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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.

This paper presents a closed-loop continuous-time subspace identification method using prior information. Based on a rational inner function, a generalized orthonormal basis can be constructed, and the transformed noises have ergodicity features. The continuous-time stochastic system is converted into a discrete-time stochastic system by using generalized orthogonal basis functions. As is known to all, incorporating prior information into identification strategies can increase the precision of the identified model. To enhance the precision of the identification method, prior information is integrated through the use of constrained least squares, and principal component analysis is adopted to achieve the reliable estimate of the system. Moreover, the identification of open-loop models is the primary intent of the continuous-time system identification approaches. For closed-loop systems, the open-loop subspace identification methods may produce biased results. Principal component analysis, which reliably estimates closed-loop systems, provides a solution to this problem. The restricted least-squares method with an equality constraint is used to incorporate prior information into the impulse response following the principal component analysis. The input–output algebraic equation yielded an optimal multi-step-ahead predictor, and the equality constraints describe the prior information. The effectiveness of the proposed method is provided by numerical simulations.

This paper proposes a subspace identification method for closed-loop EIV (errors-in-variables) problems based on instrumental variables . First, a unified framework is derived, and then the reason is discussed why some existing subspace methods based on instrumental variables could be biased under closed-loop conditions. Afterwards a remedy is given to eliminate the bias by simply replacing the instrumental variable. Using orthogonal projection, the resulting instrumental variable method is very simple and easy to extend. In addition, simulation studies illustrate the effects of different instrumental variables.

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.

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.

In recent years, a considerable amount of effort has been devoted, both in industry and academia, towards the development of advanced methods of control theory with focus on its practical implementation in various fields of human activity such as space control, robotics, control applications in marine systems, control processes in agriculture and food production. Control Systems: Theory and Applications consists of selected best papers which were presented at XXIV International conference on automatic control “Automatics 2017” (September 13–15, 2017, Kyiv, Ukraine) organized by Ukrainian Association on Automatic Control (National member organization of IFAC – International Federation on Automatic Control) and National University of Life and Environmental Sciences of Ukraine. More than 120 presentations where discussed at the conference, with participation of the scientists from the numerous countries. The book is divided into two main parts, a first on Theory of Automatic Control (5 chapters) and the second on Control Systems Applications (8 chapters). The selected chapters provide an overview of challenges in the area of control systems design, modeling, engineering and implementation and the approaches and techniques that relevant research groups within this area are employing to try to resolve these. This book on advanced methods of control theory and successful cases in the practical implementation is ideal for personnel in modern technological processes automation and SCADA systems, robotics, space and marine industries as well as academic staff and master/research students in computerized control systems, automatized and computer-integrated systems, electrical and mechanical engineering.

This paper presents the order reduction for plant and controller models estimated by the joint input-output approach in the closed loop identification, named the controllability or observability intensity truncation (COIT) method in which the index for order reduction is the controllability intensity or the observability intensity. The use of these intensities depends on the estimated models obtained by the joint input-output approach. In the application of the closed loop identification with the balanced truncation (BT) and the COIT methods; a numerical example and an inverted pendulum experimental system, the COIT method appropriately reduces the order of the estimated plant and controller models and is superior to the BT method.

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, under feedback control condition, there exists some correlation between the unknown disturbances and future inputs which offers the fundamental limitation for utilizing standard open-loop 2-D CRSD systems subspace identification methods. In other words, the existing open-loop subspace approaches will result in biased estimates of plant parameters from closed-loop data. In this study, based on orthogonal projection and principal component analysis, novel 2-D CRSD subspace identification methods are developed, which are applicable to both open-loop and closed-loop data. Additionally, the whiteness external excitation case is discussed and subsequently modified instrument variables are adopted to improve the proposed subspace algorithm. An illustrative example of the injection molding process and several numerical examples are used to validate consistency and efficiency of the proposed subspace approaches for 2-D CRSD systems.

For the identification problem of closed-loop subspace model, we propose a zero space projection method based on the estimation of correlation function to fill the block Hankel matrix of identification model by combining the linear algebra with geometry. By using the same projection of related data in time offset set and LQ decomposition, the multiplication operation of projection is achieved and dynamics estimation of the unknown equipment system model is obtained. Consequently, we have solved the problem of biased estimation caused when the open-loop subspace identification algorithm is applied to the closed-loop identification. A simulation example is given to show the effectiveness of the proposed approach. In final, the practicability of the identification algorithm is verified by hardware test of UAV servo system in real environment.

A novel promising bootstrap subspace system identification algorithm for both open and closed loop systems is presented. An outline of the SSARX algorithm by Jansson (2003) is given and a modified SSARX algorithm is presented. Some methods which are consistent for closed loop subspace system identification presented in the literature are discussed and compared to a recently published subspace algorithm which works for both open as well as for closed loop data, i.e., the DSR_e algorithm as well as the new bootstrap subspace method presented in this paper. Experimental comparisons are performed by Monte Carlo simulations.