Question 26.6: Consider the following MIMO system Go(s) = [1-s/(s+1)² s+3/(...
Consider the following MIMO system
G_{o}(s)=\begin{bmatrix} \frac{1-s}{(s+1)^{2}} & \frac{s+3}{(s+1)(s+2)} \\ \frac{1-s}{(s+1)(s+2)} & \frac{s+4}{(s+2)^{2}} \end{bmatrix} =G_{oN}(s)[G_{oD}(s)]^{-1}I (26.7.1)
where
G_{oN}(s)=\begin{bmatrix} (1-s)(s+2)^{2} & (s+1)(s+2)(s+3) \\ (1-s)(s+2)(s+3) & (s+1)^{2}(s+4) \end{bmatrix} (26.7.2)
G_{oD}(s)=(s+1)^{2}(s+2)^{2} (26.7.3)
(i) Determine the location of RHP zeros and their directions.
(ii) Evaluate the integral constraints on sensitivity that apply without enforcing dynamic decoupling and obtain bounds on the sensitivity peak.
(iii) Evaluate the integral constraints on sensitivity that apply if dynamic decoupling is required and obtain bounds on the sensitivity peak.
(iv) Compare the bounds obtained in parts (ii) and (iii).
(i) The zeros of the plant are the roots of det(G_{oN}(s)), i.e. the roots of −s^{6} − 11s^{5} − 43s^{4} − 63s^{3} + 74s + 44. Only one of these roots, that located at s = 1, lies in the RHP. Thus z_{o} = 1, and
d\Omega (z_{o},\omega )=\frac{1}{1+\omega ^{2}}d (26.7.4)
We then compute G_{o}(1) as
G_{o}(1)=\begin{bmatrix} 0 & \frac{2}{3} \\ 0 & \frac{5}{9} \end{bmatrix} (26.7.5)
from where it can be seen that the dimension of the null space is µ_{z} = 1 and the (only) associated (left) direction is h^{T} = [5 − 6]. Clearly this vector has two nonzero elements, so we could expect that there will be additional design trade-offs arising from decoupling.
(ii) Applying Theorem 24.2 on page 759 part (ii) for r = 1 and r = 2 we obtain respectively
\frac{1}{\pi }\int_{\infty }^{\infty }{\ln \left|5[S_{o}(j\omega )]_{11}-6[S_{o}(j\omega )]_{21}\right| } \frac{1}{1+\omega ^{2}}d\omega \geq \ln \left|5\right| (26.7.6)
\frac{1}{\pi }\int_{\infty }^{\infty }{\ln \left|5[S_{o}(j\omega )]_{12}-6[S_{o}(j\omega )]_{22}\right| } \frac{1}{1+\omega ^{2}}d\omega \geq \ln \left|6\right| (26.7.7)
If we impose design requirements, as in Lemma 24.4 on page 760, we have, for the interacting MIMO loop, that
\left\|[S_{o}]_{11}\right\|_{\infty }+\frac{6}{5}\left\|[S_{o}]_{21}\right\|_{\infty } \geq (\frac{1}{\epsilon _{11}+\frac{6}{5} \epsilon _{21} } ) ^{\frac{\psi (\omega c)}{\pi -\psi (\omega c)} } (26.7.8)
\left\|[S_{o}]_{22}\right\|_{\infty }+\frac{6}{5}\left\|[S_{o}]_{12}\right\|_{\infty } \geq (\frac{1}{\epsilon _{22}+\frac{6}{5}\epsilon _{21} } ) ^{\frac{\psi (\omega c)}{\pi -\psi (\omega c)} } (26.7.9)
(iii) If we require dynamic decoupling, expressions (26.7.6) and (26.7.7) simplify, respectively to
\frac{1}{\pi } \int_{\infty }^{\infty }{\ln \left|[S_{o}(j\omega )]_{11}\right|\frac{1}{1+\omega ^{2}}d\omega } \geq 0 (26.7.10)
\frac{1}{\pi } \int_{\infty }^{\infty }{\ln \left|[S_{o}(j\omega )]_{22}\right|\frac{1}{1+\omega ^{2}}d\omega } \geq 0 (26.7.11)
And, with dynamic decoupling, (26.7.8) and (26.7.9), simplify to
\left\|[S_{o}]_{11}\right\|_{\infty } \geq (\frac{1}{\epsilon _{11}} )^{\frac{(\omega c)}{\pi -\psi (\omega c)} } (26.7.12)
\left\|[S_{o}]_{22}\right\|_{\infty } \geq (\frac{1}{\epsilon _{22}} )^{\frac{(\omega c)}{\pi -\psi (\omega c)} } (26.7.13)
(iv) To quantify the relationship between the magnitude of the bounds, in the coupled and the decoupled situations, we use an indicator κ_{1d}, formed as the quotient between the right hand sides of inequalities (26.7.8) and (26.7.12) , i.e.
κ_{1d}\overset{\triangle }{=} \left(1+\frac{6 }{5}\lambda _{1\epsilon } \right)^{-\frac{\psi (\omega c)}{\pi -\psi (\omega c)} } where \lambda _{1\varepsilon }\overset{\triangle }{=} \frac{\epsilon _{21}}{\epsilon_{11}} (26.7.14)
Thus, \lambda _{1\epsilon } is a relative measure of interaction in the direction from channel 1 to channel 2.
The issues discussed above are captured in graphical form in Figure 26.10.
In Figure 26.10 we show a family of curves, each corresponding to a different bandwidth ω_{c}. Each curve represents, for the specified bandwidth, the ratio between the bounds for the sensitivity peaks, as a function of the decoupling indicator, \lambda _{1\epsilon}. We can summarize our main observations, as follows
a) When \lambda _{1\epsilon} is very small, there is virtually no effect of channel 1 into channel 2 (at least in the frequency band [0, ω_{c}]), then the bounds are very close (κ_{1d} ≈ 1).
b) As \lambda _{1\epsilon} increases, we are allowing the off diagonal sensitivity to become larger than the diagonal sensitivity in [0, ω_{c}]. The effect of this manifests itself in κ_{1d} < 1, i.e. in bounds for the sensitivity peak which are smaller than for the decoupled situation.
c) If we keep \lambda _{1\epsilon} fixed, and we increase the bandwidth, then the advantages of using a coupled system also grow.
Also note that the left hand sides of (26.7.8) and (26.7.12) are different. In particular, (26.7.8) can be written as
\left\|[S_{o}]_{11}\right\|_{\infty } \geq \left(\frac{1}{\epsilon _{11}+\frac{6}{5}\epsilon _{21} } \right)^{\frac{\psi (\omega c)}{\pi -\psi (\omega c)} } -\frac{6}{5} \left\|[S_{o}]_{21}\right\|_{\infty }\geq \left(\frac{1}{\epsilon _{11}+\frac{6}{5}\epsilon_{21} } \right)^{\frac{\psi (\omega c)}{\pi -\psi (\omega c)} }-\frac{6}{5}\epsilon _{21} (26.7.15)
