L2 Stability of Switched Networks

• L2 stability is defined by the output energy being bounded by the input energy and initial state across all switching sequences.
• The QSR-dissipativity framework unifies passivity and small-gain concepts, employing LMIs for tractable stability verification in dynamic networks.
• A common storage function, built from local LMIs and interconnection LMIs, ensures scalable stability analysis and avoids combinatorial complexity.

The $L_2$ stability of switched networks concerns the input-output behavior of large-scale interconnected dynamic systems whose interconnection topology changes over time according to a switching signal. In this context, $L_2$ stability means that the energy of the output remains bounded by a function of the energy of the input and the initial condition, uniformly over all possible switching sequences. A prominent framework for analyzing such stability is QSR-dissipativity, which generalizes passivity and small-gain theory. Recent research has established explicit, computationally tractable conditions under which a network of QSR-dissipative agents—interconnected via arbitrary piecewise-constant switching topologies—remains $L_2$-stable, and has introduced methods to avoid the combinatorial growth of complexity traditionally associated with multiple-switching Lyapunov analyses (Jang et al., 23 Nov 2025).

1. Switched Network Models and Interconnection Structure

A switched network consists of $N$ agents, each modeled as a control system: $$G_p:\ \begin{cases} \dot{x}_p(t) = f^p(x_p(t), u_p(t)),\ y_p(t) = h^p(x_p(t), u_p(t)), \end{cases} \quad p \in \{1,\dots,N\},$$ with $x_p \in \mathbb{R}^{n_p}$, $u_p \in \mathbb{R}^{m_p}$, $y_p \in \mathbb{R}^{\ell_p}$. The interconnection among agents is defined by a switching signal $\sigma: \mathbb{R}_+ \to \{1,2,\ldots,M\}$, which selects one of $M$ possible adjacency matrices $H_i$ at each time. The global agent-level variables are concatenated as $x = \col(x_1,\ldots,x_N)$, $u = \col(u_1,\ldots,u_N)$, $y = \col(y_1,\ldots,y_N)$, with $u(t) = e(t) + H_{\sigma(t)} y(t)$, where $e(t)$ models exogenous inputs. This formalism captures arbitrary interconnection switching and is applicable to both continuous- and discrete-time dynamic networks.

2. QSR-Dissipativity of Agents

QSR-dissipativity characterizes each agent $G_p$ by the existence of a differentiable storage function $V_p(x_p) \ge 0$ satisfying, for all $t \ge t_0$,

$$V_p(x_p(t)) - V_p(x_p(t_0)) \le \int_{t_0}^t [\, y_p(\tau)^\top Q_p y_p(\tau) + 2 y_p(\tau)^\top S_p u_p(\tau) + u_p(\tau)^\top R_p u_p(\tau)\,] d\tau,$$

where $Q_p$, $S_p$, and $R_p$ are real constant matrices of appropriate dimensions. This unifies standard dissipativity (for general $Q$, $S$, $R$), passivity ($Q=0$, $S=\frac{1}{2}I$, $R=0$), and $L_2$-gain conditions, and therefore allows for general interconnection analysis.

3. From Local Dissipativity to Global $L_2$ Stability

For the network, block-diagonal aggregations $Q = \diag(Q_1,\dots,Q_N)$, $S = \diag(S_1,\dots,S_N)$, and $R = \diag(R_1,\dots,R_N)$ are constructed. For each topology $i \in \{1,\dots,M\}$, the key matrix is

$$\Phi_i = Q + S H_i + H_i^\top S^\top + H_i^\top R H_i.$$

The main result (Theorem 1 in (Jang et al., 23 Nov 2025)) states that if each agent $G_p$ is QSR-dissipative and $\Phi_i \prec 0$ for all $i$, the closed-loop switched network is $L_2$-stable under arbitrary switching. Explicitly, there exist $q > 0$, $r > 0$, and a common storage function $V(x) = \sum_{p=1}^N V_p(x_p)$ such that, for any input $e \in L_2$ and any time $T \ge 0$,

$$V(x(T)) - V(x(0)) \le -q \int_0^T \|y(t)\|^2 dt + r \int_0^T \|e(t)\|^2 dt,$$

implying the network $L_2$-gain estimate

$$\|y\|_{2,[0,T]} \le \sqrt{\frac{r}{q} \|e\|_{2,[0,T]} + \sqrt{\frac{1}{q} V(x(0))}}.$$

This approach is notable for its independence from dwell-time or mode-dependent Lyapunov arguments.

