Stochastic Predictive Control with Two-Level Chance Constraint Optimization
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Solution Overview
Problem
Stochastic nonlinear model predictive control for systems with uncertainty faces high computational costs due to the need for solving large-scale nonlinear optimization problems in real-time, especially when dealing with model uncertainties and external disturbances, which is challenging for embedded systems with limited resources.
Innovation Solution
The approach involves formulating probabilistic chance constraints using a tightening method with a back-off coefficient, covariance matrices, and affine feedback gains, allowing for approximate linearization-based covariance propagation, and employing an inexact derivative-based optimization algorithm that reduces memory and computational complexity by eliminating covariance matrices and using adjoint gradient computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If direct optimal control methods with discretization of continuous-time differential equations are used for stochastic predictive control, then the probabilistic chance constraints can be formulated and solved, but the computational cost and memory requirements become prohibitively high for real-time applications on embedded hardware
Solution Approach 1:
The patent changes the parameterization of control actions from general functions to polynomial basis functions of degree m, reducing the number of optimization variables. It also transforms the continuous-time covariance propagation into a discrete-time approximation using numerical integration, changing the computational parameters to be suitable for embedded hardware while maintaining the essential stochastic constraints satisfaction.
Solution Approach 2:
The patent extracts and eliminates the covariance matrices from the nonlinear optimization problem by using an inexact derivative-based algorithm. Instead of computing and storing full covariance matrices during optimization, it uses adjoint gradient computations that only require gradient information, significantly reducing memory requirements while still ensuring probabilistic constraints are met.
2Productivity
If sequential quadratic programming (SQP) method with linearization of discretized nonlinear dynamics is performed at each iteration, then the nonlinear optimization problem can be solved, but the Jacobian evaluation and matrix factorizations require excessive computational time for real-time control
Solution Approach 1:
The patent uses cheap first-order gradient approximations instead of expensive second-order derivative computations. The adjoint gradient method provides sufficient descent direction information for convergence without requiring accurate Hessian matrices, trading off some convergence speed for dramatically reduced computational cost per iteration that is feasible on embedded hardware.
Solution Approach 2:
The patent performs only a single iteration of the SQP method rather than pursuing full convergence, which would require many expensive iterations. This partial action approach provides sufficient improvement to the control solution while keeping the computational time within real-time constraints, accepting that the solution may not be fully optimal but is adequate for practical control.
3Measurement precision
If polynomial chaos expansions with many expansion terms are used for time-varying uncertainties, then the stochastic system dynamics can be accurately represented, but the computational complexity increases significantly
Solution Approach 1:
The patent uses a low-degree polynomial parameterization (degree m) for control actions rather than high-degree polynomial chaos expansions. This partial approach provides sufficient accuracy for the control application while keeping the number of parameters manageable for real-time computation on embedded systems with limited resources.
Data Source
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AI summary
A predictive controller controls a system under uncertainty subject to constraints on state and control variables of the system. At each control step, the predictive controller solves an inequality constrained nonlinear dynamic optimization problem including probabilistic chance constraints representing the uncertainty to produce a control command, and controls an operation of the system using the control command. The predictive controller solves the dynamic optimization problem based on a two-level optimization that alternates, until a termination condition is met, propagation of covariance matrices of the probabilistic chance constraints within the prediction horizon for fixed values of the state and control variables with optimization of the state and control variables within the prediction horizon for fixed values of the covariance matrices.