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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 applic ...
, 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 Linear system#Definition, linearity and Time-invariant system, ...
(LTI) is exponentially stable if and only if the system has
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s (i.e., the poles 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 (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, call ...
). 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, models the system's output for each possible ...
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 Eucli ...
centered on the origin of the complex plane. 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 Lambda (; uppe ...
. Exponential stability is a form of asymptotic stability, valid for more general
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 ...
.


Definition

Consider the system \dot = f(t, x), \ x(t_0) = x_0, where f is piecewise continuous in t and Lipschitz in x. Assume without loss of generality that f has an equilibrium at the origin x=0. This equilibrium is exponentially stable if there exist c, k, \lambda > 0 such that \, x(t) \, \leq k \, x(t_0) \, e^, for all x(t) displays an exponential rate of decay.


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), then any resulting oscillations in the output will decay at an exponential rate, and the output will tend asymptotically 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 impulse (). More generally, an impulse response is the reac ...
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. 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''.


See also

* Marginal stability *
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 applic ...
*
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 equation ...


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