. means the conjugate transpose of a matrix . N, Nc, and Nc¯stand for the net, clockwise, and counterclockwise encirclements of a closed complex curve around the origin 0j0. By definition, N=NcNc¯. In particular, N=0means that the number of clockwise encirclements of around the origin is equal to that of counterclockwise encirclements.

2. Preliminaries and properties in FCO-LTI systems

2.1 Preliminaries to fractional-order calculus

Based on [13, 15], fractional-order calculus can be viewed as a generalization of the regular (integer-order) calculus, including integration and differentiation. The basic idea of fractional-order 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 half-order derivative. The exact definition formula for the so-called r-order calculus was well established then by Riemann and Liouville in the form of

αDtrft=1Γnrdndtnαtfτtτrn+1E1

where α0and r0are real numbers while n1is an integer; more precisely, n1r<nand nis the smallest integer that is strictly larger than r. Γnris the gamma function at nr; by ([16], p. 160), Γnr=0eττnr1and it is convergent for each nr>0.

Basic facts about fractional-order calculus are given as follows [13]:

• If ftis analytical in t, then αDtrftis analytical in tand r.

• If r0is an integer and n=r+1, then αDtrftreduces to the r+1th-order derivative of ftwith respect to t; namely, αDtrft=dr+1ft/dtr+1.

• If r=0and thus n=1, the definition formula for αDtrft=αDt0ftyields the identity relation αDt0ft=ft.

• Fractional-order differentiation and integration are linear operations. Thus

αDtraft+bgt=aαDtrftft+bαDtrgt.

• Under some additional assumptions about ft, the following additive index relation (or the semigroup property) holds true:

αDtr1αDtr2ft=αDtr2αDtr1ft=αDtr1+r2ft

• If fktt=α=0for k=0,1,,mwith mbeing a positive integer, then fractional-order derivative commutes with integer-order derivative:

dmdtmαDtrft=αDtrdmdtmft=αDtr+mft

The fractional-order 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

L0Dtrft=0est0Dtrftdt=srFsk=0n1sk0Dtrftt=0E2

where Fs=Lftand sis the Laplace transform variable. Under the assumption that the initial conditions involved are zeros, it follows that L0Dtrft=srFs. To simplify our notations, we denote 0DtrftSby Dtrftin the following if nothing otherwise is meant.

2.2 Definition and features of FCO-LTI state-space equations

A scalar fractional-order linear time-invariant system can be described with a fractional-order state-space equation in the form of

Dtrxt=Axt+Butyt=Cxt+DutE3

where xt=x1txntTRn, utR, and ytRare the state, input, and output vectors, respectively. In accordance with xt, utand yt, ARn×n, BRn×1, CR1×n, and DR1×1are constant matrices. We denote:

DtrxtDtrnx1tDtr1xntRn

For simplicity, we employ rto stand for the fractional-order indices set rnr1with 0ri<1with a little abuse of notations. The corresponding transfer function follows as

Gs=Cdiagsrnsr1A1B+D=bmsβm+bm1sβm1++b1sβ1ansαn+an1sαn1++a1sα1E4

which is the fractional-order transfer function defined from Usto Ys. In (4), diagsrnsr1Cn×nstands for a diagonal matrix. Also, akk=1nand bkk=1mare constants, while αkk=1nand βkk=1mare nonnegative real numbers satisfying

αn>αn1>>α10βm>βm1>>β10

In the following, the fractional-order polynomial

Δsrnr1=detdiagsrnsr1A=ansαn+an1sαn1++a1sα1E5

is called the characteristic polynomial of the state-space equation (3).

We note by complex analysis ([16], p. 100) that sαis well-defined and satisfies

sα=eαlogs,sC\0,αCE6

where logsis the principal branch of the complex logarithm of sor the principal sheet of the Riemann surface in the sense of π<argsπ. In view of (6), we see that Δsrnr1as fractional-order polynomial and Gsas fractional-order fraction are well-defined only on C\0for all ri,αi,βiC, whenever at least one of ri, αi, and βiis a fraction number. Both are well-defined on the whole complex plane C, if all ri, αi, and βiare integers.

