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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ }

\title{Robust Design of Control System for Multi-input Multi-output
Plant with Interval Uncertainty}

\author{F. M. Akhmedjanov, V. G. Krymsky}

\maketitle
\auffil{The authors are with the 
Ufa State Aviation
Engineering University,
12 K.Marx Street,
Ufa, Bashkortostan,
450000 Russia,
email root@conver.bashkiria.su.}

\section{Formulation of the Problem}
Recent widespread
interest in robust design of control systems subject to parameter
perturbations caused a considerable part of the research activity orientation
toward the methods based on interval technique. These methods are
designed to analyze systems with {\it interval uncertainty}, i.e.,
systems  
for which we do not know the exact values of some parameters in the
their description, only intervals of possible values of these
parameters [1]. 
The use of these 
methods 
allows us to obtain important results regarding stability and control quality
provision for systems with interval uncertainty.

The majority of the current results correspond to the case when we
already know the characteristics of the 
closed-loop system (CLS). For example, this is the assumption behind the
famous
Kharitonov's theorems [2]. In the real world, however,
we only know the interval uncertainty in the description of the plant
itself. So, to apply methods based on characteristics of the CLS, we must 
compute the intervals characteristics of the CLS from the interval
characteristics of the plant.  

These computations can be performed directly: 
by applying naive interval computations (or a similar
technique) to the corresponding formulas. 
However, this approach has two major drawbacks:
\begin{itemize}
\item The corresponding computations often take too much time.
\item This application of naive interval computations 
often leads to an overestimation of the interval characteristics of
CLS. As a result, we may discard perfectly good control because, unlike
the actual intervals,  the
overestimated intervals may lie within the required bounds. 
\end{itemize}
In this paper, we describe better interval techniques. 

\section{Mathematical Description of the Problem} 
Let us consider a linear time-invariant closed-loop system with multi-input
multi-output (MIMO)
dynamical plant. For such a plant, CLS behavior 
can be described by the equations
$${\bf{Y}}(s)={\bf{H}}(s){\bf{R}}(s){\bf{E}}(s),\hskip
8pt{\bf{E}}(s)={\bf{G}}(s)-{\bf{Y}}(s),\eqno{(1)}$$
where:
\begin{itemize}
\item
${\bf{Y}}(s)=\Vert Y_{i}(s)\Vert_{N\times{1}},$, 
${\bf{G}}(s)=\Vert G_{i}(s)\Vert_{N\times{1}},$ and
${\bf{E}}(s)=\Vert E_{i}(s)\Vert_{N\times{1}}$ are Laplace transforms
of the actual output, of the desired output, and of 
the difference between them
(called {\it control error});
\item
${\bf H}(s)=\Vert H_{ij}(s) \Vert_{N\times{N}}$ and
\item[]${\bf R}(s)=\Vert R_{ij}(s) \Vert_{N\times{N}}$ are
$N\times N$ transfer matrices of the plant and of the controller;
\item $s$ is a Laplace transform variable.
\end{itemize}
Each of the functions $H_{ij}{(s), i,j}\in\lbrace\overline{1,N}\rbrace$,
is a ratio of two interval polynomials, i.e., polynomials whose
coefficients are known only with interval uncertainty; in
mathematical terms, the
coefficients of these polynomials 
belong to the set of all interval numbers
$$\Bbb I(\Bbb R^{+})=\lbrace{x\hskip
2pt\vert\underline{x}}{\le}{x{\le}\overline{x},\hskip 4pt0<}
\underline{x}{\leq}\overline{x}\rbrace.$$
So, for each coefficient $\alpha_l$, only its 
bounds $\underline{\alpha}_{l}, \overline{\alpha}_{l}$ of
$\alpha_l, l\in\lbrace\overline{1,k}\rbrace$ are known.

