Linear Time-invariant Dynamical Systems Research Articles

Intervehicle Transmission Rate Control for Cooperative Active Safety System

We propose an intervehicle communication framework for the cooperative active safety system (CASS) whose operation is based on the dissemination of each vehicle's state information through a wireless network. Such a CASS requires each subject vehicle to be aware of its surroundings, particularly of the motion and position of other vehicles in its proximity. In this paper, we assume that all vehicles are equipped with onboard communication devices. In such situations, the wireless channel is simultaneously shared by a large number of vehicles, and one of the most difficult challenges in designing CASS is to maintain real-time tracking accuracy of neighboring vehicles while avoiding network congestion and failure. To address this issue, we analyze the problem that multiple scalar linear time-invariant dynamical systems track each other over a multiaccess channel, and then, we propose a rate adaptation algorithm to distributively control the self-information broadcast behavior of each vehicle. The proposed algorithm uses a closed-loop control concept and accounts for the lossy channel. Simulation results show that, if the message generation rate is dynamically adjusted in an on-demand fashion, more accurate and robust tracking performance can be achieved under various traffic conditions.

State–space solution to the ℋ︁−/ℋ︁∞ fault‐detection problem

SUMMARYIn this paper we give an optimal state–space solution to the ℋ︁−/ℋ︁∞ fault‐detection (FD) problem for linear time invariant dynamic systems. An optimal ℋ︁−/ℋ︁∞ FD filter minimizes the sensitivity of the residual signal to disturbances while maintaining a minimum level of sensitivity to faults. We provide a state–space realization of the optimal filter in an observer form using the solution of a linear matrix inequalities optimization problem. We also show that, through the use of weighting filters, the detection performance can be enhanced and some assumptions can be removed. Two numerical examples are given to illustrate the algorithm. Copyright © 2011 John Wiley & Sons, Ltd.

Characterizing and Computing the ${\cal H}_{2}$ Norm of Time-Delay Systems by Solving the Delay Lyapunov Equation

It is widely known that the solutions of Lyapunov equations can be used to compute the H2 norm of linear time-invariant (LTI) dynamical systems. In this paper, we show how this theory extends to dynamical systems with delays. The first result is that the H2 norm can be computed from the solution of a generalization of the Lyapunov equation, which is known as the delay Lyapunov equation. From the relation with the delay Lyapunov equation we can prove an explicit formula for the H2 norm if the system has commensurate delays, here meaning that the delays are all integer multiples of a basic delay. The formula is explicit and contains only elementary linear algebra operations applied to matrices of finite dimension. The delay Lyapunov equations are matrix boundary value problems. We show how to apply a spectral discretization scheme to these equations for the general, not necessarily commensurate, case. The convergence of spectral methods typically depends on the smoothness of the solution. To this end we describe the smoothness of the solution to the delay Lyapunov equations, for the commensurate as well as for the non-commensurate case. The smoothness properties allow us to completely predict the convergence order of the spectral method.

Direct data-driven filter design for uncertain LTI systems with bounded noise

This paper investigates the filter design problem for linear time-invariant dynamic systems when no mathematical model is available, but a set of initial experiments can be performed where also the variable to be estimated is measured. Instead of using the initial experimental data to identify a model on the basis of which a filter is designed, these data are used to directly design a filter. Assuming norm-bounded disturbances and noises, a Set Membership formulation is followed. For classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem, where the aim is the minimization of the induced norm from disturbances to the estimation error. A method is proposed for designing almost-optimal linear filters with finite impulse response, whose worst-case filtering error is at most twice the lowest achievable one. In the H ∞ SISO case, an efficient technique is presented, that allows the evaluation of bounds on the guaranteed worst-case filtering error of the designed filter. Numerical examples illustrate the effectiveness of the proposed solution.

