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Hybrid system

A hybrid system is a dynamical system that integrates both continuous and discrete dynamics, where continuous behavior is governed by differential equations representing smooth evolution over time, and discrete behavior involves abrupt changes such as state transitions triggered by events or conditions.^[1]^[2] This combination arises naturally in cyber-physical systems, where computational elements interact with physical processes, enabling the modeling of complex real-world phenomena that neither purely continuous nor purely discrete models can fully capture.^[3]^[4] Key characteristics of hybrid systems include their heterogeneous nature, featuring time-driven continuous flows and event-driven discrete jumps, often formalized using structures like hybrid automata—a graph-based model with nodes representing continuous modes and edges denoting discrete transitions guarded by conditions on continuous variables.^[1]^[5] Stability analysis adapts tools like Lyapunov functions to account for both flow stability within modes and jump stability across transitions, while verification methods employ formal logics such as differential dynamic logic (dL) to prove safety properties amid nondeterminism and uncertainty.^[3]^[1] Modeling approaches also encompass piecewise affine systems for linear dynamics and timed automata for real-time constraints, facilitating simulation and control design.^[6] Hybrid systems find extensive applications in safety-critical domains, including automotive control (e.g., engine management and collision avoidance), robotics (e.g., obstacle navigation), air traffic management, and train control systems, where they ensure reliable coordination between software controllers and physical actuators.^[3]^[1] In chemical processes and manufacturing, they model reactions with switching regimes, while in biology, they describe gene regulatory networks with discrete events like binding.^[1] Tools such as MATLAB/Simulink for simulation and KeYmaera X for theorem proving support their analysis and implementation.^[1]^[7] Recent advancements emphasize scalable verification and data-driven identification; for instance, differential refinement logic extends dL to relate multiple hybrid systems, enabling modular proofs for large-scale cyber-physical applications like the FAA's ACAS X collision avoidance system, which has verified billions of scenarios.^[3] Symbolic regression techniques now identify hybrid models from observational data with high accuracy, as demonstrated in power converters and fluid tank systems, bridging system identification with control theory.^[2] These developments underscore hybrid systems' growing role in autonomous systems and AI-integrated control, addressing challenges in safety and adaptability.^[2]^[3]

Fundamentals

Definition and Characteristics

A hybrid system is a dynamical system that combines continuous-time evolution, characterized by smooth flows governed by differential equations, with discrete events, manifested as instantaneous jumps or transitions. In such systems, state trajectories alternate between periods of continuous dynamics, where variables evolve according to equations like \dot{x} = f(x), and discrete changes triggered by conditions such as guard crossings or external inputs.^[8] This integration arises in cyber-physical systems where physical processes interact with computational logic, enabling the modeling of complex real-world phenomena.^[9] To understand hybrid systems, it is essential to consider their building blocks: continuous dynamical systems and discrete dynamical systems. Continuous dynamical systems describe states evolving smoothly over time via ordinary differential equations, such as \dot{x} = f(x), representing phenomena like mechanical motion or chemical reactions without abrupt changes.^[8] Discrete dynamical systems, conversely, update states at specific instants, often through mappings like x_{k+1} = g(x_k), capturing event-based or sequential behaviors such as digital logic or sampled-data control.^[9] Hybrid systems build upon these by interleaving both, where continuous evolution occurs within discrete modes, and transitions between modes reset or alter the continuous dynamics. Key characteristics of hybrid systems include the presence of continuous state variables that follow differential equations within defined domains and discrete variables or events that initiate mode switches upon meeting guard conditions.^[8] These systems can exhibit distinctive hybrid phenomena, such as Zeno behavior, involving an infinite number of discrete transitions accumulating in finite time; chattering, marked by rapid, oscillatory switching between modes due to conflicting dynamics; and sliding modes, where trajectories are confined to a switching surface, blending continuous and discrete influences through equivalent dynamics.^[9] Such properties arise from the interaction of flows and jumps, often leading to non-smooth or non-unique solutions not observed in purely continuous or discrete settings.^[8] In distinction from purely continuous systems, which lack discrete interruptions and thus cannot model abrupt resets, or purely discrete systems, which overlook smooth evolution between events, hybrid systems unify both to capture event-driven disruptions in ongoing flows, such as impacts or control switches in physical processes.^[9] This synthesis allows for richer behavioral descriptions, essential in applications like robotics and automotive control, though it introduces challenges in analysis due to potential discontinuities.^[8]

