Problem Of Linear Systems Research Articles

Consensus of Singular Linear Multiagent Systems Via Hybrid Control

In this article, the consensus problem of singular linear multiagent systems (SLMASs) is investigated. Based on the properties of Drazin inverse, the considered SLMAS is transformed into an equivalent differential-algebraic system. To address this problem, an event-triggered mechanism is adopted in the existing works. However, it is pointed out in this article that this mechanism is not so effective for this problem. The reason is that the discontinuity of event-triggered controller can cause the algebraic subsystem to malfunction at the triggering instants. To avoid this phenomenon, a hybrid control strategy is adopted. The main characteristic of this strategy is that it is organized by an event-triggered controller and an impulsive controller. To show the validity of theoretical results, an example is given.

Dixon resultant theory for stability analysis of distributed delay systems and enhancement of delay robustness

This study scrutinizes the stability problem of linear time-invariant feedback control systems with a constant-coefficient, partial delay distribution from a new perspective, which is built on an equivalence between the system of interest and the one with two lumped delays. We aim to determine all the potential stability changing curves (PSCC) of the system in the domain of delays in order to make a non-conservative stability assessment. First, we propose the Dixon resultant-based frequency sweeping procedure to calculate the so-called kernel and offspring hypersurfaces (KOH) of the system. The superiority in the computational efficiency of this Dixon-type method is revealed by comparison with the Sylvester-type one. Second, we specifically tackle the standing root case for the singularity at the zero root, leading to what we call the standing root boundary (SRB). Then, we claim that the union of the KOH and SRB constitutes the PSCC of the system. With these, the stability map of the system is then created using the Cluster Treatment of Characteristic Roots paradigm. Furthermore, we declare the delay robustness is enhanced by the proposed control law. Finally, we demonstrate the effectiveness of the presented procedures over two example case studies by the Quasi-Polynomial mapping-based Root-finder routine as well as the Simulink-based simulation.

Adaptive Kalman Filtering for Recursive Both Additive Noise and Multiplicative Noise

This paper focuses on adaptive Kalman filtering problem for linear systems with unknown covariances of both dynamic multiplicative noise (multiplicative measurement noise) and additive noises (additive process and measurement noises). A recursive-noise adaptive Kalman filter is proposed to estimate both states and covariances of noises by using the varaitional Bayesian (VB) inference and an indirect method. First, we characterize inverse Wishart priors for both measurement noise covariance and process noise covariance and employ the Student's t-distribution to represent the likelihood function, which is non-Gaussian and affected by mixing multiplicative noise and additive measurement noise. Then, an adaptive Kalman filtering for recursive both noise covariance matrices and dynamic state, is proposed following VB inference. Performance analysis for VB procedures and the proposed filter is provided to ensure the convergence and stability. A target tracking example is provided to validate the effectiveness of the proposed filtering algorithm.

A State of the Art in Simultaneous Localization and Mapping (SLAM) for Unmanned Ariel Vehicle (UAV): A Review

Abstract For the past decade, the main problem that has attracted researchers’ attention in aerial robotics is the position estimation or Simultaneous Localization and Mapping (SLAM) of Unmanned Aerial Vehicles (UAVs) where the GPS signal is poor or denied. This article reviews the strengths and weaknesses of existing methods in the field of aerial robotics. There are many different techniques and algorithms that are used to overcome the localization and mapping problem of these UAVs. These techniques and algorithms use different sensors, such as Red Green Blue-Depth (RGB_D), Light Detecting and Ranging (LIDAR), and Ultra-wideband (UWB). The most common technique is used, i.e., probability-based SLAM, which uses two algorithms: Linear Kalman Filter (LKF) and Extended Kalman Filter (EKF). LKF consists of five phases and this algorithm is just used for linear system problems. However, the EKF algorithm is used for non-linear systems. Aerial robots are used to perform many tasks, such as rescue, transportation, search, control, monitoring, and different military operations because of their vast top view. These properties are increasing their demand as compared to human service. In this paper, different techniques for the localization of aerial vehicles are discussed in terms of advantages and disadvantages, practicality and efficiency. This paper enables future researchers to find the suitable SLAM solution based on their problems; either the researcher is dealing with a linear problem or a non-linear problem.

