Stability And Optimal Control Of Discrete-time Antilinear Systems Withmultiplicative Noise

The main purpose of this dissertation is to find out the motion characteristics and quadratic optimal control for the antilinear discrete-time systems which are subject to multiplicative noise. Firstly, the definitions of stability, stabilizability, observability and detectability for this kind of systems are put forwarded. Then, the relationship between the anti-Lyapunov equation and the stability of systems is established. Finally, finite time and infinite time quadratic control problems are discussed and solved completely by solving the corresponding anti-GDRE(antilinear Generalized Difference Riccati Equation) and anti-GARE(antilinear Generalized Algebra Riccati Equation).When investigating the motion characteristics of this kind of systems, the definition of asymptotic mean square stability is given based on the definition of asymptotic mean square stability of linear stochastic systems. And the definitions of detectability and observability are also given based on the ideas that “detectability is equivalent to that all the unstable models produce some non-zero outputs” and “observability is equivalent to that all non-trivial solutions(not only the unstable ones) caused by some non-zero outputs”. When addressing the stability problem, the discrete-time anti-Lyapunov equation is obtained and it is proved that the system is asymptotic mean square stable, if and only if the anti-Lyapunov equation has a positive definite solution. Furthermore, the positive definite solution is proved to be unique by means of real representation. It is also showed that observability and detectability have an important application in solving the anti-Lyapunov equation and the anti-GARE. If the system is asymptotic mean square stable and observable, the corresponding anti-Lyapunov equation has a positive definite solution. If the system is stabilizable and the relevant system is observable, the corresponding anti-GARE has a positive semi-definite solution.The main idea of linear quadratic optimal control is to design a state feedback controller to minimize the cost function. Based on the same idea, the quadratic optimal control problems of antilinear stochastic systems are put forward. For finite time and infinite time quadratic optimal control problems, the anti-GDRE and the anti-GARE are obtained respectively through dynamic programming. Then the optimal controller can be expressed and the minimum value of the cost function can be given by solving the anti-GDRE or the anti-GARE. In order to generalize the quadratic control problems, the condition that the weight matrix Q and R in the cost function are positive definite is removed. And the definitions of well-posedness and attainability of the optimal problems are proposed. It is turned out that for these quadratic optimal problems the well-posedness is equivalent to the attainability. Finally, the connection between the well-posedness and a certain matrix inequality is established. It is proven that the attainability is also equivalent to the solvability of the corresponding anti-GDRE.