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\begin{opening}
\title{WAVELET APPROACH TO POLYNOMIAL MECHANICAL PROBLEMS}
\author{A.N. Fedorova}
\author{M.G. Zeitlin}
\institute{Institute of Problems of
Mechanical Engineering,\\
Russian Academy of Sciences,
Russia, 199178, St.~Petersburg,\\
V.O., Bolshoj pr. 61, e-mail: zeitlin@math.ipme.ru}
\end{opening}
\runningtitle{WAVELET APPROACH}
\begin{document}
\begin{abstract}
We give wavelet description for nonlinear optimal
dynamics (energy minimization in high power electromechanical system).
We consider two cases of our general construction.
In a particular case we have the solution as a series on shifted
Legendre polynomials, which is parametrized by the solution
of reduced algebraical system of equations.
In the general case we have the solution as a multiresolution
expansion from wavelet analysis. In this case the solution
is parametrized by solutions of two reduced algebraic
problems, one as in the first case and the second is some linear
problem, which is obtained from one of the next wavelet
constructions: Fast Wavelet Transform, Stationary
Subdivision Schemes, the method of Connection
Coefficients.
\end{abstract}
{\it 1.~Introduction.}
Many important physical and mechanical problems are reduced to
the solving of the systems of nonlinear differential equations
with polynomial type of nonlinearities. In this paper and related
paper in this volume we consider applications of methods of wavelet
analysis to such problems. Wavelet analysis is a relatively novel set
of mathematical methods, which
gives us possibility to work with well-localized bases in functional
spaces and with general type of operators (including pseudodifferential)
in such bases. Many examples may be found in papers [9]--[18].
Now we apply our approach to a case of constrained variational problem:
the problem of energy minimization in electromechanical systems.
We consider a synchronous electrical machine and a mill as load
(in this approach we can consider instead of a mill any
mechanical load with polynomial approximation for the mechanical
moment).
We consider the problem of ``electrical economizer''
as an optimal control problem. As result of the first stage
we give the explicit time description of the optimal dynamics
for that electromechanical system. As result of the second stage
we give the time dynamics of our system via construction based
on the set of switched type functions (Walsh functions), which
can be realized on the modern thyristor technique. In this
paper using the method of analysis of dynamical
process in the Park system [1], which we
developed in ref. [9], [10], we consider the optimal control
problem in that system. As in [9], [10], our goal
is to construct explicit time soluti\-ons, which can
be used directly in microprocessor control systems. Our
consideration is based on the Integral
Variational Method, which was developed in [21]. As
we shall see later,
we can obtain explicit time dependence for all
dynamical variables in our optimal control problem. It is
based on the fact that optimal control dynamic in our case is
given by some nonlinear system of equations which is the
extension of initial Park system. Moreover, the equations
of optimal dynamics also is the system of Riccati type (we use
the quadratic dependence of the mechanical moment).
It should be noted that this system of equations is not the
pure differential system but it is the mixed
differential-algebraic or functional system of equations [19].
In \S 2 we consider the description of our variational approach, which can be
generalized in such a way that allows us to consider it in Hamiltonian
(symplectic) approach [15].
In \S 3 we consider the form of explicit time solutions.
Our initial dynamical problem (without control)
is described by the system of 6 nonlinear differential
equations, which has the next Cauchy form (for definitions see
[9], [10])
\begin{eqnarray}
{di_k\over
dt}&=&\sum_\ell A_{\ell}i_\ell+\sum_{r,s}A_{rs}i_ri_s+
A_k(t) \nonumber
\end{eqnarray}
where $A_\ell, A_{rs}(\ell,r,s=\overline{1,6})$ are constants,
$A_k(t), (k=\overline{1,5})$ are
explicit functions of time,
$A_6(i_6,t)=a+di_6+bi^2_6$ is analytical approximation for the
mechanical moment of the mill.
At initial stage of the solution of optimal control problem
in both methods we need to select from initial set of dynamical
variables ${i_1,\dots ,i_6}$ the controlling and the controllable
variables. In our case we consider ${i_1,i_2}$ as the controlling
variables. Because we consider the energy optimization, we use the
next general form of energy functional in our electromechanical
system
$\displaystyle
Q=\int^t_{t_0}[K_1(i_1,i_2)+K_2(\dot i_1,\dot i_2)]dt,
$
where $K_1,K_2$ are quadratic forms.
