Analysis Of Linear Control Systems Research Articles

Stability Analysis of Linear Control Systems by Wall’s Continued Fraction Expansion

Stability Analysis of Linear Control Systems by Wall’s Continued Fraction Expansion

A combined method for stability analysis of linear time invariant control systems based on Hermite‐Fujiwara matrix and Cholesky decomposition

AbstractIn this paper, we have developed an integrative method for checking the stability of linear time‐invariant (LTI) systems as well as nonlinear continuous‐time ones. In our method, we first apply the iterative Faddeev–Leverrier algorithm to obtain the characteristic polynomial of the LTI system. Subsequently, the associated Hermite‐Fujiwara matrix will be evaluated by a particularly efficient technique for the calculation of the Bézoutian matrices. The positive‐definiteness of the Hermite‐Fujiwara form, as the stability criterion, is next tested by performing the Cholesky decomposition. Our method is extended to assess the local stability of nonlinear continuous‐time systems with the help of the Jacobian matrix concept. The proposed method is demonstrated to approximately be 2.2 times faster than the classical Hurwitz algorithm in average, at least for matrices with dimensions less than 40, according to a performed central processing unit (CPU) time analysis. For the sake of illustration, four numerical examples are given, including dynamical models for a real‐world hydrolysis reactor and a well‐mixed bioreactor.

Sufficient Conditions for Observability in Electrocardiography Problem

The aim of this paper, is to study electrical cardiography problem. Thus, we construct the state space system of this model as mathematical model. Moreover, we present some definitions and results which is described some concepts of linear control system analysis related to this problem. More precisely, the sufficient conditions which characterize the observability notion of linear dynamical controlled system are presented and discussed. Finally, we prove that, the electrical cardiography model is completely observable system over finite time .

Adaptability analysis of linear continuous control systems with reference model

The problem of analyzing the tuning parameters of linear continuous controllers related to the class of state models is studied. To do this, algorithms for adjusting the controller are used, which are based on estimates of the plant parameters. Within the framework of this study, the adaptability property of the system is considered, and the primary goal is to determine the possibility of finding an exact solution to the problem of tuning the control loop for a given reference transient. Moreover, the study extends to a whole class of input actions, such as piecewise continuous functions of time. Based on the analysis of the convergence properties of the identification residual in a closed control system, additional simplified conditions for the adaptability of control systems using an identifier and a reference model are put forward, which supplement the already existing knowledge on this topic.

Reinventing personality from a Volterra nonlinear model of motivation

Multidimensional Laplace Transforms (MLT) is a generalisation of the Laplace transforms widely used in the analysis of linear control systems. The use of MLT allows the response of continuous nonlinear systems to be expressed as a power series of the stimulus. MLT was used in modelling animal motivation. The two distinct motivational mechanisms of priming and homeostasis are combined in a single mathematically rigorous nonlinear dynamic model. Projecting the model to simulate human motivation led to an age-dependent cybernetic personality model. The model comprises four complex dimensions, where sentences are classified to reflect states of motivation. The Age Trend Classification (ATC) is based on empirical evaluation of funniness scores over adult age: a sentence is classified according to changes in its average scores, either as falling, rising, peaking or constant. In contrast, psychometric personality models assume stable human traits; age dependence in adulthood is often considered insignificant. In this respect the age dependent artificial personality is a re-invention of and radically different from current psychological models. This paper explains the nonlinear roots of the motivation model, how the complex motivational dimensions reconcile with psychological traits and suggests a cybernetic hypothesis of human aging. Statistical analysis compares the consistency of Big Five personality with ATC funniness scores, the analysis shows that Cronbach's Alpha coefficients of raw funniness scores are higher in value than Big Five traits, and that the correlation coefficients between raw ATC funniness and Big Five scores show that nearly all ATC correlations with Extroversion and Neuroticism are closely distributed. The high consistency of ATC scores supports the reliability of funniness score data as a source of personality measures, the nearly uniform correlation with two of five personality traits suggests the reliance on such empirical measures as a criterion to select sentences with strong links to personality.

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Matrix equations have its own important in the field of control system engineering particularly in the stability analysis of linear control systems and the reduction of nonlinear control system models. There are certain conditions where the classical matrix equation are not well equipped to handle the uncertainty problems such as during the process of stability analysis and reduction in control system engineering. In this study, an algorithm is developed for solving fully fuzzy matrix equation particularly for ~ A ~X ~B 􀀀 ~X = ~ C, where the coefficients of the equation are in near-zero fuzzy numbers. By modifying the existing fuzzy multiplication arithmetic operators, the proposed algorithm exceeds the positive restriction to allow the near-zero fuzzy numbers as the coefficients. Besides that, a new fuzzy subtraction arithmetic operator has also been proposed as the existing operator failed to satisfy the both sides of the nearzero fully fuzzy matrix equation. Subsequently, Kronecker product and V ec-operator are adapted with the modified fuzzy arithmetic operator in order to transform the fully fuzzy matrix equation to a fully fuzzy linear system. On top of that, a new associated linear system is developed to obtain the final solution. A numerical example and the verification of the solution are presented to demonstrate the proposed algorithm.

