Explicit solution. Properties of vector norms

[james1234]

I have an equation (constraint) which I wish to solve explicitly in terms of g_d (or more precisely a scaling factor of the vector g_d) but I am unsure how to manipulate the equation

σ_i(G) = |u_i^Hg_d|-1

Background: (please feel free to skips this. Much of this inforomation is I'm sure irrelevant):

Looking at the above equation, g_d defines the maximum disturbance acting on a reduced order model of my system (G).

Here σ is a vector of the maximum singular values of the system (for those not familiar with singular values of transfer function matrices one might think of a plot of the singular values as simply the bode plot of a multivariable system which accounts for interaction between the respective channels (off diagonal terms))

G is my scaled system. In this instance G is simply a 4x4 transfer function matrix (4 inputs / 4 outputs)

u_i^H is the hermitian transpose (conjugate transpose) of the i'th column of the matrix U (where U has been obtained from the singular value decomposition G = UƩV^H)

Finally, just to make clear the form of the 'disturbance'. g_d is simply a column of the TF matrix Gd (corresponds to a single disturbance ~ input). I.e. while G is a 4x4 TF matrix which defines the magnitude and phase of the system between the outputs and control inputs of the system. Gd is the magnitude and phase between these same outputs and several disturbance inputs (perturbations to particular states) and has the same dimension as G.

Attempt at a solution:

As the system 'G' has already been scaled (the maximum input and output vector of the TF matrix have a euclidean norm of 1). I would now like to obtain a suitable scaling factor for the disturbance Gd which satisfies the above constraint

What I know ~
The scaled matrix Gd is equal to De^-1*Gd_unscaled*Dd, where De and Dd are the diagonal matrices used to scale Gd_unscaled
the elements of De are known
The singular values of G, (σ_i(G)), are known
u_i<sup>H is also known

Hence re-writting the above constraint I therefore have

σ_i(G) = |u_iHDe-1g_dDd_i|-1

where Dd_i represents the element corresponding to the ith column of Gd and is the only unknown

What I would like to obtain.. σ_i(G) = |u_iHDe-1g_dDd_i|-1

(σ_i(G)+1) = |u_iH||De-1||g_d||Dd_i||g_d|-1|De-1|-1|u_iH|-1(σ_i(G)+1) = |Dd_i|=d_i

Looking at the properties of the vector norm, I gather that the simple manipulation I have performed isn't pheasible and that the solution will not satisfy the initial constraint.
http://www.uAlberta.ca/MATH/gauss/fcm/LinAlg/InRn/DotPrdct/NrmPrprts.htm

If anyone can suggest a (simple! approach for re-arranging the equation (keeping in mind that gd and ui are vectors ~ ui is a vector of complex numbers and gd is a vector of proper rational functions of j*omega) I would be most grateful. I am unaware of any method for manipulating the equation to obtain a simple solution for d_i.

Cheers!</sup>

 

[james1234]

Finally, resolved Thanks to all who took a look.
I wanted to delete the post or mark it as solved to push it further down the 'list'. I don't appear to able to do either..

Cheers!