Linear Discrete Time-invariant Systems Research Articles

Positive State Controllability of Discrete Linear Time-Invariant Systems

Abstract Positive state controllability is the controllability of systems where the state is positive and the input remains in ℝn. Under some conditions, we established a relation between the reachability map of systems with only the positive state and the reachability map of a related positive system where the state and input are both positive. Using this connection, necessary and sufficient conditions are obtained for the positive state reachability of discrete linear time-invariant (LTI) systems, and then we deduced the positive state controllability. These conditions are evaluated over some numerical examples that support the theoretical results.

An anomaly affected discrete LTI systems: a moving horizon approach for estimating position and temperature measurements

A real-time system may be subjected to various anomalies that can affect the quality of the observations. The main motivation of the article arises from the need in addressing challenges posed by the presence of anomalies in discrete linear time-invariant (LTI) systems with a focus on the estimation processes, in the context of position and temperature measurements. The proposed approach leverages the properties of discrete LTI systems and takes advantage of the predictive capabilities of the moving horizon strategy (MHS). It operates recursively updating estimates of new measurement while fairly considering its past estimates that occur within the window of the moving horizon. The estimation framework will be designed to handle disturbances and provide robust estimates, to ensure the effectiveness of the system. In order to validate the proposed approach simulation studies were conducted on different and only scenarios in different order LTI system. Comparative studies with different estimation techniques demonstrate the capability of the proposed approach in terms of performance and efficiency. The proposed approach can be applied to systems with changing system dynamics. Future research may be conducted to utilize this strategy in other domains to mitigate anomalies while enhancing performance.

Inherent attack tolerance properties of model predictive control under DoS attacks

We consider the inherent attack tolerance properties of resilient model predictive control (R-MPC) for cyber–physical systems (CPSs) modeled by discrete linear time-invariant (LTI) systems subjected to limited disruptions. In this paper, the relationship between the maximum allowable duration of Denial-of-Service (DoS) attacks, namely attacks that disrupt both sensor to controller (S–C) and controller to actuator (C–A) communication channels, and the upper bound of disturbances is deduced at length. Moreover, to achieve robust recursive feasibility and closed-loop stability of the system, we discuss that MPC with state, control and terminal constraints has a certain degree of inherent attack tolerance concerning the DoS attack duration and parameters like system matrices and the nominal terminal set. Finally, numerical simulation is given to substantiate the effectiveness of the proposed approach.

Design of Wide‐Beam Leaky‐Wave Antenna Arrays Based on the Bilinear Transformation of IIR Digital Filters and the Z Transform

Abstract In the pioneering work, the radiation diagrams of leaky wave antenna arrays can achieve attenuation nulls and gains at specific angles by manually placing the zeros and the poles in the Z domain of the corresponding discrete linear time‐invariant (LTI) system. This handcrafted design procedure does not allow radiation diagrams with wide beams since the interaction between poles involved in the wide beam and their corresponding leaky modes cannot be easily handled. To overcome this limitation, this paper describes a novel method for designing radiation diagrams of leaky‐wave antenna arrays based on the theory of IIR discrete filters. The proposed method relies on the design of discrete filters with the prototypes of analog low‐pass filters defined by Butterworth and Chebyshev type I polynomials, whose roots along with the bilinear transformation provide the location of the poles and the zeros of the discrete LTI system and, therefore, the parameters of the leaky‐wave antenna array. Results with different designs and a comparison with other approaches show the utility and effectiveness of this novel method to design wide‐beam leaky‐wave antenna arrays.

