Linear Time-invariant Systems Research Articles

Recovery of signals and images using test measurements

Mathematical test measurement methods for recovering signals and images detected and/or transmitted by linear time-invariant (LTI) systems are proposed and analyzed. The use of test measurements gives a possibility to avoid the time-consuming analysis of complex LTI systems and the determination of their impulse responses. Recovery of continuous signals and discrete data using both direct methods and Fourier transforms are considered, the features and areas of application of these methods are described. The regularization technique used in the proposed methods allows to solve ill-posed and ill-conditioned problems in signal and data recovery. Methods can also be used to simulate the response of LTI systems to a given action. The numerical experiments on recovery the one-dimensional signals and two-dimensional images are considered.

Zonotopic Kalman Filter-Based Interval Estimation for Discrete-Time Linear Systems With Unknown Inputs

This letter proposes an unknown input zonotopic Kalman filter-based interval observer for discrete-time linear time-invariant systems. In such contexts, a change of coordinates decoupling the state and the unknown inputs is often used. Here, the dynamics are rewritten into a discrete-time linear time-invariant descriptor system by augmenting the state vector with the unknown inputs. A zonotopic outer approximation of the feasible state set is then obtained with a prediction-correction strategy using the information from the system dynamics, known inputs and outputs. Bounds for both the state and unknown inputs are obtained from this zonotopic set. The efficiency of the proposed interval observer is assessed with numerical simulations.

Design of a set of stabilizing P and PI controllers for LTI TITO system with multiple time-delays

This paper presents a simple decoupled design technique for two-input-two-output (TITO) linear time-invariant (LTI) systems with multiple time-delays. The output of each decoupled system corresponding to each channel’s input is controlled by designing a set of stabilizing P and PI controllers based on the Hermite-Biehler theorem. The proposed approach is restricted to the class of TITO LTI systems with multiple time-delays present in the numerator part of each block of the transfer function matrix. An illustrative example is given to demonstrate the performance and effectiveness of the design technique.

Parametric Robust Generalized Minimum Variance Control

This work presents a hybridization of the Generalized Minimum Variance control (GMVC) with Parametric Robust Control (RPC) for Linear Time Invariant (LTI) Uncertain Systems. A linear parametric uncertainties model is used to describe the system dynamic behavior. The controller synthesis is based on PID controller with a low-pass filter and formulated as a convex optimization problem which takes in account the desired closed-loop performance and the uncertainties on the model parameters. Then, the robust controller parameters, that represent the best solution for the all possible models and assign the closed-loop poles within the desired region in the s-plane, are transferred to GMVC by Tustin’s method, resulting in a parametric robust generalized minimum variance (PRGMV) controller, which was compared to two other approaches, carrying out several simulation essays in Matlab environment. In order to emphasize its performances and its efficiency, the performance index and sensitivity analysis were assessed. The results confirmed that the proposed controller ensures better robustness and performance for reference tracking and disturbance rejection.

Reset-and-Hold Control of Systems With Time Delay

Reset control is a kind of hybrid control which is capable of overcoming fundamental limitations on the performance intrinsic to linear time invariant (LTI) systems. In this work, we develop a novel control strategy for first order LTI plants with time delay, based on the proportional-integral plus Clegg integrator (PI+CI) controller, a hybrid extension of the proportional-integral (PI) controller, augmented with a new mechanism for keeping its state constant during a given interval of time (reset-and-hold strategy). This strategy is capable of producing an approximation of a flat response which greatly improves the performance in comparison with non-hybrid linear strategies, extending previous results developed for first order systems without delay. Well-posedness and closed-loop stability of the resulting control system are analyzed under the Hybrid Inclusions (HI) framework, and a set of sufficient stability conditions are provided. Furthermore, a case study is developed that showcases the possibilities of this new approach.

A Survey on Reachable Set Techniques for Fault Recoverability Assessment

The development of any fault-tolerant control solution is based on the strong assumption that fault situations can be accommodated. This paper provides a survey of four reachable set techniques to assess the fault recoverability property for constrained linear time invariant (LTI) systems by means of ellipsoid, zonotope, polytope and support function representations. These techniques are next applied to an angular velocity spacecraft model. A discussion is finally made to assess the computational complexity for the four algorithms.