4. Linear Matrix Inequality (LMI) Conditions and Computation

When each agent is linear time-invariant (LTI), given by

$$\dot{x}_p = A_p x_p + B_p u_p,\quad y_p = C_p x_p + D_p u_p,$$

QSR-dissipativity with quadratic storage $V_p(x_p) = x_p^\top P_p x_p$ is equivalent to

$$\begin{bmatrix} A_p^\top P_p + P_p A_p - C_p^\top Q_p C_p & P_p B_p - C_p^\top S_p - C_p^\top Q_p D_p \ * & - (D_p^\top Q_p D_p + D_p^\top S_p + S_p^\top D_p + R_p) \end{bmatrix} \preceq 0.$$

For each mode $i$, the interconnection LMI is $\Phi_i \prec 0$. Only $N$ local agent LMIs and $M$ interconnection LMIs must be solved, yielding a common storage $V(x)$ that works for all modes. This computational architecture avoids the exponential complexity of a search over mode-dependent Lyapunov functions. In large networks, this enables tractable verification and synthesis.

5. Construction of Common Storage Functions

The existence of a mode-independent global storage function $V(x) = \sum_{p=1}^N V_p(x_p)$ is an essential merit of this approach. Each $P_p$ is determined solely by the local agent's dissipativity LMI and is independent of the changing topology. No global Lyapunov search is required over all switching sequences; the complexity scales linearly with agent count for the local LMIs and with the number of topologies for the interconnection LMIs. This eliminates the "combinatorial explosion" associated with multiple-mode Lyapunov analyses, as previously encountered in switched network stability literature.

6. Numerical Example: UAV Swarm Under Switching

A benchmark example demonstrates the algorithm on a swarm of nine quadrotors, each linearized to a 12-state dynamic model,

$$\dot{x}_p = A_p x_p + B_p u_p, \quad u_p = -K_p\left( x_p + \sum_{q=1}^9 (H^c_{\sigma(t)})_{pq}\, x_q \right),$$

where the control gain $K_p$ is computed via LQR and the switching signal selects among $M=4$ communication topologies. The $N=9$ agent-level LMIs ($12 \times 12$ each) and $M=4$ interconnection LMIs were solved in under 2 seconds (MOSEK+YALMIP), yielding bounds $q \approx 0.12$, $r \approx 1.1$, and an $L_2$-gain $\gamma \approx 3.0$. Time-domain simulation with random $L_2$ disturbances, 15 arbitrary switchings over $180$ seconds, confirmed

$$\|y\|_{2,[0,180]} \le 3.0\|e\|_{2,[0,180]} + O(\sqrt{V(x(0))}),$$

with all states remaining bounded (Jang et al., 23 Nov 2025).

7. Distributed Design and Algorithmic Procedure

A distributed procedure certifying $L_2$ stability under arbitrary switching proceeds as follows:

1. For each agent $p$, select $V_p(x_p) = x_p^\top P_p x_p$ and solve the agent's QSR-dissipativity LMI for $(Q_p, S_p, R_p)$.
2. Assemble $Q = \diag(Q_p)$, $S = \diag(S_p)$, $R = \diag(R_p)$.
3. For each topology $i = 1,\ldots, M$, compute $$\Phi_i = Q + S H_i + H_i^\top S^\top + H_i^\top R H_i$$ and check $\Phi_i \prec 0$.
4. If feasible, define $q = (-\max_i \lambda_{\max}(\Phi_i))/2 > 0$, $r = \max_i \|R + \cdots\|$, and set $\gamma = \sqrt{r/q}$; $V(x)$ is the common storage.
5. No search over all switchings and no dwell-time or multiple-mode Lyapunov constraints are required.

This method is tractable and scalable for large-scale dynamically interconnected networks, as demonstrated in the referenced simulations and analyses (Jang et al., 23 Nov 2025).