Backstepping controller design for a class of stochastic nonlinear systems with Markovian switching

A more general class of stochastic nonlinear systems with irreducible homogenous Markovian switching are considered in this paper. As preliminaries, the stability criteria and the existence theorem of strong solutions are first presented by using the inequality of mathematic expectation of a Lyapunov function. The state-feedback controller is designed by regarding Markovian switching as constant such that the closed-loop system has a unique solution, and the equilibrium is asymptotically stable in probability in the large. The output-feedback controller is designed based on a quadratic-plus-quartic-form Lyapunov function such that the closed-loop system has a unique solution with the equilibrium being asymptotically stable in probability in the large in the unbiased case and has a unique bounded-in-probability solution in the biased case.     

On hybrid control of a class of stochastic non-linear Markovian switching systems

In this paper, we address the problem of hybrid control for a class of stochastic non-linear Markovian switching systems. First, a hybrid controller is introduced for the systems. Then under some appropriate assumptions, the stabilization condition for the systems under pure impulsive control is given. Further under impulsive control, the output feedback stabilization problem of the systems is discussed and linear output feedback controllers are designed. Finally a numerical example is provided to illustrate the effectiveness of the proposed methods.     

Delay-segment-dependent robust stability for uncertain discrete stochastic Markovian jumping systems with interval time delay

This article deals with the problem of robust stochastic stability for a class of uncertain discrete stochastic Markovian jumping systems with time-varying interval delay. By constructing a parameter-dependent Lyapunov–Krasovskii functional and checking its difference in two subintervals, respectively, some novel delay-segment-dependent stability criteria for the addressed system are derived. Two simulation examples are given to show effectiveness of the proposed method.     

Sliding mode control for Itô stochastic systems with Markovian switching

This paper deals with the problem of sliding mode control (SMC) for a class of nonlinear uncertain stochastic systems with Markovian switching. By introducing some specified matrices, the connections among the designed sliding surfaces corresponding to every mode are established. Furthermore, the present sliding mode controller including the transition rates of modes can cope with the effect of Markovian switching. By means of linear matrix inequalities (LMIs) with equality constraint, sufficient conditions are derived such that the sliding motions on the specified sliding surfaces are stochastically stable with γ-disturbance attenuation level. Finally, a numerical example is given to illustrate the applicability of the present method.     

Controllability of multi-agent systems with periodically switching topologies and switching leaders

This paper considers controllability of multi-agent systems with periodically switching topologies and switching leaders. The concept of m-periodic controllability is proposed, and a criterion for m-periodic controllability is established. The effect of the duration of subsystems on controllability is analysed by utilising a property of analytic functions. In addition, the influence of switching periods on controllability is investigated, and an algorithm is proposed to search for the fewest periods to ensure controllability. A necessary condition for m-periodic controllability is obtained from the perspective of eigenvectors of the subsystems’ Laplacian matrices. For a system with switching leaders, it is proved that switching-leader controllability is equivalent to multiple-leader controllability. Furthermore, both the switching order and the tenure of agents being leaders have no effect on the controllability. Some examples are provided to illustrate the theoretical results.     

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.     

Stabilization of stochastic systems with hidden Markovian jumps

This paper considers the adaptive control of discrete-time hybrid stochastic systems with unknown randomly jumping parameters described by a finite-state hidden Markov chain. An intuitive yet longstanding conjecture in this area is that such hybrid systems can be adaptively stabilized whenever the rate of transition of the hidden Markov chain is small enough. This paper provides a rigorous positive answer to this conjecture by establishing the global stability of a gradient-algorithm-based adaptive linear-quadratic control.     

Fault Detection and Isolation of discrete-time Markovian jump linear systems with application to a network of multi-agent systems having imperfect communication channels

This paper deals with the problem of Fault Detection and Isolation (FDI) for discrete-time Markovian Jump Linear Systems (MJLSs). A geometric property related to the unobservable subspace of an MJLS is first presented and the concept of unobservability subspace is introduced. Sufficient conditions for designing an H_∞-based FDI algorithm for MJLSs subject to input disturbances and measurement noise are presented and developed. Our proposed approach is then applied to the problem of fault detection and isolation in a network of multi-agent systems when imperfect communication channels exist among the agents. A discrete-time communication link with a stochastic packet dropping effect is considered based on the Gilbert–Elliott model and the entire network is modeled as a discrete-time MJLS. Simulation results are presented for formation flight of satellites to demonstrate and verify the effectiveness and performance capabilities of our proposed FDI algorithm.     

Robust Stochastic Stability for a Class of Singular Systems with Uncertain Markovian Jump and Time‐Varying Delay

This paper deals with the problem of the robust stochastic stability for a class of singular systems with uncertain Markovian jump and time‐varying delay. Sufficient conditions on the stochastic stability are presented. The obtained results are formulated in terms of strict linear matrix inequalities. A numerical example is provided to show the effectiveness of the proposed approaches.     

Cluster consensus of high-order multi-agent systems with switching topologies

This paper investigates the cluster consensus problems of generic linear multi-agent systems with switching topologies. Sufficient criteria for cluster consensus, which generalise the results in existing literatures, are derived for both state feedback and observer-based control schemes. By using an averaging method, it is shown that cluster consensus can be achieved when the union of the acyclic topologies contains a directed spanning tree within each cluster frequently enough. We also provide a principle to construct digraphs with inter-cluster cyclic couplings that promote cluster consensus regardless of the magnitude of inter-agent coupling weights. Finally, numerical examples are given to demonstrate the effectiveness of the proposed approaches.     

