Sum-of-squares Optimization Research Articles

Block Factor-Width-Two Matrices and Their Applications to Semidefinite and Sum-of-Squares Optimization

Semidefinite and sum-of-squares (SOS) optimization are fundamental computational tools in many areas, including linear and nonlinear systems theory. However, the scale of problems that can be addressed reliably and efficiently is still limited. In this paper, we introduce a new notion of block factor-width-two matrices and build a new hierarchy of inner and outer approximations of the cone of positive semidefinite (PSD) matrices. This notion is a block extension of the standard factor-width-two matrices, and allows for an improved inner-approximation of the PSD cone. In the context of SOS optimization, this leads to a block extension of the scaled diagonally dominant sum-of-squares (SDSOS) polynomials. By varying a matrix partition, the notion of block factor-width-two matrices can balance a trade-off between the computation scalability and solution quality for solving semidefinite and SOS optimization problems. Numerical experiments on a range of large-scale instances confirm our theoretical findings.

Region of attraction Analysis for Adaptive Control of Wing Rock System

Abstract This paper introduces a numerical method for region of attraction (ROA) estimation of the adaptive control system in polynomial form. A classical adaptive controller with sigma modification on suppressing the wing rock motion is revisited. Using the nonlinear analysis technique based on the sum of squares optimization, ROA of the origin is computed with respect to the potential measurement error. The obtained result gives a solid guarantee on the allowable initial conditions where the adaptive controller is still able to suppress the wing rock motion. It provides confidence for the adaptive controller operating under some unforeseen conditions in practice.

Polynomial Lyapunov Functions for Synchronization of Nonlinearly Coupled Complex Networks.

In this article, we search for polynomial Lyapunov functions beyond the quadratic form to investigate the synchronization problems of nonlinearly coupled complex networks. First, with a relaxed assumption than the quadratic condition, a synchronization criterion is established for nonlinearly coupled networks with asymmetric coupling matrices. Compared with the existing synchronization criteria, our results are less conservative and have a wider application. Second, the synchronization problem for polynomial networks is characterized as the sum-of-squares (SOS) optimization one. In this way, polynomial Lyapunov functions can be obtained efficiently with SOS programming tools. Furthermore, it is shown that the local synchronization of certain nonpolynomial networks can also be analyzed by using the SOS optimization method through the Taylor series expansion. Finally, three numerical examples are presented to verify the effectiveness and less conservatism of our analytical results.

An improved sum‐of‐squares based approach to fuzzy tracking control design of nonlinear systems

AbstractIn this paper, using a more general Lyapunov function, less conservative sum‐of‐squares (SOS) stability conditions for polynomial‐fuzzy‐model‐based tracking control systems are derived. In tracking control problems the objective is to drive the system states of a nonlinear plant to follow the system states of a given reference model. A state feedback polynomial fuzzy controller is employed to achieve this goal. The tracking control design is formulated as an SOS optimization problem. Here, unlike previous SOS‐based tracking control approaches, a full‐state‐dependent Lyapunov matrix is used, which reduces the conservatism of the stability criteria. Furthermore, the SOS conditions are derived to guarantee the system stability subject to a given H∞ performance. The proposed method is applied to the pitch‐axis autopilot design problem of a high‐agile tail‐controlled pursuit and another numerical example to demonstrate the effectiveness and benefits of the proposed method.

Sparse Sum-of-Squares Optimization for Model Updating Through Minimization of Modal Dynamic Residuals

This research investigates the application of sum-of-squares (SOS) optimization method on finite element model updating through minimization of modal dynamic residuals. The modal dynamic residual formulation usually leads to a nonconvex polynomial optimization problem, the global optimality of which cannot be guaranteed by most off-the-shelf optimization solvers. The SOS optimization method can recast a nonconvex polynomial optimization problem into a convex semidefinite programming (SDP) problem. However, the size of the SDP problem can grow very large, sometimes with hundreds of thousands of variables. To improve the computation efficiency, this study exploits the sparsity in SOS optimization to significantly reduce the size of the SDP problem. A numerical example is provided to validate the proposed method.

An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontag's control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches.

Integrating NOE and RDC using sum-of-squares relaxation for protein structure determination.

We revisit the problem of protein structure determination from geometrical restraints from NMR, using convex optimization. It is well-known that the NP-hard distance geometry problem of determining atomic positions from pairwise distance restraints can be relaxed into a convex semidefinite program (SDP). However, often the NOE distance restraints are too imprecise and sparse for accurate structure determination. Residual dipolar coupling (RDC) measurements provide additional geometric information on the angles between atom-pair directions and axes of the principal-axis-frame. The optimization problem involving RDC is highly non-convex and requires a good initialization even within the simulated annealing framework. In this paper, we model the protein backbone as an articulated structure composed of rigid units. Determining the rotation of each rigid unit gives the full protein structure. We propose solving the non-convex optimization problems using the sum-of-squares (SOS) hierarchy, a hierarchy of convex relaxations with increasing complexity and approximation power. Unlike classical global optimization approaches, SOS optimization returns a certificate of optimality if the global optimum is found. Based on the SOS method, we proposed two algorithms-RDC-SOS and RDC-NOE-SOS, that have polynomial time complexity in the number of amino-acid residues and run efficiently on a standard desktop. In many instances, the proposed methods exactly recover the solution to the original non-convex optimization problem. To the best of our knowledge this is the first time SOS relaxation is introduced to solve non-convex optimization problems in structural biology. We further introduce a statistical tool, the Cramér-Rao bound (CRB), to provide an information theoretic bound on the highest resolution one can hope to achieve when determining protein structure from noisy measurements using any unbiased estimator. Our simulation results show that when the RDC measurements are corrupted by Gaussian noise of realistic variance, both SOS based algorithms attain the CRB. We successfully apply our method in a divide-and-conquer fashion to determine the structure of ubiquitin from experimental NOE and RDC measurements obtained in two alignment media, achieving more accurate and faster reconstructions compared to the current state of the art.

