Editing Marginal stability

{{Short description|Dynamical system which is neither asymptotically stable nor unstable}}
{{Refimprove|date=August 2014}}

In the theory of [[dynamical systems]] and [[control theory]], a [[linear system|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,<ref name="FranklinPowell2014">{{cite book|author1=Gene F. Franklin|author2=J. David Powell|author3=Abbas Emami-Naeini|title=Feedback Control of Dynamic Systems|edition=5|year=2006|publisher=Pearson Education|isbn=0-13-149930-0}}</ref> 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 analysis|regression]] results regarding effects of the [[independent variable]]s upon a [[dependent variable]], unless appropriate techniques are used to convert the system to a stable system.

==Continuous time==
A [[homogeneous differential equation|homogeneous]] [[continuous time|continuous]] [[linear time-invariant system]] is marginally stable [[if and only if]] the real part of every [[Pole (complex analysis)|pole]] ([[eigenvalue]]) in the system's [[transfer-function]] is [[non-positive]], one or more poles have zero real part, and all poles with zero real part are [[simple root]]s (i.e. the poles on the [[complex plane|imaginary axis]] are all distinct from one another). In contrast, if all the poles have strictly negative real parts, the system is instead asymptotically stable. If the system is neither stable nor marginally stable, it is unstable.

If the system is in [[state space representation]], marginal stability can be analyzed by deriving the [[Jordan normal form]]:<ref>{{cite web |url=http://www.cds.caltech.edu/~murray/amwiki/index.php/Linear_Systems |title=Linear Systems |work=Feedback Systems Wiki |author=Karl J. Åström and Richard M. Murray|publisher=Caltech|accessdate=11 August 2014}}</ref> if and only if the Jordan blocks corresponding to poles with zero real part are scalar is the system marginally stable.

==Discrete time==
A homogeneous [[discrete time]] linear time-invariant system is marginally stable if and only if the greatest magnitude of any of the poles (eigenvalues) of the transfer function is 1, and the poles with magnitude equal to 1 are all distinct. That is, the transfer function's [[spectral radius]] is 1. If the spectral radius is less than 1, the system is instead asymptotically stable.

A simple example involves a single first-order [[linear difference equation]]: Suppose a state variable ''x'' evolves according to

:<math>x_t=ax_{t-1}</math>

with parameter ''a'' > 0. If the system is perturbed to the value <math>x_0,</math> its subsequent sequence of values is <math>ax_0, \, a^2x_0, \, a^3x_0, \, \dots .</math> If ''a'' < 1, these numbers get closer and closer to 0 regardless of the starting value <math>x_0,</math> while if ''a'' > 1 the numbers get larger and larger without bound. But if ''a'' = 1, the numbers do neither of these: instead, all future values of ''x'' equal the value <math>x_0.</math> Thus the case ''a'' = 1 exhibits marginal stability.

==System response==
A marginally stable system is one that, if given an [[dirac delta function|impulse]] of finite magnitude as input, will not "blow up" and give an unbounded output, but neither will the output return to zero. A bounded offset or oscillations in the output will persist indefinitely, and so there will in general be no final steady-state output. If a continuous system is given an input at a frequency equal to the frequency of a pole with zero real part, the system's output will increase indefinitely (this is known as pure resonance<ref>{{cite web |url=http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/pure-resonance/ | title= Pure Resonance|publisher=MIT|accessdate=2 September 2015}}</ref>). This explains why for a system to be [[BIBO stability|BIBO stable]], the real parts of the poles have to be strictly negative (and not just non-positive).

A continuous system having imaginary poles, i.e. having zero real part in the pole(s), will produce sustained oscillations in the output. For example, an undamped second-order system such as the suspension system in an automobile (a [[mass–spring–damper]] system), from which the damper has been removed and spring is ideal, i.e. no friction is there, will in theory oscillate forever once disturbed. Another example is a frictionless [[Pendulum (mathematics)|pendulum]]. A system with a 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''&nbsp;=&nbsp;0 means ''w''&nbsp;=&nbsp;0&nbsp;rad/sec). An example of such a system is a mass on a surface with friction. When a sidewards impulse is applied, the mass will move and never returns to zero. The mass will come to rest due to friction however, and the sidewards movement will remain bounded.

Since the locations of the marginal poles must be ''exactly'' on the imaginary axis or unit circle (for continuous time and discrete time systems respectively) for a system to be marginally stable, this situation is unlikely to occur in practice unless marginal stability is an inherent theoretical feature of the system.

==Stochastic dynamics==

Marginal stability is also an important concept in the context of [[stochastic dynamics]]. For example, some processes may follow a [[random walk]], given in discrete time as 

:<math>x_t=x_{t-1}+e_t,</math>

where <math>e_t</math> is an [[i.i.d.]] [[errors and residuals in statistics|error term]]. This equation has a [[unit root]] (a value of 1 for the eigenvalue of its [[characteristic equation (of difference equation)|characteristic equation]]), and hence exhibits marginal stability, so special [[time series]] techniques must be used in empirically modeling a system containing such an equation.

Marginally stable [[Markov chain|Markov processes]] are those that possess [[Markov_chain#Properties|null recurrent]] classes.

==See also==
*[[Lyapunov stability]]
*[[Exponential stability]]

==References==
{{reflist}}

{{DEFAULTSORT:Marginal Stability}}
[[Category:Dynamical systems]]
[[Category:Stability theory]]