Razumikhin and Krasovskii stability of impulsive stochastic delay systems via uniformly stable function method

This paper generalizes Razumikhin-type theorem and Krasovskii stability theorem of impulsive stochastic delay systems. By proposing uniformly stable function (USF) in the form of impulse as a new tool, some properties about USF and some novel pth moment decay theorems are derived. Based on these new theorems, the stability theorems of impulsive stochastic linear delay system are acquired via the Razumikhin method and the Krasovskii method. The obtained results enhance the elasticity of the impulsive gain by comparing the previous results. Finally, numerical examples are given to demonstrate the effectiveness of theoretical results

Stability analysis and control of discrete-time systems with delay

The research presented in this thesis considers the stability analysis and control of discrete-time systems with delay. The interest in this class of systems has been motivated traditionally by sampled-data systems in which a process is sampled periodically and then controlled via a computer. This setting leads to relatively cheap control solutions, but requires the discretization of signals which typically introduces time delays. Therefore, controller design for sampled-data systems is often based on a model consisting of a discrete-time system with delay. More recently the interest in discrete-time systems with delay has been motivated by networked control systems in which the connection between the process and the controller is made through a shared communication network. This communication network increases the flexibility of the control architecture but also introduces effects such as packet dropouts, uncertain time-varying delays and timing jitter. To take those effects into account, typically a discrete-time system with delay is formulated that represents the process together with the communication network, this model is then used for controller design While most researchers that work on sampled-data and networked control systems make use of discrete-time systems with delay as a modeling class, they merely use these models as a tool to analyse the properties of their original control problem. Unfortunately, a relatively small amount of research on discrete-time systems with delay addresses fundamental questions such as: What trade-off between computational complexity and conceptual generality or potential control performance is provided by the different stability analysis methods that underlie existing results? Are there other stability analysis methods possible that provide a better trade-off between these properties? In this thesis we try to address these and other related questions. Motivated by the fact that almost every system in practice is subject to constraints and Lyapunov theory is one of the few methods that can be easily adapted to deal with constraints, all results in this thesis are based on Lyapunov theory. In Chapter 2 we introduce delay difference inclusions (DDIs) as a modeling class for systems with delay and discuss their generality and advantages. Furthermore, the two standard stability analysis results for DDIs that make use of Lyapunov theory, i.e., the Krasovskii and Razumikhin approaches, are considered. The Krasovskii approach provides necessary and sufficient conditions for stability while the Razumikhin approach provides conditions that are relatively simple to verify but conservative. An important conclusion is that the Razumikhin approach makes use of conditions that involve the system state only while those corresponding to the Krasovskii approach involve trajectory segments. Therefore, only the Razumikhin approach yields information about DDI trajectories directly, such that the corresponding computations can be executed in the low-dimensional state space of the DDI dynamics. Hence, we focus on the Razumikhin approach in the remainder of the thesis. In Chapter 3 it is shown that by considering each delayed state as a subsystem, the behavior of a DDI can be described by an interconnected system. Thus, the Razumikhin approach is found to be an exact application of the small-gain theorem, which provides an explanation for the conservatism that is typically associated with this approach. Then, inspired by the relation of DDIs to interconnected systems, we propose a new Razumikhin-type stability analysis method that makes use of a stability analysis result for interconnected systems with dissipative subsystems. The proposed method is shown to provide a trade-off between the conceptual generality of the Krasovskii approach and the computationally convenience of the Razumikhin approach. Unfortunately, these novel Razumikhin-type stability analysis conditions still remain conservative. Therefore, in Chapter 4 we propose a relaxation of the Razumikhin approach that provides necessary and sufficient conditions for stability. Thus, we obtain a Razumikhin-type result that makes use of conditions that involve the system state only and are non-conservative. Interestingly, we prove that for positive linear systems these conditions equivalent to the standard Razumikhin approach and hence both are necessary and sufficient for stability. This establishes the dominance of the standard Razumikhin approach over the Krasovskii approach for positive linear discrete-time systems with delay. Next, in Chapter 5 the stability analysis of constrained DDIs is considered. To this end, we study the construction of invariant sets. In this context the Krasovskii approach leads to algorithms that are not computationally tractable while the Razumikhin approach is, due to its conservatism, not always able to provide a suitable invariant set. Based on the non-conservative Razumikhin-type conditions that were proposed in Chapter 4, a novel invariance notion is proposed. This notion, called the invariant family of sets, preserves the conceptual generality of the Krasovskii approach while, at the same time, it has a computational complexity comparable to the Razumikhin approach. The properties of invariant families of sets are analyzed and synthesis methods are presented. Then, in Chapter 6 the stabilization of constrained linear DDIs is considered. In particular, we propose two advanced control schemes that make use of online optimization. The first scheme is designed specifically to handle constraints in a non-conservative way and is based on the Razumikhin approach. The second control scheme reduces the computational complexity that is typically associated with the stabilization of constrained DDIs and is based on a set of necessary and sufficient Razumikhin-type conditions for stability. In Chapter 7 interconnected systems with delay are considered. In particular, the standard stability analysis results based on the Krasovskii as well as the Razumikhin approach are extended to interconnected systems with delay using small-gain arguments. This leads, among others, to the insight that delays on the channels that connect the various subsystems can not cause the instability of the overall interconnected system with delay if a small-gain condition holds. This result stands in sharp contrast with the typical destabilizing effect that time delays have. The aforementioned results are used to analyse the stability of a classical power systems example where the power plants are controlled only locally via a communication network, which gives rise to local delays in the power plants. A reflection on the work that has been presented in this thesis and a set of conclusions and recommendations for future work are presented in Chapter 8