Thus, our functional is the quadratic functional on the
variables $i_1,i_2$ and its derivatives.
Moreover, we may consider the optimization problem with some
constraints which are motivated by technical reasons [9], [13].
Then after standard manipulations from the theory of
optimal control, we reduce the problem of energy minimization to
some extended nonlinear system of equations.
As a result of the solution of equations of optimal dynamics we
have:
1). the explicit time dependence of the controlling
variables $u(t)=\{ i_1(t),i_2(t)\}$
which give
2). the optimum of corresponding functional of the energy and
3). explicit time dynamics of the controllable variables
$\{ i_3,i_4,i_5,i_6\} (t)$.
The obtained solutions are given in the next form:
$ \displaystyle
i_k(t)=i_k(0)+\sum^N_{i=1}\lambda_k^iX_i(t),
$
where in our first case (\S 3) we have $X_i(t)=Q_i(t)$,
where $Q_i(t)$ are shifted Legendre polynomials [21] and
$\lambda_k^i$ are the roots of reduced algebraic system
of equations.
In our second case (\S 4)--the general wavelet case $X_i(t)$ correspond
to multiresolution expansion in the basis of compactly
supported wavelets and $\lambda_k^i$ are the roots of
corresponding algebraic Riccati systems with
coefficients, which are given by Fast Wavelet Transform
(FWT) [2], Stationary Subdivision Schemes(SSS) [6],
the method of Connection Coefficients (CC) [23].
Giving \quad the controlling variables in the explicit form, we
have optimal, according to energy, dynamics in
our electromechanical systems. Evidently the technical
realization of controlling variables as
an arbitrary continuous functions of time is impossible, but we
can replace them by their re-expansions on the base of switching
type functions, which can be realized now on the modern
thyristor technique. We considered this
re-expansion in [9], [13], where we used Walsh and
Haar functions [3] as a base set of switching type
functions. This is a special case of general sequency analysis
[20]. About practical realization of expansions from \S 4
on the base of general wavelet packet basis [4] see
[18].
{\it 2.~Polynomial Dynamics.}
Our problems may be formulated as the systems of ordinary differential
equations
%\begin{eqnarray}
$
{dx_i}/{dt}=f_i(x_j,t), \ (i,j=1,...,n)
$
%\end{eqnarray}
with fixed initial conditions $x_i(0)$, where $f_i$ are not more
than polynomial functions of dynamical variables $x_j$
and have arbitrary dependence of time. Because of time dilation
we can consider only next time interval: $0\leq t\leq 1$.
Let us consider a set of
functions
%\begin{eqnarray}
$
\Phi_i(t)=x_i{dy_i}/{dt}+f_iy_i
$
%\end{eqnarray}
and a set of functionals
%\begin{eqnarray}
$
F_i(x)=\int_0^1\Phi_i (t)dt-x_iy_i\mid^1_0,
$
%\end{eqnarray}
where $y_i(t) (y_i(0)=0)$ are dual variables.
It is obvious that the initial system and the system $F_i(x)=0$
are equivalent.
In the second paper in this volume we consider the symplectization of this approach.
Now we consider formal expansions for $x_i, y_i$:
\begin{eqnarray}
x_i(t)=x_i(0)+\sum_k\lambda_i^k\varphi_k(t)\quad
y_j(t)=\sum_r \eta_j^r\varphi_r(t),
\end{eqnarray}
where because of initial conditions we need only $\varphi_k(0)=0$.
Then we have the following reduced algebraical system
of equations on the set of unknown coefficients $\lambda_i^k$ of
expansions (1):
\begin{eqnarray}
\sum_k\mu_{kr}\lambda^k_i-\gamma_i^r(\lambda_j)=0
\end{eqnarray}
Its coefficients are
%\begin{eqnarray}
$
\mu_{kr}=\int_0^1\varphi_k'(t)\varphi_r(t)dt,\quad
\gamma_i^r=\int_0^1f_i(x_j,t)\varphi_r(t)dt.
