Based on the key idea of higher-order singular value decomposition[1] (HOSVD) in tensor algebra, Baranyi and Yam proposed the concept of HOSVD-based canonical form of TP functions and quasi-LPV system models.[2] [3] Szeidl et al.[4] proved that the TP model transformation[5] [6] is capable of numerically reconstructing this canonical form.
Related definitions (on TP functions, finite element TP functions, and TP models) can be found here. Details on the control theoretical background (i.e., the TP type polytopic Linear Parameter-Varying state-space model) can be found here.
A free MATLAB implementation of the TP model transformation can be downloaded at https://web.archive.org/web/20120229061018/http://tptool.sztaki.hu/ or at MATLAB Central http://www.mathworks.com/matlabcentral/fileexchange/25514-tp-tool.
Assume a given finite element TP function:
N | |
f(x)=l{S}\boxtimes | |
n=1 |
wn(xn),
where
x\in\Omega\subsetRN
wn(xn)
n=1,\ldots,N
l{S}
N | |
l{S}=l{A}\boxtimes | |
n=1 |
Un.
Then,
N | |
f(x)=l{S}\boxtimes | |
n=1 |
wn(xn)=
N | |
\left(l{A}\boxtimes | |
n=1 |
Un\right)
N | |
\boxtimes | |
n=1 |
wn(xn),
that is:
N | |
f(x)=l{A}\boxtimes | |
n=1 |
\left(wn(xn)Un\right)=
N | |
l{A}\boxtimes | |
n=1 |
w'n(xn),
where weighting functions of
w'n(xn),
wn(xn)
Un
l{A}
N | |
f(x)=l{A}\boxtimes | |
n=1 |
wn(xn),
f(x)
w | |
n,in |
(xn)
in=1,\ldots,rn
in
n
n=1,\ldots,N
wn(xn)
\forall
bn | |
n:\int | |
an |
\tilde{w}n,i(pn)\tilde{w}n,j(pn)dpn=\deltai,j, 1\leqi,j\leqIn,
where
\deltai,j
\deltaij=1
i=j
\deltaij=0
i ≠ j
{l{A}} | |
in=i |
{l{A}} | |
in=i |
{l{A}} | |
in=j |
n,i
j:\left\langle {l{A}} | |
in=i |
,{l{A}} | |
in=j |
\right\rangle =0
i ≠ j
\left\| {l{A}} | |
in=1 |
\right\|
\geq\left\| {l{A}} | |
in=2 |
\right\|
\geq … \geq\left\| {l{A}} | |
in=rn |
\right\|>0
n=1,\ldots,N+2
n
f(x)
\left\| {l{A}} | |
in=i |
\right\|
(n) | |
\sigma | |
i |
n
l{A}
{l{A}}
n
f(x)
n
rankn(f(x))
n