Discrete Linear Time-invariant Research Articles

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.

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.

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.

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.

Monotone convergence rate of norm‐optimal‐gain‐arguable iterative learning control for LDTI systems

AbstractFor a class of repetitive linear discrete‐time‐invariant (LDTI) systems, this paper exploits a norm‐optimal‐gain‐arguable iterative learning control (NOGAILC) scheme. For the scheme, the sequential performance index is additive quadratic norms of the tracking error vector and the incremental input vector, while the argument is the iteration‐time‐variable learning‐gain vector. To balance the importance of the incremental input vector and the tracking error vector, the sequential tuning factors are concerned. By means of matrix theory, the learning‐gain vector and the convergence rate are explicated that indicate that both the tracking error and the optimal performance index are linearly monotonically convergent and the NOGAILC with a smaller tuning factor delivers a faster convergence. Further, for the system parameters existing uncertainties, a quasi‐NOGAILC profile is established and the time‐domain rigorous derivation conveys that the quasi‐scheme is linearly monotonically convergent for the parameters uncertainties within a wider range. Numerical simulations testify the validity and the effectiveness.

Practical Computational Approach for the Stability Analysis of Fuzzy Model-Based Predictive Control of Substrate and Biomass in Activated Sludge Processes

This paper presents a procedure for the closed-loop stability analysis of a certain variant of the strategy called Fuzzy Model-Based Predictive Control (FMBPC), with a model of the Takagi-Sugeno type, applied to the wastewater treatment process known as the Activated Sludge Process (ASP), with the aim of simultaneously controlling the substrate concentration in the effluent (one of the main variables that should be limited according to environmental legislations) and the biomass concentration in the reactor. This case study was chosen both for its environmental relevance and for special process characteristics that are of great interest in the field of nonlinear control, such as strong nonlinearity, multivariable nature, and its complex dynamics, a consequence of its biological nature. The stability analysis, both of fuzzy systems (FS) and the very diverse existing strategies of nonlinear predictive control (NLMPC), is in general a mathematically laborious task and difficult to generalize, especially for processes with complex dynamics. To try to minimize these difficulties, in this article, the focus was placed on the mathematical simplification of the problem, both with regard to the mathematical model of the process and the stability analysis procedures. Regarding the mathematical model, a state-space model of discrete linear time-varying (DLTV), equivalent to the starting fuzzy model (previously identified), was chosen as the base model. Furthermore, in a later step, the DLTV model was approximated to a local model of type discrete linear time-invariant (DLTI). As regards the stability analysis itself, a computational method was developed that greatly simplified this difficult task (in a local environment of an operating point), compared to other existing methods in the literature. The use of the proposed method provides useful conclusions for the closed-loop stability analysis of the considered FMBPC strategy, applied to an ASP process; at the same time, the possibility that the method may be useful in a more general way, for similar fuzzy and predictive strategies, and for other complex processes, was observed.

Static output feedback stabilization of discrete time linear time invariant systems based on approximate dynamic programming

This study proposes a method for the static output feedback (SOF) stabilization of discrete time linear time invariant (LTI) systems by using a low number of sensors. The problem is investigated in two parts. First, the optimal sensor placement is formulated as a quadratic mixed integer problem that minimizes the required input energy to steer the output to a desired value. Then, the SOF stabilization, which is one of the most fundamental problems in the control research, is investigated. The SOF gain is calculated as a projected solution of the Hamilton-Jacobi-Bellman (HJB) equation for discrete time LTI system. The proposed method is compared with several examples from the literature.

On Topological Properties of the Set of Stabilizing Feedback Gains

This article presents a fairly complete account on various topological and metrical aspects of feedback stabilization for single-input single-output (SISO) continuous and discrete time linear time-invariant (LTI) systems. In particular, we prove that the set of stabilizing output feedback gains for a SISO system with n states have at most [n/2] connected components. Furthermore, our analysis yields an algorithm for determining intervals of stabilizing gains for general continuous and discrete LTI systems; the proposed algorithm also computes the number of unstable roots in each unstable interval.

