Lyapunov-Based Mean Square Stability

Updated 9 August 2025

Lyapunov-based mean square stability is defined as the boundedness of the second moment of a system’s state, ensuring energy constraints amid stochastic perturbations.
The analysis employs Lyapunov methods, Markov chain models, and auxiliary systems to derive necessary and sufficient stability conditions for networked control.
Adaptive event-triggered policies and spectral radius thresholds enable practical scheduling designs that counteract transmission delays and non-Gaussian errors.

Lyapunov-based mean square stability is a foundational concept in the analysis and design of stochastic and networked control systems, quantifying the boundedness of the second moment of the state via Lyapunov-theoretic methods. In its modern applications—especially for event-based networked control—this property delineates precise criteria under which the mean-square of state trajectories or estimation errors remains bounded despite random transmission delays, network contention, non-Gaussian estimation error densities, and stochastic triggering mechanisms. The Lyapunov framework provides both necessary and sufficient stability conditions, often represented via operator inequalities, linear or nonlinear matrix recursions, and Markov models. The most advanced formulations adapt to system-specific constraints such as contention resolution in wireless networks, event scheduling strategies, and nontrivial packet loss behaviors.

1. Formal Definition of Lyapunov Mean Square Stability

Mean square stability, in the Lyapunov sense, requires that the second moment (i.e., the expected squared norm) of the state (or error) vector is uniformly bounded over time. Formally, for a discrete-time system $x_k$ :

$\limsup_{k \to \infty} E[x_k^\top x_k] \leq \zeta$

for some finite $\zeta > 0$ , where $E[\cdot]$ denotes expectation over the system and network-induced randomness. Equivalently, in state estimation contexts, mean square stability is expressed in terms of the estimation error covariance $P^{(k|k)}$ :

$\limsup_{k \to \infty} \operatorname{tr}\{ E[P^{(k|k)}] \} \leq \varpi$

for a finite $\varpi < \zeta$ . These criteria guarantee energy-boundedness for each control loop or estimator, irrespective of persistent stochastic perturbations or queuing delays.

2. Markov Model-Based Stability Criteria

In event-based networked control systems sharing a wireless medium (Ramesh et al., 2014), states are indexed both by system mode (idle, event, transmission, etc.) and by delay $d$ since last successful data transmission. The system evolution is captured by a Markov chain model, in which each system mode is paired with a delay counter. The critical mechanism for analysis is the recursive computation of mode–delay steady-state probabilities $\pi_{S,d}$ , especially for the idle state probabilities $\pi_{I,d}$ . The Bianchi assumption (conditional independence across loops) allows the busy-channel probability to be treated as a constant $p$ in steady-state, enabling decoupled analysis of each control loop.

The sufficient condition for Lyapunov mean square stability of system $j$ is given by:

$\limsup_{d \to \infty} \frac{\pi_{I,d+1}}{\pi_{I,d}} < \frac{1}{1 + \rho(A_j)^2}$

where $\rho(A_j)$ is the spectral radius of the system's state transition matrix. This inequality ensures that the probability tail for being in higher-delay idle-states drops off rapidly enough to compensate for amplification introduced by plant instability and transmission delays. When $\rho(A_j)$ is larger (more unstable), the allowable idle-state probability ratio is correspondingly more restrictive.

3. Auxiliary System and Upper Bounds for Variance Evolution

Event-triggering policies induce non-Gaussian, often truncated error distributions, complicating direct calculation of mean square stability. To circumvent these difficulties, an auxiliary system is constructed whose state density $\hat\phi_{I,d}$ is defined recursively as a (spread-out) convolution:

$\hat\phi_{I,d} = \frac{1}{|A|} (\hat\phi_{I,d-1} * \phi_N)$

where $\phi_N$ is the process noise distribution. By majorization arguments, the true estimation error density is dominated (in the convex ordering sense) by this auxiliary density, yielding a strict variance upper bound for each delay:

$\operatorname{tr}\{ E[P_d] \} \leq \operatorname{tr}\{ \hat P_d \}$

with the recursion:

$\hat P_d = \rho(A)^2\,\hat P_{d-1} + \Sigma_w, \quad \hat P_0 = \Sigma_w$

where $\Sigma_w$ is the process noise covariance. The overall state variance is then a weighted blend:

$\operatorname{tr}\{ E[P^{(k|k)}] \} = \sum_{d=0}^\infty \pi_{I,d} \operatorname{tr}(P_d)$

Mean square boundedness follows if the idle-state probabilities $\pi_{I,d}$ decay fast enough to counteract variance growth with delay.

