ICCC01LD.tex
% Contribution for ICCC2001 in Poland.
% Autor: Lubomir Dorcak, e-mail: dorcak@tuke.sk
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\begin{center}
\vspace*{30mm}

\normalsize {\textbf{STATE-SPACE CONTROLLER DESIGN FOR THE
                     FRACTIONAL-ORDER REGULATED SYSTEM}}   \\

%\vspace{9mm}
\vspace{5mm}
\normalsize
\textbf{\v{L}ubom\'{\i}r DOR\v{C}\'AK$^{\mbox{1}}$},
\textbf{Ivo PETR\'A\v{S}$^{\mbox{1}}$}  \\
\textbf{Imrich KO\v{S}TIAL$^{\mbox{1}}$},
\textbf{J\'an TERP\'AK$^{\mbox{1}}$}  \\
$^{\mbox{1}}$
Department of Informatics and Process Control \\
BERG Faculty, Technical University of Ko\v{s}ice \\
B. N\v{e}mcovej 3, 042 00 Ko\v{s}ice, Slovak Republic \\
e-mail:
\textit{\{lubomir.dorcak,ivo.petras,imrich.kostial,jan.terpak\}@tuke.sk} \\
%$^{\mbox{2}}$
%If any ... \\
\end{center}
%\vspace{9mm}
\vspace{3mm} % !!!!!!!!!!!!

\hspace{-6.5mm}
\textbf{Abstract:}
 In this paper we will present a mathematical description and analysis
of a fractional-order regulated system in the state space and the
state-space controller design based on placing the closed-loop poles on
the complex plane. Presented are the results of simulations
%of this system. The possibility of investigating the stability of such
%systems is also considered.
and stability investigation
of this system.
\vspace{3.0mm}

\hspace{-6.5mm}
\textbf{Key words:} fractional-order regulated system, fractional calculus,
                    model, state space
%\vspace{7.5mm}
\vspace{-2.0mm}


\section*{\normalsize{1. \hspace{1.2mm}Introduction}}
% Development
%\hspace{7.0mm}
\hspace{5.0mm}
Research of the \, fractional-order \, derivatives continued in the last
decades not only as a \, mathematical branch \, \cite{Oldham, Samko,
Podlubny6, PodlubnyMat}, \, etc., \, but in many applied domains, \, e.g.
\cite{Westerlund, Podlubny5} and in control theory too
\cite{Outstaloup} -
%, Axtell,
%Kalojanov, Podlubny4, Dorcak2, Dorcak3, Dorcak4, Petras1,
%Dorcak5, PetrasDorcak1, Matignon, Vinagre1,
\cite{Dorcak}, etc.
In works \cite{Outstaloup, Axtell, Kalojanov} the first generalizations
of analysis methods for fractional-order control systems were made
($s$-plane, frequency response, etc.). The following of the
above-mentioned works were oriented to the methods of fractional-order
system parameters identification, methods of fractional-order
controllers synthesis, methods of stability analysis, methods of control
of chaotic fractional-order systems, and so on.

%\hspace{1.5mm}
 In works \cite{Matignon, Vinagre1} was presented a state space
model described in vector and matrix relations expressing the
fractional-order derivatives
\begin{eqnarray} \label{r1}
        {\bf x}^{(\alpha)}(t) =
           {\bf A}~{\bf x}(t) + {\bf B}~u(t), \nonumber \\
         y(t) = {\bf C}~{\bf x}(t),
   \hspace{1.5em} t \geq 0. \hspace{0.2em}
\end{eqnarray}
This description is convenient only for simple models of systems
with only one fractional-order derivation, or for spatial type of
systems.
In work \cite{Dorcak} we proposed a state space model of the linear
time-invariant one dimensional system which expresses the first
derivatives in the state space equations and has the \,
{\it {classical \, state \, space interpretation}}
for the fractional-order system too. On the
right side of these equations we can then transfer more than one
fractional-order derivatives of the state space variables. A disadvantage
of this expression is that in time domain we cannot express the state
space equations in vector and matrix relations as in
%the
 previous
description. But we can do this in $s$-plane.