Historical Development

The study of hybrid systems emerged in the late 1980s and early 1990s from the intersection of control theory, computer science, and dynamical systems, driven by the need to model embedded systems and real-time computing where discrete events interact with continuous dynamics.^[10] This period saw initial efforts to formalize interactions between digital controllers and physical processes, motivated by applications in safety-critical domains such as manufacturing and transportation.^[11] Early theoretical foundations built on prior work in nonlinear dynamics from the 1970s and 1980s, but the modern field coalesced around addressing verification challenges in these integrated systems.^[12] Key milestones in the 1990s included the development of hybrid automata by Thomas A. Henzinger and collaborators, introduced in 1993 as a computational model for specifying and verifying hybrid behaviors through finite-state machines augmented with continuous variables.^[13] Claire Tomlin advanced the field with applications to air traffic management, contributing game-theoretic approaches to controller design in collaboration with researchers like Shankar Sastry. Foundational publications on verification appeared around 1995, stemming from workshops that compiled algorithmic techniques for reachability analysis in hybrid models.^[14] These efforts established rigorous mathematical frameworks, emphasizing decidability and simulation for practical analysis. Influential figures such as Pravin Varaiya formalized early hybrid models for intelligent vehicle-highway systems, integrating stochastic control with discrete events in the mid-1990s. George Pappas contributed to hierarchical and distributed hybrid control, particularly in multi-agent systems during the late 1990s.^[15] Xenofon Koutsoukos developed supervisory control theories for hybrid systems, focusing on abstraction and compositionality in the early 2000s. The launch of the annual Hybrid Systems: Computation and Control (HSCC) workshop in 1998 facilitated key collaborations and disseminated high-impact research across academia and industry.^[16] The field evolved in the 2000s toward cyber-physical systems, expanding hybrid models to encompass networked computation and physical actuation on larger scales, as seen in initiatives like the U.S. National Science Foundation's CPS program starting in 2008. By the 2020s, integration with machine learning has enabled adaptive hybrid control, where data-driven methods approximate unknown dynamics while preserving safety through formal verification, as demonstrated in learning-based supervisory frameworks for uncertain environments.^[17]

Modeling Formalisms

Hybrid Automata

Hybrid automata serve as a primary formal modeling framework for hybrid systems, capturing the interplay between discrete events and continuous evolution through a structured mathematical model. Introduced as a generalization of timed automata, they enable precise specification of systems where control modes switch discretely while underlying variables follow continuous dynamics governed by differential equations.^[13] The structure of a hybrid automaton H is defined by a tuple (Q, X, Init, Inv, f, E, G, R), where Q is a finite set of locations representing discrete states or modes, X \subseteq \mathbb{R}^n is the continuous state space, Init \subseteq Q \times X specifies initial conditions, and Inv: Q \to 2^X assigns invariants—constraints that the continuous state must satisfy while in a location. Continuous dynamics within each location q \in Q are described by a vector field f(q, x): Q \times X \to \mathbb{R}^n, often in the form \dot{x} = f_q(x). Discrete transitions are given by a set E \subseteq Q \times Q of edges, each equipped with a guard G(e) \subseteq X that enables the transition when the continuous state enters it, and a reset map R(e, x): X \to 2^X that updates the continuous state upon switching locations.^[8]^[18] Formally, the semantics of execution alternate between continuous flows and discrete jumps. In a flow phase within location q, the trajectory x(t) satisfies the differential equation

\dot{x}(t) = f_q(x(t))

for t in some interval, while remaining in Inv(q); the evolution ceases when an invariant boundary is approached or a guard on an outgoing edge is met. A jump then occurs instantaneously: if x(t) \in G(e) for edge e = (q, q'), the state updates to x(t^+) \in R(e, x(t)) and the location changes to q'. This process defines a hybrid trajectory as a piecewise differentiable function over time, combining smooth continuous segments with discrete resets at event times.^[13]^[8] Hybrid automata are graphically depicted as directed multigraphs, with nodes for locations annotated by their invariants and flow equations (e.g., \dot{x} = -0.1x), and directed edges for transitions labeled by guards (e.g., x > 0) and resets (e.g., x' = 0). This visualization facilitates intuitive understanding of mode switches and state constraints.^[18] Variants of hybrid automata adapt the model to specific needs. Timed automata restrict flows to constant rates \dot{x} = 1 for clock variables, with resets to zero and guards/invariants as linear constraints on clocks, enabling decidable verification for real-time systems. Untimed automata abstract away time durations, focusing solely on qualitative sequences of jumps and flows. Probabilistic hybrid automata extend the framework by incorporating probability distributions over resets in transitions (e.g., p_1: x' = g_1(x) + \dots + p_k: x' = g_k(x)) or stochastic differential equations in flows, to model uncertainty in hybrid behaviors.^[18]^[19]

Piecewise and Switched Systems

Piecewise systems model hybrid dynamics by partitioning the state space into a finite number of polyhedral regions, where within each region i, the continuous evolution follows affine dynamics of the form \dot{x} = A_i x + B_i u + c_i, with x \in \mathbb{R}^n the state, u \in \mathbb{R}^m the input, and A_i, B_i, c_i constant matrices and vectors specific to region i.^[20] This partitioning allows representation of nonlinear behaviors through linear approximations in local domains, making piecewise systems a foundational formalism for approximating general hybrid systems.^[21] When trajectories reach boundaries between regions, where the vector field is discontinuous, standard solutions may cease to exist, necessitating generalized solution concepts such as Filippov solutions.^[22] Filippov solutions extend the dynamics via differential inclusions: on a boundary separating regions with vector fields f_1 and f_2, the solution velocity is any point in the convex hull \text{co}\{f_1(x), f_2(x)\}, enabling sliding motions along the boundary if the fields point toward opposite sides.^[22] This framework ensures existence and uniqueness under mild conditions, capturing phenomena like chattering or equilibrium sliding in control applications.^[22] Switched systems, another key modeling approach, describe dynamics where a switching signal \sigma: [0, \infty) \to \{1, 2, \dots, M\} selects the active subsystem from a family of vector fields, yielding \dot{x} = f_{\sigma(t)}(x), with each f_j typically smooth.^[23] Switching can be arbitrary (any measurable \sigma(t)), state-dependent ( \sigma(x) based on current state), or autonomous (governed by additional continuous dynamics triggering switches).^[23] These systems generalize linear time-invariant models by allowing mode transitions over time or state, accommodating uncertainties or deliberate control actions.^[23] Central to both piecewise and switched systems are concepts like multiple equilibria, where subsystems may possess distinct stable points, leading to complex overall behavior under switching.^[23] Boundary crossing rules distinguish transversal crossings, where trajectories pierce the boundary with nonzero relative velocity, from tangential ones, where the flow is parallel and may induce sliding or sticking.^[24] For systems with uncertain or adversarial switching, stability analysis often employs convex hull representations, bounding trajectories within the convex combination of subsystem flows to derive robust guarantees.^[23] Unlike hybrid automata, which rely on discrete locations and event-driven transitions, piecewise and switched systems emphasize continuous spatial or temporal partitions of the dynamics, facilitating optimization-based formulations such as mixed-integer programming for control synthesis.^[20] This focus on regional affine approximations supports efficient computational tools for verification and design in engineering contexts.^[21]

