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

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.

Control Techniques for a Class of Fractional Order Systems

“…The nonlinear component g(z) verifies the inequality ( 26) with µ = 2 × 10 −3 . The dynamic system is described by the following parameters [34]: the equivalent inertia is J = 14 × 10 −5 kg. m 2 , the viscous and elastic coefficients of the joint tissues throughout phalange musculoskeletal system and glove are c ν = 0.408 × 10 −2 Nm.s.rad −1 , c e = 0.27 × 10 −2 Nm.rad −1 , respectively and the damping coefficient is c ν0 = 0.324 × 10 −2 Nm.s.rad −1 .…”

“…The control of multi-system architectures defined by FOS models is discussed in [29][30][31]. Frequency criteria for the control of Linear FOS are presented in [32][33][34]. A comprehensive review of literature related to the industrial use and integration of FOPID control is presented in [35].…”

“…the system ( ) is evaluated by direct measuring of the angular position = and by estimation of fractional variables , by using the Gr nwald-Letnikov's formula, The nonlinear component ( ) verifies the inequality (26) with = 2 × 10 . The dynamic system is described by the following parameters[34]: the equivalent inertia is = 14 × 10. , the viscous and elastic coefficients of the joint tissues throughoutphalange musculoskeletal system and glove are = 0.408 × 10 . . )…”

Elementary Derivation of the Nyquist Criterion for Fractional-Order Feedback Systems

“…To check whether this condition is satisfied, some generalisations of the criteria usually adopted to analyse the stability of integer-order systems have been suggested. In this regard, mention can be made of [11], [12], [13], where frequency-like methods are applied to the characteristic (pseudo) polynomial of a fractional-order system, of [14], where reference is made to systems in Lure's form, of [15], where non-commensurate delay systems are considered, and of [16], [17] that present adaptations of the classical algebraic criteria of Routh and Jury. Quite understandably, these contributions naturally pertain to the broader, and by now well-established, line of research on root clustering [18], [19], [20], robust and D-stability [21], [22], [23].…”