Predictive Controller With Low-Rank Preconditioning for Real-Time MPC
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Solution Overview
Problem
Existing real-time optimal control methods for dynamical systems face challenges with high computational complexity due to cubic or quadratic scaling with the number of states and controls, making them unsuitable for embedded hardware with limited resources, especially when dealing with inequality constraints.
Innovation Solution
An iterative preconditioned solver is developed that scales linearly with the control horizon length and quadratically with the number of states and controls, using a block-structured preconditioning matrix and low-rank updates to efficiently solve the Karush-Kuhn-Tucker equations, reducing computational complexity and memory requirements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If direct linear algebra routines are used to solve the optimization problem, then the solution is obtained with cubic computational complexity scaling with the number of states and controls, but this makes the controller unsuitable for embedded hardware with limited computational resources
Solution Approach 1:
The patent segments the solution process into two parts: (1) an iterative solver that handles the linear system solution with reduced complexity, and (2) a preconditioning step that prepares the system matrix. This segmentation allows the use of iterative methods with O(Nm²) complexity instead of direct methods with O(Nm³) complexity, making the controller suitable for embedded hardware while maintaining numerical accuracy through the preconditioning approach.
Solution Approach 2:
The patent changes the parameter of the solution method from direct linear algebra routines to iterative methods with preconditioning. This parameter change transforms the computational complexity from cubic to quadratic scaling with the number of states and controls, while the preconditioning ensures that the iterative method converges to the accurate solution.
2Productivity
If first order optimization methods are used after eliminating state variables, then the computational complexity scales quadratically with the horizon length, but this quadratic complexity is still undesirable for some real-time applications
Solution Approach 1:
The patent applies preliminary action by using a preconditioning step before the iterative solver. The preconditioner is constructed in advance to transform the system matrix into a form that is easier to solve iteratively. This preliminary preparation enables the iterative method to converge faster and achieve O(Nm²) complexity, improving real-time performance while reducing computational time compared to unpreconditioned iterative methods.
3Device complexity
If iterative solvers are used to solve the saddle point linear systems, then the asymptotic computational complexity is improved, but the solvers tend to converge poorly without preconditioning
Solution Approach 1:
The patent introduces a preconditioner as an intermediary between the original system matrix and the iterative solver. The preconditioner transforms the system matrix into a form with more favorable spectral properties, which acts as a mediator to improve the convergence behavior of the iterative solver. This intermediary step ensures that the iterative method converges reliably while maintaining the improved O(Nm²) computational complexity.
Data Source
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AI summary
A predictive controller for controlling a system subject to constraints including equality and inequality constraints on state and control variables of the system, includes an estimator to estimate a current state of the system using measurements of outputs of the system and a controller to solve, at each control step, a matrix equation of necessary optimality conditions to produce a control solution and to control the system using the control solution to change a state of the system. The matrix equation includes a block-structured matrix having a constraint Jacobian matrix of the equality constraints of the system. The controller determines the control solution iteratively using two levels of iterations including a first level of iterations that selects active inequality constraints for each point of time within a control horizon, updates the constraint Jacobian matrix, with a low-rank update for a change in the set of active inequality constraints, to include the equality constraints and the active inequality constraints, and updates a preconditioning matrix, with a low-rank factorization update, in response to the low-rank update of the constraint Jacobian matrix. The second level of iterations solves the matrix equation with the updated constraint Jacobian matrix using the updated preconditioning matrix to produce the control solution.