Continuous Linear Systems Research Articles

Composition of Behavioural Assume–Guarantee Contracts

The growing complexity of modern engineering systems necessitates a method for design and analysis that is inherently modular. Methods based on using contracts for system design have successfully tackled this issue for a variety of system classes, but mostly in the context of discrete software systems. Motivated by this, we present assume-guarantee contracts for continuous linear dynamical systems with inputs and outputs. Such contracts serve as system specifications through two aspects. The assumptions specify the dynamic behaviour of the environment of the system, which provides inputs for it, while the guarantees specify the desired dynamic behaviour of the output of the system when interconnected with a relevant environment. This is formalized by utilizing the behavioural approach to system theory. We define and characterize notions of contract implementation and contract refinement, where the latter is used to compare contracts. We also define and characterize two notions of contract composition that allow one to reason about two types of system interconnections: series and feedback. The properties of refinement and composition allow contracts to be used for modular design and analysis.

Necessary and sufficient conditions for crisp state-feedback design for fuzzy linear systems

In this paper, we examine the uncertain fuzzy continuous time linear systems. There is fuzzy uncertainty in the dynamics of the investigated system in the form of uncertain parameters represented by fuzzy granular numbers. A state feedback is designed to guarantee the stability of the closed-loop system with crisp gains and a necessary and sufficient condition is proposed to guarantee the stability. Furthermore, by using the proposed control input, the fuzzy eigenvalues of the closed-loop system are placed in a way that they have the greatest possibility of occurring in the desired poles. The state feedback gain which is presented in this paper is a crisp number and can be applied to real systems in practical applications. The proposed method ensures closed-loop stability whereas previous methods have presented fuzzy gains which are defuzzified to be applied to a real system without guaranteeing the stability. At the end, two practical simulation examples are given to verify the validity of the theoretical results.

Ostensible Metzler Linear Uncertain Systems: Goals, LMI Synthesis, Constraints and Quadratic Stability

This paper deals with the design problem for a class of linear continuous systems with dynamics prescribed by the system matrix of an ostensible Metzler structure. The novelty of the proposed solution lies in the diagonal stabilization of the system, which uses the idea of decomposition of the ostensible Metzler matrix, preserving the incomplete positivity of the system during the synthesis. The proposed approach creates a unified framework that covers compactness of interval system parameter representation, Metzler parametric constraints, and quadratic stability. Combining these extensions, all of the conditions and constraints are expressed as linear matrix inequalities. Implications of the results, both for design and for research directions that follow from the proposed method, are discussed at the end of the paper. The efficiency of the method is illustrated by a numerical example.

Fractal dimensions of continuous piecewise linear iterated function systems

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

Qualitative sign stability of linear time invariant descriptor systems

This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed.

Достаточные условия существования Н∝ -наблюдателя состояния линейных непрерывных динамических систем

<p>The article deals with the problem of finding the observer of the state vector of linear continuous non-stationary dynamical systems with uncertainty of the initial conditions, limited external influences and measurement errors over a finite time interval. Sufficient conditions for the existence of an observer are formulated and proved on the basis of the expansion principle. Relationships are obtained for finding the parameters of the observer and the worst laws of change in external influences and measurement errors. As a limiting case, the problem of observer synthesis for stationary linear dynamical systems on a semi-infinite time interval is considered. Two applied problems of estimating the aircraft state vector based on the results of incomplete and inaccurate measurements are solved.</p>

Ob opredelenii koeffitsientov kvadratichnoy funktsii Lyapunova s zadannymi svoystvami v sluchae kratnykh korney kharakteristicheskogo uravneniya

For continuous and discrete linear autonomous systems, we discuss the possibility of choosing the coefficients of the quadratic Lyapunov function that ensure the validity of the condition of sign-negativity of its first derivative (first difference) with a given margin in the case of multiple roots of the characteristic equation.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

$\mathcal {H}_{2}$ and $\mathcal {H}_\infty$ Suboptimal Distributed Filters Design for Linear Systems

This paper investigates the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> suboptimal distributed filtering problems for continuous time linear systems. We consider a linear system monitored by a number of filters, where each of the filters receives only part of the measured output of the system. Each filter can communicate with the other filters according to an a priori given strongly connected weighted directed graph. The aim is to design filter gains that guarantee the <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> norm of the transfer matrix from the disturbance input to the output estimation error to be smaller than an a priori given upper bound, while all local filters reconstruct the full system state asymptotically. We provide a centralized design method for obtaining such <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> suboptimal distributed filters. The proposed design method is illustrated by a simulation example.

