Hybrid Singularly Perturbed Systems

• Hybrid singularly perturbed dynamical systems are models combining continuous-time evolution with discrete switching events, featuring slow and fast state components governed by a small parameter.
• They utilize mode-dependent block-diagonalization and time-scale analysis to decompose dynamics, enabling precise stability evaluation through auxiliary reduced, pure-fast, and enriched slow systems.
• LMI-based verification and Lyapunov techniques provide rigorous stability criteria useful for designing controllers and observers in networked and switched control architectures.

A hybrid singularly perturbed dynamical system is a class of dynamical system characterized by the interplay between continuous-time evolution and discrete (impulsive or switching) events, where the state is partitioned into components evolving on disparate time scales—typically labeled "slow" and "fast." The singular perturbation manifests through a small parameter $\epsilon>0$ that scales the fast subsystem's dynamics, and the hybrid nature arises from switches or jumps that can alter the continuous dynamics, state dimension, or partitioning between slow and fast variables. These systems are central to modern networked control, multi-rate sampled-data systems, and switched/impulsive control architectures, and they necessitate specialized tools for stability and performance analysis.

1. Canonical Formulation of Hybrid Singularly Perturbed Systems

The general form of a hybrid singularly perturbed system is defined on $\mathbb{R}^d$ with a state $X(t)$ subject to both continuous evolution (flows) and discrete events (impulses or mode switches). At each switching time $t_k$ the system selects a mode $\sigma_k = (\ell_k, P_k, \Lambda_k, R_k)$, with $\ell_k \in \{1,\ldots,d-1\}$ partitioning the coordinates, $P_k \in GL(d)$ a mode-dependent invertible matrix, $\Lambda_k \in \mathbb{M}_d$ a system matrix, and $R_k \in \mathbb{M}_d$ the jump map. The small parameter $\epsilon>0$ modulates the time-scales via the diagonal matrix

$$E_{\ell_k}^\epsilon = \mathrm{diag}(\underbrace{1,\ldots,1}_{\ell_k}, \underbrace{\epsilon,\ldots,\epsilon}_{d-\ell_k}),$$

leading to the impulsive-switched system \begin{align*} E_{\ell_k}^\epsilon P_k \dot{X}(t) &= \Lambda_k X(t), \qquad t \in [t_k, t_{k+1}), \ X(t_k) &= R_{k-1} X(t_k^-). \end{align*} On each interval, the first $\ell_k$ coordinates of $P_k X$ are slow (order-one rate), while the remaining $d - \ell_k$ are fast ($O(1/\epsilon)$ rate). The system can accommodate switching between arbitrary slow/fast partitions and mode-dependent coordinate changes (Haidar et al., 3 Jul 2025).

2. Mode-Dependent Block-Diagonalization and Time-Scale Analysis

Analysis proceeds via a Tikhonov-type block-diagonalization, enabled under the "D--Hurwitz" assumption (Hurwitz fast-submatrix $D_k$ of $\Lambda_k P_k^{-1}$). The matrix $\Lambda_k P_k^{-1}$ is decomposed:

$$\Lambda_k P_k^{-1} = \begin{pmatrix} A_k & B_k \ C_k & D_k \end{pmatrix},$$

and an invertible transformation $T_k$ brings it into a block-upper-triangular form, which enables state transformation into slow ($x$) and fast ($z$) coordinates:

$$\begin{pmatrix} x(t) \ z(t) \end{pmatrix} = T_k X(t), \quad t \in [t_k, t_{k+1}).$$

In these coordinates, the system dynamics are \begin{align*} \dot{x} &= (A_k - B_k D_k^{-1} C_k - B_k Q_k) x + B_k z, \ \dot{z} &= \frac{1}{\epsilon} D_k z + (D_k^{-1} C_k + Q_k) B_k x, \end{align*} where $Q_k$ arises from the block-diagonalization.

To characterize the fast dynamics, a stretched time $s = t/\epsilon$ is introduced, yielding the fast-transient (boundary-layer) system:

$$\dot{\hat{x}} = 0, \quad \dot{\hat{z}} = D_k \hat{z},$$

revealing that $x$ is frozen while $z$ rapidly contracts to equilibrium if $D_k$ is Hurwitz (Haidar et al., 3 Jul 2025).

3. Auxiliary Single-Scale Reduced and Enriched Dynamics

Three auxiliary switched systems, all evolving on a single (non-singularly perturbed) time-scale, form the core of modern stability analysis for hybrid singularly perturbed systems:

• Reduced Slow System $(\bar\Sigma_\tau)$: Eliminates fast transients. The dynamics are

$$\dot{\bar{x}} = M_k \bar{x}, \quad \bar{x}(t_k) = J_k \bar{x}(t_k^-),$$

with $M_k = A_k - B_k D_k^{-1}C_k$ and $J_k$ the projected jump map.

