| Recent advances in technology have led to dynamical systems with increasing com-plexity, which in turn demand for more efficient and reliable control systems. Due to the requirements of specific behaviors and stringent performances, Markovian jump linear systems (MJLSs), as a class of stochastic hybrid systems, have been extensively studied. Differing from the nondeterministic switching in switched systems, the modes evolution in MJLSs is determined by a Markov process or chain. It is known that MJLSs find many applications in industrial processes, which are subject to abrupt variations in their structures and parameters, due to random failures or repairs of components, changing sub-system interconnections, sudden environmental disturbances, etc. As a crucial factor, the transition probabilities (TPs) in the jumping process determine the system behavior. How-ever, the phenomenon of defective mode information has been caused due to the physical equipment constraints, the complexity and uncertainty of the external environment in the process of modeling or information transmission. Therefore, it is challenging to develop analysis and synthesis for MJLSs with defective mode information. On the other hand, it is well known that time-delays are frequently encountered in various practical control systems, where aging components, signal transmission delay, and measurement lag usual-ly appear. It has been well recognized that time-delay is an important source of instability and poor performance of a control system. Therefore, the analysis and design for MJLSs with defective mode information and time-varying delay are research subjects of great both theoretical and practical significance.In this thesis, some novel approaches will be developed to solve the robust filtering and controller design problems for several kinds of MJLSs, for instance, MJLSs with defective mode information and time-varying delay and semi-Markovian jump systems (S-MJLSs). The main attention will be focused on the following three cases:(1) defective mode information, which simultaneously contains the exactly known, partially unknown and uncertain TPs, and time-varying transition rates (TRs);(2) time-varying delay with an interval-like type, which means that both the lower and upper bounds of the time-varying delay are available;(3) analysis and synthesis criteria with less conservatism. It is noted that a common analysis approach is to employ stochastic processes to describe the sequences, switches, and statistic properties of stochastic jump systems. Invoking standard results from stochastic systems theory, some new techniques on the analysis and synthesis of MJLSs is proposed. The merit of the proposed approaches lies in their less design conservatism, which is realized by utilizing some more advanced techniques such as new delay-dependent criteria, more powerful relaxation techniques with different filter (controller) structures and filtering (controller) schemes, and new matrix inequality linearisation methods.Firstly, we investigate the problem of delay-dependent robust (?)∞filter design for a class of discrete-time and continuous-time Markovian jump linear systems (MJLSs) with defective mode information and time-varying delay via an input-output (10) approach. For the discrete-time case, the considered systems with defective mode information si-multaneously covers the MJLSs with the exactly known, partially unknown and uncertain TPs, which is more general. By employing two-term constant delay to estimate the time-varying state-delay, which results in the reformulated systems of feedback interconnected form. The stochastic scaled small gain (SSSG) theorem is further used to reduce the analysis and synthesis problem of the original time-varying system to that of the constant delay system. Based on a new Markovian Lyapunov-Krasovskii functional (MLKF), and by fully considering the properties of transition probability matrices (TPMs) and the con-vexification of uncertain domains, a novel bounded real lemma is firstly derived. It is shown that by using a linearisation technique, the corresponding full-and reduced-order (?)∞filter design is cast into a convex optimisation problem in terms of linear matrix in-equalities. For the continuous-time case, the defective mode information under considera-tion refers to the exactly known and partially unknown TRs. Following the similar model reformulation above-mentioned, and constructing a suitable continuous-time Markovian Lyapunov-Krasovskii formulation of SSSG condition, a new delay-dependent (?)∞per-formance analysis criterion is obtained, and the filter synthesis is further developed. Via solving a set of linear matrix inequalities (LMIs), the desired filters can therefore be con-structed.Secondly, attention goes to the delay-dependent robust (?)∞dynamic output feedback controller synthesis for a class of discrete-time MJLSs with defective mode information and time-varying delay in an IO framework. Based on a new discrete-time Markovian Lyapunov-Krasovskii formulation of SSSG condition, and by exploiting the properties of TPMs and the convexification of uncertain domains, delay-dependent (?)∞performance analysis conditions are constructed. Via some linearisation techniques, two approaches, which are based on traditional bounded inequalities and Projection lemma, respective-ly, are proposed to solve the dynamic output feedback controller design problem, which guarantee the stochastic stability of the resulting closed-loop systems with a prescribed robust (?)∞performance level.Thirdly, we restrict our attention to extend the results given in the previous chapter-s to delay-dependent robust memory (?)∞state-feedback controller synthesis for a class of continuous-time MJLSs with defective mode information and time-varying delay. D-iffering from the former defective case, the concerned systems are with such defective mode information in the Markov stochastic process, which simultaneously involves the exactly known, partially unknown and polytopic-type uncertain transition rates. In order to render the unknown diagonal elements numerically tractable, the lower bounds for the unknown diagonal elements are introduced to construct a convex combination. Thanks to this convex combination, together with an augmented approach, the delay-dependent stability analysis criterion and performance analysis condition are obtained. Based on this performance analysis condition combined with some new linearisation techniques, the explicit expressions of the desired memory controller gains can be characterized in terms of strict LMIs.Then, to further relax the conditions of constant TRs on the stochastic process, atten-tion is focused on robust and non-fragile (?)∞filter design for a class of continuous-time semi-Markovian jump linear systems (S-MJLSs) with norm-bounded uncertainties via a singular system approach. By a singular model transformation approach and a semi-Markovian Lyapunov function, a sufficient condition for robust (?)∞performance analysis is firstly derived, and then the non-fragile filter synthesis is developed. It is shown that by a linearisation technique combined with sojourn-time partitioning idea, a unified frame-work can be developed to synthesize both the full-order and reduced-order filters.Finally, we turn to the issue on robust and non-fragile (?)∞static output feedback (SOF) controller synthesis for a class of continuous-time SMJLSs with norm-bounded uncertainties. Based on a semi-Markovian Lyapunov function together with the property of cumulative distribution function, a new bounded real lemma is derived. Via some ma-trix inequality linearisation procedures combined with sojourn-time decomposition idea, two approaches, namely, the convex linearisation approach and iterative approach, are de-veloped to solve the robust and non-fragile SOF controller synthesis problem. It is shown that the controller gains can be obtained by solving a set of strict LMIs or a sequential minimization problem subject to LMI constraints. |
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