Sampled-Data Switching Law

• Sampled-data switching law is a framework defining rules for switching control modes at discrete instants using quantized state data.
• It employs Lyapunov-based analysis, dwell-time conditions, and quantization error bounds to ensure stability and robustness of hybrid systems.
• This approach addresses mode mismatches between plant and controller, enabling practical stabilization despite coarse quantization and limited state information.

A sampled-data switching law specifies the rules by which control modes or actions are selected in a system where switching decisions can only be enacted at discrete sampling instants. In modern control theory, especially for switched linear and nonlinear systems with quantization or uncertainty, sampled-data switching laws must guarantee stability, boundedness, and safety under constraints on information, sampling, and switching. These laws are tightly linked to dwell-time conditions, mode-mismatch quantification, Lyapunov function techniques, and quantization error analysis. The resulting closed-loop hybrid system must accommodate potentially coarse quantization, restricted state information, and varying controller–plant mode synchrony.

1. System Architecture and Mode Information Constraints

Sampled-data switching laws operate in contexts where both the plant and controller may switch among distinct modes (indexed by $\sigma \in \mathcal{P}$) but the controller only receives mode information at sampling instants. The general continuous-time plant model is

$$\dot{x}(t) = A_{\sigma(t)} x(t) + B_{\sigma(t)} u(t),\quad x \in \mathbb{R}^n,\ u \in \mathbb{R}^m,$$

with mode $\sigma(t)$ piecewise constant. System state and mode are typically sampled at period $T_s>0$. After sampling, the controller uses the last sampled mode and quantized state: $$u(t) = K_{\sigma([t]^-)} Q(x([t]^-)),\quad [t]^- = \lfloor t/T_s\rfloor T_s,$$ where $Q$ is a static quantizer partitioning $\mathbb{R}^n$ into finite cells. Between samples, the plant may switch mode, but the controller remains synchronized to the previous sampled mode (Wakaiki et al., 2015, Wakaiki et al., 2014).

This architecture induces mode mismatch intervals: for times $t$ such that $\sigma(t) \neq \sigma([t]^-)$, the controller and plant operate with different dynamics and feedback gain.

2. Quantization: Sector Bounds and State Boundedness

Quantization is modeled by static partitions $\{ \mathcal{Q}_j \}$ with representatives $q_j$. The mapping $Q(x)$ is designed so that

$\|Q(x)-x\| \leq \delta_q \|x\|$

for some sector-type bound $\delta_q$ on quantization error (Wakaiki et al., 2015, Wakaiki et al., 2014). In the admissible region (often defined by ellipsoidal bounds derived from a Lyapunov function), this error impacts the system's growth rate and is crucial for stability analysis.

Coarser quantization (larger $\delta_q$) increases the growth rate $D_P$ during mode-mismatch, requiring longer dwell times and tighter mismatch time bounds for robust stabilization.

3. Mode-Mismatch Time and Dwell-Time Characterization

The concept of total mode-mismatch time,

$$\mu(\tau_1, \tau_2) = \int_{\tau_2}^{\tau_1} \chi(s) ds = |\{t \in [\tau_2,\tau_1): \sigma(t) \neq \sigma([t]^-)\}|,$$

quantifies the measure of intervals where plant and controller modes differ (Wakaiki et al., 2015, Wakaiki et al., 2014). Stability conditions are formulated by bounding $\mu(\cdot)$:

• Average mismatch: $\mu(t,0) \leq L t$, where $L < C_P / (C_P + D_P)$.
• Local mismatch after a switch: $\mu(t,T_0) \leq f(\kappa) + L(t-T_0)$, with $f(\kappa) = 2\ln\kappa/(C_P+D_P)$.

In practice, these bounds are enforced via dwell-time constraints: if the switching signal's dwell-time satisfies $\tau_d = n T_s$ with $n > 1 + D_P/C_P$, then $\mu(t,0) \leq t/n$ and stability criteria are satisfied (Wakaiki et al., 2015, Wakaiki et al., 2014).

