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The Fourier Transform and Linear Systems
Published in Taan S. ElAli, ®, 2020
This says that the output of a linear time-invariant system, if subject to a sinusoidal input, will have a steady-state solution equal to the magnitude of the input signal multiplied by the magnitude of the transfer function of the system, evaluated at the frequency of the input signal and shifted by the phase angle of the transfer function evaluated at the input frequency as well. Some notes on Fourier transform are worth mentioning:The magnitude spectrum of the Fourier transform of a signal x(t) is even and the phase spectrum is odd.Shifting the signal x(t) in time does not alter its magnitude.Time compression of a signal x(t) corresponds to frequency expansion of X(w).Time expansion of a signal x(t) corresponds to frequency compression of X(w).A time-limited signal x(t) has a Fourier transform, X(w), which is not band limited (by band limited we mean the frequency band).If X(w) is band limited, then x(t) is not time limited.
R
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
relaxation (1) a general computational technique where computations are iterated until certain parameter measurements converge to a set of values. (2) the response of a linear time invariant system can be represented as the sum of the zero-input response (system response to a zero input function) plus the zero-state response (system response to an input function when the system is in the zero state). Relaxation is the process of putting a system into its zero-state, i.e. all initial conditions are zero and there are no internal energy stores. A system is considered relaxed if it is in the zero state. See relaxation labelling, optimization. relaxation labelling an iterative mathematical procedure in which a system of values is processed, e.g. by mutual adjustment of adjacent or associated values, until a stable state is attained. Especially useful for achieving consistent optimal
Frequency-Response Design Methods
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
In most of the work in previous chapters the input signals used were functions such as an impulse, a step, and a ramp function. In this chapter, the steady state response of a system to a sinusoidal input signal (sinusoid) is considered. It will be observed that the response of a linear time-invariant system to a sinusoidal input signal is an output sinusoidal signal at the same frequency as the input. However, the magnitude and phase of the output signal differ from those of the input sinusoidal signal, and the amount of difference is a function of the input frequency. Hence, the frequency response of a system is defined as the steady state response of the system to a sinusoidal input signal. The sinusoid is a unique input signal, and the resulting output signal for a linear system is sinusoidal in the steady state.
Stealth identification strategy for closed loop system structure
Published in International Journal of Systems Science, 2020
Hong Wang-jian, Ricardo A. Ramirez-Mendoza
In this section, we want to replace nonlinear controller by an equivalent linear time invariant controller , which is in some sense the best possible linear approximation of the unknown nonlinear controller. The problem studied here is that of approximating and unknown controller by one linear time invariant system . If is a linear time invariant system, then the output of to the input is given as .
Minimum information rate for observability of linear systems with stochastic time delay
Published in International Journal of Control, 2019
Issues of the type discussed are motivated by several pieces of work in the literature. The research on the interplay among coding, estimation, and control was addressed by Wong and Brockett (1999). A high-water mark in the study of quantised feedback using data-rate limited feedback channels is known as the data-rate theorem. It states that for a linear time-invariant system which is open-loop unstable, a controller can be designed to stabilise it if and only if the data rate R around the closed feedback loop satisfies the data-rate inequality: (bits/sample) where A denotes the system matrix composed by only unstable modes. The intuitively appealing result was proved (Baillieul, 2001).