Predictive Control Optimization With Active-Set Interior Point Solving
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Solution Overview
Problem
Existing predictive control methods face challenges in real-time implementation due to the computational complexity of solving block-sparse structured dynamic optimization problems, particularly in embedded systems with limited resources, where interior point methods are computationally expensive and active-set methods require warm starting complications.
Innovation Solution
An active-set identification strategy combined with an inexact Newton-type optimization technique is employed to improve the numerical conditioning and reduce computational cost of interior point methods, using low-rank block-sparse matrix factorization and tailored warm starting procedures, and classifying inequality constraints to prevent numerical ill-conditioning.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If interior point methods are used to solve the optimization problem, then the solution is reliable and can handle a large class of optimization problems, but the computational cost per iteration is high due to low-rank factorization updates
Solution Approach 1:
The patent segments the optimization problem by identifying and separating active constraints from inactive constraints. By classifying constraints based on the magnitude of Lagrange multipliers, the method divides the full optimization problem into smaller subproblems that can be solved more efficiently, reducing the computational burden while maintaining solution reliability.
Solution Approach 2:
The patent changes the parameter representation by introducing a classification parameter (active vs. inactive constraints) based on Lagrange multiplier magnitudes. This parameter change allows the method to switch between different solution strategies: full interior point method for reliability and reduced method for computational efficiency, depending on the constraint activity status.
2Use of energy by moving object
If active-set methods are used to reduce computational cost per iteration, then the computational cost is lower, but warm starting becomes more complicated and the method is restricted to linear or quadratic programming
Solution Approach 1:
The patent creates a hybrid method that combines the universality of interior point methods (able to handle general convex and non-convex problems) with the computational efficiency of active-set methods. By using constraint classification based on Lagrange multipliers, the method can apply reduced computational strategies to active constraints while maintaining the ability to handle general optimization problems through the interior point framework.
3Productivity
If warm starting is applied to active-set solvers to reduce average computational cost, then the average computational cost decreases, but the implementation complexity increases
Solution Approach 1:
The patent performs preliminary classification of constraints into active and inactive sets based on Lagrange multiplier magnitudes before solving the optimization problem. This preliminary action enables warm starting by pre-identifying which constraints are likely to remain active, allowing the solver to start from a better initial guess and reduce the number of iterations required, thereby reducing average computational cost.
4Reliability
If interior point methods are used in real-time embedded systems, then the solution is robust, but the computational resources required exceed the limited availability in embedded systems
Solution Approach 1:
The patent segments the computational workload by identifying active constraints and solving only the relevant subproblem rather than the full optimization problem. This segmentation reduces the number of variables and constraints that need to be processed in each iteration, thereby reducing the computational resources (memory and processing power) required while maintaining solution robustness for the active constraints.
Data Source
AI summary
A control system for controlling an operation of a machine subject to constraints including equality and inequality constraints on state and control variables of the system iteratively solves an optimal control structured optimization problem (OCP), such that each iteration outputs primal variables and dual variables with respect to the equality constraints and dual variables and slack variables with respect to the inequality constraints. For a current iteration, the system classifies each of the inequality constraints as an active, an inactive or an undecided constraint based on a ratio of a slack variable to a dual variable of the corresponding inequality constraint determined by a previous iteration, finds an approximate solution to the set of relaxed optimality conditions subject to the equality constraints and the active and undecided inequality constraints, and update the primal, dual, and slack variables for each of the equality and inequality constraint.