# Stability theory

The mathematical stability theory deals with the development of disturbances that occur as a deviation from certain states of dynamic systems . Such a state can be a position of rest or a certain orbit , e.g. B. a periodic orbit. A system is unstable when a small disturbance leads to large and emerging deviations.

In addition to its theoretical importance, stability theory is used in physics and theoretical biology as well as in technical areas, e.g. B. in technical mechanics or control engineering .

The approaches to solving the problems of stability theory are ordinary and partial differential equations .

## Mathematical concepts of stability

For the characterization of the stability of the rest position of a dynamic system there are several stability terms, each with slightly different meaning: ${\ displaystyle {\ dot {\ vec {x}}} = f ({\ vec {x}})}$

• A position of rest is called Lyapunov -stable if a sufficiently small disturbance always remains small. More precisely formulated: For each there is a such that for all times and all trajectories with the following applies: .${\ displaystyle {\ vec {x}} _ {R}}$${\ displaystyle \ varepsilon> 0}$${\ displaystyle \ delta (\ varepsilon)> 0}$${\ displaystyle t \ geq 0}$ ${\ displaystyle {\ vec {x}} (t)}$${\ displaystyle \ | {\ vec {x}} (0) - {\ vec {x}} _ {R} \ | <\ delta (\ varepsilon)}$${\ displaystyle \ | {\ vec {x}} (t) - {\ vec {x}} _ {R} \ | <\ varepsilon}$
• A rest position is attractive when one is such that each trajectory with all exists and satisfies the threshold condition:${\ displaystyle {\ vec {x}} _ {R}}$${\ displaystyle \ eta> 0}$${\ displaystyle {\ vec {x}} (t)}$${\ displaystyle \ | {\ vec {x}} (0) - {\ vec {x}} _ {R} \ | <\ eta}$${\ displaystyle t \ geq 0}$${\ displaystyle \ lim _ {t \ to \ infty} {\ vec {x}} (t) = {\ vec {x}} _ {R}.}$
• A rest position is called asymptotically stable if it is Lyapunov stable and attractive.
• A position of rest is called neutral stable or marginally stable if it is stable but not asymptotically stable.

In the case of discrete systems that are described by difference equations , the rest position is at the same time the fixed point of the recursion equation and similar stability definitions are common. ${\ displaystyle {\ vec {x}} _ {k + 1} = f ({\ vec {x}} _ {k})}$${\ displaystyle {\ vec {x}} _ {k + 1} = f ({\ vec {x}} _ {k})}$

## Stability analysis of linear time-invariant systems

Meaning of the poles and the conjugate complex pole pairs in the left and right s-half plane

In the case of continuous linear time-invariant systems , the stability can be read from the transfer function through the position of the poles in the s-plane (denominator polynomial of the Laplace transfer function):

• Asymptotic stability: if all poles are in the left half-plane,
• Limit stability: if no pole lies in the right half-plane and at least one single pole, but not a multiple pole, lies on the imaginary axis of the half-plane,
• Instability: otherwise (if at least one pole lies in the right half-plane or if at least one multiple pole lies on the imaginary axis of the s-plane).

In discrete linear time-invariant systems , the stability can be read from the position of the poles in the z-plane (denominator polynomial of the z- transfer function ).

• Asymptotic stability: if all poles are in the unit circle,
• Limit stability: if at least one pole is on the unit circle,
• Instability: otherwise (if at least one pole is outside the unit circle in the z-plane).

## Lyapunov's direct method and Lyapunov function

Lyapunov developed the so-called direct or second method in 1883 (the first method was linearization, see below) in order to check the stability properties mentioned above on specific systems. For this purpose, one first defines the orbital derivative for a dynamic system of form and a real-valued differentiable function${\ displaystyle {\ dot {\ vec {x}}} = f ({\ vec {x}})}$ ${\ displaystyle V ({\ vec {x}})}$

${\ displaystyle {\ dot {V}} ({\ vec {x}}): = \ left \ langle \ operatorname {grad} \, V ({\ vec {x}}), {\ dot {\ vec { x}}} \ right \ rangle = \ left \ langle \ operatorname {grad} \, V ({\ vec {x}}), f ({\ vec {x}}) \ right \ rangle}$.

