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The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of the right-hand side of a differential equation without finding its solutions. Initially, the Lyapunov function method was limited by a regular class of ODE with continuous right-hand sides. The later evolution of the ODE theory and its applications had required extensions of this method to differential equations with discontinuous right-hand sides, functional differential equations, PDEs, and evolution systems.

The high-order sliding mode (HOSM) is discussed here as an alternative methodology to the conventional sliding mode control to design feedback control, observers, and to handle the chattering problem. Here, the principle ideas of HOSM methodology are presented. Similar to the second-order sliding mode, HOSM implies that sliding mode occurs in the manifold of a smaller order after a finite time interval.

This chapter introduces linear dilations in normed vector spaces and studies their properties. The related notions (such as a generator of a dilation) are considered. Some monotonicity properties of dilations in Banach, Hilbert and Euclidean spaces are studied. Geometrical structures induced by linear dilations (such as homogeneous cones, spheres and balls) are presented. A class of linear dilations in function spaces is studied. All results are supported with examples in finite-dimensional and infinite-dimensional spaces.

A homogeneity-based approach to the stability, regularity, and robustness analysis of dynamical systems in finite-dimensional and infinite-dimensional spaces is introduced in this chapter. In particular, a dilation symmetry of solutions of homogeneous evolution systems is established; a simple way for global expansion of regularity of solutions is provided; a homogeneity-based finite-time and fixed-time stability analysis is developed; the existence of a homogeneous Lyapunov function for any stable homogeneous evolution system is proven; a homogeneous Lyapunov function for a stable homogeneous ODE is characterized by a quadratic-like functional \({\mathbb {R}}^n\rightarrow {\mathbb {R}}\); a robustness (input-to-state stability) of homogeneous evolution systems is studied.

An optimal control design is the well-known problem of mathematical control theory where a geometry of a system plays an important role. Being a kind of symmetry, the homogeneity is expected to be useful for optimization purposes. In this chapter, we study homogeneous optimal control problems in \({\mathbb {R}}^n\) and revise the conventional tools of an optimal control design (the Bellman function method and the Pontryagin maximum principle) under the assumption about a homogeneity of the system model and the cost functional.

In this chapter, the Lyapunov stability tools for sliding modes analysis are considered. The notion of finite-time stabilization is introduced, and the finite-time suitability analysis is discussed based on the homogeneity property of the considered controllable system. The realization of fixed-time stability is also presented.

In this chapter, we propose a sliding-mode controller for a class of scalar bilinear systems with delay in both the input and the state. Such systems have shown to be adequate for input–output modeling and control of a class of turbulent flow systems. Since the sliding dynamics is infinite dimensional and described by an integral equation, we show that the stability and robustness analysis is simplified by using Volterra operator theory.

Discretization of continuous-time models is considered. Definition of SM for discrete-time systems is presented. The behavior under uncertainties of discrete-time systems, controlled by SM, is briefly discussed.

This chapter considers ordinary differential equations and inclusions, which are common mathematical models of control systems. In particular, elements of the theory of differential equations with discontinuous right-hand sides are presented. Problems of the existence and uniqueness of solutions as well as their continuous dependence on parameters are briefly studied. Some issues of modeling of control systems with uncertainties and disturbances are discussed.

The concept of stability introduced in the famous thesis of Lyapunov [1] is one of the central notions of the modern control theory. Many problems of state estimation and control can be reduced to a stability analysis or to a stabilization of solutions of certain dynamical models. This chapter surveys stability notions for systems modeled by differential laws (ODEs, PDEs, evolution equations, and inclusion) and discusses some related convergence properties important for the analysis of homogeneous control systems.

The theory of evolution equations proposes a unified approach to a modeling and an analysis of dynamical systems governed by generalized differential laws. A state vector x in this case is an element of an infinite-dimensional Banach (or Hilbert) space \({\mathbb {B}}\).

This brief chapter introduces a reader in the problem of SMC designing the class of infinite-dimensional systems governed by a linear parabolic PDE containing uncertainties. The distributed unit signal is considered as a robust controller. The Lyapunov-Krasovskii functional is suggested for the analysis of the corresponding closed-loop dynamics.