Examples

Bouncing Ball

The bouncing ball serves as a canonical example of a hybrid system, where the ball's motion combines continuous evolution under gravity with discrete events at impacts with the ground. The system is described by the position y \geq 0 (vertical height) and velocity v \in \mathbb{R} of the ball, assuming a point mass under constant gravitational acceleration g > 0. In the continuous phase, while y > 0, the dynamics follow the free-fall equations \dot{y} = v and \dot{v} = -g, resulting in parabolic trajectories. A discrete transition occurs instantaneously upon impact when y = 0 and v < 0, where the post-impact velocity is given by the reset map v^+ = -e v with restitution coefficient $0 < e \leq 1, modeling partial energy loss during the bounce.^[8]^[9] The overall trajectory consists of alternating continuous parabolic arcs during flight, interrupted by instantaneous velocity reversals at each bounce. If e = 1 (perfectly elastic collisions with no energy loss), the inter-bounce intervals are constant, leading to infinitely many bounces over infinite time without Zeno behavior, resulting in periodic motion. For $0 < e < 1, however, the system exhibits Zeno behavior, characterized by infinitely many bounces accumulating in finite time, as the inter-bounce intervals decrease geometrically with ratio e, after which the solution converges to rest at y = 0, v = 0, with the ball remaining on the ground.^[8]^[9] This system can be modeled as a hybrid automaton featuring two modes: a "flying" mode for airborne motion and a "bouncing" mode for the instantaneous impact event. The flying mode has the continuous vector field f(y, v) = (v, -g) and invariant y > 0; the transition to the bouncing mode is triggered by the guard condition y = 0 \wedge v < 0, followed by the reset map v^+ = -e v (with y^+ = 0) and an immediate return to the flying mode. Such modeling captures the hybrid structure without requiring explicit time in the bouncing mode, as the impact is idealized as instantaneous.^[8]^[9] Key phenomena include progressive energy dissipation, where the kinetic energy after each bounce is scaled by e^2 < 1, leading to decreasing maximum heights h_n = h_0 e^{2n} (with initial height h_0) and inter-bounce times t_n = \frac{2 |v_{n-1}|}{g} that form a geometric series converging to a finite total duration before rest. These computations highlight the system's convergence properties, with the total time to rest given by t_\infty = \frac{2 v_0}{g (1 - e)} for initial upward velocity v_0 > 0.^[8]^[9]

Thermostat Control

The thermostat control system exemplifies a controlled hybrid system in which continuous thermal dynamics are supervised by discrete switching logic to maintain room temperature within desired bounds. The room temperature \theta evolves according to the linear differential equation \dot{\theta} = -a(\theta - \theta_a) + b u, where a > 0 represents the heat loss coefficient, \theta_a is the ambient temperature outside the room, b > 0 is the heating power coefficient, and u \in \{0, 1\} is the discrete heater input (0 for off, 1 for on). This model captures Newton's law of cooling for heat diffusion toward the ambient, augmented by a binary heating term when the heater is active.^[25]^[5] The hybrid behavior arises from the interplay between this continuous dynamics and discrete events triggered by temperature thresholds, incorporating hysteresis to avoid rapid oscillations known as chattering. Specifically, the heater turns on (u switches to 1) when \theta \leq \theta_{\min} and turns off (u switches to 0) when \theta \geq \theta_{\max}, with \theta_{\min} < \theta_{\max} defining the hysteresis band (e.g., 19°C and 21°C for a setpoint around 20°C). In the off mode (u = 0), temperature decays toward \theta_a; in the on mode (u = 1), it rises due to the added heating term. This setup prevents infinite switching at a single threshold by enforcing a minimum dwell time in each mode, ensuring finite switching frequency.^[26]^[1] As a switched system, the thermostat can be modeled with two modes corresponding to the heater states, where the continuous flow is governed by the above equation parameterized by u, and guards \theta \leq \theta_{\min} (off to on) and \theta \geq \theta_{\max} (on to off) dictate transitions without state resets. The overall behavior manifests as limit cycles, with temperature periodically oscillating between \theta_{\min} and \theta_{\max} after transients, representing a stable periodic orbit in the hybrid state space. This exemplifies bang-bang control, where the input u saturates at extremes (0 or 1) to optimally track the setpoint under the linear dynamics.^[9]^[25] The switching frequency, inversely related to the hysteresis width \theta_{\max} - \theta_{\min}, influences system efficiency: narrower bands increase frequency, potentially reducing energy efficiency in practical implementations due to transient heat losses during heater startups and shutdowns, though the idealized model assumes instantaneous switching. For example, with a = 0.1, b = 2, \theta_a = 15^\circC, \theta_{\min} = 19^\circC, and \theta_{\max} = 21^\circC, the period of the limit cycle is approximately 5.4 time units.^[1]^[26]

