0. Question
tl;dr: just see section 2. c)
I'd like to know how a non-linear system of first order ODEs (non-linear dynamical system) which was linearized at a point which is not the equilibrium of the dynamical system, can be discretized exactly.
I know that for a linear system linearized at an equilibrium state but struggle with the exact discretization of the $0$-order term of the Taylor approximation.
The resources I could find so far, all cover only the case of discretizing a (linearized) linear system without the $0$-order term of the Taylor linearization.
Below I have composed where I am and where I struggle at the linearization.
1. Findings: General Linearization
Be $f$ a non-linear time-invariant dynamical system, $x\in \mathbb{R}^n$ the state vector, $u\in \mathbb{R}^m$ an arbitrary input vector: $$ \dot{x}(t)=f(x(t),u(t)) $$ Let the tuple $(\bar{x},\bar{u})$ be the linearization point and $x_{\delta}(t) = x(t) - \bar{x}$ and $u_{\delta}(t) = u(t) - \bar{u}$ be the delta-state resp. the delta-input, the linearization of $f$ (ignoring Taylor terms of order 2 and higher) is given as $$ \dot{x}(t) = A x_{\delta}(t) + B u_{\delta}(t) + \bar{f} \\ \text{with} \quad A = \left. \frac{\partial}{\partial x} f(x,u) \right|_{\substack{x = \bar{x} \\ u = \bar{u}}}, \quad B = \left. \frac{\partial}{\partial u} f(x,u) \right|_{\substack{x = \bar{x} \\ u = \bar{u}}} \quad \text{and} \quad \bar{f} = f(\bar{x},\bar{u}) \text{.} \\ $$ Note: $\bar{f} \neq 0$ as $(\bar{x},\bar{u})$ is not constrained to be an equilibrium point.
2. a) Findings: Exact Discretization of linear system
A linear system of the form $$ \dot{x}(t) =A x(t) + B u(t) $$ can be linearized exactly as $$ x[k+1]=A_d x[k] + B_d u[k]\\ \text{with} \quad A_d = e^{A_{l}T} \quad \text{and} \quad B_d = \int\limits_{0}^{T} e^{A_{l}\tau} B_{l} d \tau \text{.} $$
2. b) Struggling: Exact Discretization of linearized system
Suppose I have this linearized system, $$ \dot{x}(t) =A x(t) + B u(t) + \bar{f} $$ I now struggle to with finding a term $f_d$ analogous to $A_d$ and $B_d$ for a discrete linear system of the form $$ x[k+1]=A_d x[k] + B_d u[k] + f_d $$
2. c) Guess for approach: Solve differential equation of linear system??
I also know the solution of differential equation of the linear state space system $$ \dot{x} = Ax(t) + Bu(t), \quad x(t_0) = x_0 $$ which is $$ x(t) = e^{A(t-t_0)}x_0 + \int_{t_0}^{t} e^{A(t-\tau)}Bu(\tau) d\tau \text{.} $$ and very likely must have been used in the exact linearization of the system in section 2a.
I'm not sure, but maybe solving the differential equation of this system $$ \dot{x}(t) = A x(t) + B u(t) + \bar{f} $$ might yield to a term for $f_d$? Would this be the right approach (and if yes, how would you approach to solve this ODE)?
Another course for mistakes might be messing up the notation of the delta-states and -inputs because when discretizing a linearized system, this linearized system is in its delta-notation as written in 1.
I appreciate any comments and thank you for your help!
The solution of the initial value problem $$ \dot x= Ax+f(t),\; x(t_0)=x_0 $$ is equal to $$ x(t)=e^{A(t-t_0)}x_0+\int\limits_{t_0}^te^{A(t-\tau)} f(\tau)\,d\tau. $$ Using this fact, we can obtain $$ x[k+1]=e^{AT}x[k]+\int\limits_{t_0}^te^{A(t-\tau)} (Bu[k]+\bar f)\,d\tau $$ $$ =e^{AT}x[k]+\int\limits_{0}^Te^{A(T-\tau)} B\,d\tau\,\cdot u[k]+ \int\limits_{0}^Te^{A(T-\tau)}\bar f \,d\tau $$ $$ =e^{AT}x[k]-\int\limits_{0}^Te^{A(T-\tau)} B\,d(T-\tau)\,\cdot u[k]- \int\limits_{0}^Te^{A(T-\tau)} \bar f \,d(T-\tau)\, $$ $$ =e^{AT}x[k]+\int\limits_{0}^Te^{A\tau} B\,d\tau\,\cdot u[k]+ \int\limits_{0}^Te^{A\tau} \bar f\,d\tau. $$ Hence, $$ f_d= \int\limits_{0}^Te^{A\tau} \bar f\,d\tau. $$
(Let me also note that this result is likely unuseful to control the original nonlinear system. Since $\bar f\ne 0$, $x[k+1]$ may differ a lot from $x[k]$ and $\bar x$, thus $A_d$, $B_d$, $f_d$ that we have found for the point $\bar x$ can become inadequate)