Linear Time-invariant Dynamical Systems Research Articles

ON EFFICIENCY OF EKF FOR PARAMETER ESTIMATION OF LTI SYSTEM FROM NON-STATIONARY ACCELERATION RESPONSES

Parameters of the linear time invariant (LTI) dynamic system using extended Kalman filter (EKF) are identified in this work. The efficiency of EKF for parameter estimation of LTI system is studied. For this purpose, a three-story steel frame is used in the laboratory, and the recorded ground motion is applied to measure the acceleration response at different floor levels. Using these responses, the EKF-based predictor-corrector algorithm is used to identify the modal parameters. It has been observed that the EKF-based identification scheme can converge to different system matrices (i.e., mass and stiffness) in different experiments for the same structure. However, their eigen values (i.e., natural frequency and mode) remain the same.

Fault Identification in Distributed Sensor Networks Based on Universal Probabilistic Modeling.

This paper proposes a holistic modeling scheme for fault identification in distributed sensor networks. The proposed scheme is based on modeling the relationship between two datastreams by means of a hidden Markov model (HMM) trained on the parameters of linear time-invariant dynamic systems, which estimate the specific relationship over consecutive time windows. Every system state, including the nominal one, is represented by an HMM and the novel data are categorized according to the model producing the highest likelihood. The system is able to understand whether the novel data belong to the fault dictionary, are fault-free, or represent a new fault type. We extensively evaluated the discrimination capabilities of the proposed approach and contrasted it with a multilayer perceptron using data coming from the Barcelona water distribution network. Nine system states are present in the dataset and the recognition rates are provided in the confusion matrix form.

Adaptive disturbance attenuation via logic-based switching

The problem of attenuating unknown and possibly time-varying disturbances acting on a linear time-invariant dynamical system is addressed by means of an adaptive switching control approach. Given a family of pre-designed stabilizing controllers, a supervisory unit infers in real-time the potential behavior of each candidate controller and selects the one providing the best potential performance. To this aim, a set of test functionals is defined, which is shown to enjoy favorable inference properties under certain assumptions on the nature of the disturbances. Both persistent-memory and finite-memory test functionals are analyzed. Further, an implementation of the switching controller is proposed which always guarantees stability of the feedback loop, even if the disturbance characteristics are such that the switching is persistent. Simulation results are provided to show the effectiveness of the proposed method.

H ∞ and Sliding Mode Observers for Linear Time-Invariant Fractional-Order Dynamic Systems With Initial Memory Effect

The H∞ and sliding mode observers are important in integer-order dynamic systems. However, these observers are not well explored in the field of fractional-order dynamic systems. In this paper, the H∞ filter and the fractional-order sliding mode unknown input observer are developed to estimate state of the linear time-invariant fractional-order dynamic systems with consideration of proper initial memory effect. As the first result, the fractional-order H∞ filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the H∞ norm. Based on the extended bounded real lemma, the H∞ filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractional-order systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractional-order sliding-mode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractional-order H∞ filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.

A new estimation of the lower error bound in balanced truncation method

For a single-input/single-output (SISO) linear time-invariant dynamical system, the standard H∞-norm lower error bound of balanced truncation method is ‖G(s)−Gr(s)‖H∞≥σr+1, where σi,i=1,…,n, are the Hankel singular values of system in decreasing order. In this paper we provide a new estimation of the lower error, namely ‖G(s)−Gr(s)‖H∞≥max{σd,2|∑i∉Jsiσi|}, where si is the sign associated with the Hankel singular value σi in Ober’s canonical form. The subset J and the index d in the above inequality will be introduced in the paper. We show by means of an example that the new bound may be relevant in deciding which states need to be kept in the balanced truncation method, and that using the standard result does not always yield the best approximation.

