Problem Of Linear Systems Research Articles

Optimal networked state estimation in the presence of multi-channel correlated packet losses

This paper studies the optimal estimation problem for linear time-invariant systems over lossy communication networks subject to multi-channel packet losses. In its full generality, we assume that the random packet losses are spatially correlated across different channels. The problem is addressed by employing a modified Kalman filter to estimate the system’s states, and the primary issue under study is the convergence of the state estimate under a mean-square criterion. This convergence is shown to be equivalent to the existence of a stabilizing solution to a modified algebraic Riccati equation. We provide a necessary and sufficient condition to characterize the existence of the stabilizing solution, and show that the solution can be computed by solving a set of linear matrix inequalities. Additionally, from a robust convergence perspective, we consider further packet losses whose probabilities are unknown and uncertain. We develop a matrix measure referred to as the critical matrix to characterize the maximal tolerance for the modified Kalman filter to converge. Specifically, we show that the mean-square estimation error is bounded if and only if the critical matrix satisfies a positive semi-definiteness condition.

Extended affine bessel summation inequalities: Applications to stability analysis of linear discrete-time systems with time-varying delays

This paper concerns the stability analysis problems of linear discrete-time systems with time-varying delays. For developing less conservative stability criteria obtained with the Lyapunov-Krsoavkii approach, numerous Lyapunov-Krasovskii functional (LKF) have been constructed by utilizing various summation quadratic functions. Thus, summation inequalities have been essential methods to develop convex stability criteria which guarantee the negative forward difference of LKFs. To further reduce the conservatism of stability criteria, this paper proposes extended affine Bessel summation inequalities. The proposed summation inequalities provide affine upper bounds of an extended summation quadratic function that contains a systems state variable, its forward difference, and their correlated terms. Further, this paper provides notes on the correlation among several summation inequalities including the proposed ones. These notes also prove that an increase in the degree of the developed affine Bessel summation inequalities only reduces conservatism. Two numerical examples effectively demonstrate the reduction of the conservatism due to the proposed summation inequalities in terms of stability regions which are expressed as delay bounds.

Delay compensation of linear systems with multiple distributed input delays via memoryless feedback

Time-delay systems are widely used in industrial applications, such as chemical process and aerospace. The predictor-based feedback method is one of the effective ways to deal with such systems. This paper studies the delay compensation problem for linear systems with multiple distributed input delays. A novel state vector associated with the future state of the system is constructed, which reduces the original linear time-delay system to a linear delay-free system. Memory feedback controller is then designed, and necessary and sufficient conditions for the stability of the closed-loop system are acquired in accordance with the stability of an integral delay system. To avoid the difficult implementation problem of memory feedback controller, the memoryless feedback controller is proposed instead to compensate the multiple distributed input delays. Finally, the proposed controllers are applied to the design of the linearized liquid-level control resonant circuit system.

Distributed Adaptive Output Consensus of Unknown Heterogeneous Non-Minimum Phase Multi-Agent Systems

This article addresses the leader-following output consensus problem of heterogeneous linear multi-agent systems with unknown agent parameters under directed graphs. The dynamics of followers are allowed to be non-minimum phase with unknown arbitrary individual relative degrees. This is contrary to many existing works on distributed adaptive control schemes where agent dynamics are required to be minimum phase and often of the same relative degree. A distributed adaptive pole placement control scheme is developed, which consists of a distributed observer and an adaptive pole placement control law. It is shown that under the proposed distributed adaptive control scheme, all signals in the closed-loop system are bounded and the outputs of all the followers track the output of the leader asymptotically. The effectiveness of the proposed scheme is demonstrated by one practical example and one numerical example.

$\mathcal {H}_{2}$ 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.

