Tracking and Regulation I: Output Regulation

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

The use of manipulators can improve sustainable energy utilization efficiency and increase sustainable manufacturing practices for solar tracking systems and manufactures, and thus it is significant to guarantee a high tracking accuracy for manipulators. In this paper, an error-tracking adaptive iterative learning control (AILC) method is proposed for a constrained flexible-joint manipulator (FJM) with initial errors. Due to the existence of the repeated positioning drift, the accuracy of the actual manipulator and the sustainable energy utilization efficiency are affected, which motivates the error-tracking approach proposed in this paper to deal with the repeat positioning problem. The desired error trajectory is constructed, such that the tracking error can follow the desired error trajectory without arbitrary initial values and iteration-varying tasks. Then, the system uncertainties are approximated by the capability of fuzzy logic systems (FLSs), and the combined adaptive laws are designed to update the weight and the approximating error of FLSs. Considering the safety operation of the flexible-joint manipulator, both input and output constraints are considered, a quadratic-fractional barrier Lyapunov function (QFBLF) is constructed, such that the system output is always within the constrained region. Therefore, the proposed method can guarantee the output tracking accuracy of manipulators under arbitrary initial values and iteration-varying tasks and keep the system output within the constraints to improve the transient performance, such that the energy utilization and accessory manufacturing efficiency can be improved. Through the Lyapunov synthesis, it is proved that the tracking error can converge to zero as the number of iterations goes to infinity. Finally, comparative simulations are carried out to verify the effectiveness of the proposed method.

The semi-global stabilization problem and output regulation problem of singular linear systems subject to input saturation are addressed. A reduced-order normal system is obtained by a standard coordinate transformation. It is further shown that the controller that solves the stabilization (output regulation) problem of the reduced-order normal systems also solves the stabilization (output regulation) problem of the original singular systems.

At present, problems related to data mining are aris-ing in various poorly formalizable areas of research(genetics, archaeology, medicine, etc.). Such problemsare characterized by insufficient knowledge about theobjects under study, which hampers the development ofmathematical models for them; by a large number ofmultitype (quantitative and qualitative) factors and arelatively small amount of data; and by the requirementthat analysis results be represented in a form clear toexperts in applied areas.A promising approach to solving such problems isbased on the class of logical decision functions. Theyare frequently represented in the form of a decisiontree. A detailed description of this class and referencescan be found in [1].The construction of decision functions with a mini-mal risk of erroneous predictions is an important task inmethods based on logical decision functions and inother data analysis techniques. It is well known that thecomplexity of the class of decision functions (thecapacity characteristic of the class [2]) is a importantfactor affecting the quality of decisions. To achieve thebest quality, we need a certain tradeoff between thecomplexity of the class and the accuracy of decisionsobtained on the training sample. Thus, the problemarises of choosing an optimal complexity of the class.In numerous applications, empirical data (trainingsample) are supplemented with expert knowledge,which is not tightly related to a distribution model(expert knowledge can be specified as upper bounds forthe risk; they can express constraints on the class of dis-tributions or the class of decision functions, specifypreference rules used in decision making, etc.). Whenan optimal complexity of the class is chosen, availableempirical data and expert knowledge have to be jointlytaken into account.This problem can be solved within the framework ofthe Bayesian learning theory. The idea behind thisapproach is to use a priori knowledge about the prob-lem in question so that each possible distribution (strat-egy, the state of nature) is assigned a certain weight.This weight reflects the expert’s intuitive confidence inwhether the unknown true distribution coincides withthat under consideration.In this paper, we propose a Bayesian recognitionmodel based on a finite set of events. The model is usedto develop and analyze methods for constructing logi-cal decision functions. While choosing an optimal com-plexity of the class, the model takes into account empir-ical data and expert knowledge.In pattern recognition, the task is to predict the pat-tern’s index for an arbitrary object of the populationdescribed by a set of variables. The prediction is madeby analyzing a training sample consisting of the valuesof these variables, together with the index of the corre-sponding pattern for each object. The variables can beof different types; i.e., some of them can be quantita-tive, while the others can be qualitative. As a rule, theproblem is solved in a class of decision functions inwhich an optimal function is sought according to agiven criterion. The class of logical decision functionsis defined on a set of partitions of the feature space intoa finite number of subdomains described by conjunc-tions of simple predicates. The number of subdomainsdetermines the degree of complexity of a logical func-tion.A Bayesian pattern recognition model on a finite setof events is defined by formulating certain statementsthat avoid the local metric properties of the featurespace (the transition from points of the space to events,where an event is understood as that of the original vari-ables taking values from a certain subdomain of thepartition). Moreover, the pattern recognition problem isconsidered on the values of a discrete unordered vari-able, the concept of a learning method (a mapping from

