Second-order Linear Systems Research Articles

Star-shaped control-vector sets of second-order systems with PWM-type input

This paper deals with second-order continuous-time linear time-invariant systems by means of a pulse-width modulation (PWM)-type input. In general, it is difficult to analyze and design a PWM system because of its nonlinearity in terms of the duty ratio. In this paper, it is shown that the nonlinearity appearing in the discrete-time model at the sampling instants can be eliminated by varying the location of the pulse as well as the duty ratio.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

AC Tie-Line Power Oscillation Mechanism and Peak Value Calculation for a Two-Area AC/DC Parallel Interconnected Power System Caused by LCC-HVDC Commutation Failures

With the rapid development of ultra-high-voltage (UHV) AC/DC, especially the step-by-step upgrading of the UHV DC transmission scale, security presents new challenges. Commutation failure (CF) is a common fault in line commutated converter (LCC) high-voltage direct current (HVDC) power systems. Once failure happens, it may cause power oscillations in a system. In this paper, taking a two-area AC/DC parallel interconnected power system as the example, based on the impulse response model of second-order linear system, the mechanism of power oscillation on the AC tie-line caused by CF are clarified. It is proved that the peak value of the AC tie-line power oscillation is mainly determined by the DC power and the equivalent CF duration, the frequency and damping ratio of dominant area oscillation mode. Meanwhile, the peak time is mainly determined by the oscillation frequency. Finally, the correctness and effectiveness of the algorithms are verified by a simulation analysis of an extended IEEE-39-bus AC/DC parallel interconnected power system. These research results can provide a basis for the arrangement of the operating modes and the formulation of control measures for interconnected power grids.

Some insights on rightmost spectral values assignment for time delay systems

This paper investigates the stability and stabilization of some generic dynamic second-order linear time-invariant system including a single delay in the mathematical representation. As a first result, an appropriate stability criterion based on the manifold defined by the coexistence of the maximal number of negative spectral values is derived. Second, such ideas are exploited in the context of delayed output feedback by an appropriate “partial” pole placement guaranteeing simultaneously the stability in closed-loop and an appropriate exponential decay rate of the corresponding solution for the closed-loop system. To perform such an analysis, the argument principle is explicitly used. An illustrative example completes the presentation.

ℋ∞ Control Tunning to Guarantee the Output Performance of LTI Second-Order Systems

Settling time and input/output specifications are general for optimal operation of control systems and for their avoiding irreversible damages. We present an ℋ∞ control design whose parameters are tuned, not only to achieve the robustness property but also to meet the step time response characteristic of a linear time-invariant second-order system. We present explicit formulas in terms of the settling time and overshoot response. Simulation and experimental evidence corroborate the results.

Fuzzy Controllers With Neural Network Predictor for Second-Order Linear Systems With Time Delay

Artificial intelligence methods are widely applied in advanced control practices such as fuzzy calculation-based intelligent control method, neural network-based predictive control method, etc. Control performances are expected to be improved such as: faster control speed, smaller steady-state error, and less repeated manual tuning workloads in harmful environments for engineers. Main works are as follows: Firstly, change ratio-based fuzzy adjusted PID (FA-PID) method is improved. The adjusted parameters of FA-PID are the multiplication result of the change ratioa at the current control cycle and the control parameters at the time of previous adjustment cycle. Secondly, prediction of back propagation neural networks-based fuzzy-PID (BPNN-F-PID) and prediction of back propagation neural networks-based FA-PID (BPNN-FA-PID) are improved, in which the adjusted control parameters are calculated according to the predicted output of control system. Thirdly, comparative simulations of all the above methods are implemented, a series of better effects are found. Effects are as follows: The improved controllers have better performance such as faster control speed, better ability of anti-interference, restrained overshoot and smaller steady-state error.

