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Pole-Zero Stability
Published in Samir I. Abood, Digital Signal Processing, 2020
Similar to the analog system, the digital system requires that all poles plotted on the z-plane must be inside the unit circle. We summarize the rules for determining the stability of a DSP system as follows: If the outermost pole(s) of the z-transfer function H(z) describing the DSP system is(are) inside the unit circle in the z-plane pole-zero plot, then the system is stable.If the outermost pole(s) of the z-transfer function H(z) is(are) outside the unit circle in the z-plane pole-zero plot, the system is unstable.If the outermost pole(s) is(are) first-order pole(s) of the z-transfer function H(z) and on the unit circle in the z-plane pole-zero plot, then the system is marginally stable.If the outermost pole(s) is(are) multiple-order pole(s) of the z-transfer function H(z) and on the unit circle in the z-plane pole-zero plot, then the system is unstable.The zeros do not affect system stability.
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
From the point of view of the transfer function, stable systems have closed-loop transfer functions with only left half-plane poles, where a pole is defined as a value of s that causes F(s) to be infinite, such as a root of the denominator of a transfer function. Unstable systems have closed-loop transfer functions with at least one right half-plane pole and/or poles of multiplicity greater than one on the imaginary axis. Marginally stable systems have closed-loop transfer functions with only imaginary axis poles of multiplicity one and left half-plane poles. Stability is the most important system specification. An unstable system cannot be designed for a specific transient response or steady-state error requirement. Physically, instability can cause damage to a system, adjacent property, and human life. Many times, systems are designed with limit stops to prevent total runaway.
Stability
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Allen R. Stubberud, Stephen C. Stubberud
While the differentiation operator violates the asymptotic stability definition, since the impulse response is not bounded, it does satisfy the BIBO definition. In any case, it is generally considered stable. For an integrator, the impulse response is a constant and thus bounded and in the limit is a constant, as described in Oppenheim et al. [4]; however, for a step input, which is bounded, the output grows without bound. The same would occur if the input is a sinusoid of the same frequency as any imaginary axis (unit circle for discrete-time systems) poles of a system. Since such systems only “blow up” for a countable finite number of bounded inputs, such systems are often considered stable. However, for any input along the imaginary axis (unit circle), the output of these systems will neither decay to zero nor even to a stable value. Systems such as these are referred to as marginally stable. For these systems the roots of the polynomial, or eigenvalues of the state transition matrix, that do not meet the criteria for absolute stability, lie on the imginary axis (zero real-value part) for a continuous-time system or lie on the unit circle (magnitude equal to 1) for a discrete-time system. While some consider such systems as stable, others consider them unstable because they violate the definition of absolute stability.
On privacy vs. cooperation in multi-agent systems
Published in International Journal of Control, 2018
Vaibhav Katewa, Fabio Pasqualetti, Vijay Gupta
First, assume Q > 0. For i = 1, 2, …, N we have 0 < γ1λi(Q) ≤ γ1λ1(Q) < 2. Since λi(A1) = 1 − γ1λi(Q), the above condition is equivalent to −1 < λi(A1) < 1. Thus, all eigenvalues of A1 lie inside the unit circle and a steady- state solution of Equation (3) is achieved. Assume now that Q has a 0 eigenvalue. Then, by assumption A.1, b1 = 0 and A1 has a single eigenvalue at 1 and all other eigenvalues lie inside the unit circle. Thus, the linear system in Equation (3) is marginally stable and a finite steady-state solution is achieved. Further, it can be easily observed that the steady-state solution of Equation (3) satisfies the first-order optimality condition ∇Jco(x) = Qx + r = 0N of problem P. Thus, it minimises Jco(x) and is an optimum of P.
Stability analysis of a controlled mechanical system with parametric uncertainties in LuGre friction model
Published in International Journal of Control, 2018
Yun-Hsiang Sun, Yuming Sun, Christine Qiong Wu, Nariman Sepehri
The and values are adjusted as 99.998% of the nominal σ0 value and 101.407% of the nominal σ1 value, respectively. This means and . The initial conditions for system states and the time length are consistent with the one used in Section 3.2. Figures 10 and 11 show that the perturbed system displays sustained oscillations of fixed amplitude and fixed period in each system state with no external excitation throughout the trial. Such oscillations are limit cycles which are unique features of nonlinear system. Unlike the marginally stable linear system that has the amplitude governed by its initial conditions, the amplitude of the self-sustained oscillations in limit cycle are independent of the initial conditions.
A unified framework of cell population dynamics and mechanical stimulus using a discrete approach in bone remodelling
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Diego Quexada, Salah Ramtani, Olfa Trabelsi, Kalenia Marquez, Dorian Luis Linero Segrera, Carlos Duque-Daza, Diego Alexander Garzón Alvarado
In Figure 6, the results of normal bone remodeling show great similarity with other works on bone remodeling, (see e.g. Hambli et al. 2016; Peyroteo et al. 2019) after the system reaches a steady state (marginally stable due to the oscillations from Komarova’s model), this oscillations are asynchronous as seen in Figure 7. Moreover, the main trabecular groups and low-density zones such as ward’s triangle can be seen to some extent. In both pathological cases, these structural patterns can also be seen, with the variation in magnitude and mass turnover period. Overall, the structural patterns have great similarity to those found in other works (see Hambli 2014; Peyroteo et al. 2019).