Problem Of Linear Systems Research Articles

Dual Domain Decomposition Method for High-Resolution 3D Simulation of Groundwater Flow and Transport

The high-resolution 3D groundwater flow and transport simulation problem requires massive discrete linear systems to be solved, leading to significant computational time and memory requirements. The domain decomposition method is a promising technique that facilitates the parallelization of problems with minimal communication overhead by dividing the computation domain into multiple subdomains. However, directly utilizing a domain decomposition scheme to solve massive linear systems becomes impractical due to the bottleneck in algebraic operations required to coordinate the results of subdomains. In this paper, we propose a two-level domain decomposition method, named dual-domain decomposition, to efficiently solve the massive discrete linear systems in high-resolution 3D groundwater simulations. The first level of domain decomposition partitions the linear system problem into independent linear sub-problems across multiple subdomains, enabling parallel solutions with significantly reduced complexity. The second level introduces a domain decomposition preconditioner to solve the linear system, known as the Schur system, used to coordinate results from subdomains across their boundaries. This additional level of decomposition parallelizes the preconditioning of the Schur system, addressing the bottleneck of the Schur system solution while improving its convergence rates. The dual-domain decomposition method facilitates the partition and distribution of the computation to be solved into independent finely grained computational subdomains, substantially reducing both computational and memory complexity. We demonstrate the scalability of our proposed method through its application to a high-resolution 3D simulation of chromium contaminant transport in groundwater. Our results indicate that our method outperforms both the vanilla domain decomposition method and the algebraic multigrid preconditioned method in terms of runtime, achieving up to 8.617× and 5.515× speedups, respectively, in solving massive problems with approximately 108 million degrees of freedom. Therefore, we recommend its effectiveness and reliability for high-resolution 3D simulations of groundwater flow and transport.

At Most Five Limit Cycles in a Class of Discontinuous Piecewise Linear Systems with Three Zones

In this paper, we study the maximum number of limit cycles where planar discontinuous piecewise differential systems can exist formed by three regions separated by a nonregular line. We show that such discontinuous piecewise linear systems can have at most five limit cycles, two of which are of the four intersection points type, the third one is of the two intersection points type and the other two are of three intersection points type. In a setting, we have solved the extended 16th Hilbert problem for these discontinuous piecewise linear differential systems.

Fast and Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems

Fast and Accurate Randomized Algorithms for Linear Systems and Eigenvalue Problems

On Robustness for the Skolem, Positivity and Ultimate Positivity Problems

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

Fully distributed adaptive event-triggered bipartite containment control of linear multiagent systems with actuator faults

This paper studies a new type of fully distributed double event-triggered bipartite containment (DETBC) problem for linear multiagent systems (MASs). Meanwhile, multiple leaders with non-zero inputs and time-varying multiplicative and additive actuator faults (TMAAFs) are considered. First, in contrast to traditional event-triggered mechanisms (ETMs), the proposed novel double ETMs can guarantee that information transfer between agents and controller updates proceeds asynchronously. Some adaptive parameters are introduced in the trigger functions to enhance the dynamic adjustment ability of double ETMs. The trigger threshold depends on the relative state-dependent, which is independent of continuous information from neighbors. Then, a fault-tolerant DETBC protocol is designed to solve actuator fault problems. Moreover, compared with other relevant works, our control protocol is fully distributed, which avoids reliance on global information. Finally, a simulation experiment is conducted to validate the feasibility of the designed protocol.

On solving terminal approach and evasion problems for linear discrete-time systems under state constraints

The paper is devoted to elaboration of polyhedral techniques for solving two control problems for linear discrete-time systems with uncertainties under state constraints, namely, the terminal approach problem and the terminal evasion one. Such problems arise in systems with two controls, where the aim of the first is to steer the trajectory onto a given terminal set at a given instant without violating the state constraints, the aim of the other is opposite. It is assumed that the terminal set is a parallelepiped, the controls are bounded by parallelotope-valued constraints, and the state constraints are given in the form of so-called zones. We present techniques for solving both problems basing on polyhedral (parallelotope-valued or parallelepiped-valued) tubes. The techniques for solving the approach problem were proposed by the author earlier, but here additional properties of them are investigated. In particular, for the case without state constraints, guaranteed estimates are found for the trajectory that ensure that it is inside the tube. Convenient sufficient conditions are given to guarantee the obtaining of nondegenerate cross-sections during the calculations. For the evasion problem, a common solution scheme is considered, and then polyhedral techniques are proposed. The whole parametric families of external and internal polyhedral estimates for the solvability tubes for both problems are presented and compared. An illustrative example is given.

