Closed-loop Subspace Identification Method Research Articles

An Introduction of a CCA Weighting Matrix to a Closed-Loop Subspace Identification Method

Abstract This paper introduces a CCA (canonical correlation analysis) weighting matrix to an estimation method of the innovations model previously proposed by the authors. A numerical simulation illustrates that the CCA weighting reduces the covariance of the estimate and that the proposed method gives similar or better performance compared to Closed-Loop MOESP and PBSID. Especially, the design parameter called “past horizon” in the proposed method can be set small compared to Closed-Loop MOESP and PBSID. It is also analyzed how the bias of the estimate is reduced.

Modal-Decomposition-Dependent State-Space Modeling and Modal Analysis of a Rigid-Flexible, Coupled, Multifreedom Motion System: Theory and Experiment

In a multiple degrees of freedom motion system with rigid-flexible coupling, the flexible internal dynamics have a significant negative impact on performance because the degrees of freedom are coupled. Inspired by this problem, a multi-input multioutput state-space model based on modal coordinates is proposed to decouple the rigid body and flexible modes. The closed-loop subspace identification method based on orthogonal is utilized to develop an unbiased standard state-space model. Based on the similarity principle, a modal analysis method is proposed to transfer the standard state-space model to the proposed modal-decomposition-dependent state-space model. Controllability and observability criteria are met to guarantee the minimum realization of the proposed state-space model. Finally, a modeling and modal analysis experiment is conducted on a developed wafer stage of lithographic tool. The results verify the effectiveness and accuracy of the proposed modeling method, as well as the controllability and observability of the proposed model.

Recursive Parsimonious Subspace Identification for Closed-Loop Hammerstein Nonlinear Systems

In this paper, a recursive closed-loop subspace identification method for Hammerstein nonlinear systems is proposed. To reduce the number of unknown parameters to be identified, the original hybrid system is decomposed as two parsimonious subsystems, with each subsystem being related directly to either the linear dynamics or the static nonlinearity. To avoid redundant computations, a recursive least-squares (RLS) algorithm is established for identifying the common terms in the two parsimonious subsystems, while another two RLS algorithms are established to estimate the coefficients of the nonlinear subsystem and the predictor Markov parameters of the linear subsystem, respectively. Subsequently, the system matrices of the linear subsystem are retrieved from the identified predictor Markov parameters. The convergence of the proposed method is analyzed. Two illustrative examples are shown to demonstrate the effectiveness and merit of the proposed method.

Control Performance Assessment for ILC-Controlled Batch Processes in a 2-D System Framework

In this paper, control performance assessment (CPA) is studied for batch processes controlled by iterative learning control (ILC). A 2-D linear quadratic Gaussian (LQG) benchmark is proposed to assess the performance of ILC in a 2-D framework. Based on the 2-D theory, an ILC-controlled batch process is first converted into a 2-D Roesser model. Subsequently, in order to assess the control performance of the converted 2-D system, the conventional LQG tradeoff curve is upgraded to the LQG performance assessment tradeoff surface. However, the complete knowledge of the system model is required to obtain the LQG tradeoff surface. For system without accurate model knowledge, a novel data-driven CPA method is further proposed. In this case, a novel 2-D closed-loop subspace identification method is proposed to identify the converted 2-D Roesser system. Based on the identified model, the LQG tradeoff surface can be obtained and utilized to assess the control performance. Overall, several simulation examples verified the feasibility and effectiveness of the proposed method.

Improved closed-loop subspace identification with prior information

It has been proven that combining open-loop subspace identification with prior information can promote the accuracy of obtaining state-space models. In this study, prior information is exploited to improve the accuracy of closed-loop subspace identification. The proposed approach initially removes the correlation between future input and past innovation, a significant obstacle in closed-loop subspace identification method. Then, each row of the extended subspace matrix equation is considered an optimal multi-step ahead predictor and prior information is expressed in the form of equality constraints. The constrained least squares method is used to obtain improved results, so that the accuracy of the closed-loop subspace can be enhanced. Simulation examples are provided to demonstrate the effectiveness of the proposed algorithm.

