# Stability

Stability is one of the most important properties of dynamic systems, especially when designing efficient control systems. Therefore a great deal of effort has been focused towards proper definition and testing of system stability and feasible stabilization of unstable processes.

## Definition

Although all people naturally understand the concept of stability and are able to describe what stable behaviour is and what is not (see the classical example of a ball in the gravitational field below), a proper mathematical definition is not so straightforward. Generally, stability can be formulated as ability to recover from perturbations - short-time disturbances or non-zero initial conditions, i.e. ability to return to an equilibrium position which can be different from the previous one.

There are many definitions of stability according to the properties of a system to be analyzed (linear / nonlinear, continuous-time / discrete-time, time-invariant / time-variant, ...). Most used are probably the so-called BIBO stability and Lyapunov stability.

• BIBO stability, i.e. "Bounded Input - Bounded Output" stability, defines a stable system when a bounded input signal into the system produces a bounded output signal from the system.
• Lyapunov stability states, simply speaking, that a system is stable if its output and all states are bounded and converge asymptotically to zero from sufficiently small initial conditions.

## Testing

When it comes to the testing of stability, again there are many methods (both numerical and graphical) and usage of a particular method depends on the properties of a system to be analyzed (linear / nonlinear, continuous-time / discrete-time, time-invariant / time-variant, ...) and on a type of description at hand.

From a number of sources in English where the stability topic is discussed let us choose at least the following ones:

[1]  Willems, J.L. Stability Theory of Dynamical Systems. New York: Wiley, 1970.

[2]  Parks, P.C. A.M. Lyapunov's stability theory - 100 years on. IMA Journal of Mathematical Control & Information, vol. 9, no. 4,

pp. 275-303, 1992.

[3]  Skogestad, S., Postlethwaite, I. Multivariable Feedback Control: Analysis and Design. Chichester: Wiley, 2005.

[4]  Åström, K.J., Murray, R.M. Feedback Systems: An Introduction for Scientist and Engineers. Princeton University Press, 2008.

[5]  Doyle, J.C., Francis, B.A., Tannenbaum, A.R. Feedback Control Theory. Dover Publications, 2009.

## Stabilization

Unstable processes can be stabilized by feedback. There are many sources devoted to the issue of control system design for unstable processes. Some starting interesting publications are listed here:

[6]  Padma Sree, R., Chidambaram, M. Control of unstable systems. Oxford: Alpha science Int. Ltd., 2006.

[7]  Stein, G. Respect the unstableIEEE Control systems magazine, vol. 23, no. 4, pp. 12-25, 2003.

[8]  Skogestad, S., Havre, K., Larsson, T. Control limitations for unstable plants. In Proceedings of the 15th Triennial World

Congress, IFAC, Barcelona, Spain, pp. 485-490, 2002.

[9]  Middleton, R.H. Trade-offs in linear control system design. Automatica, no. 27, pp. 281-292, 1991.

## More...

Besides "yes/no" stability testing it is often important to test and ensure certain measure of stability, i.e. relative stability which gives answer to the question how far the system is from the instability. In control engineering the so called gain and phase margins are frequently used for this purpose. Details on this subject can be found in most books focused on control engineering, e.g. [3], [4], [5] listed above.

Next important term in control engineering is the so called robust stability which is useful for the case we require to test/achieve stability not only for one system but for a certain class of systems. Typically a nominal system and some its neighbourhood, which is useful in the case of uncertain process models. Interested readers are referred to classical books devoted to the robust system design:

[10]  Barmish, B.R. New Tools for Robustness of Linear Systems. Macmillan, 1994.

[11]  Bhattacharyya, S.P., Chapellat, H., Keel, L.H. Robust Control - The Parametric Approach. Prentice-Hall, 1995.