|
In control theory, a continuous linear time-invariant system is asymptotically stable if and only if the system's transfer function has poles (or, equivalently, eigenvalues) only with strictly negative real parts. That is, the poles are in the left half of the complex plane. For control theory in psychology and sociology, see control theory (sociology). ...
In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ...
A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...
A discrete linear time-invariant system is asymptotically stable if and only if the poles (or eigenvalues) of its transfer function lie strictly within the unit circle centered on the origin of the complex plane.
Practical consequences
An asymptotically stable system is one that, if given a finite input, will not "blow up" and give an unbounded output. Moreover, if the system is given a fixed, finite input (that is, a step), then any resulting oscillations in the output will decay, 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. The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...
The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...
In the theory of dynamical systems, a linear time-invariant system is marginally stable if every eigenvalue in the systems transfer-function is non-positive, and all eigenvalues with zero real value are simple roots. ...
Example asymptotically stable systems
 The impulse responses of two asymptotically stable systems The graph on the right shows the impulse response of two similar systems. The green curve is the response of the system with impulse response  , while the blue represents the system  . Although one response is oscillatory, both return to the original value of 0 over time. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
The Impulse response from a simple audio system. ...
Real-world example Imagine putting a marble in a laddle. 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. The Dirac delta or Diracs delta, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0 and the value zero elsewhere. ...
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 asymptotically stable over a certain range of inputs.
See also |
Feeds |
Contact