Computing compositional proofs of Input-to-Output Stability using SOS optimization and [formula omitted]-decidability

We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations.In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated.By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced–with bounded error–by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. The BFs can then be used to obtain bounds on the error between the outputs of the original system and its reduced approximation. To this end, we illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable Hodgkin–Huxley-type approximation. In addition to a detailed performance evaluation of BFComp, our case study also presents workarounds for systems with non-polynomial vector fields, which are not amenable to standard SOS optimizers.

Stabilization of Recurrent Fuzzy Systems via Sum of Squares-based Hybrid Control

This paper presents an approach for stabilization of equilibria in recurrent fuzzy systems. This type of dynamic fuzzy systems being defined via linguistic rules can be interpreted as interpolation between constant gradients, and therefore as hybrid dynamical system. It is shown that the latter viewpoint allows for a precise description of the system dynamics, but on the other hand lacks transparency. In order to render a given equilibrium of the recurrent fuzzy system globally asymptotically stable, local polynomial controllers are computed via sum of squares optimization to allow only for deterministic mode transitions on a micro level. In addition, the controlled recurrent fuzzy system can then be interpreted as finite deterministic automaton, thus allowing for analysis of system properties on a more abstract macro level. Relaxations are proposed in cases where recurrent fuzzy systems may not be rendered deterministic and the method is applied to two examples.

Closed-Form Estimates of the Domain of Attraction for Nonlinear Systems via Fuzzy-Polynomial Models

In this paper, the domain of attraction of the origin of a nonlinear system is estimated in closed form via level sets with polynomial boundaries, iteratively computed. In particular, the domain of attraction is expanded from a previous estimate, such as a classical Lyapunov level set. With the use of fuzzy-polynomial models, the domain of attraction analysis can be carried out via sum of squares optimization and an iterative algorithm. The result is a function that bounds the domain of attraction, free from the usual restriction of being positive and decrescent in all the interior of its level sets.

THE DOMAIN OF ATTRACTION FOR A SEIR EPIDEMIC MODEL BASED ON SUM OF SQUARE OPTIMIZATION

This paper is estimating the domain of attraction for a class of susceptible-exposed-infectious-recovered (SEIR) epidemic dynamic mo- dels by using sum of squares optimization. First, the stability is analyzed for the equilibriums of SEIR model, and the domain of attraction in the endemic equilibrium is estimated by using sum of squares optimization. Finally, a numerical example is examined.

Pseudopolynomial algorithms for certain computationally hard vector subset and cluster analysis problems

We analyze several NP-hard problems related to clustering and searching, in a given set of vectors in a Euclidean space, for a subset of vectors of fixed size. An important data mining problem related to sum of squares optimization reduces to these problems. We show pseudopolynomial algorithms that are guaranteed to find an optimum in these problems in case when vector components have integer values and the dimension is fixed.

Estabilidad de Sistemas No Lineales Basada en la Teoría de Liapunov

El comportamiento dinámico de los sistemas no lineales es mucho más rico que el de los lineales y su análisis mucho más complicado. Para el análisis de estabilidad, las técnicas basadas en la teoría de Liapunov tienen un lugar destacado. En este articulo se revisa parte de esta teoría incluyendo las técnicas de estimación de la cuenca de atracción. También se repasan los resultados que han aparecido en los últimos años sobre la aplicación a este campo de los métodos numéricos de optimización de suma de cuadrados.

Field Calibration and Validation of Solute Transport Models for the Unsaturated Zone

AbstractData from two solute transport field experiments, one obtained from soil solution samplers and the other from soil coring, are used to illustrate the parameter estimation problem for two models describing area‐averaged vertical transport of mobile solutes. The two models, the lognormal transfer function model (TFM) and the convection dispersion equation (CDE), were selected because of their simplicity and because they represent different hypotheses for the governing mechanisms determining longitudinal solute spreading on the field scale. The two parameters of each model are estimated using three calibration procedures: sum of squares optimization, method of moments, and maximum likelihood. The three calibration procedures yield different estimates for the parameters using a given data set, and different trends in parameter variation at different depths or times in a given experiment. Calculated uncertainty in the parameter estimates was high enough that no conclusion could be drawn about which model better described the data from the solution sampler experiment, but the TFM provided a better description of the coring experimental data after calibration than the CDE. It was concluded that future experiments on field scale solute transport through the unsaturated zone will have to monitor movement below 5 m in order to yield a data set suitable for distinguishing between different models.