A discretization of continuous-time models is an important issue for the digital implementation of homogeneous control algorithms as well as for the numerical simulation of homogeneous controlled processes. This is a nontrivial task even in the finite-dimensional space. The explicit Euler discretization of a stable homogeneous system with nonzero degree is never globally asymptotically stable, but the implicit Euler discretization (being stable) does not preserve a convergence rate of the original continuous-time system. In other words, the classical discretization schemes are not appropriate for homogeneous models. This chapter introduces a new methodology of a consistent discretization, which allows all important properties of the continuous-time homogeneous system to be preserved in its discrete-time counterpart. The developed scheme is shown to be useful for the discretization of the homogeneous control algorithm designed in Chap. 9.

A discord is a refinement of the concept of an anomalous subsequence of a time series. The task of discovering discords is applied in a wide range of subject areas involving time series: medicine, economics, climate modeling, and others. In this paper, we propose a novel parallel algorithm for discord discovery using Intel MIC (Many Integrated Core) accelerators in the case when time series fit in the main memory. We achieve parallelization through thread-level parallelism and OpenMP technology. The algorithm employs a set of matrix data structures to store and index the subsequences of a time series and to provide an efficient vectorization of computations on the Intel MIC platform. Moreover, the algorithm exploits the ability to independently computing Euclidean distances between subsequences of a time series. The algorithm iterates subsequences in two nested loops; it parallelizes the outer and the inner loops separately and differently, depending on both the number of running threads and the cardinality of the sets of subsequences scanned in the loop. The experimental evaluation shows the high scalability of the proposed algorithm.

In automated health services based on text and voice interfaces, there is a need to be able to understand what the user is talking about, and what is the attitude of the user towards a subject. Typical machine learning methods for text analysis require a lot of annotated data for the training. This is often a problem in addressing specific and possibly very personal health care needs. In this paper, we propose an active learning algorithm for the training of a text classifier for a conversational therapy application in the area of health behavior change. A new active learning algorithm, Query by Embedded Committee (QBEC), is proposed in the paper. The methods are particularly suitable for the text classification task in a dynamic environment and give a good performance with realistic test data.

40 years ago W. Haupin stressed that anode overvoltages on the carbon materials have a scatter more than 300 mV under the same current density. This is a reason to attempt to find out the reason for greater differences because decreasing the overvoltage promises high energy saving. Experiments in lab.cells in galvanostatic conditions have been conducted to determine the overvoltages for the smelter anodes used in Sayanogorsk and Boguchany smelters (Russia) (more than 80 curves currents-overvoltages are received). Overvoltages are compared with other carbon block properties. Recommendations to use these values as a parameter of carbon block quality were made.

This chapter analyses the theory of state and law proposed by N. Korkunov, the Russian legal scholar. The theory is discussed in the context of the ideas developed by the St. Petersburg school of legal philosophy, which was heavily influenced by Korkunov’s works. Korkunov created a realistic conception of law, based on a synthesis of positivism, sociology and psychology. Thus, Korkunov not only saw the normative (regulatory) dimension of law, but also its social dimension, manifested in legal relationships, as well as its psychological dimension, which provides what we would call the axiological legitimisation of law. In fact, this thinker anticipated the subsequent attempts at an integrated understanding of law that could reconcile various theoretically valid approaches and underlie a “universal definition of law covering all the legal phenomena”.

In this chapter, we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantized. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimization problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented to illustrate the applicability of the proposed design method.

In this chapter, we formulate our main problem and discuss some necessary mathematical concepts related to feedback control design for nonlinear systems under sample-data output measurements. Then we present a theoretical analysis of an extended version of the invariant ellipsoid method. Then two feedbacks are analyzed: a linear feedback proportional to the current state estimate obtained by a Luenberger-type estimator; and a full-order linear dynamic controller governed by a linear ordinary differential equation with available sample data as input.

In this chapter, a new sliding mode control design algorithm for a linear and a class of nonlinear quasi-Lipschitz disturbed systems is presented. It is based on the appropriate selection of a sliding surface via the invariant ellipsoid method. The designed control guarantees minimization of unmatched disturbance effects to system motions in a sliding mode. The theoretical results are verified by numerical simulations. Additionally, a methodology for the design of sliding mode controllers for linear systems subjected to matched and unmatched perturbations is proposed. It is considered that the control signal is applied through a first-order low-pass filter. The technique is based on the existence of an attracting (invariant) ellipsoid such that the convergence to a quasiminimal region of the origin using the suboptimal control signal is guaranteed. The design procedure is given in terms of the solution of a set of matrix inequalities. Benchmark examples illustrating the design are given.