Analysis and Verification

Stability Analysis

Stability analysis in hybrid systems extends classical notions from continuous and discrete dynamics to account for both continuous flows and discrete jumps. A compact set A is uniformly asymptotically stable for a hybrid system if it is uniformly stable—meaning that for every \epsilon > 0, there exists \delta > 0 such that solutions starting within \delta of A remain within \epsilon for all future times, uniformly over initial times—and uniformly attractive, where solutions converge to A within any \epsilon after a time bounded uniformly by initial distance.^[27] This property ensures robust long-term behavior despite mode switches. A Lyapunov characterization requires a continuous, positive definite function V: \mathbb{R}^n \to \mathbb{R}_{\geq 0} such that a \|x\|^2 \leq V(x) \leq b \|x\|^2 for constants a, b > 0, with V nonincreasing along flows (\dot{V}(x) \leq 0) and at jumps (V(x^+) \leq V(x)).^[27] For asymptotic stability, stricter decrease is imposed, such as the Lie derivative satisfying \dot{V}(x) \leq -\alpha(\|x\|) along flows for some class \mathcal{K}_\infty function \alpha, and V(x^+) - V(x) \leq -W(x) at jumps with W(x) > 0.^[27] Key techniques leverage mode-specific analysis, such as multiple Lyapunov functions, where a distinct V_i is associated with each discrete mode i, ensuring decrease within modes and compatibility across jumps (e.g., V_j(x^+) \leq \mu V_i(x) for \mu < 1).^[28] Hybrid invariance principles extend LaSalle's theorem to hybrid settings, where trajectories with \dot{V} \leq 0 and V(x^+) \leq V(x) converge to the largest weakly invariant subset of the level set where the decrease functions vanish, certifying asymptotic stability if this subset is contained in A.^[29] For exponential convergence, conditions like \dot{V}(x) \leq -\alpha \|x\|^2 along flows and V(x^+) \leq \rho V(x) with $0 < \rho < 1 at jumps guarantee uniform exponential stability.^[27] Challenges arise from phenomena unique to hybrid dynamics, such as Zeno equilibria, where solutions exhibit infinitely many jumps in finite time, converging to a point without further evolution; stability requires a Lyapunov function with non-positive flow derivative, non-negative event conditions, and order-preserving reset maps ensuring the Poincaré map contracts distances to the equilibrium.^[30] Sliding modes, occurring when flows are confined to switching surfaces, complicate analysis by introducing Filippov-type differential inclusions. Converse Lyapunov theorems address necessity, asserting that if a hybrid system is asymptotically stable with respect to measures \omega_1, \omega_2, then a smooth Lyapunov function exists satisfying the decrease conditions, with robustness under small perturbations if the system is KLL-stable (satisfying a class-\mathcal{K}LL estimate).^[31] Advanced tools include extensions of the Matrosov theorem, using nested auxiliary functions V_i alongside a Lyapunov-like V_0 to establish uniform global asymptotic stability when direct decrease fails; specifically, under uniform global stability, if \nabla V_i \cdot f \leq u_{c,i}(x) and V_i(g(x)) - V_i(x) \leq u_{d,i}(x) with nested sign conditions on the u's (e.g., if u_{c,k} = 0 for k \leq j, then u_{c,j+1} \leq -\beta(\|x\|) for \beta > 0), convergence to the origin follows without invoking invariance sets.^[32] For interconnected hybrid systems, dissipativity theory provides compositional stability guarantees; a system is QSR-dissipative with supply rate \omega(u,y) = \begin{bmatrix} y^T & u^T \end{bmatrix} \begin{bmatrix} Q & S \\ S^T & R \end{bmatrix} \begin{bmatrix} y \\ u \end{bmatrix} if a storage function decreases accordingly, implying Lyapunov stability for Q \leq 0 and enabling feedback interconnections to preserve dissipativity and thus stability.^[33] Recent advancements as of 2025 include data-driven methods for stability analysis, such as using recurrent neural networks to learn Lyapunov functions and verify stability for complex hybrid systems where traditional methods are computationally intensive. These approaches integrate machine learning with classical theory to handle high-dimensional systems and provide certificates of stability.^[34]

