Chapter 6: Introduction of Uniform Model Structures for Partial, Temporal, and
6.1. The Orthogonal Group as Source of Uniform Structures in CM
6.2.2. Complementary Tuning Possibilities in the Cumulative Control
From a purely mathematical point of view Symplectic matrices form a Lie Group more or less similar to the Orthogonal Group. By the use of special generators of this group, each of its elements can be “parameterized” by continuous parameters in the form of simple closed analytical expressions describing simultaneous exponential stretches and shrinks, conventional and hyperbolic rotations in the tangent space [J2]. On the above basis the following model strategy can be elaborated. Instead using the original matrix relation A=SB its slight modification can be introduced in the form as
P*A=SPB (6.2.5)
in which P* is a special Symplectic matrix leaving the first column of A invariant. In similar way, P also is a Symplectic matrix leaving the first column of B invariant. It is evident, that (6.2.5) corresponds to the same control requirement as the original equation A=SB, but the resulting Symplectic transformation S will differ from the original one. The difference between the two controls consists in different dealing
with the antiorthogonal subspaces of the 1st columns of A ad B, respectively. Since the first columns of the A(n) and B(n) matrices of the nth control step may contain components from the antiorthogonal subspaces of the 1st columns of the preceding A(n-1) and B(n-1) matrices, slight tuning of the P* and P matrices can improve the control quality since it can “reveal” and “trace” tendencies in the variation of the control task.
For constructing appropriate P and P* matrices their exponential series expression can be used. For instance, let u correspond to the first column of matrix B, and G be one of the generators of the Symplectic Group:
u expedient to systematically study the structure of the generators of the Symplectic Group.
For this purpose the standard technique for constructing the generators of a Lie Group considering almost unit transformations can conveniently be used:
(
I+ξG) (
ℑI+ξG)
T =ℑ+ξ(
G+GT)
ℑ+O( )
ξ2 =ℑ. (6.2.7)In the 1st order approximation according to ξ≠0 (6.2.7) satisfied if Gℑℑℑℑ+ℑℑℑGℑ T=0. Since ℑℑ
ℑℑT= -ℑℑℑℑ this means that ℑℑGℑℑ T= -ℑℑℑℑTGT= -(Gℑℑℑℑ)T, therefore the matrix (Gℑℑℑ)=S must be ℑ symmetric. This immediately reveals the dimension of the linear space of the generators: for an n DOF mechanical system S has the dimensions of 2n×2n that may have 2n+(4n2-2n)/2=2n2+n linearly independent elements. Considering S in a block-diagonal structure it must consist of the symmetric real A and B matrices, and an arbitrary H matrix as
in which the minus sign was introduced for later convenience, A and B together have 2[n+(n2-n)/2]=n2+n independent elements due to their symmetry, and H has n2 independent elements that altogether is 2n2+n. Since ℑℑℑℑ2= -I, Gℑℑℑℑ2= -G=Sℑℑℑℑ, so from (6.2.8) it is obtained that
It s worth noting that for achieving block-diagonal transformations not mixing the q and p components, in (6.2.9) the condition A=0, B=0 has to be met, and we have the n2 linearly independent components of the arbitrary real H.
A relatively lucid description of the generators can be achieved if we observe that GT in (6.2.9) essentially has the same structure as G, therefore if G is generator then GT is a generator, too:
Really, in the upper left block of GT an arbitrary matrix stands, and in the lower right block minus one times its transpose can be found. In the upper right and lower left
Utilizing the fact that the generators form a linear space, symmetric and skew symmetric generators can be constructed of (6.2.9) and (6.2.10) as
( ) ( ) ( )
symmetric components of H, and in the skew symmetric part (6.2.12) we have the (n2-n)/2 linearly independent skew symmetric components of H, that altogether is n2.It is very easy to find conveniently applicable basis vectors in the space of the generators that consist of 0 and ±1 matrix elements, and lead to simple exponential series that can be expressed in closed analytical form.
Consider at first the symmetric generators! For instance in the case of n=1 the generators that have elements only in the main diagonal of the symmetric matrices yield as e.g.
These generators generate simultaneous stretches and shrinks strictly in the main diagonals. In similar manner, if we have nonzero elements in the nondiagonal parts we easily obtain that
( ) ( )
since the 2nd power of this generator just yields the unit matrix, therefore the proper powers of variable t can be recognized in the appropriate matrix elements. So these generators generate hyperbolic rotations in the main block diagonals. In close analogy with that hyperbolic rotations can be generated between the nondiagonals as e.g. by
( ) ( )
since the 3rd power of such generators is just identical with their 1st power therefore it is very easy to recognize the appropriate power series of t in the matrix elements.
