Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Increasing penetration of renewable energy introduces significant uncertainty into power systems. Traditional simulation-based verification methods may not be applicable due to the unknown-but-bounded feature of the uncertainty sets. Emerging set-theoretic methods have been intensively investigated to tackle this challenge. The paper comprehensively reviews these methods categorized by underlying mathematical principles, that is, set operation-based methods and passivity-based methods. Set operation-based methods are more computationally efficient, while passivity-based methods provide semi-analytical expression of reachable sets, which can be readily employed for control. Other features between different methods are also discussed and illustrated by numerical examples. A benchmark example is presented and solved by different methods to verify consistency.

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“…or written as K : (g 1 (x) ≥ 0 ∧ · · · ∧ g m (x) ≥ 0) for short. Then the following theorem can be used to verify the positivity [69].…”

Section: B Positivity For Algorithmic Solutionsmentioning

confidence: 99%

Review on set-theoretic methods for safety verification and control of power system

Zhang

Tomsovic

et al. 2020

IET Energy Systems Integration

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“…In [28,27], the authors present LP formulations, based on Handelman representations of polynomials, to compute Lyapunov functions. Consequently, the computation avoids semi-definite programming, which enables SOS optimization, and is therefore more robust to numerical errors.…”

Section: Related Workmentioning

confidence: 99%

Computing bisimulation functions using SOS optimization and δ -decidability over the reals

Murthy

Islam

Smolka

et al. 2015

Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control

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We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations.In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated.By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced-with bounded error-by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. We illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable HodgkinHuxley-type approximation.

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“…Details may be found in the long version [1] For the stabilization problem (Problem 2), we replace the negative definiteness condition by finding both c ∈ C and θ s.t. the following condition is satisfied:…”

Section: Reduction To Polynomial Optimizationmentioning

confidence: 99%

“…(b) Tight bounds: 0 ≤ B I,δ (y) ≤ n k=1 B I k ,δ k ( I k δ k ), ∀I ≤ δ; (c) Linear "Pascal's triangle"-like recurrences that connect lower degree Bernstein polynomials with higher degree ones [1]. The overall idea of using Bernstein polynomial in a POP over a compact feasible region is as follows:…”

Section: Bernstein Polynomial Relaxationsmentioning

confidence: 99%

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Stability and stabilization of polynomial dynamical systems using Bernstein polynomials

Sassi

Sankaranarayanan

2015

Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control

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In this work, we examine relaxations for the stability analysis and synthesis of stabilizing controllers for polynomial dynamical systems. It is well-known that such problems can be naturally solved using a reduction to polynomial optimization problems. The Sum of Squares (SOS) programming relaxation further relaxes these polynomial optimization problems to convex Semi-Definite Programming (SDP) problems. However, their application, in practice, to formal verification and correct-by-construction synthesis has been made harder due to numerical stability issues encountered while solving SDPs. Our work proposes a new approach to relaxations based on Bernstein polynomials to yield linear programming (LP) relaxations for polynomial optimization problems. This allows us to find polynomial Lyapunov functions that certify the (asymptotic) stability of a system. The approach is also extended to synthesizing a stabilizing feedback control law for a controlled system that will stabilize the closed-loop dynamics to a specified equilibrium.

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Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Cited by 31 publications

References 53 publications

Review on set-theoretic methods for safety verification and control of power system

Review on set-theoretic methods for safety verification and control of power system

Computing bisimulation functions using SOS optimization and δ -decidability over the reals

Stability and stabilization of polynomial dynamical systems using Bernstein polynomials

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