6. Special applications
6.2 Non-linear distortion in cascaded MIMO systems
( ) ( ) ( ) , , ( )
( ) ( ) ( ) ( )
( 1 3 1 111 1 2 1 2
1 2
k M
M k
M
M k
k U k U k U L
L k k H l
U l H l Y l Y l
Y
∑ ∑
−
= =−
+
= +
= (6.1.2)
Multiplication of three or more cubic terms means contribution of order higher than ε2and such terms will be omitted from further consideration. We have:
Theorem 6.1.1: The Best Linear Approximation FRF of the cascade of two cubic Wiener-Hammerstein systems is:
1 1
( ) { ( ) ( )} (1 1 ) ( ) ( )
BLA ( )
Q l E Z l U l K C G l H l
= = +L l (6.1.3)
where L(l)=S1(l)R2(l). (6.1.4)
For small levels of ε1=a3 a1, ε2=b3 b1, K ≈1+O(ε), C1≈O(ε2), (ε = max(ε1, ε2)), and (6.1.3) yields: QBLA( )l ≈GBLA( )l HBLA( )l ≅G l H l1( ) 1( ), (6.1.5) i.e. the usual product expression for the cascaded systems holds also in the case of weak non-linear systems in those frequency bands, where the coherence function is high (see Fig. 6.1.2).
Observe however a small bias K due to the presence of non-linearity (ε1, ε2 = 0.1). In those frequency bands using the best linear approximation is a sound modeling strategy, which provides the proper view of the dynamics of the cascade.
Proof: In Appendix A.6
Fig. 6.1.2 Cascading weakly non-linear systems: the FRF of the cascade of linear components, i.e. R1 S1 R2 S2 (solid black), the product GBLA HBLA (gray o), and the QBLA averaged from N=10 (noisy solid black). In the frequency bands, where the coherence functions of the cascade components are high (see Figs. 2.5.1-2.5.2), the linear approximation is sufficient.
6.2 Non-linear distortion in cascaded MIMO systems
The simplified SISO cascade problem will be now investigated in the fully blown MIMO setting. Cascading is an elementary way to build complex systems and to model and solve practically important questions. Here we name only two:
(1) The excitation signals are applied through non-ideal actuators and/or are distorted by the non-linear loads. Should we accept the measurement results as they are or should we redesign the excitations to counter the effects of the actuators and the loads? [117-118]
(2) Linear FRF approximation to a non-linear system is theoretically valid only for the particular choice of input signals used during the measurements. This limits the usability of the linear approximations, because in different applications the inputs will generally be different. If however the new inputs do not differ much, the original linear approximation should hold, or shouldn’t it? (see also Sect 3.6 for this problem)
These questions will be modeled as follows. What is the deterioration of the measurement quality, if the ideal excitation signal is distorted by passing through a non-linear system?
The measurement set-up is presented in Fig. 6.2.1. Ideally the designed reference signal R should be applied directly to the input of the system V. The Best Linear Approximation FRF
] ˆ [
, m k r r= G
G of this system can be estimated from the Y - R measurements. We call this FRF the reference estimate. (Note that in the kernels G...,mkthe upper index, or indices, refer to the input signals and the multiple equals the order of the kernel, the lower index refers to the output signal and will be left out, if unambiguous).
In practice the reference signal undergoes distortions before reaching the excited input. Thus the FRF characteristics ˆ [ˆm]
Gk
=
G are estimated from the Y - U data, where U is the output of the system U modeling the distortions. We assume that the distorted signals U can be directly measured. The question now is when is the estimated Gˆ a fair approximation to the Best Linear Approximation Gˆr? The answer is not trivial because Gˆ depends upon the actual excitation!
To tackle the problem we assume that both cascaded systems are MIMO Volterra systems limited to be at most 3rd order, to enumerate the distorting effects. For the input signals random phase multisines (2.2.17) and the orthogonal multisines will be used but results are valid also for other random excitations [15*, 16*, 21* 163].
Fig. 6.2.1 The ideal and the real measurement set-up.
