Giorgio Valmorbida

Antonis Papachristodoulou, James Anderson, Giorgio Valmorbida et al.

The release of SOSTOOLS v4.00 comes as we approach the 20th anniversary of the original release of SOSTOOLS v1.00 back in April, 2002. SOSTOOLS was originally envisioned as a flexible tool for parsing and solving polynomial optimization problems, using the SOS tightening of polynomial positivity constraints, and capable of adapting to the ever-evolving fauna of applications of SOS. There are now a variety of SOS programming parsers beyond SOSTOOLS, including YALMIP, Gloptipoly, SumOfSquares, and others. We hope SOSTOOLS remains the most intuitive, robust and adaptable toolbox for SOS programming. Recent progress in Semidefinite programming has opened up new possibilities for solving large Sum of Squares programming problems, and we hope the next decade will be one where SOS methods will find wide application in different areas. In SOSTOOLS v4.00, we implement a parsing approach that reduces the computational and memory requirements of the parser below that of the SDP solver itself. We have re-developed the internal structure of our polynomial decision variables. Specifically, polynomial and SOS variable declarations made using sossosvar, sospolyvar, sosmatrixvar, etc now return a new polynomial structure, dpvar. This new polynomial structure, is documented in the enclosed dpvar guide, and isolates the scalar SDP decision variables in the SOS program from the independent variables used to construct the SOS program. As a result, the complexity of the parser scales almost linearly in the number of decision variables. As a result of these changes, almost all users will notice a significant increase in speed, with large-scaleproblems experiencing the most dramatic speedups. Parsing time is now always less than 10% of time spent in the SDP solver. Finally, SOSTOOLS now provides support for the MOSEK solver interface as well as the SeDuMi, SDPT3, CSDP, SDPNAL, SDPNAL+, and SDPA solvers.

Giorgio Valmorbida, James Anderson

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration.

Aditya Gahlawat, Giorgio Valmorbida

We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs.

Ross Drummond, Pablo R Baldivieso-Monasterios, Giorgio Valmorbida

Model predictive control (MPC) for linear systems with quadratic costs and linear constraints is shown to admit an exact representation as an implicit neural network. A method to "unravel" the implicit neural network of MPC into an explicit one is also introduced. As well as building links between model-based and data-driven control, these results emphasize the capability of implicit neural networks for representing solutions of optimisation problems, as such problems are themselves implicitly defined functions.

Giorgio Valmorbida, Dhruva Raman, James Anderson

We consider the effect of parametric uncertainty on properties of Linear Time Invariant systems. Traditional approaches to this problem determine the worst-case gains of the system over the uncertainty set. Whilst such approaches are computationally tractable, the upper bound obtained is not necessarily informative in terms of assessing the influence of the parameters on the system performance. We present theoretical results that lead to simple, convex algorithms producing parametric bounds on the $\mathcal{L}_2$-induced input-to-output and state-to-output gains as a function of the uncertain parameters. These bounds provide quantitative information about how the uncertainty affects the system.