---
title: Sampled-Data Switching Law
url: https://www.emergentmind.com/topics/sampled-data-switching-law
type: topic
---

# Sampled-Data Switching Law

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 ([1509.00540], [1403.4691]).
 
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 ([1509.00540], [1403.4691]). 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 ([1509.00540], [1403.4691]). 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 ([1509.00540], [1403.4691]).

## 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 $D_P = D/\lambda_{\min}(P)$ ([1509.00540], [1403.4691]). 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 ([1509.00540]).

## 5. Synthesis and Implementation Strategies

Sampled-data switching synthesis proceeds as follows ([1403.4691], [1509.00540]):
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 ([1403.4691]) 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 ([1509.00540]). 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 ([1904.00616]), observer-based switching with nonlinearities ([2511.01672]), and systems with uniform actuator quantization ([2208.05694]). 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.

Source: https://www.emergentmind.com/topics/sampled-data-switching-law