18 research outputs found
PIETOOLS: A Matlab Toolbox for Manipulation and Optimization of Partial Integral Operators
In this paper, we present PIETOOLS, a MATLAB toolbox for the construction and
handling of Partial Integral (PI) operators. The toolbox introduces a new class
of MATLAB object, opvar, for which standard MATLAB matrix operation syntax
(e.g. +, *, ' e tc.) is defined. PI operators are a generalization of bounded
linear operators on infinite-dimensional spaces that form a *-subalgebra with
two binary operations (addition and composition) on the space RxL2. These
operators frequently appear in analysis and control of infinite-dimensional
systems such as Partial Differential equations (PDE) and Time-delay systems
(TDS). Furthermore, PIETOOLS can: declare opvar decision variables, add
operator positivity constraints, declare an objective function, and solve the
resulting optimization problem using a syntax similar to the sdpvar class in
YALMIP. Use of the resulting Linear Operator Inequalities (LOIs) are
demonstrated on several examples, including stability analysis of a PDE,
bounding operator norms, and verifying integral inequalities. The result is
that PIETOOLS, packaged with SOSTOOLS and MULTIPOLY, offers a scalable,
user-friendly and computationally efficient toolbox for parsing, performing
algebraic operations, setting up and solving convex optimization problems on PI
operators
A Variation on a Random Coordinate Minimization Method for Constrained Polynomial Optimization
In this paper, an algorithm is proposed for solving constrained and unconstrained polynomial minimization
problems. The algorithm is a variation on random coordinate descent, in which transverse steps are seldom taken.
Differently from other methods available in the literature, the proposed technique is guaranteed
to converge in probability to the global solution of the minimization problem, even when the objective polynomial is nonconvex.
The theoretical results are corroborated by a complexity
analysis and by numerical tests that validate its efficiency
Operator-Theoretic Characterization of Eventually Monotone Systems
Monotone systems are dynamical systems whose solutions preserve a partial
order in the initial condition for all positive times. It stands to reason that
some systems may preserve a partial order only after some initial transient.
These systems are usually called eventually monotone. While monotone systems
have a characterization in terms of their vector fields (i.e. Kamke-Muller
condition), eventually monotone systems have not been characterized in such an
explicit manner. In order to provide a characterization, we drew inspiration
from the results for linear systems, where eventually monotone (positive)
systems are studied using the spectral properties of the system (i.e.
Perron-Frobenius property). In the case of nonlinear systems, this spectral
characterization is not straightforward, a fact that explains why the class of
eventually monotone systems has received little attention to date. In this
paper, we show that a spectral characterization of nonlinear eventually
monotone systems can be obtained through the Koopman operator framework. We
consider a number of biologically inspired examples to illustrate the potential
applicability of eventual monotonicity.Comment: 13 page
Towards Scalable Synthesis of Stochastic Control Systems
Formal control synthesis approaches over stochastic systems have received
significant attention in the past few years, in view of their ability to
provide provably correct controllers for complex logical specifications in an
automated fashion. Examples of complex specifications of interest include
properties expressed as formulae in linear temporal logic (LTL) or as automata
on infinite strings. A general methodology to synthesize controllers for such
properties resorts to symbolic abstractions of the given stochastic systems.
Symbolic models are discrete abstractions of the given concrete systems with
the property that a controller designed on the abstraction can be refined (or
implemented) into a controller on the original system. Although the recent
development of techniques for the construction of symbolic models has been
quite encouraging, the general goal of formal synthesis over stochastic control
systems is by no means solved. A fundamental issue with the existing techniques
is the known "curse of dimensionality," which is due to the need to discretize
state and input sets and that results in an exponential complexity over the
number of state and input variables in the concrete system. In this work we
propose a novel abstraction technique for incrementally stable stochastic
control systems, which does not require state-space discretization but only
input set discretization, and that can be potentially more efficient (and thus
scalable) than existing approaches. We elucidate the effectiveness of the
proposed approach by synthesizing a schedule for the coordination of two
traffic lights under some safety and fairness requirements for a road traffic
model. Further we argue that this 5-dimensional linear stochastic control
system cannot be studied with existing approaches based on state-space
discretization due to the very large number of generated discrete states.Comment: 22 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1407.273