Stability Of Linear Systems Research Articles

On the analysis of positivity and stabilization of fractional multidimensional discrete linear systems

On the analysis of positivity and stabilization of fractional multidimensional discrete linear systems

Novel Stability and Stabilization Conditions of Linear Fractional-Order Time-Delay Systems Using Free Matrix Approach

The stability and stabilization problem of linear fractional-order time-delay systems is investigated. By introducing a free matrix, the linear fractional-order time-delay systems is transformed to a delay-free system interconnected with uncertainties. New stability and stabilization criteria are derived based on the small gain theorem. The criteria are formulated as linear matrix inequalities and are computationally efficient. Since the free matrix is utilized, the new conditions are less conservative than the existing ones. Two numerical examples indicate that the proposed criteria are less conservative than existing ones.

The generalized matrix equation for systematic evaluation of oral Russian self-study

Abstract Generalized Lyapunov matrix equations appear in the fields of controllability analysis and model reduction of bilinear systems, stability analysis and optimal stabilization of linear stochastic systems, etc. The author studies the numerical solution of the generalized Lyapunov matrix equation combined with the evaluation of the Russian spoken language self-study system. Taking the empirical analysis as the research object, we selected 120 oral test samples from 12 colleges and universities in the 2020 national four-level unified test of Russian majors, compared with the teaching syllabus and the examination syllabus, the data were analyzed from the perspectives of speech speed, vocabulary, grammar and vocabulary errors, and a series of important conclusions were drawn, according to the specific problems exposed by the students in the oral test, the current situation of oral Russian teaching is discussed from the aspects of oral teaching materials, oral teachers and oral teaching methods.

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

Global Asymptotic Stability of Submodule Capacitor Voltage Self-converging Modular Multilevel Converters Based on Characteristic Root Method

This study presents global asymptotic stability verification of submodule capacitor voltage self-converging modular multilevel converters (SC-MMCs) according to the characteristic root method, which constructs a submodule switching state matrix for voltage balancing control. The mathematical model of MMC is established in form of the state-space equation. It is proved that SC-MMCs have the characteristic of global asymptotic stability at a certain time from the perspective of switching state matrix characteristic roots, according to the theories of Lyapunov stability and continuous-time piecewise linear system stability. Then, the submodule capacitor voltage self-convergence over a period of time is deduced based on the capacitor voltage deviation vector. Under the constraints of linearly independent switching state matrix row vectors, submodule capacitor voltage self-convergence without real-time voltage calculation and feedback can be achieved. The global asymptotic stability of SC-MMCs and the efficacy of submodule capacitor voltage self-convergence based on switching state matrix are testified by simulations.

Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems.

Forward Action to Stabilize multiple Time-Delays MIMO Systems

In this paper, the problem of closed-loop stability of linear time-invariant systems with multiple time-delays is studied. We prove that stability of closed-loop configurations of such systems with unitary feedback can be guaranteed by means of a forward control action. The case of single-input single-output (SISO) systems is first considered, and then, an extension to the more general case of multi-input multi-output (MIMO) systems with multiple time-delays is dealt with. Numerical examples illustrating the theoretical results are also discussed.

Robust Stabilization of Linear Time-Delay Systems under Denial-of-Service Attacks.

This research examines new methods for stabilizing linear time-delay systems that are subject to denial-of-service (DoS) attacks. The study takes into account the different effects that a DoS attack can have on the system, specifically delay-independent and -dependent behaviour. The traditional proportional-integral-derivative (PID) acts on the error signal, which is the difference between the reference input and the measured output. The approach in this paper uses what we call the PID state feedback strategy, where the controller acts on the state signal. Our proposed strategy uses the Lyapunov-Krasovskii functional (LKF) to develop new linear matrix inequalities (LMIs). The study considers two scenarios where the time delay is either a continuous bounded function or a differentiable and time-varying function that falls within certain bounds. In both cases, new LMIs are derived to find the PID-like state feedback gains that will ensure robust stabilization. The findings are illustrated with numerical examples.

Fixed-time stabilization of periodic linear systems and its application to the elliptical spacecraft rendezvous

A smooth periodic delayed feedback (SPDF) control scheme is proposed for the fixed-time stabilization problem of linear periodic systems subject to input delay. By investigating the monodromy matrix of the periodic system, it is proved that the SPDF controller can achieve the fixed-time stabilization of linear periodic systems with arbitrarily long yet bounded input delays under the condition that the original system is uniformly completely controllable. The proposed controller is continuously differentiable and smooth. The SPDF control scheme is then applied to the elliptical spacecraft rendezvous problem. The effectiveness of the established method is verified on numerical simulations.

Asymptotic Properties of a Class of Systems with Linear Delay

Sufficient conditions for the asymptotic stability of linear systems of differential equations with linear delay are obtained. On the basis of these conditions, some systems of linear differential equations are studied, and one of them is stabilized on an infinite time interval.