4. Lyapunov Function Techniques and Ultimate Boundedness

Stability of sampled-data switched systems with quantization is typically analyzed using a common quadratic Lyapunov function $V(x) = x^\top P x$, $P \succ 0$, computed through randomized LMI or gradient-sampling algorithms [Ishii–Tempo, Liberzon–Tempo].

Mode-by-mode, two rates are computed: $$\dot V(x) \leq -C \|x\|^2 \quad\text{(no mismatch)},\qquad \dot V(x) \leq D \|x\|^2 \quad\text{(mismatch)},$$ with normalized versions $C_P = C/\lambda_{\max}(P)$ and $D_P = D/\lambda_{\min}(P)$ (Wakaiki et al., 2015, Wakaiki et al., 2014). Trajectories are shown to ultimately enter and remain within a prescribed inner ellipsoid: $V(x) \leq (\kappa r)^2 \lambda_{\min}(P),$ given that $L$ and mismatch-wise bounds are enforced. Explicit exponential estimates for state norm are derived: $$\|x(t)\| \leq \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}\, e^{-0.5(C_P-L(C_P+D_P))t} \|x(0)\|.$$ This result ensures ultimate boundedness and robust stabilization under sampled, quantized feedback and mode-mismatch (Wakaiki et al., 2015).

5. Synthesis and Implementation Strategies

Sampled-data switching synthesis proceeds as follows (Wakaiki et al., 2014, Wakaiki et al., 2015):

1. Compute mode-dependent stabilizing gains $\{K_p\}$ such that $(A_p+B_pK_p)$ is Hurwitz.
2. Fix a sampling rate $T_s$ (sufficiently small for sector bounds).
3. Design quantizer $Q$ to ensure sector bounds $\|Q(x)-x\| \leq \delta_q\|x\|$ over admissible region.
4. Solve for common $P \succ 0$ and constants $C, r, R$ via randomized algorithms.
5. Compute mismatch constants, $D$, and normalized rates.
6. Choose mismatch fraction $L < C_P/(C_P+D_P)$, tuning parameter $\kappa$, and dwell time $n > 1+D_P/C_P$.
7. Implement sampled-data feedback: controller uses last sampled and quantized state, mode information only at sample instants.

Numerical illustration (Wakaiki et al., 2014) confirms exponential convergence and boundedness: For two-mode $2\times 2$ systems, with $T_s=0.025$ and logarithmic quantization, $D_P \approx 16.8$, $C_P \approx 0.275$, and dwell time $T_d \approx 2.1$ s yield robust boundedness.

6. Practical Implications and Limitations

These sampled-data switching laws rigorously account for quantization, mode-mismatch, and dwell-time constraints. Mode-mismatch, if left unbounded, can destabilize the system even when individual modes are stabilized (Wakaiki et al., 2015). Explicit exponential bounds provide practical guarantees for state boundedness in terms of sampling, quantizer design, and dwell-time selection.

However, coarse quantization (large $\delta_q$) necessitates longer dwell-times to maintain stability. The randomized Lyapunov function computation is tractable, but further refinements or special rare-event switching cases may require tailored analysis.

A plausible implication is that in multi-mode, high-dimensional systems, mode-mismatch time analysis and dwell-time enforcement become increasingly important, and optimization of quantizer partitions and sampling rates offers leverage for tighter attractors and performance.

7. Connections to Broader Research

The sampled-data switching law paradigm elucidated above has been extended and generalized, with similar structures and stability proofs appearing in works on cascaded switched systems with event-based sampling (Zhang et al., 2019), observer-based switching with nonlinearities (Katz et al., 3 Nov 2025), and systems with uniform actuator quantization (Ferrante et al., 2022). Each of these contributions leverages dwell-time, sector-bound, and Lyapunov-based criteria to ensure practical stability and boundedness under sampled-data constraints. The approach is a core component in modern robust control and hybrid system theory for quantized, intermittently-observed, or switched systems.