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Wavelets transform for nonlinear control of multibody systems
Journal of the Franklin Institute 338 (2001) 321–334
Wavelets transform for nonlinear control of multibody systems V.F. Poterasu Technical University, ‘‘Gh. Asachi’’, Iasi, Romania
Abstract The paper solves using wavelet transform a nonlinear control problem of multibody systems. After, we present a wavelet collocation method to solve a nonlinear time-evolution problem, discrete wavelet transform and the algorithm for interpolation functions and the derivative matrix. We consider a nonlinear quadratic control for an inverted pendulum as a numerical example. The results, using Haar wavelets in algebrization of problem, are accurate compared with those from the literature. The method can be extended for any nonlinear control problem obtaining better results than other procedure for localized phenomena. # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Wavelets; Nonlinear control; Multibody dynamics; Collocation method; Inverted pendulum
1. Introduction Wavelet approximation has attracted much attention as a potential numerical technique for the solution of PDE or integral equations. Issued from signal processing and having a lot of applications in pattern recognition, turbulence, geophysics, etc. this technique finds now more and more studies in mechanics. Because of their advantageous properties of localization in both space and frequency domains, wavelets seem to be a great candidate for adaptive schemes for solutions which vary dramatically both in space and time and develop singularities. However, in order to take advantage of the properties of wavelet approximations we have to find an efficient way to deal with the non linearity and general boundary conditions in PDE. Most of problems in multibody dynamics, robotics, fluid E-mail address: [email protected] (V.F. Poterasu). 0016-0032/01/$20.00 # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 9 0 - 9
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dynamics, electromagnetism, etc. which involve solutions with quite different scales are governed by nonlinear PDE with complicated boundary conditions. Among the important books in the wavelets domain are the following: Dahmen, Kurdila, Oswald (Eds.) [1], Daubechies [2], Chui [3], Meyer [4], Wickerhauser [5], Strang, Nguyen [6]. One of the important scientists specialized in interpolating method with wavelets is Bertoluzza [7,8] that underline the fact when expressing a function in Vj in terms of the nodal basis, generally no matter how smooth it is, all the coefficients will be needed in order to get a good approximation while, when expressing the same function in the wavelet basis, in order to get an approximation of the same order, one would usually need only a subset of the coefficients (essentially the ones corresponding to those basis functions whose center is close to singularities). In other words, the nodal basis naturally corresponds to taking a uniform discretization and the wavelet basis to a nonuniform one. That means it is advantageous to use different windows having high-order approximation in smoothness regions and at the same time grid refinement near singularities. Another advantage is to use different kinds of wavelets in various domains as Haar, Daubechies, Mallat, mexican-hat, etc. Beylkin [9], Beylkin and Keiser [10] solve non-linear PDE using adaptive pseudowavelet approach. The basic mechanism of refinement in wavelet-based algorithms is very simple. Due to the vanishing moments of wavelets [2], we know that (for a given accuracy) the wavelet transform of a function ‘‘automatically’’ places significant coefficients in the neighborhood of large gradient present in the function. We simply remove coefficients below the given accuracy threshold. This combination of a basis expansion and adaptive-thresholding is the foundation of adaptive pseudo-wavelet approach. This paper deals with the use of wavelets transforms for the nonlinear control of multibody systems. The nonlinear equations describe the dynamic model of mechatronic systems, which need to be controlled and present an example concerning an inverted pendulum.
2. Wavelet collocation method to solve a nonlinear time evolution problem The key component of this collocation method is the so-called ‘‘discrete wavelet transform’’ (DWT), which maps a solution between the physical space and the wavelet coefficient space. The wavelet decomposition is based on a new cubic spline wavelet for H02 ðIÞ where I is a bounded interval. In order to treat the boundary conditions, an extraboundary scaling function jðxÞ and a boundary wavelet jb ðxÞ have been used. A special ‘‘pointwise orthogonality’’ of the wavelet functions cj; k ðxÞ results in 0ðN log NÞ operations for DWT transform where N is the total number of unknowns. Thus, the nonlinear term in differential equations can be easily treated in the physical space and the derivatives of those nonlinear terms then computed in the wavelet space.
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323
As a result, collocation method will provide the flexibility of handling nonlinearity (and also the implementation of various boundary conditions), which usually are not sheared by Galerkin-type wavelet methods and finite element methods. 2.1. Scaling functions jðxÞ; jb ðxÞ, and wavelet functions cðxÞ; cb ðxÞ In order to generate a multiresolution for Sobolev space H02 ðIÞ we consider two scaling functions, an interior scaling function jðxÞ and a boundary scaling function jb ðxÞ ! 4 4 1X jðxÞ ¼ ð1Þ ð1Þ j ðx jÞ3þ ¼ N4 ðxÞ; 6 j¼0 j 3 3 3 3 3 jb ðxÞ ¼ 32x2þ 11 12xþ þ 2ðx 1Þþ 4ðx 2Þþ :
N4 is the fourth order B-spline and any real number n. ( xn if x50; n xþ ¼ 0 otherwise: In a pair they satisfy the following two-scale relationship: ! 4 X 4 23 jðxÞ ¼ jð2x kÞ; j k¼0 jb ðxÞ ¼ b1 jb ð2xÞ þ
2 X
bk jð2x kÞ;
ð2Þ
ð3Þ
ð4Þ
ð5Þ
k¼0
here b1 ¼ 3=4; b0 ¼ 3=8; b1 ¼ 17=4; b2 ¼ 13=4. To construct a wavelet decomposition of Sobolev space H02 ðIÞ under the inner product we consider the following two-wavelet functions: 3 cðxÞ ¼ 37jð2xÞ þ 12 7 jð2x 1Þ 7jð2x 2Þ 2 V1 ;
ð6Þ
6 jb ðxÞ ¼ 24 13jb ð2xÞ 13jð2xÞ 2 V1 :
ð7Þ
It can be verified that cðnÞ ¼ jb ðnÞ ¼ 0;
for all n 2 Z:
As usual we define the dilation and translation of these two functions cj; k ðxÞ ¼ jb ð2 j x kÞ c1b; j ðxÞ ¼ cb ð2 j xÞ
j50; k ¼ 0; . . . ; nj 3;
crb; j ðxÞ ¼ cb ð2 j ðL xÞÞ;
ð8Þ ð9Þ
where nj ¼ 2 j L. For the sake of simplicity, we adopt the following notations: cj;1 ðxÞ ¼ c1b; j ðxÞ
cj; n2 ðxÞ ¼ crb; j ðxÞ:
ð10Þ
So, when k ¼ 1; nj ¼ 2; cj; k ðxÞ will denote the two boundary wavelet functions, not the usual translation and dilations of cðxÞ.
