Accelerated Iterative Algorithms For Solving Coupled Lyapunov Equations Of Markovian Jump Linear Systems

Markov jump system is a kind of stochastic systems with lots of modes. It occurs based on the facts that the structure of actual systems will change variedly due to sudden environmental changes, internal component failure or repair. Many systems, such as systems involve both random and sequential control of decision-making,electric systems and communication systems, have this kind of mode. Markov jump system has won wide concern of the community because of its broad application. Traditional control theory and methods can’t apply to the research of Markov jump system, because of its special hybrid structure. Therefore, the research of Markov jump system is a challenging work.The most commonly method that used to solve Markov jump Lyapunov equations is Kronecker product. But it requires more computational efforts and more storage space in implementation especially in high dimensional systems. To solve the problem, we put forward two accelerated iterative algorithms: implicit accelerated iterative algorithm and accelerated gradient iterative algorithm. The implicit accelerated iterative algorithm uses the latest estimates to update the final solution, which makes the real-time of the solution better and faster. Gradient acceleration iterative algorithm is proposed based on some gradient iterative algorithms we often used, when solving Sylvester matrix. Compared to other algorithms, the algorithm in this thesis required less storage capacity and less computation. Similarly, accelerated gradient iterative algorithm uses the latest estimates to update the final solution. We compare those two algorithms in the subsequent examples analysis. Compared to the general iterative algorithm, implicit accelerated iterative algorithm has a great advantage on the convergence rate with an ideal error. Compared to general gradient iterative algorithm, the convergence speed of accelerated gradient iterative algorithm is faster and its error is smaller in condition of appropriate step size. We use the usual three-dimensional system example to verify two algorithms proposed in the thesis.