Robust stability analysis of uncertain multiorder fractional systems: Young and Jensen inequalities approach

Abstract

Summary Robust stability analysis of multiorder fractional linear time‐invariant systems is studied in this paper. In the present study, first, conservative stability boundaries with respect to the eigenvalues of a dynamic matrix for this kind of systems are found by using Young and Jensen inequalities. Then, considering uncertainty on the dynamic matrix, fractional orders, and fractional derivative coefficients, some sufficient conditions are derived for the stability analysis of uncertain multiorder fractional systems. Numerical examples are presented to confirm the obtained analytical results.

Stability and Stabilization of Fractional‐Order Systems with Different Derivative Orders: An LMI Approach

“…where , , and are defined in (11) and (12). According to (22) and (24), and assuming matrix ( ) ( ) in the form of (19), the inequality (22) can be written as follows…”

“…This kind of system representation is used to analyze linear electrical circuits composed of resistors, supercondensators (ultracapacitors), coils, and voltage (current) sources [10], and formulate the problem of model reference adaptive control of fractional-order systems presented in [11]. Moreover, relaxation processes, viscoelastic materials models, and diffusion phenomena can be easily modeled by multi-order fractional differential equations [12]. In [2,13,14] non-integer order of the proposed controllers are different from the plant orders, which results in multi-order closed-loop systems.The stability of fractional-order feedback systems was investigated in [15], by adapting classical root locus plot analysis to some viscoelastic structures.…”

Differential flatness‐based ADRC scheme for underactuated fractional‐order systems

“…It is shown that fractional‐order control (FOC) owns a higher degree of freedom than integer ones in controller parameters selection, and therefore owns a greater potential in system performance improvement. As we become more demanding on model accuracy and control performance, FOS theory and fractional‐order controller design have developed rapidly in recent years. In terms of control schemes for FOSs, many schemes are proposed to deal with the internal or external disturbance, among them, fractional‐order proportion integration differentiation (PID) and CRONE have been widely used, but both schemes have limitation.…”

Robust stability analysis of fractional‐order interval systems with multiple time delays

“…10,11 Papers were published to introduce some criteria for the robust stability of fractional-order systems based on the concepts of value set, Young and Jensen inequalities, and principal characteristic equation. [12][13][14][15][16][17] Moreover, fractional-order controllers have been applied to control a diversity of dynamical processes, including both fractional-order and integer-order systems to increase the robustness. [18][19][20] On the other hand, modeling real world systems often includes some uncertainties.…”

“…Recently, using fractional‐order calculus for modeling physical systems widely has been spread because it can describe the behavior of many systems more accurately than its integer‐order counterpart, including delay systems . Papers were published to introduce some criteria for the robust stability of fractional‐order systems based on the concepts of value set, Young and Jensen inequalities, and principal characteristic equation . Moreover, fractional‐order controllers have been applied to control a diversity of dynamical processes, including both fractional‐order and integer‐order systems to increase the robustness .…”