Dynamical Systems

What is a dynamical System?

How can we abstract the concept of system? We can think about it as a black box with a set of inputs and a set of outputs. Our system will generate a correlation between the stream of input and outputs. So we can try to define our system describing the way in which it relates inputs and outputs.

A useful way to do so is to define a function that describes the evolution of the state of our system. But how to define the state of the system? It can be described with a set of state variables at a specific time (e.g. temperature, position, velocity). From this perspective, the system time evolution can be characterized by a function that tell us how the state variables evolve.

System (1).svg

In the above diagram, we can recognise:

  • the input function $\mathbf{u}(\ldots,t) $: it expresses the evolution of the input values to the system. In general, its evolution in time determines the state of the system $s_{t_i}$ at a time $t_i$. It is important to notice that in general, the state of our system $s_{t_i}$ at some time $t_i$ depends on the time evolution of the input function $u(t)$ and not only on its value $u_{t_i}$ at time $t_i$.
  • the state transition function $\mathbf{s}(\mathbf{u}(\ldots,t),t) $: it describes how the state of the system (state variables) evolves depending on the input history and evolution (input function) and according to the system characteristics. 
  • the output function $\mathbf{y}(\ldots,s_{t_i},t)$: it gives the time evolution of the system outputs. Generally, it is an algebraic function of the state of the system $s_{t_i}$ at time $t_i$. 

The values assumed by the state variables in a generic time $t$ contain all the information on the past history of the system necessary to assess its the future trend (state evolution and output values). The concept of state allows us to avoid to take into consideration all the system history before time $t$. State variables are not uniquely defined: typically they exist infinite different ways to define the state of a system.

Continuous vs Discrete Systems

The input function $u(t)$ gives the inputs values at each time $t_i$, so we can write 

$$ \mathbf{u}(t) : T\to U $$

that means: the function $u$ associate at each element of the oriented time set $T$ one element of the input set $U$. 

We can have two cases:

  • $T \subseteq \mathbb{R}$: in this case, the system is called time-continuous;
  • $T \subseteq \mathbb{N}$: in this case, the system is called time-discrete;

In the same way, this distinction can be done from the space point of view. If for instance the state variable that describes the system is defined only at some points in space, we say that the system has a space-discrete, by contrast, if the state variable is defined in the continuous space the system is space-continuous. 

How to evaluate the state transition Function?

Usually, the state transition function is not known a priori and so, its computation becomes the key to determine how the system behave. Since the behaviour of the system is driven by some laws (i.e. physical, mathematical, chemical etc), the idea is to take advantage of these laws to find an expression for $s()$. Unfortunately, often is not possible to find an analytical expression for $s()$ because most of the time these laws are expressed by intricated integrodifferential equations that are impossible to solve analytically. 

example: conduction in a solid system, numerical evaluation

Imagine that you want to study the thermal state of a block of metal. In this case, you can recognise:

  • the input function $u$ is the boundary temperature field that changes in time;
  • the state function $s$ is the temperature variation over time of your metal;
  • the output function can be set as equal to the state function.

We do not know the explicit expression for $s()$ but what we can do is to use one of the fundamental law of physics to predict the evolution of $s()$ . In this case, we need the energy conservation law in a very simplified form: the law of conduction. Its expression is

$$ \frac{\partial T}{\partial t} = \alpha \nabla^2 T $$

 With $T$ the temperature and $\alpha$ the thermal diffusion coefficient. In the case in which the temperature at the boundaries is constant, the temperature of the system relaxes toward the steady state. In the latter case, the temporal derivative must be zero and this leads to the following equation ( in the 2D case):

$$ \frac{\partial T}{\partial x^2} + \frac{\partial T}{\partial y^2} = 0.$$ 

We propose to solve the above equation numerically, by discretizing the domain into a given number of cells with $ T (i, j) $ the temperature on the cell with $ (i, j) $ coordinates, and with the following numerical scheme:

$$T(i,j)_{k+1} = \frac{1}{4}\left[ T(i+1,j)_k + T(i-1,j)_k + T(i,j+1)_k + T(i,j-1)_k \right],$$

where $k$ is the number of the step or iteration. Taylor's expansion shows the correspondence between the two equations. After a sufficiently large number of iterations, if the value difference between $T_{k + 1}$ and $ T_k$ is sufficiently small, the scheme is said to converge.