Existence Conditions for ODE Functional Observer Design of Descriptor Systems

This letter studies the design of functional observers for linear time-invariant descriptor systems which are not necessarily square and regular. A novel characterization for the existence of functional observers is presented. The observer is realized by a system of ordinary differential equations (ODEs). The existence of ODE observers is proved by means of a set of simple rank conditions on the system coefficient matrices. The proposed conditions are less restrictive than the existing ones. The derivation is purely algebraic, and the algorithm is illustrated by designing an observer-based controller.

Stability Conditions for Remote State Estimation of Multiple Systems over Multiple Markov Fading Channels

We investigate the stability conditions for remote state estimation of multiple linear time-invariant (LTI) systems over multiple wireless time-varying communication channels. We answer the following open problem: what is the fundamental requirement on the multi-sensor-multi-channel system to guarantee the existence of a sensor scheduling policy that can stabilize the remote estimation system? We propose a novel policy construction and analytical framework and derive the necessary-and-sufficient stability condition in terms of the LTI system parameters and the channel statistics.

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.

Data-driven approximation and reduction from noisy data in matrix pencils frameworks

This work aims at tackling the problem of learning surrogate models from noisy time-domain data by means of matrix pencil-based techniques, namely the Hankel and Loewner frameworks. A data-driven approach to obtain reduced order state-space models from time-domain input-output measurements for linear time-invariant (LTI) systems is proposed. This is accomplished by combining the aforementioned model order reduction (MOR) techniques with the signal matrix model (SMM) approach. The proposed method is illustrated by a numerical example consisting of a high-order building model.

Data-driven MPC of descriptor systems: A case study for power networks

Recently, data-driven predictive control of linear systems has received wide-spread research attention. It hinges on the fundamental lemma by Willems et al. In a previous paper, we have shown how this framework can be applied to predictive control of linear time-invariant descriptor systems. In the present paper, we present a case study wherein we apply data-driven predictive control to a discrete-time descriptor model obtained by discretization of the power-swing equations for a nine-bus system. Our results show the efficacy of the proposed control scheme and they underpin the prospect of the data-driven framework for control of descriptor systems.

Leader-Following Cluster Consensus as a Graphical Differential Game With a Nash Equilibrium Solution

This paper formulates the leader-following cluster consensus problem of a homogeneous linear time-invariant multi-agent system over a directed graph as an N-player graphical differential game. The agents utilize distributed state feedback control laws based on the available information to calculate their best response. In general, the absence of centralized information prevents the best responses of agents from constituting a Nash equilibrium solution. We show that the best responses depending on the unique solution of an algebraic Riccati equation constitute a Nash equilibrium for the proposed graphical differential game with a modified cost function. A numerical example illustrates the Nash equilibrium solution by showing that the agents can not improve their performance by altering their responses while solving the leader-following cluster consensus problem.

Decentralized Robust LMI-based Voltage Control Strategy for Autonomous Inverter-Interfaced Multi-DG Microgrids

This article's objective is to propose a decentralized one Degree of Freedom (DoF) robust control system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance criteria for optimal voltage control of an uncertain inverter-interfaced AC microgrid in the islanding mode of operation. The islanded microgrid system comprises multiple Distributed Generation (DG) units connected with general topology. The proposed one DoF voltage control system provides robust performance alongside robust stability against various sources of uncertainty, such as local load variations, Plug-and-Play (PnP) operation of the DG units, topology variations, and different subsequent changes. To that end, first, each DG unit is modeled as a Linear Time-Invariant (LTI) system with a new polytopic type uncertainty. Next, a Linear Matrix Inequality (LMI) based state-feedback control system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance criteria is implemented on the LTI polytopic islanded microgrid system. The controller's decentralized structure excludes the need for communication channels, and the controller's robust performance characteristic eliminates the necessity of pre-filter design. Simulating multiple scenarios in MATLAB/SimPowerSystems Toolbox confirms the developed controller's performance in different scenarios, including voltage tracking, local load changes, PnP operation, and topology changes.

