Identification Of Linear Time-invariant Systems Research Articles

Closed-loop identification of stabilized models using dual input–output parameterization

This paper introduces a dual input–output parameterization (dual IOP) for the identification of linear time-invariant systems from closed-loop data. It draws inspiration from the recent input–output parameterization developed to synthesize a stabilizing controller. The controller is parameterized in terms of closed-loop transfer functions, from the external disturbances to the input and output of the system, constrained to lie in a given subspace. Analogously, the dual IOP method parameterizes the unknown plant with analogous closed-loop transfer functions, also referred to as dual parameters. In this case, these closed-loop transfer functions are constrained to lie in an affine subspace guaranteeing that the identified plant is stabilized by the known controller. Compared with existing closed-loop identification techniques guaranteeing closed-loop stability, such as the dual Youla parameterization, the dual IOP neither requires a doubly-coprime factorization of the controller nor a nominal plant that is stabilized by the controller. The dual IOP does not depend on the order and the state-space realization of the controller either, as in the dual system-level parameterization. Simulation shows that the dual IOP outperforms the existing benchmark methods.

Parsimonious system identification from fragmented quantised measurements

Quantisation is the process of mapping an input signal from an infinite continuous set to a countable set with a finite number of elements. It is a non-linear irreversible process, which makes the traditional methods of system identification no longer applicable. In this work, we propose a method for parsimonious linear time invariant system identification when only quantised observations, discerned from noisy data, are available. More formally, given a priori information on the system, represented by a compact set containing the poles of the system, and quantised realizations, our algorithm aims at identifying the least order system that is compatible with the available information. The proposed approach takes also into account that the available data can be subject to fragmentation. Our proposed algorithm relies on an ADMM approach to solve a quasi-norm objective problem. Numerical results highlight the performance of the proposed approach when compared to the minimisation in terms of the sparsity of the induced solution.

Identification of Linear Time-Invariant Systems: A Least Squares of Orthogonal Distances Approach

This work describes the parameter identification of servo systems using the least squares of orthogonal distances method. The parameter identification problem was reconsidered as data fitting to a plane, which in turn corresponds to a nonlinear minimization problem. Three models of a servo system, having one, two, and three parameters, were experimentally identified using both the classic least squares and the least squares of orthogonal distances. The models with two and three parameters were identified through numerical routines. The servo system model with a single parameter only considered the input gain. In this particular case, the analytical conditions for finding the critical points and for determining the existence of a minimum were presented, and the estimate of the input gain was obtained by solving a simple quadratic equation whose coefficients depended on measured data. The results showed that as opposed to the least squares method, the least squares of orthogonal distances method experimentally produced consistent estimates without regard for the classic persistency-of-excitation condition. Moreover, the parameter estimates of the least squares of orthogonal distances method produced the best tracking performance when they were used to compute a trajectory-tracking controller.

Kernel-Based Identification of Incrementally Input-to-State Stable Nonlinear Systems

Methods based on Reproducing Kernel Hilbert Spaces (RKHS) have proven to be a valuable tool for the identification of linear time-invariant systems in both discrete- and continuous-time. In particular, unlike most other techniques, they enable to systematically confer a priori desirable properties, such as stability, on the estimated models. However, existing RKHS methods mainly target impulse responses and, hence, do not extend well to the context of nonlinear systems. In this work, we propose a novel RKHS-based methodology for the identification of discrete-time nonlinear systems guaranteeing that the identified system is incrementally input-to-state stable (δISS). We model the identified system using a predictor function that, given past input and output samples, yields the output prediction at the next time instant. The predictor is selected from an RKHS by solving a constrained optimization problem that guarantees its δISS properties. The proposed approach is validated via numerical simulations.

Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.

Non-Asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories

This paper considers the problem of linear time-invariant (LTI) system identification using input/output data. Recent work has provided non-asymptotic results on partially observed LTI system identification using a single trajectory but is only suitable for stable systems. We provide finite-time analysis for learning Markov parameters based on the ordinary least-squares (OLS) estimator using multiple trajectories, which covers both stable and unstable systems. For unstable systems, our results suggest that the Markov parameters are harder to estimate in the presence of process noise. Without process noise, our upper bound on the estimation error is independent of the spectral radius of system dynamics with high probability. These two features are different from fully observed LTI systems for which recent work has shown that unstable systems with a bigger spectral radius are easier to estimate. Extensive numerical experiments demonstrate the performance of our OLS estimator.

