Time-invariant Discrete-time Systems Research Articles

Data-driven distributed output consensus control for multi-agent systems with unknown internal state

This paper discusses the topic of optimal output consensus control of linear time-invariant discrete-time multi-agent systems (MASs). The attainment of optimal output consensus control for MASs hinges upon the resolution of the interconnected Hamilton–Jacobi-Bellman equation, a task typically precluded by the inherent intractability of analytical solutions. Furthermore, most real-world systems are too complex to obtain the internal states of systems. To address these issues, a modified deep Q-learning network is constructed using current and historical system data rather than a precise model of the system. First, reconstructing the internal state of each agent using an adaptive distributed observer based on output feedback prevents the system instability brought on by the augmented systems. Then the local error system of the agent can be redefined. Based on the redefined error system, a data-driven adaptive dynamic programming (ADP) method is introduced, realized by using the actor–critic neural network structure. In addition, an experience replay strategy is proposed to reduce the propagation of estimation bias and improve the learning speed. Finally, the comparative numerical simulations substantiate the efficacy of the proposed algorithm in a quantifiable manner.

Output null controllability for linear time-invariant structured discrete-time systems: A graph theoretic condition

In this paper, we consider a linear time-invariant discrete-time system and study the output null controllability problem, i.e., the problem of steering the output to zero in a finite number of steps. We assume that we only know the structure of the system, i.e., the zero/nonzero location in the system matrices. Hence, we consider a structural version of the output null controllability problem. We represent the structure of the system by means of a directed graph and present a graph theoretic sufficient condition for the problem to be generically solvable. Here generically solvable means that the problem is solvable for almost all systems with the same structure. We illustrate the conditions using an example.

Nonlinear Sampled-data Systems—A lifting framework

This short note gives a new framework for dealing with nonlinear sampled-data systems. We introduce a new idea of lifting, which is well known for linear systems, but not successfully generalized to nonlinear systems. This paper introduces a new lifting technique for nonlinear, time-invariant systems, which are different from the linear counterpart as developed in Bamieh et al. (1991); Yamamoto (1994) etc. The main difficulty is that the direct feedthrough term effective in the linear case cannot be generalized to the nonlinear case. Instead, we will further lift the state trajectory, and obtain an equivalent time-invariant discrete-time system with function-space input and output spaces. The basic framework, as well as the closed-loop equation with discrete-time controller, is given. As an application of this framework, we give a representation for the Koopman operator derived from the given original nonlinear system.

Robust Interval Observer for Systems Described by the Fornasini–Marchesini Second Model

This letter proposes a novel robust interval observer for a two-dimensional (treated as a synonym for a double-indexed system) linear time-invariant discrete-time system described by the Fornasini-Marchesini second model. This system is subject to unknown but bounded state disturbances and measurement noise. Built on recent interval estimation strategies designed for one-dimensional systems, the proposed observer is based on the introduction of weighting matrices which provide additional degrees of freedom in comparison with the classical structure relying on a change of coordinates. Linear matrix inequality conditions for the exponential stability and peak-to-peak performance of a two-dimensional system described by the Fornasini-Marchesini second model are then proposed, and applied to the design of a robust interval observer. Numerical simulation results are provided to show the efficiency of the proposed estimation strategy.

An Event-Triggered Watermarking Strategy for Detection of Replay Attacks

The problem of detecting replay attacks in linear time-invariant discrete-time systems is considered in this paper. In the same spirit of watermarking techniques that apply a distinctive signature to the plant\’s signals, we propose an event-triggered control scheme, that is purposely designed to generate a unique sequence of switching intervals, by computing an appropriate input value to be held constant while the communication is not triggered. We provide a detailed undetectability characterization in the time domain and design a controller that achieves the desired behavior. Our proposed method results in a control scheme that makes it hard for an attacker to satisfy undetectability conditions, and, as a result, a standard observer-based residual generator can be employed to reveal replay attacks. We finally validate the method using a numerical example.

Trackability for Discrete-Time Linear Time-Invariant Systems: A Brief Review and New Insights

Abstract Trackability is a known system property and has been studied extensively in the literature both from a theoretical standpoint and from the standpoint of constructing inversion-based tracking controllers. There is also a significant body of literature on closely related ideas focusing on existence of tracking control inputs but using varied terminology and tools. In this paper, we start with the existence question, given an linear time-invariant (LTI) discrete-time system, when does there exists control inputs to enable the system to track arbitrary reference commands. We adopt an algebraic approach that then allows us to define the sets of all trackable output reference commands and untrackable output reference commands, and subsequently permitting us to determine conditions for existence of a right inverse of the system based on these sets and a simple rank test. These results permit us to then gain insights into tracking behavior of systems that are not trackable. We discuss in some detail the relationship of these rank tests with other results in the literature. We also define three indices that indicate the expected tracking behavior of any given system (whether trackable or not). Furthermore, we also present a Venn diagram explaining in detail the connections between trackability and other fundamental system properties like controllability, observability and output controllability, while discussing several facts that elaborate these connections. The presented work is expected to provide a framework for receiving new theoretical insights on the topic of tracking control and trackability.

