Linear Discrete Time-varying Systems Research Articles

Iterative learning control approach for linear discrete time-varying systems with different initial time points

Different from the existing iterative learning control (ILC) works for dynamical systems with iteration-varying trial lengths, in which the terminal time points are not identical, this paper presents an ILC approach for linear discrete time-varying (LDTV) systems with different initial time points. A piecewise ILC law is designed by using the segment compensation techniques in time axis. Through a rigorous mathematical analysis, it is theoretically proved that as the mathematical expectation of the iterative initial state equals to the desired initial state, the ILC tracking error at the maximum universal time interval can be driven into a bounded region in mathematical expectation sense. Particularly, the mathematical expectation of the ILC tracking error at the maximum universal time interval can even be driven to zero under a certain initial state condition. Numerical simulations are provided to verify the effectiveness of the proposed ILC scheme.

Unbiased minimum-variance estimation and dynamic event-driven disturbance rejection control for discrete time-varying systems

This paper studies a dynamic event-driven disturbance rejection control problem for a class of linear discrete time-varying systems with unknown external disturbances, random process and measurement noises. To obtain the accurate estimates of both the system states and external disturbances, simultaneously, an unbiased minimum-variance time-varying extended state observer is designed, and the stability of the estimation error system is analyzed. Then, a dynamic event-driven disturbance rejection controller is designed by minimizing the upper bound of the cost function of the closed-loop system over a finite horizon. It can be shown that, by applying the proposed control approach, not only the transmission frequency of the control signal is reduced, but also the external disturbances are compensated actively. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed control approaches.

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.

Design of distributed recursive filters based on data compression for sensor networks

This paper is concerned with the distributed filtering problem based on data compression for linear discrete time-varying systems over sensor networks. Two kinds of sensors are considered: one kind only has a measuring capability and communicates measurements, while the other adds a calculating capability and communicates estimates. At each sensor node with calculating capability, a weighted measurement fusion algorithm is employed to compress the measurements of the sensor and its neighbor nodes with only measuring capability; and a weighted state fusion algorithm is employed to compress the estimates from neighbor nodes with calculating capability. Using compressed data, an optimal distributed recursive filter with a Kalman consensus filter form is devised in the linear unbiased minimum variance (LUMV) criterion. Cross-covariance matrices (CCMs) of the filtering errors between sensor nodes are derived. To avoid calculating CCMs, a suboptimal distributed recursive filter is presented, where a covariance intersection fusion algorithm is employed to compress the estimates from neighbor nodes with calculating capability. The filtering and consensus gains are solved to minimize an upper bound of covariance matrix (CM) of the filtering error. It is proved that the filtering error is exponentially bounded in the mean square. Simulations illuminate the effectiveness of proposed algorithms.

Distributed consensus of discrete time-varying linear multi-agent systems with event-triggered intermittent control.

The consensus problem of discrete time-varying linear multi-agent systems (MASs) is studied in this paper. First, an event-triggered intermittent control (ETIC) protocol is designed, aided by a class of auxiliary functions. Under this protocol, some sufficient conditions for all agents to achieve consensus are established by constructing an error dynamical system and applying the Lyapunov function. Second, in order to further reduce the communication burden, an improved event triggered intermittent control (I-ETIC) strategy is presented, along with corresponding convergence analysis. Notably, the difference between the two control protocols lies in the fact that the former protocol only determines when to control or not based on the trigger conditions, while the latter, building upon this, designs new event trigger conditions for the update of the controller during the control stage. Finally, two numerical simulation examples are provided to demonstrate the effectiveness of the theoretical results.

Controllability and observability of discretized satellite magnetic attitude control system

<abstract><p>In this paper, two different discrete schemes of the second-order linear time-varying system represented by the linearized satellite magnetic attitude control motion equation are obtained by Euler method. Then, the controllability and observability conditions of a new discrete second-order linear time-varying system are proposed and the validity of these conditions is further verified by some numerical examples. Next, the theoretical results are applied to investigate the controllability and observability of the discretized satellite magnetic control system. Different periods $ \tau $ are chosen to investigate the effect on the controllability and observability of the resulting discrete system. The corresponding conclusions are obtained.</p></abstract>

Distributed State and Input Estimation Over Sensor Networks Under Deception Attacks

This paper is concerned with the joint state and input estimation problem for linear discrete time-varying systems over peer-to-peer sensor networks, in which deception attacks are taken into account. By resorting to singular value decomposition, a local estimator structure is proposed to jointly estimate the system states and unknown inputs. Then, a distributed state estimator is constructed by fusing local estimates and covariance matrices. In addition, a sufficient condition is provided to ensure the uniformly bounded estimation error (in mean square) in each node. Finally, a numerical example is provided to show the effectiveness of the proposed estimation algorithm.

An approximate minimum mean-square error estimator for linear discrete time-varying systems: Handling Try-Once-Discard protocol

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying stochastic systems under communication constraints. In order to mitigate the undesired phenomena of data collisions, a Try-Once-Discard (TOD) protocol is utilized to regulate the signal transmissions over the sensor-to-estimator communication channel, where the TOD protocol is based on the scheduling rule described by a specific switching function. Under the introduced TOD protocol, the approximate minimum mean-square error (MMSE) state estimation problem is investigated and a recursive algorithm is developed for the MMSE estimator design whose computational complexity is comparable to that of the conventional Kalman filter. Furthermore, some conditions are established for the boundedness of the estimation error covariance. A numerical example is presented to illustrate the effectiveness of the proposed MMSE estimator.

