Linear Time-invariant System Model Research Articles

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

An effective controller design strategy for higher order systems

The recent methods that guarantee the stability of the reduced-order models are Mihailov stability, improved Pade approximation and truncation, improved generalized pole clustering, moment matching and salp swarm optimization. Further, these methods could overcome the limitations such as non-uniqueness, pole clustering, gain adjustment and difficulty to maintain the dominant roots in the lower order system for non-minimum higher order plants. This research emphasizes the design of the proposed compensating algorithm by using an additional open loop zero for the stable reduced-order models of large-scale single-input single-output linear time-invariant continuous time systems. The results of the proposed algorithm are compared with the existing compensating methods and the design is validated and illustrated numerically.

Model Predictive Control in Partially Observable Multi-Modal Discrete Environments

Autonomous systems operate in environments that can be observed only through noisy measurements. Thus, controllers should compute actions based on their beliefs about the surroundings. In these settings, we design a Model Predictive Controller (MPC) based on a continuous-state Linear Time-Invariant (LTI) system model operating in a discrete-state environment described by a Hidden Markov Model (HMM). Environment constraints are modeled as chance constraints and environment observations can be asynchronous with system state measurements and controller updates. We show how to approximate the solution of the MPC problem defined over the space of feedback policies by optimizing over a trajectory tree, where each branch is associated with an environment measurement. The proposed approach guarantees chance constraint satisfaction and recursive feasibility. Finally, we test the proposed strategy on navigation examples in partially observable environments, where the proposed MPC guarantees chance constraint satisfaction.

Clustering of Linear Time-Invariant Systems: A Bayesian Nonparametric Method

In this paper, we consider the clustering of linear time-invariant (LTI) systems according to the similarity of dynamics. There are two main difficulties for clustering of LTI systems: one is to determine the labels of LTI systems and the number of clusters, and another one is to build models of LTI systems. To address these two difficulties simultaneously, a Bayesian nonparametric method based on Dirichlet process mixture models (DPMM) and kernel-based regularization method is proposed. To be specific, the Dirichlet process mixtures of Bayesian linear regression models (DPM-BLRM) is proposed to determine the label of each LTI system according to the probability that the system belongs to each cluster, and adapt the number of clusters to the complexity of data. In addition, DPM-BLRM incorporates the prior knowledge of LTI systems to deal with the possible large variance of model parameters estimation when the collected data is short or has low signal-to-noise ratio, where the carrier of prior knowledge is kernel matrix. Then Markov chain Monte Carlo (MCMC) method is used for Bayesian inference for the DPM-BLRM. The simulation results illustrate the efficiency of the proposed method.

Characterization and correction of diffusion gradient-induced eddy currents in second-order motion-compensated echo-planar and spiral cardiac DTI.

Very high gradient amplitudes played out over extended time intervals as required for second-order motion-compensated cardiac DTI may violate the assumption of a linear time-invariant gradient system model. The aim of this work was to characterize diffusion gradient-related system nonlinearity and propose a correction approach for echo-planar and spiral spin-echo motion-compensated cardiac DTI. Diffusion gradient-induced eddy currents of 9 diffusion directions were characterized at b values of 150 s/mm2 and 450 s/mm2 for a 1.5Tesla system and used to correct phantom, ex vivo, and in vivo motion-compensated cardiac DTI data acquired with echo-planar and spiral trajectories. Predicted trajectories were calculated using gradient impulse response function and diffusion gradient strength- and direction-dependent zeroth- and first-order eddy current responses. A reconstruction method was implemented using the predicted -space trajectories to additionally include off-resonances and concomitant fields. Resulting images were compared to a reference reconstruction omitting diffusion gradient-induced eddy current correction. Diffusion gradient-induced eddy currents exhibited nonlinear effects when scaling up the gradient amplitude and could not be described by a 3D basis alone. This indicates that a gradient impulse response function does not suffice to describe diffusion gradient-induced eddy currents. Zeroth- and first-order diffusion gradient-induced eddy current effects of up to -1.7rad and -16 to +12 rad/m, respectively, were identified. Zeroth- and first-order diffusion gradient-induced eddy current correction yielded improved image quality upon image reconstruction. The proposed approach offers correction of diffusion gradient-induced zeroth- and first-order eddy currents, reducing image distortions to promote improvements of second-order motion-compensated spin-echo cardiac DTI.

A compute-bound formulation of Galerkin model reduction for linear time-invariant dynamical systems

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. Finally, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

Adaptive Controller and Observer Design Using Open and Closed-Loop Reference Models for Linear Time-Invariant Systems With Unknown Dynamics

This article presents an output feedback controller and observer design approach for linear time-invariant systems with unknown dynamics. The presented method uses an open-loop reference model to generate the desired trajectory and a closed-loop reference model as an observer. The controller only uses the observer states. Lyapunov-based stability proofs show that the error states converge asymptotically to zero and that all other signals are uniformly stable. Furthermore, bounds are proven on the transient behavior.