$
%\end{eqnarray}
Now, when we solve system (2) and determine
unknown coefficients from formal expansion (1) we therefore
obtain the solution of our initial problem.
It should be noted if we consider only truncated expansion (1) with N terms
then we have from (2) the system of $N\times n$ algebraical equations and
the degree of this algebraical system coincides
with degree of initial differential system.
So, we have the solution of the initial nonlinear
(polynomial) problem in the form
\begin{eqnarray}
x_i(t)=x_i(0)+\sum_{k=1}^N\lambda_i^k X_k(t),
\end{eqnarray}
where coefficients $\lambda_i^k$ are roots of the corresponding
reduced algebraical problem (2).
Consequently, we have a parametrization of solution of initial problem
by solution of reduced algebraical problem (2). But in general case,
when the problem of computations of coefficients of reduced algebraical
system (2) is not solved explicitly as in the quadratic case, which we
shall consider below, we have also parametrization of solution (1) by
solution of corresponding problems, which appear when we need to calculate
coefficients of (2).
As we shall see, these problems may be explicitly solved in wavelet approach.
{\it 3.~The solutions.}
Next we consider the construction of explicit time
solution for our problem. The obtained solutions are given
in the form (3),
where in our first case we have
$X_k(t)=Q_k(t)$, where $Q_k(t)$ are shifted Legendre
polynomials and $\lambda_k^i$ are roots of reduced
quadratic system of equations. In wavelet case $X_k(t)$
correspond to multiresolution expansions in the base of
compactly supported wavelets and $\lambda_k^i$ are the roots of
corresponding general polynomial system (2) with coefficients, which
are given by FWT, SSS or CC constructions. According to the
variational method to give the reduction from
differential to algebraical system of equations we need compute
the objects $\gamma ^j_a$ and $\mu_{ji}$,
which are constructed from objects:
\begin{eqnarray}
\sigma_i&\equiv&\int^1_0X_i(\tau)d\tau=(-1)^{i+1},\\
\nu_{ij}&\equiv&\int^1_0X_i(\tau)X_j(\tau)d\tau=
\sigma_i\sigma_j+\frac{\delta_{ij}}{ (2j+1)},\nonumber\\
\mu_{ji}&\equiv&\int
X'_i(\tau)X_j(\tau)d\tau=\sigma_jF_1(i,0)+F_1(i,j),\nonumber\\
& &F_1(r,s)=[1-(-1)^{r+s}]\hat{s}(r-s-1),\quad
\hat{s}(p)=\left\{ \begin{array}{ll}
1, & \quad p\geq 0\\
0, & \quad p < 0
\end{array}
\right. \nonumber \\
\beta_{klj}&\equiv&\int^1_0X_k(\tau)X_l(\tau)X_j(\tau)
d\tau=\sigma_k\sigma_l\sigma_j+ \nonumber\\
& & \alpha_{klj}+
\frac{\sigma_k\delta_{jl}}{ 2j+1}+
\frac{\sigma_l\delta_{kj}}{ 2k+1}+
\frac {\sigma_j\delta_{kl}}{ 2l+1}, \nonumber\\
\alpha_{klj}&\equiv&\int^1_0X^\ast_kX^\ast_lX^\ast_jd\tau=
\frac{1}{ (j+k+l+1)R\bigl(1/2(i+j+k)\bigr)}\times\nonumber\\
& & R\bigl(1/2(j+k-l)\bigr)
R\bigl(1/2(j-k+l)\bigr)
R\bigl(1/2(-j+k+l)\bigr),\nonumber
\end{eqnarray}
if $j+k+l=2m , m\in{\it Z}$, and
$\alpha_{klj}=0$ if $ j+k+l=2m+1 $;
$ R(i)=(2i)!/(2^ii!)^2$, $Q_i=\sigma_i+P_i^\ast$, where the
second equality in the formulae for $\sigma, \nu, \mu, \beta,\alpha$
hold for the first case.
{\it 4.~ Wavelet computations.}
Now we give construction for
computations of objects (4) in the wavelet case.