Robust Output Regulation and Reinforcement Learning-Based Output Tracking Design for Unknown Linear Discrete-Time Systems

In this article, we investigate the optimal output tracking problem for linear discrete-time systems with unknown dynamics using reinforcement learning (RL) and robust output regulation theory. This output tracking problem only allows to utilize the outputs of the reference system and the controlled system, rather than their states, and differs from most existing works that depend on the state of the system. The optimal tracking problem is formulated into a linear quadratic regulation problem by proposing a family of dynamic discrete-time controllers. Then, it is shown that solving the output tracking problem is equivalent to solving output regulation equations, whose solution, however, requires the knowledge of the complete and accurate system dynamics. To remove such a requirement, an off-policy RL algorithm is proposed using only the measured output data along the trajectory of the system and the reference output. By introducing reexpression error and analyzing the rank condition of the parameterization matrix, we ensure the uniqueness of the proposed RL-based optimal control via output feedback.

Distributed H∞ consensus of multiagent systems with nonuniform time-varying communication delays

This article addresses the distributed H∞ consensus control problem of general linear multiagent systems with time-varying communication delays and external disturbances. The communication network is subject to arbitrarily bounded and nonuniform time-varying communication delays. Based on the leader-following control mode, a novel distributed state feedback control protocol is proposed to achieve state consensus over undirected fixed topology while guaranteeing a prescribed disturbance rejection objective. By means of Lyapunov analysis and some inequality techniques, admissible upper bounds of nonuniform time-varying delays and H∞ consensus criteria are acquired. As opposed to the existing result for the similar problem, our result has its distinct advantages in that it can accommodate arbitrarily bounded nonuniform time-varying communication delays, and has greater delay margin and better consensus performance. Moreover, the consensus problem for the multiagent systems over connected undirected switching topology is discussed. Finally, a numerical example is provided to show the effectiveness and superiority of these obtained results.

Inverse Scattering Problem for Linear System of Four-Wave Interaction Problem on the Half-Line with a General Boundary Condition

Inverse Scattering Problem for Linear System of Four-Wave Interaction Problem on the Half-Line with a General Boundary Condition

Finite-time contractive boundedness analysis and controller synthesis for interval positive linear systems under [formula omitted]-gain performance

This paper investigates the finite-time contractive boundedness (FTCB) analysis and controller synthesis problems for interval positive linear systems under L1-gain performance. Firstly, both FTCB and L1-gain FTCB definitions are extended to positive systems with disturbances. Subsequently, FTCB and L1-gain FTCB criteria are established for the considered positive systems by using the linear copositive Lyapunov function approach. Furthermore, the state feedback controller is designed to guarantee the positivity and L1-gain FTCB of the resulting closed-loop systems, and a convex optimization-based algorithm is given to derive the optimal L1-gain and controller parameter. Finally, a numerical example and an application example about pest control are used to verify the effectiveness of the proposed conditions.

Submodularity of optimal sensor placement for traffic networks

The need for monitoring the state of a traffic network versus the costly installation and maintenance of roadside sensors constitutes the tough sensor placement problem in designing transportation networks. Placement problems naturally lie in the category of subset selection problems, which are known to be inherently combinatorial, and therefore, finding their exact solution is intractable for large problems. Due to this intractability, numerous heuristics have been proposed in the literature for approximately solving placement problems for traffic networks. Among these approaches, it has been observed that greedy algorithms normally outperform other heuristics. In this paper, we show the mathematics of why greedy algorithms are appropriate proxies for solving these subset selection problems; similar to placement problems for linear systems, placement problems for traffic networks also normally have a submodular structure. In this work, we analyze the problem of road sensor placement for transportation networks under different information structures available: when no vehicle routing information is available, when vehicles’ routings are known, and when it is necessary to maximize the number of origin–destination (O/D) traffic flows that are monitored with a set of sensors. We show that in all these cases, the placement problem has a submodular monotone structure. It is well known that the submodularity and monotonicity of discrete optimization problems can be leveraged to derive greedy algorithms that approximate the optimal solution. Consequently, our result is of great practical importance since by exploiting the submodularity and monotonicity of a problem, we show that it is possible to use polynomial-time greedy algorithms to approximate the combinatorial optimization problem with guaranteed optimality bounds for large problems, which are intractable to solve otherwise. Our results shed light upon the success of heuristic greedy algorithms that have been developed in some of the literature for solving placement problems at scale. To demonstrate the applicability of submodular optimization for solving placement problems, we first compare the performance of our polynomial-time approximation algorithm with the true optimum in an example traffic network which is small enough for finding the exact optimal solution with enumerating all possible subsets. Then, we investigate and validate our submodular approach in a case study involving a large-scale traffic network in Berkeley, California, where finding the exact optimal solution is intractable. Submodularity of the placement problem in these scenarios provides a powerful computational tool which can be further extended to other placement problem formulations that can become a reference for solving similar problems in the transportation literature.