Students' creative thinking ability in solving linear equation system problems based on brain domination

Students' creative thinking processes need to be developed to support them in solving complex mathematical problems, but this is not easy. This study aims to describe students' creative thinking skills in solving linear equation system problems based on brain dominance. This research is a qualitative descriptive research with a case study approach. Subjects were categorized into right brain dominance and left brain dominance by using a brain dominance questionnaire. The research instruments were in the form of tests and interview guidelines. Data validation uses method triangulation and data is analyzed through data reduction, data presentation, data verification, and drawing conclusions. The results showed that the subject of right brain dominance in fluency gave 3 responses, in flexible thinking the subject gave 3 different ideas, in original thinking the subject gave a unique response, and in elaboration the subject did not show a coherent answer. While the left brain dominant subject in fluent thinking (fluency) gave 4 responses, in flexible thinking the subject gave 3 general ideas, in original thinking the subject did not show a unique response, and in detailed thinking (elaboration) the subject showed a flurry of answers.

Metacognitive skills of student in solving problems of two-variable linear equation systems in terms of self-regulated learning

It is empirically known that metacognitive skill contributes to self-regulated learning. This study aims to describe students’ metacognitive skills in three levels of self-regulated learning, particularly in solving problems of a two-variable linear equation system. Three indicators were used in metacognition skills, namely planning indicators, monitoring indicators, and evaluation indicators. Employing a descriptive qualitative approach, this study involved All students of grade VIII A of the academic year 2021/2022 from a state junior school as subjects involving each category for high, medium, and low self-regulated learning. The data were collected using questionnaires, tests, interviews, documentation, and observation. A triangulation method, The data were then analyzed through condensation, data presentation, and drawing conclusions. is the current study found that high learning students used their metacognitive skills well and met three indicators, namely planning indicators, monitoring indicators, and evaluation indicators. Students with moderate self-regulated learning were able to meet two indicators, including planning indicators and monitoring indicators, while those with low self-regulated learning only met planning indicators.

State observers for the time discretization of a class of impulsive mechanical systems

AbstractIn this work, we investigate the state observer problem for linear mechanical systems with a single unilateral constraint, for which neither the impact time instants nor the contact distance is explicitly measured. We propose to attack the observer problem by transforming and approximating the original continuous‐time system by a discrete linear complementarity system (LCS) through the use of the Schatzman–Paoli scheme. From there, we derive a deadbeat observer in the form of a linear complementarity problem. Sufficient conditions guaranteeing the uniqueness of its solution then serve as observability conditions. In addition, the discrete adaptation of an existing passivity‐based observer design for LCSs can be applied. A key point in using a time discretization is that the discretization acts as a regularization, that is, the impacts take place over multiple time steps (here two time steps). This makes it possible to render the estimation error dynamics asymptotically stable. Furthermore, the so‐called peaking phenomenon appears as singularity within the time discretization approach, posing a challenge for robust observer design.

Reach control problem for a class of convex differential inclusions on simplices

Abstract This paper focuses on the reach control problem for systems expressed as a class of convex differential inclusions. The purpose is to find affine feedback for the trajectories of the systems to reach and leave an unrestricted facet of a given simplex in a finite time. It is proved that the condition for strong reachability is sufficient and necessary. The sufficient condition for weak reachability is obtained based on the solvability of the reach control problem for linear affine systems. At last, algorithms are designed and numerical examples are given to verify the validity of the results.

Event-Triggered Consensus of Multiagent Systems With Time-Varying Communication Delay

In this article, the consensus problem of linear multiagent systems (MASs) is investigated, where an event-triggered mechanism combined with sampler is introduced to rationally utilize the network bandwidth and the communication energy. Different from most existing results on the event-triggered consensus of MASs, a time-varying communication delay (CD) with a more general form is constructed in this article. With the help of a new class of Lyapunov functional and advanced inequalities, some novel criteria for achieving the asymptotic consensus of the considered MASs under different CD cases are obtained. For demonstrating the superiority of the proposed method from the conservatism and the practicability, a compared example and an application of spacecraft formation flying are proposed, respectively.

