Search Results (9)

This paper introduces a novel field-oriented control (FOC) strategy for an open-end stator three-phase winding induction motor (OEW-TP-IM) using dual space vector modulation-pulse width modulation (SVM-PWM) inverters. This configuration reduces common mode voltage at the motor’s terminals, enhancing efficiency and reliability. The study presents a backstepping control approach combined with a mean value theorem (MVT)-based observer to improve control accuracy and stability. Stability analysis of the backstepping controller for key control loops, including flux, speed, and currents, is conducted, achieving asymptotic stability as confirmed through Lyapunov’s methods. An advanced observer using sector nonlinearity (SNL) and time-varying parameters from convex theory is developed to manage state observer error dynamics effectively. Stability conditions, defined as linear matrix inequalities (LMIs), are solved using MATLAB R2016b to optimize the observer’s estimator gains. This approach simplifies system complexity by measuring only two line currents, enhancing responsiveness. Comprehensive simulations validate the system’s performance under various conditions, confirming its robustness and effectiveness. This strategy improves the operational dynamics of OEW-TP-IM machine and offers potential for broad industrial applications requiring precise and reliable motor control. Full article

This paper concerns control BVPs, driven by ODEs $x^{\prime }\left(t\right)=u\left(t\right)$, using controls $u^0\left(·\right)$ $\&u^1\left(·\right)$ in $L^1\left(\left(a,b\right),{\mathbb{R}}^2\right)$. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write $(u^1-u^0)\left(·\right)=v\left(·\right)\left(1,f\left(·\right)\right)$ for some $f\left(·\right)$. Given a relaxed non bang-bang solution $\overline{x}\left(·\right)\in W^{1,1}\left(\left[a,b\right],{\mathbb{R}}^2\right)$, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution $\widehat{x}\left(·\right)$ having lower first-coordinate ${\widehat{x}}_1\left(·\right)\le {\overline{x}}_1\left(·\right)$? Being the answer always yes in dimension $d=1$, hence without $f\left(·\right)$, as proved by Amar and Cellina, for $d=2$ the problem is to find out which functions $f\left(·\right)$ “are good”, namely “allow such 1-lower bang-bang solution $\widehat{x}\left(·\right)$ to exist”. The aim of this paper is to characterize “goodness of $f\left(·\right)$” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given $f\left(·\right)$ is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good. Full article

This work investigates the stability conditions for linear systems with time-varying delays via an augmented Lyapunov–Krasovskii functional (LKF). Two types of augmented LKFs with cross terms in integrals are suggested to improve the stability conditions for interval time-varying linear systems. In this work, the compositions of the LKFs are considered to enhance the feasible region of the stability criterion for linear systems. Mathematical tools such as Wirtinger-based integral inequality (WBII), zero equalities, reciprocally convex approach, and Finsler’s lemma are utilized to solve the problem of stability criteria. Two sufficient conditions are derived to guarantee the asymptotic stability of the systems using linear matrix inequality (LMI). First, asymptotic stability criteria are induced by constructing the new augmented LKFs in Theorem 1. Then, simplified LKFs in Corollary 1 are proposed to show the effectiveness of Theorem 1. Second, asymmetric LKFs are shown to reduce the conservatism and the number of decision variables in Theorem 2. Finally, the advantages of the proposed criteria are verified by comparing maximum delay bounds in four examples. Four numerical examples show that the proposed Theorems 1 and 2 obtain less conservative results than existing outcomes. Particularly, Example 2 shows that the asymmetric LKF methods of Theorem 2 can provide larger delay bounds and fewer decision variables than Theorem 1 in some specific systems. Full article

The general delay Hopfield neural network is studied. We consider the case of time-varying delay, continuously distributed delays, time-varying coefficients, and a special type of a Riemann–Liouville fractional derivative (GRLFD) with an exponential kernel. The kernels of the fractional integral and the fractional derivative in this paper are Sonine kernels and satisfy the first and the second fundamental theorems in calculus. The presence of delays and GRLFD in the model require a special type of initial condition. The applied GRLFD also requires a special definition of the equilibrium of the model. A constant equilibrium of the model is defined. An inequality for Lyapunov type of convex functions with the applied GRLFD is proved. It is combined with the Razumikhin method to study stability properties of the equilibrium of the model. As a partial case we apply quadratic Lyapunov functions. We prove some comparison results for Lyapunov function connected deeply with the applied GRLFD and use them to obtain exponential bounds of the solutions. These bounds are satisfied for intervals excluding the initial time. Also, the convergence of any solution of the model to the equilibrium at infinity is proved. An example illustrating the importance of our theoretical results is also included. Full article