The goal of robust design is to find controller transfer matrix $\bf
R(s)$ that would 
provides the desired performance of CLS for all possible combinations
of MIMO plant parameters $\alpha_{l}{, l}\in\lbrace\overline{1,k}\rbrace$,
from the given intervals. Typical examples of performance criteria
include:
\begin{itemize}
\item system stability and
\item required level of control
quality.
\end{itemize}

\section{Two Approaches to Solving The Robust Control Problem}
There are two basic techniques for solving 
traditional control problems:
\begin{itemize}
\item {\it Algebraic-polynomial} techniques that is based on Laplace
transforms.
\item {\it Frequency domain} methods that are based on Fourier
transforms.
\end{itemize}
Similar methods can be applied to design of robust controllers. 
These methods will be described in the following two sections.

\section{Algebraic Polynomial Approach}
The corresponding method consists of the four stages:

\subsection{The first stage of the proposed method}
The goal of the beginning (structural) stage is to choose
the orders of the polynomials
that are numerators and denominators of 
$R_{ij}(s),\hskip 2pt{i,j}\in\lbrace\overline{1,N}\rbrace$.
To find these orders, we formulate inequalities that reflect 
the requirement that the controlled plant must satisfy, such as:
\begin{itemize}
\item the possibility to physically implement the controller;
\item dynamic accuracy;
\item structural stability.
\end{itemize}
The interconnections between subsystems of the 
controller subsystems 
are also taken into account on this stage. 

As an example, let us consider the conditions that describe structural
stability of CLS.
For this example, we will need the following notations:
\begin{itemize}
\item 
For a rational function $F$, by $m[F]$, we will denote the order of
its numerator.
\item 
For a rational function $F$, by $n[F]$, we will denote the order of
its numerator.
\item 
For a polynomial $P$, by $deg[P]$, we will denote its order.
\end{itemize}

Let
$${\bf W}(s)={\bf H}(s){\bf R}(s)={\bf Q}(s)/\lbrack
s^{\sigma}\cdot T(s)\cdot Z(s)\rbrack\eqno{(2)}$$
be an open-loop system transfer matrix. Here,
\begin{itemize}
\item $s^{\sigma}\cdot T(s)$
is the common denominator for
$R_{ij}(s),\hskip 5pt i,j\in\lbrace\overline{1,N}\rbrace$;
\item $Z(s)$ is the common denominator
for $H_{ij}(s),$
$i,j\in\lbrace\overline{1,N}\rbrace$;
\item ${\bf Q}(s)$ is a polynomial matrix; 
\item $\sigma$ is a positive integer 
number which is chosen to provide the
desired control accuracy.
\end{itemize}
Then characteristic polynomial $B(s)$ of the CLS can be described by
an expression
$$B(s)=\sum\limits_{l=0}^{N}P_{l}(s),\eqno{(3)}$$
where:
\begin{itemize}
\item
$$P_{0}(s)=\lbrack s^{\sigma}\cdot T(s)\cdot Z(s)\rbrack^{N},\eqno{(4)}$$
\item 
$$P_{l}(s)=D_{l}({\bf Q})\cdot \lbrack
s^{\sigma}\cdot T(s)\cdot Z(s)\rbrack^{N-l},\eqno{(5)}$$
\item $D_{l}({\bf Q})$ denotes a sum of all principle
minors of $l-$th order of the matrix ${\bf Q}(s)$. 
\end{itemize}
To obtain structural stability of CLS, it
is necessary for $B(s)$ to contain all degrees of $s$ from zero to
maximal value. 
Using (3), (4), and (5) one can form the following
inequality that describes this requirement:
$$m\lbrack R_{j^{*}i^{*}}\rbrack\ge\sigma+deg\lbrack
Z\rbrack-m\lbrack H_{i^{*}j^{*}}\rbrack-1,\eqno{(6)}$$
where
$i^{*},j^{*}$ are such that
$$m\lbrack H_{i^{*}j^{*}}\rbrack=\max\lbrace m\lbrack
H_{ij}\rbrack\rbrace.\eqno{(7)}$$
Inequality (6) thus describes the requirement of structural stability.

In [3], we describe how to find the values 
$m\lbrack R_{ij}\rbrack, n\lbrack
R_{ij}\rbrack,\hskip 5pt i,j\in\lbrace\overline{1,N}\rbrace$
that satisfy all inequalities (corresponding to all the requirements).
Namely, we form a cost function that describes how the cost of the
system depends on the choice of the degrees, and we 
look for the values that minimize this (linear)
cost function under the condition that all inequalities are satisfied.

