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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
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     {\flushleft\it Том 20, стр. 144--154 (2023) \hspace{65mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
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         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\title[Gyroscopic Stabilization Of Vibrational System]{Optimal Gyroscopic Stabilization \\ Of Vibrational System: Algebraic Approach}
\author{{A.V. CHEKHONADSKIKH}}%
\address{Alexander  Vasilievich Chekhonadskikh
\newline\hphantom{iii} Novosibirsk State Technical University,
\newline\hphantom{iii} K Marx av., 20,
\newline\hphantom{iii} 630073, Novosibirsk, Russia}%
\email{Chekhonadskikh@corp.nstu.ru}%

\thanks{\sc Chekhonadskikh, A.V.,
Optimal Gyroscopic Stabilization Of Vibrational System: Algebraic Approach}
\thanks{\copyright \ 2023 Chekhonadskikh, A.V.}
\thanks{}
\thanks{\it Received  Month, Date, 2023, published  Month, Date,  2023.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } The paper deals with LTI vibrational systems with positive definite stiffness matrix $K$ and symmetric damping matrix $D$. Gyroscopic stabilization means the existence of gyroscopic force with a skew-symmetric matrix $G$, such that a closed loop system with damping matrix\\ $D+G$ is asymptotically stable. The feature of characteristic polynomial in the case  predetermines such stabilization as a low order control design. Assuming the nesessary condition of gyroscopic stabilization is fulfilled, we pose the problem of achieving relative stability maximum using a  stabilizer $G$. The stability maximum value is determined by a matrix $D$ trace, but its reachability is connected with system pole locations according to critical root diagrams. We illustrate root diagrams and root polynomials techniques to an optimal gyroscopic stabilizer design by examples of dimension 3--5.\medskip

\noindent{\bf Keywords:} vibrational system, gyroscopic stabilizer, low order control, rightmost poles, relative stability, critical root diagram.
 \end{quote}
}


\section{Introduction}

Let multidimensional vibrational system be described by a matrix differential equations of a form 

$$X'' + DX' + KX = 0,$$

where  $X(t)$ is a state variable vector, $D=D^T$  is indefinite damping matrix and $K=K^T$  is positive definite stiffness matrix. 
 In outlining the problem, we will follow [1--2]. In general case vibrational systems are unstable; but if matrix  $D$ is also positive definite, then the system is asymptotically stable. The gyroscopic stabilization tends to stabilize a system without further dissipation or damping by adding gyroscopic forces $GX'$ with some skew-symmetric matrix $G$  (see e.g. [3]). The reason is the ‘negative damping’ effect in models with vibrations which are induced by friction, [4--7]. This is similar to the effect on which Galileo Galilei have said in 1638 that «a glass of water may be made to emit a tone merely by the friction of the fingertip upon the rim of the glass»; then Benjamin Franklin in 1761 designed an ‘armonica’, where sound was produced by vibration of rotating glass bowls touched by moistened fingers of a performer, [8].  
\begin{definition} Let us call the matrix $G = - G^{T}$ a $\textit{gyroscopic stabilizer}$ for a vibrational system, if the solution $X$ of an equation $$X'' + (D + G)X' + KX = 0$$  is asymptotically stable. In this case this system is $\textit{gyroscopically stabilizable}$. \end{definition}
  A necessary condition for gyroscopic stabilization is well-known: both traces of matrices   $D$ and  $K^{-1}D$ must be positive; for short proof see [1]. In [6] the authors ask whether this condition is also sufficient. In the case  $n=2$ they proved it using Lyapunov matrix equation approach, and for $n > 2$ gyroscopic stabilizability was shown only under additional conditions. The sufficiency of this conditions in cases  $n = 3, 4$ was proven in [1] using inverse eigenvector technique; the sufficiency hypothesis has been tested for dimensions $n > 4$  on a large number of examples. 
 Prof. T. Damm in his lecture on gyroscopic stabilization has formulated several problems including a sufficiency proof for dimensions   $n > 4$ (Lehrstuhl fuer Angewandte Mathematik, Universitaet Bayreuth, Wuerzburg, June 1, 2012).