Confidence set-based analysis of minimal detectable fault under hybrid Gaussian and bounded uncertainties

This paper proposes a confidence set-based computational method of minimal detectable fault (MDF) based on the confidence set-separation condition between the healthy and faulty residual sets for discrete linear time-invariant systems. The state-estimation-error dynamics for the analysis of MDF under hybrid random and bounded uncertainties is divided into two sub-dynamics. The first sub-dynamics is only affected by the bounded uncertainties. Under the precondition of Schur stability, an outer-approximation of minimal robust positively invariant set with any given precision is obtained. While the second sub-dynamics is only affected by the random uncertainties following the Gaussian distributions. It is proved that the behavior of the second sub-dynamics at steady stage also follows a certain Gaussian distribution, which can be bounded by confidence zonotopes given a proper confidence level. MDF for actuator and sensor faults can be obtained by solving a non-convex optimization problem to minimize the magnitude of fault subjected to the residual set-separation constraints, which is equivalent to compute a distance from the origin to hyperplane along a fixed direction by exploiting the geometric property. At the end of this paper, a two-link manipulator model and a vehicle model are used to verify the effectiveness of our proposed method.

Axiomatic Foundations of Anisotropy-Based and Spectral Entropy Analysis: A Comparative Study

An axiomatic development of control systems theory can systematize important concepts. The current research article is dedicated to the investigation and comparison of two axiomatic approaches to the analysis of discrete linear time-invariant systems affected by external random disturbances. The main goal of this paper is to explore axiomatics of an anisotropy-based theory in comparison with axiomatics of a spectral entropy approach in detail. It is demonstrated that the use of the spectral entropy approach is mathematically rigorous, which allows one to prove that the minimal disturbance attenuation level in terms of an anisotropy-based control theory provides the desired performance that is not only for ergodic signals. As a result, axiomatics of the spectral entropy approach allows one to rigorously prove that anisotropy-based controllers can be used to guarantee the desired disturbance attenuation level, not only for stationary random sequences, but also for a wider set of input random signals.

A Single-Tube Robust Model Predictive Control Method Based on ε-Approximation

This paper aims to solve the problem of the robust model predictive control, the contradiction of the system robustness, and the conservative terminal constraint set range. A robust model predictive control (RMPC) method based on an ε-approximation single-tube set is proposed. We construct a single-tube RMPC structure for linear discrete time invariant systems with additive disturbances. To this structure, we add the state estimation and state feedback to improve the convergence rate. Furthermore, we use the ε-approximation to estimate the terminal constraint set with less conservatism, thus improving the robustness. Then, we conduct a stability analysis of the ε-approximation single-tube RMPC system. The simulation results demonstrate the stability and interference allowance advantages of the proposed method.

Minimal Detectable and Isolable Faults of Active Fault Diagnosis

This article aims to compute guaranteed minimal detectable and isolable faults of set-based active fault diagnosis methods for discrete linear time-invariant systems. First, guaranteed detectable and isolable faults of set-based active fault diagnosis methods are defined in this article. Second, an augmented vector is defined to integrate the effect of both inputs and faults such that the couplings of inputs and faults can be handled effectively. Third, guaranteed minimal detectable and isolable faults are computed by solving a mixed integer quadratic fractional programming problem under a proposed theoretic framework. At the end of this article, an example is used to illustrate the effectiveness of the proposed solution.

Algorithm for adaptive observation based on method of instrumental variables

When the input signal and the output value of the object of control cannot be measured accurately, the state vector is estimated. The instrumental variables (IVs) method is a commonly used parameter estimation method [1-10]. The task of adaptive observation is to create state observers containing parameter estimators. In adaptive observers, the matrices A and b or c (depending on the chosen canonical state-space representation form) are assumed to be unknown. In the monitoring process, parameter estimation is performed, the unknown matrices are determined, and then the state vector is calculated. The paper aims to present a non-recurrent adaptive observation algorithm for SISO linear time-invariant (LTI) discrete systems. The algorithm is based on the instrumental variables (IVs) method, and the adaptive state observer (ASO) estimates the parameters, the initial and the current state vectors of the discrete system. The algorithm's workability and effectiveness are proved by using simulation data in MATLAB/Simulink.

