Linear Time-varying Systems Research Articles

Metzler asymptotic stability of initial time linear time-varying real-order systems

Determining the asymptotics of many time-varying systems associated with real orders remains a challenging issue in qualitative asymptotic theory. This paper introduces a new concept of Metzler asymptotic stability to linear time-varying real-order systems. The design class is considered in standard and block matrix forms whenever the initial time is defined on the real axis. The non-negativity of such systems is discussed by introducing a time-dependent Metzler matrix. New theoretical conditions are developed that guarantee the asymptotic stability of such systems. It is shown that, to ensure the Metzler asymptotic stability of such systems; it suffices to construct a suitable linear non-negative asymptotic stable system where the coefficient matrix should be Metzler. Examples illustrate the potential practical and promising applicability of the introduced results.

Prescribed-time robust ZNN models for solving equality and inequality systems

At present, there are few studies on solving time-variant linear equality and inequality systems (TVLEIS) under noise interference, and the numerical algorithm has limitations in solving the TVLEIS problems. Therefore, to determine the online solution of the TVLEIS in a complex environment, two prescribed-time robust zeroing neural network (PTRZNN) models are proposed, investigated, and verified in this paper. The PTRZNN models have a faster convergence rate and superior robustness compared with other zeroing neural network models activated by common activation functions. In addition, the detailed theoretical derivation of the prescribed-time convergence and robustness of the PTRZNN models is provided. The effectiveness and superiority of the PTRZNN models for determining the TVLEIS are further demonstrated by simulation results. It is worth mentioning that the design idea of the PTRZNN models is applied to the multi-agent system, which shows the practical value of the PTRZNN models.

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.

Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving

For human–machine collaborative driving conditions, a hierarchical chassis multi-objective cooperative control method is proposed in this paper. Firstly, based on the phase plane theory, vehicle dynamics analysis is carried out to complete the definition of vehicle stability region. Secondly, based on the linear time-varying (LTV) system model, a cooperative control strategy combining fuzzy control with model predictive control (MPC) is proposed in the upper layer. In this strategy, the assisted driving weight adjustment coefficient and the stability weight adjustment coefficient are obtained by fuzzy mapping combining human–machine cooperation index and the vehicle stability region, respectively, and the optimization objectives of MPC are designed based on the above coefficients. In the lower layer torque allocation strategy, the stability weight adjustment coefficient is introduced to achieve multi-objective optimization of tire load rate and energy efficiency. For energy efficiency optimization, an optimal energy efficiency point-based tracking method is proposed to avoid nonlinearity caused by the introduction of motor loss models. Simulation analysis results show that the proposed strategy can effectively alleviate human–machine conflicts and improve vehicle handing stability. It also can achieve smaller tire load rate optimization through torque allocation and can reduce energy consumption by approximately 8% compared with the inter-axle torque allocation strategy. This study helps to promote the improvement of the comprehensive performance of assisted driving vehicles in human–machine cooperation, handling stability, and energy-saving torque distribution.

Computing dispersion diagrams and forced responses of arbitrarily varying waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.

Quadratic filtering for non-Gaussian stochastic parameter systems with buffer-aided strategy

This paper is concerned with the quadratic filtering problem for a class of linear discrete time-varying systems with non-Gaussian stochastic parameters and intermittent measurements. In order to improve the filtering performance, a buffer-aided strategy is used to store the historical measurement data which is sent to the filter via a shared communication channel at each transmission instant. The main objective of this paper is to design a quadratic filter to estimate the states of the considered non-Gaussian stochastic systems by using the buffer-aided strategy. Specifically, an augmented system is constructed by combining the original system states and their second-order Kronecker powers. As such, the considered quadratic filtering problem amounts to the Kalman filtering problem for the augmented system. The filtering error covariance (FEC) conditioned on the transmission intervals is firstly derived recursively and then is minimized by choosing the suitable filtering gains. Moreover, sufficient conditions are deduced to guarantee the boundedness of both the FEC and the conditional FEC. Finally, two illustrative examples are provided to demonstrate the effectiveness of the proposed filtering method.

Distributed observers for LTV systems: A distributed constructibility gramian based approach

The paper addresses the problem of designing distributed observers for discrete linear time-varying (LTV) systems with distributed sensor nodes. It is shown, under the conditions of collective observability, strong connectivity of the sensor communication network, and invertibility of the state transition matrix, that the resulting observer yields exponential stability of the estimation errors with a pre-defined convergence rate and, in certain situations, requires the exchange of fewer data than other methods. It is shown that for linear time-invariant (LTI) systems this method yields fixed gains that can be computed in a few steps of a distributed algorithm with the underlying communication network. A design example is given where the asymptotic performance of the proposed observer is shown to be similar to that obtained using a distributed Kalman filtering approach.