\section{Problem statement}

The authors in [9--10]  were finding maximal relative stability of LTI control systems through the appropriate value choice of a controller parameters vector  $\bar{p}$. Characteristic polynomial $\chi{(s)}$   of a close loop system has coefficients depending on control parameters $\bar{p}$ -- or, in our case, on elements of stabilizer matrix  $G$. The difficulty of low order control design is that a number of control parameters or  their entering into characteristic polynomial coefficients do not allow to achieve an arbitrary system pole location; therefore, one must search for an optimal one.

Hurwitz stability function  $H(\bar{p})$ is expressed directly through $m$  system poles $z_{1},...,z_{m}$ (i.e. roots of a characteristic equation): $$H(\bar{p})=H(G)= max Re(z_{1},...,z_{m}).$$
It is negative to relative stability of a system.

 Thus Hurwitz function should be minimized by choosing optimal vector $\bar{p}$  value. As a rule, extrema and subextrema of function $H(\bar{p})$ and similar objective functions [10] are achieved in those cases when system poles are accumulated in the  maximum possible number on a right boundary of their location domain. This number depends on a vector  $\bar{p}$ dimension (i.e. a number of free matrix stabilizer $G$ elements); all variants of a mutual poles location on their right boundary are represented by critical root diagrams [9, 11]. Using root polynomial technique, the authors examined, whether each pole location is realized and for what vector  $\bar{p}$ value it is achieved.
Minimization problems were solved in [9, 10] for the controlled plant composed of three massive bodies with elastic connections; its equations coincide with the vibrational system ones for  $n = 3$.
Here we consider the maximum possible relative stability of a vibrational system with a gyroscopic stabilizer and effective maximum finding technique by similar means as those used in [9,10]. This can be done in general as the method description. Therefore, here we will show applications of the latter to some examples of dimensions 3--5.


\section{Theoretical preface}
 
Turning from differential equations of a vibrational system to its Laplace operator form, we obtain a polynomial matrix equation: $L(s)X = (Is^2 + (D + G)s + K)X = 0$. All the matrix properties here are as indicated above, $X$  is Laplace transform of a 	state vector $X(t)$ ; $I$ is the identity matrix. Hence the characteristic equation is $$\chi(s)=det(Is^2 + (D + G)s + K) = 0.$$
 
 System poles are denoted as above $z_{1},...,z_{2n}$.  The following statement represents an analog of Vieta theorem:

\begin{proposition}
(a) $z_{1}+...+z_{2n} = -trD$; (b) $z_{1}\cdot...\cdot z_{2n} = |K|$.
\end{proposition}

\begin{proof}
(a) Let's find the polynomial $\chi(s)$  coefficient $\chi_{2n-1}$   of the monomial\\ $\chi_{2n-1}\cdot s^{2n-1}$ , it is negative to the pole sum. From the skew-symmetry of the matrix $G$  it follows that $trD = tr(D+G)$. We expand the determinant by the $1^{st}$ row: $detL(s)= l_{1 1}L_{1 1}+l_{1 2}L_{1 2}+...+l_{1 n}L_{1 n}$  (here  $L_{1 k}$ is an element $l_{1 k}$ adjunct) and see, that for  $k>1$ the degrees in the variable $s$ of all terms $l_{1 k}L_{1 k}$ do not exceed $2n-2$  For the first term we get by induction $$l_{1 1}L_{1 1}=(s^2 +d_{1 1}s)L_{1 1}=(s^2 +d_{1 1}s)\cdot (s^{2n-2} +D_{1 1}s^{2n-3} +...),$$ 
where $D_{1 1}$ is reduced matrix $D$  without $1^{st}$ row and $1^{st}$ column.  So, \\ $trD = -z_{1}-...-z_{2n}$.  
  The statement (b) is obvious.
\end{proof}