Fig. 6.2.2 Channel distortions for U1, for N
= 2. Similar scheme is valid for signal U2.
The distorting system M is the sum of an ideal system L0 (phase distortion only), a linear distortion system E (cross channel distortions) and a non-linear distortion system N, Fig.
6.2.2.
Generally the linear distortion between the reference inputs R and the distorted inputs U will be modeled as:
where N is the input dimension of the cascade and l is a discrete frequency.
The ideal system L0 can contain deterministic phase shifts. It will not influence the results because the random phases of the chosen excitation signals are random and uniformly distributed on the unit circle. The “ideal” channel is then:
reference R and actual input U:
[
( )]
[ ( )] (without linear terms, which are accounted for by E):∑
In the frequency domain the model is:
∑
frequency lines. E.g. for N = 2, the model of a particular distorted excitation Uk contains Nk11, Nk12, and Nk22 2nd order kernels, and Nk111, Nk112, Nk122, and Nk222 3rd order kernels. The model is both general and simple. Effects of even and odd order non-linearities can be analyzed and it can be extended to higher order models, if required. We will also assume that the kernels can be written as:The measured system is a MIMO Volterra system of at most 3rd order (with linear terms):
∑
In the frequency domain the system model is:
∑
frequency lines. The kernels Gk…are assumed to be of O(1) order.
The main result can be stated then as:
Theorem 6.2.1: If the cascaded MIMO systems in Fig. 6.2.1-6.2.2 are weak non-linear systems in the sense that:
max km 1, max k 1, max( , ) 1.
ε= ε << δ = δ << ξ = ε δ << (6.2.14) then the 1st order perturbation (in the introduced distortions) of the measured Best Linear Approximation of the system Y=V[U] is:
1 1 2
BLA LIN BIAS BLA
BLA BIAS BLA BLA
O
[ m , ] (1)
G , due to the superposition of the linear distortions in the inputs;
1
0 ( )
BIAS BLA − =O δ
G N L , where NBLA is the Best Linear Approximation of the non-linear part of the distorting system, is due to the superposition of the non-linear distortions in the inputs;
1
( 0) ( )
BIAS − BLA − =O ξ
G I M L , where MBLA=L0+ +E NBLA;
1=O( )ε
H comes from the distortion of the non-linear kernels in the measured system caused by the linear mixing of the excitation channels;
2 =O( )δ
H is caused by the interaction of the non-linear kernels between the two (measured and distorting) systems.
Proof: In Appendix A.7
The analysis of (6.2.15) shows that weak distortions do not cumulate and that the degradation in the measured FRF can be accounted for by 1st order distortions, if the overall distortion level is low. Furthermore, the effects of feed-forward and cross-channel distortions can be separated under such assumptions.
A weakly non-linear system does not make this much damage to the random multisines, and that they are still suitable to gain insight into the behavior of the measured system. But what about the orthogonal multisines?
If RN(l) is an orthogonal random multisine:
with wkm entries of an orthogonal or unitary matrix:
N linear distortions:
)
Taking into account (6.2.3) and retaining only the first order terms, we have:
)}
On the other hand when the non-linear distortions are also present in the system U[R]:
) ](
[ ) ( )]
( ) ( [ )
(l 0 l l N l N l
N L E R N R
U = + + , (6.2.21)
then retaining only the first order expressions:
)}
](
[ ) ( ) ( Re{
2 )}
( ) ( Re{
2 )
( )
(l HN l N N N l H0 l 0 l N l H N l
N U I E L L R N R
U = + + (6.2.22)
the last term brings in also the randomness (when the frequencies in the non-linear kernels and the frequency l are not paired [30*]), which means that the variance of the FRF estimate increases comparing to the ideal orthogonal case.