On the Reachability and Controllability of Positive Linear Time-Invariant Dynamic Systems with Internal and External Incommensurate Point Delays

On the Reachability and Controllability of Positive Linear Time-Invariant Dynamic Systems with Internal and External Incommensurate Point Delays

Quasi-random-sampling high dimensional model representations for the construction of reduced discrete time state space dynamic models

In the context of real time model-based applications, complex high fidelity models may be computationally too expensive. Model order reduction and system identification techniques have been employed to transform complex models into equivalent reduced order models. However, most of the literature on model order reduction concerns linear time invariant dynamic systems, and the research into non linear model reduction is still on early stage. In this paper, we present a novel approach using quasi random sampling – high dimensional model representation (QRS-HDMR) to generate reduced discrete time state space dynamic models. The approach has the advantages of being able to handle the high dimensional case and produce affine discrete state space models, readily usable in control engineering applications.

On a Time-Varying Parameter Adaptive Self-Organizing System in the Presence of Large Outliers in Observations

In the previous papers (Pupeikis, 2000; Genov et al., 2006; Atanasov and Pupeikis, 2009), a direct approach for estimating the parameters of a discrete-time linear time-invariant (LTI) dynamic system, acting in a closed-loop in the case of additive noise with contaminating outliers uniformly spread in it, is presented. It is assumed there that the parameters of the LQG (Linear Quadratic Gaussian) controller are unknown, as well as known beforehand, too. The aim of the given paper is development of a minimum variance control (MVC) approach for a closed-loop discrete-time linear dynamic system when slowly or suddenly time-varying coefficients of the transfer function of such a system as well as that of a minimum variance (MV) controller are not known and ought to be estimated. The recursive parametric identification of an open-loop system and determination of the coefficients of the MV controller are performed in each current operation by processing observations in the case of additive noise at the output with contaminating outliers uniformly spread in it. The robust recursive technique, based on the S-algorithm, with a version of Shweppe's GM-estimator and with discounting previous data, used in the estimation, by introducing a constant as well as time-varying forgetting factors in the abovementioned estimator, is applied here in the calculation of estimates of the parameters of a dynamic system. Then, the recursive parameter estimates are used in each current iteration to determine unknown parameters of the MV controller. Afterwards, the current value of the MV control signal is found in each operation, and it is used to generate the output of the system, too. The results of numerical simulation by computer are presented and discussed.

Identification of dynamical systems with fractional derivative damping models using inverse sensitivity analysis

The problem of identifying parameters of time invariant linear dynamical systems with fractional derivative damping models, based on a spatially incomplete set of measured frequency response functions and experimentally determined eigensolutions, is considered. Methods based on inverse sensitivity analysis of damped eigensolutions and frequency response functions are developed. It is shown that the eigensensitivity method requires the development of derivatives of solutions of an asymmetric generalized eigenvalue problem. Both the first and second order inverse sensitivity analyses are considered. The study demonstrates the successful performance of the identification algorithms developed based on synthetic data on one, two and a 33 degrees of freedom vibrating systems with fractional dampers. Limited studies have also been conducted by combining finite element modeling with experimental data on accelerances measured in laboratory conditions on a system consisting of two steel beams rigidly joined together by a rubber hose. The method based on sensitivity of frequency response functions is shown to be more efficient than the eigensensitivity based method in identifying system parameters, especially for large scale systems.

Nonconcurrent Error Correction in the Presence of Roundoff Noise

This paper analyzes the effects of roundoff noise on our ability to nonconcurrently detect and identify transient faults that corrupt state variables during the operation of a fault-tolerant discrete-time linear time-invariant (LTI) dynamic system. Our analysis leads to two decoding algorithms, i.e., one based on the Peterson-Gorenstein-Zierler algorithm and the other based on singular-value decomposition techniques. We analytically obtain bounds on the roundoff noise level (equivalently, the precision) at which both algorithms can guarantee the correct determination of the number of errors. Our simulations verify our analysis and suggest that our bounds can be very tight for certain choices of design parameters. The developments in this paper can be used to provide guidance about the design of fault-tolerant systems and have immediate implications for digital implementations of LTI dynamic systems (e.g., digital filters) because such implementations unavoidably have to deal with finite-precision effects.

