## Classification of systems

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Definition of system

System can be considered as a physical entity which manipulates one or more input signals applied to it. For example a microphone is a system which converts the input acoustic (voice or sound) signal into an electric signal. A system is defined mathematically as a unique operator or transformation that maps an input signal in to an output signal. This is defined as y(t) = T[x(t)] where x(t) is input signal, y(t) is output signal, T[] is transformation that characterizes the system behavior.

There are many classifications of systems based on parameter used to classify them. They are

a) Linear, non linear systems

b) Time variant, time invariant systems

c) Stable, unstable systems

d) Causal, non causal systems

e) Continuous time, discrete time systems

f) Invertible and noninvertible systems

g) Dynamic and static systems.

Linear, nonlinear systems

A linear system is one which satisfies the principle of superposition and homogeneity or scaling.

Consider a linear system Ϭ characterized by the transformation operator T[]. Let x1, x2 are the inputs applied to it and y1, y2 are the outputs. Then the following equations hold for a linear system

y1 = T[x1], y2 = T[x2]

Principle of homogeneity: T [a*x1] = a*y1, T [b*x2] = =b*y2

Principle of superposition: T [x1] + T [x2] = a*y1+b*y2

Linearity: T [a*x1] + T [b*x2] = a*y1+b*y2

Where a, b are constants.

Linearity ensures regeneration of input frequencies at output. Nonlinearity leads to generation of new frequencies in the output different from input frequencies. Most of the control theory is devoted to explore linear systems.

Time variant, time invariant systems

A system is said to be time variant system if its response varies with time. If the system response to an input signal does not change with time such system is termed as time invariant system. The behavior and characteristics of time variant system are fixed over time.

In time invariant systems if input is delayed by time t0 the output will also gets delayed by t0. Mathematically it is specified as follows

y(t-t0) = T[x(t-t0)]

For a discrete time invariant system the condition for time invariance can be formulated mathematically by replacing t as n*Ts is given as

y(n-n0) = T[x(n-n0)]

Where n0 is the time delay. Time invariance minimizes the complexity involved in the analysis of systems. Most of the systems in practice are time invariant systems.

Note: In describing discrete time systems the sampling times n*Ts are mentioned as n i.e. a discrete signal x(n*Ts) is indicated for simplicity as x(n).

Stable, unstable systems

Most of the control system theory involves estimation of stability of systems. Stability is an important parameter which determines its applicability. Stability of a system is formulated in bounded input bounded output sense i.e. a system is stable if its response is bounded for a bounded input (bounded means finite).

An unstable system is one in which the output of the system is unbounded for a bounded input. The response of an unstable system diverges to infinity.

Causal, non causal systems

The principle of causality states that the output of a system always succeeds input. A system for which the principle of causality holds is defined as causal system. If an input is applied to a system at time t=0 s then the output of a causal system is zero for t<0. If the output depends on present and past inputs then system is casual otherwise non casual.

A system in which output (response) precedes input is known as Non causal system. If an input is applied to a system at time t=0 s then the output of a non causal system is non zero for t<0.  Such systems are referred as non-anticipative as the system output does not anticipate future values of input. Non causal systems do not exist in practice.

Continuous time, discrete time systems

A system which deals with continuous time signals is known as continuous time system. For such a system the outputs and inputs are continuous time signals.

Discrete time system deals with discrete time signals. For such a system the outputs and inputs are discrete time signals.

Invertible and non-invertible systems

A system is said to be invertible if distinct inputs lead to distinct outputs. For such a system there exists an inverse transformation (inverse system) denoted by T-1[] which maps the outputs of original systems to the inputs applied. Accordingly we can write

TT-1 = T-1T = I

Where I = 1 one for single input and single output systems.

A non-invertible system is one in which distinct inputs leads to same outputs. For such a system an inverse system will not exist.

Dynamic and static systems

In static system the outputs at present instant depends only on present inputs. These systems are also called as memory less systems as the system output at give time is dependent only on the inputs at that same time.

Dynamic systems are those in which the output at present instant depends on past inputs and past outputs. These are also called as systems with memory as the system output needs to store information regarding the past inputs or outputs.