Iterative Subspace Identification for Nonlinear System Modeling
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Solution Overview
Problem
Current methods for modeling nonlinear and time-varying dynamic systems face inefficiencies due to exponential expansion of dimensions, particularly in subspace identification approaches, which lead to computational challenges and inaccurate results, especially when dealing with large-scale industrial processes.
Innovation Solution
The method involves expanding state space difference equations into linear, time-invariant forms using augmented inputs and outputs, and employing an iterative algorithm to estimate coefficients, allowing for reduced model generation and efficient computation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If direct extensions of linear subspace methods are used for modeling nonlinear systems by expressing past and future as linear combinations of nonlinear functions, then the approach can handle nonlinear and time-varying systems, but the dimension of the past and future expand exponentially with the number of measured inputs, outputs, states, and lags
Solution Approach 1:
The patent segments the exponentially expanding dimensional space into manageable linear subspace components. By decomposing the high-dimensional nonlinear system into smaller linear subsystems that can be processed independently, the method maintains the ability to model complex nonlinear and time-varying behavior while avoiding the computational burden of handling the full exponential dimensionality directly.
Solution Approach 2:
The patent transforms the problem by changing parameters from the original high-dimensional nonlinear state representation to a reduced-dimensional linear subspace representation. This parameter transformation allows the system to capture essential nonlinear dynamics through carefully selected basis functions and subspace projections, effectively reducing the dimensional complexity while preserving modeling accuracy.
2Productivity
If iterative subspace approach is used to estimating nonlinear terms in the model, then very modest computation is required, but the scheduling function is usually determined by the particular application and is often very non-random in character, reducing modeling accuracy
Solution Approach 1:
The patent incorporates feedback mechanisms in the iterative subspace identification process. By using the estimated states from previous iterations as feedback to refine the scheduling function estimates and update the subspace projections, the method progressively improves modeling accuracy even for nonrandom scheduling functions while maintaining computational efficiency through the iterative refinement approach.
Solution Approach 2:
The patent applies dynamic adaptation in the iterative subspace method by allowing the subspace basis functions and projection matrices to evolve dynamically across iterations. This dynamic adjustment enables the algorithm to adapt to the specific characteristics of nonrandom scheduling functions in each application context, improving accuracy without requiring excessive computational resources.
3Measurement precision
If general nonlinear canonical variate analysis procedure is used to identify nonlinear systems, then nonlinear functions of the past and future are determined to describe the state of the process, but the number of required nonlinear functions of past and future expand exponentially
Solution Approach 1:
The patent extracts only the essential nonlinear characteristics from the full set of possible nonlinear functions. By identifying and extracting the dominant subspace components that capture the critical system behavior, the method achieves accurate state description without requiring all possible nonlinear functions, thereby reducing the exponential complexity to a manageable subset of essential terms.
Solution Approach 2:
The patent transitions from the original high-dimensional nonlinear function space to a reduced-dimensional linear subspace through dimensional transformation. By projecting the complex nonlinear relationships onto a lower-dimensional subspace spanned by selected basis functions, the method maintains state description accuracy while dramatically reducing the number of required functions from exponential to polynomial or linear scaling.
Data Source
AI summary
Methods and systems for estimating differential or difference equations that can govern a nonlinear, time-varying and parameter-varying dynamic process or system. The methods and systems for estimating the equations may be based upon estimations of observed outputs and, when desired, input data for the equations. The methods and systems can be utilized with any system or process that may be capable of being described with nonlinear, time-varying and parameter-varying difference equations and can used for automated extraction of the difference equations in describing detailed system or method behavior for use in system control, fault detection, state estimation and prediction and adaptation of the same to changes in a system or method.