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Phenomenological Creep Models of Fibrous Composites (Probabilistic Approach)
Published in Leo Razdolsky, Phenomenological Creep Models of Composites and Nanomaterials, 2019
Tom Caughey was a pioneer in the development of the stochastic equivalent linearization procedure for estimating the mean and variance of a non-linear system to random variables. The stochastic equivalent linearization procedure or statistical linearization procedure was almost simultaneously introduced more than fifty years ago by three independent investigators: Booton [9], Kazakov [10], and Caughey [11]. In mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamic systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology. Linearization is an effective method for approximating the output of a function y = f(x) at any x = a based on the value and slope of the function at x = b, given that y = f(x) is continuous on [a,b] (or [b,a]) and that a is close to b. In, short, linearization approximates the output of a function near x = a. The concept of local linearity applies to the most of points arbitrarily close to x = a, and the slope M should be, most accurately, the slope of the tangent line at x = a.
Nonlinear Systems Analysis and Modeling
Published in Naim A. Kheir, Systems Modeling and Computer Simulation, 2018
An equilibrium point of a system is defined as a special state in which the velocity ẋ of the state is identically equal to zero. In other words, each equilibrium point is a root of the algebraic equation f(t, x) = 0. For a full-ranked linear time-invariant system, x˙=Ax, where A is an n × n nonsingular constant matrix (i.e., the determinant of A does not vanish), it can be shown that this system has only one isolated equilibrium point at x = 0. Although x˙=Ax can have more than one equilibrium point if A is singlular, all points except the origin x = 0 are the continuum of equilibrium points. Therefore, any linear system has no more than one isolated equilibrium point.
Modeling of Dynamic Systems
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
The strategy is to determine the equilibrium points and then linearize the system. An equilibrium point is a state in which the system would remain if it were unperturbed by external disturbances. An equilibrium point can be unstable (an egg standing on its end), neutrally stable (a ball on a table), or stable (a book lying on a table). For a system under feedback control, an equilibrium point is called an operating point. A linearized model can be used to approximate a nonlinear system near an equilibrium point of the nonlinear system by a procedure called small signal linearization. The resulting linear system has an equilibrium point at zero that corresponds to the equilibrium point of the nonlinear system. While linearized models are only an approximation of the nonlinear system, they are convenient to analyze and they give considerable insight into the behavior of the nonlinear system near the equilibrium point. For example, if the zero equilibrium point of the linear system is stable, then the equilibrium of the nonlinear system is locally stable. The approximate linearized model of the system will be considered.
A cooperative control strategy for yaw rate and sideslip angle control combining torque vectoring with rear wheel steering
Published in Vehicle System Dynamics, 2022
The increasing complexity of vehicles – due to multiple actuators/controllers calls for new tools helping system engineers to assess the impact of the additional control systems on vehicle safety and performance, as well as their integration with the existing ones. Phase portraits are proven to be a very effective tool for evaluating the impact of control actions on enhancing vehicle handling performance and stability [3]. These plots in fact provide a graphical illustration of vehicle nonlinear dynamics, which is particularly useful in systems, which are described by two dominant states, such as vehicles running a turn on a planar surface. As a matter of fact, while ground vehicles have many degrees of freedom, the fundamental planar instabilities critical to control problems such as yaw rate control or stabilising a drifting vehicle arise from the yaw and sideslip dynamics [18,29,30,31]. In these cases, plotting the phase portraits allows to map equilibrium points locations and types and the range of possible trajectories in response to constrained or nonlinear control inputs. Phase portrait analysis is particularly informative when it is crucial accounting for nonlinear dynamics and it is difficult to display state trajectories from the equations of motion alone.
Vehicle control synthesis using phase portraits of planar dynamics
Published in Vehicle System Dynamics, 2019
Carrie G. Bobier-Tiu, Craig E. Beal, John C. Kegelman, Rami Y. Hindiyeh, J. Christian Gerdes
In developing new vehicle control systems, designers must meet dual goals of generating stable, useful, closed-loop dynamics and of providing comfort and confidence to vehicle occupants. Phase portraits are a powerful, visual medium that can assist designers by providing graphical illustrations of the vehicle dynamics. For cases in which there are two dominant states of interest, plotting the phase portrait provides insight into equilibrium point locations and types, the effect of changing parameters, and the resulting range of possible trajectories for a set of constrained or nonlinear control inputs. The phase portrait is particularly informative in cases where it is vital to consider nonlinearities in the dynamics and cases in which it is difficult to visualise the resulting state trajectories from the equations of motion alone.
Derivative-order-dependent stability and transient behaviour in a predator–prey system of fractional differential equations
Published in Letters in Biomathematics, 2019
Z. M. Alqahtani, M. El-Shahed, N. J. Mottram
The stability of the equilibria can be determined by considering the boundedness of the solution trajectories and the local stability of each equilibrium point. For boundedness results we refer the reader to Kar et al. (2010), in which it is proved that the system of Equations (3) and (4) are uniformly bounded for trajectories that start in the physically relevant region X>0, Y >0. For a fractional derivative system of nonlinear equations, the stability of the various equilibrium points may be investigated using a direct Lyapunov method, to determine the ‘Mittag–Lefler stability’, as described in Li et al. (2009), or by considering a linearization of the system with the Mittag–Lefler functions acting as eigenfunctions of the system, the equivalent to exponential eigenfunctions in integer-derivative systems, as consider by, for example Alidousti, Khoshsiar Ghaziani, and Bayati Eshkaftaki (2017), Matignon (1996), and Cresson and Szafranska (2017). In this latter approach, determining whether an equilibrium point is stable is undertaken by considering the eigenvalues of the Jacobian matrix, at the equilibrium point . For our system in (5) and (6) the Jacobian matrix is