Abstract The verification of real-time codes implemented in embedded systems is a very important problem, as many of these systems, such as autopilot systems and medical devices, are safety critical and need guarantees of their proper operations (Edwards et al. in Proc IEEE 85(3):366–390, 1997, [1]). These embedded systems are interacting with the physical world, and thus continuous variables, such as time clocks, have to be taken into consideration. Hence, modeling such a system as a hybrid system becomes a natural choice. A typical question is whether certain properties, such as safety (e.g., bad things never happen), liveness (e.g., good things eventually happen), fairness (e.g., all nodes get a chance to transmit), and other properties, hold true or not for a given hybrid system model. Such problems are commonly known as formal verification problems. In this chapter, we will focus on the model checking approach in the formal verification literature since model checking can be fully automated and has been gaining a lot of attention from both academia and industry. Other approaches, such as deductive approaches, will be briefly mentioned at the end of this chapter.

Abstract Hybrid dynamical systems contain both time-driven continuous dynamics and event-driven discrete dynamics that interact with each other, and their interactions determine the behavior of the system over time. To fully understand the tight interactions between discrete and continuous dynamics, we need a unified modeling framework. Various models have been developed for hybrid systems; they can be broadly classified into two groups according to their origins. On the one hand, computer scientists look at hybrid systems primarily as discrete (computer) programs interacting with the continuous environment. They extend their computational models, such as finite state machines, automata, and Petri nets, from discrete systems to hybrid systems by embedding the continuous-variable dynamics into these discrete models. On the other hand, researchers from the areas of dynamical systems and control regard hybrid systems as a collection of differential/difference equations with discontinuous or multi-valued variables. They extend the models and methodologies for traditional continuous-variable systems, such as those described by ordinary differential/difference equations, by including discrete variables to describe the jumping or switching phenomena. Both approaches have their own strengths. For instance, the first group of approaches has traditionally been able to deal with complex discrete dynamics emphasizing analysis results (verification) and simulation methodologies, while the second group typically deals with complex continuous-variable dynamics and mainly concerns itself with stability, robustness, optimality, and synthesis issues.

Youla parameterization is an effective method to control stable processes. The controller is designed in open loop to provide good reference signal tracking. Extending the structure with IMC–Internal Model Control disturbance rejection is ensured. Application of this method is shown.

If a system is characterized by its state equation, it is straightforward to design a controller based on this system representation. The control signal is created by feeding back the state variables of the system through constant gains. These state feedback vector can be calculated to ensure the prescribed location of the poles of the control system. If the state variables are not measurable, they have to be estimated and then state feedback is realized from the estimated state variables. Properties of control systems with state feedback are discussed.

In a sampled-data control circuit the regulator can be designed in the frequency domain taking into account that by sampling and applying a zero order hold the system behaves as if additional dead-time had occurred in the system. Discrete PID controllers considering this effect are designed with pole cancellation technique.