3,507 research outputs found
Block Factor-width-two Matrices and Their Applications to Semidefinite and Sum-of-squares Optimization
Semidefinite and sum-of-squares (SOS) optimization are fundamental
computational tools in many areas, including linear and nonlinear systems
theory. However, the scale of problems that can be addressed reliably and
efficiently is still limited. In this paper, we introduce a new notion of
\emph{block factor-width-two matrices} and build a new hierarchy of inner and
outer approximations of the cone of positive semidefinite (PSD) matrices. This
notion is a block extension of the standard factor-width-two matrices, and
allows for an improved inner-approximation of the PSD cone. In the context of
SOS optimization, this leads to a block extension of the \emph{scaled
diagonally dominant sum-of-squares (SDSOS)} polynomials. By varying a matrix
partition, the notion of block factor-width-two matrices can balance a
trade-off between the computation scalability and solution quality for solving
semidefinite and SOS optimization. Numerical experiments on large-scale
instances confirm our theoretical findings.Comment: 26 pages, 5 figures. Added a new section on the approximation quality
analysis using block factor-width-two matrices. Code is available through
https://github.com/zhengy09/SDPf
Neural Lyapunov Control
We propose new methods for learning control policies and neural network
Lyapunov functions for nonlinear control problems, with provable guarantee of
stability. The framework consists of a learner that attempts to find the
control and Lyapunov functions, and a falsifier that finds counterexamples to
quickly guide the learner towards solutions. The procedure terminates when no
counterexample is found by the falsifier, in which case the controlled
nonlinear system is provably stable. The approach significantly simplifies the
process of Lyapunov control design, provides end-to-end correctness guarantee,
and can obtain much larger regions of attraction than existing methods such as
LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality
solutions for challenging control problems.Comment: NeurIPS 201
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
Data-driven computation of invariant sets of discrete time-invariant black-box systems
We consider the problem of computing the maximal invariant set of
discrete-time black-box nonlinear systems without analytic dynamical models.
Under the assumption that the system is asymptotically stable, the maximal
invariant set coincides with the domain of attraction. A data-driven framework
relying on the observation of trajectories is proposed to compute
almost-invariant sets, which are invariant almost everywhere except a small
subset. Based on these observations, scenario optimization problems are
formulated and solved. We show that probabilistic invariance guarantees on the
almost-invariant sets can be established. To get explicit expressions of such
sets, a set identification procedure is designed with a verification step that
provides inner and outer approximations in a probabilistic sense. The proposed
data-driven framework is illustrated by several numerical examples.Comment: A shorter version with the title "Scenario-based set invariance
verification for black-box nonlinear systems" is published in the IEEE
Control Systems Letters (L-CSS
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