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Fundamentals of biology and thermodynamics
Published in Mohammad E. Khosroshahi, Applications of Biophotonics and Nanobiomaterials in Biomedical Engineering, 2017
(Lyapunov stability gives a definition of asymptotic stability for more general dynamical systems, and all exponentially stable systems are also asymptotically stable). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. Remembering that the definition of asymptotic is that it is a line that approaches a curve but never touches it. Consider a simple scalar equation y′(t) = ay (t). The solution is, of course, y (t) = y0 eat, where y0 = y (0). In particular, y (t) ≡ 0 is a solution. What happens if we start at some point other that 0? If a < 0, then every solution approaches 0 as t → ∞. We say that the zero solution is (globally) asymptotically stable. Figure 5.7 shows the graphs of a few solutions and the direction field of the equation, i.e., the arrows have the same slope as the solution that passes through the tail point.
Robot Dynamics and Control
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
Exponential stability is a strong form of stability; in particular, it implies uniform, asymptotic stability. Exponential convergence is important in applications because it can be shown to be robust to perturbations and is essential for the consideration of more advanced control algorithms, such as adaptive ones. A system is globally exponentially stable if the bound in equation (4.34) holds for all x0 ∈ ℝn. Whenever possible, we shall strive to prove global, exponential stability.
Advanced Control Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
Another measure employed hand in hand with the exponential stability in defining the long term Lyapunov stability is the exponential stability. The concept of exponential stability is an expression of the measure of the rate at which a perturbed equilibrium goes back to its original state x(t0) = 0. If the perturbed state vector of the system converges back to its equilibrium faster than an exponential function, the equilibrium is said to have an exponential stability. This type of stability is stronger in a sense than the asymptotic stability, because it not only gives the convergence property of the equilibrium that is expressed by the asymptotic stability, but also goes further to express how fast the convergence will occur. Mathematically, the exponential stability can be described as follows: If there exists some strictly positive numbers α and λ such that ‖s(t0+t,x(t0))‖≤α‖x(t0)‖e−λt∀t,t0≥0∀x(t0)∈Brwhere Br is the range or the radius of the Lyapunov stability, then the equilibrium x(t0) is said to be exponentially stable. This gives a stronger and explicit boundary on the state of the system at any time t ≥ t0.
Exponential stabilisation for nonlinear PDE systems via sampled-data static output feedback control
Published in Cyber-Physical Systems, 2021
Dongxiao Hu, Xiaona Song, Mi Wang, Junwei Lu
Based on the above analysis, in this paper, we study the exponential stabilisation problem for a nonlinear PDE system via designing a fuzzy sampled-data controller. We organise the paper as following: First, we introduce the nonlinear PDE system, and remodel it based on T-S fuzzy model. Then, the closed-loop systems with time-varying delays are given by proposing a sampled-data static output feedback fuzzy controller. Next, sufficient conditions are obtained in form of linear matrix inequality by Lyapunov function approach, which satisfied the exponential stabilisation and dissipative performance for the closed-loop PDE systems, and the controller gain can be obtained by solving spatial differential linear matrix inequalities. Subsequently, a simulation study is given to verify the effectiveness of the proposed method. Finally, conclusions of this paper are given. Briefly, the most contributions of this study are classified in the follows: 1) A fuzzy distributed sampled-data controller is introduced which can ensure the system is exponential stable and satisfies dissipative. 2) In contrast to asymptotical stability, exponential stability can ensure that the dynamic system converges fast enough to obtain a better response. Therefore, we consider the exponential stability problem for the nonliear PDE system.
On the two-dimensional tidal dynamics system: stationary solution and stability
Published in Applicable Analysis, 2020
Lyapunov stability is a very mild requirement on equilibrium points, which does not require that trajectories starting close to the origin tend to the origin asymptotically. The uniform asymptotic stability is a concept which guarantees that the equilibrium point is not losing stability. The notion of exponential stability is far stronger than asymptotic stability and it assures a minimum rate of decay, that is, an estimate of how fast the solutions converge to its equilibrium. In particular, exponential stability implies uniform asymptotic stability. In this section, we prove that the stationary solution is uniformly Lyapunov stable and also make a remark on the exponential stability.
Structural stability, asymptotic stability and exponential stability for linear multidimensional systems: the good, the bad and the ugly
Published in International Journal of Control, 2018
Olivier Bachelier, Thomas Cluzeau, Ronan David, Francisco José Silva Álvarez, Nader Yeganefar, Nima Yeganefar
In every case, we have thus proved that ‖x(i + 1, j + 1)‖ ≤ ε which implies and the solution is attractive: see Equation (2) of Definition 3.3. Asymptotic stability does not imply structural/exponential stability: