Linear Time-varying Systems Research Articles

Adaptive Fading-Memory Receding-Horizon Filters and Smoother for Linear Discrete Time-Varying Systems

In this paper, an adaptive fading-memory receding-horizon (AFMRH) filter is proposed by combining the receding-horizon structure and the adaptive fading-memory method. In the recent finite horizon, state error covariance is adapted with an adaptive fading factor; then the process noise covariance matrix adaption is realized by adjusting the properties of systems. An AFMRH fixed-lag smoother is also proposed by combining the proposed AFMRH filtering algorithm and a Rauch–Tung–Striebel smoothing algorithm to improve the estimation accuracy. Because the proposed AFMRH filter and smoother are reduced to the optimal receding-horizon (RH) filter and smoother when all measurements have the same weight, the proposed adaptive RH estimators could provide a more general solution than the optimal RH filter and smoother. To reduce the complexity and improve the estimation performance of the proposed RH estimators, an adaptive horizon adjustment method and a switching filtering algorithm based on an adaptive fading factor are also proposed. In particular, the proposed adaptive horizon adjustment method is designed to be computationally efficient, which makes it suitable for online and real-time applications. Through computer simulation, the performance and adaptiveness of the proposed approaches were verified by comparing them with existing fading-memory approaches.

Delay-dependent criteria for robust stability and stabilization of fractional-order time-varying delay systems

In this paper, the robust stability and stabilization problems for fractional-order time-varying delay systems are investigated. Firstly, a new fractional-order Razumikhin theorem is given. By using the proposed fractional-order Razumikhin theorem, novel delay-dependent stability conditions for both nominal and uncertain linear fractional-order time-varying delay systems are derived. The results are in form of linear matrix inequalities, which are convenient for application and calculation. Then, the obtained stability conditions are utilized to derive a state feedback stabilization controller. To tackle the computational difficulty of the controller design method, a local optimization algorithm is proposed. Finally, three examples are provided to illustrate that the proposed results are valid and less conservative.

Improved state augmentation method for buffeting analysis of structures subjected to non-stationary wind

Extreme winds such as hurricanes and thunderstorms often present non-stationary characteristics, having time-varying mean wind speeds and non-stationary wind fluctuations. When concerning the wind-induced vibrations under non-stationary wind, the excitation will be a non-stationary process, and the wind-structure coupled system can be represented by a linear time-varying (LTV) system. The aim of this study is to present a state augmentation method to investigate the non-stationary buffeting of a model bridge tower subjected to non-stationary wind with consideration of the aeroelastic damping. Based on the theory of stochastic differential equations and Itô’s lemma, the statistical moments of the non-stationary buffeting response are derived through solving a first-order ordinary differential equation system. The proposed method is validated by comparisons with the Monte Carlo method and the pseudo excitation method. The result shows that the state augmentation method has higher accuracy and efficiency than the well-accepted time–frequency techniques.

Linear time-varying state transition matrix for spacecraft relative dynamics on highly elliptical orbits

Space exploration is a topic of great interest in the space community and worldwide. For instance, the race to the Lunar surface now includes not only government agencies, but also commercial entities and organizations. Such missions will require robotic spacecraft to perform formation flying, rendezvous and docking to reduce costs of the overall mission and establish space stations within Lunar orbits. Therefore, an accurate and computationally efficient dynamics model, to be used within a spacecraft’s guidance, navigation and control system, to calculate and control the desired relative motion will be required. This paper presents a new, linear time-varying, solution to spacecraft relative motion on Lunar orbits that includes solar radiation pressure, third-body, and J2 perturbations. The solution is obtained by using Hamiltonian dynamics in combination with the Peano–Baker solution for linear time-varying systems, which allows for the inclusion of short and long periodic variations of each perturbation within the dynamical model. The solution resulted in a 75% decrease in computational time when compared to a numerical simulator with the same perturbing effects.

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

Distributed Ellipsoidal Intersection Fusion Estimation for Multi-Sensor Complex Systems.