This contribution deals with a mathematical description and analysis
of a fractional-order regulated system in the state space and the
state-space controller design based on placing the closed-loop poles
on the complex plane.


\vspace{-1.0mm}

\section*{\normalsize{2. \hspace{1.2mm}Definition of the system}}
\vspace{-2.0mm}

%\hspace{7.0mm}
\hspace{5.0mm}
        For the definition of the control system we consider
a simple unity feedback control system illustrated in Fig.1,
where $G_{s}(s)$ denotes the transfer function of the controlled
system and $G_{r}(s)$ is the controller transfer function, both
integer- or fractional-order.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.68]{obr1.eps}
\vspace{-2.0mm}
\caption{Unity-feedback control system}
         \label{fig:FBCS}
\end{figure}
        The differential equation of the above closed regulation
system for the transfer function of the controlled system
$G_{s}(s)=1/(a_2s^{\alpha}+a_1s^{\beta}+a_0)$
and the fractional-order $PD^{\delta}$ controller
$G_{r}(s)=K+T_ds^{\delta}$
has the form
\vspace{-0.8mm}
\begin{equation} \label{r2}
     a_{2}\, y^{(\alpha)}(t) +
     a_{1}\, y^{(\beta)}(t) +
     T_{d}\, y^{(\delta)}(t) +
     (a_{0}+K)\, y(t) =
     K\, w(t) +
     T_{d}\, w^{(\delta)}(t)
\end{equation}
\vspace{-2.2mm}
where  $\alpha, \beta, \delta$ are generally  real
numbers and $a_0, a_1, a_2, K, T_d$ are  arbitrary
constants.

\vspace{1.0mm}

\section*{\normalsize{3. \hspace{1.2mm}Fractional-order~control~system
                          with $PD^{\delta}$~controller~in~the~state~space}}
\vspace{-1.0mm}

%\hspace{7.0mm}
\hspace{5.0mm}
 Consider the system described by differential equation (\ref{r2}).
After its modification and substitution of state space variables
$x(t) = x_1(t), \dot{x}(t) = x_2(t), \ddot{x}(t) = \dot{x_2}(t)$
we can derive the following state space model equivalent to model
(\ref{r2})
\vspace{-2.0mm}
\begin{eqnarray} \label{r4}
    \dot{x_1}(t) = x_2(t)~,
     \hspace{10.1cm}
     \nonumber \\
    \dot{x_2}(t) = - \frac{a_0+K}{a_2}~x_1^{(2-\alpha)}(t)
                   - \frac{T_d}  {a_2}~x_2^{(1+\delta-\alpha)}(t)
                   - \frac{a_1}  {a_2}~x_2^{(1+\beta-\alpha)}(t)
                   + \frac{1}    {a_2}~w^{(2-\alpha)}(t)~,
     \hspace{0em} \\
        y  (t) =   K~x_1(t)
                 + T_d~x_2^{(\delta-1)}(t)~,
     \hspace{1.5em} t \geq 0. \hspace{17.1em}
     \nonumber
\vspace{-1.0mm}
\end{eqnarray}
Of course, we can make other alternative state space models for
the same system.
%
For the approximation of the fractional-order derivatives on the
right-hand side of equations (\ref{r4}) we can take the relation
(\ref{rr4}) from e.g. \cite{Samko, Podlubny6, Dorcak2}
\vspace{-2mm}
\begin{equation} \label{rr4}
   y^{(\alpha)}(t) \approx
%   \; _{(t-L)}D^{\alpha}_{t}y(t)\ = \;
   h^{-\alpha} \sum_{j=0}^{N(t)}b_{j}y(t-j h ) \; ,
\end{equation}
where $L$ is "memory length", $h$ is
%the
 time step of calculation,
$N(t) = \min \left\{
           \left[
              \frac{\mbox{$t$}}{\mbox{$h$}}
           \right] , \;
           \left[
              \frac{\mbox{$L$}}{\mbox{$h$}}
           \right]
         \right\} , $
$[z]$ is the integer part of $z$,
$b_j = (-1)^j {{\alpha} \choose j}$,
and $ {{\alpha} \choose j} $ is binomial coefficient. To calculate
$b_j$ it is convenient to use the following recurrent relation
\begin{equation} \label{rr6}
       b_0 = 1
         \ , \ \
       b_j = (1- \frac {1+\alpha} {j}) \ b_{j-1}
\end{equation}
  After the above-mentioned fractional-order derivatives discretisation
(\ref{rr4}) and discretisation of the first derivatives on the left-hand
side of equations (\ref{r4}) we obtained the simple Euler methods for
solving the state space model
\vspace{-1mm}
\begin{eqnarray} \label{rrr4}
    x_{1,k+1} = x_{1,k} + h \, x_{2,k}~,
     \hspace{8.0cm} \nonumber \\
    x_{2,k+1} = x_{2,k} + h \, \Bigl(
                -\frac{a_0+K}{a_2}~h^{\alpha-2}
                                      \sum_{j=0}^{k}b_j x_{1,k-j}
                -\frac{T_d}  {a_2}~h^{\alpha-\delta-1}
                                      \sum_{j=0}^{k}c_j x_{2,k-j}
     \hspace{0.5cm} \nonumber \\
                -\frac{a_1}  {a_2}~h^{\alpha-\beta-1} \hspace{0em}
                                      \sum_{j=0}^{k}d_j x_{2,k-j}
                 \hspace{0.2em} + \hspace{0.3em}
                 \frac{1}    {a_2}~h^{\alpha-2} \hspace{0.5em}
                                      \sum_{j=0}^{k}b_j w_{k-j}
                            \, \Bigr)
     \hspace{1.2em} \\
        y_k  =   K~x_{1,k} +
                 T_d~h^{(1-\delta)}
                                      \sum_{j=0}^{k}e_j x_{2,k-j}
     \hspace{3.5em} k \geq 0. \hspace{9.0em}
     \nonumber
\end{eqnarray}
%\vspace{-1mm}
From these equations we can compute state trajectories of the
fractional-order control system described above.



