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Nonlinear Systems Analysis and Modeling
Published in Naim A. Kheir, Systems Modeling and Computer Simulation, 2018
For a linear system, this stability definition in the Lyapunov sense can be shown to be equivalent to the BIBO (bounded-input/bounded-output) stability. If in a standard linear time-invariant system ẋ = Ax and the output takes the state vector x, then it is BIBO stable at x = 0 if all eigenvalues of the matrix A have nonpositive real parts. However, if all the eigenvalues of A have strictly negative real parts, conventionally referred to as A is Hurwitz, then the system is asymptotically stable. Therefore, a linear time-invariant system has a straightforward criterion to determine its stability. For a nonlinear system, however, because there is no matrix A and there can be multiple equilibrium points in nature, the stability issue is much more complicated than in the linear case.
General Properties of Linear Circuits and Systems
Published in Herbert J. Carlin, Pier Paolo Civalleri, Wideband Circuit Design, 2018
Herbert J. Carlin, Pier Paolo Civalleri
In a time invariant system, the physical properties remain invariant with time, hence if a signal is time delayed, i.e., t replaced by t − τ, then the response is similarly time delayed. Thus if a time invariant system starts in a completely deenergized state when excited, its response will always be the same measured from the time of initiation of the input signal. These properties can be precisely phrased in terms of the operators which reflect the physical properties of the system. At the outset we assume that the functions of time under discussion are members of a function space which is appropriately defined. For example, the space might consist of all real absolutely integrable functions. In general the members of the space can be combined according to the usual rules of arithmetic. In the discussions that follow when we say all “admissible functions” or simply “all functions” we mean functions in the defined space. The operator maps functions in the defined space (the domain of the operator) into the space of the transformed functions (the range of the operator). The domain and range need not coincide. Furthermore, although our interest is essentially confined to LTI operators, it should be noted that the concepts of domain and range apply to all operators, linear or not.
Fuzzy Systems—Sets, Logic, and Control
Published in Thrishantha Nanayakkara, Ferat Sahin, Mo Jamshidi, Intelligent Control Systems with an Introduction to System of Systems Engineering, 2018
Thrishantha Nanayakkara, Ferat Sahin, Mo Jamshidi
Almost any linear or nonlinear system under the influence of a closed-loop crisp controller has one type of stability test or another. For example, the stability of a linear time-invariant system can be tested by a wide variety of methods such as Routh-Hurwitz, root locus, Bode plots, Nyquist criterion, and even via traditionally nonlinear systems methods of Lyapunov, Popov, and circle criterion. The common requirement in all these tests is the availability of a mathematical model, either in time or frequency domain. A reliable mathematical model for a very complex and large-scale system may, in practice, be unavailable or not feasible. In such cases, a fuzzy controller may be designed based on expert knowledge or experimental practice. However, the issue of the stability of a fuzzy control system remains and must be addressed. The aim of this section is to present an up-to-date survey of available techniques and tests for fuzzy control system stability.
Improved stability criteria for linear time-varying systems on time scales
Published in International Journal of Control, 2020
Xiaodong Lu, Xianfu Zhang, Zhi Liu
As is the most significant and fundamental problem in system theory, stability analysis of continuous and discrete systems has received continued attention in the literatures during the past decades (Agarwal, 2000; Bellman, 1953; Harris & Miles, 1980; Jungers, Ahmadi, Parrilo, & Mardavij, 2017; Khalil, 2002; Li & Wu, 2016; Li, Zhang, & Song, 2017; Rugh, 1996; Zhou, 2016; Zhou & Zhao, 2017). As a special kind of dynamical system, linear time-invariant system has been extensively investigated and its stability analysis can be totally characterised by the locations of the eigenvalues of the system matrix (Rugh, 1996; Szyda, 2010) or by the Lyapunov's second method (Kalman & Bertram, 1960). However, these two methods are not quite effective for linear time-varying (LTV) system since its stability analysis cannot be linked with the locations of the eigenvalues of system matrix (Rugh, 1996) and the construction of Lyapunov function is not a simple task (Harris & Miles, 1980; Rugh, 1996). So, the stability test of LTV systems is still a challenging problem and has been listed as the first open problem in mathematical control theory (Aeyels & Peuteman, 1999).