A system is described by the differential equation, ty' (t) ? 8y(t) = x(t). Classify the system as to linearity, time invariance and BIBO stability.
What will be an ideal response?
Homogeneity:
If we multiply the first equation by K, we get
To satisfy this equation the derivative of “y” times “t” must be of the same functional form as “y” itself. This is satisfied by a homogeneous solution of the form,
If there is no excitation, but the zero-excitation response is not zero, the response will increase without bound as time increases.
BIBO Unstable
Causality:
The system equation can be rewritten as
So the response at any time depends on the excitation at times and not on any future values.
Causal
Memory:
The response at any time t =t0 depends on the excitation at times t
System has memory.
Invertibility:
The system equation,
ty'(t)-8y(t) = x(t)
expresses the excitation in terms of the response and its derivatives. Therefore the excitation is uniquely determined by the response.
Invertible.
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