## 4 Description of Adaptive Controller  
The pole placement strategy is applicable to plant models of the form of (8). We make the following assumptions:  
* The orders \(nA,nB\), the delay d and the coefficients of \(A(z^{-1})\) and \(B(z^{-1})\) are known,
* \(A(z^{-1})\) and \(B(z^{-1})\) do not have any common factors.  
A general form of a two-degree of freedom digital controller is given by:  
\[G_{R}(z^{-1})=\frac{R(z^{-1})}{S(z^{-1})}, \tag{14}\]  
where \(R(z^{-1})\), \(S(z^{-1})\) correspond to polynomials:  
\[R(z^{-1})=r_{0}+r_{1}z^{-1}+\cdots+r_{nR}z^{-nR} \tag{15}\]  
\[S(z^{-1})=1+s_{1}z^{-1}+\cdots+s_{nS}z^{-nS}. \tag{16}\]  
Disturbances transfer function of closed system is presented below:  
\[\frac{y(t)}{v(t)}=\frac{S(z^{-1})}{A(z^{-1})S(z^{-1})+z^{-d}B(z^{-1})R(z^{-1})}. \tag{17}\]  
The characteristic equation of the expression (17) determines poles location of the closed-loop controller, which are responsible for dynamics of the control system. It is important to designing the parameters of characteristic equation \(D(z^{-1})\) ensure stable operation of the system by placing the poles inside the unit circle. The controller parameters can be calculated by solving Diophantine equations shown below.  
\[D(z^{-1})=A(z^{-1})S(z^{-1})+z^{-d}B(z^{-1})R(z^{-1}). \tag{18}\]  
In order to effectively solve (18) one uses the fact that the polynomial equation takes the matrix form  
\[\mathbf{M}*\mathbf{x}=\mathbf{d}, \tag{19}\]  
where \(\mathbf{M}\) is Sylvester matrix (controllability matrix), \(\mathbf{x}\) is a vector of controller parameters  
\[\mathbf{x}^{T}=[1,\,s_{1},\,\,\ldots,\,s_{nS},\,r_{0},\,\,\ldots,\,r_{nR}] \tag{20}\]  
and \(\mathbf{d}\) is a vector of designed characteristic equation parameters.  
\[\mathbf{d}^{T}=[1,\,d_{1},\,\,\ldots,\,d_{nD},\,0,\,\,\ldots,\,0]. \tag{21}\]\(M\) is a square matrix \(r\times r\) with \(r=nA+nB+d\) given by [24]  
\[\overbrace{\left[\begin{array}{ccccccccc}&\overset{nB+d}{0}&&\\ \overset{nA}{1}&\overset{1}{0}&\overset{\ddots}{\cdots}&0&\\ \overset{nB}{1}&\overset{\ddots}{\ddots}&\vdots&\overset{b}{1}{0}&\overset{ \prime}{0}&\\ \overset{nB}{2}&\overset{\ddots}{\ddots}&\overset{\ddots}{0}&\overset{b}{2}{0}& \overset{\ddots}{\ddots}&\ddots\\ &&\overset{\ddots}{1}&&&\overset{\ddots}{\ddots}&0\\ \vdots&&&a_{1}&\vdots&&&b^{\prime}_{1}\\ \overset{nA}{nA}&&&a_{2}&\overset{b}{b}{}^{\prime}_{nB}&&&b^{\prime}_{2}\\ 0&\ddots&&&0&\ddots\\ \vdots&\ddots&\ddots&\vdots&\vdots&\ddots&\ddots&\vdots\\ 0&\cdots&0&a_{nA}&0&\cdots&0&b^{\prime}_{nB}\\ \end{array}\right]}nA+nB+d, \tag{22}\]  
where \(b^{\prime}_{i}=0\) for \(i=0,\ 1,\ \ldots,\ d;b^{\prime}_{i}=b^{\prime}_{i-d}\) for \(i\geq d+1\).  
The vector \(x\) containing the coefficients of polynomials \(R(z^{-1})\) and \(S(z^{-1})\) is obtained by inverting the matrix \(M\), which is nonsingular if \(A(z^{-1})\) and \(B(z^{-1})\) are relatively prime [24].  
\[x=M^{-1}d. \tag{23}\]  
If the aim is to eliminate vibrations, the polynomial \(D(z^{-1})\) should be a simple as possible. This condition is achieved when \(D(z^{-1})=1\). As a solution to the system of linear equations (19) obtained coefficients of the digital controller described by (15) and (16), which allow us to determine the control signal:  
\[\begin{split} u(t)=& r_{0}y(t)+r_{1}y(t-1)+\cdots+r_{nR }y(t-nR)+\\ &-s_{1}u(t-1)-\cdots-s_{nS}u(t-nS).\end{split} \tag{24}\]