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Nonlinear dynamical systems may, under certain conditions, be represented by a bilinear system. The paper is concerned with the construction of the controllability and observability gramians for the corresponding bilinear system. Such gramians form the core of model reduction schemes involving balancing.

The paper examines certain properties of the bilinear system and identifies parameters that capture important information relating to the behaviour of the system.

Novel approaches for the determination of approximate constant gramians for use in balancing‐type model reduction techniques are presented. Numerical examples are given which indicate the efficacy of the proposed formulations.

The systems under consideration are restricted to the so‐called weakly nonlinear systems, i.e. those without strong nonlinearities where the essential type of behaviour of the system is determined by its linear part.

The suggested methods lead to an improvement in the accuracy of model reduction. Model reduction is a vital aspect of modern system simulation.

The proposed novel approaches for model reduction are particularly beneficial for the design of controllers for nonlinear systems and for the design of radio‐frequency integrated circuits.

This paper aims to study the Gramian solutions and solitonic interactions of a (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, which models the nonlinear and dispersive long gravity waves traveling along two horizontal directions in the shallow water of uniform depth.

Pfaffian technique is used to construct the Gramian solutions of the (2 + 1)-dimensional BKK system. Asymptotic analysis is applied on the two-soliton solutions to study the interaction properties.

N-soliton solutions in the Gramian with a real function ζ(y) of the (2 + 1)-dimensional BKK system are constructed and proved, where N is a positive integer and y is the scaled space variable. Conditions of elastic and inelastic interactions between the two solitons are revealed asymptotically. For the three and four solitons, elastic, inelastic interactions and soliton resonances are discussed graphically. Effect of the wave numbers, initial phases and ζ(y) on the solitonic interactions is also studied.

Shallow water waves are studied for the applications in environmental engineering and hydraulic engineering. This paper studies the shallow water waves through the Gramian solutions of a (2 + 1)-dimensional BKK system and provides some phenomena that have not been studied.

One of the fundamental problems in control systems engineering is the problem of sensors and actuators placement. Decisions in this context play a key role in the success of control process. The methods developed for optimal placement of the sensors and actuators are known to be computationally expensive. The computational burden is significant, in particular, for large-scale systems. The purpose of this paper is to improve and extend the state-of-the-art methods within this field.

In this paper, a new technique is developed for placing sensor and actuator in large-scale systems by using restricted genetic algorithm (RGA). RGA is a kind of genetic algorithm which is developed specifically for sensors and actuator placement.

Unlike its other counterparts, the proposed method not only supports unstable systems but also requires significantly lower computations. The numerical investigations have confirmed the advantages of the proposed methods which are clearly significant, in particular, in dealing with large-scale unstable systems.

The proposed method is novel, and compared to the methods which have already been presented in literature is more general and numerically more efficient.

This paper presents the application in circuit simulation of a method for model reduction of nonlinear systems that has recently been developed for chemical systems. The technique is an extension of the well‐known balanced truncation method that has been applied extensively in the reduction of linear systems. The technique involves the formation of controllability and observability gramians either by simulated results or by measurement data. The empirical gramians are subsequently employed to determine a subspace of the full state‐space that contains the most significant dynamics of the system. A Galerkin projection is used to project the system onto the subspace to form a lower‐dimensional nonlinear model. The method is applied to a nonlinear resistor network which is a standard example for exemplifying the effectiveness of a nonlinear reduction strategy.

The purpose of this paper is to present the embedding of linear and nonlinear order reduction methods in a theoretical framework for handling hierarchically interconnected dynamical systems.

Based on the component connection modeling (CCM), a modified framework called mCCM for describing interconnected dynamic systems especially with hierarchical structures is introduced and used for order reduction purposes. The balanced truncation method for linear systems and the trajectory piecewise linear reduction scheme are used for the order reduction of systems described within the mCCM framework.

