Obtaining [V(z)] from the adaptive process of Fig. 10.16 and obtaining [Ĉ(z)] from the algebraic process of Eq. (10.26) gives the required MIMO controller. Although this technique may seem to be a bit roundabout, it does work!

In order to obtain [Ĉ_k(z)] with Eq. (10.26), an inverse of [M(z)] is needed. Since this inverse is to become a factor of [Ĉ_k(z)], it is necessary that [M(z)]^-1 be stable. Since [M(z)]is chosen by the system designer, it should be chosen so that [M(z)]^-1 is stable.

This restriction can be relaxed if a sufficiently delayed stable inverse of [M(z)]is used.¹ Referring to Eq. (10.26), the delay incorporated in [M(z)]^-1 could be compensated for by using a correspondingly time advanced form of [M(z)]. This can be readily obtained if the reference model [M(z)] has sufficient transport delay in all of its impulse responses. Without sufficient transport delay in [M(z)], one could simply redefine [M(z)] to have this delay. The end result would be an overall system response that would be like a delayed form of the original [M(z)].

Analysis of the performance of adaptive inverse control for MIMO systems proceeds in like manner to that for SISO systems as explained in Chapter 6. The overall system error consists of a sum of four components:

i.  Plant output disturbance

ii.  Dither noise filtered through the plant

iii.  System error due to truncation of [(z)] and/or [Ĉ(z)]

iv.  Dynamic system error

Calculation of noise at the plant output due to plant disturbance and dither noise is straightforward. The effects of system error due to truncation of [(z)] or [Ĉ(z)] can be made small by making the involved adaptive filters sufficiently long. We assume that truncation effects are negligible. The dynamic system error is due to the effects of noise in the weights of [Ĉ(z)], originating from noise in the weights of [(z)].

This component of error can be calculated in an analogous fashion to the SISO case given by Eqs. (5.26)–(5.41) and Eqs. (6.6)–(6.11). From Eq. (10.30), one can write

It is required that [(z)]^-1 be stable. For this to be, it may be necessary to make [(z)]^-1 a delayed inverse. Continuing,

An error vector can be defined as follows

where I(z) is the z-transform of the command input vector Combining (10.32) with (10.33), we obtain

Equation (10.34) is completely analogous to the SISO Eq. (5.37). Figure 5.15 shows, for the SISO case, how the dynamic system error forms as the command input signal propagates through the system and encounters the weight noise ΔP(z). For the MIMO case, Fig. 10.17 shows the formation of the dynamic system error.

Figure 10.17 MIMO dynamic system error due to fluctuation in [(z)].

The variance of all components of the dynamic system error vector can be computed in an analogous way with a method like the SISO calculation. A simple formula analogous to the SISO formula in Eq. (5.40) can be obtained if special conditions exist, such as

a.  The power levels on all channels at point B in Fig. 10.17 are the same. This is equivalent to the requirement that the power levels on all output channels of the controller be equal.

b.  All filters of [(z)] have the same number of weights n. All filters of [Ĉ(z)] have the same number of weights n.

c.  The values of µ for the adaptive filters of [(z)] are equal.

d.  The plant disturbance power levels are the same on all channels.

At the plant output, all channels have the same dynamic system error variance. For a single plant output channel,

Using scheme C to obtain [(z)], optimal dither levels can be determined for the special case described above. Referring to the SISO derivations of Chapter 6, the overall system error power (including plant output disturbance, dither output noise, and dynamic system error) on a single-output channel of the MIMO system can be expressed as

This relation is analogous to Eq. (6.6), and all definitions made in connections with (6.6) are relevant here. Other special assumptions are necessary for (10.36) to apply, and they are

e.  Dither power is equal on all channels.

f.  All the filters of [(z)] have essentially equal sum of squares of the impulses of their impulse responses.

For the special case above, simple relations can be obtained for optimal dither power. With a given choice of time constant τ for adaptation of [(z)] the overall system error power for a single-output channel can be expressed as

This equation is based on the SISO formula of Eq. (6.6), and all definitions made relative to (6.6) apply here. Adaptation of [(z)] is done by scheme C. Differentiating (10.37) with respect to and setting the derivative to zero yields the optimal dither power:

This equation is analogous to the SISO relation of Eq. (6.9).

The minimum overall system error power can be obtained by analogy to Eq. (6.12). Using the optimal dither with scheme C, the result for a single-output channel is

¹ It can be shown that the delayed inverse of any [M(z)] will be stable, with enough delay.