The mathematical model of a system is traditionally associated with differential or difference equations, typically derived from physical laws governing the dynamics of the system under consideration. Consequently, most of the control theory and tools have been developed for such systems, in particular for systems whose evolution is described by smooth linear or nonlinear state transition functions. Recent technological innovations have caused a considerable interest in the study of dynamical processes of a heterogeneous continuous and discrete nature, denoted as hybrid systems, characterized by the interaction of continuous-time models (governed by differential or difference equations), and of logic rules and discrete event systems (described by temporal logic, finite state machines, if-then-else rules, etc.) and discrete components (on/off switches, digital circuitry, software code, etc.).
Hybrid systems switch between many operating modes where each mode is governed by its own characteristic dynamical laws. Mode transitions are triggered by variables crossing specific thresholds (state events), by the elapse of certain time periods (time events), or by external inputs (input events). Typical hybrid systems are embedded systems, constituted by dynamical components governed by logical/discrete decision components. Complex systems organized in hierachical way, where for instance discrete planning algorithms interact with continuous control algorithms, are another example of hybrid systems. In these systems, a hierarchical organization helps manage the complexity of the system, as higher levels in the hierarchy require less detailed models (=abstractions) of the functioning of the lower levels.
As an example of hybrid control problem consider the design of a cruise control system that commands the gear shift, the engine torque, and the braking force in order to track a desired vehicle speed while minimizing fuel consumption and emissions. Designing a control law that optimally selects both the discrete inputs (gears) and continuous inputs (torque and brakes) requires a hybrid model that includes the continuous dynamics of the power train, the discrete logic of the gearbox, and consumption/emission maps.
A gasoline engine has also a natural hybrid representation: the power train, gas flow, and thermal dynamics are continuous processes, while the pistons have four modes of operation which can be described as a discrete event process or a finite state machine. These two heterogeneous processes interact tightly, as the timing of the transitions between two phases of the pistons is determined by the continuous dynamics of the power train, which, in turn, depends on the torque produced by each piston.
The interest in hybrid systems has considerably grown over the last few years. It is partly because of the theoretical challenges involved in the study of hybrid systems, but also because of their impact on applications in several industrial contexts, such as manufacturing, communication networks, aerospace, robotics, traffic control, chemical processes, and the aforementioned automotive applications. Currently, the synthesis of control schemes for hybrid systems is approached with heuristic rules, usually driven by engineering insight and experience, with a consequently long design and verification process. The interest of the control community is motivated by several clearly discernible trends in industry which point toward an extended need for new tools to design control/supervisory schemes for hybrid systems and to analyze their stability, safety, and performance.
The main research efforts in the study of hybrid systems can be categorized in the following sub-areas:
- Synthesis of control schemes, that govern both discrete and continuous actuators and are designed on the basis of a hybrid model of the system, and on-line monitoring schemes that are able to estimate unmeasured quantities and the occurrence of faults (a fault can be interpreted as a switch of dynamic mode).
- Formal verification of safety properties, which aims at certifying that for a given set of initial conditions and a class of exogenous disturbances the hybrid system never enters unsafe areas, and provides counterexamples that can be for instance used by the designer as test-cases in a detailed simulator of the system.
- Stability, robustness, and performance analysis. There is a rich theory and powerful tools for linear systems, nonlinear continuous systems, and for discrete systems, but still a substantial need for results for hybrid systems.
- System theoretical properties, such as existence, uniqueness, persistence of trajectories, observability, reachability, have to be deeply investigated.