 Robust performance

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.^[ref]

An original example uses a stable scalar family with a bounded control disturbance $d$:The solution and a direct bound areThe 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 $|x(t)|\le 0.75D$. 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 $k\ge 0$. The corresponding worst-case gain becomesThe0.75gain requirement therefore holds for every allowed $a$ exactly when $k\ge 1/3$. 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.