Let f be a function on a set of variables V. For each x e V, let c(x) be the cost of reading the value of x. An algorithm for evaluating f is a strategy for adaptively identifying and reading a set of variables U ⊆ V whose values uniquely determine the value of f. We are interested in finding algorithms which minimize the cost incurred to evaluate f in the above sense. Competitive analysis is employed to measure the performance of the algorithms. We study two variants of the above problem. First we consider the classical setting in which one assumes that the algorithm knows the cost c(x), for each x e V. For the case where f is a monotone boolean function which is representable by a threshold tree, we provide a polynomial time algorithm with the best possible competitive ratio γfc for each fixed cost function c(·). Remarkably, the best known result for the same class of functions is a pseudo-polynomial algorithm with competitiveness 2γfc. For the class of game tree functions our polynomial time algorithm attains, for each fixed cost function, the same competitiveness as the best known algorithm for the same class of functions, which instead runs in pseudo-polynomial time.In the second part of the paper, we study a novel variant of the problem. Here, we assume that the cost function is not known in advance and some preemption is allowed in the reading operations. This model has applications, e.g., when reading a variable coincides with obtaining the output of a job on a CPU and the cost is the CPU time. In such a case, it is reasonable to assume that no exact knowledge of the cost is available. We define a new algorithm for this problem based on the solution of a Linear Program. We show the optimality of our algorithm for the class of monotone boolean functions representable by AND-OR-trees. We also show a sub-optimal implementation for general monotone boolean functions.

The problem of global robust output regulation for a class of switched nonlinear systems with nonlinear exosystems is investigated in this paper where not all subsystems are stabilizable. First, as an extension of the concept of non‐switched internal model, this paper defines a switched internal model, which together with the plant is called the switched augmented system. Also, we show that the problem of output regulation for the switched nonlinear system can be converted into a global robust regulation problem of the switched augmented system. Second, based on the average dwell time method, the global robust regulation problem under a class of switching signals with average dwell time is solved, which leads to the solution of the problem of global robust output regulation. Finally, an example is provided to demonstrate the effectiveness of the proposed design approach.

Fluctuating hydrodynamics and concentration fluctuations in ternary mixtures

The holomorphic embedding (HE) method is a novel recursive method for power flow analysis. Using the fixed flat start, the HE guarantees to yield the high-voltage solution and provides unequivocal signals if no solution exists. The fast and flexible HE (FFHE) method is adaptive with flexible initial values. However, the FFHE cannot start from arbitrary initial values because it is not globally convergent. This paper studies the convergence performance of the FFHE and finds their regularity. To further improve the practicality of this method, this paper finds a numerically common domain of convergence for cases with different scales, ranging from 2 to 2383 buses, using the FFHE. The proposed common domain of initial values is applicable for a variety of cases using FFHE.

As key electrical devices, large reactors and converter transformers play an important role in the operation of ultra high voltage direct current (UHVDC) transmission [1]. Meanwhile, the existence of rich harmonics in the exciting current or driving voltage leads to distorted flux densities in the saturated iron core. Consequently the magnetic force due to the leakage flux not only exhibits special spectrum in the frequency domain but also threatens the normal operation of electrical devices e.g. vibration and deformation of windings [2–3]. Therefore it is important to investigate the influence of magnetic force on iron core and windings when electrical devices such as reactors work under harmonic magnetizations. The time-stepping method starting from arbitrary initial value is usually used for the computation of time-periodic nonlinear problems. However, the main drawback is that many periods are required to be stepped through for steady state solution, especially when a much larger time constant exists in the nonlinear system to be analyzed. Numerical methods in time domain and frequency domain have been presented and developed to solve the steady state nonlinear magnetic field efficiently [4–5] by combining the fixed point technique with the finite element method. In this paper, the harmonic-balance system of equations is established first by approximating all time-periodic variables by complex series. Thereupon, decomposition of the system of equations is achieved according to diagonal dominant characteristic of the reluctivity matrix so that harmonic solutions can be decoupled and computed separately and in parallel. This method is used to compute the magnetic force in a gapped-core model under harmonic magnetizations based on the harmonic solutions of magnetic field.