Bearing (6) in mind, our questions are (i) under what conditions Δsrnr0is holomorphic and has only isolated zeros and (ii) under what conditions Gsis meromorphic and has only isolated zeros and poles?

To address (i), let us return to (6) and observe for any sC\0and αCthat

ddssα=ddseαlogs=ddsk=01k!αlogsk=k=01k!ddsαlogsk=k=01k1!αlogsk1αddslogs=αk=11k1!αlogsk1s1=αeαlogss1=αeαlogselogs=αeα1logs=αelogsα1=αsα1

where we have used ddslogs=s1for any s0(see [27], p. 36). The deductions above actually say by the definition ([16], p. 8) that each term in Δsrnr1on C\0is holomorphic. Thus, Δsrnr1is holomorphic on C\0.

To see under what conditions Δsrnr1(respectively, Gs) possesses only isolated zeros and poles, let us assume that there exists 0<r1such that

αn=knr>αn1=kn1r>>α1=k1rβm=lmr>βm1=lm1r>>β1=l1rE7

where kn>kn1>>k10, lm>lm1>>l10, and knlmare integers. Then, if we set z=sr, then Gsand Δsrnr1can be re-expressed as

Gzr=bmzlm+bm1zlm1++b1zl1anzkn+an1zkn1++a1zk1Δzr=anzkn+an1zkn1++a1zk1E8

Obviously, Gzris a so-called rational function on the z-complex plane and possesses only finitely many isolated zeros and poles. More precisely, Gzris a special meromorphic function that is determined up to a multiplicative constant by prescribing the locations and multiplicities of its zeros and poles ([16], p. 87); or equivalently it is in the form of a complex fraction with regular-order polynomials as its nominator and denominator. Also, Δzris a regular-order polynomial that is holomorphic on the whole z-complex plane and has finitely many isolated zeros. Thus, Gzrand Δzrcan be written in the form of

Gzr=bmlz+zlμlankz+pkνkΔzr=ankz+pkνkE9

Under the assumption that (7) and suppose that Gzrand Δzrhave no zero, zeros, and zero singularities, it holds that zl0,pk0C. In addition, μl1,νk1are integers such that lμl=lmand kνk=kn.

Based on (6) and (9), the s-domain relationships can be rewritten by

Gs=bmlsr+zlμlanksr+pkνkΔsrnr1=anksr+pkνkE10

By (10), it is not hard to see that Gs=0as srzl(or equivalently by (6), sexplogzlr) and Gsas srpk(or equivalently, sexplogpkr). The latter says specifically by Corollary 3.2 of [16] that s=explogpk/ris a pole of Gs. Then, Gsis holomorphic on C\explogpk/rkand has only finitely many isolated zeros and poles. It follows by ([16], pp. 86–87) that Gsis meromorphic on C\0. Confined to the discussion of this paper, Gsmay or may not be rational.

In the sequel, when the assumption (7) is true and Δzrand Gzrhave no zeros and poles at the origin, then Δsrnr1and Gsare well-defined on C\0with respect to fractional commensurate order r. Only fractional commensurate systems are considered in this study.

2.3 Closed-loop configuration with FCO-LTI systems

Consider the feedback system illustrated in Figure 1, in which we denote by ΣGand ΣH, respectively, an FCO-LTI plant and an FCO-LTI feedback subsystem that possess the following fractional-order state-space equations.

Figure 1.

FCO-LTI feedback configuration.

ΣG:Dtrx=Ax+Bey=Cx+De,ΣH:Dtqζ=Λζ+Γμη=Θζ+ΠμE11

where ARn×n, BRn×m, CRl×n, and DRl×m, respectively, are constant matrices, while ΛRp×p, ΓRp×l, ΘRm×p, and ΠRm×lare constant.

