Abstract
In real-world applications, dynamic systems often encounter abrupt changes in parameters or structures due to various uncertainties, such as environmental disturbances, equipment malfunctions, and erratic subsystem interconnections, etc. Markov jump systems, an important class of stochastic systems whose dynamics switch between several operating modes according to a Markov chain, provide an effective mathematical framework for describing such systems with abrupt parametric and structural changes. The research on Markov jump systems has received enormous interest in the past decades, and a great deal of theoretical results have been obtained. Most existing results on Markov jump systems commonly rely on some ideal assumptions, for example, i) system mode can always be correctly detected, and ii) the transition probabilities of the Markov chain are fixed (time-invariant) and fully known. Although these ideal assumptions can significantly simplify theoretical analysis, they do not always hold in real-world applications, thus limiting the application of those theoretical findings.The so-called mismatched mode (asynchronous) phenomenon occurs when system mode cannot always be correctly detected. This phenomenon can be effectively characterized by hidden Markov model, in which the relationship between the system mode and detected mode is formulated via conditional probabilities. Additionally, hidden Markov model provides a unified analysis/design framework for Markov jump systems because it includes synchronous, asynchronous, and mode-independent schemes as special cases. Therefore, the asynchronous control of Markov jump systems with mismatched modes received enormous interest in recent years, and a great deal of theoretical results have been proposed for Markov jump systems. However, those results commonly focus on time-invariant and fully known transition probabilities, whereas the asynchronous control problems of Markov jump systems with complex (i.e., time-varying and/or imperfectly known) transition probabilities have not been fully studied yet, which need further investigation.
Motivated by the aforementioned discussions, this thesis investigates the asynchronous control problems of discrete-time Markov jump linear systems with mismatched modes and complex transition probabilities. The main results of this thesis can be divided into three parts, summarized as follows:
1. Asynchronous control of homogeneous Markov jump linear systems. This part considers the asynchronous stabilization and H∞ control problems of Markov jump linear systems with imperfectly known transition and observation probabilities. More specifically, a hidden Markov model is used to characterize the mismatched mode phenomenon, and a general description is used to characterize the imperfectly known transition and observation probabilities. Additionally, novel nonconservative probability decoupling principles are developed so that several equivalent stability conditions are obtained. For the stabilization problem, it is shown that the asynchronous control law can be designed by one of the stability conditions in the form of linear matrix inequalities. Subsequently, the asynchronous H∞ control of networked Markov jump systems under denial-of-service attacks is further considered, and several stability conditions with H∞ performance are obtained by means of a piecewise homogeneous Lyapunov functional. It is shown that the asynchronous control law can be designed by solving an LMI-based convex optimization algorithm.
2. Asynchronous control of piecewise homogeneous Markov jump linear systems. This part studies the asynchronous stabilization and passivity-based control problems for a class of piecewise homogeneous Markov jump linear systems with stochastically switching transition probabilities subjected to a higher-level Markov chain. A dual hidden Markov model is introduced to characterize the mismatched modes phenomenon, and all four probability matrices of the dual hidden Markov model are assumed to be partly known. By means of a dual-mode-dependent Lyapunov functional, stochastic stability and passivity conditions are obtained for the concerned systems, respectively. Additionally, nonconservative probability decoupling principles are developed to deal with the simultaneous existence of unknown probabilities in the four probability matrices. It is shown that the proposed stability conditions can recover some results in the literature without sacrifice of any conservatism, and the obtained passivity conditions are less conservative than two results in the literature for homogeneous Markov jump systems.
3. Asynchronous control of nonhomogeneous Markov jump linear systems. This part considers the dissipativity-based asynchronous control problem of nonhomogeneous Markov jump systems with polytopic time-varying transition probabilities. A nonhomogeneous Markov model, whose transition and observation probabilities are assumed to evolve in two convex polytopes, respectively, is introduced to characterize the mismatched mode phenomenon. By means of dual-parameter-dependent Lyapunov functional and a nonconservative probability decoupling principle, several sufficient conditions are obtained for the (Q,S,R)-γ-dissipativity of the concerned system. It is shown that the results obtained under the dual-parameter-dependent Lyapunov functional are less conservative than those under the single-parameter-dependent and parameter-independent ones.
| Date of Award | 8 Sept 2025 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Gang Gary FENG (Supervisor) & Lu LIU (Co-supervisor) |