Stability concept of linear control system

Stability of Linear Control Systems

1.1 Definitions of Stability

In the Web Dictionary of Cybernetics and Systems (by F. Heylighen) there are some

definitions for the term ‘stability’ of a control system as follows:

1. Stability: The tendency of the variables or components of a system to remain within defined

and recognizable limits despite the impact of disturbances. (Young, p. 109) 

2. Expanded or global stability: The ability of a system to persist and to remain qualitatively

unchanged in response either to a disturbance or to fluctuations of the system caused by a

disturbance. This idea of stability combines the concepts of traditional stability and Holling's

new concept of resilience. (Holling) 

3. Stability: The capacity of an object or system to return to equilibrium after having been

displaced. Note with two possible kinds of equilibrium one may have a static (linear) stability

of rest or a dynamic (non-linear) stability of an endlessly repeated motion. (Iberall) 

4. Stability: System is stable if, when perturbed, it returns to its original state. The more

quickly it returns, the more stable it is.

1.2 Concepts of Stability

Stability is probably the most important consideration when designing control systems. One

of the most important characteristics of a control system is that the output must follow the

desired signal as exactly as possible. The stability of a system is determined by the form of

the response to any input or disturbance. Absolute stability refers to whether a system is stable

or unstable. In a stable system the response to an input will arrive at and maintain some useful

value. In an unstable system the output will not settle at the desired value and may oscillate or

increase towards some high value or a physical limitation. For example, a system is stable if

the response to an impulse input approaches zero as time approaches infinity.

In the context of linear control systems, stability would be intuitively reasonable to define a

linear system to be stable if its output is bound for every bounded input, namely bounded

input bounded output (BIBO) stability. The system is said BIBO stable. This definition states

that stability is dependent on the system itself. The properties or dynamic behaviours of a

system are characterised by its transfer function G(s).

The input to a system will not affect or determine its stability. The components of the system

provide its characteristics, and hence determine stability. The solution to the differential

equation describing a system is made up of two terms, a transient response and a steady-state

response. For stability the transient response terms must all die away as time progresses. The

coefficients of the exponential terms must therefore be negative real numbers or complex

numbers with negative real parts.

The stability or instability of a closed-loop control system is determined by the poles of its

transfer function. The system is stable if the function response y(t) remains bounded as the

time (t) tends to infinity. For most control engineering purposes an even stronger concept of

stability if required: 2

Asymptotically stable: A system is said to be asymptotically stable if its response decays to

zero as t tends to infinity.

Marginally stable: as mentioned in previous section an undamped second order system has its

response that is indefinitely oscillatory. This is an example of a system that is stable but not

asymptotically stable. In fact, this system is known as marginally stable. Otherwise, the

system is said unstable.

For analysis and design purposes stability can be classified in absolute stability and relative

stability. As stated above, absolute stability refers to the condition of whether the system is

stable or unstable; it is a yes or no answer. Once the system is found to be stable, it is of

interest to determine how stable it is, and this degree of stability is a measure of relative stability.