ℒ2 gain tracking control of linear completely unknown discrete‐time networked control systems with dropout

AbstractWe introduce an online, model‐free algorithm to address the gain optimal tracking problems in discrete linear networked control systems. The algorithm is specifically proposed to handle stochastic information dropout in the feedback loop. Our goal is to design a control law that achieves the system output reference tracking while attenuating the effect of the disturbance input. We first construct an augmented system consisting of the original system and the command generator system. Then, the performance index is expressed in a quadratic form taking into account packet loss. Next, we obtain the optimal solution by solving the dropout generalized algebraic riccati equation (GARE). Finally, a ‐learning algorithm is utilized to estimate the control and disturbance feedback gains of the system, using only measurement data with unknown system dynamics in the presence of dropout. Two algorithms are tested on a numerical example to demonstrate the validity and effectiveness of the proposed methodology.

Quadratic Tracking Control of Linear Stochastic Systems with Unknown Dynamics Using Average Off-Policy Q-Learning Method

This article investigates the optimal tracking control problem for data-based stochastic discrete-time linear systems. An average off-policy Q-learning algorithm is proposed to solve the optimal control problem with random disturbances. Compared with the existing off-policy reinforcement learning (RL) algorithm, the proposed average off-policy Q-learning algorithm avoids the assumption of an initial stability control. First, a pole placement strategy is used to design an initial stable control for systems with unknown dynamics. Second, the initial stable control is used to design a data-based average off-policy Q-learning algorithm. Then, this algorithm is used to solve the stochastic linear quadratic tracking (LQT) problem, and a convergence proof of the algorithm is provided. Finally, numerical examples show that this algorithm outperforms other algorithms in a simulation.

Adaptive optimal tracking control of networked linear systems under two-channel stochastic dropouts

This work investigates the adaptive optimal tracking control problem for networked discrete-time linear systems by directly using the data transmitted via communication networks. It is shown that the concerned problem can be addressed by solving two sub-problems: an adaptive optimal control one and an adaptive output regulation one. Two novel online reinforcement learning based policy iteration and value iteration algorithms are developed, which constitute an integrated framework to learn the optimal feedback control gain and the solutions to regulator equations by directly using the data transmitted via communication networks. Furthermore, it is shown that the tracking error of the closed-loop control system is mean square asymptotically stable even in the case of network packet stochastic dropouts. Simulation results demonstrate the efficacy of the proposed approaches.

Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems

This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method.

Robust asymptotic stability analysis for fractional-order systems with commensurate time delays: The 1 < β ≤ 2 case

The robust asymptotic stability problem for fractional-order linear systems with commensurate time delays (FOSCDs) and structured perturbations is investigated in this manuscript. Based on the properties of Kronecker products and μ-analysis, the necessary and sufficient conditions are given to guarantee the asymptotic stability of FOSCDs. Then, the necessary and sufficient conditions for the robust asymptotic stability of FOSCDs with structured perturbations are investigated. To reduce the computing complexity and deal with the problems of systems with larger dimensions, a sufficient condition on the robust asymptotic stability of FOSCDs with structured perturbations is given based on the properties of Kronecker products and 2-norm transformations. Finally, several examples are illustrated to test the effectiveness of the proposed results.

Optimal Tracking Control of Heterogeneous MASs Using Event-Driven Adaptive Observer and Reinforcement Learning.

This article considers the output tracking control problem of nonidentical linear multiagent systems (MASs) using a model-free reinforcement learning (RL) algorithm, where partial followers have no prior knowledge of the leader's information. To lower the communication and computing burden among agents, an event-driven adaptive distributed observer is proposed to predict the leader's system matrix and state, which consists of the estimated value of relative states governed by an edge-based predictor. Meanwhile, the integral input-based triggering condition is exploited to decide whether to transmit its private control input to its neighbors. Then, an RL-based state feedback controller for each agent is developed to solve the output tracking control problem, which is further converted into the optimal control problem by introducing a discounted performance function. Inhomogeneous algebraic Riccati equations (AREs) are derived to obtain the optimal solution of AREs. An off-policy RL algorithm is used to learn the solution of inhomogeneous AREs online without requiring any knowledge of the system dynamics. Rigorous analysis shows that under the proposed event-driven adaptive observer mechanism and RL algorithm, all followers are able to synchronize the leader's output asymptotically. Finally, a numerical simulation is demonstrated to verify the proposed approach in theory.

On the invariant factors and the minimal polynomial assignments of linear systems

In this note, the invariant factors assignment and the minimal polynomial assignment problems for multiple-input linear systems are studied. It is found that for any invariant factors with the geometry multiplicity of each eigenvalue less than or equal to the number of the inputs, the invariant factors assignment problems are solvable if and only if the controllability indices set of the linear system satisfies certain condition. For the minimal polynomial assignment problem, a necessary and sufficient condition on the solvability is proposed. Three numerical examples are provided to illustrate the proposed results.

