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Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The mapping from the analog s-plane to digital z-plane illustrates how the natural frequency and damping ratio in the analog world manifest itself in the digital world. A pole in the s-plane at s=s0 produces a transient of the form es0t. Sampling every T seconds generates a sequence whose Z-transform has to a pole at z=z0=es0T where T is the sampling time. Figure 175.5 illustrate the z-plane that is used to locate the pole and zeros. The circle of radius one is known as the unit circle. For stability, all the poles of the discrete time system must be within this circle. The heart-shaped curves are the ones that correspond to constant damping ratio. For a damping ratio close to 1, the curves are close to the real axis. If the points corresponding to a constant damping ratio are mapped into the digital world as shown above, they produce this set of heart-shaped curves. As the damping ratio decreases towards 0, the constant damping ratio curves move toward the unit circle. Alternatively, the points on the locus of constant natural frequency are the curves that begin and end on the unit circle. The values of the natural frequency are given in steps of the 0.1 times pi divided by the sampling time.
Nonconservative LMI techniques for robust stabilisation of spatially interconnected systems
Published in International Journal of Systems Science, 2021
The notation used in this paper is fairly standard. The set of integers, natural, and non-negative real numbers are denoted by , , and , respectively. The notation is used to denote the unit circle, i.e. , where denotes the set of complex numbers. and denote the floor and ceiling operators, respectively. The identity and zero matrix are denoted by and , respectively, or just I and if their dimensions are contextually clear. For a matrix F, is the column vector generated by stacking the columns of F from the first one. For scalars a, b, c, and d, the notation is defined by The Cartesian product of sets is denoted by , and will be abbreviated as when they are identical.