Terminal bang-bang impulsive control of linear time invariant dynamic systems

For a linear time-invariant dynamic system with bang-bang impulsive actuators whose position (off, on negative, or on positive) is determined by the switching points in time, an algorithm is developed that makes it possible to use methods of modal control for solving the terminal control problem. This algorithm is used to control the rendezvous of space vehicles with constant thrust engines arranged along the axes of a Cartesian reference frame.

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.

On the analogue of the Isidori relative degree definition for linear dynamical MIMO systems

This paper deals with the inversion problem for linear time-invariant dynamical systems. Earlier in this formulation the discussed issue has been addressed and solved for invertible systems that meet the definition of Isidori relative degree. But in fact, this problem is well-posed for a wider range of linear dynamical systems. In this paper we examine a particular case based on a definition that is similar to the definition of relative degree by Isidori. Presented here columnwise relative degree and its properties allow to consider and effectively solve the inversion problem for invertible systems that meet this definition.

£1 Adaptive Control Design for Hysteresis Compensation within Piezoelectric Actuators

Hysteresis nonlinearity of piezoelectric actuator deteriorates its control performance. Unlike traditional model-based control scheme, this paper employs a novel £1 adaptive control algorithm without modeling of hysteresis to achieve a precise tracking control. The piezoelectric actuator system in question consists of models in the form of a linear time-invariant dynamic system in series with a differential hysteresis model. The hysteresis can be transformed into general uncertainties to the system. Then an £1 adaptive control algorithm can be applied. By implementing this control scheme on the piezoelectric actuator system and comparing its result to the result of a PID controller, the effectiveness of this controller is verified.

Parametric dominant pole algorithm for parametric model order reduction

Standard model order reduction techniques attempt to build reduced order models of large scale systems with similar input–output behavior over a wide range of input frequencies as the full system models. The method known as the dominant pole algorithm has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems as well as nonlinear time-delay systems. In this paper, we show that the dominant pole algorithm can be adapted for parametric systems where these parameters usually have physical meaning. There are two approaches for finding dominant poles. These algorithms are illustrated by the second order numerical examples.

A cognitive fault diagnosis system for distributed sensor networks.

This paper introduces a novel cognitive fault diagnosis system (FDS) for distributed sensor networks that takes advantage of spatial and temporal relationships among sensors. The proposed FDS relies on a suitable functional graph representation of the network and a two-layer hierarchical architecture designed to promptly detect and isolate faults. The lower processing layer exploits a novel change detection test (CDT) based on hidden Markov models (HMMs) configured to detect variations in the relationships between couples of sensors. HMMs work in the parameter space of linear time-invariant dynamic systems, approximating, over time, the relationship between two sensors; changes in the approximating model are detected by inspecting the HMM likelihood. Information provided by the CDT layer is then passed to the cognitive one, which, by exploiting the graph representation of the network, aggregates information to discriminate among faults, changes in the environment, and false positives induced by the model bias of the HMMs.

Model falsification using set-valued observers for a class of discrete-time dynamic systems: a coprime factorization approach

This article introduces a new method for model falsification using set-valued observers, which can be applied to a class of discrete linear time-invariant dynamic systems with time-varying model uncertainties. In comparison with previous results, the main advantages of this approach are as follows: The computation of the convex hull of the set-valued estimates of the state can be avoided under certain circumstances; to guarantee convergence of the set-valued estimates of the state, the required number of previous steps is at most as large as the number of states of the nominal plant; and it provides a straightforward nonconservative method to falsify uncertain models of dynamic systems, including open-loop unstable plants. The results obtained are illustrated in simulation, emphasizing the advantages and shortcomings of the suggested method. Copyright © 2013 John Wiley & Sons, Ltd.

A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients

In this article, a new technique for solving the linear time invariant dynamical system with fuzzy parameters is proposed. This approach is used to solve a fuzzy controlled system by using α-cuts and presentation of numbers in a more compact form by moving to the field of complex numbers. By considering a new feature of the Hamiltonian function and using the Pontryagin maximum principle, the solution of the fuzzy optimal controlled system is considered.