\section*{\normalsize{4. \hspace{1.2mm}Design of the $PD^{\delta}$
                                       controller~in~the~state~space}}

%\hspace{7.0mm}
\hspace{5.0mm}
After Laplace transformation we can write state space equations (\ref{r4})
in vector and matrix relations in $s$-plane
\vspace{-1mm}
\begin{eqnarray} \label{r5}
         p{\bf X}(s) =
           {\bf A}(s)~{\bf X}(s) + {\bf B}(s)~W(s), \nonumber \\
         Y(s) = {\bf C}(s)~{\bf X}(s),
   \hspace{3.3em} t \geq 0. \hspace{0.2em}
\end{eqnarray}
where matrix ${\bf A}(s)$ and vectors ${\bf B}(s)$, ${\bf C}(s)$ are
\begin{displaymath} \label{r6}
{\bf A}(s)=               \hspace{-0.2em}
\left[                    \hspace{-0.4em}
 \begin{array}{cc}
    0    &     1   \\
    \,    &    \,  \\
   -\frac{a_0+K}{a_2}s^{2-{\alpha}}&
   -\frac{a_1s^{1+{\beta}-{\alpha}}+T_ds^{1+{\delta}-{\alpha}}}{a_2}
 \end{array}              \hspace{-0.4em}
\right]\hspace{-0.2em},
{\bf B}(s)=               \hspace{-0.2em}
\left[                    \hspace{-0.4em}
 \begin{array}{c}
    0          \\
    \,        \\
    \frac{1}{a_2}s^{2-{\alpha}}
 \end{array}              \hspace{-0.4em}
\right]\hspace{-0.2em},
{\bf C}(s)=               \hspace{-0.2em}
\left[                    \hspace{-0.4em}
 \begin{array}{c}
    K          \\
    \,        \\
    T_d s^{{\delta}-1}
 \end{array}              \hspace{-0.4em}
\right]^T
\end{displaymath}
From equations (\ref{r5}) we can derive the
overall transfer function
\begin{equation} \label{r7}
       \frac{Y(s)}{W(s)} =
       \frac{{\bf C} \, adj(s{\bf I}-{\bf A}) \, {\bf B}}
            {det(s{\bf I} - {\bf A})}
\end{equation}
The characteristic equation of the closed-loop system is determined by
solving the determinant in the denominator of the transfer function,
which is a fractional-order polynomial in $s$
\begin{equation} \label{r8}
       a_2 s^{\alpha} + a_1 s^{\beta} + T_d s^{\delta} + (a_0 + K) = 0
\end{equation}
Solving this
%characteristic
equation we can find the
%system
poles of the closed-loop system with known parameters, or design the
$PD^\delta$ controller parameters for desired control system poles.

