Marginal stability
Encyclopedia
In the theory of dynamical systems, and control theory
, a continuous linear
time-invariant system
is marginally stable if and only if
the real part of every eigenvalue (or pole) in the system's transfer-function is non-positive, and all eigenvalues with zero real value are simple root
s (i.e. the eigenvalues on the imaginary axis
are all distinct from one another). If all the poles have strictly negative real parts, the system is instead asymptotically stable.
A discrete linear time-invariant system is marginally stable if and only if the transfer function's spectral radius
is 1. That is, the greatest magnitude of any of the eigenvalues (or poles) of the transfer function is 1. The values of the poles must also be distinct. If the spectral radius is less than 1, the system is instead asymptotically stable.
of finite magnitude as input, will not "blow up" and give an unbounded output. However, oscillations in the output will persist indefinitely, and so there will, in general, be no final steady-state output. If the system is given a step
as an input, the system's output will increase indefinitely, with the system effectively acting as an integrator on the input, and so a marginally stable system is not a Bounded Input/Bounded Output
system (the information in this para must be verified from other sources).
A system having imaginary poles, i.e having zero real part in the pole(s), will produce sustained oscillations in the output. For example a undamped second order system such as suspension system of your car (mass-spring-damper), from where damper has been removed and spring is ideal i.e. no friction is there, then in theory your car will oscillate forever when you will hit a bump. A system with pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (jw = 0 means w = 0 rad/sec).
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, a continuous linear
Linear system
A linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....
time-invariant system
Time-invariant system
A time-invariant system is one whose output does not depend explicitly on time.This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output....
is marginally stable if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
the real part of every eigenvalue (or pole) in the system's transfer-function is non-positive, and all eigenvalues with zero real value are simple root
Simple root
in mathematics the term simple root can refer to one of two unrelated notions:*A simple root of a polynomial is a root of multiplicity one*A simple root in a root system is a member of a subset determined by a choice of positive roots...
s (i.e. the eigenvalues on the imaginary axis
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
are all distinct from one another). If all the poles have strictly negative real parts, the system is instead asymptotically stable.
A discrete linear time-invariant system is marginally stable if and only if the transfer function's spectral radius
Spectral radius
In mathematics, the spectral radius of a square matrix or a bounded linear operator is the supremum among the absolute values of the elements in its spectrum, which is sometimes denoted by ρ.-Matrices:...
is 1. That is, the greatest magnitude of any of the eigenvalues (or poles) of the transfer function is 1. The values of the poles must also be distinct. If the spectral radius is less than 1, the system is instead asymptotically stable.
Practical consequences
A marginally stable system is one that, if given an impulseDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
of finite magnitude as input, will not "blow up" and give an unbounded output. However, oscillations in the output will persist indefinitely, and so there will, in general, be no final steady-state output. If the system is given a step
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
as an input, the system's output will increase indefinitely, with the system effectively acting as an integrator on the input, and so a marginally stable system is not a Bounded Input/Bounded Output
BIBO stability
In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...
system (the information in this para must be verified from other sources).
A system having imaginary poles, i.e having zero real part in the pole(s), will produce sustained oscillations in the output. For example a undamped second order system such as suspension system of your car (mass-spring-damper), from where damper has been removed and spring is ideal i.e. no friction is there, then in theory your car will oscillate forever when you will hit a bump. A system with pole at the origin is also marginally stable but in this case there will be no oscillation in the response as the imaginary part is also zero (jw = 0 means w = 0 rad/sec).