A parallel inertial SP-iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings and its application in linear system, differential, and signal recovery problems

A parallel inertial SP-iteration monotone hybrid algorithm for a finite family of G-nonexpansive mappings and its application in linear system, differential, and signal recovery problems

Orthotope-search-expansion-based extended zonotopic Kalman filter design for a discrete-time linear parameter-varying system with a dual-noise term

A novel algorithm utilizing an extended zonotopic Kalman filter based on the orthotope search expansion method is introduced to solve the state estimation problem in discrete-time linear parameter-varying systems when the dual noise of the systems is assumed to be bounded but unknown. First, to identify unknown parameters, an orthotope search expansion method is proposed with a measurement strip constraint. Subsequently, based on the parameter identification orthotopic set, an extended zonotopic Kalman filter algorithm is proposed to derive an estimation interval that wraps the true state of the system. Next, by minimizing the size of the zonotope, the optimal gain matrix for the extended zonotopic Kalman filter algorithm is determined. Finally, the proposed methods are demonstrated for their effectiveness and accuracy through the presentation of two simulation examples.

Output feedback H∞$$ {H}_{\infty } $$ control of linear systems with infinite distributed input delays

AbstractThis article investigates the output feedback control problem of linear systems with infinite distributed input delays. An observer‐based output feedback (OBOF) controller is proposed via the predictor feedback approach, and a sufficient condition for the existence of the OBOF controller is given in terms of a set of linear matrix inequalities. It is shown that the resulting closed‐loop system is globally exponentially stable with a prescribed performance. The obtained results include some results on the same problem for linear systems with bounded distributed input delays as special cases. Two numerical examples are finally presented to illustrate the effectiveness of the proposed controller.

Reinforcement learning for optimal control of linear impulsive systems with periodic impulses

This paper focuses on the finite- and infinite-horizon optimal control problems of linear impulsive systems with periodic impulses under the quadratic performance index. Necessary and sufficient conditions for the optimal impulsive system are derived in terms of hybrid Riccati equations by utilizing the variational method and the collocation method combined with a time-varying Lyapunov function. Different from the existing model-based impulsive control schemes, three reinforcement learning (RL)-based algorithms are proposed to solve the optimal impulsive controller and hybrid controller for the impulsive system without the exact knowledge of the system dynamics. The asymptotical stability of the impulsive control system and the convergence of the RL-based algorithms are proved rigorously. Finally, a numerical simulation illustrates the effectiveness of the proposed control methods.

Kalman filtering based on dynamic perception of measurement noise

In this work, we focus on the online state estimation problem for linear systems in non-Gaussian measurement noise. Specifically, the measurement noise is modeled as a Gaussian mixture model (GMM). Then, Kalman innovation is used to approximate the current measurement noise, and dynamically perceive its responsiveness to each sub-model of the GMM. The responsiveness from different Gaussian scales is mapped to a new cost function, while the corresponding Kalman filter algorithm is derived. The theoretical steady-state error and computational complexity analysis of the algorithm are also given. The simulation and real experimental results agree with the theoretical predictions and demonstrate the superior performance of the proposed algorithm.

Stochastic LQ optimal control for Markov jumping systems with multiplicative noise using reinforcement learning

The stochastic linear quadratic (LQ) problem of discrete-time linear Markov jumping systems with multiplicative noise is investigated in this paper. Two reinforcement learning algorithms, one model-based and one model-free, are designed. The algorithms can be readily adapted to handle nonlinear cases by employing an appropriate function approximator. In the linear case, the network structure is designed to enable the problem to be solved without succumbing to modeling errors. For the model-free algorithm, an approach based on moving averages is proposed to reduce estimation variance. A proof of convergence is also provided. Finally, the effectiveness of the proposed algorithms is demonstrated through a numerical example.

Model-Free Q-Learning for the Tracking Problem of Linear Discrete-Time Systems.

In this article, a model-free Q-learning algorithm is proposed to solve the tracking problem of linear discrete-time systems with completely unknown system dynamics. To eliminate tracking errors, a performance index of the Q-learning approach is formulated, which can transform the tracking problem into a regulation one. Compared with the existing adaptive dynamic programming (ADP) methods and Q-learning approaches, the proposed performance index adds a product term composed of a gain matrix and the reference tracking trajectory to the control input quadratic form. In addition, without requiring any prior knowledge of the dynamics of the original controlled system and command generator, the control policy obtained by the proposed approach can be deduced by an iterative technique relying on the online information of the system state, the control input, and the reference tracking trajectory. In each iteration of the proposed method, the desired control input can be updated by the iterative criteria derived from a precondition of the controlled system and the reference tracking trajectory, which ensures that the obtained control policy can eliminate tracking errors in theory. Moreover, to effectively use less data to obtain the optimal control policy, the off-policy approach is introduced into the proposed algorithm. Finally, the effectiveness of the proposed algorithm is verified by a numerical simulation.