Feedback Control with Constrained Zonotopes Under Uncertainty
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Solution Overview
Problem
Existing methods for computing robust controllable sets in high-dimensional systems are computationally inefficient and suffer from numerical issues, limiting their applicability to low-dimensional systems, which is a challenge for online control applications.
Innovation Solution
Employing constrained zonotopes to represent polyhedra, using closed-form expressions for set operations like intersection, affine transformation, and Minkowski sum, and approximating the Pontryagin difference with inner and outer-approximations to compute robust and stochastic controllable sets efficiently.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If polytope-based RC set computation is used, then exact robust controllable sets can be obtained, but computational complexity becomes prohibitive and numerical issues arise in high dimensions
Solution Approach 1:
The patent changes the representation parameters from polytope vertices and facets to constrained zonotope generators (center, shape matrix, and constraint matrices). This parameter transformation enables closed-form computation of set operations like Minkowski sum and Pontryagin difference, avoiding NP-hard vertex-facet enumeration while maintaining computational accuracy for robust controllable sets in high-dimensional spaces
Solution Approach 2:
The patent substitutes the mechanical computational process of polytope operations (vertex-facet enumeration, Fourier-Motzkin elimination) with algebraic operations on constrained zonotope parameters. This substitution replaces complex geometric algorithms with efficient matrix computations, eliminating numerical instability issues associated with high-dimensional polytope manipulations
2Measurement precision
If vertex-facet enumeration is used for Minkowski sum, then exact results are obtained, but the problem becomes NP-hard and computationally intractable
Solution Approach 1:
The patent transforms the computational parameters from geometric elements (vertices and facets) to algebraic generators (center vectors, shape matrices, constraint matrices). This parameter change enables the Minkowski sum to be computed through simple matrix addition and concatenation operations, achieving both exact results and computational efficiency without requiring NP-hard vertex-facet enumeration
3Reliability
If polyhedral computational geometry is used, then robust controllable sets can be computed, but the approach is limited to low-dimensional systems due to numerical issues
Solution Approach 1:
The patent changes the computational parameters from polyhedral coordinates to constrained zonotope generators, which represent polyhedra through linear transformations of unit hypercubes. This parameter transformation maintains numerical stability through well-conditioned matrix operations and enables the computation of robust controllable sets in high-dimensional systems, removing the dimensional limitations of traditional polyhedral methods
Solution Approach 2:
The patent implicitly operates in an extended parameter space by representing d-dimensional polyhedra using constrained zonotope generators with potentially higher-dimensional constraint matrices. This dimensional transformation allows the computation to proceed in a more favorable parameter space, avoiding the numerical pathologies that arise in direct high-dimensional polyhedral computations
Data Source
AI summary
A feedback controller collects a feedback signal indicative of the current state of the operation of the system subject to constraints and current uncertainty on the current state of the operation of the system and determines a robust controllable set for maintaining the state of the system employing closed-form expressions on a constrained zonotope defining the constraints on the state of the operation of the system and an affine transformation of the symmetric bounded set inclosing the current uncertainty into space of the constrained zonotope. The closed-form expressions include a closed-form approximation of a Pontryagin difference between the constrained zonotopic representation and the zonotopic transformation of the symmetric bounded set. The controller determines and submits a control command for controlling the operation of the system subject to the robust controllable set.