\begin{corollary}
For Hurwitz function of a vibrational system with gyroscopic stabilization it holds the inequality $H(G)\geq -trD/2n$. The value $trD/2n$ is hypothetical supremum of a system relative stability.
\end{corollary}


\begin{proposition}
The polynomial $\chi(s)$ coefficient $\chi_1$ at the variable $s$ does not include stabilizer matrix $G$ elements.
\end{proposition}

\begin{proof}
By an orthogonal transformation we reduce the matrix $K$ to a diagonal form $SKS^T=diag(\kappa_1 ...\kappa_n)=K_1$, where equalities $\kappa_j = \alpha ^2_j$ hold due to positive definiteness of matrix $K$.
 Since the determinant  $\chi (s) = detL(s)$ remains the same, we get 
 $$\chi (s) =s^{2n}det(I+S(D+G)S^Ts^{-1}+K_1 s^{-2})=s^{2n}det(I+S(D_1+G_1)S^T\zeta +K_1 \zeta^2),$$
where $ \zeta =s^{-1}$ and matrix $G_1 = SGS^T={(g^1_{i j})}_{n\times n}$  is still skew-symmetric.
  Taking out $\alpha_k$, $k=1,...,n$, as common factors of $k^{th}$ row and $k^{th}$ column, we note that the matrix ${({(g^1_{i j})}/{(\alpha_i\cdot \alpha_j)})}_{n\times n}$ is still skew-symmetric. By Proposition 1 we conclude 
 $$\chi (s) = s^{2n} \kappa_1 \cdot ... \cdot \kappa_n \cdot det(I_2+S(D_2+G_2)S^T\zeta +I \zeta^2)=$$ $$=s^{2n}\cdot|K|\cdot(\zeta^{2n}+trD_2\zeta^{2n-1}+...)=|K|+|K|\cdot trD_2s+...\;.$$
\end{proof}

\begin{corollary}
Gyroscopic stabilization for any vibrational system dimension is a low order control design problem. This is clear from the fact that the two highest and two lowest characteristic polynomial coefficients do not include gyroscopic stabilizer $G$ elements, and an arbitrarily given pole location of a closed loop system is not reachable.
\end{corollary}

\begin{proposition}
A characteristic polynomial $\chi (s)$ coefficients vector has zero derivatives with respect to gyroscopic stabilizer $G$ elements at zero values of the latter.
\end{proposition}

\begin{proof}
Since  partial derivatives are to be found at the initial value $G_0 = (0)_{n\times n}$, we can consider without loss of generality the case when only two elements $g_{12}  = -g_{21}$  of the matrix $G$ are nonzero and the matrix $D$ has a diagonal form. Let us clarify the characteristic polynomial dependence on stabilizer elements: 
$$\chi (s)=det\left(Is^2+\left(\begin{pmatrix}
  d_{1,1} & 0 & \cdots & 0 \\
  0 & d_{2,2} & \cdots & 0 \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  0 & 0 & \cdots & d_{n,n} 
 \end{pmatrix} + \begin{pmatrix}
  0 & g_{1,2} & \cdots & 0 \\
  -g_{1,2} & 0 & \cdots & 0 \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  0 & 0 & \cdots & 0 
 \end{pmatrix}\right)s+\right.$$ $$+\left. \begin{pmatrix}
  k_{1,1} & k_{1,2} & \cdots & k_{1,n} \\
  k_{1,2} & k_{2,2} & \cdots & k_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & k_{n,n} 
 \end{pmatrix} \right)=  $$

$$=det \begin{pmatrix}
  0 & g_{1,2}s & \cdots & 0 \\
  k_{1,2}-g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$+det \begin{pmatrix}
  s^2+d_{1,1}s+k_{1,1} & k_{1,2} & \cdots & k_{1,n} \\
  k_{1,2}-g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n}\\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}=$$