Can Replay Attacks Designed to Steal Water from Water Distribution Systems Remain Undetected?

Industrial Control Systems (ICS) monitor and control physical processes. ICS are found in, among others, critical infrastructures such as water treatment plants, water distribution systems, and the electric power grid. While the existence of cyber-components in an ICS leads to ease of operations and maintenance, it renders the system under control vulnerable to cyber and physical attacks. An experimental study was conducted with replay attacks launched on an operational water distribution (WADI) plant to understand under what conditions an attacker/attack can remain undetected while stealing water. A detection method, based on an input-output Linear Time-invariant system model of the physical process, was developed and implemented in WADI to detect such attacks. The experiments reveal the strengths and limitations of the detection method and challenges faced by an attacker while attempting to steal water from a water distribution system.

Discussion on the Determination Method of the Undetermined Coefficient in the Response Solution of Linear Time-Invariant Discrete System

There are some problems in the existing methods for determining the undetermined coefficients in the linear time-invariant discrete system response solution, which are mainly to determine whether the conditions for the undetermined coefficients of the general solution of the classical solution can be extended, and the ambiguity of the conditions and the extension of the conditions for determining the undetermined coefficients of the general solution of the zero input response, and the error of using condition when determining the undetermined coefficient of general solution of zero state response. In this paper, by means of the mathematical analysis of the difference equation model of a linear time-invariant discrete system, firstly, based on the existing analytical methods, the conditional extension method for determining the undetermined coefficients of the classical solution is given, and the classical solution is analyzed under the extended conditions. Then, the method to determine the undetermined coefficient in the solution of zero input and zero state response is analyzed, and the unclear conditions and errors in the existing methods to determine the undetermined coefficient are corrected. Finally, by comparing the system response obtained by the classical method with the zero input method and the zero state method, the correctness and validity of the method proposed in this paper are verified, so that the premise error will not occur when the linear time-invariant system model is used for problem analysis in practical engineering.

Robust stability analysis of LTI systems with fractional degree generalized frequency variables

Abstract A novel linear time-invariant (LTI) system model with fractional degree generalized frequency variables (FDGFVs) is proposed in this paper. This model can provide a unified form for many complex systems, including fractional-order systems, distributed-order systems, multi-agent systems and so on. This study mainly investigates the stability and robust stability problems of LTI systems with FDGFVs. By characterizing the relationship between generalized frequency variable and system matrix, a necessary and sufficient stability condition is firstly presented for such systems. Then for LTI systems with uncertain FDGFVs, we present a robust stability method in virtue of zero exclusion principle. Finally, the effectiveness of the method proposed in this paper is demonstrated by analyzing the robust stability of gene regulatory networks.

Fault indicators and unique mode-dependent state equations from a fixed-causality diagnostic bond graph of linear models with ideal switches

Analytical redundancy relations are fundamental in model-based fault detection and isolation. Their numerical evaluation yields a residual that may serve as a fault indicator. Considering switching linear time-invariant system models that use ideal switches, it is shown that analytical redundancy relations can be systematically deduced from a diagnostic bond graph with fixed causalities that hold for all modes of operation. Moreover, as to a faultless system, the presented bond graph–based approach enables to deduce a unique implicit state equation with coefficients that are functions of the discrete switch states. Devices or phenomena with fast state transitions, for example, electronic diodes and transistors, clutches, or hard mechanical stops are often represented by ideal switches which give rise to variable causalities. However, in the presented approach, fixed causalities are assigned only once to a diagnostic bond graph. That is, causal strokes at switch ports in the diagnostic bond graph reflect only the switch-state configuration in a specific system mode. The actual discrete switch states are implicitly taken into account by the discrete values of the switch moduli. The presented approach starts from a diagnostic bond graph with fixed causalities and from a partitioning of the bond graph junction structure and systematically deduces a set of equations that determines the wanted residuals. Elimination steps result in analytical redundancy relations in which the states of the storage elements and the outputs of the ideal switches are unknowns. For the later two unknowns, the approach produces an implicit differential algebraic equations system. For illustration of the general matrix-based approach, an electromechanical system and two small electronic circuits are considered. Their equations are directly derived from a diagnostic bond graph by following causal paths and are reformulated so that they conform with the matrix equations obtained by the formal approach based on a partitioning of the bond graph junction structure. For one of the three mode-switching examples, a fault scenario has been simulated.

Gradient and shim pre-emphasis by inversion of a linear time-invariant system model.