Frequency domain identification using adaptive Fourier decomposition method with polynomials

The adaptive Fourier decomposition method is an approximation technique of generalised Fourier series, it leads to fast approximations by selecting basis functions in a maximal selection criterion. In recent research studies, it has been efficiently applied in the identification of linear time-invariant systems and a new algorithm is named two-step (T-S) algorithm. In this work, some further modification is made for the T-S algorithm. The improvement is made at the first step, where polynomials are used instead of cauchy integral formula. By doing this, the algorithm becomes simpler and easier to realise. The approximation errors are analysed. Owing to the analysed results, this new T-S algorithm is only to get poles for the finite rational orthogonal basis functions but not the approximation to the systems, the coefficients are estimated by using least-squares methods. The effectiveness is examined through numerical examples that show it takes much less running times and can get comparable approximating results. Besides, the case that errors are included in the frequencies is also studied, the obtained results imply that minor errors in the frequencies would not affect the estimation.

Finite-horizon integration for continuous-time identification: bias analysis and application to variable stiffness actuators

ABSTRACT Direct identification of continuous-time dynamical models from sampled data is now a mature discipline, which is known to have many advantages with respect to indirect approaches based on the identification of discretised models. This paper faces the problem of continuous-time identification of linear time-invariant systems through finite-horizon numerical integration and least-square estimation. The bias in the least-squares estimator due to the noise corrupting the signal observations is quantified, and the benefits of numerical integration in the attenuation of this bias are discussed. An extension of the approach which combines numerical integration, least-squares estimation and particle swarm optimisation is proposed for the identification of nonlinear systems and nonlinear-in-the-parameter models, and then applied to the estimation of the torque-displacement characteristic of a commercial variable stiffness actuator driving a one-degree-of-freedom pendulum.

A Randomized Algorithm for Parsimonious Model Identification

Identifying parsimonious models is generically a “hard” nonconvex problem. Available approaches typically rely on relaxations such as Group Lasso or nuclear norm minimization. Moreover, incorporating stability and model order constraints into the formalism in such methods entails a substantial increase in computational complexity. Motivated by these challenges, in this paper we present algorithms for parsimonious linear time invariant system identification aimed at identifying low-complexity models which i) incorporate a priori knowledge on the system (e.g., stability), ii) allow for data with missing/nonuniform measurements, and iii) are able to use data obtained from several runs of the system with different unknown initial conditions. The randomized algorithms proposed are based on the concept of atomic norm and provide a numerically efficient way to identify sparse models from large amounts of noisy data.

A hybrid approach to parameter identification of linear delay differential equations involving multiple delays

ABSTRACTIn this paper, we are concerned with the parameter identification of linear time-invariant systems containing multiple delays. The approach is based upon a hybrid of block-pulse functions and Legendre's polynomials. The convergence of the proposed procedure is established and an upper error bound with respect to the L2-norm associated with the hybrid functions is derived. The problem under consideration is first transformed into a system of algebraic equations. The least squares technique is then employed for identification of the desired parameters. Several multi-delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method. It is shown that the proposed approach is also applicable to a class of nonlinear multi-delay systems. It is demonstrated that the suggested procedure provides accurate results for the desired parameters.

EKF Based Parameter Identification of LTI System from Seismic Response Measurements

Parameter identification of linear time invariant (LTI) dynamic system using extended Kalman filter (EKF) are presented in this work. The efficiency of the EKF is demonstrated using non-stationary responses. For this purpose, a three storied steel frame is used in the laboratory and the recorded ground motion is applied to measure the acceleration response at the top. Using these responses, the EKF based predictor-corrector algorithm is applied to identify the model parameters. It has been observed that EKF based identification scheme may converge to different system matrices (i.e. stiffness) for different experimental results of the same structure. However, their eigen values (i.e. natural frequency and mode shape) remain same.

Identification of Linear Time-Invariant Systems From Multiple Experiments

A standard assumption for consistent estimation in the errors-in-variables setting is persistency of excitation of the noise-free input signal. We relax this assumption by considering data from multiple experiments. Consistency is obtained asymptotically as the number of experiments tends to infinity. The main theoretical and algorithmic difficulties are related to the growing number of to-be-estimated initial conditions. The method proposed in the paper is based on analytic elimination of the initial conditions and optimization over the remaining parameters. The resulting estimator is consistent; however, achieving asymptotically efficiency is an open problem.

Identification of High-Tech Motion Systems: An Active Vibration Isolation Benchmark

The benchmark Active Vibration Isolation System (AVIS) in this paper is a complex high-tech industrial system used in vibration and motion control applications. The system is complex in the sense of high order exible dynamics and multiple inputs and outputs. The aim of this benchmark is to compare difierent black box, linear time invariant system identification algorithms. Difierent large data sets are provided, enabling the use of both frequency domain and time domain identification approaches. The idea of the benchmark is to investigate both model accuracy and numerical reliability of the computational steps. Several reference solutions are provided, which demonstrate the challenging aspects of this benchmark already in the singleinput single-output case. The benchmark data and additional info are available on the website of Tom Oomen.