Reachability and Verification

The reachability problem in hybrid systems involves computing the set of states that can be reached from a given set of initial conditions through a combination of continuous flows (evolutions governed by differential equations) and discrete jumps (transitions triggered by guards and resets).^[35] This computation is essential for verifying safety properties, such as whether the system can avoid unsafe regions. However, the reachability problem is undecidable for general classes of nonlinear hybrid systems, even in low dimensions, due to the interplay between continuous and discrete dynamics that can encode undecidable problems like the halting problem.^[36] For linear hybrid systems, decidability holds under certain restrictions, such as bounded time horizons, but nonlinear cases often require approximation techniques.^[37] To address undecidability, several computational methods have been developed for approximate reachability analysis. One approach is abstraction to finite-state models, where the continuous state space is partitioned using predicates to create a discrete quotient system whose reachability can be analyzed via model checking, preserving key properties through refinement.^[38] For linear hybrid systems, polyhedral approximations represent reachable sets as convex polyhedra, iteratively computing flow-pipe enclosures by supporting hyperplanes to bound the evolution under linear dynamics and transitions.^[39] These methods provide over-approximations that are sound for safety verification but may introduce conservatism. Another prominent technique is the level set method based on Hamilton-Jacobi reachability, which formulates the backward reachable set as the subzero level set of a value function satisfying a partial differential equation (PDE) of the form

\frac{\partial V}{\partial t} + \min_u \max_d H(x, \nabla V, u, d) = 0,

where H is the Hamiltonian incorporating system dynamics, controls u, and disturbances d, solved numerically via level set evolution to handle nonlinear hybrids.^[40] This approach excels in controlled hybrid systems by enabling synthesis of safety controllers alongside reachability. Verification tools implement these methods for practical bounded-time analysis of hybrid systems. SpaceEx supports scalable reachability for linear hybrids using support-function-based polyhedral abstractions, handling large state spaces through location-guided simulations and parallel processing.^[41] Flow* applies Taylor model methods for nonlinear hybrids, computing tight flowpipe approximations via remainder bounds and validated numerics to detect safety violations efficiently on benchmarks like navigation systems.^[42] dReach encodes bounded reachability as \delta-complete satisfiability problems over the reals, using SMT solvers like Z3 with interval arithmetic to approximate nonlinear dynamics and provide robustness guarantees against numerical errors.^[43] For probabilistic hybrid systems, statistical model checking tools like UPPAAL SMC estimate satisfaction probabilities of temporal properties through Monte Carlo simulations of stochastic differential equations and Markov chains, offering confidence bounds without exhaustive enumeration.^[44] Recent developments as of 2023 include enhanced tool support for hybrid systems verification, such as improved algorithms for safety analysis in cyber-physical systems, integrating automated theorem proving and machine learning for scalable verification.^[45] Despite these advances, reachability and verification face significant challenges. The curse of dimensionality limits scalability as state-space volume explodes with dimensions, complicating enclosure computations and requiring dimensionality reduction or decomposition strategies.^[46] Handling guards (conditions triggering jumps) and resets (discrete state updates) introduces non-convexity in reachable sets, often necessitating specialized intersection and projection operations that amplify approximation errors. Over-approximation errors can lead to false positives in safety checks, particularly in long-time horizons or under disturbances, demanding tight bounds and validation techniques to ensure reliability.^[47]

Applications

Engineering and Control

In engineering applications, hybrid systems theory enables the design of controllers for devices that exhibit discrete mode switches intertwined with continuous dynamics, such as in automotive powertrains and robotic locomotion. These controllers leverage hybrid models to manage transitions between operating regimes, ensuring stability and performance under varying conditions. For instance, switched systems concepts, where dynamics change based on discrete events, underpin many such designs by allowing seamless integration of multiple control laws.^[48] Recent advancements as of 2025 integrate hybrid systems with AI for autonomous applications, such as in self-driving vehicles where hybrid automata combine with reinforcement learning to handle mode switches in uncertain environments like urban traffic, ensuring safety through verified decision-making policies.^[49] For example, in drone navigation, hybrid models orchestrate discrete collision avoidance triggers with continuous trajectory optimization, achieving robust performance in simulations with over 95% success rates in cluttered spaces.^[50] In the automotive domain, hybrid electric vehicles (HEVs) utilize hybrid system models to orchestrate mode switches between electric motor propulsion and internal combustion engine operation, optimizing fuel efficiency and drivability. A torque-coordinated control strategy for power-split HEVs addresses mode transition stability by compensating for torque fluctuations during switches, achieving smooth shifts with reduced jerk (e.g., limiting peak jerk to under 10 m/s³ in simulations).^[48] Similarly, anti-lock braking systems (ABS) in hybrid vehicles model friction modes as discrete switches between regenerative (electric) and hydraulic braking, preventing wheel lockup while maximizing energy recovery. A supervisory sliding mode controller for hybrid electromagnetic-electrohydraulic ABS regulates slip ratios across low-μ surfaces, demonstrating superior performance over non-ABS braking with stopping distances reduced by up to 20% on icy roads.^[51] These approaches highlight how hybrid modeling captures the nonlinear interactions in braking torque distribution.^[52] Robotics benefits from hybrid systems in generating stable gaits for walking machines, where discrete foot impacts alternate with continuous swing phases. Hybrid zero dynamics (HZD) provides a framework for designing exponentially stable periodic orbits in underactuated bipeds by restricting motion to a low-dimensional zero dynamics manifold, enabling gait synthesis without full-dimensional feedback linearization. For a five-link planar walker like RABBIT, HZD controllers achieve forward locomotion at 1.05 m/s with peak torques of 47 Nm, verified through a scalar return map analysis ensuring exponential stability (e.g., eigenvalues |δ_zero| < 1).^[53] In multi-agent robotic systems, event-based control treats communication triggers as hybrid switches to achieve consensus while minimizing bandwidth use. A decentralized event-triggered strategy for linear agents guarantees convergence to the average initial state for connected graphs, with inter-event times bounded below by τ_D > 0 to prevent Zeno behavior, as proven via hybrid invariance principles.^[54] Cyber-physical systems (CPS) employ hybrid models to handle networked control challenges, such as communication delays modeled as discrete switches between delayed and instantaneous feedback modes. Delay hybrid automata extend standard hybrid automata to incorporate time delays, enabling synthesis of switching controllers that stabilize CPS under bounded perturbations, with reachability analysis ensuring safety in scenarios like vehicle platooning.^[55] Fault-tolerant designs further utilize hybrid observers to estimate and mitigate actuator faults in CPS with intermittent measurements. A hybrid observer scheme generates state and fault estimates via correction terms updated sporadically, achieving exponential input-to-state stability with a prescribed decay rate (e.g., via Lyapunov functions), as demonstrated in aircraft control simulations where estimation errors converge within 5 seconds despite 30% packet loss.^[56] Control synthesis for hybrid systems often relies on optimization techniques tailored to discrete-continuous interactions. Optimal hybrid control formulates problems as mixed-integer nonlinear programs (MINLPs), solved via safeguarded augmented Lagrangian methods that linearize subproblems while preserving integrality, offering efficient local solutions for applications like motion planning (e.g., solving in seconds for 100-stage problems with 100 binary variables).^[57] For piecewise systems, model predictive control (MPC) recasts predictions as mixed-integer quadratic programs (MIQPs), with multiparametric solutions providing explicit piecewise affine policies offline to ensure closed-loop stability. In traction control examples, such MPC reduces computational load to 20 ms per sample while tracking references with errors under 5%, leveraging Lyapunov-based feasibility guarantees. These methods prioritize high-impact contributions, such as Bemporad's MIQP formulations, which have influenced industrial implementations in automotive and process control.^[58]