Robust H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump

This paper deals with the problem of H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump. Based on the extended It^o stochastic differential formula, sufficient conditions for the solvability of these problems are obtained. Furthermore, It is shown that a desired filter can be constructed by solving a set of linear matrix inequalities. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.     

Controllability and observability of multi-agent systems with heterogeneous and switching topologies

ABSTRACT

This paper focuses on controllability and observability of multi-agent systems with heterogeneous and switching topologies, where the first- and the second-order information interaction topologies are different and switching. First, based on the controllable state set, a controllability criterion is obtained in terms of the controllability matrix corresponding to the switching sequence. Next, by virtue of the subspace sequence, two necessary and sufficient algebraic conditions are established for controllability in terms of the system matrices corresponding to all the possible topologies. Furthermore, controllability is considered from the graphic perspective. It is proved that the system is controllable if the union graph of all the possible topologies is controllable. With respect to observability, two sufficient and necessary conditions are derived by taking advantage of the system matrices and the corresponding invariant subspace, respectively. Finally, some simulation examples are worked out to illustrate the theoretical results.     

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

pth moment and almost sure exponential stability of impulsive neutral stochastic functional differential equations with Markovian switching

In this paper, the problems on the pth moment and the almost sure exponential stability for a class of impulsive neutral stochastic functional differential equations with Markovian switching are investigated. By using the Lyapunov function, the Razumikhin-type theorem and the stochastic analysis, some new conditions about the pth moment exponential stability are first obtained. Then, by using the Borel–Cantelli lemma, the almost sure exponential stability is also discussed. The results generalise and improve some results obtained in the existing literature. Finally, two examples are given to illustrate the obtained results.     

Consensus problem of networked multi-agent systems with constant communication delay: stochastic switching topology case

This article proposes an observer-based control strategy for networked multi-agent systems with constant communication delay and stochastic switching topology. First, using the system transformation method, the mean-square consensus problem of multi-agent systems can be converted into the mean-square stability problem of an equivalent system, and some equivalent conditions concerning the mean-square consensus are presented. Then, an example is given to illustrate that the connection weights should be regarded as the parameters to be designed, since they have a great effect on the mean-square consensus of multi-agent systems. By choosing appropriate connection weights, the mean-square consensus problem can be converted into the mean-square stabilisation problem of N-1 delay systems with stochastic switching signal, whose related observer-based stabilisability criteria can be established in the form of linear matrix inequalities (LMIs). Furthermore, if the LMIs are feasible, the multi-agent systems achieve mean-square consensus if and only if the union of graphs in the switching topology set has a directed spanning tree. Finally, numerical simulations are given to illustrate our results.     

Robust stabilisation and L2 -gain analysis for switched systems with actuator saturation under asynchronous switching

Robust stabilisation and L₂-gain analysis for a class of switched systems with actuator saturation are studied in this paper. The switching signal of the controllers lags behind that of the system modes, which leads to the asynchronous switching between the candidate controllers and the subsystems. By combining the piecewise Lyapunov function method with the convex hull technique, sufficient conditions in terms of LMIs for the solvability of the robust stabilisation and weighted L₂-gain problems are presented respectively under the dwell time scheme. Finally, a numerical example is given to demonstrate the feasibility and effectiveness of the proposed results.     

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

This paper focuses on the problem of delay-dependent robust stochastic stability analysis and controller synthesis for Markovian jump systems with state and input delays. It is assumed that the delays are constant and unknown, but their upper bounds are known. By constructing a new Lyapunov-Krasovskii functional and introducing some appropriate slack matrices, new delay-dependent stochastic stability and stabilization conditions are proposed by means of linear matrix inequalities (LMIs). An important feature of the results proposed here is that all the robust stability and stabilization conditions are dependent on the upper bound of the delays. Memoryless state feedback controllers are designed such that the closed-loop system is robustly stochastically stable. Some numerical examples are provided to illustrate the effectiveness of the proposed method.     

Stabilisation of a class of positive switched nonlinear systems under asynchronous switching

This paper addresses the stabilisation problem for a class of positive switched nonlinear systems under asynchronous switching, which means that the switches between the candidate controllers and the system modes are not synchronous. The continuous and discrete cases are considered respectively. Sufficient conditions are firstly provided for the existence of the asynchronous switching controllers to guarantee the closed-loop system to be positive and exponentially stable, and the corresponding admissible switching signals are presented. As a special case, the stabilisation results for positive switched linear systems under asynchronous switching are provided accordingly. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Exponential stability of stochastic systems with hysteresis switching

For stochastic systems with state-dependent switching which are motivated by active regions of subsystems, the exponential stability is studied in this paper. Distinct from most of the existing references, the existence of the solution to stochastic switched systems is not given as a priori information but can be proved under some easily verified conditions. By the aid of Dynkin’s formula, Itô’s formula and exponential martingale inequality, the criteria on moment exponential stability and almost sure exponential stability of the stochastic switched system are established based on Lyapunov-like techniques. Simulation examples are presented to illustrate the validity of the results.