What is the difference between $\dot{x} = Ax + b +Bu$ and $\dot{x} = Ax + Bu$, where b is a constant vector?

Question

As an engineer, when studying control theory, a linear system is often described as $\dot{x} = Ax + Bu$.

However, when reading about geometry of control spaces or on system abstraction on full-dimensional polytopes, the linear system is defined as $\dot{x} = Ax + b +Bu$, where b is a constant vector.

In the work I have read so far, the b does not play a significant role. So, I was wondering, why this b is introduced in the system definition? What is this representation motivated by?

Answer 1

If you read the equation as

$$\dot x-Ax=b+Bu,$$

$b$ is nothing but a constant excitation, i.e. a translation of the equilibrium point.

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As you probably know, we can get rid of it with $y:=x+A^{-1}b$, giving

$$\dot y-Ay=Bu.$$

Answer 2

I would imagine it is to create a more general system. By this I mean it is a system model that is explicitly taking into account a reference input or a fixed point the system naturally operators around. I believe this is what the comment above is alluding to, as it is technically equivalent to a shift of coordinates.

If a system, $Ax+Bu$, has a control law, $u$, designed such that $x\rightarrow 0$. Then by definition the closed loop system is stable i.e. $eig(A-BK)<0$. The similar system defined by $Ax+Bu +b$ will have $x\rightarrow b$ for a $K$ such that $eig(A-BK)<0$. Therefore, any analysis done on one system also holds for the other.