Dynamic Systems

Properties of LTI systems

Linear time invariant (LTI) dynamic systems are modeled by LTI differential equations.  Their response (or the solution of their differential equation model) must satisfy two main rules: 

1)      Their response must satisfy the principle of superposition.

2)      Their response can be expressed as the convolution of their inputs with their impulse response (the response of the system to an impulse input).  

The principle of superposition states that if the response of an LTI system to two inputs u₁(t) and u₂(t) are, respectively, y₁(t) and y₂(t).  Then the response of this system to a linear combination of these two inputs must be the same linear combination of their corresponding responses.  That is, the response of the system to the input u(t)=a₁u₁(t)+a₂u₂(t) must be y(t)=a₁y₁(t)+a₂y₂(t), where a₁ and a₂ are constant real numbers.