Biological and Physical Systems

In biological systems, hybrid models effectively capture the discrete events of gene transcription and the continuous dynamics of protein concentrations within gene regulatory networks. For example, in the lac-operon system of Escherichia coli, hybrid approaches divide the phase space into discrete regions based on thresholds for glucose and lactose concentrations, triggering switches in promoter states (e.g., bound or unbound), while continuous differential equations govern the evolution of mRNA and enzyme levels such as β-galactosidase and permease.^[59] This framework accounts for accelerated transcription rates (up to 50-fold) when glucose is low and lactose is present, mediated by the c-AMP-CAP complex, enabling precise simulation of induction dynamics.^[59] Similarly, pharmacokinetic models employ hybrid automata to represent absorption phases, where discrete transitions model the flow of drugs between gastrointestinal compartments (e.g., stomach to small intestine) via recursive equations, coupled with continuous ordinary differential equations for blood concentration profiles.^[60] Such models resolve inconsistencies in traditional first-order kinetics by incorporating formulation-dependent transit times, as demonstrated in simulations of propranolol and 5-ASA disposition.^[60] In physical systems, hybrid modeling describes the switching behavior in electronic circuits, where diodes facilitate continuous current flow under forward bias but trigger discrete state changes upon reaching reverse bias thresholds. In DC-DC buck converters, for instance, the system operates via a hybrid automaton with two modes: one where the switch is closed and the diode is reverse-biased, governed by inductor current accumulation, and another where the switch opens, activating the diode for energy transfer to the load.^[61] Guard conditions based on current limits ensure stable transitions, with multiple Lyapunov functions verifying robustness against parameter variations.^[61] Power systems further exemplify this through fault-induced mode changes, modeled by hybrid automata that switch grid-connected inverters from grid-following to grid-forming operations when voltage drops below thresholds during low-voltage ride-through events.^[62] These transitions involve state resets to clamp currents, improving transient response accuracy over purely continuous models, as evidenced by reduced root-mean-square errors in state estimation.^[62] Key modeling approaches for these natural hybrid phenomena include impulsive differential equations, which incorporate instantaneous events via Dirac delta functions, and phase-field models for gradual material transitions. Impulsive equations take the form

\dot{x}(t) = f(x(t)) + \sum_{k=1}^{\infty} \delta(t - t_k) g(x(t_k^-)),

where continuous flows f(x) are punctuated by discrete jumps g(x) at times t_k, applicable in biology for sudden perturbations like drug impulses and in physics for mechanical impacts.^[63] Phase-field models, conversely, use a continuous order parameter \phi to diffuse phase boundaries, as in the Ginzburg-Landau free energy functional F = \int \left[ f(\phi) + \frac{\epsilon^2}{2} |\nabla \phi|^2 \right] dV, enabling hybrid simulations of non-isothermal solidification where thermal diffusion couples with order parameter evolution.^[64] In binary alloys, this captures solute trapping and dendritic growth via anti-trapping fluxes, bridging microscopic and continuum scales without explicit interfaces.^[64] Hybrid interactions in these systems often yield emergent oscillations or bifurcations, enhancing dynamical complexity. In gene regulatory networks, hybrid feedback loops combining positive and negative regulation produce tunable oscillations, with asymmetry in feedback strengths modulating amplitude and frequency for robust rhythms in processes like the cell cycle.^[65] Bifurcations arise from nonlinear multi-timescale dynamics, expanding oscillatory states beyond single-feedback mechanisms.^[65] In epidemiology, hybrid SIR models treat outbreak thresholds as impulsive jumps, activating vaccination or isolation when susceptible populations exceed critical levels at fixed monitoring intervals, leading to bistable or tristable equilibria depending on control parameters.^[66] These jumps prevent endemic persistence, restoring disease-free states under optimal thresholds.^[66]

