Stability criterion

Last updated June 02, 2019

In control theory, and especially stability theory, a stability criterion establishes when a system is stable. A number of stability criteria are in common use:

Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control model for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability.

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 norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.

In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either:

In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for linear time-invariant (LTI) systems.

In signal processing and control theory, the Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh–Hurwitz stability criterion. The Jury stability criterion requires that the system poles are located inside the unit circle centered at the origin, while the Routh-Hurwitz stability criterion requires that the poles are in the left half of the complex plane. The Jury criterion is named after Eliahu Ibraham Jury.

In control system theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed by A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations.

Stability may also be determined by means of root locus analysis.

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system. This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system. The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter.

Although the concept of stability is general, there are several narrower definitions through which it may be assessed:

In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for linear signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.

In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form $, where A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable . Other names for linear stability include exponential stability or stability in terms of first approximation . If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".$

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. In simple terms, if the solutions that start out near an equilibrium point $stay near forever, then is Lyapunov stable . More strongly, if is Lyapunov stable and all solutions that start out near converge to, then is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.$

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Harry Nyquist was a Swedish-born American electronic engineer who made important contributions to communication theory.

Nyquist may refer to:

In control theory and stability theory, the Nyquist stability criterion, discovered by Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a dynamical system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system. As a result, it can be applied to systems defined by non-rational functions, such as systems with delays. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplanes.

In control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time invariant (LTI) control system. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive. The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants than computing them directly. A polynomial satisfying the Routh–Hurwitz criterion is called a Hurwitz polynomial.

Criterion, or its plural form criteria, may refer to:

In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations.

Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the "plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output.

Vasile Mihai Popov is a leading systems theorist and control engineering specialist. He is well known for having developed a method to analyze stability of nonlinear dynamical systems, now known as Popov criterion.

In control engineering, a single-input and single-output (SISO) system is a simple single variable control system with one input and one output. In radio it is the use of only one antenna both in the transmitter and receiver.

Daniel Charles Drucker was American civil and mechanical engineer and academic, who served as president of the American Society of Mechanical Engineers in the year 1973–74. and as president of the American Academy of Mechanics in 1981–82.

In electronics, the Barkhausen stability criterion is a mathematical condition to determine when a linear electronic circuit will oscillate. It was put forth in 1921 by German physicist Heinrich Georg Barkhausen (1881–1956). It is widely used in the design of electronic oscillators, and also in the design of general negative feedback circuits such as op amps, to prevent them from oscillating.

Explanatory power is the ability of a hypothesis or theory to effectively explain the subject matter it pertains to. The opposite of explanatory power is explanatory impotence.

Drucker stability 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.

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