Fractional-order transfer functions for ΣGand ΣHare given as follows:

Gs=Cdiagsrnsr1A1B+DCInsrA1B+DĜs+DCadjInsrAB+detInsrADdetInsrAGs/detInsrAHs=Θdiagsqpsq1Λ1Γ+ΠΘIpsqΛ1Γ+ΠĤs+ΠΘadjIpsqΛΓ+detIpsqΛΠdetIpsqΛHs/detIpsqΛE12

where Ĝs, Ĥs, Gs, and Hsare obvious and adjis the adjoint. Also

Insr=diagsrnsr1,Ipsq=diagsqpsq1

Now we construct the state-space equations for the open- and closed-loop systems of Figure 1. The open-loop system can be expressed by the fractional-order state-space equation:

ΣO:DtrxDtqζ=A0ΓCΛxζ+BΓDuη=ΠCΘxζ+ΠDuE13

In the closed-loop system, we can write the closed-loop state-space equation as

ΣC:DtrxDtqζ=ABΞΠCBΞΘΓCΓDΞΠCΛΓDΞΘxζ+BΞΓDΞuη=ΞΠCΞΘxζ+ΠDΞuE14

where Ξ=Im+ΠD1. To explicate the Nyquist approach, we begin with the conventional return difference equation in the feedback configuration ΣC.

By definition, the characteristic polynomial for the closed-loop system ΣCis

ΔCsrqdetInsr00IpsqABΞΠCBΞΘΓCΓDΞΠCΛΓDΞΘ=detInsrA0ΓCIpsqΛdetIn+p+InsrA0ΓCIpsqΛ1BΞΠCBΞΘΓDΞΠCΓDΞΘ=ΔOsrqdetIn+p+InsrA0ΓCIpsqΛ1BΞΠCBΞΘΓDΞΠCΓDΞΘE15

where detmeans the determinant of and

ΔOsrqdetInsrA0ΓCIpsqΛ=detInsrAdetIpsqΛ=ΔGsrΔHsqE16

with ΔGsr=detInsrAand ΔHsq=detIpsqΛ. Clearly, ΔOsrqis the characteristic polynomial for ΣO, while ΔGsrand ΔHsqare the characteristic polynomials for the subsystems ΣGand ΣH, respectively, in the feedback configuration of Figure 1.

Let us return to (15) and continue to observe that

ΔCsrq
=ΔOsrqdet(In+p+InsrA10IpsqΛ1ΓCInsrA1IpsqΛ1
B00ΓΞΠΞDΞΠDΞC00Θ)
=ΔOsrqdet(Il+m
+C00ΘInsrA10IpsqΛ1ΓCInsrA1IpsqΛ1
B00ΓΞΠΞDΞΠDΞ
=ΔOsrqdetIl+m+Ĝs0ĤsĜs,ĤsΞΠΞDΞΠDΞ
=ΔOsrqdetIm+ΠĜsΞΠĜsΞĤsGsΞ,Im+ĤsGsΞ
=ΔOsrqdetImΠĜsΞImIm+ĤsGsΞ
=ΔOsrqdetImΠĜsΞ0Im+ĤsGsΞ+ΠĜsΞ
=ΔOsrqdetIm+ĤsGsΞ+ΠĜsΞ
=ΔOsrqdetΞdetIm+ΠD+ĤsGs+ΠĜs
=ΔOsrqdetΞdetIm+HsGsE17

In deriving (17), the determinant equivalence detI1+XY=detI2+YXis repeatedly used, where Xand Yare matrices of compatible dimensions and I1and I2are identities of appropriate dimensions. By (17), we have

ΔCsrqΔOsrq=ΔCsrqΔGsrΔHsq=detIm+HsGsdetIm+ΠDE18

which is nothing but the return difference relationship for the fractional-order feedback system ΣC. By the definitions, it is clear that ΔOsrq, ΔOsrq, ΔGsr, and ΔHsqare all fractional-order. It is based on (18) that Nyquist-like criteria will be worked out. However, in order to get rid of any open-loop structure and spectrum, let us instead work with

ΔCsrq=ΔGsrΔHsqdetIm+HsGsdetIm+ΠDE19

Remark 1. Recalling our discussion in Section 2.2 and assuming that there exists a number 0<ρ1such that ΣGand ΣHare fractionally commensurate with respect to the same commensurate order ρ, it follows that Gsand Hsare meromorphic on C\0, while ΔCsrq, ΔGsr, and ΔHsqare holomorphic on the whole complex plane. These complex functional facts will play a key role for us to apply the argument principle to (18) as well as (19).