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(PDF) The Past, Present and Future of Cyber-Physical Systems
Oct 16, 2025 · This paper is about better engineering of cyber-physical systems (CPSs) through better models. Deterministic models have historically proven ...
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Hybrid automata: An algorithmic approach to the specification and ...
Jun 8, 2005 · We introduce the framework of hybrid automata as a model and specification language for hybrid systems.
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Hybrid Systems III: Verification and Control - SpringerLink
This reference book documents the scientific outcome of the DIMACS/SYCON Workshop on Verification and Control of Hybrid Systems, held at Rutgers University ...
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‪George Pappas‬ - ‪Google Scholar‬
Conflict resolution for air traffic management: A study in multiagent hybrid systems. C Tomlin, GJ Pappas, S Sastry. IEEE Transactions on automatic control 43 ( ...Missing: contributions | Show results with:contributions
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Hybrid Systems: Computation and Control - SpringerLink
Hybrid Systems: Computation and Control. First International Workshop, HSCC'98, Berkeley, California, USA, April 13 - 15, 1998, Proceedings. Conference ...
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[PDF] The Theory of Hybrid Automata y
As embedded computing becomes ubiquitous, hybrid systems are increasingly employed in safety-critical applications, thus making reliability a prime concern.Missing: emergence 1980s 1990s
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[PDF] Safety Verification for Probabilistic Hybrid Systems
In this paper, we focus on the probabilistic hybrid automata model [6], an extension of hybrid automata where the jumps involve probability distributions. This ...
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Optimization-Based Verification and Stability Characterization of ...
Bemporad, G. Ferrari-Trecate, and M. Morari. Observability and controllability of piecewise affine and hybrid systems. IEEE Trans. Automatic Control, to ...
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[PDF] Stability and Stabilization of Piecewise Affine and Hybrid Systems
In this paper we present various algorithms both for stability analysis and state-feedback design for discrete- time piecewise affine systems.
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Differential Equations with Discontinuous Righthand Sides
Download chapter PDF · Introduction. A. F. Filippov. Pages 1-2. Equations with the Right-Hand Side Continuous in x and Discontinuous in t. A. F. Filippov. Pages ...
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Switching in Systems and Control | SpringerLink
Nov 14, 2015 · Switching Control · Front Matter. Pages 73-76. Download chapter PDF · Systems Not Stabilizable by Continuous Feedback. Daniel Liberzon. Pages 77- ...
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Analysis of discrete-time piecewise affine and hybrid systems
The algorithm to obtain the discrete-time PWA representation of an MLD system and vice versa is reported in Bemporad, Ferrari-Trecate and Morari (2000a).
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[PDF] Hybrid System Control - University of Notre Dame
The hybrid system in this example consists of a typical thermostat and furnace. Assuming that the thermostat is set at 70° F, the system behaves as follows.
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[PDF] Lecture #1 Hybrid systems are everywhere: Examples - UCSB ECE
Hybrid Control and Switched Systems. Summary. Examples of hybrid systems. 1. Bouncing ball. 2. Thermostat. 3. Transmission. 4. Inverted pendulum swing-up. 5 ...
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[PDF] Stability Theory for Hybrid Dynamical Systems - UCSB ECE
We call uniformly asymptotically stable if is uniformly stable and if there exits a and for every there exists a such that for all. , and all whenever . Fig. 3.
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https://web.ece.ucsb.edu/~hespanha/ece229/references/MichelHouTAC98.pdf
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[PDF] Invariance principles for hybrid systems with connections to ...
When coupled with stability, our convergence results give new sufficient conditions for asymptotic stability. Special cases include hybrid versions of ...
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[PDF] On the Stability of Zeno Equilibria? - Aaron Ames
We provide sufficient conditions for stability of these equilibria, resulting in sufficient conditions for the existence of Zeno behavior. 1 Introduction.
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[PDF] Converse Lyapunov Theorems and Robust Asymptotic Stability for ...
Hybrid systems are ones whose trajectories can flow in continuous time and also jump at discrete instants. The system variables can be dynamical processes ( ...
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[PDF] A Nested Matrosov Theorem for Hybrid Systems - Ricardo Sanfelice
Our result also shows that. Matrosov's theorem is a reasonable alternative to LaSalle's in- variance principle for time-invariant hybrid systems to conclude.
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[PDF] Dissipativity of Hybrid Systems: Feedback Interconnections and ...
Abstract— This paper focuses on the notions of QSR dissipa- tivity and passivity in hybrid systems. The work presented in this paper can mainly be divided ...
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Reachability within Hybrid Automata
The Reachability problem for Hybrid Automata is the problem of determining, given an automaton $ H$ , whether there is a trajectory of the timed transition ...
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[PDF] Robust Undecidability of Timed and Hybrid Systems - Berkeley EECS
While several orthogonal undecidability results are known for hybrid systems, it is the rectangular reachability problem which best highlights the essential ...<|separator|>
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[PDF] On Reachability for Hybrid Automata over Bounded Time*
We also show that the problem becomes undecidable if either diagonal constraints or both negative and positive rates are allowed. 1 Introduction. The formalism ...
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[PDF] Reachability Analysis of Hybrid Systems via Predicate Abstraction
This paper presents algorithms and tools for reachability analy- sis of hybrid systems by combining the notion of predicate abstraction with recent techniques ...