Design of False Data Injection Attacks in Cyber-Physical Systems

Cyber-physical systems (CPSs) are more vulnerable to false data injection (FDI) attacks. An attacker can launch an FDI attack at any chosen location in the CPS. It is required to analyze the system’s behaviour in the presence of FDI attacks rather than analyzing attacks at a specific location. In this paper, we consider security in CPS due to FDI attacks at a single as well as multiple locations. The CPS is modeled as a discrete linear time-invariant system with white Gaussian noise. The system is equipped with a Kalman filter as a state estimator and a Chi-square detector for attack detection. We studied FDI attacks at the sensor, actuator, and physical system. Attackers may guess the system parameters in the worst-case and remain undetectable by carefully designing the attack sequences. The attacker always attempts to increase the state estimation error in the system. Based on the system model, we have identified seven kinds of FDI attacks and analyzed their effects on the system’s security. It is found that the attacker can produce bounded errors in some cases and unbounded errors in some others. Simulation of the attacks on the system model is performed through numerical examples to illustrate their effectiveness.

A Novel Fast Iterative Learning Control for Linear Discrete Systems with Parametric Disturbance and Measurement Noise

Precise industrial control technology is constantly in need of accurate and strong control. Error convergence for a typical linear system is very minimal when using a conventional iterative learning control strategy. This study develops a quick iterative learning control law to address this issue. We have presented a new PD iterative learning control approach which is basically grounded on backward error and control parameter rectification for a class of linear discrete time-invariant (LDTI) systems. We have deliberated the repetitive system, which has constraint disturbance and measurement noise. First, we have developed a form of the faster learning law along with a full explanation of the algorithm’s control factor generation process. And then, using the vector method in conjunction with the theory of spectral radius, sufficient conditions for the algorithm’s convergence are introduced for parameter estimation with no noise, parameter uncertainty but excluding the noise, parameter uncertainty with small perturbations, and noise in four different scenarios. Eventually, results show that convergence depends on the control law’s learning factor, the correction term, the factor of association, and the learning interval. Ultimately, the simulation results indicate that suggested approach has a faster error convergence as compared with classical PD algorithm.

SYNTHESIS AND ANALYSIS OF LINEAR, DISCRETE AND TIMEINVARIANT SYSTEMS, USED IN THE FIELD OF COMMUNICATIONS, USING MATLAB AND SIGNAL PROCESSING TOOLBOX

Synthesis and Analysis of Linear Discrete Time-Invariant Systems Applied in Communications Using MATLAB and Signal Processing Toolbox: The paper presents the main steps in the process of synthesizing and analyzing linear discrete time-invariant systems applied in the field of communications. The simulation results of the systems implementing pre-emphasis and de-emphasis and removing DC offset are given in the paper such as impulse responses, frequency responses and pole-zero plots using MATLAB and Signal Processing Toolbox.

Output feedback controller design for discrete LTI systems with polytopic uncertainty via direct searching of the design space

AbstractThis paper concerns static output feedback stabilization of polytopic discrete linear time‐invariant (LTI) systems. The previous related studies were mainly based on linear matrix inequality (LMI) approaches which are naturally conservative. In this paper, a novel design algorithm is presented that iteratively partitions a primary design space to subspaces. Then, by assessing stabilizability status of each generated subspace, the algorithm determines the total stabilizable parts and removes the undesired parts of the design space. Mathematical theories are developed to determine the total de‐stabilizability or stabilizability of a given subspace. These subspaces' properties are detected through checking the existence of critical polynomials (which have roots on the unit circle of the complex plane) on their exposed edges. By omitting the undesired parts of the design space, the algorithm just searches the desired parts which are far smaller than the primary design space. This strategy improves the feasibility performance of the algorithm. Some illustrating examples are provided to show the steps of the design algorithm. Furthermore, 100 random models are generated to evaluate the feasibility performance of the proposed algorithm as compared to some existing methods. The results reveal the superiority of the proposed algorithm.

Positive output controllability of linear discrete–time invariant systems

Abstract This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.

