Sum Of Squares Problem Research Articles

Sums of squares II: Matrix functions

This paper is the second in a series of three papers devoted to sums of squares and hypoellipticity of infinitely degenerate operators. In the first paper we established a sharp ω-monotonicity criterion for writing a smooth nonnegative function f that is flat at, and positive away from, the origin, as a finite sum of squares of C2,δ functions for some δ>0, namely that f is ω-monotone for some Hölder modulus of continuity ω. Counterexamples were provided for any larger modulus of continuity.In this paper we consider the analogous sum of squares problem for smooth nonnegative matrix functions M that are flat at, and positive away from, the origin. We show that such a matrix function M=[akj]k,j=1n can be written as a finite sum of squares of C2,δ vector fields if the diagonal entries akk are ω-monotone for some Hölder modulus of continuity ω, and if the off diagonal entries satisfy certain differential bounds in terms of powers of the diagonal entries. Examples are given to show that in some cases at least, these differential inequalities cannot be relaxed.Various refinements of these results are also given in which one or more of the diagonal entries need not be assumed to have any monotonicity properties at all. These sum of squares decompositions will be applied to hypoellipticity in the infinitely degenerate regime in the third paper in this series.

A Novel Path-Following-Method-Based Polynomial Fuzzy Control Design.

This article presents a novel path-following-method-based polynomial fuzzy control design. By examining the stabilization problem, the nonconvex stabilization criterion represented in terms of bilinear sum-of-squares (SOS) constraints is proposed to complement the existing convex stabilization criteria. Based on the polynomial Lyapunov function and considering the operation domain, the stabilization control is designed with a systematic region of attraction (ROA) analysis method. Since the proposed stabilization criterion remains in nonconvex form, the conservativeness caused by the transformation from nonconvex (bilinear SOS) constraints into convex (SOS) constraints can be avoided. Moreover, the restriction on the Lyapunov function candidates for the convex transformation in the literature does not exist in the proposed nonconvex stabilization criterion. The stabilization analysis for polynomial fuzzy control systems is concerned with the double fuzzy summation problem that can be treated as the copositivity problem. Therefore, the SOS-based copositive relaxation technique is applied for the proposed stabilization criterion. Since the proposed nonconvex stabilization criterion is represented in terms of bilinear SOS constraints, the path-following method is employed for solving the bilinear SOS problem. Finally, design examples are provided to demonstrate that the proposed nonconvex stabilization criterion complements the existing convex stabilization criteria.

State‐dependent switching law design with guaranteed dwell time for switched nonlinear systems

SummaryIn this article, without the help of predesigned dwell time constraints, a new state‐dependent switching law with guaranteed dwell time for switched nonlinear systems is studied. Some sufficient conditions for asymptotic stability of switched nonlinear systems are derived. Also, all the abovementioned conditions can be transformed to a set of sum of squares (SOS) constraints, which can be checked by using the bilinear SOS methodology. Meanwhile, an improved path following method is provided to solve a bilinear SOS problem. Finally, three simulation examples are given to demonstrate the effectiveness of the obtained results.

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically.

An SOS method for the design of continuous and discontinuous differentiators

ABSTRACTGiven a (differentiable) signal, it is an important task for many applications to estimate on line its derivatives. Some well-known algorithms to solve this problem include the (continuous) high-gain observers and (discontinuous) Levant's exact differentiators. With exception of the linear high-gain observers, a systematic design of the gains of nonlinear differentiators is, in general, a difficult task and an open research field. In this work, we propose a novel method for the gain-tuning of a family of homogeneous differentiators which estimate the time-derivatives of a signal in finite-time. We show that the stability analysis and the gain-tuning of such differentiators can be done under a unified Lyapunov function framework, and it is converted to a sum of squares problem, that can be solved using LMIs, much in the same spirit of the linear systems.

Exploiting Sparsity in the Coefficient Matching Conditions in Sum-of-Squares Programming Using ADMM

This letter introduces an efficient first-order method based on the alternating direction method of multipliers (ADMM) to solve semidefinite programs arising from sum-of-squares (SOS) programming. We exploit the sparsity of the coefficient matching conditions when SOS programs are formulated in the usual monomial basis to reduce the computational cost of the ADMM algorithm. Each iteration of our algorithm requires one projection onto the positive semidefinite cone and the solution of multiple quadratic programs with closed-form solutions free of any matrix inversion. Our techniques are implemented in the open-source MATLAB solver SOSADMM. Numerical experiments on SOS problems arising from unconstrained polynomial minimization and from Lyapunov stability analysis for polynomial systems show speed-ups compared to the interior-point solver SeDuMi, and the first-order solver CDCS.

Control design for discrete‐time bilinear systems using the scalarized Schur complement

SummaryIn this paper, controller design for discrete‐time bilinear systems is investigated by using sum of squares programming methods and quadratic Lyapunov functions. The class of rational polynomial controllers is considered, and necessary conditions on the degree of controller polynomials for quadratic stability are derived. Next, a scalarized version of the Schur complement is proposed. For controller design, the Lyapunov difference inequality is converted to a sum of squares problem, and an optimization problem is proposed to design a controller, which maximizes the region of quadratic stability of the bilinear system. Input constraints can also be accounted for. Copyright © 2017 John Wiley & Sons, Ltd.

