Discrete-time Linear Time-invariant Systems Research Articles

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.

Interval Estimation Methods for Discrete-Time Linear Time-Invariant Systems

This paper investigates interval estimation methods for discrete-time linear time-invariant systems. We propose a novel interval estimation method by integrating robust observer design with reachability analysis. By introducing $H_{\infty }$ design into interval estimation, the proposed method is effective in improving the accuracy of interval estimation. Moreover, the comparisons and relationships between the existing methods and the proposed method are discussed in detail. Finally, simulation results are presented to demonstrate the effectiveness of the proposed method.

Explicit model predictive controller under uncertainty: An adjustable robust optimization approach

Conventional model predictive control (MPC) involves solving an optimization problem online to determine the control actions that minimize a performance criterion function. The high computational expense associated with conventional MPC may make its application challenging for large-scale systems. Explicit MPC has been developed to solve the optimization problem offline. In this work, adjustable robust optimization is used to obtain the explicit solution to the MPC optimization problem offline for discrete-time linear time invariant systems with constraints on inputs and states. In the robust model formulation, an uncertain additive time-varying error is introduced to account for model uncertainty resulting from plant-model mismatch caused by un-measurable disturbances or process nonlinearities. The explicit solution is an optimal time-varying sequence of feedback control laws for the control inputs parameterized by the system’s states. The control laws are evaluated in a time-varying manner when the process is online using state measurements. This study shows that the resulting control laws ensure the implemented control actions maintain the system states within their feasible region for any realizations of the uncertain parameters that are within the user-defined uncertainty set. Three case studies are presented to demonstrate the proposed approach and to highlight the benefits and limitations of this method. The proposed framework advances the development of Explicit MPC by further expanding its application to large as well as nonlinear systems.

Minimal-delay FIR delayed left inverses for systems with zero nonzero zeros

This paper considers finite-time input reconstruction for discrete-time linear time-invariant systems in the case where the initial condition is unknown. There are three main results. First, a specific construction of finite-impulse-response (FIR) delayed left inverse with the minimal delay for systems with zero nonzero zeros is presented. Next, it is shown that, in the presence of an arbitrary unknown initial condition, finite-time input reconstruction is possible using a delayed left inverse H if and only if H is FIR. Finally, it is shown that a transfer function with full column normal rank has an FIR delayed left inverse with the minimal delay if and only if the system has zero nonzero zeros.

Synchronization of linear systems via relative actuation

Synchronization in networks of discrete-time linear time-invariant systems is considered under relative actuation. Neither input nor output matrices are assumed to be commensurable. A distributed algorithm that ensures synchronization via dynamic relative output feedback is presented.

A note on approximating the nearest stable discrete-time descriptor systems with fixed rank

Consider a discrete-time linear time-invariant descriptor system Ex(k+1)=Ax(k) for k∈Z+. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system Eˆx(k+1)=Aˆx(k) with rank(Eˆ)=r. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.

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.

One-Shot Verification of Dissipativity Properties From Input–Output Data

Due to their relevance in controller design and robustness analysis, we present a novel framework to find and verify dissipativity properties for discrete-time linear time-invariant systems from only one input-output trajectory. Introducing a new data-driven formulation for dissipativity, we obtain necessary and sufficient conditions for dissipativity based only on a definiteness condition on a single data-dependent matrix. Furthermore, we provide a promising relaxation in the case of measurement noise, which, as illustrated in numerous numerical examples, works remarkably well.

An Event-Triggered Output-Based Model Predictive Control Strategy

In many of today's control systems, the control loop is closed over a communication network with limited resources that requires explicit consideration of the network utilization in the controller design. At first, this paper deals with the decentralized control of discrete-time linear time-invariant systems consisting of multiple dynamically coupled subsystems subject to state and input constraints where the sensors send the measurements over a network to the controllers. A novel design for an event trigger mechanism for reducing the communication effort over the network in combination with an output-based model predictive controller is proposed. The approach is enhanced with another event trigger for a configuration where additionally the controllers send the inputs to the actuators over the network. Furthermore, it is discussed how the scheme can be extended for the control of sampled-data systems and how the network imperfections bounded time-varying delays, and sampling intervals can be considered in the design. Recursive feasibility of the control law and exponential stability of the origin are guaranteed for all considered setups. Illustrative examples demonstrate the clear reduction of the communication effort compared to a time-triggered output-based model predictive controller.

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.

Feedback control using a strategic sensor

ABSTRACT A dynamic estimation and control problem with a strategic sensor is considered. The strategic sensor may provide corrupted messages about the state measurements of a discrete-time linear time-invariant dynamical system to the system operator (or the controller). The system operator then uses this information to construct an estimate of the state of the system (and perhaps private variables of the sensor). The estimate is used to control the system to achieve the operator's desired objective. The problem is formulated as a game, which might be conflicting to that of the strategic sensor. An equilibrium of the game is computed and its properties are investigated.

Mean Square Stabilization Over SNR-Constrained Channels With Colored and Spatially Correlated Additive Noises

This note addresses the problem of stabilizing a multi-input multi-output (MIMO) discrete-time linear time invariant (LTI) system over a MIMO additive noise channel. To treat this problem, we assume that the communication link between the plant output and the controller input consists of multiple additive colored mutually correlated noise channels subject to independent signal-to-noise ratio (SNR) constraints. We derive analytical conditions for which mean square stabilization (MSS) can be achieved under such constraints. We also formulate numerical methods to test these conditions when the noise is white and correlated. Moreover, for simpler plant models, a characterization of the set of power constraints compatible with MSS is obtained. Our results show that the frequency response of the spectral factor related to the channel noise affects the minimum SNR for stability depending mostly on the unstable poles and their directions. This is aggravated by the existence of nonminimum phase zeros and higher relative degree of the plant. On the other hand, we detect that systems under highly correlated noise show lower SNR requirements for stability compared to ones with independent noise channels. Numerical simulations are provided to illustrate the theoretical results.

