This paper studies the metaheuristic optimizer-based direct identification of a multiple-mode system consisting of a finite set of linear regression representations of subsystems. To this end, the concept of a multiple-mode linear regression model is first introduced, and its identification issues are established. A method for reducing the identification problem for multiple-mode models to an optimization problem is also described in detail. Then, to overcome the difficulties that arise because the formulated optimization problem is inherently ill-conditioned and nonconvex, the cyclic-network-topology-based constrained particle swarm optimizer (CNT-CPSO) is introduced, and a concrete procedure for the CNT-CPSO-based identification methodology is developed. This scheme requires no prior knowledge of the mode transitions between subsystems and, unlike some conventional methods, can handle a large amount of data without difficulty during the identification process. This is one of the distinguishing features of the proposed method. The paper also considers an extension of the CNT-CPSO-based identification scheme that makes it possible to simultaneously obtain both the optimal parameters of the multiple submodels and a certain decision parameter involved in the mode transition criteria. Finally, an experimental setup using a DC motor system is established to demonstrate the practical usability of the proposed metaheuristic optimizer-based identification scheme for developing a multiple-mode linear regression model.
The derivation of reasonable mathematical models is the most important part of designing and analyzing control systems; thus, many theoretical and applied studies have been devoted to this research subject [
One attractive approach is to introduce switched linear and piecewise affine (PWA) models (see [
Inferring a multiple-mode model from a set of finite input-output measurements is a highly complex process that requires simultaneous estimation of both the mode transitions and the linear subsystems. The underlying identification problem is inherently nonconvex and admits multiple local solutions, so the formulated optimization problem is not well posed [
This paper studies the metaheuristic optimization-based identification of a multiple-mode system, which consists of a finite set of linear regression representations of subsystems, from a collection of input-output data. Note that the multiple-mode linear regression model is simple to describe and captures essential properties of multiple-mode models. In addition, its identification problem is identical to parameter estimation of the subsystems included in PWA systems, without any prior knowledge of their mode transition. Therefore, the investigation of the multiple-mode linear regression model can play an essential role in a variety of control engineering problems. This paper first shows how to reduce the identification problem for a multiple-mode model into an optimization problem. Then, the cyclic-network-topology-based constrained particle swarm optimizer (CNT-CPSO) is applied to solve the formulated nonconvex optimization problem, with no prior knowledge regarding the mode transitions between subsystems. Note that compared to the standard PSO, the CNT-CPSO scheme exhibits improved performance when searching for the global optimum [
The remainder of this paper is organized as follows. In Section
This section first introduces the concept of a multiple-mode linear regression model and establishes the identification issues. Then, a method for reducing the identification problem for the multiple-mode model into an optimization problem is described.
The multiple-mode linear regression model formulated as
Let If Figure
Data set
Standard linear regression model
Two-mode linear regression model
Now suppose that a collection of
Calculation of the objective function.
For the above optimization problem for multiple-mode linear regression model identification, the following theorem presents the condition under which the estimated set of optimal coefficient vectors becomes uniquely identical to the true set of coefficient vectors.
Assume that a collection of
Its proof, which is explained as follows, is self-evident. The optimal solution mentioned above lets the objective function in (
In this section, the CNT-CPSO-based direct identification scheme for multiple-mode linear regression models is described in detail. The design parameter vector is defined as
Otherwise, go to Step
Next, go to Step
Following the above identification procedure using the CNT-CPSO tool, the formulated optimization problem (
Experiments were conducted with a measured set of input-output data from the DC motor system shown in Figure
Experimental DC motor system.
The experimental DC motor system was excited with a sinusoidal electric current
Experimental data: current input
The multiple-mode linear regression model for
To find the estimate of
Model outputs corresponding to
From the parameter vectors (
Time histories of
Comparison of the output
Some remarks on the characteristics observed from the set of identified parameter vectors
This section considers an extension of the CNT-CPSO-based identification for a multiple-mode linear regression model that enables us to find not only the optimal parameters of the submodels, but also a certain decision parameter involved in the mode transition criteria. Its effectiveness is examined using a set of finite input-output measurements obtained from the experimental DC motor system described above.
The experimental results presented in the previous subsection show that mode switching,
The multiple-mode linear regression model (
Under the above problem formulation, our CNT-CPSO-based identification procedure described in Section
The identified parameter vector set
Time behavior of the angular velocity
Time histories of
Angular acceleration predicted using the identified model parameters in (
In this paper, the CNT-CPSO-based direct identification of a multiple-mode system was studied, and two new strategies were introduced. The first strategy was to introduce a multiple-mode system consisting of a finite set of linear regression representations of subsystems and then to reduce the identification problem for such a multiple-mode model into an optimization problem. The target systems include the subsystems of PWA systems, and the introduced method is applicable regardless of the mode transition mechanism of the target system. The second strategy was to adopt a metaheuristic optimizer, the CNT-CPSO algorithm, which was developed relatively recently by some of the authors of this paper. This tool plays a key role in addressing some complex difficulties arising due to the inherent ill-conditioned and nonconvex nature of the formulated optimization problem. Then, a concrete procedure for applying the CNT-CPSO-based identification methodology to develop a multiple-mode linear regression model was described in detail. This scheme requires no prior knowledge of mode transitions between subsystems and, unlike some conventional methods, can handle a large amount of data without difficulty during the identification process, which is another distinguishing feature of the proposed method. Finally, an experiment was conducted on a DC motor system to evaluate and demonstrate the practical usability of the proposed metaheuristic optimizer-based identification scheme. Using the same experimental setup, an extension of the CNT-CPSO-based identification scheme was also proposed and examined. Comparison of the experimental results verified that this extension makes it possible to simultaneously obtain both the optimal parameters of multiple submodels and the designated decision parameter involved in the introduced mode transition criteria.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03935288) and the Chung-Ang University Excellent Student Scholarship in 2016.