$$=det \begin{pmatrix}
  0 & g_{1,2}s & \cdots & 0 \\
 -g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
 0 & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$ + det \begin{pmatrix}
  0 & g_{1,2}s & \cdots & 0 \\
  k_{1,2}& s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$+det \begin{pmatrix}
  0 & k_{1,2} & \cdots & k_{1,n} \\
  -g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n}\\
  \vdots  & \vdots  & \ddots & \vdots  \\
  0 & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$+det \begin{pmatrix}
  s^2+d_{1,1}s+k_{1,1} & k_{1,2} & \cdots & k_{1,n} \\
  k_{1,2}& s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n}\\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix} =$$
$$=det \begin{pmatrix}
  0 & g_{1,2}s & \cdots & 0 \\
 -g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n} \\
  \vdots  & \vdots  & \ddots & \vdots  \\
 0 & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$+det \begin{pmatrix}
  s^2+d_{1,1}s+k_{1,1} & k_{1,2} & \cdots & k_{1,n} \\
  k_{1,2}& s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n}\\
  \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix} =$$

$$=-g^2_{12}s^2detX_{n-2}(s)+\chi^0(s),$$
  
 where $\chi^0(s)=det(Is^2+Ds+K)$, \\ $X_{n-2}(s) = \begin{pmatrix}
  s^2+d_{3,3}s+k_{3,3} &  \cdots & k_{3,n} \\
  \vdots   & \ddots & \vdots  \\
  k_{3,n} & \cdots & s^2+d_{n,n}s+k_{n,n} 
 \end{pmatrix}$, and
  $$det \begin{pmatrix}
  0 & g_{1,2}s & \cdots & 0 \\
  k_{1,2}& s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n} \\
 \vdots  & \vdots  & \ddots & \vdots  \\
  k_{1,n} & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix}+$$ $$+det \begin{pmatrix}
  0 & k_{1,2} & \cdots & k_{1,n} \\
  -g_{1,2}s & s^2+d_{2,2}s+k_{2,2} & \cdots & k_{2,n}\\
  0  & \vdots  & \ddots & \vdots  \\
  0 & k_{2,n} & \cdots & s^2+d_{n,n}s+k_{n,n}
 \end{pmatrix} = 0.$$

  So $\frac{\partial}{\partial g_{ij}} \chi_k =0$  for all the coefficients $\chi_k =0$  of the polynomial $\chi (s)$ at the starting point $G_0 = (0)$.

\end{proof}

\begin{corollary}
The point $G_0 = (0)_{n\times n}$ is  critical point of Hurwits stability function $H(G)$.\end{corollary}

\begin{proof}
Since coefficients of a characteristic polynomial $\chi (s)$ are even functions of each stabilizer element $g_{ij}$ at zero values of other ones, a dependence $z_k(g_{ij})$  of each of the poles $z_{1},...,z_{2n}$. also turns out to be even. So is Hurwitz stability function $H(G)= max Re(z_{1},...,z_{m}).$ Since a derivative of an even function at zero point is equal to zero too, then in both cases -- rightmost pole dependence on stabilizer elements $g_{i j}$ is differentiable , as well as nondifferentiable (the latter occurs at multiple roots) -- the point $G_0 = (0)$ is critical for Hurwits function.
\end{proof}

$\textit{Remark}.$ By corollary 3 numerical minimization of Hurwitz function should take as starting points some non-zero values of a stabilizer $G$. Many numerical examples have shown that the point $G_0 = (0)}$  turns out to be a local minimum for a function $H(G)$. In other words, a small gyroscopic stabilizer does not improve a relative stability value of an open loop vibrational system.



\section{Numerical examples}

Everywhere below we consider the problem of optimal gyroscopic stabilization in coordinates, where the symmetric matrix $D$ takes a diagonal form.