Consider the system described by differential equation (\ref{r2})
with system coefficients
$a_2 = 0.8, a_1 = 0.5, a_0 = 1, \alpha = 2.2, \beta = 0.9$.
The task is to determine the controller parameters
$K, T_d$ and $\delta$ for desired control system poles
$s_{1,2}=-1{\pm}6i$ and the steady state error less than 4\%.
%
Parameter $K=24$ can be computed from the equation
based on the steady state error $e_{ss}(\infty)$ of the closed-loop
control system
$K = ( 100 / e_{ss} - 1 ) a_0$.
%
If we introduce the desired system poles $s_{1,2}$ to the characteristic
equation (\ref{r8}) we obtain system of two nonlinear equations from
which we can easy derive two equations
$\delta = \arctan(2,9839) \frac{1,8098}{\pi}$  and
${T_d}^2 37^\delta = 645,2174$ \, for the calculation of the
controller parameters $T_d = 6,9407$ and  $\delta = 0,71859$.
In Fig.2 and Fig.3 are depicted the unit - step responses
of the classical numerical solution
\cite{Dorcak2}
and the numerical solution of the state space model (\ref{rrr4}).
The obtained state trajectory represents stable focal point
for the above-mentioned coefficients of the system.

\vspace *{3mm}
\begin{tabular}{cc}
%\centering
\includegraphics[scale=0.3]{obr3.eps} &
%\centering
\hspace {2mm}
\includegraphics[scale=0.3]{obr4.eps} \\ %[-3ex]
%Figure 2: \textit{Unit step responses} &
%Figure 3: \textit{State trajectories}   \\[3ex]
\vspace *{2mm}
Figure 2: Unit step response &
Figure 3: State trajectory   \\ %[3ex]
\end{tabular}

\setcounter{figure}{3}


Assuming the integer-order $PD$ controller we obtain a system of two
linear equations for direct calculation of the controller parameters
$K = 36,0854$ and  $T_d = 4,0141$.

If we increased in case of fractional-order $PD^\delta$ controller
the requirement on steady state error less than 2\%
we would obtain the following controller parameters
$K = 49$, $T_d = -79,74427$ and  $\delta = -0,55194$. That means, we
obtained the controller with weak integrator. But the characteristic
equation of such a system has one additional pole
%$s_3 = 1,9889$
$s_3 = 1,98$
and it follows from the stability analysis that such closed-loop
system is unstable. We can verify this fact also with frequency methods
for stability investi-gation of the fractional-order system
\cite{Dorcak5, PetrasDorcak1}. In this case we can change the
desired control system poles $s_{1,2}$, or we have to proceed
in a different way, e.g. as follows.



\section*{\normalsize{5. \hspace{1.2mm}Fractional-order~control~system~with
                         $PI^{\lambda}$~controller~in~the~state~space}}

%\hspace{7.0mm}
\hspace{5.0mm}
        The differential equation of the closed regulation
system for the transfer function of the controlled system
$G_{s}(s)=1/(a_2s^{\alpha}+a_1s^{\beta}+a_0)$
and the fractional-order $PI^{\lambda}$ controller
$G_{r}(s)=K+T_is^{-\lambda}$
has the form
\vspace{-0mm}
\begin{equation} \label{r10}
     a_{2}\, y^{(\alpha+\lambda)}(t) +
     a_{1}\, y^{(\beta+\lambda)}(t) +
     (a_{0}+K)\, y^{(\lambda)}(t) +
     T_i\, y(t) =
     K\, w^{(\lambda)}(t) +
     T_i\, w(t)
\end{equation}
where  $\alpha, \beta, \lambda$ are generally real
numbers and $a_0, a_1, a_2, K, T_i$ are arbitrary
constants.