Regarding the skew symmetric generators similar considerations can be done.
Consider the main block diagonals in (6.2.12) for n=1:
( ) ( )
off-main diagonals as e.g. the example below, in which( ) ( )
because the 3rd power of such generators yield their 1st power again, etc. In higher dimensional cases just the same considerations can be done with very similar generators and their power series.
Now let us return to the problem in (6.2.6) for finding appropriate generator for the condition Gu=0. For convenient utilization of the block structures in (6.2.11) and (6.2.12) let us decompose u into two sub-blocks of dimension n as u=[aT,bT]T and consider the generally n-2 dimensional orthogonal subspace of a and b. (For their specialties the a parallel to b, a=0, b=0, and a=b=0 cases are not considered since they are insignificant from the needs of the control as later it will be explained).
Let the set of orthonormal basis vectors {c(i)|i=3,4,…,n} in the orthogonal subset of a and b! It is very easy to create symmetric blocks for (6.2.11) in the form of S(ij)kl:=(c(i)kc(j)l+c(i)lc(j)k)/2 and skew symmetric blocks for (6.2.12) as A(ij)kl:=(c(i)kc(j)l-c(i)lc(j)k)/2 making arbitrary linear combinations of these matrices according to their upper pair of indices (i,j) since the linear combination of symmetric and skew symmetric matrices remain symmetric and skew symmetric, respectively. If the appropriate blocks of generator G are built up of such linear combinations the Gu=0 restriction automatically and trivially holds.
The next question is for the goal of elaborating continuous parameterization:
how can we calculate the power series of such generators in closed analytical form.
The answer is very easy if the set of n+2 vectors is considered consisting of a and b, and of the columns of the n×n dimensional unit matrix as {a, b, e(i)|i=1,2,…,n}, in which e(i)j=δij. Normally this set is linearly dependent. If we apply the Gram-Schmidt orthonormalization algorithm detailed in Table A.10.1. of Appendix A.10., n linearly independent, orthonormal unit vectors have to be obtained as {c(i)|i=1,2,…,n}, in which c(1) is parallel with a, c(2) is parallel with the component of b that is orthogonal
to a, while the remaining {c(i)|i=3,4,…,n} vectors span the n-2 dimensional subspace orthogonal to both a and b. By putting near each other the columns of the {c(i)|i=1,2,…,n} vectors, due to their orthogonality an orthogonal matrix C is obtained, that satisfying the relationships with the unit matrix I as C=CI, can be interpreted in the following manner:
( ) ( ) ( )
i.e. for obtaining the c(i) vectors the columns of the e(i) vectors have to be rotated by the orthogonal matrix C. Since the orthogonal matrices satisfy the simple relationship CTC=I, the appropriate blocks of G can be obtained from the Ce(i)e(j)TCT matrices. Now consider arbitrary blocks D, E, F, G, K, L, M, N of the dimension n×n, and consider the matrix product below
( ) ( )
from which it follows that e.g. the appropriate blocks of the exponential series of the transformed generators can be obtained from the blocks of the exponential series of the original generators multiplied by the orthogonal matrix C from the left hand side, and by CT from the right hand side. If the original generators are cleverly chosen their exponential series can easily be computed in closed analytical form as it was shown in the equations (6.2.13)-(6.2.17).
Now for the sake of a simple application example apply the above considerations for the case of n=3, in which the components of u as a and b allow only a single dimensional subspace. Since from a single vector no non-zero skew symmetric matrix can be produced, we can produce only the symmetric generators in (6.2.11) by using the unit vector e(3) and C in the form as K=Ce(3)e(3)TCT=CK0(3)
CT=KT, and this symmetric matrix can be placed either into the main diagonals and in the non main diagonals as
The first generator evidently yields the power series as
(
( ))
Regarding the second one, its 2nd and 3rd powers can directly be considered as
( ) ( ) ( ) ( )
therefore even and the odd powers can be separately accumulated as
( ) ( )
It is evident that (6.2.21) and (6.2.24) well and easily programmable, as well as the Gram-Schmidt Algorithm that can be used for a given u to construct C. In the sequel an application example will be given for the control of a robot arm the endpoint of which is connected to a dashpot producing elastic spring forces and viscous damping as external perturbations.
6.2.3. Application Example for the Use of Symplectic Transformations as the