This paper introduces an input-output bilinear Koopman realization with an optimization algorithm of lifting functions. For nonlinear systems with inputs, Koopman-based modeling is effective because the Koopman operator enables a high-dimensional linear representation of nonlinear dynamics. However, traditional approaches face significant challenges in industrial applications. Measuring all system states is often impractical due to constraints on sensor installation. Moreover, the predictive performance of a Koopman model strongly depends on the choice of lifting functions, and their design typically requires substantial manual effort. In addition, although a linear time-invariant (LTI) Koopman model is the most commonly used model structure in the Koopman framework, such model exhibit limited predictive accuracy. To address these limitations, we propose an input-output bilinear Koopman modeling in which the design parameters of radial basis function (RBF)-based lifting functions are optimized using a global metaheuristic algorithm to improve long-term prediction performance. Consideration of the long-term prediction performance enhances the reliability of the resulting model. The proposed methodology is validated in simulations and experimental tests, with the airpath control system of a diesel engine as the plant to be modeled. This plant represents a challenging industrial application because it exhibits strong nonlinearities and coupled multi-input multi-output (MIMO) dynamics. These results demonstrate that the proposed input-output bilinear Koopman model significantly outperforms traditional linear Koopman models in predictive accuracy.
A. Motivation S Ystem identification is essential for enabling system pre- diction, control, and fault detection. Traditional system identification methods, such as Auto-Regressive with eXogenous (ARX) models and Numerical Algorithms for Subspace State Space System Identification (N4SID) [1] have proven highly effective for linear systems. However, identifying models for highly nonlinear industrial systems remains challenging, which in turn complicates the control of complex industrial processes. To address this issue, data-driven approaches for nonlinear systems have been proposed, including the Koopman operator framework [2], [3], [4], deep learning methods such as recurrent neural networks (NNs) and long short-term memory (LSTM) networks [5], nonlinear ARX (NARX) models [6], and sparse identification of nonlinear dynamics (SINDy) [7], [8].
The Koopman operator is formulated as a linear operator in an infinite-dimensional space by mapping the original state space to a higher-dimensional space through lifting functions [9], [10]. Its ability to represent a nonlinear system as a linear model is a distinctive feature not found in other data-driven approaches. Although Koopman operator theory inherently deals with infinite-dimensional spaces, in practical computations, a finite-dimensional approximation of the Koopman operator can be identified from data using dynamic mode decomposition (DMD) [11], [12] and extended DMD (EDMD) [13], [14]. The resulting Koopman model can accurately capture the original nonlinear behavior by leveraging a high-dimensional linear structure. This enables the application of conventional linear control theory [2], [15], [16]. Despite these achievements, several challenges for application to industrial systems remain, including the selection of lifting functions that significantly impact prediction performance, the inherent limitations of linear model predictions, and the unavailability of measurements for the full system states.
It has been noted that LTI Koopman models may not adequately capture the control-affine dynamics of nonlinear systems, which makes accurate prediction challenging [17].
To address this limitation, bilinear Koopman realizations have been investigated. These models offer improved predictive accuracy compared to LTI Koopman models while remaining computationally more efficient than fully nonlinear Koopman approaches [18]. Theoretical foundations of bilinear Koopman realizations have been explored in [18], [19]. In addition, methods for computing bilinear lifting functions using deep NNs have been proposed [20], [21], [22]. The choice of the observation function that maps state variables to observable variables-commonly referred to as the lifting function-has a significant impact on the predictive performance of the Koopman model. According to [23], previous approaches for designing lifting functions include mechanics-inspired selections, empirical selections such as monomials and polynomials, and RBF functions with randomly assigned centers. Another proposed approach employs highly contributory basis functions derived from SINDy modeling [24], [25]. However, these methods often fail to ensure reproducibility or improve the predictive accuracy of model identification. To address these limitations, recent research has explored neural network-based approaches for lifting function design [26], [27]. Nevertheless, NN-based methods present challenges such as hyperparameter tuning, overfitting, model complexity, computational cost, and issues related to vanishing or exploding gradients. Furthermore, the orthogonality of NN-based lifting functions is not guaranteed. To date, meta-heuristic approaches such as particle swarm optimization (PSO) have not been investigated for lifting function design.
Previously, LTI and bilinear Koopman realizations typically assumed that all system states and inputs were measurable. However, in practical industrial applications, it is often infeasible to deploy sensors capable of capture every state variable. To address this limitation, a Koopman realization framework that relies solely on input-output data is required. This perspective is particularly important for industrial implementation. Moreover, prior research on Koopman operators using input-output data [2], [25], [28] has been limited, with most studies focusing exclusively on LTI Koopman formulations. Additionally, the selection of the arguments (i.e., embedded states) of the lifting function for controlled systems has not been thoroughly investigated.
Although modeling of diesel engine airpath systems has been widely investigated [29], [30], [31], achieving accurate and robust models remains highly challenging. These systems exhibit nonlinear and multivariable dynamics. The traditional approaches rely on physical modeling [30], often represented as linear parameter-varying (LPV) systems [32]. However, physical modelings suffer from limited accuracy and difficulties in para
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