Optimal control requires that your dynamic model of the system be perfect, whereas in reality your model is going to be nowhere close to the actual system. An incorrect model could even result in unstable system behavior.
Robust control on the other hand renders your control law impervious to modelling uncertainties thereby allowing for a realistic margin of error. The simplest and most effective robust control strategy is sliding mode control, SMC. In SMC, denote by $s \in \mathbb{R}$ a linear combination of the system's error in state, i.e.:
$$s = \sum_{r = 0}^{n-1}\sigma_re^{(r)} \tag{1}$$
$\begin{align}\text{where,}& \\ &e \in \mathbb{R} : e= x - x_d \text{ is the error in system state} \\ &x_d \text{ is the desired state} \\ &e^{(r)}\text{ represents the r}^{th}\text{ derivative wrt time of }e \\ &\sigma_r\text{ are chosen such that the monic polynomial formed using them as coefficients is Hurwitz}\end{align}$
The insensitivity of SMC to model uncertainties becomes clear if we consider the following problem:
Consider that the system in consideration is given by the following dynamics,
$$x^{(n)} = f(\textbf{x}) + g(\textbf{x})u\tag{2}$$
$\begin{align}\text{where,} & \\ &\textbf{x}\in\mathbb{R}^{(n-1)}: \textbf{x} = \begin{bmatrix}x \ \dot{x} \ .... \ x^{(n-1)}\end{bmatrix}^{T} \text{ is the system state} \\ &f, g \in \mathbb{R}\text{ are scalar functions assumed smooth} \\ &u \text{ is the control input}\end{align}$
For conciseness, assume that $g(\textbf{x}) = 1$, but the derivation extends trivially to the general case, and that the following relation is known,
$$\mid\text{ }f - \hat{f}\mid\text{ } \leq \text{ F} \tag{3}$$
where, $\text{ }F\text{ is a constant , }\hat{f}\text{ is our estimate on the dynamics, i.e., on f. }$
Now taking derivative of (1):
$$\begin{align}&\dot{s} = x^{(n)} - x_d^{(n)} + \sigma_{n-2}e^{(n-1)}\text{ + ...} \\ \implies\text{ } & \dot{s} = f + u - v\tag{from (2)}\end{align}$$
where, $v = x_d^{(n)} - \sigma_{n-2}e^{(n-1)}\text{ + ...}$
Let u = $v - \hat{f} -Ksgn(s)$ where K is a constant whose value will be made clear further. Now consider the Lyapunov function,
$$\begin{align} & V = \frac{1}{2}s^2 \\ \implies\text{ } & \dot{V} = s\dot{s} \\ \implies\text{ } &\dot{V} = s(f + u - v) \\ \implies\text{ } & \dot{V} = s(f - \hat{f} - Ksgn(s))\end{align}$$
if we chose, K = F + $\eta$, where $\eta > 0$ then,
$$\dot{V} \leq -\eta\mid{s}\text{ }\mid$$
Thus, by Lyapunov theory we have finite convergence of $s$ to $0$. How does this help? When $s = 0$, from (1), $$e^{(n-1)} + \sum_{r = 0}^{n-2}\sigma_re^{(r)} = 0$$
which represents a stable error dynamics, which asymptotically dies down. Thus, we have formulated a control law that does its best to "cancel" out the dynamics (by using our estimate) and then we have also taken care of the uncertainty.
There is tons of literature on SMC and it is definitely worth a read. Rate of convergence is decided by your choice of the parameters and you cannot really compare it to optimal controllers as its a whole different ballpark.
The Kalman filter is indeed widely used, but keep in mind that it's an estimator, not a controller. If the system is observable, then you can estimate the state with a KF and use it for feedback, but the KF alone does not give you any gains.
– JMJ Jul 10 '17 at 20:06