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An induced signal gain is the smallest norm-based amplification bound obtained by taking the worst case over admissible inputs. For a zero-initial-state operator $T$ satisfying $T0=0$,For a linear map this is an operator norm between the specified normed spaces. Axler's definition takes the supremum of output norm over inputs of norm at most1, which is equivalent by linear scaling.^[ref]

The norms are part of the definition. An original finite-dimensional example usesWith the same maximum-component norm at input and output, $\Vert Mu{\Vert }_{\infty }\le 2\Vert u{\Vert }_{\infty }$, and $u=(1,1)$ attains the ratio2. With the Euclidean norm, the Cauchy-Schwarz inequality gives $|u_1+u_2|\le \sqrt{2}\Vert u{\Vert }_2$, with equality at $u=(1,1)$. With the sum-of-absolute-components norm, $|u_1+u_2|\le |u_1|+|u_2|$, and $u=(1,0)$ attains ratio1. The same matrix therefore has induced gains2, $\sqrt{2}$ and1 under these three conventions.

For a time-dependent signal, the choice may additionally compare peak amplitudes or integrated squared amplitudes. The example in input-output gain derives a sharp peak-amplitude bound for a scalar dynamic system. A large response to one trial input supplies a lower bound on induced gain; proving a universal upper bound requires controlling every admissible input.

Compatible induced bounds compose. If $\Vert T_{1}u\Vert \le {\gamma }_1\Vert u\Vert$ and $\Vert T_{2}v\Vert \le {\gamma }_2\Vert v\Vert$, thenThe intermediate norms must be the same. This inequality bounds a cascade without requiring the inputs that maximize the individual blocks to be simultaneously realizable, so the product need not be sharp. In robust performance, both the signal norms and the allowed control model uncertainty must be stated before such a bound can be interpreted.

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.