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Complex Variables
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
For example, consider the simple closed curve γ of Figure 4.3. If a function f(z) has three poles and a single zero enclosed by the curve γ, then the argument principle states that the curve Γ = f(γ) must make two negative encirclements of the origin.
Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling
Published in International Journal of Control, 2020
In this paper, the feedback connections with LCP plants and controllers are attacked via complex scaling. More precisely, the harmonic return difference relation (Zhou & Hagiwara, 2005) is re-written into a complex scaling manner by separating the open- and closed-loop characteristic polynomials, with respect to prescribed scaling comparators. Based on the argument principle of complex analysis, we validate rigorously the complex scaling 2-regularised criteria for asymptotic stability in LCP systems (Zhou, 2013). The suggested stability conditions are necessary and sufficient. Different from the criteria of (Zhou & Hagiwara, 2005; Zhou et al., 2002), however, the complex scaling ones involve no open-loop Floquet factorisation and are stated with self-defined contour and locus orientation conditions. Implementation algorithms are proposed via truncating the harmonic transfer operators, in terms of finite-dimensional LTI system approximations. Moreover, the finite-dimensional criteria can be employed graphically with locus plotting, or numerically by computing complex argument incremental; the latter makes the approach be numerically tractable so that it is suitable for stabilisation as well.
Non-Singular Kernels for Modelling Power Law Type Long Memory Behaviours and Beyond
Published in Cybernetics and Systems, 2020
Images of path by the operators (and by operator ) are represented in Figure 4. Using this figure, the stability domain of model (30) can be deduced. The image of path by relation (33) is the image of path by shifted by the vector that appears in figure 5 and that links the coordinate of point to the origin of the complex plane. Then, after this shift, the origin of the complex plane is located in place of the point As the denominator of relation (32) has no root inside path the shifted origin will lie outside the image of to ensure the stability of system (30). If the origin of the complex plane is inside the image of path then according to the argument principle, the characteristic equation has one root inside path and system (30) is unstable.
A reliable graphical criterion for TDS stability analysis
Published in International Journal of Systems Science, 2020
Tiao Yang Cai, Hui Long Jin, Xiang Peng Xie
As is indicated in argument principle that, the number of zeros of in the interior of Γ, is equal to the winding number of the imaginary curve of with respect to the origin, Division of by an auxiliary polynomial, will determine a curve whose rotations will be given by: And is chosen to be a nth order polynomials with n known roots with negative real parts as in (4), thus we have Therefore, this choice of gives the result that the zeros of inside Γ equals to the number of clockwise encirclements of the origin by the locus of the function for λ varies along the Jordan curve Γ.