Classification on Boundary-Equilibria and Singular Continuums of Continuous Piecewise Linear Systems

In this paper, we show that any switching hypersurface of [Formula: see text]-dimensional continuous piecewise linear systems is an [Formula: see text]-dimensional hyperplane. For two-dimensional continuous piecewise linear systems, we present local phase portraits and indices near the boundary equilibria (i.e. equilibria at the switching line) and singular continuum (i.e. continuum of nonisolated equilibria) between two parallel switching lines. The index of singular continuum is defined. Then we show that boundary-equilibria and singular continuums can appear with many parallel switching lines.

Global dynamics of a degenerate planar piecewise linear differential system with three zones

We study the dynamics of a continuous planar piecewise linear differential system with three zones separated by two parallel lines. We provide a complete study on the maximum number of limit cycles by geometric techniques when the system is degenerate in the left and central zones. The bifurcation diagrams and global phase portraits in the Poincaré disc are given. It is shown that a degenerate continuous planar piecewise linear differential system can have scabbard bifurcation, grazing bifurcation and double limit cycle bifurcation. Moreover, we detect three new scabbard bifurcations, such as both the limit cycle with two zones and the limit cycle with three zones can be bifurcated from a singular continuum with infinite length.

Stabilization of a Scalar Continuous Time Linear System with Unknown Noise Based on Two-stage Event-triggering

Stabilization of a Scalar Continuous Time Linear System with Unknown Noise Based on Two-stage Event-triggering

Event-triggered stabilization of linear input-delay systems with state quantization and unknown time-varying delay: A Razumikhin stability theorem-based approach

Benefiting from the input-to-state stability (ISS) version of the Razumikhin stability theorem, this paper deals with the stabilization of continuous linear input-delay systems with state quantization and without any knowledge about the delay. An event-triggered delay independent truncated predictor feedback (ET-DITPF) controller with a switched feedback control law is proposed to guarantee the globally uniformly asymptotic stability (GUAS)/globally uniformly exponential stability (GUES) of the closed-loop system. As a prominent feature, this controller uses no information about the time delay (neither its value nor its upper bound). In the proposed event-triggered framework, a lower bound for each two consecutive events is derived to show the non-occurrence of the Zeno behavior. Numerical simulations confirm the effectiveness of the obtained theoretical results.

Data-driven kinematics-consistent model order reduction of fluid-structure interaction problems: application to deformable microcapsules in a Stokes flow.

In this paper, we present a generic approach of a dynamical data-driven model order reduction technique for three-dimensional fluid-structure interaction problems. A low-order continuous linear differential system is identified from snapshot solutions of a high-fidelity solver. The reduced order model (ROM) uses different ingredients like proper orthogonal decomposition (POD), dynamic mode decomposition (DMD) and Tikhonov-based robust identification techniques. An interpolation method is used to predict the capsule dynamics for any value of the governing non-dimensional parameters that are not in the training database. Then a dynamical system is built from the predicted solution. Numerical evidence shows the ability of the reduced model to predict the time-evolution of the capsule deformation from its initial state, whatever the parameter values. Accuracy and stability properties of the resulting low-order dynamical system are analysed numerically. The numerical experiments show a very good agreement, measured in terms of modified Hausdorff distance between capsule solutions of the full-order and low-order models both in the case of confined and unconfined flows. This work is a first milestone to move towards real time simulation of fluid-structure problems, which can be extended to non-linear low-order systems to account for strong material and flow non-linearities. It is a valuable innovation tool for rapid design and for the development of innovative devices.

