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We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let

From an engineering point of view, non‐linear systems are essential to the operation of control systems, because all systems actually have a non‐linear state in nature. In reality, there are many different kinds of non‐linear systems hidden by this negative definition. For successful analysis and control, the identification of non‐linear systems using unknown models is typically necessary. Till now, numerous approaches are developed for identifying non‐linear systems, but it cannot be employed with a large number of components. Moreover, system identification is typically restricted to output and input signals alone, also such systems are rarely used in reality. This is the primary justification for using non‐linear systems in this research. So, this research proposed a non‐linear model of system identification for large‐scale systems under the consideration of two systems: bilinear system and Volterra system. Therefore, a novel algorithm named Self Adaptive Penguin Search Optimization (SAPeSO) is introduced to attain the system characteristics properly and minimize the output variation. Finally, the effectiveness of the proposed work is compared with existing works in terms of various error measures. This research mainly focuses on the application‐oriented engineering problems. In particular, the Mean Absolute Error (MAE) of the proposed work for the Volterra system at 4000 samples is 18.83%, 14.05%, 8.88%, 29.72%, 19.91%, and 6.70% which is better than the existing bald eagle search (BES), arithmetic optimization algorithm (AOA), whale optimization algorithm (WOA), nonlinear autoregressive moving average with exogenous inputs‐ frequency response function + principal component analysis (NARMAX‐FRF+PCA), Global Gravitational Search Algorithm‐Assisted Kalman Filter (CGS‐KF), and sparse regression and separable least squares method (SR‐SLSM) methods, respectively. Finally, the error is minimum for the proposed model when compared with the other traditional approaches.

For a connected smooth projective curve Let As an application, when

This study introduces a two‐stage estimation framework designed to mitigate the online computational burden associated with traditional state estimation algorithms, such as the Kalman filter and extended Kalman filter, which require continuous computation of time‐varying gains. The proposed framework integrates a fixed‐gain Luenberger observer for base estimation with a lightweight neural network‐based compensation module, which is trained offline and deployed online to enhance accuracy by correcting residual errors caused by fixed‐gain limitations. To handle both linear and non‐linear systems effectively, distinct loss functions are constructed by integrating prior knowledge of the system model, enabling the network to produce physically meaningful and interpretable outputs. Numerical simulations validate the effectiveness of the proposed method, demonstrating improved estimation accuracy and reduced computational cost compared to conventional estimators. This article proposes a two‐stage estimation framework that combines a fixed‐gain Luenberger observer with an offline‐trained neural network for online error compensation. By incorporating system‐informed loss functions, the method enhances estimation accuracy for both linear and non‐linear systems while significantly reducing online computational burden.

The paper addresses the problem of accelerating predictive control of non‐linear system models augmented with Gaussian processes (GP‐MPC). Due to the non‐linear and stochastic prediction model, predictive control of GP‐based models requires to solve a stochastic optimization problem. Different model simplification methods have to be applied to reformulate this problem to a deterministic, non‐linear optimization task that can be handled by a numerical solver. As these problems are still complex, especially with exact moment calculations, real‐time implementation of GP‐MPC is extremely challenging. The existing solutions accelerate the computations at the solver level by linearizing the non‐linear optimization problem and applying sequential convexification. In contrast, this paper proposes a novel GP‐MPC solution approach that without linearization formulates a series of surrogate quadratic programs (QP‐s) to iteratively obtain the solution of the original non‐linear optimization problem. The first step is embedding the non‐linear mean‐variance dynamics of the GP‐MPC prediction model in a linear parameter‐varying (LPV) structure and rewriting the constraints in parameter‐varying form. By fixing the scheduling trajectory at a known variation (based on previously computed or initial state‐input trajectories), optimization of the input sequence for the remaining varying linear model reduces to a linearly constrained quadratic program. After solving the QP, the non‐linear prediction model is simulated for the new control input sequence and new scheduling trajectories are updated. The procedure is iterated until the convergence of the scheduling, that is, the solution of the QP converges to the solution of the original non‐linear optimization problem. By designing a reference tracking controller for a 4DOF robot arm, we illustrate that the convergence is remarkably fast and the approach is computationally advantageous compared to current solutions. The proposed method enables the application of GP‐MPC algorithms even with exact moment matching on fast dynamical systems and requires only a QP solver. This paper aims to discuss the approach of constrained modified feedback linearization model predictive control (CMFLMPC) for the spacecraft simulator. The simulation and experimental results demonstrate that the proposed hybrid controller has an insignificant calculative cost and facilitates the spacecraft to perform the regulation manoeuvre with sufficient precision in the presence of external torques and actuator saturations.