Distributed Robust Containment Control of Linear Heterogeneous Multi-Agent Systems: An Output Regulation Approach

In this paper, we consider the robust output containment problem of linear heterogeneous multi-agent systems under fixed directed networks. A distributed dynamic observer based on the leaders' measurable output was designed to estimate a convex combination of the leaders' states. First, for the case of followers with identical state dimensions, distributed dynamic state and output feedback control laws were designed based on the state-coupled item and the internal model compensator to drive the uncertain followers into the leaders' convex hull within the output regulation framework. Subsequently, we extended theoretical results to the case where followers have nonidentical state dimensions. By establishing virtual errors between the dynamic observer and followers, a new distributed dynamic output feedback control law was constructed using only the states of the compensator to solve the robust output containment problem. Finally, two numerical simulations verified the effectiveness of the designed schemes.

On regularization of the Lagrange principle in the optimization problems for linear distributed Volterra type systems with operator constraints

Regularization of the classical optimality conditions - the Lagrange principle and the Pontryagin maximum principle - in a convex optimal control problem subject to functional equality and inequality constraints is considered. The controlled system is described by a linear functional-operator equation of second kind of the general form in the space $L_2^m$. The main operator on the right-hand side of the equation is assumed to be quasi-nilpotent. The objective functional to be minimized is strongly convex. The derivation of the regularized classical optimality conditions is based on the use of the dual regularization method. The main purpose of the regularized Lagrange principle and regularized Pontryagin maximum principle is to stably generate minimizing approximate solutions in the sense of J. Warga. As an application of the results obtained for the general linear functional-operator equation of second kind, two examples of concrete optimal control problems related to a system of delay equations and to an integro-differential transport equation are discussed.

Distributed H∞ Robust Control of Multiagent Systems With Uncertain Parameters: Performance-Region-Based Approach

This article deals with the distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> robust control problem for linear multiagent systems perturbed by external disturbances and norm-bounded uncertain parameters over the Markovian randomly switching communication topologies. To tackle this problem, the distributed observer-based controller is proposed, which requires the relative information between neighbors and the absolute information of a subset of the nodes, and thus is intrinsically distributed. It is of great interest to see that the distributed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> robust control problem governed by such a controller can be converted to the stabilization with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> disturbance attenuation problems of some decoupled linear systems, whose dimensions equal those of a single node. Then, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {H}_{\infty }$ </tex-math></inline-formula> stochastic robust performance region is defined to indicate the robustness of this controller against the variation of communication topologies. It is theoretically shown that the distributed observer-based controller yields bounded and connected robust performance region. Finally, the theoretical results are verified by conducting numerical simulations and experiments.

스위칭 제어기를 이용한 선형 다개체 시스템의 일치

This paper addresses the adaptive consensus problem of linear multiagent systems using a switching controller. The proposed controller has an adaptive gain that is adjusted in a piecewise constant manner using the switching logic. Moreover, the proposed controller is fully distributed since it uses only local information from neighboring agents and needs no network topology information. The Lyapunov method is used to show that the proposed controller ensures consensus among agents and the boundedness of the adaptive gain. In addition, the proposed switching logic does not exhibit the Zeno phenomenon. Finally, a simulation example is given to validate our control method.

Distributed Control of Discrete-Time Linear Multi-Agent Systems With Optimal Energy Performance

This paper investigates the leader-based distributed optimal control problem of discrete-time linear multi-agent systems (MASs) on directed communication topologies. In particular, the communication topology under consideration consists of only one directed spanning tree. A distributed consensus control protocol depending on the information between agents and their neighbors is designed to guarantee the consensus of MASs. In addition, the optimization of energy cost performance can be obtained using the proposed protocol. Subsequently, a numerical example is provided to demonstrate the effectiveness of the presented protocol.

Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles

In this paper, a new method to manage the stabilization and control problems of n-dimensional linear systems plus dead time, which includes one, two, or three unstable poles, is proposed. The control methodology proposed in this work is an Observer-based Proportional-Integral-Derivative (PID) strategy, where an observer and a PID controller are used to relocate the original unstable open-loop poles to stabilize the resultant closed-loop system. The observer provides an adequate estimation of the delayed-free variables and the PID uses the delay-free variables estimated by the proposed observer. Also, step-tracking is achieved in the overall control scheme. Necessary and sufficient conditions are presented to ensure closed-loop stability based on the open loop parameters of the system. The observer-based PID strategy considers five to seven constant parameters to obtain a stable closed-loop system. A general procedure to implement the proposed control strategy is presented and its performance is evaluated by means of numerical simulations.