Estimation and stochastic control of nonlinear dynamic systems over the AWGN channel: Application in tele‐presence and tele‐operation of autonomous vehicles

AbstractThis paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Efficient Learning of a Linear Dynamical System With Stability Guarantees

We propose a principled method for projecting an arbitrary square matrix to the non-convex set of asymptotically stable matrices. Leveraging ideas from large deviations theory, we show that this projection is optimal in an information-theoretic sense and that it simply amounts to shifting the initial matrix by an optimal linear quadratic feedback gain, which can be computed exactly and highly efficiently by solving a standard linear quadratic regulator problem. The proposed approach allows us to learn the system matrix of a stable linear dynamical system from a single trajectory of correlated state observations. The resulting estimator is guaranteed to be stable and offers statistical bounds on the estimation error.

New conditions on finite-time stability of linear discrete-time system

This paper is concerned with the problems of set-based finite-time stability (SFTS) and set-based finite-time boundedness (SFTB) for both certain and uncertain linear time-varying systems. The concepts of SFTS and SFTB are defined. Different from existing results, sufficient conditions for SFTS and SFTB are directly derived from the basic definitions of finite-time stability (FTS) and finite-time boundedness (FTB) by using the convex hull technique rather than utilizing the weighted quadratic functions. Thus, more practical constraints on the system states can be dealt with. Furthermore, intervals, zonotopes and polytopes are employed to describe the typical compact convex sets. For linear uncertain systems, the uncertain time-varying state sets are assumed to be represented by interval matrices and matrix zonotopes, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the main results.

Self-triggered resilient stabilization of linear systems with quantized outputs

This paper studies the problem of stabilizing a self-triggered control system with quantized output. Employing a standard observer-based state feedback control law, a self-triggering mechanism that dictates the next sampling time based on quantized output is co-developed with an output encoding scheme. If, in addition, the transmission protocols at the controller-to-actuator (C–A) and sensor-to-controller (S–C) channels can be adapted, the self-triggered control architecture can be considerably simplified, leveraging a delicate observer-based deadbeat controller to eliminate the need for running the controller in parallel at the encoder side. To account for denial-of-service (DoS) in the S–C channel, the proposed output encoding and self-triggered control schemes are further made resilient. It is shown that a linear time-invariant system can be exponentially stabilized if some conditions on the average DoS duration time are met. There is a trade-off between the maximum inter-sampling time and the resilience against DoS attacks. Finally, a numerical example is presented to demonstrate the practical merits of the proposed self-triggered control schemes and associated theory.

Stabilization of Linear Time-Invariant Systems by State-Derivative Feedback

In this paper is studied the stabilization problem by state-derivative feedback for linear time-invariant continuous-time systems. In particular, explicit necessary and sufficient conditions are established for the stability of a closed-loop system, obtained by state-derivative feedback from the given linear time-invariant continuous-time system. Furthermore a procedure is given for the computation of stabilizing state-derivative feedback. Our approach is based on properties of real and polynomial matrices.

Delay-dependent Exponential Stability of Linear Stochastic Neutral Systems with General Delays Described by Stieltjes Integrals

Delay-dependent Exponential Stability of Linear Stochastic Neutral Systems with General Delays Described by Stieltjes Integrals

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

Exponential stability of linear discrete systems with multiple delays by degenerated Lyapunov–Krasovskii functionals

The problem of exponential stability of delayed discrete systems with multiple delays x(n+1)=(I+A)x(n)+∑i=1sBix(n−i),n=0,1,…is studied, where x=(x1x2…xm)T is an unknown vector, m and s are fixed positive integers, A, Bi are square constant matrices and I is a unit matrix. A new degenerated Lyapunov–Krasovskii functional is used to derive sufficient conditions for exponential stability and to derive an exponential estimate of the norm of solutions. Though often used in the study of stability, the assumption that the spectral radius of the matrix of linear terms is less than 1 is not applied here. The criterion derived is illustrated by an example and compared with previously known results.

A new stability criterion and its application to robust stability analysis for linear systems with distributed delays

Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented.

Lie-brackets-based averaging of affine systems via a time-delay approach

In this paper, we study input-to-state stability (ISS) of affine systems with a small parameter ɛ>0 and additive disturbances in the presence of state-delays. We present a time-delay approach to Lie-brackets-based averaging, where we transform the system to a time-delay (neutral type) one. The latter has a form of perturbed Lie brackets system. The ISS of the time-delay system guarantees the same for the original one. We present a direct Lyapunov–Krasovskii (L–K) method for the time-delay system and provide sufficient conditions for regional ISS. Further we apply the results to stabilization of linear uncertain systems under unknown control directions using the bounded extremum seeking controller with measurement delay. In contrast to the existing results that are all qualitative, we derive constructive linear matrix inequalities for finding quantitative upper bounds on ɛ and the time-delay that ensure regional ISS of the original system and on the resulting ultimate bound. Numerical examples illustrate the efficiency of our method.