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For any function f ðxÞ 2 H 2 ðIÞ by Sobolev embedding theorem we have f ðxÞ 2 C I ðIÞ and therefore we can define the following boundary interpolation: Ib; j f ðxÞ ¼ a1 Z1 ð2 j xÞ þ a2 Z2 ð2 j xÞ þ a3 Z1 ð2 j ½L xÞ þ a4 Z2 ð2 j ½L xÞ;
ð11Þ
such that Ib; j f ð0Þ ¼ f ð0Þ;
Ib; j f ðLÞ ¼ f ðLÞ;
ð12Þ
Ib; j f 0 ð0Þ ¼ f ð0Þ;
Ib; j f 0 ðLÞ ¼ f ðLÞ:
ð13Þ
In order to have Ib;j f satisfy conditions (13)–(14) we must take a1 ¼
f 0 ð0Þ 3 f ð0Þ; 2 jþ1 2
a3 ¼
f 0 ðLÞ 3 f ðLÞ; 2 jþ1 2
a2 ¼ f ð0Þ;
ð14Þ
a4 ¼ f ðLÞ:
ð15Þ
To preserve the correct order of accuracy for a cubic-spline approximation is better to use the following values for derivatives at the limits of interval ½0; L: 1X ck f ðkhÞ þ oðhs Þ; h k¼0
ð16Þ
1X f 0 ðLÞ ¼ ck f ðL khÞ þ oðhs Þ; h k¼0
ð17Þ
p
f 0 ð0Þ ¼
p
where h > 0 and p53. For p ¼ 3, if we take c0 ¼ 11 6;
c1 ¼ 3;
c2 ¼ 32;
c3 ¼ 13
then s ¼ 3. Correspondingly, the coefficients ak ; 14k44 for Ib; j f ðxÞ become X a1 ¼ c0k f ðkhÞ; a2 ¼ f ð0Þ; ð18Þ a1 ¼
p X
c0k f ð1 khÞ;
a3 ¼ f ðLÞ;
ð19Þ
k¼c
where c00
1 3 ¼ jþ1 c0 ; 2 h 2
c0k ¼
ck 14k4p: 2 jþ1 h
We can find an approximation fj ðxÞ for any function f ðxÞ 2 H 2 ðIÞ as close as possible, provided that j is large enough, in the form of fj ðxÞ ¼ Ib; j f þ f0 þ g0 þ g1 þ þ gi :
ð20Þ
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3. Discrete wavelet transform (DWT) 3.1. Interpolant operator IV0 in V0 H02 ðIÞ and denote the interior knots for V0 by
Consider any function f ðxÞ ð1Þ
xk
¼ k;
k ¼ 1; . . . ; L 1 ð1Þ L¼1 gk¼1
and the values of f ðxÞ on fxk ð1Þ
fk
¼ f ð1Þ ðxk Þ;
ð21Þ by
k ¼ 1; . . . ; L 1: ð1Þ
The cubic interpolant IV0 f ðxÞ of data f fk IV0 f ðxÞ ¼ cð1Þ jb ðxÞ þ IV0 f ðxÞ interpolant data ð1Þ
IV0 f ðxk
L4 X
k¼0 ð1Þ fk ; k
ðxÞg can be expressed as
ck j0;k ðxÞ þ cL3 jb ð1 xÞ:
ð22Þ
¼ l; . . . ; L I namely
ð1Þ
Þ ¼ fk
ð23Þ
and f ð1Þ ¼ Bc; where
2
7 12 61 66
6 6 6 6 6 6 B¼6 6 6 6 6 6 6 6 4
1 6 2 3 1 6
ð24Þ 3 1 6 2 3
1 6
... ... ... 1 6
2 3 1 6
1 6 2 3 1 6
1 6 1 12
7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 5
ð25Þ
We obtain the coefficients ck1 , where 14k4L 3. 3.2. Interpolant operator LWj in Wj We choose the following interpolation points in I: k þ 1:5 ðiÞ ; 14k4nj 2; xk ¼ 2j nj ¼ Dim Wj ¼ 2 j L:
ð26Þ
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334 ð1Þ
The interpolation points fxk g for V0 in (24) and fx jk g for Wj ; j50 in (26) satisfy a ‘‘point-wise’’ orthogonality condition ð jÞ
ð jÞ
cj;k ðxk Þ ¼ 1;
cj;k ðxl Þ ¼ 0; 14l4nj 2;
14l4L 1 if j ¼ 1; j50: This orthogonality condition is crucial for DWT. The interpolation IWj f ðxÞ of a function f ðxÞ 2 H02 ðIÞ in Wj; j 50, nX j 2
IWj f ðxÞ ¼
f^j;k cj;k ðxÞ;
15k4nj 2;
k¼1 ð jÞ
ð jÞ
IWj f ðxk Þ ¼ f ðxk Þ;
ð27Þ
f ð jÞ ¼ Mj f^ð jÞ :
ð28Þ
where
2
1
6 6 1 6 13 6 6 6 6 6 6 6 6 6 Mj ¼ 6 6 6 6 6 6 6 6 6 6 6 6 4
3
1 14
1
1 14
1 14
1
1 14
1 14
1
1 14
1 14
1 1 14
7 7 7 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 7 1 7 13 7 5 1
ð29Þ
The wavelet interpolation Pj f ðxÞ ¼ f^1;1 cb ðxÞ þ
4 X
f^1;k ck ðxÞ þ f^1;L3 ck ðL xÞ
k¼0
þ
" n 2 j J X X j¼0
# J X ^ fj;k cj;k ðxÞ ¼ f1 ðxÞ þ fj ðxÞ; j¼0
k¼1
where f1 ðxÞ ¼ IV0 f ðxÞ
V0
fj ðxÞ ¼
nX j 2 k¼1
f^j;k cj;k 2 Wj ; j50
ð30Þ
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and ð1Þ
Pj f ðxk
ð1Þ
Þ ¼ f ðxk
Pj f ðx jk Þ ¼ f ðx jk Þ;
Þ;
14k4L 1;
j50; 14k4nj 2:
Let us denote f ¼ ð f 1 ; f 0 ; . . . ; f J ÞT the values of f ðxÞ on all interpolation points, i.e. n 2
j f j ¼ f f ðx jk Þgk¼1 ; j50
L1 f ð1Þ ¼ f f ðx1 k Þgk¼1 ;
and f^ ¼ ð f^ð1Þ ; f^ð0Þ ; . . . ; f^ðJÞ ÞT the wavelet coefficients in expansion (30). It is a recessive algorithm to compute all the wavelet coefficients f^ and the wavelet expansion (30) to include higher-level wavelet spaces. 3.3. Algorithm for interpolation functions with wavelets Step 1: Define f1 ðxÞ ¼ IV0 f ð1Þ ¼ f^1;1 cb ðxÞ þ
L4 X
f^1;k ck ðxÞ þ f^1;L3 ck ðL xÞ;
ð31Þ
k¼0 ð1Þ
where f1 ðxÞ interpolate f ðxÞ at the interpolation points xk namely ð1Þ
f1 ðxk
ð1Þ
Þ ¼ f ðxk
; 14k4L 1,
Þ:
Step 2: Define f0 ðxÞ ¼ IV0 f ð0Þ ðIV0 f Þð0Þ ¼
nX 0 4
f0;‘ c0;‘ ðxÞ;
ð32Þ
i¼1
where 0 2 ðIV0 f Þð0Þ ¼ fIV0 f ðx0k Þgnk¼1 :
We use the ‘‘point-wise orthogonality’’ conditions of the interpolation points ð1Þ
Þ ¼ 0;
14l4n0 2;
ð1Þ
Þ ¼ 0;
14k4L l;
c0;1 ðxl
c0;1 ðxk
ð0Þ f1 ðxk Þ
þ
ð0Þ f0 ðxk Þ
ð0Þ
ð0Þ
ð0Þ
¼ lV0 f ðxk Þ þ ð fk ðlV0 f Þ0k Þ ¼ fk
ð0Þ
¼ f1 ðxk Þ:
ð33Þ
Step 3: Generally, we define for 14j4J fj ðxÞ ¼ IWj ð f ð jÞ ðPj1 f Þð jÞ Þ ¼
nX j 1
f^j;k cj;k ðxÞ;
k¼1
where ð jÞ
ð jÞ
ðPj1 f Þk ¼ Pj1 f ðxk Þ;
14k4nj 2:
We can verify that function f1 ðxÞ þ f0 ðxÞ þ þ fj ðxÞ interpolates function f ðxÞ ð1Þ ð jÞ on all interpolation points fxk g; . . . ; fxk g.