3. Main results

3.1 Nyquist contours in the z-/s-domains

As another preparation for stability analysis in fractional-order 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 z-domain as illustrated by the dashed-line in Figure 2 is the standard contour for a Nyquist-like stability criterion in terms of Δzr. The contour portions of Nzalong the two slopes actually overlap the slope lines. Clearly, the contour Nzis actually the boundary of the sector region encircled by the dashed-line. The radiuses of the two arcs in the sector are sufficiently small and large, respectively, or simply γ0and R. The sector is symmetric with respect to the real axis, whose half angle is π2r.

Figure 2.

The standard z -domain contour N z .

Secondly, the simply closed curve defined on the s-domain as illustrated by Figure 3 presents the standard contour used for a Nyquist-like stability criterion in terms of Δsrnr1. Again the contour is plotted with dashed-lines; in particular, the contour portion along the imaginary axis is actually overlapping the imaginary axis. More precisely, Nsis the boundary of the open right-half complex plane C+=sC:Res>0in the sense that

Figure 3.

The standard s -domain contour N s .

NsC+=sC:Res0,IntNs=C+

where Intdenotes the interior of a closed set. Similar to the contour Nz, the radiuses of the two half-circles in Nsare sufficiently small and large, respectively, namely, γ0and R.

Remarks about the contours Nzand Ns:

• 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 Gsand Δsrnr1(respectively, Gzrand Δzr) are well-defined merely on C\0due to the relation (6) and z=sr.

• One might suggest that in order to detour the origin, the small arc in Nzand the small half-circle in Nscan also be taken from the left-hand side around the origin so that possible sufficiency deficiency in the subsequent stability conditions can be dropped. In fact, if the origin is in the interior of Ns, Gsand Δsrnr1have no definition at the origin. Therefore, the argument principle does not apply rigorously.

3.2 Stability conditions related to FCO-LTI systems

Stability conditions in terms of the zeros distribution of Δsrnr1=0and that of Δzr=0are given by the following proposition [13, 15].

Proposition 1. Consider the fractionally commensurate system with commensurate order 0<r1defined by the fractional-order differential equation (1) or the fractional-order state-space equation (3). The system is stable if and only if all the zeros of Δsrnr1=0, denoted by skk, have negative real parts or all the zeros of Δzr=0, denoted by zll, satisfy

Argzl>π2r,zlE20

where Argis the principal branch argument of on the Riemann surface.

3.3 Stability criterion in FCO-LTI systems

In what follows, a fractional-order polynomial βsris said to be commensurately Hurwitz if βsris fractionally commensurate order rwith some 0<r1in the sense that all the roots of βsr=0possess negative real parts. More specifically, we write

βsr=cpsrlp+cp1srlp1++c1erl1

where lp>lp1>>l10are integers and Res<0for any βsr=0. It is straightforward to see by definition that a fractionally commensurate order Hurwitz polynomial must be holomorphic on the whole complex plane.

Theorem 1. Consider the fractional-order system with commensurate order 0<r1defined by the differential equation (1) or the state-space equation (3). The concerned system is stable if and only if for any γ>0sufficiently small and R>0sufficiently large, any prescribed commensurate order Hurwitz polynomial βsr, the stability locus

fsβsrsNsΔsrnr1βsrsNsE21

vanishes nowhere over Ns, namely, fsβsr0for all sNs; and the number of its clockwise encirclement around the origin is equal to that of its counterclockwise ones, namely,

N(f(sβsrsNs)=NcfsβsrsNs)Nc¯(f(sβ(sr))sNs=0

In the above, the clockwise/counterclockwise orientation of f(sβsr)sNscan be self-defined.