The goal of this contribution is to enhance the fidelity and switching speed of gradient and shim fields by advancing pre-emphasis toward broadband and full cross-term correction. The proposed approach is based on viewing gradient and shim chains as linear, time-invariant (LTI) systems. Pre-emphasis is accomplished by inversion of a broadband digital system model. In the multiple-channel case, it amounts to a matrix of broadband filters that perform concerted self- and cross-term correction. This approach is demonstrated with gradients and shims up to the third order in a 7 Tesla whole-body MR system. Pre-emphasis by LTI model inversion is first verified by studying settling speeds and response behavior without and with the correction. It is then demonstrated for rapid shim updating, achieving substantially enhanced fidelity of field dynamics and shim settling within approximately 1 ms. In single-shot echo-planar imaging (EPI) acquisitions in vivo, this benefit is shown to translate into enhanced geometric fidelity. The fidelity of gradient and shim dynamics can be greatly enhanced by pre-emphasis based on inverting a general LTI system model. Permitting shim settling on the millisecond scale, broadband multiple-channel pre-emphasis promises to render higher-order shimming fully versatile at the level of MRI sequence design. Magn Reson Med 78:1607-1622, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

A Fully Adaptive Scheme for Model Order Reduction Based on Moment Matching

A fully adaptive model order reduction scheme based on moment matching is proposed to derive the reduced-order models of linear time-invariant (LTI) systems. According to the given error tolerance, the order of the reduced-order model as well as the expansion points for the transfer function is automatically determined on the fly during the process of model reduction. In this sense, the reduced-order model is automatically obtained without assigning the number of moments and expansion points in a priori , which is a prerequisite for the standard implementation of model reduction based on moment matching. The proposed adaptive scheme is found to be efficient when it is tested on various LTI systems.

Application and Comparative Analysis of Various Classical and Soft Computing Techniques for Model Reduction of MIMO Systems

Model reduction techniques are simplification methods based on mathematical approaches employed to realize reduced models for the original high order systems. Some existing classical model reduction techniques for multivariable system are considered and compared for their performances. Interlacing property and coefficients matching (IPCM) method gives overall minimum integral square error (ISE), integral absolute error (IAE) and integral time absolute error (ITAE) values compared to other methods. Though the IPCM method is efficient, it may not guarantee for minimization of all objective functions simultaneously. In this paper, model reduction approach based on objectives like ISE, IAE and ITAE using multi-objective differential evolution (MODE) method is proposed for reducing the numerator and the denominator is reduced by interlacing property. MODE method minimizes the small, normal and large errors persisting for long time between original and reduced models. This multi-objective approach is applied for model reduction of 10th order multivariable linear time invariant power system model. Simulation results are demonstrated for single and multi-objective model reduction and compared with multi-objective particle swarm optimization (MOPSO) method to prove the validity of proposed MODE technique.

Bijective Differential Search Algorithm for Robust Design of Damping Controller in Multimachine Power System

Low frequency oscillation (LFO) is a serious threat to the interconnection of power system and its safe operation. In this paper, optimum damping performances over LFO is achieved by implementing Bijective Differential Search Algorithm (B-DSA) to large interconnected power system. Conventional two stages lead-lag compensator is optimized as the Power System Stabilizer (PSS) and Linear Time Invariant (LTI) State Space system models are used to conduct stability analysis of power system. The tuning problem of PSS in multimachine system was formulated as a multi-objectives function. The simulations are conducted in 5-AREA 16 Machine test power system for severe system fault in order to verify the robust design of damping controller. The obtained results are compared with standard DSA optimization technique. The findings show the improved damping achieved by B-DSA than DSA algorithm. The settling times achieved using B-DSA based designed PSSs are 3.74sec and 4.57sec for local mode and inter-area mode of oscillations respectively. The successful damping over oscillation modes of LFO justifies the proposed technique is efficient for the improvement of power system security in adverse condition.

Use Hausdorff metric to analyze convergence of parameter estimation in system identification

For a linear time-invariant system model, this paper analyzes the convergence of parameter estimations as the length of the input–output data tends to infinity through the prediction error method. It is known that the sequence of the criterion functions converges uniformly in the parameter with probability one as the data length tends to infinity. The parameter estimation is represented by a set in general, instead of by a single point, on which the criterion function takes its minimum. Thus a mathematical feature of the convergence problem of parameter estimation is in that we are needed, from the convergence of a sequence of functions, to infer the convergence of the sequence of their sets of minimizing arguments. The Hausdorff metric is suggested to measure the distance between sets and then is used to discuss the convergence problem here. According to the Hausdorff metric, the convergence of parameter estimation is not guaranteed in general. A condition guaranteeing such convergence is given.

