Stability and Dynamics

Abstract

In this chapter we shortly review the well known principles and definitions of stability and dynamics of systems.

1 Introduction: Definition of Stability

Generally, a system is regarded as stable when small deviations between the actual operating conditions and the operating conditions for which it was designed do not fundamentally affect the system behaviour.

Actually, most people will intuitively feel whether a system exhibits stable behaviour or not. However, for practical applications, some more specific definitions of stability are required.

2 Classifications of Stability

2.1 Stability of an Equilibrium Point

The first type of stability involves the behaviour of a system when disturbed in an equilibrium state. A typical example is that of a ball resting on a surface and subjected to gravity (see Fig. 23.1).

Fig. 23.1

Equilibrium states

In case (a) in Fig. 23.1, the ball will oscillate around its initial position with a limited amplitude (depending on the initial deviation) when it is moved away from its equilibrium position. In case (b), the ball will remain in its new position when moved away from its initial equilibrium position, as the new position is an equilibrium position too. In both cases, this behaviour can be called stable: the deviation from the initial position remains bounded and can be deliberately made small by making the initial displacement sufficiently small.

In case (c) in Fig. 23.1, the ball will definitely move away from its initial position when displaced from this position, whatever small the displacement. The equilibrium state (c) is therefore called unstable.

If friction is taken into account, in case (a) the ball will return to its original position after some damped oscillations, or even without any oscillations. This equilibrium state is therefore called asymptotically stable. In case (b), the ball will never return to its original position. This equilibrium state is stable but not asymptotically stable, and it is asymptotic stability that is required in practical systems.

Fig. 23.2

Boundaries for stability

In the previous section, we disregarded the magnitude of the deviation from the equilibrium position. Consider now the situations depicted in Fig. 23.2. In case (a), the neighbourhood of the initial position where the asymptotic stability holds is much smaller than in case (b) in Fig. 23.2. In case (c), asymptotic stability is ensured for every displacement. For practical applications, it is not sufficient that the equilibrium state is stable within a, possibly very small, neighbourhood of this equilibrium (which is called local asymptotic stability or stability in the small). The stability properties should also remain valid within a sufficient range (i.e. the extent of [asymptotic] stability) around the considered equilibrium state.

In an analogous way, the stability of a motion may be studied, i.e. the trajectory of the output of a (non-linear) system. This implies that it should be known how far the actual trajectory may deviate from the trajectory for which it was designed.

2.2 Input–Output Stability

A second class of stability concerns the behaviour of the system when subjected to a given input signal. In this case, the question is whether the outputs y of the system, regarded as a black box, correspond to the expected behaviour of the system with the signals u as inputs (see Fig. 23.3). The expected behaviour can (for example) be described as an output between prescribed boundaries $y_{min}<y<y_{max}$ for an input between given boundaries $u_{min}<u<u_{max}$. If it is possible to transform the output variable y into a variable $y'$ so that the boundaries become $-\infty$ and $+\infty$, respectively, and if the input variable u can be transformed into a variable $u'$ with as boundaries $-\infty$ and $+\infty$, respectively, we obtain an equivalent dynamic system in which a bounded input needs to yield a bounded output if input–output stability is to be achieved.

Fig. 23.3

Input–output stability

3 Mathematical Tools to Explore the Stability of a System

For a linear system, the (asymptotic) stability can be investigated by looking at the eigenvalues of the system. If all eigenvalues are in the left half of the complex plane, the system is asymptotically stable. Eigenvalues in the right half plane indicate an unstable system. If eigenvalues on the imaginary axis are found and all other eigenvalues are in the left half plane, the system is stable but not asymptotically stable. For linear systems, the extent of stability is the whole state plane. At the same time, input–output stability is ensured.

For non-linear systems, the system may be linearised. When the linearised system is asymptotically stable, there is a neighbourhood of the equilibrium state in which the non-linear system is stable. However, this neighbourhood might be very small, and to explore the extent of stability methods like Lyapunov’s method need to be applied. Note, however, that Lyapunov’s method only offers sufficient conditions for stability, not necessary conditions.

If the linearised system shows eigenvalues in the right half plane, the non-linear system is unstable as to the envisioned equilibrium point. If the linearised system has eigenvalues on the imaginary axis, no conclusions can be drawn for the non-linear system.

For some classes of non-linear systems, more specific methods exist (e.g. the circle criterion). For more details, see Ref. [39].