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