There is this topic "Linear Control Systems" with vast literature. Nevertheless, I am still able to make further thoughts to bring some new ideas to it. For instance:
Given the time derivative $\dot{x}$ equal to vector span $A x + B u + d$, for bounded disturbance $\lvert\lvert d \rvert\rvert \leq \delta$, provide a control map $K \, x + \nu$ such that state timeseries $\lvert\lvert x(t) \rvert\rvert \leq \varepsilon$ for time $t \to \infty$ hold.
I know, the linear control map $K x$ with such eigenvalues $\lambda(A+BK) \in \mathbb{C}^-$ guarantee this condition asymptotically. However, I fail to find control map candidates $\nu$ to supress disturbances.
Any control map $\nu$ that satisfies the inequality below is valid:
$$\varepsilon > \exp((A+BK)t) + \int\limits_0^t \exp((A+BK)\tau) (d(\tau) + B \nu(t)) d\tau$$
Ok, so from my understanding, you would like to have
$$ \limsup_{t\to\infty}||x(t)||\le\epsilon $$ and assume that $\epsilon$ is left free, that is, we would like to find a suitable expression for it.
Let us also assume that $d$ is a constant. Then, we get that
$$ ||x(t)||\le \int_0^t||e^{(A+BK)(t-s)}(d+B\nu)||\mathrm{d}s\le M\dfrac{1-e^{-\alpha t}}{\alpha}||d+B\nu||\le \dfrac{M}{\alpha}||d+B\nu|| $$
where $M\ge1,\alpha>0$ are such that
$$ ||e^{(A+BK)t}||\le Me^{-\alpha t},\ t\ge0. $$
Those constant can be computed by solving the Lyapunov inequality
$$(A+BK)^TP+P(A+BK)\preceq-2\alpha P$$
in $P\succ0$ and $M$ will be given by $(\lambda_{\max}(P)/\lambda_{\min}(P))^{1/2}$ where $\lambda_{\max}(P)$ and $\lambda_{\min}(P)$ are the maximum and minimum eigenvalue of $P$, respectively.
Since the terms $M/\alpha$ and $||d+B\nu||$ are independent, and only the latter depends on $\nu$. Let us just focus on this one.
So, in the end, what we would like is to find $\nu$ that solves
$$ \min_\nu\max_{||d||\le\delta}||d+B\nu||. $$
Assuming that $B$ is full-column rank, it can be shown that the optimal value is $\nu^*=0$. This makes sense since it is a neutral value that may not add to the value for $d$.
As a result, since the disturbance $d$ is not known, it is not possible to "suppress" it using $\nu$. The only thing that can be done is to attenuate it, that is having a ratio $M/\alpha$ as small as possible. You may check the literature on the $L_\infty$-gain and the $L_1$-control, which addresses such a problem.