= Robust control
\b[Robust control] designs or assesses a <controller> against a specified family of possible <controlled plants>, rather than only one nominal model. <Control model uncertainty> can represent uncertain parameters or unmodeled dynamics. <Robust stability> asks whether every allowed plant gives a stable closed loop; <robust performance> also requires a quantitative response criterion throughout the allowed family. Stability alone does not guarantee the required performance.<Robust Stability, Robust Performance, and Mu Analysis in MATLAB documentation>[Robust Stability and Robust Performance]{ref}
An original scalar example makes the quantifiers explicit. Consider a <controlled plant> with measured state $x$ and <control input> $u$:
$$
\dot x=ax+u,\qquad a\in[1,3],\qquad u=-kx.
$$
The same <feedback gain> $k$ must be chosen before knowing $a$. Its solution is $x(t)=x(0)e^{(a-k)t}$. At the nominal value $a=2$, <asymptotic stability> requires only $k>2$. Choosing $k=2.5$ stabilizes that nominal model but fails at the allowed value $a=3$, where deviations grow as $e^{0.5t}$. Uniform stability over the entire closed interval instead requires
$$
k>3.
$$
At $k=3$, the endpoint plant has $\dot x=0$ and does not decay, so the strict inequality matters.
Now specify the performance bound $|x(t)|\leq |x(0)|e^{-t}$ for every allowed $a$ and all $t\geq0$. In this example it holds exactly when $k\geq4$. Thus $k=3.1$ gives robust stability but not the requested rate of decay; the endpoint plant decays only as $e^{-0.1t}$. The performance statement is stronger than the stability statement.
The guarantee remains conditional on its model: exact state measurement, the stated parameter interval and the realizable law $u=-kx$. At $k=4$, an initial state $x(0)=2$ demands an initial input of magnitude8 in the chosen consistent units. An <actuator saturation> limit of5 invalidates that ideal law for this initial condition. Increasing feedback gain therefore cannot be treated as a cost-free route to robustness. A useful design specifies both the uncertainty family and the physical limits under which its guarantee is meant to apply.
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