In this paper, we propose a technique to solve LR -bipolar fuzzy linear system(BFLS), LR -complex bipolar fuzzy linear (CBFL) system with real coefficients and LR -complex bipolar fuzzy linear (CBFL) system with complex coefficients of equations. Initially, we solve the LR -BFLS of equations using a pair of positive ( ∗ ) and negative ( ∙ ) of two n × n LR -real linear systems by using mean values and left-right spread systems. We also provide the necessary and sufficient conditions for the solution of LR -BFLS of equations. We illustrate the method by using some numerical examples of symmetric and asymmetric LR -BFLS equations and obtain the strong and weak solutions to the systems. Further, we solve the LR -CBFL system of equations with real coefficients and LR -CBFL system of equations with complex coefficients by pair of positive ( ∗ ) and negative ( ∙ ) two n × n real and complex LR -bipolar fuzzy linear systems by using mean values and left-right spread systems. Finally, we show the usage of technique to solve the current flow circuit which is represented by LR -CBFL system with complex coefficients and obtain the unknown current in term of LR -bipolar fuzzy complex number.

A full-rank matrix ${\\bf A}\\in {\\Bbb R}^{n\\times m}$ with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries. Can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena, in particular the existence of easily verifiable conditions under which optimally sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable, but there is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to undetermined systems have energized research on such signal and image processing problems--to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical results on sparse modeling of signals and images, and recent applications in inverse problems and compression in image processing. This work lies at the intersection of signal processing and applied mathematics, and arose initially from the wavelets and harmonic analysis research communities. The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications.

This paper investigates the robust model reference tracking control problem for high‐order descriptor linear systems (HODLS) subject to norm‐bounded parameter uncertainties. The problem is divided into two subproblems: a robust proportional plus derivative state feedback stabilization (RPPDSFS) problem and a robust feed‐forward compensation (RFFC) problem. In the presence of uncertainties, by using the controller consisting of RPPDSFS part and RFFC part, all the signals of the closed‐loop system are bounded. The latter is equivalent to seeking three coefficient matrices such that a series of linear matrix equations are satisfied, and simultaneously the effect caused by uncertainties is minimized. Based on the right coprime factorization, the RFFC problem is further converted into a minimization problem with a quadratic performance index and some linear constraints. Thereafter, a linear matrix equation that determines the optimal solution is achieved. The restrictive condition that the coefficient matrix of the augmented reference system is non‐defective is relaxed, which reduces the conservativeness. A numerical example and the application of a three‐axis dynamic flight motion simulator manifest the effectiveness and the practicability of the proposed algorithm. This paper focuses on the robust model reference tracking control problem for high‐order descriptor linear systems subject to parameter uncertainties. The new robust feed‐forward compensation algorithm is proposed for high‐order descriptor linear systems, which take the first‐ or second‐order descriptor linear systems and high‐order linear systems as special cases and have a wide perspective of applications. The restrictive condition that the coefficient matrix of the augmented reference system is non‐defective is relaxed, which reduces the conservativeness.

Research on the physical memristive system is vital for realizing widespread applications of the memristor in practical engineering. In this article, a family of the asymmetric generalized passive voltage‐controlled memristive system (AGPVCMS) is proposed; also the mathematical model of the AGPVCMS is established and its remarkable memristive characteristics are analysed. Especially, the AGPVCMS only consists of the diode bridge and LR filter, however it can enhance its asymmetry as well as display various asymmetrical pinched hysteresis loops in the voltage–current plane, via expediently changing the number of the diodes on different bridge arms. In addition, the effectiveness of the AGPVCMS is verified by the consistency between theoretical and experimental results. A novel family of asymmetric generalized passive voltage‐controlled memristive systems (AGPVCMS) is proposed and analysed. Especially the AGPVCMS can enhance its asymmetry and display various asymmetrical pinched hysteresis loops in the voltage–current plane only by changing the number of the diodes on different bridge arms. Also, the asymmetrical hysteresis loop characteristics of the AGPVCMS are verified by the consistency of the theoretical mathematical model and experimental results.