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This chapter introduces the class of stochastic switching systems we discuss in this book by giving the motivation for studying it. After giving some practical systems, it also defines the problems we deal with. The contents of this book can be viewed as an extension of the class of linear time-invariant systems studied extensively in the last few decades. As will be shown by some examples, this class of systems is more general since it allows the modeling of systems with some abrupt changes in the state equation that cannot be described using the class of linear time-invariant systems. In this volume we concentrate mainly on the linear case, which has been extensively studied and reported in the literature. References [52, 12, 45, 51] and the references therein are particularly noted. But we would like to advise the reader that nonlinear models have also been introduced; we again refer the reader to [12, 45, 52] and the references therein.

Consider a linear time-invariant system with the following dynamics: where () ∈ ℝ is the state vector at time , ∈ ℝ is the initial state, and is a constant known matrix with appropriate dimension. The stability of this class of systems has been extensively studied and many interesting results can be used to check the stability of a given system of this class. Lyapunov equations or equivalent LMI conditions are often used to check stability.

One of the most popular control problems, the stabilization problem consists of determining a control law that forces the closed-loop state equation of a given system to guarantee the desired design performances. This problem has and continues to attract many researchers from the control community and many techniques can be used to solve the stabilization problem for dynamical systems. From the practical point of view when designing any control system, the stabilization problem is the most important in the design phase since it will give the desired performances to the designed control system.

In the stabilization chapter we discussed the design of controllers that guarantee the stochastic stability of the closed loop for nominal and uncertain dynamical systems belonging to the class of piecewise deterministic systems we are considering in this book. In practice we are interested in more than stability and its robustness; for instance, designing a controller that rejects the effect of external disturbance that may act on the system dynamics. Among the controllers we can use to reach this goal, the linear quadratic regulator is a good candidate for stabilizing and rejecting the effect of disturbances for the class of piecewise deterministic systems. Unfortunately this approach requires special assumptions on these external disturbances that should be Gaussian with some given statistical properties that are difficult to satisfy.

In the previous chapters we assumed complete access to the state vector to compute state feedback control. But as is well known, sometimes this access is not possible for physical or cost reasons. In order to continue to use the state feedback controller, we can estimate the state vector. This technique has been used for many years and continues in many industrial applications ranging from aerospace to economics, including engineering, biology, geoscience, and management. In real-time applications, care should be taken to guarantee that the estimation dynamics be faster than those of the closed loop of the considered systems.

In the previous chapters we dealt with many classes of systems, including regular stochastic switching systems with and without Brownian disturbance. For these classes of systems we studied the stability and stabilization problems. Many stabilization techniques have been considered and most of the results we developed are in the LMI framework, which makes the results powerful and tractable.