\textbf{\textit{Example} 1.} The necessary condition holds for the following coefficient matrices of the vibrational system of dimension 3:
$$ D= \begin{pmatrix}
2 &  0 & 0 \\
0   & 1 & 0  \\
0 & 0 & -1 \end{pmatrix},  
K= \begin{pmatrix}
2 &  1 & 0 \\
1   & 2 & 1  \\
0 & 1 & 2} \end{pmatrix}, 
 K^{-1}D= \begin{pmatrix}
3/2 &  -1/2 & -1/4 \\
-1   & 1 & 1/2 \\
1/2& -1/2  & -3/4 \end{pmatrix}.$$

It is unstable, because the system poles $z_k$ set contains a complex pair\\ $0.3552 \pm 1.2882 i $.
Obviously $z_{1}+...+z_{6}=-2$ and  $max Re(z_{1},...,z_{6})\geq -\frac{1}{3}$. 
There exists a one-dimensional gyroscopic stabilizer $G = \begin{pmatrix}
0 & 0 & b \\
0 & 0 & 0  \\
-b & 0 & 0 \end{pmatrix}$, which provides a suboptimal result with $b \approx 1.658$:\\
$$max Re(z_{1},...,z_{6})\approx Re(z_{3},...,z_{6})\approx -0.3241 \approx -\frac{1}{3} .$$
 
By introducing into the stabilizer the second parameter $c=g_{2 3} = -g_{3 2}$ , we can try to achieve a theoretical maximum of relative stability $H_{opt} (G) = max Re(z_{k}) =-1/3$. Let us seek all the poles location on the complex plane vertical: $Re z =-1/3$ , i.e. characteristic polynomial $\chi (s)$ must be equal to the \textit{root polynomial} $r(s)$(see [9, 10]):
 $$\chi (s)= s^6 +2s^5 +(b^2 +c^2 +5)s^4 +(b^2 +2c^2 +6)s^3 +(2b^2 -2bc +2c^2 +8)s^2 +7s +4=$$
$$=(s^2 +\frac{2}{3}s + y_1)\cdot (s^2 +\frac{2}{3}s + y_2)\cdot (s^2 +\frac{2}{3}s + y_3), $$
here $y_k =\frac{1}{9} + Im^2 z_{2k-1,\: 2k}$ are \textit{root coordinates} [9].
We equate coefficients and solve the polynomial equations with respect to $b, c, y_{1,2,3}$. From the system we get the poles $z_{1,2}\approx -\frac{1}{3} \pm 0.6488i, z_{3,4}\approx -\frac{1}{3} \pm 1.4537i, 
z_{5,6}\approx -\frac{1}{3} \pm 1.8080i$.   
The correction $c$ is rather small: $с \approx -0.092$; the parameter $b$ is almost of the same value: $b \approx 1.569$.

A decrease in the matrix $D$ trace allows achieving the maximum stability according to prop. 1 by using one more stabilizer coordinate $a=g_{1 2} = -g_{2 1}$. E.g. let vibrational system matrices be the following:
$$  D= \begin{pmatrix}
0.1 &  0 & 0 \\
0   & 1 & 0  \\
0 & 0 & -1 \end{pmatrix},  
G= \begin{pmatrix}
0 &  a & b \\
-a   & 0 & c  \\
-b & -c & 0} \end{pmatrix}. $$

Here the polynomial equation $\chi (s)=r(s)$ leads to 5 equations for coefficients at degrees $s^0\; -\;s^4$, because the coefficients at $s^5$ and $s^6$ are equal a priori. Therefore, one can choose some free variable and assign different values to it, finding five other unknowns and obtaining one-dimensional manifold, at each point of which the system relative stability is maximal(see [12]). For example, we can set one parameter $c = -0.2833$  and find the rest ones: $y_1 \approx 0.11565, y_2 \approx 0.9292, y_3 \approx 37.2233, \\a \approx -3.9337, b \approx -4.2092$.
All the poles are located on one complex vertical  $Re z = -trD/6 = -1/60$:
$Im z_{1,2}\approx \pm 0.3397i, Im z_{3,4}\approx  \pm 0.9638i,Im z_{5,6}\approx  \pm 6.1011i$.

