I have two questions about control theory. I am looking for an appropriate theory to my master thesis.
Let $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})$, $\mathbf{f}: \mathcal{X} \times \mathcal{U} \to \mathcal{X}$ a function of $\mathcal{C}^{\infty}$. Then:
Is it possible to convert the proposed system into the sampled form $\mathbf{x}_{k+1} = \mathbf{f}_{\rm{T}}(\mathbf{x}_k, \mathbf{u}_k)$, $\rm{T}$ the sample time to convert the system into .
Given $\mathbf{x}_{k+1} = \mathbf{f}_T(\mathbf{x}_k, \mathbf{u}_k)$, is there a map $\mathbf{x}_k = \mathbf{\gamma}(\mathbf{z}_k, \mathbf{u}_k)$ $\mathbf{\gamma}: \mathcal{X} \times \mathcal{U} \to \mathcal{U}$ and diffeomorfic map $\phi: \mathcal{X} \to \mathcal{X}$ to convert the system into $\mathbf{z}_{k+1} = \mathbf{A} \mathbf{z}_{k} + \mathbf{B} \mathbf{u}_{k}$.
The method must be in a systematic computational form to be implemented. Adopt the necessary assumptions e.g. $\mathbf{f}$ is lipschitz or the system must be in the afine form. If an well stablished tool as well as paper script is available, please, I ask for make me know it.
Best regards
In order to derive the discrete time system, we need to be able to predict $x_{k+1}$ from $x_k$ and $u_k$. For a linear time-invariant system $\dot{x}=Ax+Bu$, we are able to derive a closed form solution $$x(t) = \exp((t-t_0)A)x(t_0)+\int_{t_0}^{t}\exp((\tau-t_0))Bu(\tau)d\tau$$ and use this to determine the discrete time version of the system. This can be achieved by setting $t=t_0+T_\text{sampling}\to k+1$ and $t_0\to k$ It is evident that we can only find the exact solution for a very limited set of nonlinear systems. Hence, in general, we will not be able to find the exact discrete time version of the system. For nonlinear systems we have
$$\dot{x} = f(x,u) \implies x(t) = x(t_0) +\int_{t_0}^tf(x(\tau),u(\tau))d\tau.$$
If you are able to solve the integral you can quickly come up with the discrete time version of the system. As Arastas proposed you can try to approximate the integral, but that is just an approximation and not an exact version of the discrete time system.
It should be clear that we will not able able to always find such a solution but the paper Exact Linearization of Discrete-Time Nonlinear Systems Using State Space Transformation (Jayaraman & Chizeck, 1995) has an example of a system for which this can be achieved. The system is
$$ x_{1,k+1} = (1+x_{1,k})x_{2,k}+u_k\\ x_{2,k+1} = \dfrac{x_{1,k}}{1+u_k+(1+x_{1,k})x_{2,k}}. $$
the transformation (I will drop the $k$ in the index) is
$$ z_1 = (1+x_1)x_2\\ z_2 = x_1. $$
By this, you can transform the system into the linear time-invariant system
$$ z_{1,k+1} = z_{2,k}\\ z_{2,k+1} = z_{1,k}+u_k. $$
The authors of the paper also describe a method for systematically deriving such transformations.