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

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

Discrete and intersample analysis of systems with aperiodic sampling

International audienceThis article addresses the stability analysis of linear time invariant systems with aperiodic sampled-data control. Adopting a difference inclusion formalism, we show that necessary and sufficient stability conditions are given by the existence of discrete-time quasi-quadratic Lyapunov functions. A constructive method for computing such functions is provided from the approximation of the necessary and sufficient conditions. In practice, this leads to sufficient stability criteria under LMI form. The inter-sampling behavior is discussed there: based on differential inclusions, we provide continuous-time methods that use the advantages of the discrete-time approach. The results are illustrated by numerical examples that indicate the improvement with regard to the existing literature

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

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

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Sampled-data control of linear systems subject to input saturation : a hybrid system approach

In this work, a new method for the stability analysis and synthesis of sampled-data control systems subject to variable sampling intervals and input saturation is proposed. From a hybrid systems representation, stability conditions based on quadratic clockdependent Lyapunov functions and the generalized sector condition to handle saturation are developed. These conditions are cast in semidefinite and sum-of-squares optimization problems to provide maximized estimates of the region of attraction, to estimate the maximum intersampling interval for which a region of stability is ensured, or to produce a stabilizing controller that results in a large implicit region of attraction, through the maximization of an estimate of it.Neste trabalho é proposto um novo método para a análise da estabilidade de sistemas de controle amostrados aperiodicamente e com saturação na entrada, e também para a síntese de controladores estabilizantes. A partir de uma representação por sistemas híbridos, condições de estabilidade baseadas em funções quadráticas de Lyapunov dependentes do clock e na condição de setor generalizada para o tratamento de saturação são desenvolvidas para o sistema amostrado em questão. Essas condições são incorporadas como restrições em problemas de otimização. Os problemas de otimização são baseados em programação semidefinida e em programação sum-of-squares, e têm o objetivo de obter estimativas maximizadas da região de atração do sistema, estimativas do intervalo de amostragem máximo para o qual uma dada região de estados iniciais seja uma região de estabilidade, ou para produzir controladores (dados por ganhos estáticos estabilizantes) que resultem em uma região de atração implicitamente grande, através da maximização da estimativa dessa região de atração