On Recursive Calculation of M- and GM-Estimates by Direct Identification in LQG Control Systems

In the previous papers (Pupeikis, 2000; Genov et al., 2006), a direct approach for estimating the parameters of a discrete-time linear time-invariant (LTI) dynamic system, acting in a closed-loop in the case of additive correlated noise with contaminating outliers uniformly spread in it, is presented. It is assumed here that the parameters of the LQG (Linear Quadratic Gaussian Control) controller are known beforehand. The aim of the given paper is development of a parametric identification approach for a closed-loop system when the parameters of an LTI system as well as that of LQG controller are not known and ought to be estimated. The recursive techniques based on an the M- and GM- estimator algorithms are applied here in the calculation of the system as well as noise filter parameters. Afterwards, the recursive parameter estimates are used in each current iteration to determine unknown parameters of the LQG-controller, too. The results of numerical simulation by computer are discussed.

Direct Data-Driven Filter Design For Uncertain LTI Systems With Bounded Noise

This paper investigates the filter design problem for linear time-invariant dynamic systems when no mathematical model is available, but a set of initial experiments can be performed where also the variable to be estimated is measured. Instead of using the initial experimental data to identify a model on the basis of which a filter is designed, these data are used to directly design a filter. Assuming norm-bounded disturbances and noises, a Set Membership formulation is followed. For classes of filters with exponentially decaying impulse response, approximating sets are determined that guarantee to contain all the solutions to the optimal filtering problem, where optimality refers to the minimization of the induced norm from disturbances to the estimation error. A method is proposed for designing almost-optimal linear filters with finite impulse response, whose worst-case estimation error is at most twice the lowest achievable one. A numerical example illustrates the effectiveness of the proposed solution.

On calculation of recursive M- and GM-estimates in LQG control systems

The aim of the given paper is development of a parametric identification approach for a closedloop system when the parameters of a discrete-time linear time-invariant (LTI) dynamic system as well as that of LQG (Linear Quadratic Gaussian) controller are not known and ought to be calculated. The recursive techniques based on an the maximum likelihood(M) and generalized maximum likelihood(GM) estimator algorithms are applied here in the calculation of the system as well as noise filter parameters. Afterwards, the recursive parameter estimates are used in each current iteration to determine unknown parameters of the LQG-controller, too. The results of numerical simulation by computer are discussed.

Observer-based fault detection and isolation filter design for linear time-invariant systems

In this paper we consider a model-based fault detection and isolation problem for linear time-invariant dynamic systems subject to faults and disturbances. We use a state observer scheme that cancels the system dynamics and defines a residual vector signal that is sensitive only to faults and disturbances. We then design a stable fault detection and isolation filter such that the ℋ∞-norm of the transfer matrix function from disturbances to the residual is minimised (for fault detection) subject to the constraint that the transfer matrix function from faults to residual is equal to a pre-assigned diagonal transfer matrix (for isolation of possibly simultaneous occurring faults). Our solution is given in the form of linear matrix inequalities using state-space techniques, as well as a model matching problem using matrix factorisation techniques. A numerical example is given to illustrate the efficiency of the fault detection and isolation filter.

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249–256]. We use recently derived results to provide an alternative proof of this result and to derive extensions.

Multicommodity Eulerian-Lagrangian Large-Capacity Cell Transmission Model for En Route Traffic

A new paradigm for building an Eulerian-Lagrangian cell transmission model for air traffic flow is developed. It is based on an aggregation of track data and is applied to the full National Airspace System in the United States. The Eulerian-Lagrangian model is based on a multicommodity network flow model constructed from historical air traffic data. The flow model is reduced to a linear time invariant dynamical system, in which the state is a vector of aggregate aircraft counts. This model is called a large-capacity cell transmission model in reference to the cell transmission model in highway traffic, which inspired this work. The predictive capabilities of the model are successfully validated against recorded air traffic data by showing an accurate match between predicted sector counts (based on filed flight plans) and measured sector counts. The problem of controlling the sector aircraft count is posed as an integer program in which the dynamical system appears in the constraints. To improve the computational time for solving the problem, the integer program is relaxed to a linear program. The computational results show that a high portion of solutions of the linear program are integers, making this method appropriate for high-level traffic flow management.