This paper aims to develop a sliding mode control (SMC) approach with elementary compensation for input matrix uncertainty in affine systems. As a multiplicative uncertainty regarding the control inputs, input matrix uncertainty adversely modifies the control effort and even further causes the instability of systems. To solve this issue, a sliding mode control algorithm is developed based on a two-step design strategy. The first step is to design a general sliding mode controller for the system without input matrix uncertainty. In the second step, a control term is specially designed to compensate for input matrix uncertainty. In order to realize the elementary compensation for input matrix uncertainty, this term is obtained by solving a nonlinear vector equation which is derived from the Lyapunov function inequality. Theorems and lemmas based on the convex cone theory are proposed to guarantee the existence and uniqueness of the solution to the vector equation. Additionally, an algorithmic process is proposed to solve the vector equation efficiently. In the simulation part, the proposed controller is applied to two systems with different structures and compared with two state-of-the-art SMC algorithms. The comprehensive simulation results demonstrate that the proposed method is able to provide the closed-loop system with a competitive performance in terms of convergence level, overshoot reduction and chattering suppression. Full article

This paper is concerned with the problem of static output feedback control for a class of continuous-time nonlinear time-delay semi-Markov jump systems with incremental quadratic constraints. For a class of time-delay semi-Markov jump systems satisfying incremental quadratic constrained nonlinearity, an appropriate mode-dependent Lyapunov–Krasovskii functional is constructed. Based on the matrix transformation, projection theorem and convex set principle, the mode-dependent static output feedback control laws are designed. The feedback control law is given in the form of a linear matrix inequality, which is convenient for a numerical solution. Finally, two practical examples are given to illustrate the effectiveness and superiority of the proposed method. Full article

An infinite-bound stabilization of a system modeled as singularly perturbed bilinear systems is examined. First, we present a Lyapunov equation approach for the stabilization of singularly perturbed bilinear systems for all ε∈(0, ∞). The method is based on the Lyapunov stability theorem. The state feedback constant gain can be determined from the admissible region of the convex polygon. Secondly, we extend this technique to study the observer and observer-based controller of singularly perturbed bilinear systems for all ε∈(0, ∞). Concerning this problem, there are two different methods to design the observer and observer-based controller: one is that the estimator gain can be calculated with known bounded input, the other is that the input gain can be calculated with known observer gain. The main advantage of this approach is that we can preserve the characteristic of the composite controller, i.e., the whole dimensional process can be separated into two subsystems. Moreover, the presented stabilization design ensures the stability for all ε∈(0, ∞). A numeral example is given to compare the new ε-bound with that of previous literature. Full article

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given. Full article

H-theorem states that the entropy production is nonnegative and, therefore, the entropy of a closed system should monotonically change in time. In information processing, the entropy production is positive for random transformation of signals (the information processing lemma). Originally, the H-theorem and the information processing lemma were proved for the classical Boltzmann-Gibbs-Shannon entropy and for the correspondent divergence (the relative entropy). Many new entropies and divergences have been proposed during last decades and for all of them the H-theorem is needed. This note proposes a simple and general criterion to check whether the H-theorem is valid for a convex divergence H and demonstrates that some of the popular divergences obey no H-theorem. We consider systems with n states A_i that obey first order kinetics (master equation). A convex function H is a Lyapunov function for all master equations with given equilibrium if and only if its conditional minima properly describe the equilibria of pair transitions A_i ⇌ A_j . This theorem does not depend on the principle of detailed balance and is valid for general Markov kinetics. Elementary analysis of pair equilibria demonstrate that the popular Bregman divergences like Euclidian distance or Itakura-Saito distance in the space of distribution cannot be the universal Lyapunov functions for the first-order kinetics and can increase in Markov processes. Therefore, they violate the second law and the information processing lemma. In particular, for these measures of information (divergences) random manipulation with data may add information to data. The main results are extended to nonlinear generalized mass action law kinetic equations. Full article