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Introduction to Dynamical Systems Theory
Published in Nandan K. Sinha, N. Ananthkrishnan, Advanced Flight Dynamics with Elements of Flight Control, 2017
Nandan K. Sinha, N. Ananthkrishnan
Note that the notion of stability introduced here refers to a steady state. In systems with multiple steady states, some of them may be stable as defined here, whereas some others may be unstable. It makes no sense, in general, to talk about stability of the dynamical system itself. Stability as a concept is usually applied to each individual steady state. However, in cases where the dynamical system has only a single steady state, and this always happens for linear systems, the notion of stability is sometimes loosely carried over from that of the lone steady state to that of the system itself. Thus one may come across statements about a linear dynamical system being stable or unstable. However, strictly, and to be safe, it is better to ensure that the notion of stability here is used only to describe a particular steady state.
B
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
bounded-input bounded-output stability a linear dynamic system where a bounded input yields a bounded zero-state response. More precisely, let be a bounded-input with as the least upper bound (i.e., there is a fixed finite constant such that for every t or k), if there exists a scalar
A recursive algorithm for estimating multiple models continuous transfer function with non-uniform sampling
Published in International Journal of Systems Science, 2018
Anísio Rogério Braga, Walmir Caminhas, Carmela Maria Polito Braga
A linear dynamic system with input u(t) and output y(t), where t is time, is conventionally described by the ordinary linear differential equation where the rates of change and y(t) depend on u(t) and its rates of change. A linear dynamic system can also be represented by the difference equation where i is an integer sequence of sampling time intervals. When the sampling interval is a constant, h, an approximation of the rates of change of (26), known as δ-operator, which is applicable to both continuous-time and discrete-time systems, is defined as with an extension to higher-order derivatives as follows: Note that the definition (28) uses a backward shift operator q−1 instead of q, which is the usual definition proposed by Middleton and Goodwin (1990). The reason to use the definition (28) for the δ-operator is to make a unique mapping function to transform G(p) into G(λ) and vice-versa, besides providing a straightforward way to convert any discrete-time polynomial into continuous-time polynomials.