Adaptive control of linear singular time-invariant single-input single-output systems with external point delay

This paper is devoted to the robust adaptive control of solvable single-input single-output impulse-free linear time-invariant singular dynamic systems of known orders and unknown parameterizations subject to single external point delays. The control law is of pole-placement type and based on input/output measurements and parametrical estimation only. The parametrical estimation algorithm incorporates adaptation dead zones to prevent against potential instability caused by disturbances and unmodeled dynamics. The Weierstrass canonical form is investigated in detail concerning the properties of controllability and observability via testable conditions of the given arbitrary state-space realization of the same order. Robust closed-loop stability is proved under similar conditions to those used for (nonsingular) standard systems.

On the External Positivity of Linear Time-Invariant Dynamic Systems

This brief is devoted to present a general method to characterize precisely the external positivity property of linear time-invariant continuous-time dynamic systems which is based on the Post formula for Laplace inversion.

Analysis of periodic solutions in piecewise linear feedback systems via a complementarity approach

Periodic solutions in piecewise linear feedback systems in Luré form, composed by a linear time invariant dynamical system closed in feedback through a static piecewise linear mapping are analyzed. A procedure for representing the closed loop system in complementarity form, based on resistors-diode-sources equivalent circuits of the piecewise linear characteristic is presented. Conditions for the existence of discrete–time periodic solutions in terms of solvability of a suitable static linear complementarity problem are obtained. Numerical results for a chaotic circuit show that the proposed approach can be also used to predict periodic solutions of continuous–time systems in Luré form provided that consistency of the discretization is assumed.

MULTILEVEL CONTROLLER FOR LINEAR INTERCONNECTED DYNAMICAL SYSTEMS WITH CONSTRAINTS

In this paper, a class of linear time invariant interconnected dynamical systems with constraints on the states and/or control inputs is considered. In this class, the interconnections are considered as perturbations acting in contradiction to the autonomy of the individual subsystems. A multilevel controller is designed to control the behavior of this system while satisfying system constraints. The upper level part of the controller is designed to balance or weaken the effect of the interactions, while the lower level decentralized controllers are designed to insure optimal subsystem performance as well as to satisfy system constraints. An example is given to illustrate the performance of the developed procedure.

An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems

In this paper, we propose a model reduction algorithm for approximation of large-scale linear time-invariant dynamical systems. The method is a two-sided projection combining features of the singular value decomposition (SVD)-based and the Krylov-based model reduction techniques. While the SVD-side of the projection depends on the observability gramian, the Krylov-side is obtained via iterative rational Krylov steps. The reduced model is asymptotically stable, matches certain moments and solves a restricted H2 minimization problem. We present modifications to the proposed approach for employing low-rank gramians in the reduction step and also for reducing discrete-time systems. Several numerical examples from various disciplines verify the effectiveness of the proposed approach. It performs significantly better than the q-cover [A. Yousouff, R.E. Skelton, Covariance equivalent realizations with applications to model reduction of large-scale systems, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 22, Academic Press, 1985, pp. 273–348; A. Yousouff, D.A. Wagie, R.E. Skelton, Linear system approximation via covariance equivalent realizations, Journal of Math. Anal. and Appl. 196 (1985) 91–115] and the least-squares [S. Gugercin, A.C. Antoulas, Model reduction of large scale systems by least squares, Linear Algebra Appl. 415(2–3) (2006) 290–321] methods that have a similar projection structure to the proposed method. Also, in terms of both the H2 and H∞ error measures, its performance is comparable to or sometimes better than that of balanced truncation. Moreover, the method proves to be robust with respect to the perturbations due to usage of approximate gramians.