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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] |
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] |
Pole (complex Analysis)
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of a function, zero of the function and is holomorphic function, holomorphic (i.e. complex differentiable) in some neighbourhood (mathematics), neighbourhood of . A function is meromorphic function, meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass–spring–damper
The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. This form of model is also well-suited for modelling objects with complex material behavior such as those with nonlinearity or viscoelasticity. As well as engineering simulation, these systems have applications in computer graphics and computer animation. Derivation (Single Mass) Deriving the equations of motion for this model is usually done by summing the forces on the mass (including any applied external forces F_\text): :\Sigma F = -kx - c \dot x +F_\text = m \ddot x By rearranging this equation, we can derive the standard form: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = u where \omega_n=\sqrt\frac; \quad \zeta = \frac; \quad u=\frac \omega_n is the undamped natural frequency and \zeta is the damping ratio. The homogeneous equation for the mass spring system is: :\ddot x + 2 \zeta \omega_n \dot x + \omega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BIBO Stability
In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for 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. A signal is bounded if there is a finite value B > 0 such that the signal magnitude never exceeds B, that is :For discrete-time signals: \exists B \forall n(\ , y \leq B) \quad n \in \mathbb :For continuous-time signals: \exists B \forall t(\ , y(t), \leq B) \quad t \in \mathbb Time-domain condition for linear time-invariant systems Continuous-time necessary and sufficient condition For a continuous time linear time-invariant (LTI) system, the condition for BIBO stability is that the impulse response, h(t) , be absolutely integrable, i.e., its L1 norm exists. : \int_^\infty \left, h(t)\\,\mathordt = \, h \, _1 \in \mathbb Discrete-time sufficient condition For a discrete time LTI system, the condition f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Difference Equation
In mathematics (including combinatorics, linear algebra, and dynamical systems), a linear recurrence with constant coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable (mathematics), variable—that is, in the values of the elements of a sequence. The polynomial's linearity means that each of its terms has degree of a polynomial, degree 0 or 1. A linear recurrence denotes the evolution of some variable over time, with the current discrete time, time period or discrete moment in time denoted as , one period earlier denoted as , one period later as , etc. The ''equation solving, solution'' of such an equation is a function of , and not of any iterate values, giving the value of the iterate at any time. To find the solution it is necessary to know the specific values (known as ''initial conditions'') of of the iterates, and normally these are the iterates that are ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Radius
''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Mark Clyne, with Max Martini, Emily Mortimer, Clayne Crawford, and Bruce Greenwood in supporting roles. The film is set in a civil war-ridden Moldova as invisible entities slaughter any living being caught in their path. The film was released worldwide on December 9, 2016 on Netflix. On February 1, 2017, Netflix released a prequel graphic novel of the film called ''Spectral: Ghosts of War'' which was made available digitally through the website ComiXology. Plot DARPA researcher Mark Clyne is sent to a United States, US United States Armed Forces, military Air base, airbase on the outskirts of Chișinău, to consult his created line of hyperspectral imaging goggles issued to United States Army, US Army United States Army Special Forces, S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Time
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as being unchanged throughout each non-zero region of time ("time period")—that is, time is viewed as a discrete variable. Thus a non-time variable jumps from one value to another as time moves from one time period to the next. This view of time corresponds to a digital clock that gives a fixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, each variable of interest is measured once at each time period. The number of measurements between any two time periods is finite. Measurements are typically made at sequential integer values of the variable "time". A discrete signal or discrete-time signal is a time series consisting of a sequence of quantities. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Normal Form
\begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ \hphantom\lambda_n 1\hphantom\\ \hphantom\hphantom\lambda_n \end Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number ''λi'' on its main diagonal, and 1s directly above the main diagonal. The ''λi''s are the eigenvalues of the matrix; they need not be distinct. In linear algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space Representation
State most commonly refers to: * State (polity), a centralized political organization that regulates law and society within a territory **Sovereign state, a sovereign polity in international law, commonly referred to as a country **Nation state, a state where the majority identify with a single nation (with shared culture or ethnic group) ** Constituent state, a political subdivision of a state ** Federated state, constituent states part of a federation *** U.S. state * State of nature, a concept within philosophy that describes the way humans acted before forming societies or civilizations State may also refer to: Arts, entertainment, and media Literature * '' State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * '' Our State'', a monthly magazine published in North Carolina and formerly called ''The State'' * The State (Larry Niven), a fictional future governm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vector (geometry), vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates: the magnitude or ' of the product is the product of the two absolute values, or moduli, and the angle or ' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane or Gauss plane. Notational conventions Complex numbers In complex analysis, the complex numbers are customarily represented by the symbol , which can be sepa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Root
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebrai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
