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

Small-signal modeling of current-mode controlled modular DC-DC converters using the state-space algebraic approach

Small-signal models are useful tools to preliminary understand the dynamics of interconnected systems like modular dc-dc converters which find a wide range of industrial applications. This work proposes a state-space-based averaged small-signal model in symbolic form for a peak current-mode controlled parallel-input/parallel-output buck converter operating in the continuous-conduction mode. In modeling the converter power-stage each module is independently represented. For modeling the current-mode control the state-space algebraic approach is used to incorporate the current-mode control-law into the power-stage equations. For each module two parasitic elements in addition to the current-loop sampling action are included in the derivation. Furthermore, the control-to-output voltage transfer functions are presented in symbolic form for two cases of interest: the first when the converter has two non-identical modules to study the effect of inductor mismatch, and the second when the converter is composed of n-connected identical modules to assess the effect of varying the number of modules. All responses from PSIM cycle-by-cycle simulations are in good agreement with the mathematical model predictions up to half the switching frequency

A resilience approach to the design of future moon base power systems

This paper proposes a novel approach to the design of complex engineering systems which maximise performance, and global system resilience. The approach is applied to the system level design of the power system for future Moon bases. The power system is modelled as a network, where each node represents a specific power unit: energy storage, power distribution, power generation, power regulation. The performance and resilience of each power unit is defined by a mathematical model that depends on a set of design (control) and uncertain variables. The interrelationship among nodes is defined by functional links. The combination of multiple interconnected nodes defines the performance and resilience of the whole system. An optimisation procedure is then used to find the optimal values of the design parameters. The optimal solution maximises global system resilience where an optimal resilient solution is either robust, i.e. it is not subject to disruptive failures, or recovers from failures to achieve a functioning state, albeit different from the starting one, after a contingency occurs. The power system supports a Lunar base developed within the ESA-lab initiative, IGLUNA, led by the Swiss Space Centre. The power system, developed at the University of Strathclyde as part of the PowerHab project, is composed of nine interconnected elements: a hydrogen fuel cell energy storage system, a thermal mass storage system, a lithium-ion battery storage system, a constellation of solar power satellites (SPS) working in conjunction with a microwave wireless power transmission system, a reflecting satellite constellation and a ground-based solar power array. Distinct space and ground segments are identifiable, with orbit, AOCS and reflecting satellite nodes cooperating to provide optimal performance of the SPS constellation. The ground segment encompasses the ground-based solar array, energy storage systems, Lunar habitation module and the power transmission lines connecting these elements. Power generation is predominantly supplied by the ground-based array, with the SPS constellation and energy storage systems complementing this source; as well as providing redundancy and a reliable power supply during the Lunar night period

Model Prediction-Based Approach to Fault Tolerant Control with Applications

Abstract— Fault-tolerant control (FTC) is an integral component in industrial processes as it enables the system to continue robust operation under some conditions. In this paper, an FTC scheme is proposed for interconnected systems within an integrated design framework to yield a timely monitoring and detection of fault and reconfiguring the controller according to those faults. The unscented Kalman filter (UKF)-based fault detection and diagnosis system is initially run on the main plant and parameter estimation is being done for the local faults. This critical information\ud is shared through information fusion to the main system where the whole system is being decentralized using the overlapping decomposition technique. Using this parameter estimates of decentralized subsystems, a model predictive control (MPC) adjusts its parameters according to the\ud fault scenarios thereby striving to maintain the stability of the system. Experimental results on interconnected continuous time stirred tank reactors (CSTR) with recycle and quadruple tank system indicate that the proposed method is capable to correctly identify various faults, and then controlling the system under some conditions