Controllability and observability of linear time-varying fractional systems

We consider linear time-varying control problems involving the Caputo fractional derivative. By introducing the controllability Gramian matrix, we prove that the invertibility of the Gramian is a necessary and sufficient condition for controllability. Furthermore, we prove equivalence between controllability and observability of the associated adjoint problem. Using methods from linear control theory for systems with integer-order derivatives, and adapting them to the fractional setting, we derive the form of the optimal control in weighted $$L^2$$ space as well as of the optimal control in $$L^{\infty }$$ . We provide an example to illustrate the obtained results.

Kernel-based regularized iterative learning control of repetitive linear time-varying systems

For data-driven iterative learning control (ILC) methods, both the model estimation and controller design problems are converted to parameter estimation problems for some chosen model structures. It is well-known that if the model order is not chosen carefully, models with either large variance or large bias would be resulted, which is one of the obstacles to further improve the modeling and tracking performances of data-driven ILC in practice. An emerging trend in the system identification community to deal with this issue is using regularization instead of the statistical tests, e.g., AIC, BIC, and one of the representatives is the so-called kernel-based regularization method (KRM). In this paper, we integrate KRM into data-driven ILC to handle a class of repetitive linear time-varying systems, and moreover, we show that the proposed method has ultimately bounded tracking error in the iteration domain. The numerical simulation results show that in contrast with the least squares method and some existing data-driven ILC methods, the proposed one can give faster convergence speed, better accuracy and robustness in terms of the tracking performance.

Iterative Learning Identification and Control for Discrete-time Linear Time-varying Systems with Iteration Varying Lengths

Iterative Learning Identification and Control for Discrete-time Linear Time-varying Systems with Iteration Varying Lengths

Singular value decomposition based learning identification for linear time‐varying systems: From recursion to iteration

AbstractSystem identification is a critical task in various engineering applications such as motion control, signal processing and robotics. In this article, the identification of linear time‐varying (LTV) systems that perform tasks repetitively over a finite‐time interval is investigated. Traditional LTV system identification typically adopts recursive algorithms in the time domain, which are incapable of tracking drastic‐varying parameters and are subject to estimation lag and numerical instability. To address these issues, this article proposes the utilization of an iteration axis in addition to the time axis for estimating repetitive time‐varying parameters. Specifically, the proposed approach involves an estimation algorithm for the time‐varying parameters based on a recursive least squares (RLS) method along the iteration axis, as well as an update algorithm for the covariance matrix based on singular value decomposition (SVD) to enhance numerical stability. Additionally, a bias compensation method based on noise variance estimation is introduced for the sake of eliminating estimation error induced by measurement noise. Numerical comparisons with existing methods are conducted to demonstrate the effectiveness and superiority of the proposed method.

On consistency and stability of distributed Kalman filter under mismatched noise covariance and uncertain dynamics

This paper considers the state estimation problem for a class of discrete-time linear time-varying systems over a peer-to-peer sensor network under mismatched noise covariance and uncertain dynamics. To deal with the inconsistency and instability of distributed Kalman filter for the problem, we propose a general consistent distributed Kalman filter framework by considering compensation to the mismatched noise covariance and the uncertain dynamics. The estimated error covariance of the distributed filter is proven to be monotonic with respect to compensation parameters. Futhermore, we show the design methods of compensation matrices to ensure consistency of the proposed filter in each sensor. Then, stability of the proposed distributed Kalman filter is demonstrated. Simulation examples show the effectiveness of our methods.

Topological equivalence of linear time-varying control systems

Abstract In this paper, we mainly studied the topological equivalence of linear time-varying (LTV) control system $\dot{x}\left ( t\right )=A(t)x(t)+B(t)u(t)$ defined on an interval $I \subset \mathbb{R}^{+}$. After giving a new definition of the topological equivalence, we investigated the local equivalence of LTV control systems under two new hypotheses. These hypotheses were made by the local behavior of Krylov indices (which turned out to be controllability indices for the linear time-invariant (LTI) control systems). It was found out that Krylov indices play an important role in the classification problem of LTV control systems. Compared with our former work on the topological equivalence of LTI control systems, new methods and techniques were taken to deal with new difficulties occurred for LTV control systems.

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.