Identification of Linear Time-Invariant Systems under Periodic Disturbance with its Estimation

In this paper, we propose a new method for identification of linear time-invariant systems and estimation of unknown periodic disturbances. We assume that a disturbance to be considered here is unknown except its period. In this method, by stacking not only the unknown coefficients of a mathematical model but also the unknown disturbance as a parameter vector, we modify the equation error method so as to treat the unknown disturbance in the identification scheme. In particular, we also explain how the periodicity of the disturbance is utilized. In order to show the validity of this method, we give an illustrative numerical example.

Identification of LTI Time-Delay Systems with Missing Output Data Using GEM Algorithm

This paper considers the parameter estimation for linear time-invariant (LTI) systems in an input-output setting with output error (OE) time-delay model structure. The problem of missing data is commonly experienced in industry due to irregular sampling, sensor failure, data deletion in data preprocessing, network transmission fault, and so forth; to deal with the identification of LTI systems with time-delay in incomplete-data problem, the generalized expectation-maximization (GEM) algorithm is adopted to estimate the model parameters and the time-delay simultaneously. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Identification of Linear Time-Invariant Systems via Constrained Step-Based Realization

A constrained step-based realization algorithm is developed to produce linear, time-invariant, state-space system estimates. To match a priori knowledge of the system behavior, the eigenvalues of the estimate are required to be stable, real, and positive; the step response is required to have no undershoot or overshoot; and the steady-state gain is required to match a known value. The standard step-based realization method is augmented to become a convex optimization problem subject to a linear-matrix inequality that constrains eigenvalue location, and a subsequent convex optimization problem is developed to constrain time-domain behavior. Simulation results motivate the need for such constraints and are used for comparison with familiar alternative methods. Although the procedure is applied only to step-response data, it may be generalized to constrain eigenvalues to convex regions of the complex plain and is applicable to all subspace identification methods.

ADAPT-lpv Software for Identification of Nonlinear Parameter-Varying Systems

The ADAPT-lpv software for the identification of linear parameter-varying (LPV) and/or nonlinear systems is a direct and simple extension of the canonical variate analysis (CVA) method as implemented in the ADAPTx software for identification of linear time-invariant systems. The computational structure and problem size is very similar to ADAPTx except that the matrix row dimension (number of lagged variables of the past) is multiplied by the effective number of parameter-varying functions. This is in contrast with the exponential explosion in the number of variables using current subspace methods for LPV systems. Compared with current methods, initial results indicate much less computation, maximum likelihood accuracy, and better numerical stability. The method can automatically remove a number of redundancies in the nonlinear models producing near minimal state orders and polynomial degrees by hypothesis testing. There is detailed discussion of the computational methods and structure of the software modules. The software is demonstrated on a widely studied aircraft flutter problem.

Fast multi-order computation of system matrices in subspace-based system identification

Subspace methods have proven to be efficient for the identification of linear time-invariant systems, especially applied to mechanical, civil or aeronautical structures in operation conditions. Therein, system identification results are needed at multiple (over-specified) model orders in order to distinguish the true structural modes from spurious modes using the so-called stabilization diagrams. In this paper, new efficient algorithms are derived for this multi-order system identification with subspace-based identification algorithms and the closely related Eigensystem Realization Algorithm. It is shown that the new algorithms are significantly faster than the conventional algorithms in use. They are demonstrated on the system identification of a large-scale civil structure.

Identification of linear time-invariant systems

An algorithm for the identification of a stationary linear system from sampled data on a transient process is presented. The algorithm allows the original problem to be decomposed into two problems: finding the roots of the characteristic polynomial and determining the amplitudes of the respective modes. A special objective function permits increasing the accuracy of identification

Identification of linear time-invariant systems based on initial condition responses

Forced responses are commonly used for system identification, but only initial condition responses may be available when a system has no controllable inputs or forced responses cannot be practically measured. It is therefore meaningful to analyse whether initial condition responses can be used to estimate the system matrix. In this work, the identifiability of the nth order (n > 1) linear time-invariant systems from initial condition responses is analysed. Systems with only one measurable state (OMS systems) and those with n measurable states (NMS systems) are considered. Analysis indicates that one initial condition response is not sufficient to uniquely determine the system matrix of an OMS system and n initial condition responses are necessary. The identifiability of an OMS system with data from n independent initial condition responses is equivalent to that of an NMS system with only one initial condition response. Explicit formulations and a non-iterative algorithm are developed for both OMS and NMS systems.