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Operational Amplifiers
Published in Michael Olorunfunmi Kolawole, Electronics, 2020
The z-transform is the discrete-time counterpart of the Laplace transform. Like Laplace transform the z-transform allows insight into the transient behavior, the steady state behavior, and the stability of discrete-time systems. A working knowledge of the z-transform is essential to the study of digital filters and systems. This chapter does not attempt to delve deeply into the intricacies and application of z-transform beyond giving the reader a brief explanation of the relationship between Laplace and z-transform from the Laplace transform of a discrete-time signal. By definition, the Laplace transform X(s), of a continuous-time signal x(t), is given by the integral X(s)=∫0∞x(t)e−stdt
z-Transform
Published in David C. Swanson, ®, 2011
Given a complete mathematical expression for a discrete time-domain signal, why transform it to another domain? The main reason for time–frequency transforms is that many mathematical reductions are much simpler in one domain than the other [1]. The z-transform in the digital domain is the counterpart to the Laplace transform in the analog domain. The z-transform is an extremely useful tool for analyzing the stability of digital sequences, designing stable digital filters, and relating digital signal processing operations to the equivalent mathematics in the analog domain. The Laplace transform provides a systematic method for solving analog systems described by differential equations. Both the z-transform and the Laplace transform map their respective finite-difference or differential systems of equations in the time or spatial domain to much simpler algebraic systems in the frequency or wavenumber domains, respectively. However, the relationship between the z-domain and the s-domain of the Laplace transform is not linear, meaning that the digital filter designer will have to decide whether to match the system poles, zeros, or impulse response. As will be seen later in this chapter, one can warp the frequency axis to control where and how well the digital system matches the analog system. We begin by that assuming time t increases as life progresses into the future, and a general signal of the form est, s = σ + jω, is stable for σ ≤ 0. A plot of our general signal is shown in Figure 2.1.
Digital Filters and z-Transform
Published in Francis F. Li, Trevor J. Cox, Digital Signal Processing in Audio and Acoustical Engineering, 2019
Since the bk coefficients represent a discrete impulse response, it should be possible to obtain the filter frequency response by using an appropriate transform to the frequency domain. In the Tutorial question 1, you actually do this using a Fourier transform. However, in the case of digital filters, the z-transform will turn out to be even more useful. (But the Fourier and z-transforms are closely related). The advantage of the z-transform is it readily enables the frequency response and stability of digital filters to be explored through the pole-zero representation.
Impedance Parameters Estimation of An RLCM Ladder Network Using Subspace and Similarity Transformation Approach
Published in IETE Journal of Research, 2023
The equivalent electrical network of various physical systems, such as electrical, mechanical, and thermal systems are modeled using ladder networks [1]. Substantial interest has been shown to determine the equivalent input resistance of large or infinite ladder networks, especially the resistive ladders [2–7]. For instance, methods based on finite difference [2], Fibonacci sequence [3], Fourier transform [6] and Z-transform [7] have been applied. Explicit expressions are also derived to determine the node voltage, mesh currents, and equivalent resistance between two arbitrary nodes for finite and infinite ladder networks [8,9]. However, for this, an accurate model having impedance parameters resembling the actual physical systems is required. The accurate modeling of any system depends on how accurately the impedance parameters of the ladder network are estimated. One such complex ladder network is that of an electrical power transformer, whose impedance parameters consist of resistances, self and mutual inductances, and shunt and series capacitances [10–12]. Determining these impedance parameters, based solely on the measured input–output data, is a challenging task [13]. Thus, the primary objective of this work is to estimate the impedance parameters of the complex ladder network of a transformer winding directly from the measured input–output data. The approach given in this paper is not restricted to transformers alone but can be applied to any other device that can be modeled as a multi-element homogenous ladder network.
Multidimensional realisation theory and polynomial system solving
Published in International Journal of Control, 2018
Philippe Dreesen, Kim Batselier, Bart De Moor
For one-dimensional systems it is well-known that the Laplace transform or the Z-transform (Kailath, 1980) can be used to relate a polynomial formulation with the system description. This connection is central in systems theory and its applications. For multidimensional systems, the analysis is more difficult as it involves multivariate polynomials, and hence the tools of (computational) algebraic geometry or differential algebra (Buchberger, 2001; Hanzon & Hazewinkel, 2006a). Nevertheless, several multidimensional models and their properties have been studied extensively (Attasi, 1976; Bose, 2016; Bose, Buchberger, & Guiver, 2003; Fornasini, Rocha, & Zampieri, 1993; Gałkowski, 2001; Kaczorek, 1988; Kurek, 1985; Livšic, 1983; Livšic, Kravitsky, Markus, & Vinnikov, 1995; Oberst, 1990; Roesser, 1975), and applications in identification (Ramos & Mercère, 2016) and control (Rogers et al., 2015) are known.
Perfect tracking control of discrete-time quadratic TS fuzzy systems via feedback linearisation
Published in International Journal of Systems Science, 2019
Xiaojun Ban, Liwei Ren, Zhibin Yan, Hao Ying
There are different ways to determine the stability of Equation (6). The easiest way is to use the z-transform method: the linearised system is strictly stable if all the roots of its corresponding z-transform equation are inside the unit circle: If any root of Equation (10) is outside the unit circle, the linearised system is unstable, and so is the original nonlinear system (Equation (2)) with at the origin. If at least one of the roots is on the unit circle and the rest are all inside, then one cannot conclude anything from the first-order approximation, which is the limitation of the Lyapunov linearisation method and is not considered in this paper.