Proof of Theorem 1. By introducing any fractional-order commensurately Hurwitz polynomial βsrinto Δsrnr1, we obtain

Δsrnr1βsr=fsβsrE22

Under the given assumption about the concerned characteristic polynomial and the fact that βsris holomorphic, we can assert that fsβsris well-defined in the sense that it is meromorphic and without singularities at the origin. This says in particular that the argument principle applies to (22) as long as fsβsr0over sNs. Apparently, Eq. (22) holds even if there are any factor cancelations between Δsrnr1and βsr. This says that all unstable poles in IntNsNs, if any, remain in the left-hand side of (22).

Bearing these facts in mind, let us apply the argument principle to (22) counterclockwisely with Nsbeing the Cauchy integral contour, and then the desired assertion in terms of (25) follows.

More precisely, since f(sβsr)sNsvanishes nowhere over sNs, we conclude readily that

NzΔsrnr1Nz(β(sr))=N(f(sβ(sr))sNs)E23

where Nzdenotes the zero number of in IntNsNsand Ndenotes the net number of the locus encirclements around the origin.

Note that all the roots of βsrare beyond IntNs. It follows that Nβsr=0. Together with N=NcNc¯, it follows by (17) that

NzΔsrnr1=NcfsβsrsCγ)Nc¯(f(sβ(sr))sCγ

The above equation says that NzΔsrnr1=0if and only if N(f(sβsr)sCγ)=0. The desired results are verified if we mention that NzΔsrnr1=0is equivalent to the assertion that Δsrnr1=0has no roots in IntNsNs. This says exactly that Δsrnr1=0has no roots in IntNsNs, and thus the concerned system is stable.

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 βsrcan be arbitrarily prescribed so that no existence issues exist. In addition, the stability locus is not unique. The polynomial βsractually provides us additional freedom in frequency-domain analysis and synthesis.

• When the stability locus with respect to the infinite portion of Nsis concerned, it is most appropriate to let the commensurate degree of βsr, namely, c-degβsr=lp, satisfy lp=kn, although βsrcan be arbitrary as long as it is fractionally commensurate Hurwitz. For example, in the sense of (25), if c-degβsr>kn, the stability locus approaches the origin 0j0as s. It may be graphically hard to discern possible encirclements around 0j0; if c-degβsr<kn, the stability locus contains portions that are plotted infinitely far from 0j0. When c-degβsr=knand let βsr=Lβsrwith L>0be constant and βsrmonic, it follows from (25) that

limsfsβsr=limsΔsrnr1Lβ'sr=1/L<E24

Since fsβsris continuous in s, (27) says that fsβsrMover sNsfor some 0<M<and f(sβsr)sNscan be plotted in a bounded region around the origin. Thus, no prior frequency sweep is needed when dealing with the stability locus (25).

• Each and all the conditions in Theorem 1 can be implemented only by numerically integrating fsβsrfor computing the argument incremental fsβsralong the Cauchy integral contour Ns, and then checking if fsβsr/2π=0holds. In this way, numerically implementing Theorem 1 entails no graphical locus plotting. This is also the case for the following results.

• The clockwise/counterclockwise orientation of f(sβsr)sNscan be self-defined. This is also the case in all the subsequent results.

Next, a regular-order polynomial αzris said to be πr-sector Hurwitz if all the roots of αzr=0satisfy (20). More specifically, we write

αzr=cpzlp+cp1zlp1++c1zl1

where lp>lp1>>l10are integers and Argz<0for any αzr=0. It is easy to see that a πr-sector Hurwitz polynomial is holomorphic.

Theorem 2. Consider the fractional-order system with commensurate order 0<r1of the differential equation (1) or the state-space equation (3). The system is stable if and only if for any γ>0small and R>0large sufficiently, any prescribed πr-sector Hurwitz polynomial αzr, the stability locus

gzαzrzNzΔzrαzrzNzE25

vanishes nowhere over Nz, namely, gzαzr0for all zNz; and the number of its clockwise encirclement around the origin is equal to that of its counterclockwise ones, namely, N(g(zαzrzNz)=0.

Proof of Theorem 2. Repeating those for Theorem 1 but in terms of gzαzrrather with fsβsr, while the contour Nsis replaced with Nz.

