Analytic Linearization for Singular Mass Matrix Systems

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Solution Overview

Problem

The standard approach to linearizing dynamic systems with singular mass matrices is expensive and time-consuming, relying on perturbation techniques that provide approximate solutions.

Innovation Solution

The method uses analytic Jacobians to generate a state-space representation of dynamic systems with singular mass matrices, eliminating the need for perturbation techniques and enabling efficient linearization.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Measurement precision

If perturbation techniques are used to linearize systems with singular mass matrices, then a state-space representation can be obtained, but the process becomes expensive and time-consuming

Engineering Contradiction:

Improvelinearization accuracyVSAvoidlinearization time

Core Design Contradiction:

Measurement precisionVSLoss of time

Solution Approach 1:

The patent replaces the mechanical perturbation-based linearization process with an analytical approach using symbolic mathematics. Instead of applying small perturbations to state and input variables and measuring resulting changes, the invention directly computes analytic Jacobians of the system equations, eliminating the need for iterative perturbation measurements and significantly reducing computation time while maintaining accuracy

Inventive Principle:

Principle #28Mechanics substitution (Replace mechanical system)

Solution Approach 2:

The patent performs preliminary symbolic differentiation to derive closed-form expressions for the Jacobian matrices before numerical evaluation. By pre-computing the analytic Jacobians symbolically, the system avoids the need for repeated perturbation trials during the linearization process, thereby reducing overall computation time while preserving measurement precision

Inventive Principle:

Principle #10Preliminary action

2Measurement precision

If perturbation techniques are used to obtain state-space representation, then the representation can be derived, but the complexity of the process increases

Engineering Contradiction:

Improvemodel representation accuracyVSAvoidlinearization process complexity

Core Design Contradiction:

Measurement precisionVSDevice complexity

Solution Approach 1:

The patent substitutes the complex iterative perturbation methodology with a direct analytical computation approach. By using symbolic mathematics to compute Jacobians directly from the system equations, the invention simplifies the overall process while maintaining the ability to handle singular mass matrices, thereby reducing procedural complexity without sacrificing model accuracy

Inventive Principle:

Principle #28Mechanics substitution (Replace mechanical system)

3Ease of operation

If standard linearization approaches are used for singular mass matrices, then descriptor form is obtained, but state-space representation cannot be directly achieved

Engineering Contradiction:

Improveease of working with state-space representationVSAvoidlinearization accuracy

Core Design Contradiction:

Ease of operationVSMeasurement precision

Solution Approach 1:

The patent performs preliminary symbolic analysis of the system equations to directly derive the state-space representation in closed form. By pre-computing the analytic Jacobians and using them to construct the state-space matrices A, B, C, and D directly, the invention achieves both ease of operation with state-space models and high linearization accuracy, avoiding the intermediate descriptor form

Inventive Principle:

Principle #10Preliminary action

Data Source

PatentUS7962317B1Analytic linearization for system design

Publication Date: 2011.06.14 MATHWORKS INC

AI summary

A method and apparatus may linearize a model representing a dynamic system without using perturbation techniques. The model may include a differential-algebraic system of equations to represent the dynamic system. The mass matrix of the model may be singular. The linear model of the system may be generated in a state-space representation using the analytic Jacobians of the model.