This paper investigates the problem of distributed ellipsoidal intersection (DEI) fusion estimation for linear time-varying multi-sensor complex systems with unknown input disturbances and measurement data transmission delays. For the problem with external unknown input disturbance signals, a non-informative prior distribution is used to model the problem. A set of independent random variables obeying Bernoulli distribution is also used to describe the situation of measurement data transmission delay caused by network channel congestion, and appropriate buffer areas are added at the link nodes to retrieve the delayed transmission data values. For multi-sensor systems with complex situations, a minimum mean square error (MMSE) local estimator is designed in a Bayesian framework based on the maximum a posteriori (MAP) estimation criterion. In order to deal with the unknown correlations among the local estimators and to select the fusion estimator with lower computational complexity, the fusion estimator is designed using ellipsoidal intersection (EI) fusion technique, and the consistency of the estimator is demonstrated. In this paper, the difference between DEI fusion and distributed covariance intersection (DCI) fusion and centralized fusion estimation is analyzed by a numerical example, and the superiority of the DEI fusion method is demonstrated.

Adaptive State Observer for Linear Time-Varying System with Partially Unknown State Matrix and Input Matrix Parameters

In this paper the problem of adaptive state observer synthesis for linear time-varying SISO (single-input-single-output) dynamical system with partially unknown parameters was considered. It is assumed that the input signal and output variable of the system are measurable. It is also assumed that the state matrix of the plant contains known variables and unknown constants when the input matrix (vector) is unknown. Observer synthesis is based on GPEBO (generalized parameter estimation based observer) method proposed in [1]. Observer synthesis provides preliminary parametrization of the initial system and its conversion to a linear regression model with further unknown parameters identification. For identification of the unknown constant parameters classical estimation algorithm — least squares method with forgetting factor — was used. This approach works well in cases, when the known regressor is " frequency poor" (i.e. the regressor spectrum contains r/2 harmonics, where r is a value of the unknown parameters) or does not meet PE (persistent excitation) condition. To illustrate performance of the proposed method, an example is provided in this paper. A time-varying second-order plant with four unknown parameters was considered. Parametrization of the initial dynamical model was made. A linear static regression with six unknown parameters (including unknown state initial conditions vector) was obtained. An adaptive observer was synthesized and the simulation results were provided to illustrate the purpose reached. The main difference with the results, that were published earlier in [2], is the new assumption that not only does the state matrix of the linear time-varying system contain unknown parameters, but input matrix (vector) contains unknown constant coefficients.

Asymptotic stability of switched linear time-varying systems based on top-floor function

This paper considers the problem of asymptotic stability for switched linear time-varying (SLTV) systems. First, some stability conditions for SLTV systems are given by using infinite integrals. Then, based on the results obtained, two stability conditions are proposed by combining the methods of top-floor function and average dwell time. Moreover, using strict top-floor function, a stability condition is also provided when some subsystems are unstable. With the help of top-floor function, the stability problem of SLTV systems can be simplified and solved by using the technique of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the theoretical results.

State for linear discrete time-varying systems, with applications to dissipative systems

We study the notion of state for a class of linear discrete time-varying systems and provide a characterization of polynomial operator in the shift used to define state trajectory from both external and auxiliary trajectories. We then characterize lossless systems and show that a storage function for a dissipative system can be written as a quadratic function of the state.

Hierarchic Controllability Analysis in High-Dynamic Guidance for Autonomous Vehicle Landing

Controllability analysis is significant for guidance law optimization and control system synthesis during atmospheric landing of hypersonic flight vehicles. This article concerns the problem of quantifying the controllability and developing a novel controllability measure, i.e., the degree of controllability (DOC), for linear (or linearized) time-varying systems, and conducts the hierarchically quantitative controllability analysis in high-dynamic guidance for autonomous vehicle landing. In particular, a novel DOC criterion is derived via analyzing the characteristics of singular values and the condition number of the controllability Gramian. The proposed DOC criterion can offer a deep insight into the controllability of both the system and each state subspace, and thus, it possesses a clear physical meaning in practice. Furthermore, a certain normalization and modification method is designed to make the criterion more applicable in aerospace engineering. To validate the derived criterion, the scenario of 3-DOF Mars entry guidance is simulated based on a reference-trajectory tracking scheme with active disturbance rejection control algorithm. According to the simulation results, the optimality of different reference trajectories can be evaluated and compared via the DOC criterion. Therefore, the DOC criterion can function as an indicator to analyze the influence of trajectory parameters on the landing process from the perspective of controllability analysis.