Model predictive control with closed-loop re-identification

In this work, we address the problem of handling plant-model mismatch by designing a subspace identification based MPC framework that includes model monitoring and closed-loop identification components. In contrast to performance monitoring based approaches, the validity of the underlying model is monitored by proposing two indexes that compare model predictions with measured past output. In the event that the model monitoring threshold is breached, a new model is identified using an adapted closed-loop subspace identification method. To retain the knowledge of the nominal system dynamics, the proposed approach uses the past training data and current input, output and set-point as the training data for re-identification. A model validity mechanism then checks if the new model predictions are better than the existing model, and if they are then the new model is utilized within the MPC. The effectiveness of the proposed method is illustrated through simulations on a nonlinear polymerization reactor.

Closed‐loop subspace identification methods: an overview

In this study, the authors present an overview of closed-loop subspace identification methods found in the recent literature. Since a significant number of algorithms has appeared over the last decade, the authors highlight some of the key algorithms that can be shown to have a common origin in autoregressive modelling. Many of the algorithms found in the literature are variants on the algorithms that are discussed here. In this study, the aim is to give a clear overview of some of the more successful methods presented throughout the last decade. Furthermore, the authors retrace these methods to a common origin and show how they differ. The methods are compared both on the basis of simulation examples and real data. Although the main focus in the literature has been on the identification of discrete-time models, identification of continuous-time models is also of practical interest. Hence, the authors also provide an overview of the continuous-time formulation of the identification framework.

Global data-driven modeling of wind turbines in the presence of turbulence

This paper presents a practical approach to identify a global model of a wind turbine from operational data, while it operates in a turbulent wind field with a varying mean wind speed and under closed-loop control. The approach is based on the realization that the nonlinearities are dominated by the aerodynamics of the rotor, which change with the operating condition. The dynamics of a wind turbine can be decomposed into a nonlinear static part, governed by the torque and thrust characteristics of the rotor, and a linear time-invariant dynamic part. The multi-input–multi-output linear dynamics are estimated using a recent closed-loop subspace identification method. The practical applicability of the algorithm is demonstrated by applying it to data obtained from the NREL CART 3 research turbine.

An Extended Closed-loop Subspace Identification Method for Error-in-variables Systems

A closed-loop subspace identification method is proposed for industrial systems subject to noisy input-output observations, known as the error-in-variables (EIV) problem. Using the orthogonal projection approach to eliminate the noise influence, consistent estimation is guaranteed for the deterministic part of such a system. A strict proof is given for analyzing the rank condition for such orthogonal projection, in order to use the principal component analysis (PCA) based singular value decomposition (SVD) to derive the extended observability matrix and lower triangular Toeliptz matrix of the plant state-space model. In the result, the plant state matrices can be retrieved in a transparent manner from the above matrices. An illustrative example is shown to demonstrate the effectiveness and merits of the proposed subspace identification method.

A local approach to LPV-identification of a Twin Rotor MIMO System

AbstractA local approach to LPV (Linear Parameter Varying) identification of a Twin Rotor MIMO System (TRMS) is considered via grey-box modeling. A white-box model of the TRMS is derived from the first principles, and black-box models at local operating points are given by a closed loop subspace identification method. Local black-box models that have consistent states are obtained by means of a frequency domain subspace method with the knowledge of the white-box model, and an LPV model is computed by interpolating the consistent local state-space models.

Effect of Varying Controller Parameters in Closed-Loop Subspace Identification

It is well known that subspace identification methods that assume open loop data without correlations between the input and noise, may give biased estimates when applied to closed loop data. The effect of the controller gain parameters on the quality of the identified model is studied when closed loop data are used. Several subspace identification methods (both open loop and closed loop methods), and different simulated data sets ranging from ideal 2 × 2 linear systems, to a fairly realistic nonlinear debutanizer process simulator, are considered. The results show that up to a point, higher controller gain during the identification experiment gives more accurate models than with lower controller gain, for both open and closed loop subspace identification methods. It is observed that the sensitivity to the controller gain is very small for the closed loop subspace method tested for the ideal cases when its assumptions are satisfied. An explaination for this is that in this case there will be no bias, while the open loop methods may have a bias that depends on the controller parameters. Another interesting observation is that for the debutanizer example, the nonlinearities seem to lead to biased estimates also with the closed loop subspace method, and the choice of controller gain appears to be just as important as the use of a closed loop subspace identification method for the accuracy of the estimates.

