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What are the general forms of first and second order dynamic systems?

by CraigH
Tags: dynamic, forms, order, systems
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CraigH
#1
Apr11-14, 05:15 PM
P: 202
What is the general form of a first and second order dynamic system, in laplace and time space?

I have some of the information, but a lot of it is missing and I can't find a clear answer on the web.

The general form of a second order dynamic system is:

$$\frac{d^2x(t)}{dt^2}+2\zeta\omega_n\frac{dx(t)}{dt}+\omega_n^2x(t)=f( t)\:\:\:\:\:\:\:[1]$$

where

x(t) - is the output, e.g current
f(t) - is the input, e.g a voltage signal
zeta - is the damping coefficient
wn - is the natural frequency


An example of a second order dynamic system is a RLC circuit. If a resistor, capacitor, and inductor are connected to a power source with voltage at time t equal to f(t), the summed voltage across the three components always equals f(t) at any time t.

$$Ri(t)+\frac{1}{C}\int_0^ti(t)dt+L\frac{d\,i(t)}{d\,t}=f(t)\:\:\:\:\:\ :[2]$$

So for example if f(t) is a step input from 0 to 6V, the equation will be:

$$Ri(t)+\frac{1}{C}\int_0^ti(t)dt+L\frac{d\,i(t)}{d\,t}=6\:\:\:\:\:\:\:[3]$$

If this equation is rearranged so that it is in the general form shown in equation [1], then you will be able to find out the natural frequency of the system (omega_n) and the damping coefficient of the system (zeta).

The general form of a first order dynamic system in laplace space is:

$$G(s)=\frac{k(s+a)}{s+b}\:\:\:\:\:\:\:[4]$$

where

G(s) - is the transfer function (output/input)
k - is the "gain" of the system
a - is the zero of the system, which partially determines transient behavior
b - is the pole of the system, which determines stability and settling time

An example of a first order dynamic system is a RC circuit. The voltage across the resistor plus the voltage across the capacitor equal the voltage of the source f(t) at any time t.

$$Ri(t)+\int_0^ti(t)dt=f(t)\:\:\:\:\:\:\:[5]$$

So again, for a step input from 0 to 6V, the equation will be:

$$Ri(t)+\int_0^ti(t)dt=6\:\:\:\:\:\:\:[6]$$

In laplace space this is

$$RI(s)+\frac{1}{Cs}I(s)=\frac{6}{s}\:\:\:\:\:\:\:[7]$$

rearrange to general form:

$$I(s)*(Rs+\frac{1}{c})=6\:\:\:\:\:\:\:[8]$$

$$I(s) =\frac{6}{Rs+\frac{1}{c}}\:\:\:\:\:\:\:[9]$$

$$I(s) =\frac{\frac{6}{R}}{s+\frac{1}{Rc}}\:\:\:\:\:\:\:[10]$$

Since the input was the step input 6, you can see that the steady state gain is equal to 6/R, the pole is equal to 1/RC, and there are no zeros. You could do the inverse laplace transform of this equation to find out the instantaneous current i(t) at any time t.

Now my questions:

- I have shown the general form of a first order dynamic system in laplace space, and the general form of a second order dynamic system in time space, but what is the general form of a first order system in time space, and a second order system in laplace space? I have definitely seen these written somewhere but I can't find them in my notes or online.

- How would you go from equation [3], to the general form (time space) as shown in equation [1]?

Any answer is much appreciated, even if it only answers one of the questions, or partially answers a question. Thanks for reading!
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