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Generalized bilinear Koopman realization from input-output data for multi-step prediction with metaheuristic optimization of lifting function and its application to real-world industrial system

Shuichi Yahagi, Ansei Yonezawa, Heisei Yonezawa, Hiroki Seto, Itsuro Kajiwara

TL;DR

Addressing prediction and control of nonlinear industrial systems with limited sensing, the paper develops an input-output generalized bilinear Koopman realization whose lifting functions are optimized via a metaheuristic algorithm. It employs time-delay coordinates to form an IO embedding and designs lifting through radial-basis-function centers optimized by PSO, while carefully restricting lifting arguments to avoid input-induced spurious dynamics. The approach is validated on a diesel engine airpath system through simulation and engine bench experiments, showing clear improvements over LTI and standard bilinear Koopman realizations, with $R^2>0.9$ in long-horizon predictions. This work expands the applicability of Koopman-based modeling to industrial settings by enabling reliable, data-driven, long-horizon predictions under partial observability and without exhaustive state sensing.

Abstract

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.

Generalized bilinear Koopman realization from input-output data for multi-step prediction with metaheuristic optimization of lifting function and its application to real-world industrial system

TL;DR

Addressing prediction and control of nonlinear industrial systems with limited sensing, the paper develops an input-output generalized bilinear Koopman realization whose lifting functions are optimized via a metaheuristic algorithm. It employs time-delay coordinates to form an IO embedding and designs lifting through radial-basis-function centers optimized by PSO, while carefully restricting lifting arguments to avoid input-induced spurious dynamics. The approach is validated on a diesel engine airpath system through simulation and engine bench experiments, showing clear improvements over LTI and standard bilinear Koopman realizations, with in long-horizon predictions. This work expands the applicability of Koopman-based modeling to industrial settings by enabling reliable, data-driven, long-horizon predictions under partial observability and without exhaustive state sensing.

Abstract

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.
Paper Structure (27 sections, 2 theorems, 39 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 2 theorems, 39 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work
  4. Contribution and novelty
  5. Article organization
  6. Notation
  7. Data-driven modeling via Koopman operator
  8. Koopman operator for non-autonomous system
  9. Finite-dimensional Koopman operator
  10. LTI Koopman form
  11. Bilinear Koopman form
  12. Learning of Koopman matrix
  13. Proposed methodology
  14. Time-delay coordinates
  15. IO generalized bilinear Koopman form
  16. ...and 12 more sections

Key Result

Theorem 1

Consider the IO generalized bilinear Koopman realization (eq:io_general_bilinear) with the state-transition structure in (eq:transision_time-delay), where $z_k(=\psi_x(\cdot))$ represents “state’’ variables updated by a time-invariant linear map $A$, and the new input $w_k$ is injected via $B_0$ and

Figures (6)

  • Figure 1: Diesel engine airpath system.
  • Figure 2: The learning and test data in simulation.
  • Figure 3: The long-term predictive performance of $y_1$ and $y_2$ for test data in simulation.
  • Figure 4: Engine bench test. Fuel type:diesel, Arrangement cylinders: 4 in line, Maximum power:110 kW (150 PS)3600 rpm, Maximum torque: 350 Nm (35.69 kgm)1800-2600 rpm (net), Combustion type: Direct injection with water-cooled 4-valve DOHC (double overhead camshaft).
  • Figure 5: The learning and test data in experimental test.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Corollary 1