Finite-Time State Observer for a Class of Linear Time-Varying Systems With Unknown Inputs

A finite-time observer for a class of linear time-varying systems with bounded unknown inputs is presented in this note. The design of the observer exploits structural properties of the system and, through a linear operator, allows applying the robust exact differentiator in a block form without requiring additional stabilizing terms. The proposed observer provides an exact estimate of the states after a finite transient time despite the possible instability of the system and the effects of bounded unknown inputs.

Noise-Tolerant Zeroing Neural Dynamics for Solving Hybrid Multilayered Time-Varying Linear Equation System

Resource allocation problem in a wireless sensor network can be formulated as time-varying optimization, which can be further converted as time-varying linear equation system (TVLES). Hybrid multilayered time-varying linear equation system (HMTVLES) involving hybrid multilayers and time-variation characteristic is a complicated and challenging problem. Recently it has been solved by zeroing neural dynamics (ZND) method under ideal conditions, i.e., without noises. However, noises are ubiquitous, immanent and unavoidable in real-time systems. In this work, we propose a noise-tolerant zeroing neural dynamics (NTZND) model for solving HMTVLES. It can deal with different kinds of noises such as constant noise, linear-increasing noise, and random noise. Theoretical analyses guarantee the precision of NTZND model in the presence of different kinds of noises. In addition, a general NTZND model is proposed based on general activation function. Besides, classical ZND method and gradient neural dynamics (GND) method are also investigated and compared. Numerical experimental results are presented to verify the theoretical results of proposed models.

On distributed fusion estimation with stochastic scheduling over sensor networks

The paper deals with the distributed fusion estimation for linear time-varying systems over sensor networks, in which stochastic sensor scheduling and unknown exogenous inputs are taken into account. In the stochastic sensor scheduling, expensive and cheap channels are used to respectively transmit the high-precision data and the low-precision quantized data. Based on the stochastic scheduling scheme, a recursive minimum mean square error (MMSE) estimator is proposed against the unknown inputs. Then, a distributed fusion estimator is presented by combining local estimates and covariances from all sensors, relying on the covariance intersection (CI) fusion rule. Sufficient conditions are established to ensure that the proposed fusion estimator is stable with the stochastically ultimately bounded estimation error. Finally, a target tracking example is given to show the effectiveness of the proposed method.

A novel strategy for identification and control of a class of linear systems

Control of time-varying systems has become increasingly popular in recent years, as various domains such as aviation, automobile and manufacturing industries require a quick and precise response in the presence of uncertainties. Adaptive control, a popular strategy is used to control systems with unknown or time-varying properties. Classic adaptive control is insufficient to cope with time-varying parameters to produce desired results with minimal parametric error. Furthermore, for these strategies to work, identification model must be initialised closer to the plant. Else the transient response of the plant is likely to grow unbounded. This paper mainly concentrates on employing adaptive control for a type of linear time-invariant (LTI) and linear time-varying (LTV) systems. The time variations in LTV plant parameters are either slow, rapid or fast. This research in turn proposes a two-stage adaptive scheme which swiftly determines the compact region in which plant parameters reside. This method in turn ensures atleast one model to be initialised closer to the plant which significantly reduces parametric estimation error. The uniform boundedness of identification error, tracking error as well as the parametric error of LTI and LTV systems, are deduced. Sufficient examples are also considered to demonstrate the efficacy of the proposed two-stage scheme.