Order reduction of linear time invariant large-scale system by preserving the impact of dominant poles in the reduced model

ABSTRACT This paper introduces a new reduction method for linear time-invariant continuous and discrete-time systems. The method is created considering only the effect of the dominance pole to get the reduced-order model. The denominator coefficient of reduced order model (ROM) is obtained by conserving the dominant poles of a higher order system while the numerator polynomial of the reduced model is found by comparing the original system transfer function with the transfer function of desired reduced order model. Five standard numeric examples are examined from literature considering single input single output (SISO) and discrete systems. To prove the validity, accuracy and simplicity of the proposed method, the results obtained are compared with other recent established and well-known techniques like Truncation and Routh Stability methods.

Complexity of constrained sensor placement problems for optimal observability

This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations: (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed k>2, the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most k, is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of outputs in order to maximize the number of states of the system that are structurally observable (i.e., to maximize the size of the observable subgraph) is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a (1−1e)-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly 400 states in its linearized model.

Flexible optimal Kalman filtering in wireless sensor networks with intermittent observations

This paper is concerned with the distributed Kalman filtering over the wireless sensor networks (WSNs) in the presence of intermittent observations and different sensing states, where only task nodes are required to estimate the state of a linear time-invariant discrete-time system. A class of flexible binary values is used to develop the adaptability of flexible optimal Kalman filtering (FOKF) for variable sensing states. Based on the minimum error covariance trace principle, two classes of FOKFs have optimal collaborative estimation via their own and community observations, including the original FOKF and the FOKF with uncertain noise variance. The performance analysis of these two types of filters show that they have high estimation accuracy, strong robustness, low energy consumption and user-friendliness. The proposed algorithms are applied to estimate and track the position of a moving target in WSNs. The simulation illustrates that the proposed filters have superior performance, compared with the existing algorithms.

Statistical and Nonstatistical Methods for Estimating the Impulse Characteristic of a Dynamical System

We consider the problem of estimating the impulse characteristic of a linear time-invariant discrete-time dynamical system, using data on its input and output signals. We propose versions of estimation methods for asymptotically stable dynamical systems in the context of various approaches to the presentation of parametric uncertainty, including approaches based on the interval analysis and the fuzzy set theory. A comparative analysis and numerical illustration of the methods are presented.

Non-prehensile manipulation of a devil-stick: planar symmetric juggling using impulsive forces

Juggling a devil-stick can be described as a problem of non-prehensile manipulation. Assuming that the devil-stick remains confined to the vertical plane, the problem of juggling the stick between two symmetric configurations is considered. Impulsive forces are applied to the stick intermittently and the impulse of the force and its point of application are modeled as control inputs to the system. The dynamics of the devil-stick due to the impulsive forces and gravity is described by half-return maps between two Poincare sections; the symmetric configurations are fixed points of these sections. A coordinate transformation is used to convert the juggling problem to that of stabilization of one of the fixed points. Inclusion of the coordinate transformation in the dynamic model results in a nonlinear discrete-time system. A dead-beat design for one of the inputs simplifies the control problem and results in a linear time-invariant discrete-time system. Standard control techniques are used to show that symmetric juggling can be achieved from arbitrary initial conditions.

Set-Valued Feedback Control and Its Application to Event-Triggered Sampled-Data Systems

Motivated by challenges in feedback control with event-triggered control input updates, this article studies an ellipsoidal set-valued feedback control problem for linear time-invariant (LTI) discrete-time (DT) systems, and its application to event-triggered control systems. First, a state feedback controller based on the set-valued information is designed for LTI DT systems with disturbances. Then, the stability of the closed-loop system is analyzed, and the state of the closed-loop system is shown to be uniformly ultimately bounded in an ellipsoidal set. Based on the proposed set-valued feedback controller, we consider the case when the event-triggering condition (ETC) is described by an ellipsoidal set for an event-triggered control system. To ensure the requirements on control performance and communication rates simultaneously, an optimal ETC design problem is formulated and its optimal solution is provided. Finally, simulation examples are given to show the effectiveness of the proposed method.

Numerical computation and new output bounds for time-limited balanced truncation of discrete-time systems

In this paper, balancing based model order reduction (MOR) for large-scale linear discrete-time time-invariant systems in prescribed finite time intervals is studied. The first main topic is the development of error bounds regarding the approximated output vector within the time limits. The influence of different components in the established bounds will be highlighted. After that, the second part of the article proposes strategies that enable an efficient numerical execution of time-limited balanced truncation for large-scale systems. Numerical experiments illustrate the performance of the proposed techniques.