Regression model-based adaptive receding horizon control of soft open points for loss minimization in distribution networks

This study proposes a method of using a soft open point (SOP) to flexibly connect different distribution networks. A novel operation strategy of SOP using a model predictive control (MPC) framework that adheres to the adaptive receding horizon control rule is developed to effectively respond to volatile renewable energy resources. Notably, the proposed method models the plant corresponding to voltage and network losses as a linear time-variant system, which provides better performance than the conventional MPC method in reducing network losses and improving voltage profile. To minimize the need for network observation of distribution system operators, we propose a process of deriving voltage-to-power and network loss-to-power sensitivity using a polynomial regression model. The effectiveness of the proposed strategy is verified on the modified IEEE 33-bus test systems, and its superiority is demonstrated through performance comparison with the existing general MPC method.

Nonnegativity, Convergence and Bounds of Non-homogeneous Linear Time-Varying Real-Order Systems with Application to Electrical Circuit System

Nonnegativity, Convergence and Bounds of Non-homogeneous Linear Time-Varying Real-Order Systems with Application to Electrical Circuit System

Robust output-feedback orbital stabilization for underactuated mechanical systems via high-order sliding modes

The manuscript deals with the robust orbital stabilization for a class of disturbed Euler–Lagrange systems with one degree of underactuation. The proposed strategy relies on the virtual holonomic constraints approach, using incomplete state measurements. First, a high-order sliding-mode extended observer estimates the state and the disturbances affecting the input channel. Then, proposing a new set of coordinates, the so-called virtual holonomic constraints, a robust output partial-feedback linearization approach takes the system into a double integrator with a particular zero dynamics. Thus, considering the general integral of motion of the zero dynamics, the orbital stabilization is reduced to stabilize a linear time-varying system. The resulting control law is a continuous signal. Therefore, the robustness to disturbances is addressed without the tarnishing effects of chattering. The closed-loop stability analysis is done using the Lyapunov theory. The feasibility of the method is illustrated experimentally in a cart-pendulum system.

Encrypted Finite-Horizon Energy-to-Peak State Estimation for Time-Varying Systems Under Eavesdropping Attacks: Tackling Secrecy Capacity

This paper is concerned with the problem of finite-horizon energy-to-peak state estimation for a class of networked linear time-varying systems. Due to the inherent vulnerability of network-based communication, the measurement signals transmitted over a communication network might be intercepted by potential eavesdroppers. To avoid information leakage, by resorting to an artificial-noise-assisted method, we develop a novel encryption-decryption scheme to ensure that the transmitted signal is composed of the raw measurement and an artificial-noise term. A special evaluation index named secrecy capacity is employed to assess the information security of signal transmissions under the developed encryption-decryption scheme. The purpose of the addressed problem is to design an encryption-decryption scheme and a state estimator such that: 1) the desired secrecy capacity is ensured; and 2) the required finite-horizon <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l_{2}-l_{\infty}$</tex> performance is achieved. Sufficient conditions are established on the existence of the encryption-decryption mechanism and the finite-horizon state estimator. Finally, simulation results are proposed to show the effectiveness of our proposed encryption-decryption-based state estimation scheme.

On the Dynamics of the Tavis–Cummings Model

The purpose of this paper is to present a comprehensive study of the Tavis-Cummings model from a system-theoretic perspective. A typical form of the Tavis-Cummings model is composed of an ensemble of non-interacting two-level systems (TLSs) that are collectively coupled to a common cavity resonator. The associated quantum linear passive system is proposed, whose canonical form reveals typical features of the Tavis-Cummings model, including $\sqrt{N}$- scaling, dark states, bright states, single-excitation superradiant and subradiant states. The passivity of this linear system is related to the vacuum Rabi mode splitting phenomenon in Tavis-Cummings systems. On the basis of the linear model, an analytic form is presented for the steady-state output state of the Tavis-Cummings model driven by a single-photon state. Master equations are used to study the excitation properties of the Tavis-Cummings model in the multi-excitation scenario. Finally, in terms of the transition matrix for a linear time-varying system, a computational framework is proposed for calculating the state of the Tavis-Cummings model, which is applicable to the multi-excitation case.

THUẬT TOÁN ĐIỀU KHIỂN THEO ĐẦU RA HỆ THỐNG TUYẾN TÍNH KHÔNG DỪNG, SỬ DỤNG CÁC PHƯƠNG PHÁP NHẬN DẠNG THAM SỐ

The problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. For synthesis of the observer of state variables, a new approach is proposed, providing the possibility of obtaining monotonous estimates of convergence with the regulation of transition time. The main idea of the synthesis of observers is based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. Then the DREM algorithm is used to estimate the state variables. Some simulations on Matlab/Simulink prove the correctness of the theory built, ensuring the good quality of the response processes.