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For j ¼ J we have Pj f ðxÞ ¼ f1 ðxÞ þ f0 ðxÞ þ þ fj ðxÞ which satisfy the required interpolation condition mentioned above. The wavelet coefficient in absolute value j fj; k j depends upon the local regularity of f ðxÞ in the neighborhood of the abscissa 2 j k. More precisely, if 2j k 2 ða; bÞ the decay of j fj;k j depends upon. The Lipschitz regularity of f ðxÞ over the interval ½a; b as the resolution 2 j increases. This property of the wavelet coefficients allows us to detect the location of the function and, then provide a general knowledge of the distribution of wavelet basis functions whose coefficients are larger in magnitude than a given threshold. 3.4. Derivative matrix We consider the derivative matrix, which approximates the first-differential operator Lu ¼ ux
ð34Þ
with the boundary conditions uðLÞ ¼ 0:
ð35Þ
We can write the wavelet interpolation uj ðxÞ of ðUÞ for function uðxÞ as a linear combination of I0; jþ1 f ðxÞ and basis in VJþ1 , namely uj ðxÞ ¼ Ib; jþ1 uðxÞ þ u^1 jb; jþ1 ðxÞ þ
i4 X
u^k jjþ1; k ðxÞ þ u^L3 jb; jþ1; k ðL xÞ; ð36Þ
k¼0
where Ib;Jþ1 uðxÞ is defined in (12). With the transformation x ¼ 2Jþ1 , Eq. (36) becomes uJ ðxÞ ¼ uJ ðxÞ ¼ Ib;0 uðxÞ þ u^1 jb ðxÞ þ
0 L 4 X
k¼0
where L0 ¼ 2Jþ1 . Using the notations u0 ¼ ðuð1Þ; uð2Þ; . . . ; uðL0 1ÞÞT ; u ¼ ðuð0Þ; ðuÞT ; . . . ; uðL0 ÞÞT ; u^ ¼ ð^ u1 ; u^0 ; . . . ; u^L0 3 Þ and Eq. (24) we have u^ ¼ B1 ðu0 ub Þ; where the vector ub is defined by ub ¼ ðIb;0 uð1Þ; 0; . . . ; 0; Ib;0 uðL0 1ÞÞT
u^k jb ðxÞ þ u^L0 3 jb ðL0 xÞ;
ð37Þ
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
329
and Ib;0 uð1Þ ¼ 16ðc00 ; c01 ; c02 ; c03 ; 0; . . . ; 0Þu0 ¼ g1 u0 ; Ib;0 uðL0 1Þ ¼ 16ð0; . . . ; 0; c03 ; c02 ; c01 ; c00 Þu0 ¼ g2 u0 : Therefore, u^ ¼ B1 ðI ½g1 ; 0; . . . ; g2 ÞT ;
u0 ¼ B1 Gu0 :
I is the ðL0 1Þ ðL0 1Þ identity matrix. To obtain approximation of the derivatives of uðxÞ, we differentiate Eq. (37), with respect to, x and evaluate at xk ¼ k; 04k4L0 u0J ðxk Þ ¼ ðIb;0 uÞðxk Þ þ u^1 j0b ðxÞ þ
0 L 4 X
u^k j0k ðxk Þ u^L0 3 j0b ðL0 xk Þ
k¼0
¼ u01 ðxk Þ þ u02 ðxk Þ;
04k4L0 ;
ð38Þ
where u01 ðxk Þ denotes the first term in the first equation and u02 ðxk Þ the rest. Recalling Lb;0 uðxÞ in (11) and coefficients ak in (18) and (19) with h ¼ 1 and j ¼ J þ 1; L ¼ L0 , we have 2 3 d1 Pp 3 2 3 2 0 0 u1 ð0Þ 2 k¼0 ck uðkÞ 3uð0Þ 6 7 d2 7 7 6 7 6 6 7 7 6 6 u0 ð1Þ 7 6 Pp 6 1 0 7 6 7 607 6 1 c uðkÞ k¼0 k 2 7 6 7 6 7 6 7 6 7 6 . 7 6 7 6 .. 7 6 . 7 6 0 7 6 7 6 .. 7 6 7 6 7 6 7 6 7 6 7 6 . 7 6 . u ¼ Du; ð39Þ 7¼6 7 ¼ 6 .. 7 6 .. 7 6 7 6 .. 7 6 7 6 7 6 . 7 6 7 6 7 6 .. 7 6 0 7 6 7 6 . 7 6 7 6 7 6 . 7 6 P 7 6 7 6 . 7 6 p 1 0 0 c uðL kÞ 7 607 6 . 7 6 k k¼0 2 5 4 5 6 7 4 7 P 4 d3 5 p 2 k¼0 c0k uðL0 kÞ þ 3uðL0 Þ u01 ðL0 Þ d4 where the form L0 þ 1 dimension vectors d1 ¼ ð2c00 3; 2c01 ; 2c02 ; 2c3 ; 0; . . . ; 0Þ; d2 ¼ 12ðc00 ; c01 ; c02 ; c3 ; 0; . . . ; 0Þ; d3 ¼ 12ð0; . . . ; 0; c3 ; c02 ; c01 ; c00 Þ; d4 ¼ ð0; . . . ; 0; 2c3 ; 2c02 ; 2c01 ; 2c00 3Þ:
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
Using (24) we have 2 6 6 6 3 6 2 0 6 u1 ð0Þ 6 6 u0 ð1Þ 7 6 7 6 6 1 7 6 6 6 .. 7 6 6 . 7 6 7 6 6 7 6 6 6 .. 7 ¼ 6 6 . 7 6 7 6 6 6 . 7 6 6 . 7 6 4 . 5 6 6 6 6 u01 ðLÞ 6 6 6 4
0
0
0
1 4
1 2
0
12
0
1 2
0
12
0
.. . ..
.
0 .. .
.. .
.. .
.. .
12 0
0
.. .
.. .
.. .
0
0
0
0
0
1 2
12
14
0
0
...
