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

Designing the Model Predictive Control for Interval Type-2 Fuzzy T-S Systems Involving Unknown Time-Varying Delay in Both States and Input Vector

In this paper, the model predictive control is designed for an interval type-2 Takagi-Sugeno (T-S) system with unknown time-varying delay in state and input vectors. The time-varying delay is a weird phenomenon that is appeared in almost all systems. It can make many problems and instability while the system is working. In this paper, the time-varying delay is considered in both states and input vectors and is the sensible difference between the proposed method here and previous algorithms, besides, it is unknown but bounded. To solve the problem, the Razumikhin approach is applied to the proposed method since it includes a Lyapunov function with the original nonaugmented state space of system models compared to Krasovskii formula. On the other hand, the Razumikhin method act better and avoids the inherent complexity of the Krasovskii specifically when large delays and disturbances are appeared. To stabilize output results, the model predictive control (MPC) is designed for the system and the considered system in this paper is interval type-2 (IT2) fuzzy T-S that has better estimation of the dynamic model of the system. Here, online optimization problems are solved by the linear matrix inequalities (LMIs) which reduce the burdens of the computation and online computational costs compared to the offline and non-LMI approach. At the end, an example is illustrated for the proposed approach

Hierarchical Optimization-Based Model Predictive Control for a Class of Discrete Fuzzy Large-Scale Systems Considering Time-Varying Delays and Disturbances

Altres ajuts: Acord transformatiu CRUE-CSICIn this manuscript, model predictive control for class of discrete fuzzy large-scale systems subjected to bounded time-varying delay and disturbances is studied. The considered method is Razumikhin for time-varying delay large-scale systems, in which it includes a Lyapunov function associated with the original non-augmented state space of system dynamics in comparison with the Krasovskii method. As a rule, the Razumikhin method has a perfect potential to avoid the inherent complexity of the Krasovskii method especially in the presence of large delays and disturbances. The considered large-scale system in this manuscript is decomposed into several subsystems, each of which is represented by a fuzzy Takagi-Sugeno (TS) model and the interconnection between any two subsystems is considered. Because the main section of the model predictive control is optimization, the hierarchical scheme is performed for the optimization problem. Furthermore, persistent disturbances are considered that robust positive invariance and input-to-state stability under such circumstances are studied. The linear matrix inequalities (LMIs) method is performed for our computations. So the closed-loop large-scale system is asymptotically stable. Ultimately, by two examples, the effectiveness of the proposed method is illustrated, and a comparison with other papers is made by remarks

Fuzzy control turns 50: 10 years later

In 2015, we celebrate the 50th anniversary of Fuzzy Sets, ten years after the main milestones regarding its applications in fuzzy control in their 40th birthday were reviewed in FSS, see [1]. Ten years is at the same time a long period and short time thinking to the inner dynamics of research. This paper, presented for these 50 years of Fuzzy Sets is taking into account both thoughts. A first part presents a quick recap of the history of fuzzy control: from model-free design, based on human reasoning to quasi-LPV (Linear Parameter Varying) model-based control design via some milestones, and key applications. The second part shows where we arrived and what the improvements are since the milestone of the first 40 years. A last part is devoted to discussion and possible future research topics.Guerra, T.; Sala, A.; Tanaka, K. (2015). Fuzzy control turns 50: 10 years later. Fuzzy Sets and Systems. 281:162-182. doi:10.1016/j.fss.2015.05.005S16218228

Stability and stabilization of sampled-data control for lure systems

Este trabalho apresenta um novo método para a análise de estabilidade e estabilização de sistemas do tipo Lure com controle amostrado, sujeitos a amostragem aperiódica e não linearidades que são limitadas em setor e restritas em derivada, em ambos contextos global e regional. Assume-se que os estados da planta estão disponíveis para medição e que as não linearidades são conhecidas, o que leva a uma formulação mais geral do problema. Os estados são adquiridos por um controlador digital que atualiza a entrada de controle em instantes de tempo discretos e aperiódicos, mantendo-a constante entre dois instantes sucessivos de amostragem. A abordagem apresentada neste trabalho é baseada no uso de uma nova classe de looped-functionals e em uma função do tipo Lure generalizada, que leva a condições de estabilidade e estabilização que são escritas na forma de desigualdades matriciais lineares (LMIs) e quasi-LMIs, respectivamente. Com base nestas condições, problemas de otimização são formulados com o objetivo de computar o intervalo máximo entre amostragens ou os limites máximos do setor para os quais a estabilidade assintótica da origem do sistema de dados amostrados em malha fechada é garantida. No caso em que as condições de setor são válidas apenas localmente, a solução desses problemas também fornece uma estimativa da região de atração para as trajetórias em tempo contínuo do sistema em malha fechada. Como as condições de síntese são quasi-LMIs, um algoritmo de otimização por enxame de partículas é proposto para lidar com as não linearidades envolvidas nos problemas de otimização, que surgem do produto de algumas variáveis de decisão. Exemplos numéricos são apresentados ao longo do trabalho para destacar as potencialidades do método.This work presents a new method for stability analysis and stabilization of sampleddata controlled Lure systems, subject to aperiodic sampling and nonlinearities that are sector bounded and slope restricted, in both global and regional contexts. We assume that the states of the plant are available for measurement and that the nonlinearities are known, which leads to a more general formulation of the problem. The states are acquired by a digital controller which updates the control input at aperiodic discrete-time instants, keeping it constant between successive sampling instants. The approach here presented is based on the use of a new class of looped-functionals and a generalized Luretype function, which leads to stability and stabilization conditions that are written in the form of Linear Matrix Inequalities (LMIs) and quasi-LMIs, respectively. On this basis, optimization problems are formulated aiming to compute the maximal intersampling interval or the maximal sector bounds for which the asymptotic stability of the origin of the sampled-data closed-loop system is guaranteed. In the case where the sector conditions hold only locally, the solution of these problems also provide an estimate of the region of attraction for the continuous-time trajectories of the closed-loop system. As the synthesis conditions are quasi-LMIs, a Particle Swarm Optimization (PSO) algorithm is proposed to deal with the involved nonlinearities in the optimization problems, which arise from the product of some decision variables. Numerical examples are presented throughout the work to highlight the potentialities of the method