transfer function - 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 cant 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:

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  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.

  
  


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  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!

Answer

You want to use the following property of Laplace transform:

$$\mathscr{L}\left({\frac{dx(t)}{dt}}\right)(s) = s\mathscr{L}\left(x(t)\right)-x(0)$$

This allows you to easily move between differential equations and polynomial equations.

Time domain to frequency domain: Take your first equation for example

$$\frac{d^2 x(t)}{dt^2} + 2 \zeta \omega_n \frac{dx(t)}{dt} + \omega_n^2 x(t) = f(t).$$

If we denote the Laplace transform of $x(t)$ by $X(s)$ and $f(t)$ by $F(s)$ and apply Laplace transform to this equation then this property implies

$$s^2 X(s) + 2 \zeta \omega_n sX(s) + \omega_n^2 X(s) = F(s)$$ where for simplicity I assume that $x(0) = f(0) = 0$.

The transfer function is defined as the ratio:

$$\frac{X(s)}{F(s)} = \frac{1}{s^2+2 \zeta \omega_n s + \omega_n^2}$$

Frequency domain to time domain: Lets try the example $G(s)=\frac{1}{s^3+1}$ then we have by definition that

$$G(s)F(s) = X(s)$$ which implies $$F(s) = s^3X(s)+ X(s).$$

Taking inverse Laplace transform we find that

$$f(t) = \frac{d^3 x(t)}{dt^3} + x(t).$$

Hopefully this allows you to see the pattern in general.