Linear system security—Detection and correction of adversarial sensor attacks in the noise-free case

We address the problem of attack detection and attack correction for multi-output discrete-time linear time-invariant dynamical systems under sensor attack. More specifically, we focus on the situation where adversarial attack signals are added to some of the system’s output signals. Algorithms for sensor attack detection and correction are presented using the notion of ‘security index’ which characterizes the system’s vulnerability against such attacks. The results are illustrated by examples involving multiple sensors.

Mean-Square Stabilization Over Gaussian Finite-State Markov Channels

This paper studies the mean-square stabilization problem of discrete-time linear time-invariant systems over Gaussian finite-state Markov channels, which suffer from both signal-to-noise ratio constraint and correlated channel fading modeled by a Markov process. The existence of a fundamental limitation for mean-square stabilization is first established. Then, sufficient stabilization conditions under a time-division multiple access (TDMA) communication scheme are derived in terms of the stability of a Markov jump linear system. Moreover, we present a necessary and sufficient condition for mean-square stabilization of 2-D systems controlled over power-constrained Markov lossy channels. Furthermore, improved sufficient stabilization conditions are derived based on an adaptive TDMA communication scheme for general high-dimensional systems, which achieves a larger stabilization region than the TDMA communication scheme.

Discrete Razumikhin-type stability theorems for stochastic discrete-time delay systems

By using the Razumikhin-type technique, for stochastic discrete-time delay systems, this paper establishes the discrete Razumikhin-type theorems on the pth moment stability, the global pth moment stability and the pth moment exponential stability, respectively. The almost sure exponential stability is also investigated by using the pth moment exponential stability and the Borel–Cantelli lemma. As the applications of t he established theorems, stability of a special class of stochastic discrete-time delay systems, synchronization of the stochastic discrete-time delay dynamical networks and stabilization of a stochastic discrete-time linear delay time invariant system are examined.

A trust-region method for optimal H2 model reduction of discrete-time dynamical systems

ABSTRACTIn this paper, we present an optimal model reduction method on the basis of the trust-region technique for linear time-invariant discrete-time dynamical systems. First, based on the poles and residues, the error norm for single-input and single-output discrete-time systems is investigated, which leads to the error gradient and Hessian. Next, for multiple-input and multiple-out discrete-time systems, the gradient and Hessian of the error norm are accordingly derived. Then, the error gradient and Hessian are employed to establish the trust-region method for optimal model reduction. Moreover, it is shown that the proposed method can produce a decreasing sequence. The construction of the state space realization of the reduced order system is studied concerning the divisions of the resulting residues. Model reduction is investigated for nonlinear discrete-time system. Finally, an illustrative example is used to demonstrate the performance.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.

Closed-Form Impulse Responses of Linear Time-Invariant Systems: A Unifying Approach [Lecture Notes

In many signal processing applications, filtering is accomplished through linear time-invariant (LTI) systems described by linear constant-coefficient differential and difference equations since they are conveniently implemented using either analog or digital hardware [1]. An LTI system can be completely characterized in the time domain by its impulse response or in the frequency domain by its frequency response, which is the Fourier transform of the system's impulse response. Equivalently, using the Laplace transform [or the z-transform in the case of discrete-time (DT) systems] as a generalization of the Fourier transform, any continuous-time (CT) or DT LTI system can be characterized by its transfer function (or system function) in the s-domain or the z-domain, respectively. In this article, we explain how to find impulse responses of LTI systems described by differential and difference equations directly in the time domain without resorting to any transform methods or recursive procedures.

An upper bound of mean-square error in state estimation with quantized measurements

In this paper, we study the state estimation for a linear time-invariant (LTI) discrete-time system with quantized measurements. The quantization law under consideration has a time-varying data rate. To cope with nonlinearities in quantization laws and to analyse stability in the state estimation problem, a Kalman-filter-based sub-optimal state estimator is developed and an upper bound of its estimation error covariance is minimized. It turns out that, to guarantee the convergence of the upper bound, the averaged data rate of the quantization law must be greater than a minimum rate. This minimum data rate for the quantization law is presented in terms of the poles of the system and design parameters in the state estimator. Numerical examples are presented to illustrate the results in this work.

<i>H</i><sub>2 </sub>Analysis of LTI Discrete-Time Systems via Conversion to Externally Positive Systems

In this paper, we explore H2 analysis techniques of general, not necessarily positive, discrete-time linear time-invariant (LTI) systems by means of positive system theory. While externally positive systems are a particular class of systems whose impulse responses take only nonnegative values, we often encounter such impulse responses even in the analysis of general LTI systems. A typical example is the computation of the H2 norm where we focus on squared impulse responses. In view of this observation, we first clarify that we can construct an SISO externally positive system whose impulse response is given by the square of the impulse response of a given SISO LTI system. This system conversion allows us to reduce the H2 norm computation problem of a general LTI system to the l∞-induced norm computation problem (or l1 problem in short) of the resulting externally positive system. It also turns out that this problem reduction enables us to derive various alternative formulas for the H2 norm of general discrete-time LTI systems.