Stability markers in control theory

The linear dynamic systems stability criterion, allowing one to identify the defective/failed coefficients in the characteristic polynomial at each time quantum, is presented for the first time.


Introduction
Is there a great probability to generate Hurwitz's polynomial in a random way?And, fundamentally, is it possible to achieve Hurwitz's features with a selection ( ) for the high order systems?It is appropriate to concentrate the answer to such questions on the extent/volume assessing of the two clusters, stability and instability, as the interaction of extreme opposites.
It should be admitted that the potential of the socalled stability criteria [1,2] by Hermite-Biller's theorem about the radicals alternation, Sturm's method for the rational function index calculation, Routh's algorithm, Hurwitz and Lienard-Schipar's determinantal conditions, quadratic forms based on Schur's method, Chebotarev's theorem, a generalization for the entire functions of Hermite-Biler's theorem, Pontryagin's theorem on stable quasipolynomials, and others has been exhausted for formulated problems solving and it is quite insignificant despite of the apparent theoretical harmony and algorithmic/computational clutters.Is it necessary to confirm the stability or instability dynamical systems?Out-of-date.With the same success, the step response ( ) h t step(sys) is the essence of the answer to this question, in addition geometrically.And a map of the radicals pzmap(sys) distribution is more informative (step(sys) and pzmap(sys) the computational procedures of Control System Toolbox in MATLAB).But, in fact, besides the ascertaining of the instability one should clearly understand what to do with non Hurwitz's polynomial?When a patient needs emergency medical care, it is necessary to provide it qualifiedly, but not contemplating the obvious fact to convince yourself that it is really required.
In [3,4] authors propose a method for computing the distance of a stable polynomial to the set of unstable ones (both in the Hurwitz and in the Schur case).The method is based on the reformulation of the problem as the structured distance to instability of a companion matrix associated to a polynomial.
and points the recovery ways of each coefficient , so it is the important issue.There are no similar solutions in the automatic control theory.

Material and researches results
Definition.There is the 2 n × pieces set spectrum of the geometric averages : gmean locations of the instant dynamic solutions ( ) , fig. 1, formed with a serial dropping of the polynomial (1) highest (2) and youngest (3) degrees.New solutions of the so-called enclosed polynomials of ( ) , n i j − orders define other combinations of valid and complex conjugated radicals on the complex plane.In their turn, these radicals are geometrically averaged with the own "universe brick" of the encloses on the right: of the encloses on the left:   The horizontal and vertical numbering of the ( ) t P matrix (4) cells will be associated with the degrees of the polynomial (1), beginning from the highest left top one accordingly, thereby the localization of each cell ij P will be attached to the spectrum of the geometric averages: forming two lower triangular Р matrices, ( ) ( ) In other words, the scaling constants of cells then, for example, the polynomial of the hybrid enclose first iteration in ( ) , A p t determines the corresponding cell of the Р -matrix ( 6): the matrix of the radii scales of enclosed invariants ( ) ( ) ( ) MATLAB P -matrices filling procedure is presented below: function pmatrix=buildpmatrix(den) [m,n]=size(den); pmatrix=zeros(n+1); den_fliplr=fliplr(den); for i=1:n for j=0:n-i pmatrix(j+1+i,j+1)=nthroot(abs(den_fli plr(n+1-j-i)/den_fliplr(n+1-j)),i); end end Theorem.The nonstationary dynamic system with the transfer function Each element ij P of the lower triangular matrix ( )

P
is a linear dynamic system stability marker in the graphical interpretation of the adjacent and neighbouring 1 ij± P relationship (Fig. 2).The columns reciprocal geometry, but not their absolute values, is the essential event.The mentioned facts in a formulaic interpretation correspond to: for the i -th for the i -th and for the ( ) 1 n + rows, and > and the ( ) columns, and also The violation of any one surely results the dynamic system in the dominant instability cluster.The limits/prohibitions plenty of the each marker ij Р location in the Р matrix guarantees the adjustment mechanism existence of shallow and more profound system.The adjustment mechanism of shallow system is identified with the conditions of rows, columns and subdiagonals of the main diagonal, the adjustment mechanism of more profound system is defined with the subdiagonals conditions of its secondary diagonal.

Conclusion
The linear dynamic systems stability criterion, allowing one to identify the defective/failed ( ),

Fig. 1 .
Fig. 1.A distribution map of radicals, geometric averages ( ) 0 i R still non-stationary in the text here and further, time dependence symbol is omitted because of the matrix overcharging.The first column and the last row of P -matrices are those investments on the right (2) and on the left (3), which are separated by a dotted line.The other elements ij P should be associated with the so-called hybrid investments, mentally leaving any adjacent components with k p at the same time.Definition.The linear non-stationary dynamic system is characterized by ( ) the selected transfer function polynomial denominator ( ) dimension of the subdiagonal cells of Р - matrices [ ] / sec rad , all cells of Р -matrices (6), (7) are the frequencies, that can be seen obviously, if we present the transfer function(5) with the normalized polynomials

n
is stable at each time quantum k t if the component ij P − of the denominator polynomial ( ) , B p t P -matrix is under the lack of growth, as well as the same component ij P of each superdiagonal 3 n − pieces of her side diagonal, calculated at the same time quantum k t is under decreasing (11).

Fig. 2 .
Fig. 2. Probability to generate this harmony in P -matrix of the solutions space with a selection of random samples in the primary space -is equal to 0!

1 n
last but one subdiagonals of the main diagonal.The same phenomenon occurs in component ij of the main and secondary diagonals, of each row 2 n − pieces and each column − of the denominator polynomial ( ) , B p t P -matrix is under the lack of growth, as well as the same component ij P of each superdiagonal 3 n − pieces of her side diagonal (Fig.3 the balance of phase transition and getting to the cluster stability is delicate.Is there any probability to randomly generate a polynomial a stable according to Hurwitz' criteria?This probability is equal to zero with the growth of the polynomial ( ) , B p t order!It is similar to the probability of life appearance in the Universe.As, for example, can fifty-one randomly assigned component *49 interdependent locations of the markers?

Fig. 3 .
Fig. 3.About the lack of growth of the components ij P of each

Fig. 4 .
Fig. 4.About the lack of growth of the components ij P of the

Fig. 5 .
Fig. 5.About the lack of growth of the components ij P of the

Fig. 6 .Fig. 7 .
Fig. 6.About the non-decreasing of the components ij P of

Fig. 8 .
Fig. 8.There is a special radicals geometry in each cell of Р - matrix.Dominant of the neighboring components 1 ij± Р = at each time