¼ H u^ ¼ HB1 Gu0 ¼ ð0; HB1 G; 0Þ:
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð40Þ
Using (39) and (40), we find 3 2 Dx uJ ð0Þ 7 6 6 Dx uJ ð1Þ 7 7 6 ux ¼ 6 7 ¼ D0 u; .. 7 6 . 5 4 0 Dx uJ ðL Þ where the derivative matrix D0 is defined by 3 3 2 2 Dx uJ ð0Þ Dx uJ ðx0 Þ 7 7 6 6 6 Dx uJ ðx1 Þ 7 6 Dx uJ ð1Þ 7 7 7 6 6 ux ¼ 6 7 ¼ 2Jþ1 6 7 ¼ 2Jþ1 D0 u; .. 7 7 6 6 ... . 5 5 4 4 Dx uJ ðxL0 Þ Dx uJ ðL0 Þ x¼
i 2Jþ1
;
ð41Þ
04i4L0 :
4. Nonlinear quadratic control of multibody systems The multibody systems for example an open-chain manipulator are generally described by Lagrange equations of mixed type d2 s ds ds MðsÞ 2 þ FT l ¼ Q s; þ T s; u; ð42Þ dt dt dt
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
ds F s; ¼ 0; dt
331
ð43Þ
where s are the co-ordinates describing the multibody system, u the vector of input control variables, M is the mass matrix, Q are the generalized forces, F are the kinematic (holonomic or nonholonomic) constraints, F is the Jacobian corresponding to the kinematic constraints, l are the Lagrange multiplies, u the control input (forces) acting through transmissions T. This description must be based on the independent co-ordinates q. The co-ordinates must be independent, e.g. the independent co-ordinates q are selected from s s ¼ rðqÞ;
ds dq ¼R ; dt dt
d2 s d2 q d dq ¼ R þ R : dt2 dt2 dt dt
ð44Þ
The state vector is x ¼ ½q; dq=dt. Then the equations of motion can be transformed into the state-space model in general form 8 < dx ¼ f ðxÞ þ gðxÞu; dt ð45Þ : y ¼ hðxÞ: We consider the quadratic performance index of control which is to be synthesized in the infinite horizon control Z 1 J¼ ðxT Qx þ uT RuÞ dt ð46Þ 0
with the decomposition of the dynamic system f ðxÞ ¼ AðxÞx;
ð47Þ
where the matrices Q and R are positive definite, the matrices A and g are analytic-valued functions, the control function uðxÞ is finite, the pair of matrices ðAðxÞ; gðxÞÞ is controllable and stabilizable for each x in the linear system sense rank½gðxÞ; AðxÞgðxÞ; . . . ; An1 ðxÞgðxÞ ¼ n; the state vector x is fully measured then exists the control u ¼ KðxÞx;
ð48Þ
which minimizes the performance index. The nonlinear gain matrix KðxÞ is determined as KðxÞ ¼ R1 gT ðxÞPðxÞ;
ð49Þ
where PðxÞ is the solution of the Riccati equations AT ðxÞPðxÞ þ PðxÞAðxÞ þ Q PðxÞgðxÞR1 gT ðxÞPðxÞ ¼ P0 ðxÞ:
ð50Þ
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
The equations of the motion can be transformed into the state-space model dq _ ¼ q; dt dq_ _ þ ðRT MRÞð1Þ RT Tu: ¼ ðRT MRÞð1Þ ðRT g RT M qÞ dt
ð51Þ
The control synthesis based on the computation of decomposition and solution on Riccati equations by means of wavelet interpolation developed in previous chapters may be provided. Thus, we discretize the interval of time motion into a number of points, xk and each subinterval in dyadic points 2 j . Eq. (50) may be considered similarly with (34) where functions AðxÞ are the derivative of function PðxÞ. After Eq. (41) P0 ðxÞ ¼ 2Jþ1 D0 PðxÞ
where x ¼
i 2Jþ1
and PJ ðxÞ ¼ Ib; Jþ1 PðxÞ þ P^1 jb; Jþ1 ðxÞ þ
0 L 4 X
P^k jb; Jþ1 ðxÞ þ P^L0 3
k¼0
þ P^L0 3 jb; Jþ1 ðL0 xÞ; where j is the scaling function expressed in (4) and Ib;j PðxÞ in (11) depending on coefficients ai . The filter coefficients ci are determined from (22) an algebraic system when IV0 AðxÞ is developed in scale functions. The interpolation points in I are given in (26). Thus, we obtain an algebraic system solving PðxÞ and, respectively the control u for the suitable matrices Q and R. From the state equations system (51) it _ results in the generalized and velocity co-ordinates q, respectively, q.
Fig. 1. Inverted pendulum.
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
333
5. Numerical example We consider the well-known inverted pendulum illustrated in Fig. 1 with the independent co-ordinates x1 ; x2 ¼ dx1 =dt; x3 ; x4 ¼ dx3 =dt. The stabilized motion for initial conditions ½0; 0:25; 1:48; 0:25 is in Fig. 2 (the initial angle is 84:88). If we
Fig. 2. Nonlinear control of inverted pendulum.
Fig. 3. LQR control of inverted pendulum.
334
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
linearize the equations of motion in origin with the initial conditions ½0; 0:25; 1; 1; 0:25 the LQR model is unstable, Fig. 3. We consider in the computations Haar wavelets and j ¼ 8 for a subinterval. For the computations of Figs. 1–3 we used MATLAB, Toolbox on wavelets.
6. Conclusions We can solve the nonlinear control problems by means of wavelets with good results. The example is presented for a quadratic problem of inverse pendulum in order to compare the results with those from literature, but it is possible to be extended to any nonlinear problems. Also, we may use in order to improve the results other wavelets functions like Daubechies, Morlet, Mallat, etc. We need not to use the decomposition from (47) proposed by Valasek, Steinbauer. We may localize easily the peaks, singularities, resonance, etc.
References [1] W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methodes for Partial Differential Equations, Academic Press, New York, 1997. [2] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, USA, 1992. [3] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1997. [4] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, MA, 1992. [5] M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, A.K. Peters Ltd., Wellesley, MA, 1994. [6] G. Strang, T. Nguyen, Wavelets and Filter Banks, Cambridge Press, Wellesley, 1997. [7] S. Bertoluzza, An adaptive collocation method based on interpolating wavelets, in: W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methods for PDE, Academic Press, New York, 1997, pp. 109–135. [8] S.G. Beylkin, Wavelets and fast numerical algorithms, Proceedings of Symposium on Mathematics, vol. 47, 1993, pp. 89–111. [9] G. Beylkin, J.M. Keiser, An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, in: W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methods for PDE, Academic Press, New York, 1997, pp. 137–197. [10] V. Stejskal, M. Valasek, Kinematics and Dynamics of Machinery, Marcel Dekker Inc., New York, 1996.
Further Reading S.P. Banks, K.J. Mhana, Optimal control and stabilisation for nonlinear systems, SIAM J. Math. Control Inform. 9 (1992) 179–196. J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1992. V.F. Poterasu, Optimal control of smart composite plates for distributed sensors using FEM and wavelets, in: U. Gabbert (Ed.), Modelling and Control of Adaptive Mechanical Structures, VDI, 11, 268, VDI Verlag, Dusseldorf, 1998, pp. 267–277. J. Angeles, G. Hommel, P. Kovach (Eds.), Computational Kinematics, Kluwer Academic Pub., Dordrecht, 1993.
