Closed-loop Subspace Identification Research Articles

Closed-Loop Subspace Identification for Stable/ Unstable Systems Using Data Compression and Nuclear Norm Minimization

This paper provides a subspace method for closed-loop identification, which clearly specifies the model order from noisy measurement data. The method can handle long I/O data of the target system to be noise-tolerant and determine the model order via nuclear norm minimization. First, the proposed method compresses the long data by projecting them to an appropriate low dimensional subspace, then obtains a low order model whose order is specified by a combination of data compression and nuclear norm minimization. Its effectiveness is demonstrated through detailed numerical examples.

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.

Minimal realization of an unstable plant under feedback

Abstract This paper formulates a problem of finding a minimal realization of an unstable plant under feedback, given an infinite string of input-output data of the plant. It is assumed that the plant and noise models have non-common state matrices and that the plant is stabilized by a strictly proper linear-time-invariant controller of the finite order. A minimal realization of the innovation form has been estimated using closed-loop subspace identification algorithms, whereas that of the plant is determined in this paper. A matrix is proposed to avoid a numerical problem of a traditional algorithm applied to an unstable plant.

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.

Multivariable Constrained Adaptive Predictive Control based on Closed-loop Subspace Identification

In order to deal with nonlinear, time-varying, and multivariable constrained characteristics in closed-loop industrial processes, a multivariable constrained adaptive predictive control (CAPC) method based on closed-loop subspace identification is proposed. The state-space model is obtained through the closed-loop subspace identification algorithm, which is regarded as the system model. The algorithm is implemented online to update the R matrix with a receding window. By comparing the prediction errors before and after updating, it considers whether or not to update the system model. The model is then used to design the model predictive controller, which involves the solution of a quadratic program solving multivariable constraints. This paper presents a comparison between the performance of the proposed control method when applied to a 2-CSTR system, and that of an open-loop subspace CAPC method. The superiority of the proposed method is illustrated by the simulation results.

Closed-loop Subspace Identification of Dual-rate Non-uniformly Sampled System under MPC with Zone Control

Frequent changes in process dynamics make re-identification of a dynamic model prerequisite for sustainable application of model predictive control. When the process needs to comply with a particular operating range for product specification or safety requirement, the model should be re-identified in closed-loop. In addition to potentially poor exciting signal for the identification, another challenge is that many industrial processes are multi-rate systems whose variables have different sampling intervals. This paper proposes a re-identification method for dual-rate non-uniformly sampled systems under closed-loop with a MPC controller by lifting the original system. The proposed identification method provides accurate and realistic model compared to the model used before the identification is conducted. We also compare the identified model and the existing model by applying to MPC.

Improved closed-loop subspace identification based on principal component analysis and prior information

Subspace identification is a very useful tool for estimating a state-space model for a dynamic system. However, most of the subspace identification methods (SIMs) can only provide consistent estimations when the quality of the data is good. This problem can be solved by integrating prior information about a system into an identification procedure. In this paper, we propose a new approach for closed-loop SIMs based on principal component analysis (PCA) utilizing prior information. After performing the PCA procedure, we use the constrained least squares (CLS) approach with an equality constraint to incorporate prior information into the impulse response. The simulation results reveal that the proposed methods are more accurate and stable in model identification.

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.

Pole Assignment Control of MIMO Motion Systems With Flexible Structures and Its Application to an Ultraprecision Wafer Stage

With high velocity/acceleration and nanometer level motion accuracy requirements, the flexible dynamic behavior of ultraprecision wafer stage motion systems will significantly induce structural deformation, deteriorate control performance, and consequently, influence the final lithography accuracy. Inspired by this challenge, a new pole assignment control is proposed in this paper for multiinput–multiout (MIMO) systems with flexible structures. Specifically, through a closed-loop subspace identification and a modal approach, the modal state-space model is expressed as a diagonal subspace form that presents each rigid and flexible mode in these individual subspaces. Based on this model, a full-state feedback control architecture with a MIMO pole assignment control algorithm is developed to realize the modes’ control aims. The proposed control strategy is applied to a developed ultraprecision wafer stage. Comparative experiments were conducted on a single-input–single-output control, a MIMO control method with linear quadratic regulator control, and the proposed MIMO pole assignment algorithm. The results validate that the proposed scheme could achieve better disturbance rejection performance, transient performance, and excellent tracking accuracy in practical ultraprecision applications.

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.

