Hybrid systems have received a wide interest from the systems and control community in the last decades thanks to their ability to describe, in a rigorous mathematical formalism, systems characterized by the presence and interaction of discrete and continuous dynamics. These two are also referred to as the “cyber” and “physical” parts of what are called cyber-physical systems (CPS). Hybrid modeling techniques thus play a fundamental role for the design, analysis, and control of tomorrow's CPS, thanks to their ability to describe complex and nonlinear phenomena via the interaction of several simpler models, which not only facilitates the task of modeling hybrid systems from data, but also the following tasks for which the models are used, such as performance analysis, controller synthesis, fault detection, and diagnosis. This special issue collects 13 innovative and advanced contributions in the field of hybrid modeling and control, which include new algorithms for identification and analysis of hybrid systems, new methods for observer and control design, and contributions considering real-world problems (such as real-time detection of packet-loss rate in networked control systems and optimal scheduling of refrigerator systems) that can be solved by leveraging on a hybrid modeling framework. The first two contributions are dedicated to data-driven modeling of two instances of hybrid models, namely switching and piecewise affine (PWA) models. In particular, the first paper Identification of switched autoregressive systems exogenous from large noisy data sets by S. Hojjatinia, C. Lagoa, and F. Dabbene considers the problem of identification of switching autoregressive linear models affected by measurement and process noise. The proposed algorithm provides a consistent estimate of the model parameters and is able to handle large-size data sets. In the second contribution Identification of hybrid and LPV models via piecewise affine regression using mixed integer programming by M.D. Mejari, V. Naik, D. Piga, and A. Bemporad the authors present a two-stage regularized moving-horizon algorithm for PWA regression which simultaneously estimates the model parameters and iteratively clusters the training samples. Thanks to the ability of PWA functions to approximate any (smooth) nonlinear function, the proposed algorithm can also be used for data-driven modeling of more general nonlinear dynamical systems. The third paper Stochastic approximation with non-decaying gain: error bounds and data-driven gain-tuning by J. Zhu and J. Spall addresses the problem of optimizing a (noise-corrupted) time-varying objective function through a recursive stochastic approximation algorithm with nondecaying gains. A detailed analysis of the tracking performance of the algorithm is also provided. The proposed approach is applied to detect jumps/changes in hybrid systems. Furthermore, the results of the paper look promising for recursive identification of the parameters of hybrid (switching) models. The fourth paper Minimality and identifiability of discrete-time SARX systems by M. Petreczky, L. Bako, S. Lecoeuche, and K.M.D. Motchon introduces the concept of minimality and identifiability of switching linear autoregressive models with exogenous input, and it provides necessary and sufficient conditions for minimality and identifiability. The fifth paper Hybrid dynamical systems with hybrid inputs: definition of solutions and applications to interconnections by P. Bernard and R.G. Sanfelice provides new definitions of solutions for hybrid dynamical models (both in continuous and discrete time) excited by “hybrid” (noncontinuous) inputs. An algorithm that automatically performs the construction of solutions is given and issues related to its numerical implementation are also discussed. Stability analysis is discussed in the next two contributions. In particular, the sixth paper Analysis of pulse width modulation controlled systems based on a piecewise-affine description by P. Massioni, L. Bako, G. Scorletti, and A. Trofino analyzes asymptotic stability of feedback loops with pulse width modulation (PMW) control schemes. By keeping the duty cycle as a parameter, the closed-loop system is described in terms of PWA models with parameter-varying matrices. This description allows the authors to use well-known and well-settled tools developed for stability analysis of PWA models. Sum of squares (SOS) relaxations are then used to find Lyapunov-like function and to determine an estimate of the region of attraction via semidefinite programming. The seventh paper Analysis and design of event-triggered control algorithms using hybrid systems tools by J. Chai, P. Casau, and R.G. Sanfelice exploits tools from the theory of hybrid systems for analysis of the behavior of closed-loop systems with event-triggered controllers. Control synthesis is also considered. Design of state observers with Luenberger-like structure for linear hybrid systems is addressed in the eighth paper State estimation for linear hybrid systems with periodic jumps and unknown inputs by H. Ríos, J. Davila, and A. Teel. The case of periodic jumps and unknown inputs is considered and the observer design is formulated in terms of linear matrix inequalities (LMIs). To handle the unknown inputs, a sliding-mode observer is added to the Luenberger-like observer structure, which provides exponential convergence to zero of the state estimation error despite certain class of unknown inputs. Control design is addressed in the following three contributions. The ninth paper Robust constrained control of piecewise affine systems through set-based reachability computations by R. Desimini and M. Prandini proposes a method to design a robust static state-feedback controller for uncertain PWA systems with input constraints. This controller guarantees that the closed-loop system satisfies some finite-horizon robust specifications given in terms of constraints on the admissible value of the state. The 10th paper Necessary conditions for a class of optimal multiprocess with state constraints by K. Cortez, M. de Pinho, and A. Matos addresses the problem of designing optimal multiprocess control systems, which can be seen as a special case of hybrid controllers. In order to show the effectiveness of the proposed approach, an example concerning path planning for autonomous underwater vehicle (AUV) is considered, where the AUV should be driven from a fixed initial position to a target, and in between it should visit a particular space region. The 11th contribution Direct data-driven design of switching controllers by V. Breschi and S. Formentin is focused on data-driven (model-free) control design, where a PWA controller is synthesized based on input-output data generated by the plant to be controlled, without the time-consuming and costly intermediate step of deriving a model of the plant. Statistical properties of the noise corrupting data are taken into account in the control design. Real-world control problems are discussed in the last two contributions of the special issue. In particular, the paper Packet loss detection in networked control systems by M. Baglietto, G. Battistelli, and P. Tesi employs the hybrid modeling paradigm to describe the behavior of a network control system with packet loss, where the operating mode of the hybrid model depends on the transmission outcomes. Using this hybrid formulation, a moving-horizon estimation algorithm is designed to detect in real time and directly from sensor data if the packet-loss rate exceeds a prescribed critical threshold. The paper MINLP-based hybrid strategy for operating mode selection of TES-backed-up refrigeration systems by G. Bejarano, D. Rodríguez, J.M. Lemos, M. Vargas Villanueva, and M. Ortega ends the volume and is focused on optimal scheduling of a vapor-compression refrigeration cycle and a thermal energy storage. The scheduling problem is posed in terms of a hybrid nonlinear model predictive control problem, with economic objectives and constraints defined according to energy price forecast, cooling demand satisfaction, limits on the storable cold energy, and power feasibility. We hope that this special issue will be able to highlight the power and the importance of the hybrid modeling paradigm to researchers and practitioners, underlining what current challenges and open problems are in the field, to ultimately stimulate further research and applications in this area.

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