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

In this chapter, square integrable time functions of Lebesgue measure are approximated via hybrid functions and such approximations are compared with similar approximations using BPF and Legendre polynomials. For handling discontinuous functions, a modified method of approximation is suggested in hybrid function domain. This modified approach, named HFm approach, seems to be more accurate than the conventional HF domain technique, termed as HFc approach. The mean integral square errors (MISE) for both the approximations are computed and compared. Finally, error estimates for the SHF domain approximation and TF domain approximation are derived. The chapter contains many tables and graphs along with six illustrative examples.

Function approximation via hybrid functions

In this chapter, parameter estimation of the transfer function of a linear system has been done employing many non-sinusoidal orthogonal function sets, e.g., block pulse functions, non-optimal block pulse functions, triangular functions, hybrid functions and sample-and-hold functions. A comparative study of the parameters estimated by different methods are made with focus on estimated errors. One numerical example has been studied extensively and ten figures and fourteen tables have been presented as illustration.

System Identification: Parameter Estimation of Transfer Function

Function Approximation via Block Pulse Function and Related Functions

This chapter discusses HF domain identification of time varying systems in state space. Both homogeneous and non-homogeneous systems are treated. Illustration has been provided with the support of three examples, ten figures and two tables.

Srimanti Roychoudhury

Papers

Function approximation via hybrid functions

System Identification: Parameter Estimation of Transfer Function

Function Approximation via Block Pulse Function and Related Functions

System Identification Using State Space Approach: Time Invariant Systems

System Analysis and Identification Using Convolution and “Deconvolution” in BPF Domain