Control of Dynamic Objects in the Conditions of Uncertainty in the Point Sliding Mode

There is development of the well-known sliding mode, which in the classical formulation didn’t find the development to be applied to control systems discussed. Alternatively, there is method of organizing one of the uniformity of the sliding mode called the "point sliding mode" proposed. The distinctive feature of this mode is that here the control gaps occur at time-equal points of the switching line (hyperplane) which allows the origin of coordinates for a finite number of switches. The possibility of changing the time interval between these points makes it possible to obtain various modes: a finite mode, in which a given point is reached from any initial state in one switch, and in this mode the switch line is "isochronous"; point sliding mode in which a given point is reached in a finite number of switchings; limit mode, when the length of time intervals tend to zero, and the switching frequency to infinity. Considering this feature the concept of "degree of slip" is introduced. It is shown that in the case of forced movement in the SPS, a sliding motion is observed, which does not allow for ensuring invariance with respect to external disturbances. There are two ways to eliminate the forced component of the movement offered. One of the advantages of using a point sliding mode is that, in order to improve performance, it is not necessary to use a boundary layer, which is realized by entering various logical conditions into the control algorithm. The practical significance of a point sliding mode lies in the fact that, with a small switching frequency, it is possible to maintain the quality indices of an undefined object within an acceptable interval. The studies were conducted for onedimensional second-order linear systems (SISO). Results can be generalized for higher order multidimensional systems. Solution of model problems on MATLAB / Simulink allows us to make a number of positive conclusions that are of great practical importance in terms of expanding the area of use of skipping modes, especially in relation to the management of undefined objects.

Receptance-Based Dominant Eigenvalues Computation of Controlled Vibrating Systems with Multiple Time-Delays Using a Contour Integral Method

The computation of dominant eigenvalues of second-order linear control systems with multiple time-delays is tackled by using a contour integral method. The proposed approach depends on a reduced characteristic function and the associated characteristic matrix comprised of measured open-loop receptances. This reduced characteristic function is derived from the original characteristic function of the second-order time delayed systems based on the reasonable assumption that eigenvalues of the closed-loop system are distinct from those of the open-loop system, and has the same eigenvalues as those of the original. Then, the eigenvalues computation is equivalent to solve a nonlinear eigenvalue problem of the associated characteristic matrix by using a contour integral method. The proposed approach also utilizes the spectrum distribution features of the retarded time-delay systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

No Spillover Eigenvalues Assignment of Second-Order Systems with Dense Force Actuators Matrices Using Brauer’s Theorem

This paper presents an approach for eigenvalue assignment in second-order linear systems with no spillover property. Second-order differential equations arise from dynamical modeling of vibrating structures by finite element or lumped parameter first principles approach in several practical problems. Certain structures can face practical issues when subjected to external perturbation forces, as resonance or flutter type vibrations. The control of excessive vibrations can be attempted by techniques of active vibration control using linear feedback. To change only a few eigenvalues and eigenvectors that cause excessive vibrations, the requirement of no spillover property is a somewhat attractive issue. Furthermore, only the part of the eigenstructure whose eigenvalues must be reassigned is necessary to be known for an efficient parametrization of the feedback matrices. Brauer’s theorem, a milestone result of linear algebra, as well as some recent related results, is applied here to achieve partial eigenvalue assignment using dense force actuator (influence) matrices. The proposal can be applied to general systems with no restriction on the mass, damping, and stiffness with symmetry or definiteness. The procedures to implement the proposal are synthesized in a step-by-step form, and some numerical examples are given to illustrate its application.

Master-slave synchronization of hyperchaotic systems through a linear dynamic coupling.

The development of synchronization strategies for dynamical systems is an important research activity that can be applied in several different fields from locomotion control of multilimbed structures to secure communication. In the presence of chaotic systems, synchronization is more difficult to accomplish and there are different techniques that can be adopted. In this paper we considered a master-slave topology where the coupling mechanism is realized through a second-order linear dynamical system. This control scheme, recently applied to chaotic systems, is here analyzed in the presence of hyperchaotic dynamics that represent a more challenging scenario. The possibility to reach a complete synchronization and the range of allowable coupling strength is investigated comparing the effects of the dynamical coupling with a standard configuration characterized by a static gain. This methodology is also applied to weighted networks to reach synchronization regimes otherwise not obtainable with a static coupling.