On almost sure stability of hybrid stochastic systems with mode-dependent interval delays

This note develops a criterion for almost sure stability of hybrid stochastic systems with mode-dependent interval time delays, which improves an existing result by exploiting the relation between the bounds of the time delays and the generator of the continuous-time Markov chain. The improved result shows that the presence of Markovian switching is quite involved in the stability analysis of delay systems. Numerical examples are given to verify the effectiveness

Lyapunov methods for time-invariant delay difference inclusions

Motivated by the fact that delay difference inclusions (DDIs) form a rich modeling class that includes, for example, uncertain time-delay systems and certain types of networked control systems, this paper provides a comprehensive collection of Lyapunov methods for DDIs. First, the Lyapunov–Krasovskii approach, which is an extension of the classical Lyapunov theory to time-delay systems, is considered. It is shown that a DDI is KL-stable if and only if it admits a Lyapunov–Krasovskii function (LKF). Second, the Lyapunov–Razumikhin method, which is a type of small-gain approach for time-delay systems, is studied. It is proved that a DDI is KL-stable if it admits a Lyapunov–Razumikhin function (LRF). Moreover, an example of a linear delay difference equation which is globally exponentially stable but does not admit an LRF is provided. Thus, it is established that the existence of an LRF is not a necessary condition for KL-stability of a DDI. Then, it is shown that the existence of an LRF is a sufficient condition for the existence of an LKF and that only under certain additional assumptions is the converse true. Furthermore, it is shown that an LRF induces a family of sets with certain contraction properties that are particular to time-delay systems. On the other hand, an LKF is shown to induce a type of contractive set similar to those induced by a classical Lyapunov function. The class of quadratic candidate functions is used to illustrate the results derived in this paper in terms of both LKFs and LRFs, respectively. Both stability analysis and stabilizing controller synthesis methods for linear DDIs are proposed

On input-to-state stability of stochastic retarded systems with Markovian switching

This note develops a Razumikhin-type theorem on pth moment input-to-state stability of hybrid stochastic retarded systems (also known as stochastic retarded systems with Markovian switching), which is an improvement of an existing result. An application to hybrid stochastic delay systems verifies the effectiveness of the improved result

Effects of Delay on the Functionality of Large-scale Networks

Networked systems are common across engineering and the physical sciences. Examples include the Internet, coordinated motion of multi-agent systems, synchronization phenomena in nature etc. Their robust functionality is important to ensure smooth operation in the presence of uncertainty and unmodelled dynamics. Many such networked systems can be viewed under a unified optimization framework and several approaches to assess their nominal behaviour have been developed. In this paper, we consider what effect multiple, non-commensurate (heterogeneous) communication delays can have on the functionality of large-scale networked systems with nonlinear dynamics. We show that for some networked systems, the structure of the delayed dynamics allows functionality to be retained for arbitrary communication delays, even for switching topologies under certain connectivity conditions; whereas in other cases the loop gains have to be compensated for by the delay size, in order to render functionality delay-independent for arbitrary network sizes. Consensus reaching in multi-agent systems and stability of network congestion control for the Internet are used as examples. The differences and similarities of the two cases are explained in detail, and the application of the methodology to other technological and physical networks is discussed

Delay-dependent exponential stability of neutral stochastic delay systems (vol 54, pg 147, 2009)

In the above titled paper originally published in vol. 54, no. 1, pp. 147-152) of IEEE Transactions on Automatic Control, there were some typographical errors in inequalities. Corrections are presented here

Stability, observer design and control of networks using Lyapunov methods

We investigate different aspects of the analysis and control of interconnected systems. Different tools, based on Lyapunov methods, are provided to analyze such systems in view of stability, to design observers and to control systems subject to stabilization. All the different tools presented in this work can be used for many applications and extend the analysis toolbox of networks. Considering systems with inputs, the stability property input-to-state dynamical stability (ISDS) has some advantages over input-to-state stability (ISS). We introduce the ISDS property for interconnected systems and provide an ISDS small-gain theorem with a construction of an ISDS-Lyapunov function and the rate and the gains of the ISDS estimation for the whole system. This result is applied to observer design for single and interconnected systems. Observers are used in many applications where the measurement of the state is not possible or disturbed due to physical reasons or the measurement is uneconomical. By the help of error Lyapunov functions we design observers, which have a so-called quasi ISS or quasi-ISDS property to guarantee that the dynamics of the estimation error of the systems state has the ISS or ISDS property, respectively. This is applied to quantized feedback stabilization. In many applications, there occur time-delays and/or instantaneous jumps of the systems state. At first, we provide tools to check whether a network of time-delay systems has the ISS property using ISS-Lyapunov-Razumikhin functions and ISS-Lyapunov-Krasovskii functionals. Then, these approaches are also used for interconnected impulsive systems with time-delays using exponential Lyapunov-Razumikhin functions and exponential Lyapunov-Krasovskii functionals. We derive conditions to assure ISS of an impulsive network with time-delays. Controlling a system in a desired and optimal way under given constraints is a challenging task. One approach to handle such problems is model predictive control (MPC). In this thesis, we introduce the ISDS property for MPC of single and interconnected systems. We provide conditions to assure the ISDS property of systems using MPC, where the previous result of this thesis, the ISDS small-gain theorem, is applied. Furthermore, we investigate the ISS property for MPC of time-delay systems using the Lyapunov-Krasovskii approach. We prove theorems, which guarantee ISS for single and interconnected systems using MPC