On the Control Design for Linear Time–Invariant Systems with Moments Constraints of Disturbances in Anisotropy–based Theory

Abstract In this paper, the design procedure of anisotropic suboptimal dynamic output regulator for linear discrete time-invariant (LDTI) system is considered. The mean anisotropy level of exogenous disturbance is bounded by the certain positive value. Additional constraints of exogenous disturbance are connected with the first and the second moments. The anisotropic suboptimal regulator design problem is formulated in terms of solvability of linear matrix inequalities (LMI) under convex constraints.

Optimal control of a wind turbine with digital fluid power transmission

Digital fluid power (DFP) technology may lead to a paradigm shift in large-scale transmission systems in, e.g., wind and wave energy. Therefore, the development of applicable control algorithms is of major importance, but is complicated by the non-smooth behavior of the DFP displacement machines. The power throughput of a full stroke operated digital displacement machine is quantized by the number of pressure chambers. The dynamics of each pressure chamber may be described by highly nonlinear continuous differential equations, whereas the input is discretely updated and binary (active or inactive). This paper contributes with a feedback control strategy for a digital displacement machine, where the binary inputs are handled by a pulse density modulator. The paper presents a linearization method of handling the many nonlinearities and thereby enabling the use of Discrete Linear Time Invariant (DLTI) control theory. The control strategy is validated for control of a digital fluid power wind turbine transmission, where both a deterministic and a stochastic optimal controllers are synthesized. The study is based on the NREL 5-MW reference wind turbine, where its model is combined with a nonlinear model of the DFP transmission and full-field flow wind profiles are used for a realistic performance evaluation scenario. By simulation, it is found that the performance of the optimal controllers using the DFP transmission is similar to that of the NREL controller using a conventional transmission.

Stochastic iterative learning control for discrete linear time-invariant system with batch-varying reference trajectories

In this paper, we present adaptive iterative learning control (ILC) schemes for discrete linear time-invariant (LTI) stochastic system with batch-varying reference trajectories (BVRT). If reference trajectories change every batch, ILC shows a different convergence property from that of the identical reference trajectory. First, we derive the convergence property and propose deterministic adaptive ILC combined with iterative learning identification for LTI system with BVRT. If the state noise and measurement noise exist, convergence rate and tracking performance are degraded because the controller considers the difference arising from the noise as tracking error. To deal with such a problem, we propose two approaches. The first is based on a batch-domain Kalman filter, which uses the difference between the current output trajectory and the next reference trajectory as a state vector, while the second is based on a time-domain Kalman filter. In the second approach, the system is identified at the end of each batch in an iterative fashion using the observer/Kalman filter identification (OKID). Then, the stochastic problem is handled using Kalman filter with a steady-state Kalman gain obtained from the identification. Therefore, the second approach can track the reference trajectories of discrete LTI stochastic system using only the input–output information. Simulation examples are provided to show the effectiveness of the proposed schemes.

Defining a Pseudo-Metric Topology on Linear Dynamic Systems

This paper defines a (pseudo) metric topology on the space of stable discrete linear time-invariant (LTI) dynamic systems. The article seeks to present a solution to the problem of comparing two LTI systems or, more formally, to find a metric for the space of stable discrete LTI dynamic systems. To this effect, by comparing the performance of two Kalman filters designed for two dynamic systems, a distance-like pseudo-norm between two systems is developed. The defined metric topology can be exploited to select the closest model, among several possible models Ps, all of which are known, to an observed data sequence modeled as P*. Numerical simulations are provided illustrating the efficacy of the metric derived.

