Can someone please guide me towards any web link or provide an answer which explains in simple words what state space modelling (often heard in control theory) is and how is it different from other modelling techniques such as transfer functions?
I have tried to google my query but almost all the google results presented the content difficult to grasp.
where $\mathbf{Y}$ is the output vector and $\mathbf{U}$ is the vector of inputs (people often use X symbol than U for transfer function. But I use U for uniformity).
This model tells you the relationship between the input and output when the system is stimulated with a specific frequency. For example $$Y(s)=\frac{1}{1+5s}U(s)$$ means $$(1+5s)Y(s)=U(s)$$ You can easily transfer it to a time/domain as $$y(t)+5\frac{d}{dt}y(t)=u(t)$$ However, keeping it as original form, it tells you the frequency relationship. You just need to know complex numbers. Let's say we stimulate the system with frequency $f=3 Hz$. This corresponds to $\omega=2\pi f=6\pi \frac{rad}{s}$. The frequency variable $s=j\omega=j6\pi$ is a complex number and $j$ refers to the imaginary part (mathematicians show it by $i$ but electrical engineers show it by $j$ not to be confused by electric current). Now, $\frac{Y(j\omega)}{U(j\omega)}=\frac{1}{1+5j\times6\pi}$. Using complex number mathematics (if you have advanced calculator of use wolfram website) you can find it in polar system: $$\frac{Y(j\omega)}{U(j\omega)}=0.0332595\angle -86.1859 ^{\circ}$$ In another word, for a $3 Hz$ input frequency, the amplitude of output is $0.0332595$ times of the amplitude of input and its phase is $-86.1859 ^{\circ}$ behind. This means that, stimulating the system with $$u(t)=\cos(6\pi t)$$ will result in output of $$y(t)=0.0332595 \cos(6\pi t - 86.1859\times\frac{\pi}{180})$$.
If you know complex numbers this calculation will be pleasant to you. If you don't, it will be difficult. Transfer functions are easy to use when working with frequency and the system is linear. They are simple in control context. (I am not sarcastic). You can easily simplify a fraction in a transfer function. You can easily tell if a system is stable or not. But this representation is useless for nonlinear or time variant systems.
For a nonlinear time variant system, it is modeled as $$ \begin{cases} \mathbf{\dot{x}}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t),t)\\ \mathbf{y}(t)=\mathbf{g}(\mathbf{x}(t),\mathbf{u}(t),t) \end{cases} $$ This model is good at time domain. For example if you know the system input and system state from $t=7 s$, then you can estimate the output at $t=7.01 s$. Such a representation is so tangible and time domain friendly. It is very general and easy for software programmers to implement. However, is the system stable? Unlike transfer function, it is not very easy to tell. Still for linear systems it is easier to check the formulation. For, nonlinear ones, investigation of the stability is a disaster. But, there is not better option.