Input-output gain

An input-output gain bounds how large a system's output can be relative to its input, after the input and output measures have been specified. For a zero-initial-state operator $T$ and chosen signal norms, its induced signal gain isA finite gain gives $\Vert y{\Vert }_{\mathrm{out}}\le \gamma \Vert u{\Vert }_{\mathrm{in}}$. The supremum is over the admissible input class, not merely a collection of trial inputs. Initial-state response must be treated separately.

For an original scalar linear time-invariant system, letUsing the time-domain L-infinity norm, the convolution solution obeysA constant input of suitable sign approaches the limiting ratio $|b|/a$, so this bound is sharp. At $a=2$ and $b=3$, the peak-amplitude gain is1.5: an input bounded by2 gives output bounded by3 from zero initial state, and the constant input approaches that value.

The signal norms change the problem. For the same stable scalar system, the peak frequency response magnitude isIt coincides numerically with the preceding gain in this example. For stable linear systems, this frequency-domain quantity is the energy-signal induced gain associated with the time-domain L2 norm; it should not generally be identified with the peak-amplitude induced gain. MathWorks's linear-model norm documentation distinguishes this peak frequency-response quantity from its separate impulse-response norm.^[ref]

The units matter as well: a gain from force input to displacement output has physical units, whereas identical normalized signal units can produce a dimensionless gain. In robust performance, the bound must hold across the allowed control model uncertainty. A nominal gain calculation alone supplies no such universal guarantee.

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.