It is shown that order reduction methods can be embedded into the context of interconnected dynamical systems with the benefit of having a further degree of freedom in form of the hierarchical level, on which the order reduction is performed.

The aspect of hierarchy within system descriptions is exploited for order reduction purposes. This distinguishes the presented approach from common methods, which already start with single large-scale systems without explicitly considering hierarchical structures.

This study aims to focus on a short review on recent results dealing with the mechanical modelling and experimental characterization of a novel class of tensegrity structures, named class θ = 1 tensegrity prisms. The examined structures exhibit six bars connected by two disjoint sets of strings.

First, the self-equilibrium problem of tensegrity θ = 1 prisms is numerically investigated for varying values of two aspect parameters and, next, their prestress stability is studied. The mechanical behavior of the examined structures in the large displacements regime under uniform compression loading is also numerically computed through a path-following procedure. Finally, the predicted constitutive response is validated through experimental tests.

The presented results highlight that the examined structures exhibit a large number of infinitesimal mechanisms from the freestanding configuration, and reveal that they exhibit tunable elastic response switching from stiffening to softening.

This multi-faceted elastic response is in agreement with previous literature results on the elastic response of minimal tensegrity prism, and suggests that such units can be usefully used as non-linear springs in next-generation tensegrity metamaterials.

The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X.

By applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation.

The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms.

So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation.

The purpose of this paper is to present a proportional-integral (PI) observer design on a linear system with stochastic noises.

The noised disturbances are modeled as independent Brownian motions for various affections, such as radiation, heat, and material fatigue. These phenomena are common in applications, such as biomolecules, nonlinear control, and biochemical networks. Under this framework, this paper proposes a new approach on a PI observer in terms of four crucial theorems, and an illustrative numerical example is given to verify the proposed design.

The results provide potential solutions for system fault tolerance and isolation.

This paper proposes a design, solvability, and controllability analysis on a PI observer in terms of four crucial theorems.

The purpose of this paper is to analyze and control the flutter vibrations of a thermoelastic functionally graded material (FGM) beam subjected to follower force using the piezoelectric sensors/actuators.

The beam is made of FGM properties which are functionally graded in the thickness direction according to the volume fraction power law distribution and change with temperature. As the two sides of the beam are located in two different temperatures, the thermoelastic effects are considered in the governing equation of motion. The beam is fixed from one end and a follower force is applied to the free end of it. An active control is applied to the system to suppress the flutter vibration of the beam.

After the simulation, the effects of the temperature gradient, magnitude of the follower force and piezoelectric lengths on the dynamic stability and the response of the system are studied. Simulation results show that the vibration of the system has been damped rapidly by applying the controller to the system.

Stability analysis and robust control of a thermoelastic FGM beam subjected to a follower force using piezoelectric sensors and actuators is the novelty of this study.

This paper aims to investigate the feasibility of using the combination of Lorentz force and aerodynamic force as a propellantless control method for spacecraft formation.

It is assumed that each spacecraft is equipped with several large flat plates, which can rotate to produce aerodynamic force. Lorentz force can be achieved by modulating spacecraft’s electrostatic charge. An adaptive output feedback controller is designed based on a sliding mode observer to account for unknown uncertainties and the absence of relative velocity measurements. Aiming at distributing the control input, an optimal control allocation method is proposed to calculate the electrostatic charge of the Lorentz spacecraft and control commands for the atmospheric-based actuators.

Numerical examples are provided to demonstrate the effectiveness of the proposed control strategy in the presence of J₂ perturbations. Simulation results show that relative motion in a formation can be precisely controlled by the proposed propellantless control method under uncertainties and unavailability of velocity measurements.

The controllability of the system is not theoretically investigated in the current work.

The proposed control method introduced in this paper can be applied for small satellites formation in low Earth orbit.

The main contribution of the paper is the proposal of the propellantless control approach for satellite formation using the combination of Lorentz force and aerodynamic force, which can eliminate the requirement of the propulsion system.