Remark 2. The proof arguments can also be understood by using the transformation z=srto (22) and then applying the argument principle to the resulting z-domain relationship. Note that z=sris holomorphic on C\0. The angle preserving property ([16], pp. 255–256) leads immediately that the stability conditions of Theorem 2 are equivalent to those of Theorem 1.

3.4 Stability criteria for closed-loop FCO-LTI systems

Based on the return difference equation (31) claimed in the feedback configuration of Figure 1, together with the argument principle, the following s-domain criterion follows readily.

Theorem 3. Consider the feedback system as in Figure 1 with the fractional-order subsystems ΣGand ΣHdefined in (22). Assume that both subsystems are fractionally commensurate with respect to a same commensurate order 0<ρ1. Then, the closed-loop system is stable if and only if for any s-domain contour Nsϵwith R>0sufficiently large and γ>0, ϵ0sufficiently small, any prescribed commensurate order Hurwitz polynomial βsr; the locus

fCsβsρsNsϵΔGsρΔHsρβsρdetIm+HsGsdetIm+ΞDsNsϵE26

satisfies: (i) fCsβsρ0for all sNsϵ; (ii) N(fC(sβsρ)sNsϵ)=0. Here, Nsϵstands for the contour by shifting Nsto its left with distance ϵ.

Proof of Theorem 3. Under the given assumptions, the return difference equation (19) is well-defined on Nsϵand IntNsϵ. Then, applying the argument principle to (19) and repeating some arguments similar to those in the proof for Theorem 1, the desired results follow readily.

To complete the proof, it remains to only show why we must work with the contour Nsϵin general, rather than the standard contour Nsitself directly. To see this, we notice that the shifting factor ϵ0is introduced for detouring possible open-loop zeros and poles on the imaginary axis but excluding the origin. Since all zeros and poles, if any, are isolated, ϵ0is always available. Furthermore, we have by (12) that

ΔGsρΔHsρdetIm+HsGsdetIm+ΞD=ΔGsρΔHsρdetΔGsρΔHsρIm+HsGsΔGmsρΔHmsρdetIm+ΞD

which says clearly that if there are imaginary zeros of ΔGsρΔHsρ, then factor cancelation will happen between ΔGsρΔHsρand detIm+HsGs. Such factor cancelation, if any, can be revealed analytically as in the above algebras, whereas it does bring us trouble in numerically computing the locus (obviously, numerical computation cannot reflect any existence of factor cancelation rigorously). Consequently, when the standard contour Nsis adopted, if there do exist imaginary open-loop poles, the corresponding stability locus cannot be well-defined with respect to Ns. When this happens, one need to know the exact positions of imaginary zeros and/or poles and modify the contour accordingly to detour them before computing the locus; in other words, working with Nsbrings sufficiency deficiency when imaginary open-loop poles exist.

On the contrary, if there exist no imaginary open-loop poles, it is not hard to see that working with Nsϵin numerically computing the locus yields no sufficiency redundance as ε0, noting that ϵ0always exists.

As a z-domain counterpart to Theorem 3, we have.

Theorem 4. Under the same assumptions of Theorem 3; the closed-loop system is stable if and only if for any s-domain contour Nzϵwith R>0large sufficiently and γ>0, ϵ0sufficiently small, any prescribed πr-sector Hurwitz polynomial αsr; the stability locus

gCzαzρzNzϵΔ˜GzρΔ˜HzραzρdetIm+H˜zG˜zdetIm+ΞDzNzϵE27

satisfies: (i) gCzαzρ0for all zNzϵ; (ii) N(gC(zαzρ)zNzϵ)=0. Here, we have

Δ˜Gzρ=ΔGsρsρ=zΔ˜H(zρ)=ΔH(sρ)sρ=zH˜z=Hssρ=zG˜z=Gssρ=z

In the above, Nzϵis the contour by shifting Nzto its left with distance ϵ.

Several remarks about Theorems 3 and 4:

• The shifted contour Nsϵreduces to the standard contour Nswhen ϵ=0. This is also the case for the shifted contour Nzϵand Nz.

• 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 fractional-order characteristic polynomials, in which transfer functions are not involved.

