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Unstable
In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability, marginally stable or exhibit limit cycle behavior. In structural engineering, a structural beam or column can become unstable when excessive compressive load is applied. Beyond a certain threshold, structural Deflection (engineering), deflections magnify stress (physics), stresses, which in turn increases deflections. This can take the form of buckling or crippling. The general field of study is called structural stability. Atmospheric instability is a major component of all weather systems on Earth. Instability in control systems In the theory of dynamical systems, a state variable in a system is said to be unstable if it evolves without bounds. A system itself is said to be unstable if at least one of its state variables is unstabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atmospheric Instability
Atmospheric instability is a condition where the Earth's atmosphere is considered to be unstable and as a result local weather is highly variable through distance and time. Atmospheric instability encourages vertical motion, which is directly correlated to different types of weather systems and their severity. For example, under unstable conditions, a lifted air parcel, parcel of air will find cooler and denser surrounding air, making the parcel prone to further ascent, in a positive feedback loop. In meteorology, instability can be described by various indices such as the Bulk Richardson Number, lifted index, K-index (meteorology), K-index, convective available potential energy, convective available potential energy (CAPE), the Showalter, and the Vertical totals. These indices, as well as atmospheric instability itself, involve temperature changes through the troposphere with height, or lapse rate. Effects of atmospheric instability in moist atmospheres include thunderstorm d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp space, Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff convergence, Gromov–Hausdorff distance. In dynamical systems, an orbit (dynamics), orbit is called ''Lyapunov stability, Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Stability
In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable. Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes further and further away from any state, without being bounded. A marginal system, sometimes referred to as having neutral stability, is between these two types: when displaced, it does not return to near a common steady state, nor does it go away from where it started without limit. Marginal stability, like instability, is a feature that control theory seeks to avoid; we wish that, when perturbed by some external force, a system will return to a desired state. This necessitates the use of appropriately designed control algorithms. In econometrics, the presence of a unit root in observed time series, rendering them marginally stable, can lead to invalid regression results re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912). Definition We consider a two-dimensional dynamical system of the form x'(t)=V(x(t)) where V : \R^2 \to \R^2 is a smooth function. A ''trajectory'' of this system is some smooth function x(t) with values in \mathbb^2 which satisfies this differential equation. Such a trajectory is called ''closed'' (or ''periodic'') if it is not constant but returns to its starting point, i.e. if there exists some t_0>0 such that x(t + t_0) = x(t) for all t \in \R. An orbit (dynamics), orbit is the ima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State (controls)
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial conditions. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buckling
In structural engineering, buckling is the sudden change in shape (Deformation (engineering), deformation) of a structural component under Structural load, load, such as the bowing of a column under Compression (physics), compression or the wrinkling of a plate under Shearing (physics), shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have ''buckled''. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column. Buckling may occur even though the Stress (mechanics), stresses that develop in the structure are well below those needed to cause Catastrophic failure, failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space (controls)
In control engineering and system identification, a state-space representation is a mathematical model of a physical system that uses state variables to track how inputs shape system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values and inputs, while outputs depend on the states and sometimes the inputs too. The state space (also called time-domain approach and equivalent to phase space in certain dynamical systems) is a geometric space where the axes are these state variables, and the system’s state is represented by a state vector. For linear, time-invariant, and finite-dimensional systems, the equations can be written in matrix form, offering a compact alternative to the frequency domain’s Laplace transforms for multiple-input and multiple-output (MIMO) systems. Unlike the frequency domain approach, it works for systems beyond just linear ones with zero initial condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control Stability theory, stability; often with the aim to achieve a degree of Optimal control, optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or Setpoint (control system), set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drucker Stability
Drucker stability (also called the Drucker stability postulates) refers to a set of mathematical criteria that restrict the possible nonlinear stress- strain relations that can be satisfied by a solid material. The postulates are named after Daniel C. Drucker. A material that does not satisfy these criteria is often found to be unstable in the sense that application of a load to a material point can lead to arbitrary deformations at that material point unless an additional length or time scale is specified in the constitutive relations. The Drucker stability postulates are often invoked in nonlinear finite element analysis. Materials that satisfy these criteria are generally well-suited for numerical analysis, while materials that fail to satisfy this criterion are likely to present difficulties (i.e. non-uniqueness or singularity) during the solution process. Drucker's first stability criterion Drucker's first stability criterion (first proposed by Rodney Hill and also cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