Proportional Local Assignability of the Dichotomy Spectrum of One-Sided Discrete Time-Varying Linear Systems

We consider a problem of assignability of the dichotomy spectrum for one-sided discrete time-varying linear systems. Our purpose is to prove that uniform complete controllability is a sufficient condition for proportional local assignability of the dichotomy spectrum.

Exponential stability of switched linear impulsive time-varying system and its application

This paper aims at the problem of exponential stability for switched linear impulsive time-varying system. By constructing two different switched discretized Lyapunov functions, some new sufficient conditions ensuring the global exponential stability of switched linear impulsive time-varying system are provided, which can be employed to the case when all subsystems are unstable. Furthermore, we apply theoretical results to the consensus of multi-agent system with switching topologies. Finally, numerical examples demonstrate the effectiveness of given results.

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.

An efficient solver of predictive control methods for motion control of autonomous vehicles with experimental verification

Real-time performance is one of the major challenges for optimal control and MPC control. In this paper, a novel general open-source C++ MPC solver based on state-of-the-art sparse QP solver OSQP is designed to solve the motion control problem of autonomous vehicles efficiently. A new linear dynamic model is proposed by replacing heading angle error with planning lateral projection speed, whose measurement is independent of the measurement of heading angle. A general local linearization method for the nonlinear tire model is proposed. Through the theory of offset-free MPC control, the steady-state value problem is solved caused by a disturbance in the time domain systematically. The algorithm in this paper is implemented in ROS, which means that the algorithm can be directly used in the prototype development of autonomous vehicles. The proposed MPC solver can solve a linear time-varying or time-invariant system, with or without disturbances. It is very efficient and can handle different kinds of hard and soft constraints. A real-time simulation platform based on Unreal Engine 4 and CarSim is designed to verify the algorithm’s effectiveness. Real-time simulations and real vehicle experiments verify the efficiency of the algorithm.

A New Approach to Finite-Horizon Optimal Control for Discrete-Time Affine Nonlinear Systems via a Pseudolinear Method

In this article, a new time-varying adaptivedynamic programming (ADP) algorithm is developed to solve finite-horizon optimal control problems for a class of discrete-time affine nonlinear systems. Inspired by the pseudolinear method, the nonlinear system can be approximated by a series of time-varying linear systems. In each iteration of the time-varying ADP algorithm, the optimal control law for the time-varying linear system is obtained. For an arbitrary initial state, it is proven that states of the time-varying linear systems converge to the states of discrete-time affine nonlinear systems. It is also shown that the iterative value functions and the iterative control laws converge to the optimal value function and the optimal control law, respectively. Finally, numerical results are presented to verify the effectiveness of the present method.

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

This research studies a class of linear, hybrid, time-varying, continuous time-systems with time-varying delayed dynamics and non-necessarily bounded, time-varying, time-differentiable delay. The considered class of systems also involves a contribution to the whole delayed dynamics with respect to the last preceding sampled values of the solution according to a prefixed constant sampling period. Such systems are also subject to linear output-feedback time-varying control, which picks-up combined information on the output at the current time instant, the delayed one, and its discretized value at the preceding sampling instant. Closed-loop asymptotic stabilization is addressed through the analysis of two “ad hoc” Krasovskii–Lyapunov-type functional candidates, which involve quadratic forms of the state solution at the current time instant together with an integral-type contribution of the state solution along a time-varying previous time interval associated with the time-varying delay. An analytic method is proposed to synthesize the stabilizing output-feedback time-varying controller from the solution of an associated algebraic system, which has the objective of tracking prescribed suited reference closed-loop dynamics. If this is not possible—in the event that the mentioned algebraic system is not compatible—then a best approximation of such targeted closed-loop dynamics is made in an error-norm sense minimization. Sufficiency-type conditions for asymptotic stability of the closed-loop system are also derived based on the two mentioned Krasovskii–Lyapunov functional candidates, which involve evaluations of the contributions of the delay-free and delayed dynamics.