= Robust performance
\b[Robust performance] requires a <control system> to meet a specified quantitative response criterion throughout its allowed <control model uncertainty>. It includes the relevant stability requirement, but adds constraints such as disturbance attenuation, tracking accuracy or decay rate. MathWorks's robust-analysis example explicitly separates stability of the uncertain loop from maintaining its prescribed performance level.<Robust Stability, Robust Performance, and Mu Analysis in MATLAB documentation>[Robust Performance]{ref}
An original example uses a stable scalar family with a bounded <control disturbance> $d$:
$$
\dot x=-ax+d(t),\qquad a\in[1,2],\qquad x(0)=0,\qquad |d(t)|\leq D.
$$
The solution and a direct bound are
$$
x(t)=\int_0^t e^{-a(t-s)}d(s)\,ds,\qquad
|x(t)|\leq\frac{D}{a}(1-e^{-at})\leq\frac Da.
$$
The constant disturbance $d(t)=D$ approaches $D/a$, so the worst-case <input-output gain> for this peak-amplitude criterion is $1/a$. At nominal $a=1.5$, the limiting response is $2D/3$, satisfying a proposed bound $|x(t)|\leq0.75D$. The allowed endpoint $a=1$ instead approaches $D$, violating that requirement despite remaining stable.
Now add state <feedback control> $u=-kx$ to the right-hand side, with $k\geq0$. The corresponding worst-case gain becomes
$$
\sup_{a\in[1,2]}\frac1{a+k}=\frac1{1+k}.
$$
The0.75gain requirement therefore holds for every allowed $a$ exactly when $k\geq1/3$. This is a statement about a stipulated peak-amplitude bound, not a numerical output from the software example.
The criterion must include its input and initial-state assumptions. A nonzero $x(0)$ contributes $e^{-(a+k)t}x(0)$, and <actuator saturation> can prevent realization of $u=-kx$. Changing the disturbance norm, measured output or uncertainty family changes the performance problem. <Robust performance> is therefore a precise guarantee about a stated comparison, rather than a claim that a controller performs well under unspecified conditions.
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