\subsection{The second stage of the proposed method}
On the second stage, we compute the (interval) coefficients
$b_0,...,b_n$ of the
characteristic polynomial $B(s)$ of CLS. This ($n-$th order) polynomial
is defined as a numerator of the expression
$$\rm{det\lbrack{{\bf I}+{\bf H}(s){\bf R}(s)}\rbrack,}\eqno{(8)}$$
where ${\bf I}$ is the ($N\times N$) identity matrix.

To following idea enables us to 
simplify (and thus, to speed up) the computations of these
coefficients $b_i$: the vector 
$\bf{b}^{0}=\Vert{b}_{i}\Vert_{(\rm{(N+1)\times{1}}}$ satisfies the
equation 
$$F\cdot \bf{b}^{0}=\bf{B}^{0},\eqno{(9)}$$
where:
\begin{itemize}
\item $\bf{B}^{0}$ is an $(n+1) -$dimensional vector of that consists of
values $B(s_1), ..., B(s_{n+1})$ computed (using (8)) 
for positive real numbers
$s_1, ..., s_{n+1}$, and    
\item 
$F=\Vert{F}_{ij}\Vert_{(n+1)\times(n+1)}$ is a 
matrix with elements
$$F_{ij}=s_{i}^{n+1-j},\quad 
{i,j\in\lbrace\overline{1,n+1}}\rbrace.\eqno{(10)}$$
\end{itemize}
Algorithms for solving the system (9) are described in [4].

\subsection{The third stage of the proposed method}
\subsubsection{The main goal of the third stage}
The third stage of any algebraic-polynomial method is usually
connected with placing the roots of $B(s)$ in the desired domain (since
we want a stable control, this domain must lie in the left half of the
complex plane). Frequently, the desired domain has the following
trapezoid shape:
\begin{itemize}
\item its 
upper and lower bounds pass through 0, and form angles
$\pm\varphi$ with the real axis;
\item the right bound is located 
at a distance of $\eta$ from the imaginary axis.
\end{itemize}
Here, $\eta$ (called {\it stability degree})
and $\varphi$ characterize settling time and damping ratio of CLS dynamic
processes.
For such domains, checking whether all the roots in the domain, is
equivalent to checking two things:
\begin{itemize}
\item Checking whether all the roots are to the left of the line
$Re(z)=-\eta$.
\item Checking whether all the roots are in the sector formed by the
angles $\pm\varphi$.
\end{itemize}       

\subsubsection{How is $\eta-$checking implemented in the existing interval
methods}
In the existing interval control methods, to check whether the roots lie
to the left of the desired line $\eta$, the following method is
applied:
\begin{itemize} 
\item
First, we substitute $s=z-\eta$ into $B(s)$, and get a new polynomial
$B^*(z)$. We use the known interval bounds of the coefficients of $B(s)$
to compute the intervals of possible values of the coefficients of
$B^*(z)$.
\item Then, we use the existing methods of checking stability of
interval polynomials (e.g., Kharitonov's method) to check whether 
this polynomial $B^*(z)$ is stable.
\end{itemize}
If this polynomial is stable, then the $\eta-$condition is satifsied for
all the roots of the original polynomial $B(s)$.

\subsubsection{The existing $\eta-$checking method often reduces the
design accuracy}
The main problem with the above-described method is as follows:
 the computation of the interval enclosures for the coefficients of
$B^*(z)$ include several interval operations. As a result, we get
intervals that are guaranteed to contain the desired coefficients, but
these intervals are an overestimate. So, to guarantee that these
overestimated intervals are in the desired domain, we must impose much
stricter requirements on the control than is really necessary. As a
result, we reduce the accuracy of the design.

\subsubsection{$\eta-$checking: our idea}
To prevent the reduction of design accuracy, we propose a new method of
$\eta-$checking that is based on checking a finite number of
polynomials. 

