By Prof. Mufeed M. Mahmoud, Prof. Jin Jiang, Prof Youmin Zhang (auth.)

Modern technological platforms depend upon refined keep watch over features to fulfill elevated functionality requisites. For such platforms, Fault Tolerant keep an eye on platforms (FTCS) must be constructed. lively FTCS are depending on a Fault Detection and identity (FDI) technique to observe procedure functionality and to discover and isolate faults within the platforms. the most target of this ebook is to check and to validate a few vital concerns in real-time energetic FTCS via theoretical research and simulation. numerous versions are offered to accomplish this target, considering sensible elements of the procedure to be managed, functionality deterioration in FDI algorithms, and obstacles in reconfigurable keep an eye on laws.

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**Additional resources for Active Fault Tolerant Control Systems: Stochastic Analysis and Synthesis**

**Example text**

Complexity arises due to the stochastic behavior of the combined AFTCS, and vitality is the natural result of practical constraints arising from application studies. Modelling AFTCS to account for practical limitations and stochastic stability of the developed AFTCS models are the prime focus of this book. To better characterize the stochastic behavior of AFTCS, diŞerent concepts and tools used to study stochastic stability for general stochastic systems are presented with some detail in next chapter.

CONDITIONS FOR STOCHASTIC STABILITY These two works concerned with the moment stability. Bucy 7- was the 0rst to real, ize that stochastic Lyapunov function must have the supermartingale property while studying nonlinear stochastic stability. A systematic treatment was carried out by Kushner 879 85- and by Khasminskii - to study stochastic systems with white noise. Other works include Wonham 559 5 -9 Khashminskii 5-9 Kats (-9 and Rabotnikov (7-. The Lyapunov function approach for stochastic systems is brie3y outlined as follows.

2. DEFINITIONS OF STOCHASTIC STABILITY are three modes of convergence: convergence in probability, convergence in the mean, and almost sure convergence, there are at lease three times as many de nitions for the stochastic stability as there are for the deterministic stability . 2 De nitions of Stochastic Stability In this book, we are concerned with the Lyapunov concept of stability In simple words, Lyapunov stability criterion is concerned with the deviation of the system state vector from a given solution when the initial conditions are close to the initial conditions of the given solution To de ne stochastic stability one needs to state the concept of Lyapunov stability for deterministic systems Without loss of generality, the equilibrium solution, x = 0, is considered as the solution whose stability property is being tested Let xo be the initial state at the initial time to, the solution with initial state xo at time t will be denoted as x(t xo to) 2 Rn Moreover, unless speci ed otherwise, the norm is de ned as: kxkm = Xn j xi jm i=1 The following de nitions were developed in .