Suppose we want to minimize a polynomial p(x) = p(x1, . . . , xn), subject to some polynomial constraints q1(x), . . . , qm(x) ≥ 0, using the Sum-of-Squares (SOS) SDP hierarachy. Assume we are in the “explicitly bounded” (“Archimedean”) case where the constraints include xi ≤ 1 for all 1 ≤ i ≤ n. It is often stated that the degree-d version of the SOS hierarchy can be solved, to high accuracy, ...

In recent years, Sum–Of–Squares (SOS) method has attracted increasing interest as a new approach for stability analysis and controller design of nonlinear dynamic systems. In this paper, we present a robust SOS/robust LMI method to design a nonlinear controller for longitudinal dynamics of a hypersonic aircraft model with parametric uncertainties. To this end, the control design problem is firs...

Magnetic resonance (MR) images can be acquired by multiple receiver coil systems to improve signal-to-noise ratio (SNR) and to decrease acquisition time. The optimal SNR images can be reconstructed from the coil data when the coil sensitivities are known. In typical MR imaging studies, the information about coil sensitivity profiles is not available. In such cases the sum-of-squares (SoS) recon...

A novel moving horizon control strategy for input-saturated nonlinear polynomial systems is proposed. The control strategy makes use of the so called sum-of-squares (SOS) decomposition, i.e. a convexification procedure able to give sufficient conditions on the positiveness of polynomials. The complexity of SOS-based numerical methods is polynomial in the problem size and, as a consequence, comp...

Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a “cardinality constraint”, as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value β, it did not necessarily actually “satisfy” the constraint “objective = β”. In this paper we...

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepe...

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for a SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is indepen...

We study how to solve sum of squares (SOS) and Lasserre’s relaxations for large scale polynomial optimization. When interior-point type methods are used, typically only small or moderately large problems could be solved. This paper proposes the regularization type methods which would solve significantly larger problems. We first describe these methods for general conic semidefinite optimization...

This paper deals with stability analysis of hybrid systems. Such systems are characterized by a combination of continuous dynamics and logic based switching between discrete modes. Lyapunov theory is a well known methodology for the stability analysis of linear and nonlinear systems in control system literature. Construction of Lyapunov functions for hybrid systems is generally a difficult task...

Lyapunov methods have been successfully applied to compute the Region of Attraction (ROA) of a system, i.e. the set of states that remain invariant for all time. A popular method for constructing Lyapunov functions is the Sum of Squares (SOS) approach. Current methods for determining stability of hybrid systems using SOS require either stability over the entire hybrid guard or knowledge of the ...