MPC Quadratic Program Solver Using Variable Ordering and Factor Updates
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Solution Overview
Problem
Existing quadratic programming solvers for model predictive control face challenges with high computational complexity and memory requirements, particularly in embedded systems with limited CPU power and memory, due to the rapid growth of possible factors with the number of optimization variables, which limits their applicability for large-scale problems.
Innovation Solution
The system employs a factorization module with a memory containing saved factors and a factor search mechanism to find the nearest stored factor, allowing for efficient factor updates with zero floating-point operations per unit time, using variable ordering to reduce the number of factors stored and computed, thereby decreasing computational complexity and memory needs.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If standard quadratic programming solvers are used for model predictive control, then solution accuracy is maintained, but computational complexity and memory requirements grow rapidly with the number of optimization variables
Solution Approach 1:
The patent applies preliminary action by precomputing and storing factors (such as Cholesky decomposition factors) that will be needed during online optimization. These precomputed factors are stored in a database and retrieved during runtime, avoiding the need to perform computationally intensive factorization operations in real-time. This resolves the contradiction by shifting computational burden from online execution to offline preparation, maintaining solution accuracy while reducing online computational complexity.
Solution Approach 2:
The patent segments the computational process into distinct phases: offline factorization and storage, and online factor retrieval and update. By dividing the quadratic programming solution process into these segments, the computationally intensive factorization is performed offline and stored, while online operations only require retrieving precomputed factors and performing simple updates. This segmentation resolves the contradiction between maintaining accurate solutions and reducing computational complexity during online execution.
2Reliability
If standard quadratic programming solvers are used for model predictive control, then solution reliability is maintained, but memory requirements increase rapidly with the number of optimization variables
Solution Approach 1:
The patent applies preliminary action by precomputing and storing factorization results (such as LDL or Cholesky factors) in an offline phase. These precomputed factors are saved in a database with significantly reduced memory footprint compared to storing complete Hessian matrices or other intermediate computational structures. During online execution, only these compact precomputed factors are retrieved and used, maintaining solution reliability while dramatically reducing memory requirements.
Solution Approach 2:
The patent uses copying by storing compressed representations (factors) of the original computational structures. Instead of storing complete Hessian matrices or full quadratic programming problem data, the system stores factorized forms that can be efficiently copied and reused. These factor copies maintain the mathematical properties needed for reliable solutions while occupying minimal memory space, resolving the contradiction between reliability and memory requirements.
3Adaptability or versatility
If the number of optimization variables is increased to handle large-scale problems, then problem coverage is improved, but the number of factors that need to be stored grows rapidly
Solution Approach 1:
The patent applies parameter changes by transforming the storage representation from complete Hessian matrices to their factorized forms (LDL or Cholesky factors). This parameter transformation reduces the memory requirements from O(n²) to O(n) where n is the number of optimization variables. By changing the storage parameter from full matrices to compact factors, the system can handle large-scale problems with increased problem coverage while the memory growth becomes linear rather than quadratic.
Solution Approach 2:
The patent applies preliminary action by performing factorization offline and storing only the essential factor structures. This preliminary processing allows the system to handle larger numbers of optimization variables because the computationally intensive and memory-intensive factorization is done in advance. The precomputed factors can be efficiently stored and reused, enabling the system to scale to larger problems without the memory requirements growing rapidly during online execution.
Data Source
AI summary
A system and approach for storing factors in a quadratic programming solver of an embedded model predictive control platform. The solver may be connected to an optimization model which may be connected to a factorization module. The factorization module may incorporate a memory containing saved factors that may be connected to a factor search mechanism to find a nearest stored factor in the memory. A factor update unit may be connected to the factor search mechanism to obtain the nearest stored factor to perform a factor update. The factorization module may provide variable ordering to reduce a number of factors that need to be stored to permit the factors to be updated at zero floating point operations per unit of time.