:''See

Lyapunov stability 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. ...

, which gives a definition of asymptotic stability for more general

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

. All exponentially stable systems are also asymptotically stable.'' In

control theory Control theory is a field of mathematics that deals with the 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 ...

, a continuous

linear time-invariant system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly define ...

(LTI) is exponentially stable if and only if the system has

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

s (i.e., the

pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...

s of input-to-output systems) with strictly negative real parts. (i.e., in the left half of the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

).David N. Cheban (2004), ''Global Attractors Of Non-autonomous Dissipative Dynamical Systems''. p. 47 A discrete-time input-to-output LTI system is exponentially stable if and only if the poles of its

transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, theoretically models the system's output for ...

lie strictly within the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...

centered on the origin of the complex plane. Exponential stability is a form of

asymptotic stability Various types of Stability theory, 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 ...

. Systems that are not LTI are exponentially stable if their convergence is bounded by

exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...

Practical consequences

An exponentially stable LTI system is one that will not "blow up" (i.e., give an unbounded output) when given a finite input or non-zero initial condition. Moreover, if the system is given a fixed, finite input (i.e., a

step Step(s) or STEP may refer to: Common meanings * Stairs#Step, Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * Steps (TV series), ''Steps'' (TV series), Hong Kong * Step (film), ' ...

), then any resulting oscillations in the output will decay at an exponential rate, and the output will tend

asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

to a new final, steady-state value. If the system is instead given a Dirac delta impulse as input, then induced oscillations will die away and the system will return to its previous value. If oscillations do not die away, or the system does not return to its original output when an impulse is applied, the system is instead marginally stable.

Example exponentially stable LTI systems

The graph on the right shows the

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...

of two similar systems. The green curve is the response of the system with impulse response

y(t) = e^

, while the blue represents the system

y(t) = e^\sin(t)

. Although one response is oscillatory, both return to the original value of 0 over time.

Real-world example

Imagine putting a marble in a ladle. It will settle itself into the lowest point of the ladle and, unless disturbed, will stay there. Now imagine giving the ball a push, which is an approximation to a Dirac delta impulse. The marble will roll back and forth but eventually resettle in the bottom of the ladle. Drawing the horizontal position of the marble over time would give a gradually diminishing sinusoid rather like the blue curve in the image above. A step input in this case requires supporting the marble away from the bottom of the ladle, so that it cannot roll back. It will stay in the same position and will not, as would be the case if the system were only marginally stable or entirely unstable, continue to move away from the bottom of the ladle under this constant force equal to its weight. It is important to note that in this example the system is not stable for all inputs. Give the marble a big enough push, and it will fall out of the ladle and fall, stopping only when it reaches the floor. For some systems, therefore, it is proper to state that a system is exponentially stable ''over a certain range of inputs''.

References

External links

Parameter estimation and asymptotic stability instochastic filtering
Anastasia Papavasiliou∗September 28, 2004 {{Differential equations topics Dynamical systems Stability theory fr:Stabilité de Lyapunov#Les stabilités

Practical consequences

Example exponentially stable LTI systems

Real-world example

See also

References

External links