Nonlinear Predictive Control with Block-Wise Jacobian Updates
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Solution Overview
Problem
Real-time predictive control of nonlinear dynamical systems is hindered by the high computational cost of solving nonlinear optimization problems, particularly due to costly Jacobian evaluations and matrix operations in SQP methods, which are exacerbated by the destruction of sparsity in Jacobian matrices and the complexity of implicit integration methods.
Innovation Solution
The implementation of a block-structured rank-one update method for the Jacobian matrix, using quasi-Newton and Broyden type updates, preserves sparsity and reduces computational complexity by eliminating intermediate variables and updating condensing matrices through matrix-vector operations and adjoint directional derivatives, without iterative procedures or matrix factorizations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If exact Jacobian evaluation is performed in SQP method, then solution accuracy is improved, but computational cost increases
Solution Approach 1:
The patent uses low-rank Jacobian approximations that are computationally inexpensive to compute and update, replacing the expensive exact Jacobian evaluation. These approximate Jacobians are updated using rank-one updates based on gradient differences, providing sufficient accuracy for real-time control while dramatically reducing computational burden.
Solution Approach 2:
The patent changes the parameter representation from full-rank Jacobian matrices to low-rank approximations characterized by fewer parameters (rank k << n). This parameter reduction allows the Jacobian to be stored and updated efficiently using only the essential directional information, reducing computational complexity while maintaining solution accuracy.
2Productivity
If block-structured sparsity is preserved in Jacobian updates, then computational efficiency is improved, but update complexity increases
Solution Approach 1:
The patent segments the Jacobian matrix into block-structured components corresponding to different time intervals or state groups. Each block is updated independently using rank-one updates, preserving the sparsity pattern. This segmentation allows efficient storage and computation by only maintaining non-zero blocks, reducing overall update complexity despite the structured approach.
Solution Approach 2:
The patent pre-computes and stores the sparsity pattern and block structure of the Jacobian matrix before optimization iterations. This preliminary action identifies which blocks are zero and which require updates, allowing the solver to skip unnecessary computations and focus only on relevant blocks, significantly improving computational efficiency.
3Stability of the object's composition
If implicit integration method is used, then numerical stability is improved, but Jacobian computation complexity increases
Solution Approach 1:
The patent introduces intermediate variables representing the implicit integration states and uses adjoint differentiation to compute their derivatives. These intermediates act as mediators that decouple the complex implicit relationships from the final Jacobian computation, allowing automatic differentiation to efficiently compute gradients without manually deriving complex expressions.
Solution Approach 2:
The patent replaces manual analytical Jacobian derivation with automatic differentiation techniques. Instead of mechanically computing partial derivatives through complex symbolic manipulation, the system uses algorithmic differentiation through the computation graph of the implicit integration method, automatically generating accurate Jacobian approximations with minimal programming effort.
Data Source
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AI summary
A control system for controlling an operation of a system with continuous- time nonlinear dynamics subject to constraints including equality and inequality constraints on state and control variables of the system, including an estimator to estimate a current state of the system using measurements of the operation of the system and a controller to iteratively solve, at each control time step, an approximation of a constrained nonlinear optimization problem to produce a control solution, wherein the approximation includes a linearization of the nonlinear dynamics of the system discretized by time intervals in the control horizon and represented using an approximation of the constraint Jacobian matrix for each time interval of the control horizon. The iterative solution procedure is based on a block- wise update formula for the approximation of the constraint Jacobian matrix and the intermediate condensing matrices using an evaluation of one or combination of the discretized dynamics of the system and at least one directional derivative of the discretized dynamics of the system. Each block in the constraint Jacobian matrix and in the intermediate condensing matrices represents one time interval in the prediction horizon and can be updated independently, based on a block- wise rank-one update formula without any iterative solution procedure and without any matrix-matrix multiplications or matrix factorizations.