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james1234
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I have an equation (constraint) which I wish to solve explicitly in terms of gd (or more precisely a scaling factor of the vector gd) but I am unsure how to manipulate the equation
σi(G) = |uiHgd|-1
Background: (please feel free to skips this. Much of this inforomation is I'm sure irrelevant):
Looking at the above equation, gd 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)
uiH 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 = UƩVH)
Finally, just to make clear the form of the 'disturbance'. gd 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*Gdunscaled*Dd, where De and Dd are the diagonal matrices used to scale Gdunscaled
the elements of De are known
The singular values of G, (σi(G)), are known
uiH is also known
Hence re-writting the above constraint I therefore have
σi(G) = |uiHDe-1gdDdi|-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.. σi(G) = |uiHDe-1gdDdi|-1
(σi(G)+1) = |uiH||De-1||gd||Ddi||gd|-1|De-1|-1|uiH|-1(σi(G)+1) = |Ddi|=di
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 di.
Cheers!
σi(G) = |uiHgd|-1
Background: (please feel free to skips this. Much of this inforomation is I'm sure irrelevant):
Looking at the above equation, gd 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)
uiH 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 = UƩVH)
Finally, just to make clear the form of the 'disturbance'. gd 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*Gdunscaled*Dd, where De and Dd are the diagonal matrices used to scale Gdunscaled
the elements of De are known
The singular values of G, (σi(G)), are known
uiH is also known
Hence re-writting the above constraint I therefore have
σi(G) = |uiHDe-1gdDdi|-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.. σi(G) = |uiHDe-1gdDdi|-1
(σi(G)+1) = |uiH||De-1||gd||Ddi||gd|-1|De-1|-1|uiH|-1(σi(G)+1) = |Ddi|=di
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 di. Cheers!
Last edited by a moderator:
Thanks to all who took a look.