Discretization of the Functionals With Prescribed Derivative

There are a number of recent results aimed at developing finite necessary and sufficient stability tests for linear time-invariant time delay systems within the framework of Lyapunov-Krasovskii functionals with prescribed derivative approach. These tests are based on the concept of Lyapunov matrix. They usually require verification of positive definiteness of a certain block matrix whose blocks correspond to the Lyapunov matrix evaluated at several discretization points. In this work, we present a new test of this kind which employs a combination of the discretized Lyapunov functionals method of K. Gu and the functionals with prescribed derivative approach. Unlike existing combinations of those techniques, we avoid discretization of the functional's derivative and use the methodology of the necessary and sufficient stability tests development instead. We show a connection between our test and some of the existing results.

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.

Eigenvalue Based Method to Maximize the Exponential Convergence Rate for Consensus Tracking with Heterogeneous Delays

This paper focuses on the consensus tracking problem for a group of single-integrator agents with heterogeneous communication and input delays, with the goal of achieving the maximum exponential convergence rate. Existing research discusses a tracking protocol consisting of a consensus-based estimator and a tracking controller for each agent, as well as stability analysis. In contrast, we examine the performance evolution of multi-agent systems for such tracking protocols in terms of their exponential convergence rate. We present a systematic frequency domain method for analyzing the rightmost eigenvalue in the open left half complex plane, which determines the exponential decay rate of the tracking error. The spectral characteristics of the linear time-invariant retarded delay system are investigated, leading to the optimization of the tracking controller and estimator gains. Simulations are given to demonstrate how the system performs when the design parameters are optimized.

Observer Design in LTI Time-Delay Systems using Partial Pole Placement with Applications

In order to estimate the state of a dynamical system, we address the problem of designing an observer for linear time-invariant (LTI) dynamical systems including a time-delay. Sufficient conditions for the existence of the proposed observer are given using partial placement of the poles for the error. Namely, we exploit the multiplicity-induced-dominance property of the characteristic function corresponding to the system's error. The effectiveness of the proposed observer design is shown in both state lag and input lag respectively through the problems of Mach number control in a wind tunnel and stabilization of the inverted pendulum on a cart by using the delay.

Addendum to “Converse Theorems for Integral Quadratic Constraints”

Consider a linear time-invariant feedback system consisting of two open-loop stable subsystems. In this note, it is established that if such feedback system is stable for one subsystem being any arbitrarily passive system, then the other subsystem must be output strictly passive.

Multiplicative Fault Detection and Isolation in Dynamic Systems Using Data-Driven K-Gap Metric based kNN Algorithm

In this paper, a fault detection and isolation scheme for multiplicative faults in dynamic systems based on data-driven K-Gap metric and k-nearest neighbour (kNN) classification is proposed. To detect multiplicative faults, the standard classification task of kNN is studied from the viewpoint of system analysis. To this end, the data-driven stable kernel representation based on input/output data is presented for feature extraction capturing the dynamic of linear time-invariant (LTI) systems. Data-driven K-Gap metric is used as an alternative tool for distance measure between two kernel subspaces in the kNN algorithm. A simulation example on the three-tank system (DTS200) demonstrates the successful detection and isolation of various multiplicative faults

A Null-Space Compensation Control for Multi-input Redundant Plant Based on Model Reference Adaptive Control

In this paper, we proposed to design a null-space compensation control system for linear time invariant systems with multi-input redundant channels. In the input redundant systems, control input vector may have null-space component of the plant parameter vector. The null-space component does not contribute to the generation of the control force which drives the plant. If a control input that does not include the null-space component can be generated, efficient control is achieved in the sense of minimizing the norm of the control input. In the proposed method, a model reference adaptive control (MRAC) method was used to design the control system that compensates for the null-space component even if the plant parameters are unknown. The effectiveness of the proposed null-space compensation control system was shown by numerical examples.