Switched QSR-Dissipativity in Networked Systems

• Switched QSR-dissipativity is a framework that generalizes classical dissipativity by using mode-dependent quadratic supply rates for switched and networked systems.
• It leverages mode-specific and common storage functions alongside LMIs to ensure input-output (L2-gain) stability under diverse switching patterns and uncertainties.
• The approach is practical for large-scale applications such as cyber-physical systems and swarmed UAV networks, where robust performance and scalability are paramount.

Switched QSR-dissipativity generalizes classical dissipativity theory to switched and networked systems that admit mode-dependent quadratic supply rates. In such frameworks, the system can transition among dynamical modes or configurations, each associated with its own supply rate defined by quadratic forms in input and output signals. This concept enables rigorous analysis of input-output stability (notably $L_2$-gain stability) for large-scale networks with dynamically changing interconnections, as well as for switched systems constrained by admissible switching patterns and uncertainties. The approach is grounded in constructing mode-dependent or common storage functions and leveraging linear matrix inequality (LMI) conditions for verification and synthesis in both continuous- and discrete-time settings.

1. QSR-Dissipativity for Non-Switched Systems

QSR-dissipativity for isolated, time-invariant systems is defined in terms of a quadratic supply rate and a differentiable storage function. For a system described by

$\dot x = f(x,u), \qquad y = h(x,u)$

with state $x\in\mathbb R^n$, input $u\in\mathbb R^m$, and output $y\in\mathbb R^\ell$, the supply rate is given by: $$w(u,y) = \begin{bmatrix}y \ u\end{bmatrix}^T \begin{pmatrix} Q & S \ S^T & R \end{pmatrix} \begin{bmatrix}y \ u\end{bmatrix} = y^TQy + 2y^TSu + u^TRu$$ where $Q$, $S$, and $R$ are suitably dimensioned symmetric matrices.

The system is QSR-dissipative if there exists a continuously differentiable function $V(x)\geq 0$ such that

$$V(x(t)) - V(x(t_0)) \leq \int_{t_0}^{t} w(u(\tau), y(\tau)) d\tau$$

for all $t_0\leq t$ (Jang et al., 23 Nov 2025).

2. Switched Systems and Switched QSR-Dissipativity

A switched system can be formalized as: $$\dot x = f_{\sigma(t)}(x,u), \qquad y = h_{\sigma(t)}(x,u)$$ where the switching signal $\sigma: [0, \infty) \to \{1, \ldots, M\}$ determines the active mode. Each mode $i$ possesses an associated quadratic supply rate $(Q_i, S_i, R_i)$.

Switched QSR-dissipativity requires the existence of storage functions $V_i(x)\geq 0$ (one for each mode) and the following bounds for each switching interval $[t_k, t_{k+1})$: $$V_i(x(t)) - V_i(x(s)) \leq \int_{s}^{t} w_i^i(u(\tau), y(\tau)) d\tau$$ for active mode $i$, and for all $j\ne i$ (cross-supply),

$$V_j(x(t)) - V_j(x(s)) \leq \int_{s}^{t} w_j^i(x, u, y, \tau) d\tau$$

(Jang et al., 23 Nov 2025). This definition accommodates distinct supply rates for each mode, permitting detailed stability and performance analysis even under arbitrary or constrained switching.

3. Input-Output Stability and Common Storage Function Theorems

Input-output ($L_2$) stability can be certified for switched QSR-dissipative systems using a common storage function under strict negativity of all mode-specific $Q_i$. Specifically, under the assumptions:

• Each mode is QSR-dissipative with storage $V_i(x) \equiv V(x)$ shared for all $i$,
• $Q_i \prec 0$ for every $i$,

there exists a quadratic supply rate $\widetilde Q = -q I$, $\widetilde S=0$, $\widetilde R = r I$ such that

$$V(x(t)) - V(x(0)) \leq -q \int_0^t y^T y d\tau + r \int_0^t u^T u d\tau$$

resulting in a finite $L_2$–gain $\sqrt{r/q}$ under arbitrary switching. The proof utilizes Young’s inequality and selection of $\epsilon_i>0$ such that $Q_i + \epsilon_i I \prec 0$ to construct the margin (Jang et al., 23 Nov 2025).

A plausible implication is that, for practical large switched networks, construction of a shared storage function circumvents the prohibitive computational cost of searching for mode-dependent Lyapunov functions.