Predictive control of uncertain systems over networks with redundant link

Networked systems using redundant channels to transmit data can effectively reduce the probability of data loss and improve system reliability and control margin. However, the structural complexity and economic cost of the system are also increased. To balance the redundancy and feasibility, the ratio of attraction domain to packet loss rate is defined as a balanced feasibility index. In this paper, single-channel packet loss is considered as Bernoulli distribution and a bounded packet loss network system control model is constructed as the arbitrary bounded packet loss control problem for redundant channel transmission network system. Therefore, the robust conditions of the closed-loop system and the constraints of the input and state are established under the framework of robust predictive control to construct the linear matrix inequality (LMI) optimization problem. Finally, to verify the effectiveness of the design method proposed in this paper, the discrete time-varying linear system and the main steam control system with redundant channels are used as study cases.

Novel Unbiased Optimal Receding-Horizon Fixed-Lag Smoothers for Linear Discrete Time-Varying Systems

This paper proposes novel unbiased minimum-variance receding-horizon fixed-lag (UMVRHF) smoothers in batch and recursive forms for linear discrete time-varying state space models in order to improve the computational efficiency and the estimation performance of receding-horizon fixed-lag (RHF) smoothers. First, an UMVRHF smoother in batch form is proposed by combining independent receding-horizon local estimators for two separated sub-horizons. The local estimates and their error covariance matrices are obtained based on an optimal receding horizon filter and the smoother in terms of the unbiased minimum variance; they are then optimally combined using Millman’s theorem. Next, the recursive form of the proposed UMVRHF smoother is derived to improve its computational efficiency and extendibility. Additionally, we introduce a method for extending the proposed recursive smoothing algorithm to a posteriori state estimations and propose the Rauch–Tung–Striebel receding-horizon fixed-lag smoother in recursive form. Furthermore, a computational complexity reduction technique that periodically switches the two proposed recursive smoothing algorithms is proposed. The performance and effectiveness of the proposed smoothers are demonstrated by comparing their estimation results with those of previous algorithms for Kalman and receding-horizon fixed-lag smoothers via numerical experiments.

Some conditions for the proportional local assignability of the Lyapunov spectrum of discrete linear time-varying systems

We investigate a problem of proportional local assignability of the Lyapunov spectrum for discrete time-varying linear systems. We propose the concept of non-multiple proportional local assignability and discuss the relations of this notion and splitness of the corresponding free system. We prove that, if a given open-loop system is uniformly completely controllable and the corresponding free system is split, then the Lyapunov spectrum of the closed-loop system has the property of non-multiple proportional local assignment. We demonstrate the role of dimensional effects by proving proportional local assignability of the Lyapunov spectrum for a uniformly completely controllable two-dimensional system assuming that the corresponding free system has the Lyapunov spectrum consisting of different numbers.

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.

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.

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.

Functional observer design using the direct method for discrete time-varying linear systems

In this article, an algorithm for designing functional state observers for linear discrete time-varying systems using the direct approach is proposed. Both the single function and the multiple functions of the states are studied. The design method is simple, and it is based neither on the solution of the generalised Sylvester equation that appears in the existence conditions nor on canonical system transformations. This algorithm is explained through a detailed example.

Sufficient conditions for assignability of nonuniform dichotomy spectrum of discrete time-varying linear systems

We consider a version of the pole placement problem for tempered one-sided linear discrete-time time-varying linear systems. We prove a sufficient condition for assignability of the nonuniform dichotomy spectrum by linear feedback. The main result is that the nonuniform dichotomy spectrum is assignable if the system is completely controllable and certain lower asymptotic bound for the controllability Gramian holds.

Optimal Jamming Strategy Against Two-State Switched System

In this letter, we investigate the security issue of a discrete linear time-varying system following a fixed switching rule. We first give a sufficient condition for the stability of this two-state switched system. In our scenario, the communication channel between the sensor and the remote estimator is interfered by a jammer with limited energy. Then we prove that the continuous jamming strategy is optimal compared with other strategies, which maximizes the estimation error in the viewpoint of the attacker. Finally, numerical examples are presented to certificate the effectiveness of the proposed optimal jamming strategy.

Finite impulse response filter based fault estimation with computational efficiency for linear discrete time-varying systems subject to multiplicative noise

The robust fault estimation problem for linear discrete time-varying (LDTV) systems subject to multiplicative noise is investigated by means of finite impulse response (FIR) filter. A novel analytical redundancy, expressed via all states of the previous time window, is originally established to construct the fault estimator. To ensure the satisfactory fault estimation accuracy in stochastic sense under the interference of random uncertainty, a new performance index in forms of matrix trace function is proposed. An easy-to-check necessary and sufficient condition is presented to obtain the optimal filter gain via minimizing the performance index at each time instant. It is analytically demonstrated that, the newly proposed fault estimation algorithm enjoys obvious computational advantages in updating the filter gain, especially as the length of the time window increases for time-varying systems. Simulation results are finally provided to verify its feasibility and superiority.

Lyapunov Spectra Behavior for Linear Time-Varying Discrete Systems Under Small Perturbations

We investigate the behavior of the Lyapunov spectrum of a linear time-varying discrete system under the action of small perturbations in order to obtain some verifiable conditions for stability and openness of the Lyapunov spectrum. To this end we introduce the concepts of broken away solutions and splitted systems. The main results obtained are a necessary condition for stability and a sufficient condition for the openness of the Lyapunov spectrum, which is given in terms of the system itself. Finally, examples of using the obtained results are presented.