Robust Stability of Networked Linear Control Systems With Asynchronous Continuous- and Discrete-Time Event-Triggering Schemes

This article presents a unified framework for the stability and performance analysis of networked linear control systems with asynchronous continuous-time or discrete-time event-triggering. In the networked system, the multiple outputs of the plant and the controller are independently transmitted via a single shared or multiple channels based on local event-triggering conditions. Regarding the restrictions of physical devices, in the event-checking procedure, an inactive period of time is allowed after each event of data sending. So the Zeno behaviors in events are effectively avoided. In terms of algebraic properties of state-space models and time delays, we present sufficient conditions for asymptotic stability and guaranteed L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> gain performance and characterize the robustness of the system in a time-delayed networked scenario with independent event-triggered sending of each output. The event-triggered approach provides the potential benefit of reduced number of requests for sending data via networks. A simulation example is given to verify the proposed methods.

A Spectral Method of the Analysis of Linear Control Systems

Abstract A spectral method of the analysis of linear control systems is considered. Within the framework of this approach the σ-entropy of input signals and the σ-entropy norm of systems are introduced. The invariance of the introduced norm makes it possible to get invariant results of σ-entropy analysis.

Spectral decompositions for the solutions of Lyapunov equations for bilinear dynamical systems

In this paper, novel spectral decompositions are obtained for the solutions of generalized Lyapunov equations, which are observed in the study of controllability and observability of the state vector in deterministic bilinear systems. The same equations are used in the stability analysis and stabilization of stochastic linear control systems. To calculate these spectral decompositions, an iterative algorithm is proposed that uses the residues of the resolvent of the dynamics matrix. This algorithm converges for any initial guess, for a non-singular and stable dynamical system. The practical significance of the obtained results is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations to the asymptotic dynamics of the perturbation energy in deterministic bilinear and stochastic linear systems. In particular, the norm of the obtained eigen-components increases when frequencies of the corresponding oscillating modes approximate each other. Thus, the proposed decompositions provide a new fundamental approach for quantifying resonant modal interactions in a large and important class of weakly nonlinear systems.

A new solution of pair matrix equations with arbitrary triangular fuzzy numbers

Pair matrix equations have numerous applications in control system engineering, such as for stability analysis of linear control systems and also for reduction of nonlinear control system models. There are some situations in which the classical pair matrix equations are not well equipped to deal with the uncertainty problem during the process of stability analysis and reduction in control system engineering. Thus, this study presents a new algorithm for solving fully fuzzy pair matrix equations where the parameters of the equations are arbitrary triangular fuzzy numbers. The fuzzy Kronecker product and fuzzy $Vec$-operator are employed to transform the fully fuzzy pair matrix equations to a fully fuzzy pair linear system. Then a new associated linear system is developed to convert the fully fuzzy pair linear system to a crisp linear system. Finally, the solution is obtained by using a pseudoinverse method. Some related theoretical developments and examples are constructed to illustrate the proposed algorithm. The developed algorithm is also able to solve the fuzzy pair matrix equation.

Reflexive periodic solutions of general periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The objective of this paper is to provide four new iterative methods based on the conjugate gradient normal equation error (CGNE), conjugate gradient normal equation residual (CGNR), and least‐squares QR factorization (LSQR) algorithms to find the reflexive periodic solutions (X1,Y1,X2,Y2,…,Xσ,Yσ) of the general periodic matrix equations urn:x-wiley:mma:media:mma5596:mma5596-math-0001 for i = 1,2,…,σ. The iterative methods are guaranteed to converge in a finite number of steps in the absence of round‐off errors. Finally, some numerical results are performed to illustrate the efficiency and feasibility of new methods.

A Framework Utilizing Augmented Reality to Enhance the Teaching–Learning Experience of Linear Control Systems

In recent years, engineering education has become more complex, more uncertain and more interdisciplinary. It is therefore important to supplement the theoretical introduction of technical topics and concepts with practical, learning by doing experiences, so as to maximize the learning outcome. When introduced to challenging technical concepts and topics through conventional diagrams and equations, such as those found in textbooks, many students have difficulty with visualization; limiting their ability to gain a deep understanding of the concept. The computer visualization (using Virtual Reality or Augmented Reality (AR)) has apparently become a problem-solving tool in engineering education. AR can overlay 3D virtual objects to a particular real-world context and is an effective way to enhance the visualization and as a result, the understanding of technical concepts. One topic commonly challenging for students to understand is the stability analysis of linear time-invariant control systems. To address this, this paper introduces the framework of a tabletop AR learning environment based on the tangible user interface (TUI) and consisting of a fixed VGA camera with a display screen.

