Preconditioned Model Predictive Control Matrix Solver
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Solution Overview
Problem
Model predictive control (MPC) methods, particularly nonlinear MPC (NMPC), face challenges in achieving fast and accurate solutions for matrix equations due to the complexity and variability of the coefficient matrix, leading to slow convergence and reduced performance in real-time control applications, especially on resource-constrained systems.
Innovation Solution
The method involves determining an approximate coefficient matrix with significant entries identified through a map, using an iterative solver with a preconditioner matrix that is sparse and not block diagonal, allowing for efficient online construction and application of the preconditioner to reduce computational complexity and improve real-time performance.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If exact Newton-type optimization schemes are used with analytic Jacobian expressions, then solution accuracy is improved, but computational complexity and implementation difficulty increase significantly
Solution Approach 1:
The patent uses finite difference approximations of the Jacobian matrix instead of exact analytic expressions. This replaces the complex, difficult-to-implement exact Jacobian with a simpler, computationally cheaper approximation that can be easily calculated using numerical differentiation, sacrificing some precision for significant gains in implementability
Solution Approach 2:
The patent substitutes the mechanical/computational process of deriving and implementing analytic Jacobian expressions with a numerical approximation approach. Instead of requiring symbolic mathematics and complex derivations, the system uses finite difference methods that replace the analytic computation mechanism with a simpler numerical differentiation process
2Ease of operation
If conventional matrix-free iterative methods like GMRES are used to solve the matrix equation, then ease of operation is improved, but convergence speed deteriorates leading to increased computational time
Solution Approach 1:
The patent applies preconditioning techniques before the main iterative solving process. By transforming the original matrix equation Ax=b into a preconditioned form M⁻¹Ax=M⁻¹b where M is a preconditioner matrix, the system prepares the problem in advance to improve convergence properties, reducing the number of iterations required while maintaining the matrix-free approach
Solution Approach 2:
The patent modifies the system parameters by introducing a preconditioner that changes the spectral properties of the coefficient matrix. This parameter transformation improves the condition number and eigenvalue distribution, leading to faster convergence of the iterative solver without requiring fundamental changes to the algorithm structure
3Measurement precision
If full coefficient matrix is used in iterative solver, then solution precision is improved, but computational resources and processing time increase
Solution Approach 1:
The patent segments the computational process into matrix-free evaluation of the coefficient function and iterative solving with preconditioning. Instead of forming and storing the full coefficient matrix, the system evaluates the coefficient function A(x)·v only when needed during matrix-vector products, dividing the computation into on-demand evaluations that reduce memory usage and computational overhead
Solution Approach 2:
The patent extracts only the essential information from the coefficient matrix by using matrix-vector products instead of explicit matrix representation. The full matrix structure is not computed or stored; only its action on vectors is evaluated through the coefficient function, removing the computational burden of handling the complete matrix while preserving the necessary mathematical properties
Data Source
AI summary
A method for a model predictive control (MPC) of a system determines entries of an approximate coefficient matrix only at locations identified in a map of locations as significant. The map of locations identifies each location of an entry in the approximate coefficient matrix as either significant or insignificant. The entries are determined using one or combination of an approximate coefficient function and an exact coefficient function. Next, the method determines a preconditioner using the approximate coefficient matrix and determines a solution vector by solving a matrix equation of the MPC with a coefficient matrix defined by an exact coefficient function at a current time step of a control using an iterative method with the preconditioner. The method generates a control signal for controlling the system using the solution vector.