Piotr Bania

Laguerre Polynomial Approximation of Fractional Order Linear Systems

This paper presents a finite dimensional approximation of fractional order linear systems and its connection with transport equation. The main results show, that the linear fractional order system can be approximated by a finite number of linear differential equations. Appropriate error estimate in C 0 norm is presented. Next, the solution of fractional order linear system is presented as a linear functional of the solution of transport equation. This result establishes a connection between the semi groups theory and fractional system theory. Considerations are illustrated with simple example of fractional oscillator.

Impulse response approximation method for bi-fractional filter

In this paper method of impulse response approximation with orthogonal Laguerre functions is used for the approximation of bi-fractional filter (BFF). This filter is a new development in the field of non-integer order signal processing. Filter with three parameters allows advanced shaping of frequency response, especially the phase characteristic. Method of impulse response approximation is a very promising one because of its great conditioning and sound theoretical basis.

Impulse Response Approximation Method for “Fractional Order Lag”

”Fractional order lag” is a system that is popular in multiple applications. In this paper, authors consider a new method for approximation of this system based on its impulse response. Certain assumptions of the approximation method are verified and algorithm is presented. Also certain problems with this system analysis are discussed, especially its realisation in the form of non-integer order differential equations.

Convergence of Laguerre Impulse Response Approximation for Noninteger Order Systems

One of the most important issues in application of noninteger order systems concerns their implementation. One of the possible approaches is the approximation of convolution operation with the impulse response of noninteger system. In this paper, new results on the Laguerre Impulse Response Approximation method are presented. Among the others, a new proof of convergence of approximation is given, allowing less strict assumptions. Additionally, more general results are given including one regarding functions that are in the joint part of and spaces. The method was also illustrated with examples of use: analysis of “fractional order lag” system, application to noninteger order filters design, and parametric optimization of fractional controllers.

Parametric optimization of PD α controller using Laguerre function approximation

In this paper the method for optimisation of PDα controllers for linear systems was investigated. A new method for approximation of performance index with infinite horizon was presented and used. Optimisation was undertaken on approximated performance index with satisfactory results for multiple test systems. Results are promising, convergence of approximant and performance index was observed and numerically illustrated. Method is easily extendable and shows promise as a general method of non-integer controller tuning.

Approximation of optimal filter for Ornstein-Uhlenbeck process with quantised discrete time observation

ABSTRACT Quantisation of signals is a ubiquitous property of digital processing. In many cases, it introduces significant difficulties in state estimation and in consequence control. Popular approaches either do not address properly the problem of system disturbances or lead to biased estimates. Our intention was to find a method for state estimation for stochastic systems with quantised and discrete observation, that is free of the mentioned drawbacks. We have formulated a general form of the optimal filter derived by a solution of Fokker–Planck equation. We then propose the approximation method based on Galerkin projections. We illustrate the approach for the Ornstein–Uhlenbeck process, and derive analytic formulae for the approximated optimal filter, also extending the results for the variant with control. Operation is illustrated with numerical experiments and compared with classical discrete-continuous Kalman filter. Results of comparison are substantially in favour of our approach, with over 20 times lower mean squared error. The proposed filter is especially effective for signal amplitudes comparable to the quantisation thresholds. Additionally, it was observed that for high order of approximation, state estimate is very close to the true process value. The results open the possibilities of further analysis, especially for more complex processes.

Bayesian fault detection and isolation using Field Kalman Filter

Fault detection and isolation is crucial for the efficient operation and safety of any industrial process. There is a variety of methods from all areas of data analysis employed to solve this kind of task, such as Bayesian reasoning and Kalman filter. In this paper, the authors use a discrete Field Kalman Filter (FKF) to detect and recognize faulty conditions in a system. The proposed approach, devised for stochastic linear systems, allows for analysis of faults that can be expressed both as parameter and disturbance variations. This approach is formulated for the situations when the fault catalog is known, resulting in the algorithm allowing estimation of probability values. Additionally, a variant of algorithm with greater numerical robustness is presented, based on computation of logarithmic odds. Proposed algorithm operation is illustrated with numerical examples, and both its merits and limitations are critically discussed and compared with traditional EKF.

Example for equivalence of dual and information-based optimal control

ABSTRACT An information-based control (IBC) has been formulated for a class of dual control problems with quadratic cost and unknown, constant parameters. Simple dual control problem has been solved by dynamic programming. Then, the solution was compared with the solution obtained by IBC. It has been shown that an appropriate choice of the intensity of learning allows to recover the optimal dual control for at least one nontrivial class of dual control problems.

Bayesian estimator of a faulty state: Logarithmic odds approach

Fault detection and isolation is crucial for efficient operation and safety of any industrial process. Methods from all the areas of data analysis are being used for this task including Bayesian reasoning and Kalman filtering. In this paper authors use the discrete Field Kalman Filter for detecting and recognising faulty conditions of the system. Proposed approach, devised for stochastic linear systems allows analysis of faults that can be expressed both as parameter and disturbance variations. It is formulated for the situations when the fault catalogue is known, but because of that very efficient algorithm can be obtained. For implementation logarithmic odds are considered to improve numerical properties. Its operation is illustrated with numerical examples and both its merits and limitations are critically discussed.

Hybrid Newton Observer in Analysis of Glucose Regulation System for ICU Patients

Glucose regulation system is one of the most important problems in ICUs. Nowadays either frequent testing and insulin shots or automatic insulin pumps are used to avoid hyperglycemia even for patients without history of diabetes. As it is impossible to directly measure the glucose level, there is a need for estimation tools ensuring appropriate quality of measurements.