A Relationship between Parameters of Two Representations of Discrete LTI ARMA System Model and the Corresponding Device Fingerprints

Device fingerprints has potential to identify the source of digital communication information. Although identifying transmitters with their fingerprints remains an arduous question, fusion identification with multiple device fingerprints is a feasible methodology. In this paper, a relationship between parameters of two representations of discrete-time linear time-invariant (LTI) autoregressive moving average (ARMA) system model is derived which has definite physical meaning. With the relationship, a novel linear device fingerprints is proposed which may enlarge the inter-class distance of the transmitters to be identified. The proposed device fingerprints can be used for fusion identification of network devices and the corresponding ARMA systems especially for that with slight differences.

A Lifting Based Approach to Observer Based Fault Detection of Linear Periodic Systems

In this note, an approach based on lifting is introduced for the observer based fault detection (FD) of linear discrete-time periodic systems. The main contribution of this note is to give an analytic expression to transform the linear time-invariant (LTI) observer based residual generator designed based on the lifted LTI reformulation of the periodic system into a periodic observer based residual generator to reduce detection delay. The result enables a systematic transfer of the existing observer based FD methods for discrete LTI systems to discrete-time periodic systems.

Kalman filtering over unreliable communication networks with bounded Markovian packet dropouts

AbstractWe address the peak covariance stability of time‐varying Kalman filter with possible packet losses in transmitting measurement outputs to the filter via a packet‐based network. The packet losses are assumed to be bounded and driven by a finite‐state Markov process. It is shown that if the observability index of the discrete‐time linear time‐invariant (LTI) system under investigation is one, the Kalman filter is peak covariance stable under no additional condition. For discrete LTI systems with observability index greater than one, a sufficient condition for peak covariance stability is obtained in terms of the system dynamics and the probability transition matrix of the Markov chain. Finally, the validity of these results is demonstrated by numerical simulations. Copyright © 2008 John Wiley & Sons, Ltd.

On the Optimization of an Adaptive State Estimation of Dynamic Systems in the Presence of Time-varying Outliers

The problem of recursive estimation of a state of dynamic systems in the presence of time-varying outliers in observations to be processed has been considered. A learning phase used in the state estimation is investigated, assuming that the observations of a noisy output signal and that of a training one are given. A technique based on robust filtering by means of a bank of parallel Kalman filters and on the procedure of optimization of the state estimation itself is used, choosing, at each time moment, a current estimate, that ensures a minimal absolute deviation from the current value of the teaching signal. An approach, based on the relation between the mean squared deviation of state estimates from the true state and innovation sequence variance as well as on the fact that both variables achieve their minimum for the same filter from the respective Kalman filter bank, is proposed here for a working phase, where a training signal will be absent. The recursive technique based on an adaptive state estimation with optimization procedure is worked out. The results of numerical simulation of the linear discrete-time invariant (LTI) system (56) by computer using a bank, consisting of Kalman filters are given (Figs. 1-5).

A Study on the Optimal Cost for the $H_infty$ Problem of Discrete Linear Periodically Time-Varying Systems

In this paper, we propose a new method to obtain the optimal cost (the infimum of the achievable $H_\infty$ norm of a closed-loop system) for discrete linear periodically time-varying (LPTV) systems. This method reduces via the lifting technique the original problem to the $H_\infty$ problems of linear time-invariant (LTI) systems without the causality constraint, and therefore the optimal cost can be easily calculated. It is one of the primary advantages over other existing methods. We also show, by applying our method to LTI systems, that the optimal cost for the $H_\infty$ problem of discrete LTI systems cannot be improved even if we use a class of noncausal controllers. An intriguing implication of this fact is further shown in the context of causal controllers.

Identification/reduction to a balanced realization via the extended impulse response gramian

A new identification/reduction algorithm for linear, discrete time-invariant (LDTI) systems is proposed which is based on the extended impulse response gramian first defined here for LDTI systems. The reduction algorithm employs singular value decomposition to retain states corresponding to dominant singular values of these gramians. The proven properties of the reduced order models include convergence to a balanced realization with conditional controllability, observability, and asymptotic stability. A suboptimal property of the model (in minimizing an impulse response error norm) is also demonstrated. The proposed technique can handle impulse response data of deterministic or stochastic nature for system identification application.