Multistep Methods for Second-Order Differential-Algebraic Equations

We consider the initial value problem for linear systems of second-order ordinary differential equations with an identically singular matrix multiplying the principal part. Sufficient conditions for the existence of a unique solution are given in terms of matrix polynomials. For such problems, multistep difference schemes are proposed. An analysis of their stability and calculations of a model example are carried out.

Fully distributed leader–follower consensus for multi-agent systems based on reduced-order adaptive feedback protocol

This paper addresses the leader–follower consensus problem of linear multi-agent systems by designing reduced-order adaptive output feedback protocol. Using relative output information among neighboring agents, node-based adaptive output feedback control protocol and edge-based adaptive output feedback control protocol are presented, respectively. For both cases, based on Lyapunov analysis and algebraic Riccati equation design, some sufficient conditions are given to achieve leader–follower consensus by applying the proposed adaptive output feedback protocols. Finally, some simulation examples are given to illustrate the validity of the theoretical results.

A data-driven fault isolation and estimation approach for unknown linear systems

This paper considers the data-driven fault isolation and estimation problem for linear time-invariant systems with unknown dynamic matrices and multiple actuator faults. In most of existing fault isolation methods, how to accurately identify the types of faults has not been solved well when the system matrices are unknown. To deal with this problem, a neural network-based fault isolation method is proposed by analyzing and extracting features of different fault models in terms of constructing sparse vectors and function libraries using the available input–output data. Then, a fault estimator is designed to estimate the fault signals within the data-driven framework, where its parameters are computed by the system’s Markov parameters and the identified types of faults. Finally, two examples are used to verify the advantages and effectiveness of the proposed fault isolation and estimation approach.

Adaptive Event-Triggered Bipartite Containment Control of Linear Multiagent Systems Under Directed Topologies

This article focuses on the adaptive event-triggered bipartite containment problem of linear multiagent systems (MASs) under the signed digraphs, where multiple leaders with zero and nonzero inputs are respectively considered. Based on the relative states of neighbor agents, the fully distributed event-triggered control protocols are proposed to make containment error converge to zero. It is noted that the adaptive control is adopted to avoid the use of the global information, i.e., the number of agents in the MAS, the eigenvalues of Laplace matrix, etc. Since the signed digraph is considered, the Laplace matrix is asymmetric, which greatly increases the design difficulty of adaptive event-triggered strategy and is challenging. To solve this problem, innovative compound triggering conditions are designed, under which neighbors not only communicate intermittently, but also control protocols are updated discontinuously only at their own triggering instants. Finally, the theoretical results obtained in this article are verified by two examples.

Dynamic Periodic Event-Triggered Consensus Protocols for Linear Multiagent Systems With Network Delay

In this article, we investigate the event-triggered consensus problem for general linear multiagent systems with bounded communication network delay. To reduce the communication frequency, we propose some dynamic periodic event-triggering strategies, under which an auxiliary dynamic variable and a weighting parameter are introduced for each agent to build its event-triggering condition. Then, a distributed adaptive consensus protocol, which includes a time-varying coupling weight for each edge, is designed based on the proposed event-triggering strategies. With the well-designed updating laws of dynamic variables, coupling weights and weighting parameters, we can theoretically guarantee the consensus of the concerned multiagent systems. Our event-triggering strategies do not require agents to continuously verify the event-triggering conditions or read their neighbors’ real-time state information. Moreover, Zeno behavior is inherently excluded as the interevent intervals are bounded from below by one period in the periodic event-triggering framework. Two numerical examples are provided to verify the effectiveness of the proposed event-triggered consensus protocols.

