Physical processes which are governed by differential equations or difference equations with discontinuous behavior can be modeled as jump systems. An important type of jump systems is the one evolving linearly among the discrete events;​ this type of systems is called jump linear systems. A common analysis approach is to employ stochastic processes to describe the sequences, switches, and statistic properties of the discrete events. In this thesis, the jump linear systems to be studied are governed by semi-Markov processes. This type of jump linear systems is called the semi-Markov jump linear system. Due to the nature of the jump linear system, it finds many applications in networked control systems, fault tolerant control systems, and other systems subject to abrupt changes. It is worthwhile to mention that the well studied Markov jump linear system is a special case of the semi-Markov jump linear system.

The thesis consists of two parts:​ The analysis and synthesis of semi-Markov jump linear systems and networked dynamic systems. In Chapter 2 and Chapter 3, the stochastic stability and optimal control for semi-Markov jump linear systems with or without time delays are investigated. In Chapter 4, a novel fault tolerant control scheme is proposed based on the semi-Markov jump linear system stability conditions. Chapter 5 to Chapter 7 discuss the networked dynamic systems analysis via jump linear system approaches.

The stochastic stability conditions for semi-Markov jump linear systems are firstly derived. The Lyapunov theory is used to establish the sufficient stability conditions by deriving the infinitesimal generator of the Lyapunov function. Since in practice, almost all the system models could not be identified precisely, robust control problems for systems with uncertainties are investigated based on the established stability conditions. Considering the potential applications on networked systems where time delays are inevitable, optimal control problems for systems with time-varying delays have been studied. In the fault tolerant control design, the semi-Markov process is ideal to characterize time-varying failure rates of the system components whose life time is not exponentially distributed. The designed controller is capable of maintaining the stability when an actuator malfunctions.

In the networked control system analysis, stochastic processes are used to model time delays and sensor scheduling rules. Network limitations are compensated by considering more historical information or planning for all possible delays that happen in the future. Both simulations and experiments show the improvements of the control performance by using the proposed techniques. A networked haptic system is investigated via the switching system approach. In the haptic system, the avatar interacts one-dimensionally with a multi-material virtual wall in the virtual environment.The random trajectory along which the avatar moves upon the wall is modeled by stochastic processes, then the multi-material virtual wall rendering is achieved.

Finally, the thesis work is summarized and two future research topics are proposed. One is on the networked control system design where delays are modeled by semi-Markov processes, and the other one is on the event-trigger scheme design for networked dynamic systems.