4. Numerical illustrations

4.1 Example description for Theorems 1 and 2

Consider a single fractional-order commensurate system [15] with the characteristic polynomial

s2r+2asr+b=0,a,bR

where the commensurate order 0<r=km1with k, m, and kmbeing positive integers as appropriately. In all the numerical simulations based on Theorem 1, the fractionally commensurate Hurwitz polynomial βsr=s+12ris employed. In all the numerical simulations based on Theorem 2, the πr-sector Hurwitz polynomial αzr=z+0.12is adopted.

In what follows, the s-domain contour Nsis defined with γ=0.01and R=100000, while the z-domain contour Nzis defined with γ=0.01and R=10000.

4.2 Numerical results for Theorems 1 and 2

The following cases are considered in terms of aand b. In each figure, the left-hand sub-figure plots the stability locus in terms of f(sβsr)sNs, or simply the s-locus, for some fixed aand, while the right-hand sub-figure presents the stability locus in terms of g(zαzr)zNz, or simply the z-locus.

a>0, b>0, and a2b. By examining the s-loci of Figure 4 graphically, no encirclements around the origin are counted in each case of r0.2,0.4,0.6,0.8,1.0; indeed, N(f(sβsr)sNs)=0for each r0.2,0.4,0.6,0.8,1.0can be verified numerically without locus plotting. Therefore, the system is stable in each case.

Figure 4.

Stability loci with a = 2 and b = 1 .

The same conclusions can be drawn by examining the z-loci of Figure 4. More precisely, we have N(f(zαzr)zNz)=0for each r0.2,0.4,0.6,0.8,1.0numerically.

a>0, b>0, and a2<b. By the s-loci of Figure 5, no encirclements around the origin are counted in each case of r0.2,0.4,0.6,0.8,1.0graphically; or N(f(sβsr)sNs)=0for each r0.2,0.4,0.6,0.8,1.0numerically. Therefore, the system is stable in each case.

Figure 5.

Stability loci with a = 1 / 2 and b = 1 .

The same conclusions can be drawn by examining the z-loci of Figure 5. More precisely, we have N(f(zαzr)zNz)=0for each r0.2,0.4,0.6,0.8,1.0numerically.

a<0, b<0, and thus a2bholds always. By the s-loci of Figure 6, one net encirclement around the origin is counted in each case of r0.2,0.4,0.6,0.8,1.0graphically; alternatively, N(f(sβsr)sNs)=1for each r0.2,0.4,0.6,0.8,1.0numerically. Therefore, the system is unstable in each case.

Figure 6.

Stability loci with a = −1 and b = −1.

The instability conclusions in each r0.2,0.4,0.6,0.8,1.0can be revealed by means of the z-loci of Figure 6. More precisely, we have N(f(zαzr)zNz)=1for each r0.2,0.4,0.6,0.8,1.0numerically.

a>0, b<0, and thus a2bholds always. By the s-loci of Figure 7, one net encirclement around the origin is counted in each case of r0.2,0.4,0.6,0.8,1.0graphically; alternatively, N(f(sβsr)sNs)=1for each r0.2,0.4,0.6,0.8,1.0numerically. Therefore, the system is unstable in each case.

Figure 7.

Stability loci with a = 1 and b = −1.

The instability conclusions in each r0.2,0.4,0.6,0.8,1.0can be drawn again by means of the z-loci of Figure 7. More precisely, we have N(f(zαzr)zNz)=1for each r0.2,0.4,0.6,0.8,1.0numerically.

a<0, b>0, and a2b. By the s-loci of Figure 8, one net encirclement around the origin is counted in the case of r=0.2graphically, or N(f(sβsr)sNs)=1for r=0.2numerically; two net encirclements around the origin are counted in each case of r0.4,0.6,0.8,1.0, or N(f(sβsr)sNs)=2for each r0.4,0.6,0.8,1.0numerically. Therefore, the system is unstable in each case of r0.2,0.4,0.6,0.8,1.0.

Figure 8.

Stability loci with a = − 2 and b = 1 .

