L2-L2-gain bounds for quadratic output systems

Abstract: We derive an explicit bound for the L2-L2-gain of linear time-invariant systems whose output is a quadratic function of the state and the input. Such systems appear naturally in many areas, for example for port-Hamiltonian systems, optimal-control, and stochastic problems. In case the output is purely quadratic in the state, the bound equals the L2-norm of the bivariate transfer function evaluated along the anti-diagonal ${(s,\,-s)\mid s\in i\mathbb R}$ of the $i \mathbb R\times i \mathbb R$ frequency domain. Further, we show how the bound can be computed by solving linear matrix equations. This result provides a practical tool for assessing and reducing quadratic-output models.

• The paper establishes computable L2-L2 bounds for purely quadratic outputs by restricting a bivariate transfer function to the frequency-domain anti-diagonal and evaluating the resulting norm through Lyapunov and Sylvester equations.
• The paper extends the analysis to bilinear outputs using auxiliary LTI-system norms, while showing that quadratic throughput terms require inputs in L2∩L∞ rather than L2 alone.
• The paper validates the bound numerically with a three-state example yielding 0.491‖u‖L2, while tested-input gains range from 0.047 to 0.361, and identifies balanced truncation and IRKA-style reduction as open problems.

Motivation and problem setting

Linear time-invariant (LTI) systems with quadratic outputs (LQO systems) arise in port-Hamiltonian modeling, linear-quadratic optimal control, and stochastic problems where the output is a variance. The paper considers systems of the form

$$\dot{x} = Ax + Bu, \qquad y = Cx + Du + M(x \otimes x) + R(u \otimes x) + P(u \otimes u),$$

with $A$ Hurwitz and $x(0)=0$, $u \in L_2(\mathbb{R}_+)$. While for stable LTI systems the $L_2$-$L_2$ gain equals the $H_\infty$-norm of the transfer function, no analogous computable expression was known for LQO systems; prior work had only produced $L_2$-$L_\infty$ bounds via a bivariate-output supremum argument, together with Gramian-based balanced truncation and IRKA-style reduction methods. The goal here is a computable constant $c_2$ such that $A$0, where $A$1 is the $A$2-norm of the linear part.

The key technical obstacle is that the bivariate-output inequality $A$3 has no analogue for $A$4-norms. The paper circumvents this by applying the association transform to pass from bivariate to univariate frequency-domain objects.

Main bound for purely quadratic outputs

For systems with $A$5 only, the central result states

$A$6

where $A$7 is the bivariate transfer function. The gain is thus an $A$8-norm of the bivariate transfer function restricted to the anti-diagonal $A$9 of the frequency domain. The proof proceeds via three steps: the association transform expresses the univariate transfer function on the imaginary axis as an integral over the bivariate one; Parseval's theorem plus Hölder inequalities convert the output $x(0)=0$0-norm into fiberwise $x(0)=0$1-norms of the bivariate frequency-domain output; and an eigendecomposition argument exploiting symmetry of $x(0)=0$2 shows the supremum over shifts $x(0)=0$3 of the fiber norm is attained on the anti-diagonal.

An important caveat is stated explicitly: although this quantity resembles an $x(0)=0$4-norm, $x(0)=0$5 is analytic in neither half-plane, so it does not belong to any Hardy space and cannot be identified with an $x(0)=0$6-norm. For single-output systems it coincides with the $x(0)=0$7-norm of the associated transfer function up to the factor $x(0)=0$8.

Sylvester-equation-based computation

The second contribution makes the bound computable without frequency-domain quadrature. An inner product on pairs of LQO transfer functions is defined by integrating traces over the anti-diagonal, and its induced norm recovers the gain bound. The main computational theorem shows this inner product equals

$x(0)=0$9

where $u \in L_2(\mathbb{R}_+)$0 solve Lyapunov equations $u \in L_2(\mathbb{R}_+)$1, and $u \in L_2(\mathbb{R}_+)$2, $u \in L_2(\mathbb{R}_+)$3 solve Sylvester equations $u \in L_2(\mathbb{R}_+)$4 and $u \in L_2(\mathbb{R}_+)$5. The derivation rests on a matrix-valued partial-fraction decomposition of $u \in L_2(\mathbb{R}_+)$6, valid because eigenvalues of $u \in L_2(\mathbb{R}_+)$7 are sums $u \in L_2(\mathbb{R}_+)$8 of Hurwitz eigenvalues, hence invertible. This yields an exact error formula between two LQO models — directly relevant to model order reduction — via the parallelogram identity on the inner product.

The author notes a limitation: unlike the $u \in L_2(\mathbb{R}_+)$9-oriented balanced truncation for LQO systems, an $L_2$0-$L_2$1-oriented balanced truncation would require expressing observability energy through these Gramian-like matrices, and no such relation was found.

Bilinear and throughput terms

For bilinear outputs $L_2$2, a direct time-domain argument using Cauchy–Schwarz inequalities gives

$L_2$3

which for $L_2$4 is exactly the $L_2$5-norm of the auxiliary LTI system $L_2$6 squared times $L_2$7. For quadratic throughput terms $L_2$8, the paper observes that $L_2$9 alone does not guarantee bounded output: one necessarily obtains an estimate involving $L_2$0. Consequently, throughput terms require the stronger assumption $L_2$1, and the resulting bound combines standard $L_2$2-norms of LTI systems built from $L_2$3.

Combining all pieces, for general LQO systems without quadratic throughput ($L_2$4), the full bound reads

$L_2$5

obtained by splitting the output into linear, bilinear, and quadratic parts and applying the triangle inequality.

Numerical validation

A three-state example with $L_2$6 equal to the vectorized identity yields the gain bound $L_2$7 computed via the Sylvester formulation. Empirical gains for eight test inputs (exponentials, Gaussians, windowed sinusoids, chirps) range from 0.047 to 0.361, all strictly below the bound, confirming validity though not tightness for the tested inputs.

Limitations and open questions

Several restrictions are acknowledged within the paper. The quadratic-throughput case requires $L_2$8 rather than $L_2$9 alone, so the resulting statement is an $H_\infty$0-$H_\infty$1 rather than a pure $H_\infty$2-$H_\infty$3 gain. The derived bound is generally not tight — the numerical example shows substantial slack — and the anti-diagonal norm is not a Hardy-space norm, which blocks direct reuse of $H_\infty$4 optimization machinery. The author explicitly leaves open whether an IRKA-type iteration minimizing the anti-diagonal $H_\infty$5-error can be constructed, and whether a balanced truncation scheme with an a priori $H_\infty$6-$H_\infty$7 error bound exists, since the inner product structure does not reveal a suitable observability Gramian.

Conclusion

The paper establishes the first computable $H_\infty$8-$H_\infty$9 gain bounds for linear systems with quadratic outputs: an explicit anti-diagonal frequency-domain norm for the purely quadratic case, a finite set of Lyapunov and Sylvester equations evaluating it exactly, and modular extensions covering bilinear outputs and, under an $L_2$0 assumption on the input, quadratic throughput terms. The results provide the quantitative foundation needed to extend $L_2$1-style model order reduction to quadratic-output models, with the reduction algorithms themselves remaining the principal open problem.