RETRACTED ARTICLE: Gauss interference ant colony algorithm-based optimization of UAV mission planning

To enhance the effectiveness of UAV task planning optimization scheme, a UAV task planning optimization strategy based on Gaussian disturbance ant colony optimization (ACO) was proposed. Firstly, UAV task planning optimization model with expansion observability was obtained through correlation function estimation and subsequent zero space projection, as well as closed-loop subspace identification model and block matrix filling mode; secondly, ACO was used for solving multi-objective optimization problem so as to find out optimal character subset. Then, ACO was used for optimizing UAV task planning optimization model with expansion observability. At last, the simulation experiment verified the performance advantage of the proposed algorithm in the UAV task planning optimization scheme.

A new excitation scheme for closed-loop subspace identification using additional sampling outputs and its extension to instrumental variable method

It is known that feedback from outputs to inputs causes non-identifiability in many subspace identification algorithms. In this paper, a new excitation method is proposed for closed-loop subspace identification. By adding an output sampling point within two adjacent control times, a difficult closed loop identification issue is recast into an open loop identification. Uncorrelation between the process inputs and the noises in closed loop is obtained, and the persistently exciting condition is proven to be satisfied based on spectral analysis and innovation state space models so that the consistent estimates with no demand for external exciting signals can be guaranteed. Simultaneously, the additional sampling scheme offers different selections of instrumental variables for the non-error-in-variable case and the error-in-variable case in some instrumental variables-based identification algorithms. A simulation is performed to test the identification efficiency of the new scheme.

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.

Closed-loop subspace identification of multivariable dynamic errors-in-variables models based on ORT

In terms of the model of errors-in-variables, this article analyses the causes of deviation based on the existing method of subspace identification in the closed-loop system; then, it puts forward another method of subspace identification with an auxiliary variable based on orthogonal decomposition. The auxiliary variables can be selected and improved by this method, improving the quality of system identification.

Closed-loop subspace identification taking initial state into account

This paper develops a closed-loop subspace identification algorithm for the systems with the stochastic part possessing zeros close to the unit circle. The proposed algorithm estimates the system under the feedback loop, taking the effect of the initial state into account. A nonlinear optimization technique is applied for computing the Kalman gain and the covariance of the innovation. Numerical simulation shows that the proposed algorithm successfully provides models by taking the initial effect.

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 algorithm based on correlation function estimates

A novel subspace identification method based on correlation function which estimates a state-space system dynamics of unknown plant operating in closed-loop experimental condition is proposed in this paper. It is shown that the cross-correlation function of the output and external input signals are equal to the cross-correlation function of the input and external signals filtered through the system dynamics since noise signal has no correlation with the external input. The proposed algorithm is developed to obtain unbiased estimates of system matrices based on time-shifted invariance of the correlation function estimates. Later the algorithm is compared to other popular subspace methods in the simulation study and the results show the effectiveness of our method in the presence of colored noise and low signal-to-noise ratios.

Closed-loop Subspace Identification Algorithms for 2-D Separable-in-Denominator Systems

Identification for closed-loop two-dimensional (2-D) causal, recursive, and separable- in-denominator (CRSD) systems in the Roesser form is discussed in this study. For closed- loop 2-D CRSD systems, in the presence of feedback control, there exists correlation between the unknown disturbances and the future inputs, which offers the fundamental limitation for utilizing the standard open-loop 2-D CRSD subspace identification methods. In other words, the existing open-loop 2-D CRSD subspace algorithms may result in biased estimates of plant parameters under closed-loop conditions. In this paper, based on the instrument variable method and principal component analysis, a novel 2-D CRSD subspace identification algorithm that are applicable to both open-loop and closed-loop data are developed. Additionally, we discuss the whiteness external excitation case, explain why the white set-point signal may influence the estimated model results, and subsequently adopt modified instrument variables to improve the proposed subspace identification method. Several numerical examples have been conducted to validate consistency and efficiency of the proposed subspace identification approaches.

Adaptive Predictive Control: A Data-Driven Closed-Loop Subspace Identification Approach

This paper presents a data-driven adaptive predictive control method using closed-loop subspace identification. As the predictor is the key element of the predictive controller, we propose to derive such predictor based on the subspace matrices which are obtained through the closed-loop subspace identification algorithm driven by input-output data. Taking advantage of transformational system model, the closed-loop data is effectively processed in this subspace algorithm. By combining the merits of receding window and recursive identification methods, an adaptive mechanism for online updating subspace matrices is given. Further, the data inspection strategy is introduced to eliminate the negative impact of the harmful (or useless) data on the system performance. The problems of online excitation data inaccuracy and closed-loop identification in adaptive control are well solved in the proposed method. Simulation results show the efficiency of this method.

Subspace Identification of Unstable Systems by MON4SID algorithm

In this paper, we consider a closed loop subspace identification problem. Here the open loop processes are unstable. By using the subspace identification algorithm, the closed loop system is first identified. The plant dynamics are extracted from the identified closed loop system. Three unstable processes are simulated and identified by the MON4SID algorithm.