Consensus Protocol for Discrete-Time Linear Multiagent Systems With Channel Fadings: A Dynamic Event-Based Approach

This article investigates the consensus problem of discrete-time linear multiagent systems with channel fadings over wireless sensor networks. The phenomena of channel fadings are described as the stochastic Rice fading model with random time-varying channel coefficients. By utilizing the local state information and constructing a dynamic auxiliary system, a novel control protocol with the dynamic discrete-time event-triggered scheme has been proposed. Sufficient conditions are obtained to guarantee the asymptotic consensus of the multiagent systems under the proposed protocol. Compared with the existing works related to discrete-time event-triggered consensus, the interevent time between any two successive events can be prolonged, and thus, the transmission data packages can be decreased immensely. Finally, the effectiveness of the proposed consensus protocol with a dynamic discrete-time event-triggered scheme is testified by numerical simulations.

Fully Distributed Fault-Tolerant Leaderless Consensus of Multi-Agent Systems Under Directed Communication Graph

This paper investigates a fault-tolerant leaderless consensus problem for general linear multi-agent systems (MASs) under a strongly connected directed communication graph. The influences of the actuator bias faults and loss of actuator effectiveness faults on MASs are considered in this paper. The designed fully distributed adaptive fault-tolerant consensus protocols (FTCPs) can make all agents' states achieve consensus, asymptotically. Using the features of the loss of actuator effectiveness faults and the strongly connected graph, novel positive definite Lyapunov functions are constructed skillfully to analyse the validity of the proposed protocols. Finally, simulation examples are presented to demonstrate the effectiveness of the proposed protocols.

Predictor-Based Extended State Observer for Disturbance Rejection Control of Multirate Systems With Measurement Delay

This article investigates the multirate disturbance rejection control problem for linear control systems with mismatched disturbance and measurement delay using predictor-based extended state observer. A new extended state observer together with output predictor is first designed to obtain the estimation values of system state and mismatched disturbance, where output predictor is used to compensate the influences of measurement delay and sampling of output. To attenuate the undesirable influence of mismatched disturbance, we then design a new sampled-data robust controller with disturbance compensation, and the updating rate of the proposed controller is allowed to be different from that of the sensor. Thanks to prediction and disturbance/uncertainty estimation and attenuation techniques, the disturbance rejection property of the resultant closed-loop control systems is enhanced despite the multirate and measurement delay. Some sufficient conditions are presented to ensure the stability property of the resultant control systems. We finally consider the application of the visual servoing control system for an inertially stabilized platform, and the experiment results verify superiorities of the predictor-based disturbance rejection control method proposed in the article.

Robust risk‐sensitive control

SummaryWe introduce arisk‐sensitivegeneralization of the mixed control problem for linear stochastic systems with additive noise. Two criteria ofexponential‐quadraticform are used to generalise the usual quadratic criteria. The solutions are found in a linear state‐feedback form for both the finite and the infinite horizon formulations in terms of coupled Riccati differential and algebraic equations. A change of measures for both criteria and completion of squares method is used to derive the solutions, and explicit sufficient conditions for the admissibility of controls are derived. An application to the problem ofrobustportfolio control in a market with random interest rate subject to a disturbance is also given.

Periodic event-based robust output regulation for a class of uncertain linear systems with inter-event analysis

This paper investigates the robust output regulation problem of uncertain linear systems by using a periodic event-triggered controller. A new type of event triggers based on local sampled signals is proposed to monitor events. An event detector only uses local information, avoiding infinitesimal inter-event times as in some continuous event-triggered schemes. A periodic event-triggered system naturally excludes Zeno behavior as the system operates at discrete-time instants. Convergence of tracking error holds for a class of decreasing triggering positive functions. When exponential functions bound the triggering function, analytical characterization of the relationship among sampling, event triggering, and inter-event behavior is established.