Robust estimation of state vector coordinates in the controlled helicopter motion problem

The problem of finding H∞ – a observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer’s gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.

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.

Impulsive time series modeling with application to luteinizing hormone data.

This work considers the estimation of impulsive time series pertaining to biomedical systems and, in particular, to endocrine ones. We assume a signal model in the form of the output of a continuous linear time-invariant system driven by a sequence of instantaneous impulses, which concept is utilized here, in particular, for modeling of the male reproductive hormone axis. An estimation method to identify the impulsive sequence and the continuous system dynamics from sampled measurements of the output is proposed. Hinging on thorough mathematical analysis, the method improves upon a previously developed least-squares algorithm by resolving the trade-off between model fit and input sparsity, thus removing the need for manual tuning of user-defined estimation algorithm parameters. Experiments with synthetic data and Markov chain Monte-Carlo estimation demonstrate the viability of the proposed method, but also indicate that measurement noise renders the estimation problem ill-posed, as multiple estimates along a curve in the parameter space yield similar fits to data. The method is furthermore applied to clinical luteinizing hormone data collected from healthy males and, for comparability, one female, with similar results. Comparison between the estimated and theoretical elimination rates, as well as simulation of the estimated models, demonstrate the efficacy of the method. The sensitivity of the impulse distribution to the estimated elimination rates is investigated on a subject-specific data subset, revealing that the input sequence and elimination rate estimates can be interdependent. The dose-dependent effect of a selective gonadotropin releasing hormone receptor antagonist on the frequency and weights of the estimated impulses is also analyzed; a significant impact of the medication on the impulse weights is confirmed. To demonstrate the feasibility of the estimation approach for other hormones with pulsatile secretion, the modeling of cortisol data sets collected from three female adolescents was performed.

Global studies on a continuous planar piecewise linear differential system with three zones

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results.

General ELLRFS-DAZN algorithm for solving future linear equation system under various noises

This work investigates the problem of future linear equation system under various noises. Firstly, a continuous advanced zeroing neurodynamic (CAZN) model under noise is developed to solve continuous linear equation system. Subsequently, by combining a general explicit linear left–right four-step (ELLRFS) formula with the CAZN model, a general ELLRFS discrete advanced zeroing neurodynamic (ELLRFS-DAZN) algorithm under noise is proposed to solve the future linear equation system. Theoretical analyses and results manifest the convergence performance of the general ELLRFS-DAZN algorithm under various noises. Moreover, numerical experimental results, including those based on a UR5 manipulator, validate the effectiveness and robustness of the general ELLRFS-DAZN algorithm under various noises. Numerical experimental results based on mobile acoustic source localization further substantiate the superiority of the general ELLRFS-DAZN algorithm under constant noise. Finally, physical experimental results based on a Kinova JACO2 manipulator substantiate the practicability of the general ELLRFS-DAZN algorithm under bounded random noise.

Ultra compact pulse shrinking TDC on FPGA

This study propose a new design of a field-programmable gate array (FPGA) based pulse shrinking time-to-digital converter (TDC), where only several configuration logic block (CLB) slices are required to achieve a resolution of around 2.38 ps. Taking advantage of the flip-flop’s asynchronous reset function, along with the characteristics of multiplexers, we propose an ultra compact design and a semi-automatic calibration method. Most importantly, a linear continuous response system for FPGA-based TDC implementation is suggested. Experiments using the Zynq Ultrascale + platform has demonstrated an on-chip resolution of 2.38 ps, DNL = [−0.23; 0.22], and INL = [−0.24; 0.27]. Further, we conducted and analyzed external measurements as well. The measurement range dropped from 210 ps to 80 ps as a result of precision degradation, while the resolution and non-linearity performance remain. This study will discuss the propagation delay, decoupling principle, the semi-auto calibration method, offset introduction, and temperature effects along with several delay tuning techniques.