The instability conclusions in each r0.2,0.4,0.6,0.8,1.0can be drawn again by means of the z-loci of Figure 8. More specifically, we have N(f(zαzr)zNz)=1for r=0.2and N(f(zαzr)zNz)=2for each r0.4,0.6,0.8,1.0numerically.

a<0, b>0, and a2<b. By the s-loci of Figure 9, no encirclements around the origin are counted in each case of r0.2,0.4,0.6graphically, or N(f(sβsr)sNs)=0for each r0.2,0.4,0.6numerically. Therefore, the system is stable in each case of r0.2,0.4,0.6. However, two net encirclements around the origin are counted in each case of r0.8,1.0graphically or N(f(sβsr)sNs)=2for each r0.8,1.0numerically. Therefore, the system is unstable in either case of r0.8,1.0.

Figure 9.

Stability loci with a = − 1 / 2 and b = 1 .

Stability in each case of r0.2,0.4,0.6and instability for either case of r0.8,1.0can be verified by the z-loci of Figure 9 as appropriately. Indeed, we have N(f(zαzr)zNz)=0for r0.2,0.4,0.6and N(f(zαzr)zNz)=2for each r0.8,1.0numerically.

Based on the numerical results, the stability/instability conclusions based on the s-loci completely coincide with those drawn based on the z-loci. This reflects the fact of Remark 2. These numerical results are also in accordance with those by [15] about the same example, which are summarized by working with solving polynomial roots. It is worth mentioning that polynomial roots are not always solvable in general. Fortunately, the suggested Nyquist-like criteria can be implemented graphically and numerically, independent of any polynomial root solution and inter-complex-plane transformation. Hence, the suggested technique is applicable more generally.

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]:

Gs=KP+KIsλ+0.0001+100KDsμsμ+100101+0.1s1+0.4s1+s,Hs=11+0.01s

where Hsis a regular-order sensor model and Gsis a cascading model consisting of a fractional-order PID in the form of KP+KI/sλ+0.0001+100KDsμ/sμ+1001, an amplifier modeled as 10/1+0.1s, an exciter modeled as 1/1+0.4s, and a generator with model 1/1+s. Fractional-order PID parametrization is addressed in [6] by means of particle swarm optimization.

In the following, we focus merely on verifying the closed-loop stability based on Theorem 3, based on the parametrization results therein. To this end, the s-domain shifting contour Nsϵis defined with R=100000, γ=0.1, and ε=0.01. To utilize Theorem 3, the fractionally commensurate Hurwitz polynomial βsr=s+1λ+μ+4is employed.

The so-called optimal controller parameters are listed in Table 1.

KPKIKDμλ
Case 11.26230.55310.23821.25551.1827
Case 21.26230.55260.23811.25591.1832

Table 1.

PID controller parameters.

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 N(fC(sβsρ)sNsε)=0. From these facts, Theorem 3 ensures that the closed-loop fractional-order system is stable. This coincides with the results in [6].

Figure 10.

Stability loci for cases 1 and 2.

5. Conclusions

Stability is one of the imperative and thorny issues in analysis and synthesis of various types of fractional-order systems. By the literature [28, 29, 30], the frequently adopted approaches are through single/multiple complex transformation such that fractional-order characteristic polynomials are transformed into standard regular-order polynomials, and then stability testing of the concerned fractional-order systems is completed by the root distribution of the corresponding regular-order 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 s-complex domain but involves no root computation at all. What is more, the main results can be interpreted and implemented graphically with locus plotting as we do in the conventional Nyquist criteria, as well as numerically without any locus plotting (or simply via complex function argument integration). This implies that the complex scaling approach is numerically tractable so that is much more applicable in fractional-order control design and parametrization. This point is significant for practical control applications involving fractional-order plants, which are our perspective topics in the future.

Acknowledgments

The study is completed under the support of the National Natural Science Foundation of China under Grant No. 61573001.

Notes

  • The fractional-order 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 .

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Jun Zhou (August 14th 2019). Nyquist-Like Stability Criteria for Fractional-Order Linear Dynamical Systems [Online First], IntechOpen, DOI: 10.5772/intechopen.88119. Available from:

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