Stability Analysis and Region-of-Attraction Estimation Using Piecewise Polynomial Lyapunov Functions: Polynomial Fuzzy Model Approach

This paper proposes sum-of-squares (SOS) methodologies for stability analysis and region-of-attraction (ROA) estimation for nonlinear systems represented by polynomial fuzzy models via piecewise polynomial Lyapunov functions (PPLFs). At first, two SOS-based global stability criteria are proposed by applying maximum-type and minimum-type PPLFs, respectively. It is known that less-conservative results can be obtained by reducing global stability to local stability, since it is usually the case for nonlinear systems that the stability cannot be reached globally. Therefore, based on the two types of PPLFs, two local stability criteria are further proposed with the algorithms that enlarge the estimated ROA as much as possible. The constraints for checking (global and local) stability and enlarging the estimated ROA are represented in terms of bilinear SOS problems. Hence, the path-following method is applied to solve the bilinear SOS problems in the proposed methodologies. Finally, some examples are provided to illustrate the utility of the proposed methodologies.

Improved Design of Nonlinear Model Predictive Controllers

One way to ensure recursive feasibility, stability and performance of Nonlinear Model Predictive Control is the combined use of a terminal region and a terminal cost. However, finding suitable combinations of the terminal cost and terminal region that guarantee closed-loop stability for nonlinear systems is in general challenging. Most existing methods are either based on the linearized system dynamics and a linear feedback, or assume that a control Lyapunov function for the system close to the origin is know. This paper proposes the use of higher order approximations of the optimal feedback and optimal cost of the infinite horizon problem via Al’brekht’Method to determine a suitable terminal region for polynomial systems. To do so, the stability conditions are reformulated in terms of a sumof-squares problem which is iteratively used to determine the terminal region. For a nonlinear chemical reactor example it is shown that the proposed approach leads to a larger terminal region and an improved performance compared to existing approaches.

Optimizing Safety Supervisors for Wind Turbines using Barrier Certificates

Abstract To avoid structural damage, a wind turbine is equipped with a safety supervisor that triggers an emergency shutdown procedure in case of internal faults or large wind gusts. This paper leverages the (compositional) barrier certificate framework for the design and optimization of such a supervisor. The problem is formulated as a sum of squares problem and is solved using semi-definite programming. Both a direct and compositional approach are successfully implemented and verified for the NREL 5MW reference turbine. In conclusion, the methods derived in this paper are indeed viable for the syntheses and optimization of safety systems for future wind turbines.

On matrix algebras associated to sum-of-squares semidefinite programs

To each semidefinite program (SDP) in primal form, we associate the matrix algebra generated by its constraint matrices. In this note, we show that this algebra is always a full matrix algebra for SDPs arising from (commutative or non-commutative) sum of squares (SOS) problems. For SDPs arising from non-commutative SOS and commutators problems, the situation is less clear. We identify an exceptional case, where the corresponding matrix algebra is not the full matrix algebra, and use it to reprove the Burgdorf–Klep non-commutative variant of Hilbert’s ternary quartics theorem: a bivariate non-commutative polynomial of degree at most is trace positive if it is a sum of four squares and commutators.

Non-linear control of an uncertain hypersonic aircraft model using robust sum-of-squares method

In recent years, sum-of-squares (SOS) method has attracted increasing interest as a new approach for stability analysis and controller design of non-linear dynamic systems. In this study, the authors present a robust SOS/robust LMI method to design a non-linear controller for longitudinal dynamics of a hypersonic aircraft model with parametric uncertainties. To this end, the control design problem is first formulated as a robust SOS problem. Then, an LMI representation of the robust SOS problem is derived using a proposed algorithm and is solved via a stochastic ellipsoid method. As the simulation results show, the designed robust controller is capable of stabilising the aircraft and following pilot commands under up to 50% variation in aerodynamic coefficients.

Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients

We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of-squares (SOS) with rational coefficients. Our new approach turns the earlier methods by Peyrl and Parrilo at SNC’07 and ours at ISSAC’08 both based on polynomial SOS, which do not always exist, into a universal algorithm for all inputs via Artin’s theorem. Furthermore, we scrutinize the all-important process of converting the numerical SOS numerators and denominators produced by block semidefinite programming into an exact rational identity. We improve on our own Newton iteration-based high precision refinement algorithm by compressing the initial Gram matrices and by deploying rational vector recovery aside from orthogonal projection. We successfully demonstrate our algorithm on (1) various exceptional SOS problems with necessary polynomial denominators from the literature and on (2) very large (thousands of variables) SOS lower bound certificates for Rump’s model problem (up to n = 18 , factor degree = 17 ).

Algorithms for multidimensional spectral factorization and sum of squares

In this paper, algorithms are developed for the problems of spectral factorization and sum of squares of polynomial matrices with n indeterminates, and a natural interpretation of the tools employed in the algorithms is given using ideas from the theory of lossless and dissipative systems. These algorithms are based on the calculus of 2n-variable polynomial matrices and their associated quadratic differential forms, and share the common feature that the problems are lifted from the original n-variable polynomial context to a 2n-variable polynomial context. This allows to reduce the spectral factorization problem and the sum of squares problem to linear matrix inequalities (LMI’s), to the feasibility of a semialgebraic set or to a linear eigenvalue problem.