Robust stability of uncertain second-order linear time-varying systems

This paper is concerned with robust stability analysis of second-order linear time-varying (SLTV) systems with time-varying uncertainties (perturbations). With the specific Lyapunov functions, a simple and neat algebraic criterion for testing uniformly asymptotic stability of SLTV systems are derived. Without transformation to a system of first-order equations, the new conditions are imposed directly on the time-varying coefficient matrices of the system. The main feature of the proposed algebraic criterion is that the uncertain coefficient matrices are time-varying and not necessarily symmetric. Finally, the proposed stability conditions are used to design the extending space structures system of the spacecraft. Simulation results are provided to illustrate the convenience and effectiveness of the proposed method.

Parametric control to second-order linear time-varying systems based on dynamic compensator and multi-objective optimization

This paper investigates the parametric design approach to second-order linear time-varying systems by using dynamic compensator and multi-objective optimization. On the basis of the solution to a type of second-order generalized Sylvester matrix equations, the generally completely parameterized expression of the dynamic compensator is established, meanwhile, the completely parametric forms of left and right eigenvectors are obtained, it also provides two groups of arbitrary parameters. With the parametric method, the closed-loop system can be converted into a linear constant one with desired eigenstructure. Simultaneously, it also considers a novel technique to multi-objective optimization. Multiple performance indexes such as regional pole assignment, low sensitivity, disturbance attenuation, robustness degree and low gains are formulated by arbitrary parameters. Based on the above indexes, a synthetic objective function which includes each performance index weighted is formulated to express the comprehensive performances of control system. By using the degrees of freedom in arbitrary parameters, a dynamic compensator can be established by solving a multi-objective optimization problem. Finally, an example of spacecraft rendezvous problem is presented to verify that the parametric approach is effective.

Stability test and dominant eigenvalues computation for second-order linear systems with multiple time-delays using receptance method

© 2019 Elsevier Ltd Stability analysis and dominant eigenvalues computation for second-order linear systems with multiple time-delays are addressed by using a reduced characteristic function and the associated characteristic matrix comprised of measured open-loop receptances. This reduced characteristic function is derived from the original characteristic function of the second-order time delayed systems based on a reasonable assumption that eigenvalues of the closed-loop system are distinct from those of the open-loop system. Then a contour integral is used to test the stability and provide the stability chart with respect to different displacement and velocity feedback time-delays, and a Newton-type method to compute the dominant eigenvalues via this characteristic function. The proposed approach also utilizes the spectrum distribution features of the retarded time-delay systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed approach.

On the instability of the Millionshchikov linear systems depending on a real parameter

The present article considers one-parameter families of second-order linear differential systems with a coefficient matrix depending on the real parameter, which is a diagonal matrix at each odd time interval of unit length. The Cauchy matrix is the rotation matrix at each odd time interval, whereas the angle is the sum of a parameter value and some real number. Earlier, it has been has proved that the upper Lyapunov exponent of each such a system, which is considered to be the function of parameter, is positive on the set of the positive Lebesque measure if the diagonal part of the coefficient matrix is independent on a parameter and separated from zero. The proof of this result essentially uses a complex matrix of special type. In recent article, the author has given another way to prove this theorem based on implementing the Parseval equality for trygonometric sums. Besides, the author considers the special case of the above systems. Now the diagonal part of the coefficient matrix is time-independent and is sufficiently big, whereas the rotation angle is defined by a maximum degree of two that divides the number of the corresponding time interval. For such a system, in the case of a continious coefficient dependence on a parameter it is proved that such a value exists, at which the corresponding system is unstable.