Wavelets transform for nonlinear control of multibody systems V.F. Poterasu Technical University, ‘‘Gh. Asachi’’, Iasi, Romania
Abstract The paper solves using wavelet transform a nonlinear control problem of multibody systems. After, we present a wavelet collocation method to solve a nonlinear time-evolution problem, discrete wavelet transform and the algorithm for interpolation functions and the derivative matrix. We consider a nonlinear quadratic control for an inverted pendulum as a numerical example. The results, using Haar wavelets in algebrization of problem, are accurate compared with those from the literature. The method can be extended for any nonlinear control problem obtaining better results than other procedure for localized phenomena. # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. Keywords: Wavelets; Nonlinear control; Multibody dynamics; Collocation method; Inverted pendulum
1. Introduction Wavelet approximation has attracted much attention as a potential numerical technique for the solution of PDE or integral equations. Issued from signal processing and having a lot of applications in pattern recognition, turbulence, geophysics, etc. this technique finds now more and more studies in mechanics. Because of their advantageous properties of localization in both space and frequency domains, wavelets seem to be a great candidate for adaptive schemes for solutions which vary dramatically both in space and time and develop singularities. However, in order to take advantage of the properties of wavelet approximations we have to find an efficient way to deal with the non linearity and general boundary conditions in PDE. Most of problems in multibody dynamics, robotics, fluid E-mail address: [email protected] (V.F. Poterasu). 0016-0032/01/$20.00 # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 6 - 0 0 3 2 ( 0 0 ) 0 0 0 9 0 - 9
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dynamics, electromagnetism, etc. which involve solutions with quite different scales are governed by nonlinear PDE with complicated boundary conditions. Among the important books in the wavelets domain are the following: Dahmen, Kurdila, Oswald (Eds.) [1], Daubechies [2], Chui [3], Meyer [4], Wickerhauser [5], Strang, Nguyen [6]. One of the important scientists specialized in interpolating method with wavelets is Bertoluzza [7,8] that underline the fact when expressing a function in Vj in terms of the nodal basis, generally no matter how smooth it is, all the coefficients will be needed in order to get a good approximation while, when expressing the same function in the wavelet basis, in order to get an approximation of the same order, one would usually need only a subset of the coefficients (essentially the ones corresponding to those basis functions whose center is close to singularities). In other words, the nodal basis naturally corresponds to taking a uniform discretization and the wavelet basis to a nonuniform one. That means it is advantageous to use different windows having high-order approximation in smoothness regions and at the same time grid refinement near singularities. Another advantage is to use different kinds of wavelets in various domains as Haar, Daubechies, Mallat, mexican-hat, etc. Beylkin [9], Beylkin and Keiser [10] solve non-linear PDE using adaptive pseudowavelet approach. The basic mechanism of refinement in wavelet-based algorithms is very simple. Due to the vanishing moments of wavelets [2], we know that (for a given accuracy) the wavelet transform of a function ‘‘automatically’’ places significant coefficients in the neighborhood of large gradient present in the function. We simply remove coefficients below the given accuracy threshold. This combination of a basis expansion and adaptive-thresholding is the foundation of adaptive pseudo-wavelet approach. This paper deals with the use of wavelets transforms for the nonlinear control of multibody systems. The nonlinear equations describe the dynamic model of mechatronic systems, which need to be controlled and present an example concerning an inverted pendulum.
2. Wavelet collocation method to solve a nonlinear time evolution problem The key component of this collocation method is the so-called ‘‘discrete wavelet transform’’ (DWT), which maps a solution between the physical space and the wavelet coefficient space. The wavelet decomposition is based on a new cubic spline wavelet for H02 ðIÞ where I is a bounded interval. In order to treat the boundary conditions, an extraboundary scaling function jðxÞ and a boundary wavelet jb ðxÞ have been used. A special ‘‘pointwise orthogonality’’ of the wavelet functions cj; k ðxÞ results in 0ðN log NÞ operations for DWT transform where N is the total number of unknowns. Thus, the nonlinear term in differential equations can be easily treated in the physical space and the derivatives of those nonlinear terms then computed in the wavelet space.
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
323
As a result, collocation method will provide the flexibility of handling nonlinearity (and also the implementation of various boundary conditions), which usually are not sheared by Galerkin-type wavelet methods and finite element methods. 2.1. Scaling functions jðxÞ; jb ðxÞ, and wavelet functions cðxÞ; cb ðxÞ In order to generate a multiresolution for Sobolev space H02 ðIÞ we consider two scaling functions, an interior scaling function jðxÞ and a boundary scaling function jb ðxÞ ! 4 4 1X jðxÞ ¼ ð1Þ ð1Þ j ðx jÞ3þ ¼ N4 ðxÞ; 6 j¼0 j 3 3 3 3 3 jb ðxÞ ¼ 32x2þ 11 12xþ þ 2ðx 1Þþ 4ðx 2Þþ :
N4 is the fourth order B-spline and any real number n. ( xn if x50; n xþ ¼ 0 otherwise: In a pair they satisfy the following two-scale relationship: ! 4 X 4 23 jðxÞ ¼ jð2x kÞ; j k¼0 jb ðxÞ ¼ b1 jb ð2xÞ þ
2 X
bk jð2x kÞ;
ð2Þ
ð3Þ
ð4Þ
ð5Þ
k¼0
here b1 ¼ 3=4; b0 ¼ 3=8; b1 ¼ 17=4; b2 ¼ 13=4. To construct a wavelet decomposition of Sobolev space H02 ðIÞ under the inner product we consider the following two-wavelet functions: 3 cðxÞ ¼ 37jð2xÞ þ 12 7 jð2x 1Þ 7jð2x 2Þ 2 V1 ;
ð6Þ
6 jb ðxÞ ¼ 24 13jb ð2xÞ 13jð2xÞ 2 V1 :
ð7Þ
It can be verified that cðnÞ ¼ jb ðnÞ ¼ 0;
for all n 2 Z:
As usual we define the dilation and translation of these two functions cj; k ðxÞ ¼ jb ð2 j x kÞ c1b; j ðxÞ ¼ cb ð2 j xÞ
j50; k ¼ 0; . . . ; nj 3;
crb; j ðxÞ ¼ cb ð2 j ðL xÞÞ;
ð8Þ ð9Þ
where nj ¼ 2 j L. For the sake of simplicity, we adopt the following notations: cj;1 ðxÞ ¼ c1b; j ðxÞ
cj; n2 ðxÞ ¼ crb; j ðxÞ:
ð10Þ
So, when k ¼ 1; nj ¼ 2; cj; k ðxÞ will denote the two boundary wavelet functions, not the usual translation and dilations of cðxÞ.
