88 U. Musella et al. Following the theory of [15], two different approaches can be followed to find the optimal convergence coefficient, both leading to the same result. The first approach is quite simple and, for discrete system, reduces to guarantee that the poles of the Closed Loop system (the roots of the so-calledReturn Difference Matrix) need all to lie inside the unit circle. The second approach, discussed here, makes use of a set of a transformed set coordinates and the tools of linear algebra. Decomposing the FRF matrix of the plant with the Singular Values Decomposition (SVD) [20] H=R QH (13.13) where is a diagonal matrix containing the Singular Values (SVs) of Hand Rand Qare two unitary matrices. The control error at the nth iteration can be transformed pre-multiplying the left and right hand sides of Eq. (13.3) byRH ξn =RHe n =RHr − QHu n (13.14) Defining • transformed error, ξn =RHe n • transformed plant, • transformed reference, ρ =RHr • transformed drives, νn =QHu n Equation (13.3) can be re-written as ξn =ρ − νn (13.15) From Eq. (13.15) it is possible to see the advantage of moving to a transformed coordinate space: the diagonal nature of de-couples the control problem. For this reason, the singular values define a basis for the MIMO system (principal coordinates). This means that • each of the transformed error signals is affected by only one drive or none. In the principal coordinates the first msignals can therefore be independently controlled. • For each channel, the specific SV represents the optimal step size for the transformed drive. For this reason [15] also refers to the SVs as Principal Gains. In the principal coordinates, the ith independent (transformed) drive update equation can be written as νi,n+1 =νi,n +ασiξi,n (13.16) Following by the definitions of the transformed quantities and the fact that the matrices Rand Qare unitary, it is worth to notice that the cost function is the same in both the physical and the transformed space J = eHe = ξHξ, hence the convergence of the process can be analyzed in the principal coordinates. Specifically for the BIBO stability it needs to be guaranteed that |νn −νopt | |ν0 −νopt | = | 1−ασ 2 i | <1 (13.17) and hence that 0 <αi < 2 σ 2 i (13.18) Naturally, to guarantee that all the drives will eventually converge, it is necessary to limit the convergence coefficient with respect to the maximum singular value, obtaining the optimal convergence coefficient αopt = 1 σ2 max (13.19)
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