Signals and Systems: Deriving differential equation from block diagram

[KingNothing]

Homework Statement

Consider the block diagram below. Find the differential equation of the system.

Homework Equations

None, other than understanding how block diagrams work.

The Attempt at a Solution

My attempt is posted below, as an image. The reason I am having such trouble is because I can only express things in terms of the signal node I have labeled 'd'. Ideally, I should be able to find an expression just involving y(t) and x(t), and their derivatives/integrals.

 

[Valkarie]

Is there any way to do this? I don't think so, but I could use a second opinion. Thanks!\begin{array}{l}\text{x(t)} \\\downarrow \;\;\;\;\; \searrow \;\;\;\;\; \nearrow \\\text{d} \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; \;\;\;\;\; y(t) \\\end{array}\begin{array}{l}\frac{d^2y(t)}{dt^2} &+& 4\frac{dy(t)}{dt} &+& 5y(t) &=& d(t) \\\end{array}

 

[Mene]

I would approach this problem by first understanding the basic principles of signals and systems. A signal is a physical quantity that varies over time, and a system is a physical device or process that operates on signals to produce an output. In this case, the block diagram represents a system that takes in an input signal x(t) and produces an output signal y(t).

To derive the differential equation of this system, we need to understand how the different blocks in the diagram affect the input signal. The first block represents a gain, which multiplies the input signal by a constant factor. The second block is a delay, which introduces a time delay in the input signal. Finally, the third block is an integrator, which integrates the input signal over time.

Using this information, we can write the following equation:

y(t) = kx(t-T) + \int x(t) dt

Where k is the gain constant, T is the time delay, and the integral represents the output of the integrator block. However, this equation still involves the signal node 'd' and does not solely depend on y(t) and x(t) as desired.

To eliminate the signal node 'd', we can use the fact that the output of the integrator block is the input signal multiplied by the time variable t. This can be expressed as:

\int x(t) dt = tx(t)

Substituting this into the previous equation, we get:

y(t) = kx(t-T) + tx(t)

We can further simplify this equation by factoring out x(t):

y(t) = (k + t)x(t-T)

Finally, we can differentiate both sides of the equation with respect to time to get the desired differential equation:

\frac{dy(t)}{dt} = (k + t)\frac{dx(t-T)}{dt}

In conclusion, by understanding the basic principles of signals and systems and carefully analyzing the block diagram, we were able to derive the differential equation of the system. This approach can be used to derive the equations for more complex systems as well.