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For any function f ðxÞ 2 H 2 ðIÞ by Sobolev embedding theorem we have f ðxÞ 2 C I ðIÞ and therefore we can define the following boundary interpolation: Ib; j f ðxÞ ¼ a1 Z1 ð2 j xÞ þ a2 Z2 ð2 j xÞ þ a3 Z1 ð2 j ½L xÞ þ a4 Z2 ð2 j ½L xÞ;
ð11Þ
such that Ib; j f ð0Þ ¼ f ð0Þ;
Ib; j f ðLÞ ¼ f ðLÞ;
ð12Þ
Ib; j f 0 ð0Þ ¼ f ð0Þ;
Ib; j f 0 ðLÞ ¼ f ðLÞ:
ð13Þ
In order to have Ib;j f satisfy conditions (13)–(14) we must take a1 ¼
f 0 ð0Þ 3 f ð0Þ; 2 jþ1 2
a3 ¼
f 0 ðLÞ 3 f ðLÞ; 2 jþ1 2
a2 ¼ f ð0Þ;
ð14Þ
a4 ¼ f ðLÞ:
ð15Þ
To preserve the correct order of accuracy for a cubic-spline approximation is better to use the following values for derivatives at the limits of interval ½0; L: 1X ck f ðkhÞ þ oðhs Þ; h k¼0
ð16Þ
1X f 0 ðLÞ ¼ ck f ðL khÞ þ oðhs Þ; h k¼0
ð17Þ
p
f 0 ð0Þ ¼
p
where h > 0 and p53. For p ¼ 3, if we take c0 ¼ 11 6;
c1 ¼ 3;
c2 ¼ 32;
c3 ¼ 13
then s ¼ 3. Correspondingly, the coefficients ak ; 14k44 for Ib; j f ðxÞ become X a1 ¼ c0k f ðkhÞ; a2 ¼ f ð0Þ; ð18Þ a1 ¼
p X
c0k f ð1 khÞ;
a3 ¼ f ðLÞ;
ð19Þ
k¼c
where c00
1 3 ¼ jþ1 c0 ; 2 h 2
c0k ¼
ck 14k4p: 2 jþ1 h
We can find an approximation fj ðxÞ for any function f ðxÞ 2 H 2 ðIÞ as close as possible, provided that j is large enough, in the form of fj ðxÞ ¼ Ib; j f þ f0 þ g0 þ g1 þ þ gi :
ð20Þ
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3. Discrete wavelet transform (DWT) 3.1. Interpolant operator IV0 in V0 H02 ðIÞ and denote the interior knots for V0 by
Consider any function f ðxÞ ð1Þ
xk
¼ k;
k ¼ 1; . . . ; L 1 ð1Þ L¼1 gk¼1
and the values of f ðxÞ on fxk ð1Þ
fk
¼ f ð1Þ ðxk Þ;
ð21Þ by
k ¼ 1; . . . ; L 1: ð1Þ
The cubic interpolant IV0 f ðxÞ of data f fk IV0 f ðxÞ ¼ cð1Þ jb ðxÞ þ IV0 f ðxÞ interpolant data ð1Þ
IV0 f ðxk
L4 X
k¼0 ð1Þ fk ; k
ðxÞg can be expressed as
ck j0;k ðxÞ þ cL3 jb ð1 xÞ:
ð22Þ
¼ l; . . . ; L I namely
ð1Þ
Þ ¼ fk
ð23Þ
and f ð1Þ ¼ Bc; where
2
7 12 61 66
6 6 6 6 6 6 B¼6 6 6 6 6 6 6 6 4
1 6 2 3 1 6
ð24Þ 3 1 6 2 3
1 6
... ... ... 1 6
2 3 1 6
1 6 2 3 1 6
1 6 1 12
7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 5
ð25Þ
We obtain the coefficients ck1 , where 14k4L 3. 3.2. Interpolant operator LWj in Wj We choose the following interpolation points in I: k þ 1:5 ðiÞ ; 14k4nj 2; xk ¼ 2j nj ¼ Dim Wj ¼ 2 j L:
ð26Þ
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334 ð1Þ
The interpolation points fxk g for V0 in (24) and fx jk g for Wj ; j50 in (26) satisfy a ‘‘point-wise’’ orthogonality condition ð jÞ
ð jÞ
cj;k ðxk Þ ¼ 1;
cj;k ðxl Þ ¼ 0; 14l4nj 2;
14l4L 1 if j ¼ 1; j50: This orthogonality condition is crucial for DWT. The interpolation IWj f ðxÞ of a function f ðxÞ 2 H02 ðIÞ in Wj; j 50, nX j 2
IWj f ðxÞ ¼
f^j;k cj;k ðxÞ;
15k4nj 2;
k¼1 ð jÞ
ð jÞ
IWj f ðxk Þ ¼ f ðxk Þ;
ð27Þ
f ð jÞ ¼ Mj f^ð jÞ :
ð28Þ
where
2
1
6 6 1 6 13 6 6 6 6 6 6 6 6 6 Mj ¼ 6 6 6 6 6 6 6 6 6 6 6 6 4
3
1 14
1
1 14
1 14
1
1 14
1 14
1
1 14
1 14
1 1 14
7 7 7 7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 7 7 7 7 1 7 13 7 5 1
ð29Þ
The wavelet interpolation Pj f ðxÞ ¼ f^1;1 cb ðxÞ þ
4 X
f^1;k ck ðxÞ þ f^1;L3 ck ðL xÞ
k¼0
þ
" n 2 j J X X j¼0
# J X ^ fj;k cj;k ðxÞ ¼ f1 ðxÞ þ fj ðxÞ; j¼0
k¼1
where f1 ðxÞ ¼ IV0 f ðxÞ
V0
fj ðxÞ ¼
nX j 2 k¼1
f^j;k cj;k 2 Wj ; j50
ð30Þ
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
and ð1Þ
Pj f ðxk
ð1Þ
Þ ¼ f ðxk
Pj f ðx jk Þ ¼ f ðx jk Þ;
Þ;
14k4L 1;
j50; 14k4nj 2:
Let us denote f ¼ ð f 1 ; f 0 ; . . . ; f J ÞT the values of f ðxÞ on all interpolation points, i.e. n 2
j f j ¼ f f ðx jk Þgk¼1 ; j50
L1 f ð1Þ ¼ f f ðx1 k Þgk¼1 ;
and f^ ¼ ð f^ð1Þ ; f^ð0Þ ; . . . ; f^ðJÞ ÞT the wavelet coefficients in expansion (30). It is a recessive algorithm to compute all the wavelet coefficients f^ and the wavelet expansion (30) to include higher-level wavelet spaces. 3.3. Algorithm for interpolation functions with wavelets Step 1: Define f1 ðxÞ ¼ IV0 f ð1Þ ¼ f^1;1 cb ðxÞ þ
L4 X
f^1;k ck ðxÞ þ f^1;L3 ck ðL xÞ;
ð31Þ
k¼0 ð1Þ
where f1 ðxÞ interpolate f ðxÞ at the interpolation points xk namely ð1Þ
f1 ðxk
ð1Þ
Þ ¼ f ðxk
; 14k4L 1,
Þ:
Step 2: Define f0 ðxÞ ¼ IV0 f ð0Þ ðIV0 f Þð0Þ ¼
nX 0 4
f0;‘ c0;‘ ðxÞ;
ð32Þ
i¼1
where 0 2 ðIV0 f Þð0Þ ¼ fIV0 f ðx0k Þgnk¼1 :
We use the ‘‘point-wise orthogonality’’ conditions of the interpolation points ð1Þ
Þ ¼ 0;
14l4n0 2;
ð1Þ
Þ ¼ 0;
14k4L l;
c0;1 ðxl
c0;1 ðxk
ð0Þ f1 ðxk Þ
þ
ð0Þ f0 ðxk Þ
ð0Þ
ð0Þ
ð0Þ
¼ lV0 f ðxk Þ þ ð fk ðlV0 f Þ0k Þ ¼ fk
ð0Þ
¼ f1 ðxk Þ:
ð33Þ
Step 3: Generally, we define for 14j4J fj ðxÞ ¼ IWj ð f ð jÞ ðPj1 f Þð jÞ Þ ¼
nX j 1
f^j;k cj;k ðxÞ;
k¼1
where ð jÞ
ð jÞ
ðPj1 f Þk ¼ Pj1 f ðxk Þ;
14k4nj 2:
We can verify that function f1 ðxÞ þ f0 ðxÞ þ þ fj ðxÞ interpolates function f ðxÞ ð1Þ ð jÞ on all interpolation points fxk g; . . . ; fxk g.
