139 research outputs found
Computation of Polytopic Invariants for Polynomial Dynamical Systems using Linear Programming
This paper deals with the computation of polytopic invariant sets for
polynomial dynamical systems. An invariant set of a dynamical system is a
subset of the state space such that if the state of the system belongs to the
set at a given instant, it will remain in the set forever in the future.
Polytopic invariants for polynomial systems can be verified by solving a set of
optimization problems involving multivariate polynomials on bounded polytopes.
Using the blossoming principle together with properties of multi-affine
functions on rectangles and Lagrangian duality, we show that certified lower
bounds of the optimal values of such optimization problems can be computed
effectively using linear programs. This allows us to propose a method based on
linear programming for verifying polytopic invariant sets of polynomial
dynamical systems. Additionally, using sensitivity analysis of linear programs,
one can iteratively compute a polytopic invariant set. Finally, we show using a
set of examples borrowed from biological applications, that our approach is
effective in practice
Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation
We extend a template-based approach for synthesizing switching controllers
for semi-algebraic hybrid systems, in which all expressions are polynomials.
This is achieved by combining a QE (quantifier elimination)-based method for
generating continuous invariants with a qualitative approach for predefining
templates. Our synthesis method is relatively complete with regard to a given
family of predefined templates. Using qualitative analysis, we discuss
heuristics to reduce the numbers of parameters appearing in the templates. To
avoid too much human interaction in choosing templates as well as the high
computational complexity caused by QE, we further investigate applications of
the SOS (sum-of-squares) relaxation approach and the template polyhedra
approach in continuous invariant generation, which are both well supported by
efficient numerical solvers
Robust Stability Analysis of Nonlinear Hybrid Systems
We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems
Constructing Attractors of Nonlinear Dynamical Systems
In a previous work, we have shown how to generate attractor sets of affine hybrid systems using a method of state space decomposition. We show here how to adapt the method to polynomial dynamics systems by approximating them as switched affine systems. We show the practical
interest of the method on standard examples of the literature
Optimization of Lyapunov Invariants in Verification of Software Systems
The paper proposes a control-theoretic framework for verification of numerical software systems, and puts forward software verification as an important application of control and systems theory. The idea is to transfer Lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications. These include but are not limited to absence of overflow, absence of division-by-zero, termination in finite time, absence of dead-code, and certain user-specified assertions. Central to this framework are Lyapunov invariants. These are properly constructed functions of the program variables, and satisfy certain properties-analogous to those of Lyapunov functions-along the execution trace. The search for the invariants can be formulated as a convex optimization problem. If the associated optimization problem is feasible, the result is a certificate for the specification.National Science Foundation (U.S.) (Grant CNS-1135955)National Science Foundation (U.S.) (Grant CPS-1135843)United States. Army Research Office. Multidisciplinary University Research Initiative (Award W911NF-11-1-0046)United States. National Aeronautics and Space Administration (Grant/Cooperative Agreement NNX12AM52A
The Maximal Positively Invariant Set: Polynomial Setting
This note considers the maximal positively invariant set for polynomial
discrete time dynamics subject to constraints specified by a basic
semialgebraic set. The note utilizes a relatively direct, but apparently
overlooked, fact stating that the related preimage map preserves basic
semialgebraic structure. In fact, this property propagates to underlying
set--dynamics induced by the associated restricted preimage map in general and
to its maximal trajectory in particular. The finite time convergence of the
corresponding maximal trajectory to the maximal positively invariant set is
verified under reasonably mild conditions. The analysis is complemented with a
discussion of computational aspects and a prototype implementation based on
existing toolboxes for polynomial optimization
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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