Computation of Least-Conservative State-Constraint Sets for Decentralized MPC With Dynamic and Constraint Coupling

We address the problem of synthesizing state-constraint sets for a fully decentralized Model Predictive Control (MPC) scheme. We consider linear time-invariant discrete time systems, with subsystems possibly coupled in both dynamics and state constraints. For each individual subsystem we employ a set-based framework to compute the state-constraint sets, that are used to synthesize local tube-based MPC controllers. The offline problem that computes the constraint sets explicitly ensures that the feasible regions of the MPC controllers are non-empty, and whenever the controllers are feasible, the overall system constraints are satisfied with the least conservativeness possible. We demonstrate the closed-loop performance of the decentralized scheme, assessed with respect to centralized MPC, using a numerical example.

Nonparametric methods for the estimation of [formula omitted] gain and gains at sample frequencies using bandpass filters

This paper describes a method of estimating the gains at sample frequencies and the ℓ2 gain for a linear time-invariant discrete-time system using a truncated step response. The gain is estimated using a bandpass filter at each sample frequency, and the filter parameter is optimized by maximizing the estimate of the ℓ2 gain near the peak gain. This filtering method is extended to multivariable systems. Another method of making an optimal bandpass filter is derived from the iterative-experiment method. The filtering method is compared with other estimation methods in terms of noise sensitivity through a series of numerical experiments.

An Alternative Setup to Study Stabilization of Networked Control Systems Over Correlated Fading Channels

This paper studies stability of control systems over a communication network with spatially correlated fading channels. We consider a multiple-input multiple-output linear time-invariant discrete-time closed-loop system in which communications are done through multiple correlated multiplicative channels. Particularly, we provide conditions that relate the stability of the above networked control system (NCS) to the stability of an auxiliary closed-loop system that replaces the multiplicative channels by additive noise channels. This provides a simple framework to analyse NCSs over correlated multiplicative channels since standard linear systems tools can be used to analyse stability as opposed to consider the multiplicative non-linearity explicitly in the analysis. Thereby, we extend existing results in the literature that contain similar conditions for uncorrelated channels only.

Decoupling of Discrete-time Dynamical Systems Through Input-Output Blending

This paper presents a subsystem decoupling method for Linear Time Invariant Discrete-time systems. The aim is to control a selected subsystem, while not affecting the remaining dynamics. The paper extends earlier continuous time results to discrete time systems over a finite frequency interval. Decoupling is achieved by suitable input and output blend vectors, such that they maximize the sensitivity of the selected subsystem, while at the same time they minimize the transfer through the undesired dynamics. The proposed algorithm is based on an optimization problem involving Linear Matrix Inequalities, where the H- index of the controlled subsystem is maximized, while the transfer through the dynamics to be decoupled is minimized by a sparsity like criteria. The present approach has the advantage that it is directly applicable to stable and unstable subsystems also. Numerical examples demonstrate the effectiveness of the method.

The Design of Educational Tool for Jury's Stability Test

Stability analysis in systems is a very important topic. Many methods are available in this area. One of these is the Jury's stability test. In this study, a software tool that analyzes the stability of user defined systems according to Jury's stability test has been developed. Stability of the linear time-invariant discrete-time systems can be realized step by step (Jury table, stability test conditions, etc.) easily and effectively according to this criterion with the software tool which has user-friendly interface, including the topic description and can be used for educational purposes.

Asymptotic Unknown Input Decoupling Observer for Discrete-Time LTI Systems

This letter addresses a new observer structure and its corresponding design for linear time invariant discrete-time systems affected by unknown inputs (UIs). We show that the performances of standard unknown input observer (UIO) can be enhanced, especially, in the presence of stable invariant zeros with slow dynamics (detectable systems). Under mild condition, the proposed UIO design avoids the drawback of the invariant zeros, at least in a time interval, and allows to arbitrary place the eigenvalues during that interval. The main idea is to relax the exact UI decoupling condition for an asymptotic decoupling condition. The convergence analysis is analyzed using the Lyapunov theory and the asymptotic convergence conditions of the state estimation error are expressed in linear matrix inequality formalism.

Distributed Kalman Filtering Over Wireless Sensor Networks in the Presence of Data Packet Drops

We study distributed Kalman filtering over the wireless sensor network, where each sensor node is required to locally estimate the state of a linear time-invariant discrete-time system, using its own observations and those transmitted from its neighbors in the presence of data packet drops. This is an optimal one-step prediction problem under the framework of distributed estimation, assuming the TCP-like protocol. We first present the stationary distributed Kalman filter (DKF) that minimizes the local average error variance in the steady state at each sensor node, based on the stabilizing solution to the corresponding modified algebraic Riccati equation (MARE). The existence of the stabilizing solution to the MARE is addressed by adopting the stability margin, which can be computed by solving a set of linear matrix inequalities. Then, the Kalman consensus filter (KCF), consisting of the stationary DKF and a consensus term of prior estimates, is studied. Finally, the performance of the stationary DKF and KCF is illustrated by a numerical example.