# 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 unstable. *IEEE 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 15 ^{th} 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.