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Computing polyhedral approximations to flow pipes for dynamic ...
Abstract: This paper presents a new approach to approximating the flows of continuous time dynamic systems from sets of initial conditions.
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Level Set Methods for Computation in Hybrid Systems
We present an implementation of an exact reachability operator for nonlinear hybrid systems. After a brief review of a previously presented algorithm for ...<|separator|>
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[PDF] SpaceEx: Scalable Verification of Hybrid Systems - [Verimag]
Experimental results demonstrate the scalability of the algorithm and the per- formance of the tool. Unlike in classical verification there are no established.Missing: Spacex | Show results with:Spacex
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Flow*: An Analyzer for Non-linear Hybrid Systems | SpringerLink
This paper describes Flow* and demonstrates its performance on a series of non-linear continuous and hybrid system benchmarks. Our comparisons show that Flow* ...
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dReach: δ-Reachability Analysis for Hybrid Systems - SpringerLink
dReach is a bounded reachability analysis tool for nonlinear hybrid systems. It encodes reachability problems of hybrid systems to first-order formulas over ...
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[PDF] Statistical Model Checking for Stochastic Hybrid Systems - arXiv
This paper presents novel extensions and applications of the UPPAAL-SMC model checker. The extensions allow for statistical model checking of stochastic ...
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[PDF] High Dimensional Reachability Analysis - UC Berkeley EECS
Jul 26, 2017 · Alleviating computation burden is in general a primary challenge in formal verification. This thesis presents ways to tackle this “curse of ...
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[PDF] A benchmark suite for hybrid systems reachability analysis?
Since in general the reachability problem is undecidable for hybrid systems, and even the one-step successors can only be computed approximately, experimental ...
[45]
Mode Transition Control of a Power-Split Hybrid Electric Vehicle ...
Nov 13, 2020 · In order to improve the mode switching stability of a power-split hybrid electric vehicle (HEV), a torque coordinated control strategy based ...Missing: automotive | Show results with:automotive
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[PDF] An Anti-Lock Braking Control System for a Hybrid Electromagnetic ...
In the present paper, we present a sliding mode type supervisory ABS control algorithm for a hybrid electromagnetic-electrohydraulic brake-by-wire system for.
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A Survey of Hybrid Braking System Control Methods - MDPI
Aug 16, 2024 · Hybrid braking control involves brake torque distribution and coordinated control of different braking systems, often designed separately.
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[PDF] Hybrid Zero Dynamics of Planar Biped Walkers
This paper presents the design of exponentially sta- ble walking controllers for general planar bipedal systems that have one degree of freedom greater than the ...
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[PDF] Event-Triggered Control for Multi-Agent Systems - DSpace@MIT
Decentralized control of large scale multi-agent systems is currently facilitated by recent technological advances on computing and communication resources.
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[PDF] Switching Controller Synthesis for Delay Hybrid Systems under ...
This paper proposes a new model, delay hybrid automata (dHA), to model delay hybrid systems and a novel approach to synthesize switching controllers for them.
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Fault estimation for cyber–physical systems with intermittent ...
A novel hybrid observer-based fault estimation scheme is proposed to generate estimations of the system state and the fault. Specifically, the correction terms ...
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### Summary of Proposed Solution Technique for Hybrid Optimal Control Problems Using Mixed-Integer Lagrangian Methods
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[PDF] Model Predictive Control of Hybrid Systems - SYSMA@IMT Lucca
MODEL: a model of the plant is needed to predict the future behavior of the plant. •. PREDICTIVE: optimization is based on the predicted future.
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[PDF] Hybrid Model of gene regulatory networks, the case of the lac-operon
The notion of gene regulatory networks comes from cellular biology. ... What we want here is to use the hybrid systems methodology, because it seems to us that ...
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Articles Preliminary Flow Modeling by Hybrid Automata Alternating ...
In this study, we propose flow modeling using hybrid automata that combine ordinary differential equations and recursive equations.
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[PDF] Application of Theory of Hybrid Systems to Control the Switching of ...
Switching between the modes is governed by an additional condition, which ensures the stability of this switching [11]. For the given DC-DC buck converter ...
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[PDF] Linear impulsive differential equations for hybrid systems modeling
This paper shows that the linear impulsive differential equations may promote a unified framework for hybrid systems modeling. 1 Introduction. Physical systems ...
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[PDF] Phase-Field Methods in Material Science and Engineering
We will see that these models constitute a hybrid between traditional phase field theory and atomistic dynamics. After motivating the derivation of phase field.
[60]
Flexible modulation of hybrid feedback loops in competitive ... - Nature
Nov 3, 2025 · The widespread occurrence of hybrid feedback across biological scales suggests an evolutionary optimization for tuneable, robust oscillations.
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Analysis of a hybrid SIR model combining the fixed-moments pulse ...
Sep 15, 2023 · A novel hybrid mathematical model is proposed to describe the susceptibles-triggered vaccination and isolation strategies at fixed monitoring moments.<|control11|><|separator|>