The stationary, periodic solution to the problem of oscillating flow of a conducting fluid, in a duct closed at one end and with periodical mass flow rate at the other end, is known to exhibit a singularity in the mean temperature at the closed end when transverse conduction in the fluid and the duct wall is taken into account, but longitudinal conduction is neglected. For instance, a solution of that type was originally suggested by Gifford and Longsworth as a prototype for pulse-tube refrigeration. Whether these stationary singular solutions are physical or mere theoretical curiosities depend upon the existence of a scenario leading to such a limit cycle. To address that question, a transient theory is formulated, using the narrow duct approximation. The results show that at least for constant fluid thermal conductivity, all singular profiles generated by the mechanism under study are linearly stable. For round tubes, the temperature profile is shown to evolve, from an arbitrary initial value, toward an equilibrium profile which results in balanced energy fluxes.

We consider a solution u(p, g, a, b) to an initial value–boundary value problem for a wave equation: and we discuss an inverse problem of determining a coefficient p(x) and a, b by observations of u(p, g, a, b)(x, t) in a neighbourhood ω of ∂Ω over a time interval (0, T) and ∂itu(p, g, a, b)(x, T0), x ∊ Ω, i = 0, 1, with T0 < T. We prove that if T − T0 and T0 are larger than the diameter of Ω, then we can choose a finite number of Dirichlet boundary inputs g1, …, gN, so that the mapping is uniformly Lipschitz continuous with suitable Sobolev norms provided that {p, aj, bj}1 ⩽ j ⩽ N remains in some bounded set in a suitable Sobolev space. In our inverse problem, initial values are also unknown, and we do not assume any positivity of the initial values. Our key is a Carleman estimate and the exact controllability in a Sobolev space of higher order.

The state vector equation for lateral buckling in finite displacement theory is formulated using only the hypothesis of the Bernoulli–Euler beam. By using an appropriate orthogonalization of the warping functions, the normally complicated calculation has been processed systematically using only matrix notation. As a numerical analysis, the lateral buckling load on the cantilever receiving a concentrated end load on the upper flange was calculated using the coefficient matrix of the first-order increment; the post-buckling behaviour was investigated with increasing load. Since the state vector equation is a higher order nonlinear equation, the original coefficient matrix was fixed with an arbitrary initial value and the solution was provided by the Runge–Kutta transfer matrix method. Subsequent calculations were pursued in the same way with the solution obtained via Runge–Kutta methods as a new initial value and then shifted to the next load condition. This theory and analysis method does not employ an assumed displacement function, such as the Ritz's method; it is therefore useful for the finite displacement analysis of a beam with arbitrary boundary conditions and intermediate support conditions.

In this paper, we introduced the extended generalized Fibonacci polynomial sequence \{$Y_{2,n}$\} and extended generalized Tribonacci polynomial \{$Y_{3,n}$\} with arbitrary initial values and established a recursive matrix and then presented some properties of these. Further, we investigated some well-known identities like Binet's formula, Catalan's identity, Cassini's identity, d'Ocagne's identity, generating function, explicit sum formula, sum of first $n$ terms for the extended generalized Fibonacci polynomial sequence and extended generalized Tribonacci polynomial sequence.

The aim of this paper is to investigate the behavior of quotients of sequences w n defined by the linear homogeneous recurrence relation $$w_{n + 2} = pw_{n + 1} - qw_n$$ (1) with arbitrary initial values w 0 = a, w 1 = b belonging to a normed field K, and p, q ∈ K. We shall write w n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts $$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$ (2.1) is defined in K. If the field K is normed and α, β are the roots of the polynomial x 2 – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q n ’s, where it is defined, converges, and $$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$ (3)

The existing Adaptive Notch Filter (ANF)-based frequency estimation methods have problems of slow convergence speed, unstable error and limited selection of iterative initial values. In this paper, a frequency estimation method based on the indirect average plain gradient algorithm is proposed. It uses a new error function applied to the second-order adaptive Infinite Impulse Response (IIR) notch filter. The proposed error function is the mean value of the weighted average of the squares of the two signals outputted by Finite Impulse Response (FIR) and IIR sections of the ANF. It has better gradient characteristics, and arbitrary initial value can be selected for the following iterative calculation. The theory and simulation show that the proposed algorithm, compared with other indirect gradient algorithm, improves the convergence speed and estimation accuracy with little addition of the computation, while it is superior in varying-frequency signal tracking performance.