Balanced Truncation For Markovian Jump Systems

Since high-order complex mathematical modeling of physical systems and pro-cesses in many areas of engineering is frequently encountered, which brings severehardship to analysis and synthesis the concerning system. Thereby, the past decadeshave witnessed extensive research on the problem of simplifying these models withrespect to certain criterion.MJSs are widely investigated to deal with these systems with variable structureswhose structures change randomly at discrete time instances governed by a Markovianprocess. The ideal knowledge on the transition probabilities is definitely expected tosimplify the system analysis and design. In fact, whether in theory or in practice,the possibilities of obtaining the completely knowledge of the transition probabilitiesare questionable and the cost is probably high. Thus, rather than the complexity ofmeasuring or estimating all the elements of the transition probabilities matrix, it issignificant and necessary from control theory perspectives to study further more gen-eral MJSs with partially unknown transition probabilities. On the other hand, timelag phenomenon is very common for the limitation of the signal transmission and in-formation processing speed in industrial chemical process and communication systemand brings negative effects to the static and dynamic performance of MJSs.Owing to the importance of the topic of balanced truncation for MJSs, the resultsin this area should be significance. In this thesis, based on the dissipation inequalities,the problems of balanced truncation for various MJSs have been investigated:Firstly, the model reduction problem for discrete-time MJSs based on balancingis investigated. Two linear matrix inequalities (LMIs) are proposed to find the grami-ans using the dissipation inequalities that is in conjunction with a suitably definedstorage function. A cost function is introduced to find the optimal transition matrixT such that the cost function is minimized. Then by balancing transformation, thereduced order model with the same structure as that of the original one is obtained bytruncating the balanced model. For the obtain reduced order model, the stochasticallystable is preserved under simultaneous balanced truncation. An upper bound of themodel reduction error is guaranteed in the sense of perturbation operator norm.Secondly, the problem of balanced truncation for discrete-and continuous-timeMJSs with partly unknown transition probabilities is considered. A cost function is also introduced to find the optimal transition matrix T such that the cost functionis minimized. By balancing transformation, the reduced order model with the samestructure as that of the original one has been obtained by truncating the balancedmodel. In the paper, we have presented the algorithms of the MJSs with the partiallyknown transition probabilities using the dissipation inequalities. The approach of themodel reduction preserves the structure of the original systems and important systemproperties like stability.Thirdly, the problem of balanced truncation for discrete-and continuous-timeuncertain MJSs with time-varying delay is studied. Based on a stochastic Lyapunovfunction, a reciprocally convex approach is developed for the time-varying delay ofthe MJSs. Instead of the dissipation inequalities, we choose to use the generalizeddissipation inequalities with a suitably defined storage function to find the generalizedgramians. The physical interpretations of the generalized gramians are similar to theordinary gramians such that it is avoided to introduce the cost function. Then bybalancing transformation, the reduced order model with the same structure as that ofthe original one is obtained by truncating the balanced model.