• Pure-Fast System $(\hat{\Sigma})$: Models the fast transients in stretched time; critical for detecting potentially destabilizing fast behavior.
• Enriched Slow System $(\tilde{\Sigma})$: Enhances the reduced slow model by allowing insertion of finite concatenations of fast flows and jumps into the jump map of the slow system,

$$\dot{\tilde{x}} = M_k \tilde{x}, \quad \tilde{x}(t_k) = \tilde{J}_k \tilde{x}(t_k^-),$$

with $\tilde{J}_k$ ranging over suitable products of jump maps and fast flows.

The stability or instability of the original hybrid SP system is sandwiched between the properties of these auxiliary systems, facilitating analysis using standard (single-scale) tools (Haidar et al., 3 Jul 2025).

4. Sharp Stability Criteria and Sandwiched Lyapunov Exponents

The maximal Lyapunov exponents $\lambda(\Sigma)$ (for a given system $\Sigma$) govern exponential stability/instability. The main theorems establish:

• Necessary Condition: $$\lambda(\bar{\Sigma}_\tau) \leq \liminf_{\epsilon \to 0} \lambda(\Sigma_{K,\tau}^\epsilon)$$. If the reduced system is unstable, so is the full system for small $\epsilon$.
• Sufficient Condition: $$\lambda(\bar{\Sigma}_\tau) \geq \limsup_{\epsilon \to 0} \lambda(\Sigma_{K,\tau}^\epsilon)$$. If the reduced system is exponentially stable, so is the full hybrid SP system for small $\epsilon$.
• No Dwell-Time Case: If the fast-transient system $\hat{\Sigma}$ is unstable, then the full system is unstable for all small $\epsilon$. Exponential stability of the enriched slow system $\tilde{\Sigma}$ also suffices for stability.

A key consequence is that, under Hurwitz conditions and for positive dwell time, the maximal Lyapunov exponents of the $\epsilon$-dependent system converge to those of the reduced system as $\epsilon \to 0$.

The proofs rely on flow comparisons (between $e^{t \Gamma_k^\epsilon}$ and $e^{t M_k}$ for "slow flow," or $e^{t/\epsilon D_k}$ for fast flow), reduction to impulsive switched system analysis, and classical converse Lyapunov theorems (Haidar et al., 3 Jul 2025).

5. Linear Matrix Inequality (LMI)-Based Verification Methods

Verifying exponential stability (ES) of the auxiliary impulsive switched systems can be efficiently performed via quadratic (or polyhedral) Lyapunov functions and standard LMI conditions. For each mode, one seeks $P \succ 0$ and a rate $\gamma > 0$ such that

$$e^{tZ_1}{}^T P e^{tZ_1} - e^{-2\gamma t} P \preceq 0, \qquad Z_2^T P Z_2 - e^{-2\gamma t} P \preceq 0,$$

where $(Z_1, Z_2)$ parameterize the flow and jump dynamics, and $t \geq \tau$ is the dwell-time. Instability is detected by examining products (of flow and jump maps) with spectral radius exceeding unity. Such LMI-based certification is directly applicable to both the reduced and enriched slow systems and underpins practical controller and observer synthesis (Haidar et al., 3 Jul 2025).

6. Illustrative Example and Applications

A two-dimensional case with two modes demonstrates these principles. When both pure-fast and enriched-slow systems are stable—for instance, for $r < 1/\sqrt{3}$ (with $r$ parametrizing the jump map)—and dwell-time and $\epsilon$ are suitably chosen (e.g., $\epsilon = 0.1$), the full singularly perturbed hybrid system exhibits exponential stability. Numerical simulation confirms that the theoretical Lyapunov bounds tightly predict the observed decay rates (Haidar et al., 3 Jul 2025).

Such hybrid SP structures and analyses are crucial in networked control systems, observer design for systems with dual time scales, and time-triggered or event-triggered feedback under communication constraints, where fast transients and impulsive resets coexist (Abdelrahim et al., 2014, Wang et al., 20 Nov 2025, Wang et al., 26 Feb 2025).

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References:

• "Stability criteria for hybrid linear systems with singular perturbations" (Haidar et al., 3 Jul 2025)
• "Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics" (Abdelrahim et al., 2014)
• "Observer Design for Singularly Perturbed Linear Networked Control Systems Subject to Measurement Noise" (Wang et al., 20 Nov 2025)
• "Stabilization of singularly perturbed networked control systems over a single channel" (Wang et al., 26 Feb 2025)