Implicit integration - the linear case

Consider the linear system:

$$\dot x = A x + B u$$ (2.1)

For the purposes of simulation, it can be discretised (with sample interval $\Delta t$) in at least two ways:

1. $\dot x \approx \frac{x_{i+1} - x_{i}}{\Delta t}$
2. $\dot x \approx \frac{x_{i} - x_{i-1}}{\Delta t}$

The former is gives rise to the forward Euler or explicit integration scheme:

$$x_{i+1} = x_{i} + \Delta t \left [ A x_{i} + B_{i} u \right ]$$ (2.2)

and the latter gives rise to the backward Euler or implicit integration scheme:

$$x_{i} = x_{i-1} + \Delta t \left [ A x_{i} + B_{i} u \right ]$$ (2.3)

which must be rewritten as:

$$x_{i} = \left [ I - \Delta t A \right ]^{-1} x_{i-1} + \Delta t B_{i} u$$ (2.4)

for the purposes of implementation.

The explicit method gives simple implementation whereas the implicit method requires matrix inversion. However, the explicit method is only stable if:

$$\Delta t < \frac{2}{\vert \lambda \vert}$$ (2.5)

where $\lambda$ is the largest eigenvalue of $A$. If this largest eigenvalue is real so $\lambda = \frac{1}{\tau}$ where $\tau$ is the smallest system time constant:

$$\Delta t < 2 \tau$$ (2.6)

If the system is stiff, that is it contains at least one small time constant relative to the dominant time constants, Euler integration is not feasible due to the very small sample interval $\Delta t$ required.

In contrast, the implicit method is stable.