4. Event-Triggering Policy Design for Stability

Event-based scheduling policies determine the probability $p_{\gamma,d}$ that an event is triggered after delay $d$ . Two principal classes are considered:

Constant-Probability Policy: $p_{\gamma,d} = p_\gamma$ uniformly for all $d > 0$ . The effective loss probability simplifies, and the idle-state probability distribution becomes geometric, allowing clean closed-form analysis.
Additive / Exponential Policies: $p_{\gamma,d}$ increases additively or exponentially with delay to reduce the risk of instability at longer network delays.

For the constant-probability policy, stability is guaranteed provided:

$p_\gamma \cdot \left( \sum_{r=1}^{r_{\max}} \left[ q_r \prod_{s=1}^{r_{\max}-1} (1 - p_\alpha q_s) \right] \right) > \frac{1}{p_\alpha \left( 1 - 1/\rho(A)^2 \right)}$

where $p_\alpha$ is the persistence probability of the CRM, $q_r$ is the network success probability at the $r$ -th transmission attempt, and $r_{\max}$ the maximum retransmissions allowed. This quantifies explicit trade-offs between network reliability, event frequency, and tolerance to plant instability.

Thresholds for event-triggering (e.g., $\Delta_d$ ) are calibrated to realize specific $p_{\gamma,d}$ values, often via inversion of cumulative distribution functions for the error process.

5. Integration of Stability Analysis and Real-Time Adaptation

Practical deployment in congested networks is assessed via simulation: when the underlying probability ratio $\pi_{I,d+1}/\pi_{I,d}$ approaches unity (i.e., network congestion or frequent unsuccessful transmissions), even highly unstable plants cannot be stabilized within the mean square sense. Conversely, for moderate values of $\rho(A)$ and well-chosen scheduling parameters, stability is demonstrated empirically.

Design procedures are hierarchical:

Select event-probabilities $p_{\gamma,d}$ and CRM parameters $p_\alpha$ to satisfy the sufficient mean square stability conditions.
Solve for the explicit event-triggering thresholds or criteria that realize the required probabilities using model-based or data-driven methods.

These strategies enable adaptive policy tuning under real-time varying network delays and contention levels, and permit each control loop to remain stable independently, at low mean square state energy.

The analysis aligns conceptually with broader Markov jump linear system stability criteria, Lyapunov operator inequalities in infinite-dimensional spaces (Xiao et al., 24 Apr 2024), and switching system mean square stability via cone-linear or homogeneous Lyapunov functions (Ogura et al., 2014). In transactional networked control, these approaches generalize to systems with random, time-varying delays modeled by Markov chains (Impicciatore et al., 2021), with Lyapunov conditions often structured around mode-dependent functionals.

The auxiliary-system and decay-rate approach developed in this work provides a direct link between the network-induced delay distribution and mean square stability, with explicit recursions and spectral radius-based thresholds. By majorization, all non-Gaussian error distributions encountered in event-triggering frameworks can be upper bounded by analytically tractable spread-out auxiliary densities, facilitating robust design and certification even in the presence of heavy-tailed or truncated errors.

7. Computational Methods and Policy Design Implications

Simulation-based “p-success maps” and shaded design regions provide practitioners with explicit charts of permissible operating points (parameter combinations) that certify Lyapunov mean square stability. This enables systematic configuration of event-triggering policies and network access schemes with quantifiable guarantees, guiding selection of event-probabilities and contention parameters under practical wireless constraints.

In multi-agent settings with $M$ nodes, scalability is maintained provided each node independently satisfies the idle-state decay condition relative to its own plant instability. Strong numerical results show the method extends to moderately unstable plants (e.g., $A = 1.5$ ) in networks of several nodes ( $M = 5$ ), with design region boundaries matching both analytical predictions and simulated system trajectories.

In summary, Lyapunov-based mean square stability provides actionable, quantitative, and robust guidelines for the stabilization of event-triggered networked control systems and the design of adaptive scheduling policies. The Markov model and auxiliary-system frameworks advanced in recent research (Ramesh et al., 2014) underpin the real-world deployment of such systems, ensuring reliability and scalable performance under stochastic network conditions.