Model reduction in commensurate fractional-order linear systems

Abstract

In this paper, some commonly used model reduction methods for integer-order systems are employed to approximate commensurate fractional-order linear systems. In comparison with the original system, the approximating model possesses a smaller inner dimension, while its fractional order is the same as that of the original system. The applied methods fall into the global reduction category, such as direct truncation and singular perturbation methods, and into the local reduction category, such as Pade approximation, partial realization, shifted Pade approximation, and rational interpolation methods. The applicability of these methods is illustrated by approximating a sample high-dimensional, commensurate, fractional-order, linear system.

Fractional order model reduction approach based on retention of the dominant dynamics: Application in IMC based tuning of FOPI and FOPID controllers

“…Up until now, fractional order systems have been studied from different points of view such as stability analysis [5][6][7], controllability and observability [8][9][10], system identification [11][12][13][14], system representation [15,16], model approximation [17][18][19], dynamical behavior analysis [20,21], and so on. Moreover, fractional order controllers have been designed and used in many practical applications and it has been shown that in some applications this type of controllers is more efficient than the traditional integer order controllers [22][23][24][25].…”

“…But since in practice, the exact model of a process usually has a complicated structure, it is required to make use of appropriate reduction methods to approximate the complicated model of the system with a simple one. In [18,27], * Corresponding author. Tel.…”

“…some methods have been suggested for approximating fractional order transfer functions by simple fractional order models. The globally and locally reduction methods proposed in [18] respectively work based on the knowledge about the state space representation of the system model and power series expansion coefficients of the system transfer function. The approximating strategies proposed in [27] are applicable just for those fractional order systems possessing an S-shaped step response.…”

Positive observer design for fractional order systems

“…Corollary 2: If the zeros of the characteristic polynomial ∆(λ) in (12) are located in the left half of complex plane then the zeros of the characteristic polynomial ∆ α (s) in (11) are located in the sector defined by φ = Π 2α . The importance of this mapping is the fact that one can consider the stability results with respect to the left half of complex plane associated with ∆(λ) (or correspondingly the eigenvalues of the matrix A) and reveal the stability of the sector φ = π 2α , which is a convex sub-region of the stability region associated with 0 < α ≤ 1 as shown in Fig 1(a).…”

Improved model reduction and tuning of fractional-order PIλDμ controllers for analytical rule extraction with genetic programming

“…Other relevant works include dominant mode based methods [36]. Advancements on the FO model reduction techniques have been illustrated in a detail manner in [37]- [38]. In this paper another optimization framework has been used which minimizes the discrepancy between the frequency responses of the higher order and reduced parameter process model in the complex Nyquist plane.…”