\textbf{\textit{Example } 2.} Consider a four-dimensional vibrational system; as above, coordinates are assumed to be those in which the matrix $D$ is diagonal:
$$ D= \begin{pmatrix}
2 &  0 & 0 & 0\\
0   & 1 & 0 &0  \\
0 & 0 & -1 &0 \\
0   & 0 & 0 &-1   \end{pmatrix},  
K= \begin{pmatrix}
2 &  1 & 1 &0 \\
1   & 2 & 1 &1 \\
1 & 1 & 2 & 1 \\
0 & 1 & 1 & 2 \end{pmatrix}. $$

It is unstable, because its eight poles contain the rightmost pair\\ $z_{1,2} \approx 0.4262 \pm 0.8087i $. 

The  necessary  condition  holds: $ tr D = trK^{-1} D = 1 $.  Gyroscopic stabilizer in general case is represented by the matrix $G =  \begin{pmatrix}
 0 & a & b & c\\
 -a   & 0 & d & e  \\
 -b & -d & 0 & f \\
 -c & -e & -f & 0   \end{pmatrix}$  .
  
So the characteristic polynomial has the form:
$$ \chi (s)=s^8+s^7+(5+a^2+b^2+c^2+d^2+e^2+f^2)s^6+(5-2a^2+d^2+e^2+3f^2)s^5+...$$
$$+[6+a(3a-2b-2c+4d-2e+2f)+b(3b-2c-4d+2e-2f)+$$ $$+c(3c-2e-2f)+4d(d-e+f)+e(3e-2f)+3f^2]s^2+4s+4. $$
  By analogy with example 1, a three-parametric stabilizer $G =  \begin{pmatrix}
 0 & 0 & b & c\\
 0   & 0 & d & 0  \\
 -b & -d & 0 & 0 \\
 -c & 0 & 0 & 0   \end{pmatrix}$  provides the requirement $Re(z) = -tr D/8$  for four complex pole pairs location. It allow us to reach the theoretical maximum of relative stability  $H_{opt}(G)=max Re(z_1,...,z_8)=-\frac{1}{8}$ at the values $b \approx -0.3901, c \approx -1.5436$  and $d \approx -1.3153$. The system pole pairs have imaginary components  $$Im(z_k)\approx\pm 0.7051, \pm 0.8684, \pm 1.3062, \pm 2.4228.$$

The number of variable stabilizer parameters allows us to set a denser root location. E.g. let us set the damping matrix $D= \begin{pmatrix}
1 &  0 & 0 & 0\\
0   & 1 & 0 &0\\
0 & 0 & 1 & 0 \\
0 & 0 & 0 &-2 \end{pmatrix}$, which satisfies the necessary conditions, and the requirement of triple complex pole pairs location on the rightmost vertical $Re(z) = -1/8$, i.e. the root polynomial is\\ $r(s) = (s^2 + 0.25s + 0.015625 + y^2)^3$. By prop.1 the fourth complex pair must be located on the same vertical too. 
Equating to zero the element $a$ and the remainder of division $rem (\chi(s), r(s))$, we obtain six polynomial equations with respect to the rest stabilizer elements and triple pole imaginary part $y$. Their solution gives the values $$b \approx 0.3836, c \approx -0.0877, d \approx 1.2909, e \approx -0.4512, f \approx -2.0864, y \approx 0.8698.$$
Hence 'almost triple' pole pair  $z_{1,…,6}\approx -0.125 \pm 0.87i$ is located approximately on the same vertical with the forth one $z_{7,8} \approx -0.125 \pm 2.9446i$.