Example 6.2.1: (for the full experiment see [19*]) For the illustration 2-dim Wiener-Hammerstein systems were driven by unit power orthogonal multisines. Such systems permit an easy manipulation of the nonlinearities (contained between input and output dynamics). Furthermore the BLA to a Wiener-Hammerstein system is proportional to its linear dynamics (if M >> 1), which means that the expected influence of the distortion will mainly change the level of GBLA and less in its frequency behavior. The linear dynamics of the Wiener-Hammerstein systems are shown in Figs. 6.2.3-6.2.5, where every linear dynamics was normalized to the unit
||.||∞ norm. The static non-linearity within systems N1 and N2 is:
u = δ2 [r12 + r22 + r1r2] + δ3 [r13 + r23 + r1r22 + r12r2], (6.2.23) and that in the system V is:
y = u1 +u2 + α2 [u12 + u22 + u1u2] + α3 [u13 + u23 + u1u22 + u12u2], (6.2.24) with suitably adjusted coefficients. The measurements are made on the output channel Y1, thus the output index k
= 1 is dropped. The comparison is based on the following measures of distortion, calculated in the pass band:
1. The αk* gain needed to scale up the measured FRF Gˆ to match (in the LSE sense) the theoretical value of GBLA (i.e. the FRF measured without any distortions):
2 1
ˆ 2
arg min F | k( ) k ( ) |
k l F BLA
a
G l a G l
α∗=
∑
= − (6.2.25)The actual measure shown is the difference ∆k = |1 - αk*| in dB. The ideal gain is 1, consequently ∆k grows steadily with the increasing distortion levels.
2. The rms value of the residual characteristic ˆ
k BLA
α∗
−
G G :
2
2 1 1
1 F | ˆk( ) k ( ) |2
k F F l F k BLA
RMS = −
∑
= G l −α∗G l (6.2.26)3. The phase error of the measured Gˆ with respect to that of the ideal GBLA: (a) its maximum value in the frequency band, and (b) the rms value of the phase residual.
In the first test only e11, e22 were set and the system V did not contain the 2nd order terms (Test A), then 2nd order terms have been added to V (Test B), then cross distortions were added to the system E (Test C), and finally non-linear distortions N were added (Test D) to the full system M, with non-linear and nonnon-linear distortions kept at the same level. The figures show results for a small (10) and a large number of averages (1000). For the x-axis in figures, the SNR of the overall distortion level was used, defined as:
2 1
2 1 11 0 1
| ) (
|
| ) ( ] [ ) (
|
∑
∑
−=
l l
l R
l R l
Dist U L
(6.2.27) The figures show the expected slow increase in the distortion measures as the level of (13) increases. Robustness is attained until the distortion level becomes large (Fig. 6.2.7-6.2.9). Nonlinear distortions produce larger errors than those caused by linear distortions of similar magnitude (Fig. 6.2.7, Fig. 6.2.9). Similarly cross channel
distortions yield larger errors than feed-forward distortions. The phase errors grow faster comparing to the averaged amplitude errors. It means that the FRF becomes locally “twisted” rather, than globally distorted over larger frequency range. These effects are better visible in the results averaged from a larger number of measurements. For a low number of averages the effects are screened partly by the variance of the nonlinear noise on the FRF estimates.
In conclusion, the best linear approximation measurements are robust under perturbed excitations modeled by weakly nonlinear MIMO Volterra system (weak nonlinear amplitude and phase disturbances). Considered that finite order Volterra systems are smooth, these results are not so unexpected. Intuitively drastic changes for low level distortions could be difficult to explain. Figures show that distortions with respect to the reference measurements grow steadily but smoothly, even for the level of distortions for which the 1st order approximation cannot be justified. Consequently the 1st order approximation, for small distortions, yields a legitimate view of the system behavior. The presented results extend the single-input single-output results obtained earlier [10*].
Finally the 1st order approximation yields tools to investigate the influence of distortions positioned in particular places in the MIMO structure.
Fig. 6.2.3 Dynamics of the system U1=N1 [R1,R2]. Fig. 6.2.4 Dynamics of the system U2=N2 [R1,R2].
Fig. 6.2.5 Dynamics of the system Y=V [U1,U2]. Fig. 6.2.6 Scaling gain |1 - αk*|.
Fig.
6.2.7. The rms value of the FRF residual.
Fig. 6.2.8. The maximum value of the phase residual.
Fig. 6.2.9. The rms value of the FRF phase residual