We use some constructions from multiresolution analysis [8]: a sequence of
successive approximation closed subspaces $V_j$:
$
...V_2\subset V_1\subset V_0\subset V_{-1}\subset V_{-2}\subset ...
$
satisfying the following properties:
%\begin{eqnarray*}
$
\displaystyle\bigcap_{j\in{\bf Z}}V_j=0$,
$\overline{\displaystyle\bigcup_{j\in{\bf Z}}}V_j=L^2({\bf R})$,
$ f(x)\in V_j <=> f(2x)\in V_{j+1}
$
%\end{eqnarray*}
There is a function $\varphi\in V_0$ such that \{${\varphi_{0,k}(x)=
\varphi(x-k)}_{k\in{\bf Z}}$\} forms a Riesz basis for $V_0$.
We use compactly supported wavelet basis: orthonormal basis
for functions in $L^2({\bf R})$. As usually $\varphi(x)$ is
a scaling function, $\psi(x)$ is a wavelet function, where
$\varphi_i(x)=\varphi(x-i)$. Scaling relation that defines
$\varphi,\psi$ are
\begin{eqnarray*}
\varphi(x)&=&\sum\limits^{N-1}_{k=0}a_k\varphi(2x-k)=
\sum\limits^{N-1}_{k=0}a_k\varphi_k(2x),\\
\psi(x)&=&\sum\limits^{N-2}_{k=-1}(-1)^k a_{k+1}\varphi(2x+k)
\end{eqnarray*}
Let be $ f : {\bf R}\longrightarrow {\bf C}$ and the wavelet
expansion is
\begin{eqnarray}
f(x)=\sum\limits_{\ell\in{\bf Z}}c_\ell\varphi_\ell(x)+
\sum\limits_{j=0}^\infty\sum\limits_{k\in{\bf
Z}}c_{jk}\psi_{jk}(x)
\end{eqnarray}
The indices $k, \ell$ and $j$ represent translation and scaling, respectively:
$
\varphi_{jl}(x)$ $=2^{j/2}\varphi(2^j x-\ell),
\psi_{jk}(x)=2^{j/2}\psi(2^j x-k)
$.
The set $\{\varphi_{j,k}\}_{k\in {\bf Z}}$ forms a Riesz basis for $V_j$.
Let $W_j$ be the orthonormal complement of $V_j$ with respect to $V_{j+1}$.
Just as $V_j$ is spanned by dilation and translations of the scaling function,
so are $W_j$ spanned by translations and dilation of the mother wavelet
$\psi_{jk}(x)$.
If in formulae (5) $c_{jk}=0$ for $j\geq J$, then $f(x)$ has an alternative
expansion in terms of dilated scaling functions only
$
f(x)=\sum\limits_{\ell\in {\bf Z}}c_{J\ell}\varphi_{J\ell}(x)
$.
This is a finite wavelet expansion, it can be written solely in
terms of translated scaling functions. We use wavelet $\psi(x)
$, which has $k$ vanishing moments
$
\int x^k \psi(x)dx=0$, or equivalently
$x^k=\sum c_\ell\varphi_\ell(x)$ for each $k$,
$0\leq k\leq K$.
Also we have the shortest possible support: scaling function
$DN$ (where $N$ is even integer) will have support $[0,N-1]$ and
$N/2$ vanishing moments.
There exists $\lambda>0$ such that $DN$ has $\lambda N$
continuous derivatives; for small $N,\lambda\geq 0.55$.
To solve our second associated linear problem we need to
evaluate derivatives of $f(x)$ in terms of $\varphi(x)$.
Let be $
\varphi^n_\ell=d^n\varphi_\ell(x)/dx^n
$.