Data-driven optimal tracking control of discrete-time linear systems with multiple delays via the value iteration algorithm

In this paper, the optimal tracking problem for discrete-time linear systems with multiple delays is studied without system dynamics. It is known that the total state of a system without specific dynamic characteristics is hard to be measured, unless such a system is equipped with massive sensors, which, however, may lead to an increase in cost and complexity for analysing. To deal with this problem and avoid adverse effects caused by using system state information, a new data-driven value iteration (DDVI) algorithm is proposed by considering three factors: past control inputs, system outputs, and external reference trajectories. Before the algorithm is proposed, a transformation is made to the original system according to the characteristics of the time-delay system, so that the time-delay number can be reduced or become a delay-free system. A novel data-driven state equation is derived from the historical data of the three factors, and then, it is adopted to solve the optimal control of multi-delay systems. Further results show that the proposed DDVI algorithm is convergent and the tracking error is asymptotically stable. Finally, simulations are provided to show the effectiveness of the controller.

State bounding estimation of positive singular discrete-time systems with unbounded time-varying delays

This paper investigates the problem of state bounding for linear positive singular discrete-time systems with unbounded time-varying delay. Our approach is based on the comparison principle and the spectral analysis of matrices. Using the proposed approach, we provide new sufficient conditions for the regularity, causality, positivity and the existence of the smallest ultimate upper bound of such systems. The conditions are given in terms of linear programming problems, which can be solved by LP optimal toolbox. Numerical examples with simulation are given to demonstrate the advantage and validity of the proposed theoretical results. The proposed method is the first trial in the state bounding problem of linear singular discrete-time systems with unbounded time-varying delays.

Synchronization analysis of linear time-varying matrix-weighted coupled systems and its applications

In this paper, we study the synchronization problem of continuous-time linear systems under time-varying matrix-weighted coupling. With the help of the notion of persistent excitation and an adapted stability result in adaptive control, we establish two sufficient exponential synchronization conditions using the connectivity of the persistent graph and the joint connectivity of the instantaneous time-varying graphs. The conditions recover those for the single-integrator consensus model with scalar-valued coupling. The results are further applied to solve the synchronization problem of continuous-time linear systems with time-varying and nonidentical coupling matrices, extending the existing conclusions in the literature. Finally, numerical examples are given to verify the derived results.

Event-triggered based practical fixed-time consensus for chained-form multi-agent systems with dynamic disturbances

This paper investigates the practical fixed-time event-triggered consensus problem of multi-agent nonholonomic systems with dynamic disturbances. The continuous communications can be avoided in the update of controller. By converting the nonholonomic chained systems into two subsystems, a switching control algorithm was proposed to deal with the coupling problem in systems. Compared with the event-triggered fixed-time consensus problem of first-order or second-order linear systems, the research of nonholonomic chained systems is more challenging due to the inherent nonlinearity of these systems. Moreover, the convergence time of fixed-time consensus problem is independent of initial states, which is more practical than finite-time consensus problem. Furthermore, it is proved that there is no Zeno behavior under the fixed-time event-triggered consensus control strategies. The effectiveness of the proposed control algorithms is verified by numerical simulations.

Solution of the linear quadratic regulator problem of black box linear systems using reinforcement learning

In this paper, a Q-learning algorithm is proposed to solve the linear quadratic regulator problem of black box linear systems. The algorithm only has access to input and output measurements. A Luenberger observer parametrization is constructed using the control input and a new output obtained from a factorization of the utility function. An integral reinforcement learning approach is used to develop the Q-learning approximator structure. A gradient descent update rule is used to estimate on-line the parameters of the Q-function. Stability and convergence of the Q-learning algorithm under the Luenberger observer parametrization is assessed using Lyapunov stability theory. Simulation studies are carried out to verify the proposed approach.