10.2.1 Adaptive MIMO Modeling Using Scheme B

The plant can be adaptively modeled, using independent dither inputs, as shown for the two-input two-output case of Fig. 10.7. Use of uncorrelated dither in the modeling process is necessary when the plant inputs alone cannot be used, that is, when they are correlated with each other or when they are not persistently exciting. The dither scheme illustrated in Fig. 10.7 is a MIMO version of scheme B, which is shown in Fig. 4.3(b) for the SISO case. A more succinct picture of the dither scheme is shown in Fig. 10.8, making use of a vector block diagram.

The modeling process illustrated in Fig. 10.7 involves the adaptation of four separate filters, ₁₁, ₁₂, ₂₁, and ₂₂. As these filters converge, their transfer functions should closely approximate the respective plant transfer functions P₁₁, P₁₂, P₂₁, and P₂₂. Note that a common error signal is used in the adaptation of₁₁ and ₁₂, and that another common error signal is used in the adaptation of ₂₁ and ₂₂. It seems that the adaptation process for these filters should be affected by their interconnections, and a simple analysis shows this is indeed the case. Of concern are issues such as stability of the adaptive process, speed of convergence, misadjustment, and noise in the weights.

Figures 10.9, 10.10, 10.11, and 10.12 have been devised as an aid in addressing these issues. Figure 10.9 shows one-half of the adaptive process illustrated in Fig. 10.7. This half operates independently of the other half and can be studied all by itself. The natural driving signal, the controller output, is not used for plant identification by scheme B. The dither signal serves this purpose. In fact, the natural driving signal acts as noise to the adaptive identification process. Figure 10.10 is equivalent to Fig. 10.9, except that here the interference due to the controller output is accounted for by an additive noise component at the plant output, and it appears in addition to the plant disturbance.

Assume that all of the filters of Fig. 10.10 are FIR and that they all are of length n time samples. Assume that the two dither signals of Fig. 10.10 are white and of equal power. Figure 10.11 shows how a single white dither input could be used under these circumstances instead of the two dither inputs of Fig. 10.10. Two dither signals are created from a single one by using a delay of n time samples. In this way, all of the signals at the taps of the tapped delay lines of ₁₁ and ₁₂ will be mutually uncorrelated in Fig. 10.11 just as they are in Fig. 10.10. The results of adaptation in both systems will therefore be the same.

The system of Fig. 10.11 can be redrawn as shown in Fig. 10.12. The two adaptive filters ₁₁ and ₁₂ are now combined to comprise a single adaptive filter of length 2n time samples. Assume for simplicity that the values of µ are set to be the same for ₁₁and ₁₂.

Figure 10.7 Modeling a two-input two-output system in accord with scheme B.

Figure 10.8 Simplified block diagram for modeling a MIMO plant based on scheme B.

Figure 10.9 A part of the two-channel MIMO modeling process.

Figure 10.10 A part of the two-channel MIMO modeling process, simplified.

The issues of stability, and convergence rate, and misadjustment and noise in the weights are the same for the system of Fig. 10.12 as they are for the system of Fig. 10.9, under the conditions assumed above. Note that the adaptive behavior in Fig. 10.9 is the same as that in Fig. 10.7. So, to understand the adaptive behavior of the system of Fig. 10.7, one needs only to analyze the behavior of the system of Fig. 10.12. This can be done using existing methodology. Formulas for time constant, stable range of µ, and misadjustment are given in Appendix B. Since the length of each of the adaptive filters ₁₁, ₁₂, ₂₁ and ₂₂ in Fig. 10.7 is n, then the length of the equivalent adaptive filter in Fig. 10.12 is twice that. If more channels are involved, and the number of MIMO channels is represented by K, then the equivalent filter length is n multiplied by K.

Using formulas (B.20) for time constant, (B.22) for stable range of µ, and (B.27) for misadjustment, the corresponding formulas for scheme B applied to a multichannel MIMO system are

Figure 10.11 Alternative dither for two-channel MIMO modeling process.

Everything stays the same, except that the equivalent filter is long for MIMO. To keep misadjustment down at a low level, it is necessary to adapt more slowly, roughly by a factor equal to K.