# Domain of Attraction: Analysis and Control via SOS by Graziano Chesi

By Graziano Chesi

The balance of equilibrium issues performs a primary position in dynamical structures. For nonlinear dynamical structures, which characterize nearly all of genuine crops, an research of balance calls for the characterization of the area of allure (DA) of an equilibrium element, i.e., the set of preliminary stipulations from which the trajectory of the approach converges to this sort of element. it truly is recognized that estimating the DA, or maybe extra trying to keep an eye on it, are very tricky difficulties as a result of the complicated dating of this set with the version of the system.

The e-book additionally bargains a concise and straightforward description of the most beneficial properties of SOS programming that are utilized in study and instructing. specifically, it introduces a variety of sessions of SOS polynomials and their characterization through LMIs and addresses ordinary difficulties reminiscent of institution of positivity or non-positivity of polynomials and matrix polynomials, picking the minimal of rational services, and fixing platforms of polynomial equations, in instances of either unconstrained and restricted variables. The suggestions awarded during this e-book come in the MATLAB^{®} toolbox SMRSOFT, which might be downloaded from http://www.eee.hku.hk/~chesi.

*Domain of Attraction* addresses the estimation and keep watch over of the DA of equilibrium issues utilizing the unconventional SOS programming scheme, i.e., optimization thoughts which were lately constructed in accordance with polynomials which are sums of squares of polynomials (SOS polynomials) and that quantity to fixing convex optimization issues of linear matrix inequality (LMI) constraints, often referred to as semidefinite courses (SDPs). For the 1st time within the literature, a way of facing those matters is gifted in a unified framework for varied situations looking on the character of the nonlinear platforms thought of, together with the instances of polynomial platforms, doubtful polynomial structures, and nonlinear (possibly doubtful) non-polynomial structures. The tools proposed during this ebook are illustrated in a number of genuine structures and simulated structures with randomly selected buildings and/or coefficients inclusive of chemical reactors, electrical circuits, mechanical units, and social versions.

The e-book additionally deals a concise and straightforward description of the most positive factors of SOS programming that are utilized in learn and educating. particularly, it introduces quite a few periods of SOS polynomials and their characterization through LMIs and addresses ordinary difficulties comparable to institution of positivity or non-positivity of polynomials and matrix polynomials, picking the minimal of rational services, and fixing platforms of polynomial equations, in situations of either unconstrained and limited variables. The recommendations offered during this publication come in the MATLAB^{®} toolbox SMRSOFT, that are downloaded from http://www.eee.hku.hk/~chesi.

**Read or Download Domain of Attraction: Analysis and Control via SOS Programming PDF**

**Similar system theory books**

This ebook is an often-requested reprint of 2 vintage texts through H. Haken: "Synergetics. An creation" and "Advanced Synergetics". Synergetics, an interdisciplinary study software initiated through H. Haken in 1969, bargains with the systematic and methodological method of the quickly transforming into box of complexity.

Strong layout brings jointly sixteen chapters via an eminent crew of authors in a variety of fields proposing elements of robustness in organic, ecological, and computational platforms. The volme is the 1st to handle robustness in organic, ecological, and computational platforms. it's an outgrowth of a brand new examine application on robustness on the Sante Fe Institute based through the David and Lucile Packard beginning.

**Self-organized biological dynamics & nonlinear control**

The becoming influence of nonlinear technological know-how on biology and medication is essentially altering our view of dwelling organisms and disorder approaches. This booklet introduces the applying to biomedicine of a large diversity of suggestions from nonlinear dynamics, equivalent to self-organization, complexity, coherence, stochastic resonance, fractals, and chaos.

This thesis analyzes and explores the layout of managed networked dynamic structures - dubbed semi-autonomous networks. The paintings ways the matter of potent keep an eye on of semi-autonomous networks from 3 fronts: protocols that are run on person brokers within the community; the community interconnection topology layout; and effective modeling of those usually large-scale networks.

- Design of Multi-Bit Delta-Sigma A/D Converters
- Stochastic Differential Equations: An Introduction with Applications
- New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems
- Perspectives in Control Engineering Technologies, Applications, and New Directions

**Extra resources for Domain of Attraction: Analysis and Control via SOS Programming **

**Example text**

F (x) is positive if μ pol ( f ) > 0. The following result provides a relationship between the SOS index and the positivity index. 7. Consider f ∈ Pn . Then, λ pol ( f ) ≤ μ pol ( f ). 62) Proof. 58) by b pol (x, m) and b pol (x, m), respectively. 58) holds. Lastly, if f = 0, one simply has that λ pol ( f ) = μ pol ( f ) = 0. Consequently, the SOS index can be used to investigate positivity of a polynomial as explained hereafter. 2. Consider f ∈ Pn . The following statements hold. 1. f (x) is nonnegative if λ pol ( f ) ≥ 0.

In particular: 1. λ pol ( f ) = λ pol (ρ f ) for all ρ > 0; 2. λ pol ( f ) = 1 for f (x) = b pol (x, i) 2 for all i ∈ N. The following result states that the SOS index always exists. 6. The SOS index λ pol ( f ) exists (and, hence, is bounded) for all f ∈ Pn . 3 SOS Polynomials 17 Proof. First, let us observe that for any F there exists z and α such that F + L(α ) − zI ≥ 0. 58) is nonempty. Second, let us observe that for any F there exists z such that F + L(α ) − zI ≥ 0 ∀α . In fact, if one suppose for contradiction that this is not true, it follows that for all z there exists α such that 0 ≤ b pol (x, m) (F + L(α ) − zI)b pol (x, m) = f (x) − z b pol (x, m) which is impossible since b pol (x, m) is positive definite and has the same degree of f (x).

A homogeneous polynomial f (x) is positive definite if and only if μhom ( f ) > 0. 4. Let f ∈ Pn be a homogeneous polynomial of even degree. The following statements hold. 1. f (x) is SOS if and only if there exists α satisfying the LMI F + L(α ) ≥ 0 where F + L(α ) = CSMRhom ( f ). 2. The SOS index λhom( f ) exists (and, hence, is bounded). 3. f (x) is SOS if and only if λhom ( f ) ≥ 0. 4. f (x) is positive semidefinite if λhom ( f ) ≥ 0. 5. f (x) is positive definite if λhom ( f ) > 0. 6. λhom ( f ) ≤ μhom( f ).