On the bicomponent contaminant transport in wetland flow with reactions

Contaminant transport in the wetland flow is subject to various reactions. Most researches on this topic have limited to the single component contaminant, relatively little has concerned the effect of competitive reaction on the transport of coupled bicomponent contaminant. This work analytically studies the bicomponent contaminant transport in the free surface wetland flow under the combined conditions of reversible competitive transfer, irreversible degradation and bed absorption, by solving the constant-coefficient second-order linear system of parabolic type and the method of concentration moments. Up to the fourth order concentration moment in the previous work (Wang and Chen, 2017a) is applied to support the fourth order expansion of Hermite polynomials to rigorously derive the solutions with high accuracy. The results demonstrate that the masses of binary components decay at different rate, and the vertical concentration distributions of binary components are tremendously non-uniform in the asymptotic dispersion stage. Three types of reactions in addition to the hydraulic dispersion exert separate control on the concentration distributions. It suggests that the peak concentration rather than the mean should be based for strict environmental implements.

Second-order event-triggered adaptive containment control for a class of multi-agent systems

This paper is primarily concerned with the event-triggered adaptive containment control for the second-order linear multi-agent systems (MASs) subject to time-varying input delays. Different from traditional methods, the triggering thresholds can be time-variable and achieved online, which are regulated by the triggering error. Then the number of transmitted data is modulated by two adaptive laws that play a crucial role in deciding whether to release the current data or not. By selecting two augmented types of Lyapunov-Krasovskii functionals (LKFs) and applying three effective inequalities, these earlier ignored information can be recollected and the field of application can be greatly enlarged. Moreover, the issues on delay-dependence respectively take uniform input delay and nonuniform input ones into consideration, in which the inter-relationship among time-delays can be involved. Finally, two MAS examples are exploited to illustrate our theoretical results.

Robust Solution for Second-Order Systems Using Displacement–Acceleration Feedback

This paper addresses the robust and minimum norm controller design for matrix second-order linear systems by means of combined displacement and acceleration feedback. First, the necessary conditions that ensure solvability are presented. Then, the parametric expressions of gain controller and eigenvector matrix are formulated on the basis of a set of free parameters. The proposed solution simultaneously makes the resulting closed-loop system numerically robust and obtains gain controllers with minimum norms. Also, the solution is general and can be applied when mass matrices are either singular or nonsingular. This is promising for better applicability in many practical applications. Finally, two examples are provided to illustrate the effectiveness of the proposed control strategy.

How Fast Is Too Fast? The Role of Perception Latency in High-Speed Sense and Avoid

In this letter, we study the effects that perception latency has on the maximum speed a robot can reach to safely navigate through an unknown cluttered environment. We provide a general analysis that can serve as a baseline for future quantitative reasoning for design tradeoffs in autonomous robot navigation. We consider the case where the robot is modeled as a linear second-order system with bounded input and navigates through static obstacles. Also, we focus on a scenario where the robot wants to reach a target destination in as little time as possible, and therefore cannot change its longitudinal velocity to avoid obstacles. We show how the maximum latency that the robot can tolerate to guarantee safety is related to the desired speed, the range of its sensing pipeline, and the actuation limitations of the platform (i.e., the maximum acceleration it can produce). As a particular case study, we compare monocular and stereo frame-based cameras against novel, low-latency sensors, such as event cameras, in the case of quadrotor flight. To validate our analysis, we conduct experiments on a quadrotor platform equipped with an event camera to detect and avoid obstacles thrown towards the robot. To the best of our knowledge, this is the first theoretical work in which perception and actuation limitations are jointly considered to study the performance of a robotic platform in high-speed navigation.

Periodic Perturbations of Linear Systems at Resonance

Second-order linear Hamiltonian systems at resonance with periodic nonlinearity is investigated. An existence result of solutions for such systems is obtained by means of variational methods, saddle point theorem, and an index theory for second-order linear Hamiltonian systems. Meanwhile, two examples and two extensions are presented.

Index Theory for Second Order Linear Hamiltonian Systems with L1 Coefficient Matrix Satisfying Generalized Periodic Boundary Value Conditions

In this paper, we will first establish an index theory for second order linear Hamiltonian systems with L1 coefficient matrix satisfying generalized periodic boundary value conditions. And then we will investigate nontrivial solutions for asymptotically linear second-order Hamiltonian systems and obtain some new results.