On Digital Realizations of Non-integer Order Filters

Circuits, Systems, and Signal Processing, Feb 2016

Typical approach to non-integer order filtering consists of analogue design and implementation. Digital realization of non-integer order systems is susceptible to problems such as infinite memory requirement and sensitivity to numerical errors. The aim of this paper is to present two efficient methods for digital realization of non-integer order filters: discrete time-domain Oustaloup approximation and Laguerre impulse response approximation. Properties of both methods are investigated with use of non-integer low-pass filter. Filters realized with presented methods are then used for filtering of EEG signal. Paper concludes with discussion of merits and flaws of both methods.

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On Digital Realizations of Non-integer Order Filters

Circuits Syst Signal Process On Digital Realizations of Non-integer Order Filters Jerzy Baranowski 0 Waldemar Bauer 0 Marta Zagórowska 0 Paweł Pia˛tek 0 B Jerzy Baranowski 0 0 Department of Automatics and Biomedical Engineering, AGH University of Science and Technology , Al. Mickiewicza 30, 30-059 Kraków , Poland Typical approach to non-integer order filtering consists of analogue design and implementation. Digital realization of non-integer order systems is susceptible to problems such as infinite memory requirement and sensitivity to numerical errors. The aim of this paper is to present two efficient methods for digital realization of noninteger order filters: discrete time-domain Oustaloup approximation and Laguerre impulse response approximation. Properties of both methods are investigated with use of non-integer low-pass filter. Filters realized with presented methods are then used for filtering of EEG signal. Paper concludes with discussion of merits and flaws of both methods. Work realized in the scope of project titled ”Design and application of non-integer order subsystems in control systems”. Project was financed by National Science Centre on the base of decision no. DEC-2013/09/D/ST7/03960. Non-integer order filter; Fractional filter; Oustaloup method; discretization; Laguerre impulse response approximation 1 Introduction Non-integer (fractional) order systems are a rapidly growing field of interest for both mathematicians and engineers. One of the intensively analyzed aspects of this domain is non-integer order filters. One can observe that the theory of such filters is relatively well grounded; however, many problems of implementation are still open. A need for efficient implementation is obvious as potential applications are numerous in areas such as telecommunication, biomedical engineering, control and many others. In this paper, the authors focus on efficient methods of creating discrete realizations of filters that do not have the burden of infinite memory and provide good representation of non-integer order filter frequency characteristics. In order to do so, two methods are proposed resulting in approximations in the form of discrete statespace systems. These methods are discrete time-domain Oustaloup approximation and Laguerre impulse response approximation (LIRA). The problem of approximating the non-integer order system with an integer order one is being analyzed from many years. Most popular approaches are based on using Oustaloup transfer function approximation or Continuous Fraction Expansion (CFE) in the domain of transfer functions. Both of these methods were used for approximating integrators. The problem is that Oustaloup method is very sensitive to high discretization frequencies and rounding errors. It was observed in earlier authors’ works [7] that it can become destabilized very easily. On the other hand, CFE method shows inferior quality in frequency characteristic representation [25]. Detailed analysis of CFE approximation in discrete time can be found in [12,44,45]. The first method used in this article, the Oustaloup method, is used in the literature in two different, yet equivalent, versions. The original approach, developed by Oustaloup [26,27], is based on approximation of fractional systems in frequency domain. This approach is widely used, e.g., [11,22,25,32] and many others. However, this method has some flaws which cannot be neglected—when discretized, it does not guarantee stability of the system (the poles of discrete system are outside unit circle) (see, e.g., [30]). In order to avoid, i.e., numerical issues induced by this method, another kind of approximation was proposed—instead of transfer function, the state-space approach is considered. State-space realization of Oustaloup transfer function was considered in, e.g., [23,36,39,40]; however, its discrete properties were not considered in this context. In this paper, the method from [7] is used. This approach allows avoiding numerical problems observed in practical realization. The proposed approach is to realize every block of the transfer function in form of a state-space system. Those first-order systems will be then collected in a single matrix resulting in full matrix realization. The results were first presented in authors’ earlier works [7] and [6]. This approach proved to be more robust to different discretization schemes. The superiority of this approach was then validated with real-time control experiments—magnetic levitation control [10] and air heater control [9]. The other method considered in this paper, Laguerre impulse response approximation (LIRA), chosen by authors to analyze fractional filters uses approximation with (orthonormal) Laguerre functions. Some works, concerning this type of approximation, are, e.g., [3,24]. The authors’ approach was developed independently in [5]. It introduces substantial improvements such as L1 convergence, estimation of approximation error and cho (...truncated)


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Jerzy Baranowski, Waldemar Bauer, Marta Zagórowska, Paweł Piątek. On Digital Realizations of Non-integer Order Filters, Circuits, Systems, and Signal Processing, 2016, pp. 2083-2107, Volume 35, Issue 6, DOI: 10.1007/s00034-016-0269-8