Stability analysis of networked linear control systems with direct-feedthrough terms

We consider networked control systems (NCSs) composed of a linear plant and a linear controller interconnected by packet-based communication channels with communication constraints. We are interested in the setup where direct-feedthrough terms are present in the plant and/or in the controller, a case that is largely ignored in the literature due to its inherent complexity and counterintuitive results in the analysis despite its relevance for important classes of controllers including Proportional–Integral (PI) regulators. This setup calls for a novel stability analysis, for which we take a renewed look at the concept of uniformly globally exponentially stable (UGES) scheduling protocols that turned out to be instrumental in earlier approaches. We provide a generalization of the UGES property, called (DP,DC)-UGES with DP∕DC being the direct-feedthrough matrices of the plant/controller, respectively, and we present generic conditions on these direct-feedthrough terms DP∕DC such that the classical UGES property of scheduling protocols implies (DP,DC)-UGES. This allows us to derive conditions leading to a maximally allowable transmission interval (MATI) such that stability of the overall NCS is guaranteed. In addition, it is shown that it is possible to get more tailored results for the well-known sampled-data (SD), round-robin (RR), and try-once-discard (TOD) protocols leading to less conservative conditions on the direct-feedthrough terms than the generic ones. We also introduce new (DP,DC)-UGES scheduling protocols, designed to handle the direct-feedthrough terms in a more effective way than existing protocols. Our results are illustrated using the example of a batch reactor.

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems Ax=b. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions (X1,X2,…,Xλ) and (Y1,Y2,…,Yλ) of general coupled periodic matrix equations ∑s=0λ−1(Ai,sXi+sBi,s+Ci,sYi+sDi,s)=Mi,∑s=0λ−1(Ei,sXi+sFi,s+Gi,sYi+sHi,s)=Ni,for i=1,2,…,λ. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.

Некоторые вопросы инвариантного анализа линейных управляемых систем

Некоторые вопросы инвариантного анализа линейных управляемых систем

Spline wavelets use for output processes analysis of multi-dimensional non-stationary linear control systems

Methods for solving systems of differential equations based on the use of wavelets have become very popular recently. Debeshy wavelets, coiflets were used among others. The disadvantage of such wavelets is that they do not have an analytical expression. Therefore, there are great difficulties in integrating and differentiating expressions containing these wavelets. The article presents an algorithm for the numerical solution of a system of differential equations with variable coefficients based on spline-wavelets on a interval. The algorithm presented generalizes the well-known method based on Haar wavelets which are a special case of spline-wavelets. The results of the article are used to analyze the output processes of multi-dimensional non-stationary linear control systems.

Solving Lyapunov equation by quantum algorithm

Lyapunov equation is one of the most basic and important equations in control theory, which has various applications in, e.g., stability analysis and robust analysis of linear control systems. Inspired by the recent progresses of quantum algorithms, we find that solving Lyapunov equation can be exponentially accelerated by quantum algorithms rather than traditional classical algorithms. Our algorithm is more efficient especially when the system matrix is sparse and has a low condition number. The results presented in this paper open up new dimensions of research in controlling classical system by quantum information processors, which has rarely been considered in the existing literature.

On robustness analysis of linear vibrational control systems

By injecting high frequency dither signals, it is possible to stabilize an inverted pendulum without any feedback. The concept of the vibrational control system is thus proposed to provide extra design freedom in stabilization or other performance indexes. Although various vibrational control algorithms have been proposed and implemented in literature, little work has been done to show their robustness with respect to disturbances and uncertainties. This paper focuses on the robustness analysis of linear vibrational control systems with additive disturbances. By applying perturbation techniques, the linear vibrational control system is shown to be input-to-state stable with respect to disturbances. When disturbances are periodic, frequency analysis technique obtains a less conservative estimate of the ultimate bound of the system, indicating that disturbances with high frequencies lead to relatively small ultimate bounds. When additive state-dependent disturbances are considered, weak averaging techniques can be used to show the robustness of the system when bounded disturbances are slow time-varying. Numerical results support the theoretic analysis.

A Novel Iterative Method To Approximate Structured Singular Values

A novel method for approximating structured singular values (also known as $\mu$-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known MATLAB function mussv, implemented in the MATLAB Control Toolbox, illustrate the behavior of the method.

An Interactive and Comprehensive Software Tool to Promote Active Learning in the Loop Shaping Control System Design

The classical loop shaping is a design procedure that explicitly involves the shaping of the open loop transfer function $L(s)$ , within a desired frequency spectrum by manipulating the poles, zeros, and gain of the controller $C(s)$ . Interactive software tools have proven as, particularly, useful techniques with high impact on control education. This kind of interactive tools has demonstrated in the past that students learn in a much more active way. This paper presents the basic functionality of the linear control system design (LCSD), an interactive tool for analysis and design of linear control systems with special emphasis on the classical loop shaping design. The software tool is implemented in Sysquake, a MATLAB-like language with fast execution and excellent facilities for interactive graphics, and is delivered as a stand-alone executable that is readily accessible to students and instructors. Several design problems are used to illustrate the main features of the LCSD tool to perform classical loop shaping.