Reducing the value $tr D$ and shifting closer to the necessary condition boundary do not affect the calculation method. Let us set $D = diag (1; 1; 1; -2.8)$, then we'll get rightmost vertical $Re(z) = -0.025$, and the solution of the equation system turns out to be $b \approx 1.4545, c \approx 2.4437, d \approx 0.498, e \approx -0.2076, f \approx 0.1972,\\ y \approx 0.8784.$ Almost triple pole pair  $z_{1,…,6} \approx -0.025 \pm 0.78i$  is located on the same vertical as the forth pair  $z_{7,8} \approx -0.025 \approx 2.9471i .$

\textbf{\textit{Example} 3.} Let five-dimensional vibrational system be defined by the following matrices $D$ and $K$: $$ D= \begin{pmatrix}
0.75 & 0 & 0 & 0 & 0\\
0 & 0.75 & 0 & 0 & 0\\
0 & 0 & 0.75 & 0 & 0\\
0 & 0 & 0 & 0.75 & 0\\
0 & 0 & 0 & 0 & -2 \end{pmatrix},  
K= \begin{pmatrix}
2 & 1 & 0.5 & 0 & 0\\
1 & 2 & 1 & 0.5 & 0\\
0.5 & 1 & 2 & 1 & 0.5\\
0 & 0.5 & 1 & 2 & 1\\
0 & 0 & 0.5 & 1 & 2  \end{pmatrix}. $$
So the necessary stabilizability conditions are satisfied: $tr D = 1, tr K^{-1}D \approx 1.0707$. 

As in example 2, we require four complex pole pairs to be located on the vertical $Re(z)=-trD/10 = -0.1$ (the fifth pair must be located approximately there too), and we assume one of them to be double:
$$r(s) = (s^2 + 0.2s + y_1)^2 (s^2 + 0.2s + y_2)(s^2 + 0.2s + y_3),\ \;\:y_k =0.01+Im^2z_{2k-1,2k}. $$
Equating to zero the remainder of division $rem (\chi(s), r(s))$ will lead to eight polynomial equations, which will be enough for unknowns $y_{1,2,3}$  and five variable stabilizer parameters: $$G= \begin{pmatrix}
0 & 0 & 0 & 0 & d\\
0 & 0 & 0 & 0 & g\\
0 & 0 & 0 & k & l\\
0 & 0 & -k & 0 & m\\
-d & -g & -l & -m & 0 \end{pmatrix}.$$
The solution yields $d\approx -0.9817, g \approx 0.8544, k\approx 1.4066, l\approx 0.6231, m \approx 1.2527$, and the poles are
$z_{1-4}\approx -0.1000 \pm 1.5335i$ (double), $z_{5,6}\approx -0.1000 \pm 0.5633,\\  z_{7,8}\approx -0.1000 \pm 0.8699$; the fifth pair is  $z_{9,10}\approx -0.1000\pm 2.6034i .$ 

\section{Conclusion}

All aforesaid allows one to appreciate how diverse and meaningful issues raise by the problem of gyroscopic stabilization of vibrational systems. Studies in this field, apparently, can give new ideas and approaches to the development of control system design in general. From the point of view of control theory, the gyroscopic stabilization problem gives an important example of a low order design, regardless of a control parameter dimension.

From the point of view of the topic itself, along with the problem of gyroscopic stabilizability, one can state the problem of an optimal gyroscopic controller design.
 
A few technical details are noteworthy. Since all solutions turn out to be vibrational (in other words, all poles form complex pairs), we may not include one of the brackets in the root polynomial — the poles included in it will necessarily fall on the same vertical; this slightly reduces the order of the computational task. In example 2 one could notice that the vibrational frequencies (or poles imaginary parts) depend weakly on decreasing matrix $D$ trace; a similar effect also appears in the five-dimensional system.