We derive the wavelet - Galerkin approximation of a
differentiated $f(x)$ as
$
f^d(x)=\sum_\ell c_l\varphi_\ell^d(x)
$
and values $\varphi_\ell^d(x)$ can be expanded in terms of
$\varphi(x)$:\\
$
\phi_\ell^d(x)=\sum\limits_m\lambda_m\varphi_m(x),\quad
\lambda_m=\int\limits_{-\infty}^{\infty}\varphi_\ell^d(x)\varphi_m(x)dx
$
The coefficients $\lambda_m$ are 2-term connection
coefficients. In general we need to find $(d_i\geq 0)$
\begin{eqnarray}
\Lambda^{d_1 d_2 ...d_n}_{\ell_1 \ell_2 ...\ell_n}=
\int\limits_{-\infty}^{\infty}\prod\varphi^{d_i}_{\ell_i}(x)dx
\end{eqnarray}
For Riccati case we need to evaluate two and three
connection coefficients
$
\Lambda_\ell^{d_1
d_2}=\int^\infty_{-\infty}\varphi^{d_1}(x)\varphi_\ell^{d_2}(x)dx,
\quad
\Lambda^{d_1 d_2
d_3}=\int\limits_{-\infty}^\infty\varphi^{d_1}(x)\varphi_
\ell^{d_2}(x)\varphi^{d_3}_m(x)dx
$.\\
According to CC method [23] we use the next construction. When $N$ in
scaling equation is a finite even positive integer the function
$\varphi(x)$ has compact support contained in $[0,N-1]$.
For a fixed triple $(d_1,d_2,d_3)$ only some $\Lambda_{\ell
m}^{d_1 d_2 d_3}$ are nonzero : $2-N\leq \ell\leq N-2,\quad
2-N\leq m\leq N-2,\quad |\ell-m|\leq N-2$. There are
$M=3N^2-9N+7$ such pairs $(\ell,m)$. Let $\Lambda^{d_1 d_2 d_3}$
be an M-vector, whose components are numbers $\Lambda^{d_1 d_2
d_3}_{\ell m}$. Then we have the first key result: $\Lambda$
satisfy the system of equations $(d=d_1+d_2+d_3)$: \\
$
A\Lambda^{d_1 d_2 d_3}=2^{1-d}\Lambda^{d_1 d_2 d_3},
\quad
A_{\ell,m;q,r}=\sum\limits_p a_p a_{q-2\ell+p}a_{r-2m+p}
$.
By moment equations we have created a system of $M+d+1$
equations in $M$ unknowns. It has rank $M$ and we can obtain
unique solution by combination of LU decomposition and QR
algorithm.
The second key result gives us the 2-term connection
coefficients:
$
A\Lambda^{d_1 d_2}=2^{1-d}\Lambda^{d_1 d_2},\quad d=d_1+d_2,\quad
A_{\ell,q}=\sum\limits_p a_p a_{q-2\ell+p}
$.
For nonquadratic case we have analogously additional linear problems for
objects (6).
Also, we use FWT [2] and SSS [6] for computing coefficients of reduced
algebraic systems.
We use for modelling D6, D8, D10 functions and programs RADAU and
DOPRI for testing [19].
As a result we obtained the explicit time solution (3) of our
problem. In comparison with
wavelet expansion
on the real line which we use now and in
calculation of Galerkin approximation, Melnikov function approach, etc also
we need to use periodized wavelet expansion, i.e. wavelet expansion
on finite interval. Also in the solution of perturbed system we have
some problem with variable coefficients.
For solving last problem we need to consider one more refinement equation
for scaling function $\phi_2(x)$:
$
\phi_2 (x)=\sum\limits^{N-1}_{k=0} a^2_k \phi_2 (2x-k)
$
and corresponding wavelet expansion for variable coefficients $b(t)$:
$
\sum\limits_k B_k^j (b)\phi_2(2^jx-k),
$
where $B_k^j (b)$ are functionals supported in a small neighborhood of
$2^{-j}k$ [7].
The solution of the first problem consists in periodizing. In this case
we use expansion into periodized wavelets
defined by
$
\phi^{per}_{-j,k}(x)=2^{j/2}\sum\limits_{Z} \phi(2^jx+2^j\ell-k)
$ [5].
All these modifications lead only to transformations of coefficients of
reduced algebraic system, but general scheme remains the same.
Extended version and related results may be found in [9]--[18].
We would like to thank Zohreh Parsa (BNL) for many discussions and
continued encouragement during and after workshop "New Ideas for
Particle Accelerators" and Institute for Theoretical Physics,
University of California, Santa Barbara for hospitality.
This research was supported in part under "New Ideas for Particle
Accelerators Program" NSF-
Grant No.~PHY94-07194.
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\end{document}