Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Walsh, Block Pulse, and Related Orthogonal Functions in Systems and Control

In this paper, we present and investigate the analytical properties of a new set of orthogonal basis functions derived from the block-pulse functions. Also, we present a numerical method based on this new class of functions to solve nonlinear Volterra–Fredholm integral equations. In particular, an alternative and efficient method based on the formalism of artificial neural networks is discussed. The efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations

In this paper, a new method is introduced to design static output tracking controllers for a class of non-linear polynomial time-delay systems. The proposed technique is based on the projection of the controlled system and the associated linear reference model that it should follow over a basis of block-pulse functions. The useful properties of these orthogonal functions such as operational matrices jointly used with the Kronecker tensor product may transform the non-linear delay differential equations into linear algebraic equations depending only on parameters of the feedback regulator. The least-squares method is then used for determination of the unknown parameters. Sufficient conditions for the practical stability of the closed-loop system are derived, and a domain of attraction is estimated. The implementation of the proposed method is illustrated on a double inverted pendulums benchmark as well as a two-degree-of- freedom mass-spring-damper system. The simulation results show the effectiven...

Static output tracking control for non-linear polynomial time-delay systems via block-pulse functions

Triangular Orthogonal Functions for the Analysis of Continuous Time Systems: Convolution Process in Triangular Function Domain and Its Use in SISO Control System Analysis

In this paper a new approach to algebraic parameter identification of the linear SISO systems is proposed. The standard approach to the algebraic parameter identification is based on the algebraic derivatives in Laplace domain as the main tool for algebraic manipulations like elimination of the initial conditions and generation of linearly independent equations. This approach leads to the unstable time-varying state-space realization of the filters for the on-line parameter estimation. In this paper, the finite difference and shift operators in combination with the frequency-shifting property of Laplace transform is applied instead of algebraic derivatives. Resulting state-space realization of the estimator filters is asymptotically stable and doesn’t require switch-of mechanism to prevent overflow of the estimator variables. The proposed method is especially suitable for applications in closed-loop on-line identification where the stable behavior of the estimators is a necessary requirement. The efficiency of the proposed algorithm is illustrated on three simulation examples.

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Frequency-shifting-based stable on-line algebraic parameter identification of linear systems

Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions

This book introduces a new set of orthogonal hybrid functions (HF) which approximates time functions in a piecewise linear manner which is very suitable for practical applications. The book presents an analysis of different systems namely, time-invariant system, time-varying system, multi-delay systems---both homogeneous and non-homogeneous type- and the solutions are obtained in the form of discrete samples. The book also investigates system identification problems for many of the above systems. The book is spread over 15 chapters and contains 180 black and white figures, 18 colour figures, 85 tables and 56 illustrative examples. MATLAB codes for many such examples are included at the end of the book.

Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions: Theory and Algorithms with MATLAB®

This paper presents a virtual laboratory for testing a single-phase power transformer using MATLAB Simulink environment. The models presented herein replicate the actual tests like no-load test, short-circuit test, load test, and short heat run test successfully well within tolerable accuracy. The rating of the transformer is taken to be 2 kVA, 230/115 V, 50 Hz which is a typical rating for transformers usually employed for practical classes in the Electrical Machine Laboratory for undergraduate courses. This model is attractive to students and they are able to gain insights into operational aspects of a single-phase power transformer. This model, in whole or part, may also be used as final year projects of engineering undergraduate students.

Virtual Laboratory for Performing Tests on a Single-Phase Power Transformer

The present work uses a new set of hybrid functions (HF) formed by the synthesis of sample-and-hold functions (SHF) and triangular functions (TF). The SHF set is efficient for analyzing sample-and-hold control systems and the TF set have been employed for obtaining piecewise linear solution of control problems. After a brief review of the basic theory of HF, the operational matrices for integration in HF domain are also briefly discussed. Finally, this HF set is employed for the analysis and synthesis of homogeneous and non-homogeneous systems described via state space. Many examples are treated and the results are compared with the exact solutions and found to be attractively close. Since the HF set works with function samples, the computational burden in the presented method are much less than traditional ones.

Analysis and Synthesis of Homogeneous / Non-Homogeneous Control Systems via Orthogonal Hybrid Functions (HF) under State Space Environment

This paper uses a set of hybrid functions (HF) formed by a combination of sample-and-hold function (SHF) set and triangular function (TF) set. The SHF set has been applied for analysing sample-and-hold control systems and the TF set has been proved to be efficient for obtaining piecewise linear solutions of control systems. In the present work, the HF set has been employed for the analysis and synthesis of homogeneous as well as non-homogeneous time-varying control systems in state space. The HF set works with function samples, and is thus useful for building an easier algorithm for time-varying system analysis. After developing necessary theories, a few examples are treated to illustrate the efficiency as well as simplicity of the approach. The results thus obtained are compared with the results obtained via traditional analysis, and relevant tables and graphs are included to justify the case for hybrid functions.

Srimanti Roychoudhury

Papers

Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions

Analysis and Identification of Time-Invariant Systems, Time-Varying Systems, and Multi-Delay Systems using Orthogonal Hybrid Functions: Theory and Algorithms with MATLAB®

Virtual Laboratory for Performing Tests on a Single-Phase Power Transformer

Analysis and Synthesis of Homogeneous / Non-Homogeneous Control Systems via Orthogonal Hybrid Functions (HF) under State Space Environment

Analysis and synthesis of time-varying systems via orthogonal hybrid functions (HF) in state space environment