\hspace{-7.0mm}
The state space model equivalent to model (\ref{r10}) has
the folloving form
\vspace{-2mm}
\begin{eqnarray} \label{r11}
    \dot{x_1}(t) = - x_2(t) + w ,
     \hspace{9.2cm}
     \nonumber \\
    \dot{x_2}(t) = + x_3(t)~,
     \hspace{9.8cm}
     \nonumber \\
    \dot{x_3}(t) = + \frac{T_i}  {a_2}~x_1^{(3-\alpha-\lambda)}(t)
                   - \frac{a_0+K}{a_2}~x_2^{(2-\alpha)}(t)
                   - \frac{a_1}  {a_2}~x_1^{(1+\beta-\alpha)}(t)
                   + \frac{K}    {a_2}~w_1^{(2-\alpha)}(t)~,
     \hspace{0em} \\
        y  (t) = x_2(t)~,
     \hspace{1.5em} t \geq 0. \hspace{24.8em}
     \nonumber
\vspace{-2mm}
\end{eqnarray}
where $\dot{x_1}(t) = \dot{z}(t)$ is the
actuating error signal and
$\dot{x_2}(t) = \dot{v_1}(t), \dot{x_3}(t) = \dot{v_2}(t)$ are
the state variables of the controlled system (Fig.4).
%
\begin{figure}[ht]
\centering
%\includegraphics[scale=0.285]{obr2.eps}
\includegraphics[scale=0.5]{obr2.eps}
\caption{Block diagram representing the state and output equations}
         \label{fig:PI}
\end{figure}
%
As above, we can derive the following characteristic equation
of the closed-loop system
\begin{equation} \label{r12}
       a_2 s^{\alpha+\lambda} + a_1 s^{\beta+\lambda} +
       (a_0 + K) s^{\lambda} + T_i = 0
\end{equation}
%
Assume the same controlled system as in the previous section.
%
%If we introduce the desired system poles $s_{1,2}=-1{\pm}6i$,
%$s_3=-10$ to the characteristic equation (\ref{r12}) we obtain
%a system of three nonlinear equations from which we can
%numericaly calculate the controller parameters
%$K = 6,9407$, $T_i = 6,9407$ and  $\lambda = 0,7186$.
%
If we introduce the desired system poles $s_1$, $s_2$, $s_3$
to the characteristic equation (\ref{r12}) we obtain
a system of three nonlinear equations from which we can
numericaly calculate the controller parameters
$K$, $T_i$ and  $\lambda$.


\vspace{-2.5mm}
\section*{\normalsize{6. \hspace{1.2mm}Conclusion}}
\vspace{-1.5mm}
%\hspace{7.0mm}
\hspace{5.0mm}
         We have presented a mathematical description of a
fractional-order control system in the state space and the
state-space controller design based on placing the closed-loop
poles on the complex plane.
In the design of a fractional-order $PD^\delta$ regulator it is
necessary to pay attention to the question of stability of the
control system. In the case of negative values of $\delta$ the
order of the system increases. In addition to the required poles,
the system can develop
%additional
new
poles that can render the system unstable.


\vspace{-2.5mm}
\section*{\normalsize{Acknowledgements}}
\vspace{-1.5mm}
%\hspace{7.0mm}
\hspace{5.0mm}
This work was partially supported by grant VEGA 1/7098/20 from the
Slovak Grant Agency for Science.


\vspace{-14mm}
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\vspace{-2mm}
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\end{thebibliography}

\end{document}



math0204189.extracted_bib
\begin{thebibliography}{99}
\vspace{-6mm}
\bibitem{Oldham}
        K.B.Oldham, J.Spanier:
        \emph{The Fractional Calculus}.
        Acad. Press, New York, 1974.
\vspace{-6mm}
\bibitem{Samko}
        S. G. Samko, A. A. Kilbas, O. I. Marichev:
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