A new recursive method for the analysis of linear time invariant dynamic systems via double-term triangular functions (DTTF) in state space environment

This paper presents a new recursive method for system analysis via double-term triangular functions (DTTF) in state space environment. The proposed method uses orthogonal triangular function sets and proves to be more accurate as compared to single term Walsh series (STWS) method with respect to mean integral square error (MISE). This has been established theoretically and comparison of error with respect to MISE is presented for clarity. A numerical example is treated to establish the proposed method. Relevant curves for the solutions of states of the dynamic system are also presented with plots of percentage error for DTTF-based analysis.

Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

On scattering passive system nodes and maximal scattering dissipative operators

There is an extensive literature on a class of linear time-invariant dynamical systems called “well-posed scattering passive systems”. Such a system is generated by an operator S S which is called a scattering passive system node. In the existing literature such a node is typically introduced by first giving a list of assumptions which imply that S S is a system node and then adding an inequality which forces this system node to be scattering passive. Here we proceed in the opposite direction: we start by requiring that S S satisfies the passivity inequality and then ask the question, what additional conditions are needed in order for S S to be a system node? The answer is surprisingly simple: A necessary and sufficient condition for an operator S S to be a scattering passive system node is that S S is closed and maximal within the class of operators that satisfy the passivity inequality. In the absence of external inputs and outputs, this condition is identical to the standard condition which characterizes the class of operators which generate contraction semigroups on Hilbert spaces.

Nearest Linear Systems with Highly Deficient Reachable Subspaces

We consider the 2-norm distance $\tau_r(A,B)$ from a linear time-invariant dynamical system $(A,B)$ of order $n$ to the nearest system $(A+\Delta A_{\ast}, B+\Delta B_{\ast})$ whose reachable subspace is of dimension $r < n$. We first present a characterization to test whether the reachable subspace of the system has dimension $r$, which resembles and can be considered as a generalization of the Popov--Belevitch--Hautus test for controllability. Then, by exploiting this generalized Popov--Belevitch--Hautus characterization, we derive the main result of this paper, which is a singular value optimization characterization for $\tau_r(A,B)$. A numerical technique to solve the derived singular value optimization problems is described. The numerical results on a few examples illustrate the significance of the derived singular value characterization for computational purposes.

Basic principles and aims of model order reduction in compliant mechanisms

Abstract. Model order reduction appears to be beneficial for the synthesis and simulation of compliant mechanisms due to computational costs. Model order reduction is an established method in many technical fields for the approximation of large-scale linear time-invariant dynamical systems described by ordinary differential equations. Based on system theory, underlying representations of the dynamical system are introduced from which the general reduced order model is derived by projection. During the last years, numerous new procedures were published and investigated appropriate to simulation, optimization and control. Singular value decomposition, condensation-based and Krylov subspace methods representing three order reduction methods are reviewed and their advantages and disadvantages are outlined in this paper. The convenience of applying model order reduction in compliant mechanisms is quoted. Moreover, the requested attributes for order reduction as a future research direction meeting the characteristics of compliant mechanisms are commented.

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the systemʼs input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.

Multi-level substructuring combined with model order reduction methods

In many fields of engineering problems linear time-invariant dynamical systems (LTI systems) play an outstanding role. They result for instance from discretizations of the unsteady heat equation and they are also used in optimal control problems. Often the order of LTI systems is a limiting factor, since it becomes easily very large. As a consequence these systems cannot be treated efficiently without model reduction algorithms. In this paper a new approach for the combination of model order reduction methods and recent multi-level substructuring (MLS) techniques is presented. Similar multi-level substructuring methods have already been applied successfully to huge eigenvalue problems up to several millions of degrees of freedom. However, the presented approach does not make use of a modal analysis like former algorithms. Instead the original system is decomposed in smaller LTI systems which are treated with recent model reduction methods. Furthermore, the error which is induced by this substructuring approach is analysed and numerical examples based on the Oberwolfach benchmark collection are given in this paper.