Subspace closed loop identification using the integration of MOESP and N4SID methods

Linear identification of time invariant systems operating in closed loop is of special interest for a large number of engineering applications. There are different techniques and methods to carry out this type of identification. For example, modifying the N4SID method, one can derive a closed loop subspace identification method. The same can be done using the MOESP method. Based on them, the MON4SID method is introduced, which estimates the extended observability matrix and the state sequence directly from a LQ decomposition, using a combination of the techniques contained in both, MOESP and N4SID. This new method uses an algorithm to identify state space model of a plant in a closed loop system, in the same way as in MOESP method. The advantage of the proposed algorithm is that it does not require any knowledge about the controller, whereas such information is essential for other methods (e.g. N4SID). The disadvantage of this algorithm is that it needs a great amount of data to obtain better estimates. A simulated process to show the performance of this algorithm is presented.

A Realization-Based Method for Identification of Closed-Loop Systems Using Measurable Disturbance

We derive a closed-loop subspace identification method based on a realization theory without using probing inputs, but with the assumption that there is a measurable disturbance that can be used as...

A New Subspace Identification Method for Closed-Loop Systems using Measurable Disturbance

In this paper, we consider a closed-loop subspace identification problem without using probing inputs; but we assume that there is a measurable disturbance which can be used as a test input for identification. Deterministic and stochastic subsystems are derived by applying the orthogonal decomposition (ORT) of the joint input-output process and realization methods. We develop a new ORT-based closed-loop subspace identification method, consisting of identification of the two subsystems. Some numerical results are included to show the applicability of the present method.

Asymptotic Variance of Closed-Loop Subspace Identification Methods

In this paper, the asymptotic properties of a version of the "innovation estimation" algorithm by Qin and Ljung as well as of a version of the "whitening filter" based algorithm introduced by Jansson are studied. Expressions for the asymptotic error as the sum of a "bias" term plus a "variance" term are given. The analysis is performed under rather mild assumptions on the spectrum of the joint input-output process; however, in order to avoid unnecessary complications, the asymptotic variance formulas are computed explicitly only for finite memory systems, i.e., of the ARX type. This assumption could be removed at the price of some technical complications; the simulation results confirm that when the past horizon is large enough (as compared to the predictor dynamics) the asymptotic expressions provide a good approximation of the asymptotic variance also for ARMAX systems

COMPARISONS OF SUBSPACE IDENTIFICATION METHODS FOR SYSTEMS OPERATING ON CLOSED-LOOP

In this paper, we analyze two recently proposed closed-loop subspace identification methods, referred to as innovation estimation method and whitening filter approach respectively. The similarity and difference between them are investigated in detail. It turns out that all closed-loop subspace identification methods can be classified as one-step, two-step, or multi-stage projection methods. A SISO closed-loop simulation shows that to identify a consistent model the whitening filter approach might require longer future and past horizons than the innovation estimation method.

Consistency analysis of some closed-loop subspace identification methods

We study statistical consistency of two recently proposed subspace identification algorithms for closed-loop systems. These algorithms may be seen as implementations of an abstract state-space construction procedure described by the authors in previous work on stochastic realization of closed-loop systems. A detailed error analysis is undertaken which shows that both algorithms are biased due to an unavoidable mishandling of initial conditions which occurs in closed-loop identification. Instability of the open loop system may also be a cause of trouble.

Closed loop identification method using a subspace approach

This paper presents a free model reduction closed loop subspace identification method for multivariable systems operating in a well posed closed loop environment. This allows to determine the impulse response of the system from the identification of the control system sensitivity functions. The rational behind the proposed method is twofold. Firstly, the general purpose design features of the subspace system identification approach and its inherent numerical robustness. Secondly, the model reduction performed in the available subspace identification is removed by a proper order selection procedure. A bias analysis is carried out to emphasize the robustness features.

SUBSPACE IDENTIFICATION OF CLOSED LOOP SYSTEMS BY STOCHASTIC REALIZATION

We develop a closed loop subspace identification method based on stochastic realization theory. Using the preliminary orthogonal decomposition of (Picci and Katayama, 1996b) we show that, under the assumption that the exogenous input is feedback-free and persistently exciting (PE), the identification of closed loop systems is divided into two subproblems: the deterministic identification of the plant and controller, and the stochastic identification of the noise filter. Subspace methods for identifying the deterministic and stochastic parts are derived by adapting the standard subspace methods to the deterministic and stochastic components. In both cases, a model reduction procedure should be applied in order to extract controllable and observable parts from non-minimal realizations. Some numerical results are included to show the applicability of the present technique.