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.