Multiparametric/explicit nonlinear model predictive control for quadratically constrained problems

Explicit model predictive control is an established methodology for the offline determination of the optimal control policy for linear discrete time-invariant systems with linear constraints. Nevertheless, nonlinearities in the form of quadratic constraints naturally appear in process models or are imposed for stability purposes in model predictive control formulations. In this manuscript, we present the theoretical developments and propose an algorithm for the exact solution of explicit nonlinear model predictive control problems with convex quadratic constraints. Our approach is based on a second-order Taylor approximation of Fiacco’s Basic Sensitivity Theorem, which allows for the existence and the analytic derivation of the optimal control actions. The complete exploration of the parameter space is founded on an active set strategy, which employs a pruning criterion to eliminate infeasible active sets. Based on that, the optimal map of solutions is constructed along with the corresponding control actions. The proposed strategy is applied to an explicit nonlinear model predictive control problem with an ellipsoidal terminal set, and comparisons with approximate solutions are drawn to demonstrate the benefits of the presented approach. Furthermore, as a practical application, the optimal operation of a chemostat in the presence of disturbances is exhibited.

Boundedness Condition for Anisotropic Norm of Linear Discrete Time-invariant Systems with Multiplicative Noise

In this paper, the anisotropy-based analysis problem for the linear discrete time-invariant system with state-multiplicative noises is considered. The mean anisotropy of stationary input sequence is assumed to be bounded by given nonnegative value. For this setting, the boundedness condition for anisotropic norm of the system is obtained in terms of Riccati equations.

FINITE-TIME STABILITY IN MEAN FOR NABLA UNCERTAIN FRACTIONAL ORDER LINEAR DIFFERENCE SYSTEMS

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order [Formula: see text] are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.

Iterative learning control realized using an iteration-varying forgetting factor based on optimal gains

Iterative learning control with forgetting factor (ILCFF) is widely used in control engineering. However, choosing the optimal parameters of ILCFF to improve system-output characteristics has been a challenging issue for controller designers. This paper proposes an iterative learning control (ILC) algorithm that involves a variable forgetting factor based on optimal gains for a class of discrete linear time-invariant systems with aperiodic disturbances. The convergence of the algorithm is analyzed, and the necessary and sufficient condition for its convergence is derived in terms of proportional–integral–derivative coefficients. A design method based on optimal gains is established to determine the algorithm coefficients and to accelerate system convergence. Furthermore, the influence of the forgetting factor on both the system-output error and the scope of the proposed algorithm is analyzed. Additionally, the most suitable system type for the application of the forgetting factor is determined. The effectiveness of the algorithm is verified by performing a theoretical analysis and a case-based simulation. The proposed iteration-varying optimal forgetting-factor-based ILC algorithm undergoes fast convergence with a small system-output error. The findings disrupt the conventional view that the use of the forgetting factor increases system-output error. In fact, in a system with small trajectory and increased disturbances, the error induced by the forgetting factor may be smaller than that of the traditional optimal ILC algorithm.

Observer-based asymptotic active fault diagnosis: A two-layer optimization framework

This paper proposes a novel asymptotic online robust active fault diagnosis (AFD) framework for discrete linear time-invariant (LTI) systems using a bank of set-valued observers (SVO), each SVO designed to match a healthy/faulty actuator mode. Different from the existing set-based AFD methods, the proposed AFD method designs an input at each time instant by solving a two-layer optimization problem. At each time instant, an input is designed such that the output sets estimated by the SVOs at the next time instant keep away from each other as far as possible. With this idea, faults can be diagnosed after a sufficiently long period by designing and injecting inputs into the system online. The key point of the proposed AFD method is to establish a logic describing the process keeping a group of output estimation sets asymptotically away from each other as far as possible, which is done by formulating a two-layer optimization problem to simultaneously optimize the center distances and sizes of output estimation sets. At the end, a case study is used to illustrate the effectiveness of the proposed method.

Dual-Mode deadbeat [formula omitted] FIR filtering for discrete-Time systems

A new receding horizon (RH) finite impulse response (FIR) filter called the dual-mode deadbeat H2 FIR filter (DMDH2FF) is proposed for state estimation in discrete linear time-invariant (LTI) systems under external disturbances. The DMDH2FF is derived to minimize the H2 norm subject to the deadbeat condition in the disturbance-to-estimate channel, and it can manage the horizon size in two operation modes. The DMDH2FF has better robustness against temporary model uncertainty compared with a conventional H2 infinite impulse response filter. The performance of the DMDH2FF is demonstrated by an F-404 turbofan engine simulation.