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For j ¼ J we have Pj f ðxÞ ¼ f1 ðxÞ þ f0 ðxÞ þ þ fj ðxÞ which satisfy the required interpolation condition mentioned above. The wavelet coefficient in absolute value j fj; k j depends upon the local regularity of f ðxÞ in the neighborhood of the abscissa 2 j k. More precisely, if 2j k 2 ða; bÞ the decay of j fj;k j depends upon. The Lipschitz regularity of f ðxÞ over the interval ½a; b as the resolution 2 j increases. This property of the wavelet coefficients allows us to detect the location of the function and, then provide a general knowledge of the distribution of wavelet basis functions whose coefficients are larger in magnitude than a given threshold. 3.4. Derivative matrix We consider the derivative matrix, which approximates the first-differential operator Lu ¼ ux
ð34Þ
with the boundary conditions uðLÞ ¼ 0:
ð35Þ
We can write the wavelet interpolation uj ðxÞ of ðUÞ for function uðxÞ as a linear combination of I0; jþ1 f ðxÞ and basis in VJþ1 , namely uj ðxÞ ¼ Ib; jþ1 uðxÞ þ u^1 jb; jþ1 ðxÞ þ
i4 X
u^k jjþ1; k ðxÞ þ u^L3 jb; jþ1; k ðL xÞ; ð36Þ
k¼0
where Ib;Jþ1 uðxÞ is defined in (12). With the transformation x ¼ 2Jþ1 , Eq. (36) becomes uJ ðxÞ ¼ uJ ðxÞ ¼ Ib;0 uðxÞ þ u^1 jb ðxÞ þ
0 L 4 X
k¼0
where L0 ¼ 2Jþ1 . Using the notations u0 ¼ ðuð1Þ; uð2Þ; . . . ; uðL0 1ÞÞT ; u ¼ ðuð0Þ; ðuÞT ; . . . ; uðL0 ÞÞT ; u^ ¼ ð^ u1 ; u^0 ; . . . ; u^L0 3 Þ and Eq. (24) we have u^ ¼ B1 ðu0 ub Þ; where the vector ub is defined by ub ¼ ðIb;0 uð1Þ; 0; . . . ; 0; Ib;0 uðL0 1ÞÞT
u^k jb ðxÞ þ u^L0 3 jb ðL0 xÞ;
ð37Þ
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
329
and Ib;0 uð1Þ ¼ 16ðc00 ; c01 ; c02 ; c03 ; 0; . . . ; 0Þu0 ¼ g1 u0 ; Ib;0 uðL0 1Þ ¼ 16ð0; . . . ; 0; c03 ; c02 ; c01 ; c00 Þu0 ¼ g2 u0 : Therefore, u^ ¼ B1 ðI ½g1 ; 0; . . . ; g2 ÞT ;
u0 ¼ B1 Gu0 :
I is the ðL0 1Þ ðL0 1Þ identity matrix. To obtain approximation of the derivatives of uðxÞ, we differentiate Eq. (37), with respect to, x and evaluate at xk ¼ k; 04k4L0 u0J ðxk Þ ¼ ðIb;0 uÞðxk Þ þ u^1 j0b ðxÞ þ
0 L 4 X
u^k j0k ðxk Þ u^L0 3 j0b ðL0 xk Þ
k¼0
¼ u01 ðxk Þ þ u02 ðxk Þ;
04k4L0 ;
ð38Þ
where u01 ðxk Þ denotes the first term in the first equation and u02 ðxk Þ the rest. Recalling Lb;0 uðxÞ in (11) and coefficients ak in (18) and (19) with h ¼ 1 and j ¼ J þ 1; L ¼ L0 , we have 2 3 d1 Pp 3 2 3 2 0 0 u1 ð0Þ 2 k¼0 ck uðkÞ 3uð0Þ 6 7 d2 7 7 6 7 6 6 7 7 6 6 u0 ð1Þ 7 6 Pp 6 1 0 7 6 7 607 6 1 c uðkÞ k¼0 k 2 7 6 7 6 7 6 7 6 7 6 . 7 6 7 6 .. 7 6 . 7 6 0 7 6 7 6 .. 7 6 7 6 7 6 7 6 7 6 7 6 . 7 6 . u ¼ Du; ð39Þ 7¼6 7 ¼ 6 .. 7 6 .. 7 6 7 6 .. 7 6 7 6 7 6 . 7 6 7 6 7 6 .. 7 6 0 7 6 7 6 . 7 6 7 6 7 6 . 7 6 P 7 6 7 6 . 7 6 p 1 0 0 c uðL kÞ 7 607 6 . 7 6 k k¼0 2 5 4 5 6 7 4 7 P 4 d3 5 p 2 k¼0 c0k uðL0 kÞ þ 3uðL0 Þ u01 ðL0 Þ d4 where the form L0 þ 1 dimension vectors d1 ¼ ð2c00 3; 2c01 ; 2c02 ; 2c3 ; 0; . . . ; 0Þ; d2 ¼ 12ðc00 ; c01 ; c02 ; c3 ; 0; . . . ; 0Þ; d3 ¼ 12ð0; . . . ; 0; c3 ; c02 ; c01 ; c00 Þ; d4 ¼ ð0; . . . ; 0; 2c3 ; 2c02 ; 2c01 ; 2c00 3Þ:
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
Using (24) we have 2 6 6 6 3 6 2 0 6 u1 ð0Þ 6 6 u0 ð1Þ 7 6 7 6 6 1 7 6 6 6 .. 7 6 6 . 7 6 7 6 6 7 6 6 6 .. 7 ¼ 6 6 . 7 6 7 6 6 6 . 7 6 6 . 7 6 4 . 5 6 6 6 6 u01 ðLÞ 6 6 6 4
0
0
0
1 4
1 2
0
12
0
1 2
0
12
0
.. . ..
.
0 .. .
.. .
.. .
.. .
12 0
0
.. .
.. .
.. .
0
0
0
0
0
1 2
12
14
0
0
...
