System Dynamical Model

System:

• Something changes (evolves) with time

Control:

• Something which influences this change

Dynamical Model:

A dynamical Model of a system, is a model which describes how a system evolves with time.

To describe how a system evolves with time, or we can say changes with time:

we can use:

• Differential Equation
• Initial condition

Where:

• It is known that the differentiation is something that describes change over time.
• And, initial condition describes a specific state of the system.

Thus, with these 2 points, I can describe the system at a certain point of time, and I can describe how this system changes with time. 

          Hence, that's it, I described the system !

Solving the dynamical model, can be done:

A) Numerically:

By discretizing the time, I can use Taylor expansion:

X((k+1)*δt) = X(k*δt) + δt * X'(k*δt) + .... 

From the initial condition, we know X(k*δt)

From the differential equation, we know X'(k*δt) in terms of X(k*δt)

Hence, we can calculate X of k+1, from which we can get X of K+2, ... etc

B) Analytically (Integration):

To get the mathematical expression from the dynamical model:

• Integrate the differential equation to get X(t) from X'(t)
• Use the initial condition to get the integration constant

Equilibrium Point:

• X'(t) = 0