Nonlinear Analysis of Vibro-Impacts for Unloaded Gear Pairs With Various Excitations and System Parameters

Torsional systems with clearance-type nonlinearities have inherent vibratory problems such as gear rattle. Such vibro-impacts generally occur on the unloaded gear pairs of a vehicle correlated with the firing excitation of an engine. This study investigates the gear rattle phenomena on unloaded gear pairs with different excitation conditions and various system parameters. First, a linear time-invariant system model with six degrees of freedom is constructed and then a numerical analysis is applied to the gear rattle motion. Smoothening factors for clutch stiffness and hysteresis are employed for the stability of numerical simulations. Second, the dynamic characteristics of vibro-impacts are studied by examining the fast Fourier transform (FFT) components of the gear mesh force in a high frequency range. The effects of various system parameters on the vibro-impacts are examined using a nonlinear system model. Finally, the vibro-impacts, in terms of “single-sided” and “double-sided” impacts, are identified in phase planes.

Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment

A new approach for stabilizing unstable reduced order models (ROMs) for linear time-invariant (LTI) systems through an a posteriori post-processing step applied to the algebraic ROM system is developed. The key idea is to modify the unstable eigenvalues of the ROM system by moving these eigenvalues into the stable half of the complex plane. It is demonstrated that this modification to the ROM system eigenvalues can be accomplished using full state feedback (a.k.a. pole placement) algorithms from control theory. This approach ensures that the modified ROM is stable provided the system’s unstable poles are controllable and observable; however, the accuracy of the stabilized ROM is not guaranteed. To remedy this difficulty and guarantee an accurate stabilized ROM, a constrained nonlinear least-squares optimization problem for the stabilized ROM eigenvalues in which the error in the ROM output is minimized is formulated. This optimization problem is small and therefore computationally inexpensive to solve. Performance of the proposed algorithms is evaluated on two test cases for which ROMs constructed via the proper orthogonal decomposition (POD)/Galerkin method suffer from instabilities.

선형모델을 이용한 수동변속기의 동적 특성 연구

자동차 수동변속기 내 래틀과 같은 회전체 진동은 엔진 연소폭발 주기와 연동하여 발생한다. 이러한 충격형 진동 특성은 선형 시간 불변 분석법을 적용하여 살펴볼 수 있다. 기어의 동적 특성을 이해하기 위해, 특정 형태의 앞 바퀴 굴림 방식 수동 변속기에 대한 연구를 수행하였다. 첫째, 동특성 고유값과 주파수 응답함수를 기반으로 자유도를 줄이는 방법을 제시하였는데, 이는 행렬의 크기를 줄이는 효과적인 역할을 한다. 둘째, 단일 질량 플라이휠과 이중 질량 플라이휠의 동적 특성을 비교하였다. 모빌리티 분석을 기반으로 한 이중 질량 플라이휠의 효과를 검토하였는데, 이로부터 진동에 의해 가진되는 충격을 회피하기 위한 기본 개념을 이해 할 수 있다. 마지막으로, 두 단계 클러치 댐퍼로부터 유효 강성 값을 도출하여 선형 불변형 시스템 모델을 연구하였다. 두 단계의 서로 다른 클러치 강성조합을 이용하여 클러치 댐퍼의 동적특성에 대한 관계를 예측할 수 있다. Torsional vibrations, such as the gear rattle of the manual transmission in vehicle systems, are correlated with the firing stroke from the engine. These vibro-impacts can be examined based upon linear time-invariant analysis. In order to understand the gear dynamics, a specific manual transmission with a front-engine front-wheel drive configuration is investigated. A method to reduce the degrees of freedom is suggested based upon the eigensolutions and frequency response functions, which will lead to the development of an efficient matrix size. The dynamic characteristics of single- and dual-mass flywheels are then compared. The effect of the dual-mass flywheel is investigated based upon the mobility analysis, which will lead to understanding of the concepts for avoiding vibro-impacts. A linear time-invariant system model is examined by employing the effective clutch stiffness from a two-stage clutch damper. Thus, the relationship between the dynamic characteristics and the clutch damper can be predicted by assuming a combination of different stage stiffness levels.

Parameterisation of stabilising controllers with precompensators

Within the framework of the factorisation approach, the author presents parameterisation methods for stabilising controllers. The parameterisations of this study are characterised by an ideal finitely generated by some strictly causal stable transfer functions. Several typical examples of the parameterisations are also presented, namely (i) all stabilising controllers such that their relative degrees are more than or equal to some fixed number in the continuous-time linear time-invariant (LTI) system model; (ii) all stabilising controllers such that they must have at least some fixed number of delay operators in the discrete-time LTI system model; (iii) all stabilising controllers such that they must have some delay operators in the multidimensional system model. The parameterisation method of this study is also applied to (iv) a system that is stabilisable but that does not admit a doubly coprime factorisation.