\subsubsection{$\varphi-$checking}
For $\varphi-$checking, we use algorithms proposed in 
[5].
In particular, the authors of [5] has shown that if
$\varphi=(p/q)\pi$,
where $p,q$ have no common divisors, then in order to be guaranteed that
all roots of all (real-valued) 
polynomials from a given interval polynomial satisfy
are inside the $\varphi-$sector, it is sufficient to check $2q$
(real-valued) polynomials. For example, for $p/q=2/3$, it is sufficient
to check the following polynomials:
$$B_{1}^{**}(s)=\overline{b}_{0}+\overline{b}_{1}s+
\underline{b}_{2}s^{2}+\overline{b}_{3}s^{3}+...,$$
$$B_{2}^{**}(s)=\overline{b}_{0}+\underline{b}_{1}s+
\underline{b}_{2}s^{2}+\overline{b}_{3}s^{3}+...,$$

$$B_{3}^{**}(s)=\overline{b}_{0}+\underline{b}_{1}s+
\overline{b}_{2}s^{2}+\overline{b}_{3}s^{3}+...,$$
$$B_{4}^{**}(s)=\underline{b}_{0}+\underline{b}_{1}s+
\overline{b}_{2}s^{2}+\underline{b}_{3}s^{3}+...,$$
$$B_{5}^{**}(s)=\underline{b}_{0}+\overline{b}_{1}s+
\overline{b}_{2}s^{2}+\underline{b}_{3}s^{3}+...,$$
$$B_{6}^{**}(s)=\underline{b}_{0}+\overline{b}_{1}s+
\underline{b}_{2}s^{2}+\underline{b}_{3}s^{3}+...,$$

For each of these polynomials, it is not necessary to compute the roots:
checking whether all the roots of a polynomial $p(s)$ are in the
$\varphi-$sector can be reduced to checking the stability of 
auxiliary polynomials 
$$p(\exp(j\varphi)s)\cdot p(\exp(-j\varphi)s).$$

\subsection{The fourth stage of the proposed method}
In the fourth (final) stage of the proposed method, we do the following:
\begin{itemize}
\item  form the system
of inequalities that describes all requirements for the desired control
(these inequalities mainly come from applying Hurwitz criterion to the
results of the third stage), and 
\item finally, determine the parameters of the desired control ${\bf
R}(s)$. 
\end{itemize}

\subsection{How to make this method faster}
One possibility to sppeed up computation is to use simple sufficient
conditions of robust stability of CLS instead of more computationally 
complicated necessary and sufficient stability conditions.
There are also several other possibilities to reduce the computations
time.

\section{Frequency Domain Methods and Algorithms} 
The possibility to use interval methods in interval domain is 
is based on the fact that Kharitonov's fundamental theorems enable us to find
the precise values of upper and lower bounds for some 
frequency domain characteristics [6]. As a result, we can:
\begin{itemize}
\item compute 
the interval frequency domain characteristics for the plant and
for the opened-loop system (OLS), and 
\item use these characteristics to 
evaluate the performance of the CLS.
\end{itemize}

For these methods, we 
have proposed an interval version of Nyquist stability criterion. This
criterion forms the 
the foundation for the design of robust CLS controller.
In addition to this criterion, we must 
analyze the connection between the interval frequency characteristics
that correspond to different subsytems, and their responses (output
signals).

This methodology also enables us to analyze systems with interval
uncertainty by running computer simulations of appropriately chosen
particular cases. 

\section{Software Implementation}
We have implemented both methods of designing robust controllers
for MIMO dynamic plants as a special software tool 
{\it Robustness-1}. This tool has already been applied o many different
practical design problems. 

{\it Robustness-1} can also help to describe the consequences of
possible subsystem failures.
For this application to be possible, 
we had to add multi-level control algorithms [7] to 
the above-described robust design methods.

\section{The Main Application}
This software tool has been successfully applied for the design of the 
robust control systems for:
\begin{itemize}
\item aircraft engines;
\item airplanes themselves;
\item power stations.
\end{itemize}
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\bibitem{Krymsky}
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\end{thebibliography}

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