A real-valued differentiable function is called a Lyapunov function (for the vector field ) if it holds for all points in the phase space . A Lyapunov function is a fairly powerful tool for proof of stability, as the following two criteria show: ${\ displaystyle V}$${\ displaystyle f}$${\ displaystyle {\ dot {V}} ({\ vec {x}}) \ leq 0}$${\ displaystyle {\ vec {x}}}$

Lyapunov's first criterion: a dynamic system is given . Conditions apply ${\ displaystyle {\ dot {\ vec {x}}} = f ({\ vec {x}})}$
1. ${\ displaystyle {\ vec {x}} _ {R}}$ is a rest position of the system,
2. ${\ displaystyle V ({\ vec {x}})}$is a Lyapunov function for ,${\ displaystyle f}$
3. ${\ displaystyle V ({\ vec {x}})}$has a strict local minimum at the point ,${\ displaystyle {\ vec {x}} _ {R}}$
then the rest position is stable.${\ displaystyle {\ vec {x}} _ {R}}$
Lyapunov's second criterion: applies in addition to the requirements of the first criterion
4. for in an environment of calm situation applies ,${\ displaystyle {\ vec {x}} \ neq {\ vec {x}} _ {R}}$${\ displaystyle {\ vec {x}} _ {R}}$${\ displaystyle {\ dot {V}} ({\ vec {x}}) <0}$
then the rest position is asymptotically stable.

The use of a Lyapunov function is called the direct method , because statements about the stability of a rest position can be obtained directly from the vector field without knowledge of the trajectories (i.e. without having to solve the differential equation). ${\ displaystyle f}$

## Lyapunov equation

In the case of linear systems , for example, a positively definite quadratic form can always be used as the Lyapunov function. It obviously satisfies the above conditions (1) and (2). Condition (3) leads to the Lyapunov equation ${\ displaystyle {\ dot {\ vec {x}}} = A {\ vec {x}}}$${\ displaystyle v ({\ vec {x}}) = {\ vec {x}} ^ {T} R {\ vec {x}}}$

${\ displaystyle A ^ {T} R + RA = -Q}$,

which is a special form of the Sylvester equation . If is positive definite, then there is a Lyapunov function. Such a function can always be found for stable linear systems . ${\ displaystyle Q}$${\ displaystyle v ({\ vec {x}}) = {\ vec {x}} ^ {T} R {\ vec {x}}}$${\ displaystyle v ({\ vec {x}}) = {\ vec {x}} ^ {T} R {\ vec {x}}}$

## Stability analysis of linear and non-linear systems

A dynamic system is given by the differential equation . ${\ displaystyle {\ dot {\ vec {x}}} = f ({\ vec {x}})}$

We consider a fault at the point in time as a deviation from the rest position : ${\ displaystyle \ delta = {\ vec {x}} (t) - {\ vec {x}} _ {R}}$${\ displaystyle t}$${\ displaystyle {\ vec {x}} _ {R}}$

• if the system is linear this perturbation can be fully expressed by the first derivative Jacobian matrix .${\ displaystyle \ mathbf {J}}$${\ displaystyle {\ vec {x}}}$
• if the system is non-linear and the disturbance is small enough, it can be " linearized ", i.e. H. the function after around Taylor-evolve .${\ displaystyle f}$${\ displaystyle \ delta}$${\ displaystyle {\ vec {x}} _ {R}}$

In both cases, the time development results from : ${\ displaystyle \ delta}$

${\ displaystyle {\ dot {\ delta}} = \ mathbf {J} ({\ vec {x}} _ {R}) \, \ delta}$

This development is therefore largely determined by the eigenvalues ​​of the Jacobi matrix. Specifically, there are the following three cases:

• The real part of all eigenvalues ​​of the Jacobi matrix is negative . Then it drops exponentially and the rest position is asymptotically stable .${\ displaystyle \ delta}$
• The real part of an eigenvalue of the Jacobi matrix is positive . Then it grows exponentially and the rest position is unstable .${\ displaystyle \ delta}$
• The greatest real part of all eigenvalues ​​of the Jacobi matrix is zero . For a linear system this means:
• if the algebraic multiplicity is equal to the geometric multiplicity for all eigenvalues ​​with vanishing real parts : marginal stability of the rest position.
• otherwise, d. H. if not , the algebraic multiplicity of geometric multiplicity is the same for all the eigenvalues with zero real part: instability of the rest position.
In the case of non-linear systems that have only been linearized around the rest position, the stability can also be determined by higher-order terms in the Taylor expansion. In this case, the linear stability theory can not make any statements .

## Endangerment of stability in construction

In construction, bars subjected to compression must be checked for stability risk (usually buckling) and, if necessary, verified according to the second-order theory . The second order theory is needed to be able to describe stability threats. In steel construction, in (steel) concrete construction as well as in wood construction, according to current standards, bars that are at risk of stability must be verified for buckling.

## Examples

An examined deformation state of the strength theory or a state of motion of the dynamics can change to another state from a stability limit to be determined. This is usually associated with non-linearly increasing deformations or movements that can lead to the destruction of supporting structures . To avoid this, the knowledge of the stability limit is an important criterion for dimensioning components.

Further examples: