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A new form of the strong law of large numbers for dependent vector sequences using the “double averaged” correlation function is presented. The suggested theorem generalizes the well-known Cramer–Lidbetter's theorem and states more general conditions for fulfilling the strong law of large numbers within the class of vector random processes generated by a non stationary stable forming filters with an absolutely integrable impulse function.

In this paper the robust adaptive control of a multicomponent nonideal distillation column is considered. A simple dynamic neural network is used to estimate the distillation column, then a adaptive controller is proposed. Because the approximation capability of dynamic neural network is limited, a sliding mode compensator is addressed. It is shown that the effect of system uncertainties can be eliminate and the boundedness of tracking error can be guaranteed.

The design of a passivity feedback equivalence controller for a class of single input-single output nonlinear systems using neural network function approximation is proposed. Radial basis functions are used to synthesize the approximation of nolinear mappings. Assuming that the uncertainty that results from this approximation is gain bounded, an adaptive technique is also used in the learning procedure of the neural network.

A neurocontrol method for robot manipulators is presented. A single-layer dynamic neural network is used to estimate the unknown manipulators, then a direct linearization controller is derived based on the neuro identifier. Because the approximation capability is limited, another PD-like controller is applied to compensate the unmodeled dynamics. The main contribution of the paper is that the boundness of the identification error and tracking error are established.

The problem addressed is the filtering of the states of a time-varying linear system subject to random disturbances and parametric unstructured uncertainties. We give an upper bound on the performance index of the proposed filter defined as the expected squared estimation error. The filter has a Kalman structure with the gain being a function of free design parameters and of a solution to an artificially constructed Riccati equation.

The problem tackled is the filtering of the states of a time-varying uncertain linear system with deterministic disturbances of limited power. The resulting filter has the structure of that of Kalman. Its gain turns to be a function of some free parameters and of the solution of two Riccati equations constructed artificially that result in a process that employs a Lyapunov function to assure a uniform finite upper bound of the squared L-2 norm of the estimation error.

This chapter discusses the behavior of learning automata for different reinforcement schemes. It describes a number of recurrent reinforcement schemes for solving the problem of adaptive control of static systems. The nonprojectional algorithm of Narendra and Shapiro, of Luce, and of Varashavskii and Vomtsova, can be used to solve learning problems associated with binary loss functions. The algorithm of Luce has the highest convergence rate. However, a high convergence rate cannot be guaranteed for all average loss functions. The reinforcement scheme of Varashavskii and Vorontsova is a modification of the algorithm of Luce. With this algorithm, the widest range of average loss functions can be considered. The Bush–Mosteller reinforcement scheme can solve the adaptive control problem only when the average loss functions of the optimal strategy is equal or tends to zero. The projectional algorithms, for solving problems with continuous loss functions in the interval (-∞, ∞) are introduced in the chapter. These algorithms are significantly more complex and require the solution of a quadratic programming problem using the projection operator at each step.

This chapter discusses reinforcement schemes for average loss function minimization. The standard problem of average loss function minimization is formulated as a linear programming problem. This formulation considers the problem of adaptive control as a minimization of a linear function on a simplex. The recurrent control algorithms are classified into two categories: nonprojectional and projectional algorithms. For every reinforcement scheme, the pseudogradient condition must be fulfilled to guarantee the property of learning. The chapter presents all known reinforcement schemes and their classification from the point of view of fulfilling of pseudogradient condition. Following the definitions of different types of automata behavior, it is shown that the majority of learning automata possess symptotically an optimal behavior only in a special class of environments. The analysis of the behavior of learning automata is carried out using the martingale theory. The reinforcement scheme is the heart of the learning automaton. It is the mechanism used to adapt the probability distribution. The reinforcement schemes can be classified on the basis of the properties that they induce in the learning automaton or on the basis of their own characteristics.

This chapter describes multilevel automata systems for the implementation of hierarchical reinforcement schemes. The advantages of multilevel systems of automata are the acceleration of the learning process and the simplification of their implementation. In a hierarchical structure of learning automata, the selection of the control action is performed in several sequential stages. The algorithms, used to modify the probability density functions are called hierarchical reinforcement schemes and the systems, used to implement these algorithms, are called multilevel systems of automata. The efficiency of such a hierarchical control strategy can be greatly improved when a priori information on the hierarchical structure of the problem is available. A priori information may significantly help to specify the structure of the multilevel system of automata in order to accelerate the convergence of the learning process. A two-level hierarchical learning system of automata consists of a hierarchy of stochastic variable-structure automata with different control actions. The first level consists of one probabilistic automaton. The second level of the hierarchy interacts with the random environment where the two-level hierarchical system of learning automata operates.

This chapter discusses the applications of learning automata. The applications of learning automata cover a wide range of problems such as absorption columns, bioreactors, communication, computers, drying furnaces, fluidized bed reactors, image processing, irrigation canals, and liquid–liquid extraction columns. Learning automata should, by collecting and processing current information regarding the environment, be capable of changing their structure and parameters as time evolves to achieve the desired goal or the optimal performance in some sense. Learning systems have made a significant impact on all areas of engineering problems arising from complexity and uncertainty. They are attractive methods for solving process control, optimization, pattern recognition, image processing, telecommunications, and scheduling. They are very simple to implement and need little prior knowledge. The only calculation required for the implementation of learning system is the relatively simple adjustment of the state probability distribution. In the control context, it is not necessary to construct a model for the process to be controlled. The number of learning systems applications has increased with the advent of highly integrated computers that made the technology cost-effective.

This chapter presents basic notions and definitions. The basic notions are: (1) controlled finite system and its dynamic characteristics, (2) control strategies and their classification, and (3) adaptive control strategy and learning automata. The chapter also presents simple control problems of static systems that are close to the problem of synthesis of adaptive neural networks using the learning automata theory. It discusses classification of problems of adaptive control of finite systems. The problem of finite system adaptive control is of major importance in the theory of adaptive systems. The objective of adaptive system theory is the development of adaptive algorithms that optimize the behavior of a system under uncertainty conditions. All of these algorithms specify how to use the current information to improve the performance of the system. This current information corresponds to a realization of some loss functions associated to the adaptive strategy.

Algorithms of the method of least squares that are intended for estimating the parameters of non-linear autoregression processes are considered. A class of nonlinear difference equations is isolated which describes autoregression processes for which MLS estimates of parameters in these equations are found to be strongly consistent.

In this work, the problem of estimating the susceptible and the recent infective, not yet in treatment, groups in a determinate male homosexual population is addressed. The estimation methodology is based on reduced order observers with proportional and integral measurement error contribution, considering measurements of male homosexual population and infective, undergoing treatment, groups. These estimates allow inference, in some sense, of the HIV transmission dynamics in order to propose a more adequate health policy. Numerical simulations were carried out showing the good performance of the estimation methodology proposed.

This paper presents a new analysis method for model reference adaptive control(MRAC) with arbitrary bounded input nonlinearities. The reduced-order plant is assumed to be of an arbitrary relative degree. In the absence of un-modeled dynamics and bounded disturbances, the tracking error depends on input nonlinearities. A new modified MRAC scheme is also proposed to improve the transient and steady state performances. The output error is used directly in the control law to generate a modified control signal to compensate possible bad performances of the robust adaptive controllers, and the ideal asymptotic properties of MRAC is not destroyed.

The nonlinear Bergman regulation model of insulin-glucose in plasma is analyzed using modern observability theory. The observability conditions are derived. The glucose concentration measurements in plasma or any its combinations with the other components are shown to provide the observability property, that is, they contain the complete information about the considered state space dynamic model. Because this model has several a priori unknown parameters and the measurable data may have noises, the dynamic neuro state observer is suggested to obtain immeasurable state estimates. Such an approach brings significant advantages with respect to the "insulin doses decision", traditionally used in the normal insulin infusion pumps: it presents the opportunity to avoid any active patient actions, the numerical simulations illustrate the effectiveness of the suggested approach.

This paper addresses the problem of optimal design for linear networked control systems (NCSs). We deal with the delayed NCSs in the continuous time-domain and propose a computational approach to optimization problems based on the hybrid LQ-type techniques. In particular, we develop an explicit connection between the networked control processes and the corresponding hybrid systems. For the constructive feedback control design procedure we derive the necessary Riccati-formalism and propose an implementable solution algorithm.

This paper focuses on the continuity and approximability properties for nonlinear affine control systems. We consider dynamical systems governed by ordinary differential equations and establish the continuity properties of the given and relaxed (in the sense of Filippov) systems with respect to controls and initial state variables. The approach based on the set-valued analysis makes it possible to study discontinuous models in the abstract setting and to obtain general theoretical results. The latter can be effectively applied to wide classes of variable structure control systems. In particular, this paper deals with applications of the above-mentioned continuity and approximability to some hybrid control systems and to the classical sliding mode control processes.

This paper addresses the robust attitude control for an uncertain flexible spacecraft with disturbances bounded by ellipsoids. An observed-state feedback is designed using the so-called invariant ellipsoid method, which is based on the local invariant set theorem. The minimum invariant ellipsoid can be considered as the influence of the uncertainties and disturbances in the controlled system. All these considerations were expressed through LMI technique as an optimization problem with linear objective and LMI constraints. The numerical solution of this problem gives the parameters for the linear controller and the Luenberger observer that guarantee a quasi-minimal invariant ellipsoid for the controlled spacecraft.

We analyze adaptive nonlinear identification and trajectory tracking using a dynamic neural network, with the same state space dimension as the system. We assume the system space state completely measurable. By means of a Lyapunov-like analysis we determine stability conditions for the identification error. We then analyze the trajectory tracking error when the adaptive controller is utilized. For the identification analysis we use an algebraic Riccati equation and for the tracking error a differential one using online adapted parameters of the neural network. The structure of our scheme is composed-by two parts: the neural network identifier and the tracking controller. As our main contributions, we establish two theorems: the first one gives a bound for the identification error, and the second one establish a bound for the tracking error.

This paper deals with a new theoretic description of some classes of the conventional sliding mode control processes. We use a variational characterization of the first-order variable structures dynamics and extend the corresponding state space formalism. A trajectory of the original systems can finally be specified as a result of a particular system optimization procedure applied to the original model. The presented variational theory of some sliding mode-type control systems is motivated by an alternative approach to the control design procedure. The mathematical tool of the sliding mode systems theory presented in our paper constitutes a formal extension of the classic Fillipov results.

This paper aims at extending some results concerning the calculation of the residence time for linear systems to continuous flow nonlinear multidimensional systems. It is shown that the solution of a nonlinear system is equivalent to the solution of a linear system with time-varying parameters. An algorithm for calculating the residence time for nonlinear systems is developed. Simulation results dealing with three examples show the feasibility and performance of this proposed algorithm. However, this algorithm can be applied to other processes as well.

We propose a new 3-D visual servoing method. By using Lyapunov theorem, we conclude that this 3-D visual servoing approach is stable with some uncertainties. The experimental comparison of 3-D vision is realized by a manipulator A465 and a pair of Pulnix TM-72EX CCD analog cameras. The 3D algorithm has been checked experimentally.

To identify Saccharomyces cereviseae fed-batch fermentation process a differential (dynamic in continuous time) neural network (DNN) is suggested. A priory information on the dynamic equations as well as on their parameters are not required to realize this approach. Based on real data, obtained from laboratory experiments, this DNN of a simple structure (6 neurons) is shown to have a high capability to identify (to estimate the states) this process after several working hours of learning.

The continuous time observer design based on differential neural networks (DNN) is presented. It is tested by the application to an immunological interaction model for HIV. To give the corresponding data interpretation and to justify the observability property the Kirschner's model is used. The suggested observer estimates the model's variables behavior providing the evolution of the number of infected and not infected lymphocyte T-CD4 cells during 10 days approximately starting from the treatment using Zidovuina (AZT). This approach does not require any model of this process to be in use. The obtained results are shown to allow the programming a high-quality medication protocol for zero-positive patients or AIDS patients.