An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918225175904256 |
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| author | Biertümpfel, Felix Hu, Bin Dullerud, Geir Seiler, Peter |
| author_facet | Biertümpfel, Felix Hu, Bin Dullerud, Geir Seiler, Peter |
| contents | This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a non-repeated, sector-bounded nonlinearity. Sufficient conditions for stability and performance can be derived using: (i) dissipation inequalities, and (ii) Quadratic Constraints (QCs) that bound the input/output pairs of the nonlinearity. Larger classes of QCs (or multipliers) reduce the conservatism of the conditions. Full-block, circle criterion multipliers define the complete set of all possible QCs for non-repeated, sector-bounded nonlinearities. These provide the least conservative conditions. However, full-block multipliers are defined by an uncountably infinite number of constraints and hence do not lead to computationally tractable solutions if left in this raw form. This paper provides a new finite-dimensional characterization for the set of full-block, circle criterion multipliers. The key theoretical insight is: the set of all input/output pairs of non-repeated sector-bounded nonlinearities is equal to the set of all incremental pairs for an appropriately constructed piecewise linear function. Our new description for the complete set of multipliers only requires a finite number of matrix copositivity constraints. These conditions have an exact, computationally tractable implementation for problems where the nonlinearity has small input/output dimensions $(\le 4)$. We illustrate the use of our new characterization via a simple example. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers Biertümpfel, Felix Hu, Bin Dullerud, Geir Seiler, Peter Systems and Control This paper provides the first finite-dimensional characterization for the complete set of full-block, circle criterion multipliers. We consider the interconnection of a discrete-time, linear time-invariant system in feedback with a non-repeated, sector-bounded nonlinearity. Sufficient conditions for stability and performance can be derived using: (i) dissipation inequalities, and (ii) Quadratic Constraints (QCs) that bound the input/output pairs of the nonlinearity. Larger classes of QCs (or multipliers) reduce the conservatism of the conditions. Full-block, circle criterion multipliers define the complete set of all possible QCs for non-repeated, sector-bounded nonlinearities. These provide the least conservative conditions. However, full-block multipliers are defined by an uncountably infinite number of constraints and hence do not lead to computationally tractable solutions if left in this raw form. This paper provides a new finite-dimensional characterization for the set of full-block, circle criterion multipliers. The key theoretical insight is: the set of all input/output pairs of non-repeated sector-bounded nonlinearities is equal to the set of all incremental pairs for an appropriately constructed piecewise linear function. Our new description for the complete set of multipliers only requires a finite number of matrix copositivity constraints. These conditions have an exact, computationally tractable implementation for problems where the nonlinearity has small input/output dimensions $(\le 4)$. We illustrate the use of our new characterization via a simple example. |
| title | An Exact, Finite Dimensional Representation for Full-Block, Circle Criterion Multipliers |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2511.20995 |