In this chapter, we consider the class of uncertain time-delay affine-controlled systems in which a delay is admitted in state variables, and we show that the attractive ellipsoid method allows us to create a feedback that provides the convergence of any state trajectory of the controlled system from a given class to an ellipsoid whose “size” depends on the parameters of the applied feedback. Finally, we present a method for numerical calculation of these parameters that provides the “smallest” zone convergence for controlled trajectories.

This chapter deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamic models under consideration are described by implicit differential equations in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique. In this book, this extension is called the attractive ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves a modified descriptor approach associated with the usual Lyapunov-type methodology. The theoretical schemes elaborated in our contribution are finally illustrated by a simple computational example.

In this chapter, a particular family of nonlinear affine control systems with a sufficiently general type of uncertainties is considered. Nonlinear uncertain systems, considered here, are governed by vector ordinary differential equations with so-called quasi-Lipschitz right-hand sides admitting a wide class of external and internal uncertainties (including discontinuous nonlinearities such as relay and hysteresis elements, time-delay blocks, and so on). Here, the simplest class of linear state-feedback controllers is analyzed. Sufficient conditions guaranteeing the boundedness of all possible trajectories of controlled systems are presented. Since bounded dynamics can always be imposed on an ellipsoid, it is suggested that the “robust-optimal” gain matrix of the designated linear feedback be selected in such a way that the “size” of this attractive ellipsoid will be minimal. Several numerical and experimental illustrative examples are considered.

This chapter deals with the development of a state estimator and adaptive controller based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities as well as external perturbations. The set of stabilizing feedback matrices is given by a specific matrix inequality including the characteristic matrix of the attractive ellipsoid that contains all possible bounded trajectories around the origin. Here we present two modifications of the AEM that allow us to use online information obtained during the process and to adjust matrix parameters participating in constraints that characterize the class of adaptive stabilizing feedbacks. The proposed method guarantees that under a specific persistent excitation condition, the controlled system trajectories converge to an ellipsoid of “minimal size” having a minimal trace of the corresponding inverse ellipsoidal matrix.

This chapter deals with a bounded control design for a class of nonlinear systems whose mathematical model may not be explicitly given. This class of uncertain nonlinear systems is governed by a system of ordinary differential equations with quasi-Lipschitz right-hand sides and contains external perturbations as well. The attractive ellipsoid method (AEM) allows us to describe the class of nonlinear feedbacks (containing a nonlinear projection operator, a linear state estimator, and a feedback matrix gain) guaranteeing the boundedness of all possible trajectories around the origin. To satisfy this property, some modifications of the AEM are introduced: basically, some sort of sample-time corrections of the feedback parameters are required. The optimization of feedback within this class of controllers is associated with the selection of the feedback parameters such that the trajectory converges within an ellipsoid of “minimal size.” The effectiveness of the suggested approach is illustrated by its application to a flexible arm system.

In this chapter, we consider three types of possible linear feedbacks using only the current output information: static feedback proportional to the output measurable signal, observer-based feedback proportional to the state estimation vector, and full-order linear dynamic controllers. For each type of possible linear feedback, we suggest that one characterize the set of all stabilizing gain-feedback matrices by a system of the corresponding linear matrix inequalities, providing the boundedness of all possible trajectories of every controlled plant from the considered class of uncertain systems. We also suggest selecting the optimal feedback gain matrix from the described class of stabilizing feedbacks as the one that minimizes the “size” of the attractive ellipsoid containing all possible bounded dynamic trajectories. The corresponding numerical procedures for designing the best feedback gain matrices are introduced and discussed for each type of considered feedback. Several illustrative examples clearly show the effectiveness of the suggested technique.

This chapter deals with the problem of robust feedback design for a class of switched systems in the presence of bounded model uncertainties as well as external perturbations. Only the output of the system is supposed to be available for a designer. We consider nonlinear dynamic models under arbitrary switching mechanisms assuming that sample-switching times are known. Online state estimates are obtained by the use of a Luenberger-like observer using only current inputs and general information on the class of model uncertainties. The stabilization issue is solved in the sense of practical stability, and it is carried out by a linear (with respect to a current state estimate) feedback switching controller subject to an average dwell time scheme. We apply the newly elaborated (extended) version of the conventional attractive ellipsoid method for this purpose. Numerically implementable sufficient conditions for the practical stability of systems are derived using bilinear matrix inequalities. The effectiveness of the proposed method is illustrated by an example of a continuous stirred tank reactor in which only the temperature (not the concentration) is available during the process.