¼ H u^ ¼ HB1 Gu0 ¼ ð0; HB1 G; 0Þ:
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ð40Þ
Using (39) and (40), we find 3 2 Dx uJ ð0Þ 7 6 6 Dx uJ ð1Þ 7 7 6 ux ¼ 6 7 ¼ D0 u; .. 7 6 . 5 4 0 Dx uJ ðL Þ where the derivative matrix D0 is defined by 3 3 2 2 Dx uJ ð0Þ Dx uJ ðx0 Þ 7 7 6 6 6 Dx uJ ðx1 Þ 7 6 Dx uJ ð1Þ 7 7 7 6 6 ux ¼ 6 7 ¼ 2Jþ1 6 7 ¼ 2Jþ1 D0 u; .. 7 7 6 6 ... . 5 5 4 4 Dx uJ ðxL0 Þ Dx uJ ðL0 Þ x¼
i 2Jþ1
;
ð41Þ
04i4L0 :
4. Nonlinear quadratic control of multibody systems The multibody systems for example an open-chain manipulator are generally described by Lagrange equations of mixed type d2 s ds ds MðsÞ 2 þ FT l ¼ Q s; þ T s; u; ð42Þ dt dt dt
V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
ds F s; ¼ 0; dt
331
ð43Þ
where s are the co-ordinates describing the multibody system, u the vector of input control variables, M is the mass matrix, Q are the generalized forces, F are the kinematic (holonomic or nonholonomic) constraints, F is the Jacobian corresponding to the kinematic constraints, l are the Lagrange multiplies, u the control input (forces) acting through transmissions T. This description must be based on the independent co-ordinates q. The co-ordinates must be independent, e.g. the independent co-ordinates q are selected from s s ¼ rðqÞ;
ds dq ¼R ; dt dt
d2 s d2 q d dq ¼ R þ R : dt2 dt2 dt dt
ð44Þ
The state vector is x ¼ ½q; dq=dt. Then the equations of motion can be transformed into the state-space model in general form 8 < dx ¼ f ðxÞ þ gðxÞu; dt ð45Þ : y ¼ hðxÞ: We consider the quadratic performance index of control which is to be synthesized in the infinite horizon control Z 1 J¼ ðxT Qx þ uT RuÞ dt ð46Þ 0
with the decomposition of the dynamic system f ðxÞ ¼ AðxÞx;
ð47Þ
where the matrices Q and R are positive definite, the matrices A and g are analytic-valued functions, the control function uðxÞ is finite, the pair of matrices ðAðxÞ; gðxÞÞ is controllable and stabilizable for each x in the linear system sense rank½gðxÞ; AðxÞgðxÞ; . . . ; An1 ðxÞgðxÞ ¼ n; the state vector x is fully measured then exists the control u ¼ KðxÞx;
ð48Þ
which minimizes the performance index. The nonlinear gain matrix KðxÞ is determined as KðxÞ ¼ R1 gT ðxÞPðxÞ;
ð49Þ
where PðxÞ is the solution of the Riccati equations AT ðxÞPðxÞ þ PðxÞAðxÞ þ Q PðxÞgðxÞR1 gT ðxÞPðxÞ ¼ P0 ðxÞ:
ð50Þ
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
The equations of the motion can be transformed into the state-space model dq _ ¼ q; dt dq_ _ þ ðRT MRÞð1Þ RT Tu: ¼ ðRT MRÞð1Þ ðRT g RT M qÞ dt
ð51Þ
The control synthesis based on the computation of decomposition and solution on Riccati equations by means of wavelet interpolation developed in previous chapters may be provided. Thus, we discretize the interval of time motion into a number of points, xk and each subinterval in dyadic points 2 j . Eq. (50) may be considered similarly with (34) where functions AðxÞ are the derivative of function PðxÞ. After Eq. (41) P0 ðxÞ ¼ 2Jþ1 D0 PðxÞ
where x ¼
i 2Jþ1
and PJ ðxÞ ¼ Ib; Jþ1 PðxÞ þ P^1 jb; Jþ1 ðxÞ þ
0 L 4 X
P^k jb; Jþ1 ðxÞ þ P^L0 3
k¼0
þ P^L0 3 jb; Jþ1 ðL0 xÞ; where j is the scaling function expressed in (4) and Ib;j PðxÞ in (11) depending on coefficients ai . The filter coefficients ci are determined from (22) an algebraic system when IV0 AðxÞ is developed in scale functions. The interpolation points in I are given in (26). Thus, we obtain an algebraic system solving PðxÞ and, respectively the control u for the suitable matrices Q and R. From the state equations system (51) it _ results in the generalized and velocity co-ordinates q, respectively, q.
Fig. 1. Inverted pendulum.
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5. Numerical example We consider the well-known inverted pendulum illustrated in Fig. 1 with the independent co-ordinates x1 ; x2 ¼ dx1 =dt; x3 ; x4 ¼ dx3 =dt. The stabilized motion for initial conditions ½0; 0:25; 1:48; 0:25 is in Fig. 2 (the initial angle is 84:88). If we
Fig. 2. Nonlinear control of inverted pendulum.
Fig. 3. LQR control of inverted pendulum.
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V.F. Poterasu / Journal of the Franklin Institute 338 (2001) 321–334
linearize the equations of motion in origin with the initial conditions ½0; 0:25; 1; 1; 0:25 the LQR model is unstable, Fig. 3. We consider in the computations Haar wavelets and j ¼ 8 for a subinterval. For the computations of Figs. 1–3 we used MATLAB, Toolbox on wavelets.
6. Conclusions We can solve the nonlinear control problems by means of wavelets with good results. The example is presented for a quadratic problem of inverse pendulum in order to compare the results with those from literature, but it is possible to be extended to any nonlinear problems. Also, we may use in order to improve the results other wavelets functions like Daubechies, Morlet, Mallat, etc. We need not to use the decomposition from (47) proposed by Valasek, Steinbauer. We may localize easily the peaks, singularities, resonance, etc.
References [1] W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methodes for Partial Differential Equations, Academic Press, New York, 1997. [2] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, USA, 1992. [3] C.K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1997. [4] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, MA, 1992. [5] M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, A.K. Peters Ltd., Wellesley, MA, 1994. [6] G. Strang, T. Nguyen, Wavelets and Filter Banks, Cambridge Press, Wellesley, 1997. [7] S. Bertoluzza, An adaptive collocation method based on interpolating wavelets, in: W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methods for PDE, Academic Press, New York, 1997, pp. 109–135. [8] S.G. Beylkin, Wavelets and fast numerical algorithms, Proceedings of Symposium on Mathematics, vol. 47, 1993, pp. 89–111. [9] G. Beylkin, J.M. Keiser, An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, in: W. Dahmen, A. Kurdila, P. Oswald (Eds.), Multiscale Wavelet Methods for PDE, Academic Press, New York, 1997, pp. 137–197. [10] V. Stejskal, M. Valasek, Kinematics and Dynamics of Machinery, Marcel Dekker Inc., New York, 1996.
Further Reading S.P. Banks, K.J. Mhana, Optimal control and stabilisation for nonlinear systems, SIAM J. Math. Control Inform. 9 (1992) 179–196. J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1992. V.F. Poterasu, Optimal control of smart composite plates for distributed sensors using FEM and wavelets, in: U. Gabbert (Ed.), Modelling and Control of Adaptive Mechanical Structures, VDI, 11, 268, VDI Verlag, Dusseldorf, 1998, pp. 267–277. J. Angeles, G. Hommel, P. Kovach (Eds.), Computational Kinematics, Kluwer Academic Pub., Dordrecht, 1993.