Along with possibilities of an algebraic approach (including using of critical root diagrams and root polynomials) we have also seen some impossibilities, first of all, limitations in numerical solving of polynomial equation systems. Not for all potentially possible mutual pole locations we managed to find the corresponding matrix $G$; it seems not so much the unrealizability of such locations as the lack of reliable methods for solving a polynomial equation system. Since the last task is ‘static’ (without free parameters), we hope for its successful solution as a software method. Due to this, the method described in the paper can be widely applicable to low-order controllers design for vibration systems of various dimensions and in different conditions.



The author is very much indebted to Prof. Tobias Damm (Technische Universitaet Kaiserslautern) for an introduction to this exceptionally rich topic; he is also pleased to express his gratitude to Prof. Oleg N. Kirillov (Northumbria University, Newcastle), from whose works he was able to extract the basic concepts of physical and mechanical processes in vibrational systems.


\bigskip
\begin{thebibliography}{1}




\bibitem{1} T. Damm, J. Homeyer,
{\it On indefinite damping and gyroscopic stabilization}, Preprints of the 18th IFAC World Congress, Milano, Italy, August 28--September 2, 2011, 7589--7593. 

\bibitem{2} T. Damm, J. Homeyer ,
{\it Gyroscopic Stabilization of $2^{nd}$-Order-Systems with Indefinite Damping}, Proc. Appl. Math. Mech. \textbf{11}(2011), 811--812. 


\bibitem{3}  A.S. Markus, L. Barkwell, P. Lancaster 
{\it Gyroscopically stabilized systems: A class of quadratic eigenvalue problems with real spectrum}, Canad. J. Math. \textbf{44}:1 (1992), 42--53. 


\bibitem{4} E. Brommundt,
{\it Ein Reibschwinger mit Selbsterregung ohne fallende Kennlinie}, Z. Angew. Math. Mech., \textbf{75}:11 (1995), 881--820.

\bibitem{5} K. Popp, M. Rudolph,
{\it Avoidance of Stick-Slip Motion by Vibration Control}, Proc. Appl. Math. Mech., {\bf 3} (2003), 120--121.
 
\bibitem{6} W. Kliem, C. Pommer,
{\it Indefinite damping in mechanical systems and gyroscopic stabilization}, Zeitschrift fuer Angewandte Mathematik und Physik, \textbf{60}:4 (2009), 785--795.

\bibitem{7} O.N. Kirillov, {\it PT -symmetry, indefinite damping and dissipation-induced instabilities}, Physics Letters A, {\bf 376} (2012) 1244--1249. 

\bibitem{8} O.N. Kirillov, {\it Subcritical flutter in the acoustics of friction}, Proc. Roy. Soc. Ser. A.,  \textbf{464}:2097 (2008), 2321--2339.

\bibitem{9} A.V. Chekhonadskikh, A.A. Voevoda,
{\it Algebraic design method of low order control systems},  Proceedings of the International Siberian conference on control and communications (SIBCON–2015),  Omsk, 21--23 May. Omsk: IEEE, 2015. 268--272.

\bibitem{10} A.V. Chekhonadskikh,{\it Root coordinates in the design of SISO control systems}, Optoelectronics, Instrumentation and Data Processing, \textbf{51}:5 (2015), 485--495.

\bibitem{11} A.V. Chekhonadskikh, {\it Some classical number sequences in control system design}, Siberian Electronic Mathematical Reports, {\bf 14} (2017), 620--628. Mode of access: http://semr.math.nsc.ru/v14/p620-628.pdf
\bibitem{12} A.V. Chekhonadskikh, {\it Optimal'nyi giroskopicheskii stabilizator mnogomernoi vibratsionnoi sistemy} [{\it An optimal gyroscopic stabilizer of a multidimensional vibration system}],  Sistemy analiza i obrabotki dannykh [Analysis and Data Processing Systems], \textbf{86}:2 (2022), 81--94.

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