4. Networks of QSR-Dissipative Agents with Switching Topologies

In large-scale networks, each agent $G_p$ is QSR-dissipative (with its own $(Q_p, S_p, R_p)$ and storage $V_p(x_p)$). Interconnection is via a time-varying adjacency matrix $H_{\sigma(t)}$, resulting in a collective system: $u = e + H_{\sigma(t)} y$ with $y$ and $e$ stacking outputs and exogenous inputs of all agents.

Defining block-diagonal $Q$, $S$, and $R$, and denoting $H_i$ as the interconnection matrix in mode $i$, global dissipativity is verified if: $$\Xi_i = Q + S H_i + H_i^T S^T + H_i^T R H_i \prec 0 \quad \forall i$$ The storage function is the sum of the local agent storages: $V(x) = \sum_{p=1}^N V_p(x_p)$ This result (Theorem 2) certifies finite $L_2$-gain for $e \mapsto y$ under arbitrary switching and allows one to check feasibility using local dissipativity LMIs and global interconnection LMIs, avoiding a centralized search for a global Lyapunov/storage function (Jang et al., 23 Nov 2025).

5. Constrained Switching, Uncertainty, and Robust Performance

Switched QSR-dissipativity extends to discrete-time systems with constrained switching, where admissible switching sequences are graph-encoded. For each subsystem label $l$ in the constrained switched linear system: $$x(t+1) = A_l x(t) + B_l w(t), \qquad z(t) = C_l x(t) + D_l w(t)$$ and every transition $(i,j,l)$ in the graph, supply rate and storage functions are enforced locally: $$V_j(A_l x + B_l w) - V_i(x) + [w; z]^T P_l [w; z] \leq -\epsilon w^T w$$ LMIs for dissipativity (and thus quadratic performance) can be formulated in several equivalent ways, facilitating both analysis and controller synthesis. Coupled LMIs of sizes ranging from $2n$ up to the sum of input/output dimensions check existence of the storage function family $\{X_i\}$ (Lang et al., 2024).

When system matrices exhibit uncertainty, robust dissipativity is encoded by adding multiplier terms to the LMI conditions and applying the S-procedure. This ensures the performance bound holds uniformly over the uncertainty set, as shown for linear fractional representations (LFR) (Lang et al., 2024).

State feedback controller synthesis is convexified by change of variables (e.g., $Z_i=K_i G_i$), yielding convex programs for selecting controller gains and guaranteeing switched QSR-dissipativity under constraints and uncertainties.

6. Numerical Example: Swarmed UAV Network

A representative example features a networked swarm of nine UAVs linearized about hover, each with state $x_p\in\mathbb{R}^{12}$ and controlled by embedded LQR ($u_p = -K_p x_p$). Each agent’s local closed-loop QSR-dissipativity is verified by solving the local LMI for $(Q_p, S_p, R_p, P_p)$.

Four distinct interconnection topologies $H_1, \dots, H_4$ encode UAV communications, with arbitrary switching among them. For each mode, the global interconnection LMI

$Q + S H_i + H_i^T S^T + H_i^T R H_i \prec 0$

is checked for feasibility with MOSEK + YALMIP. Simulations over 15 random switches in 180 s—including $L_2$ disturbances—demonstrate bounded swarming errors, certifying $L_2$-stability as predicted by theory (Jang et al., 23 Nov 2025). This suggests switched QSR-dissipativity is tractable and effective for complex, dynamical interconnection scenarios.

7. Summary and Connections

Switched QSR-dissipativity enables precise stability and performance guarantees for nonlinear, switched, or networked systems with mode-dependent quadratic supply rates. Key findings are:

• Mode-dependent dissipativity extends classical results, supporting diverse switching and interconnection patterns.
• Shared storage function existence under strict $Q_i\prec 0$ yields global $L_2$-gain bounds without excessive complexity.
• Networked systems leverage local dissipativity and simple summation for global analysis; verifying interconnection LMIs suffices.
• Constrained switching and system uncertainty are handled via coupled LMIs and multiplier methods, underlining the flexibility of the QSR-dissipativity framework.
• Practical synthesis of controllers and verification of stability is enabled via tractable convex optimization.

Noteworthy applications include real-time cyber-physical systems, networked drones, and robust controllers for time-varying and uncertain environments. Direct computation of quadratic performance bounds and stability properties depends only on local agent solves and moderate verification of LMIs, making the approach scalable for large networks and complex switching structures (Jang et al., 23 Nov 2025); (Lang et al., 2024).