# 21.7: Rank-One <mu>

Although we do not have methods for computing $$\mu$$ exactly, there is one particular situation where this is possible. This situation occurs if $$M$$ has rank 1, i.e

$M=a b^{*}\nonumber$

where $$a, b \in \mathbb{C}^{n}$$. Then it follows that $$\mu$$ with respect to $$\Delta_{0}$$ containing complex diagonal perturbations is given by

$\frac{1}{\mu(M)}=\inf _{\Delta \in \Delta_{0}}\left\{\sigma_{\max }(\Delta) \mid \operatorname{det}(I-M \Delta)=0\right\}\nonumber$

However,

\begin{aligned} \operatorname{det}(I-M \Delta) &=\operatorname{det}\left(I-a b^{*} \Delta\right) \\ &=\operatorname{det}\left(I-b^{*} \Delta a\right) \end{aligned}\nonumber

$\begin{array}{l} =\operatorname{det}\left(I-\left[\Delta_{1} \cdots \Delta_{n}\right]\left[\begin{array}{c} \overline{b_{1}} a_{1} \\ \overline{b_{2}} a_{2} \\ \vdots \\ b_{n} a_{n} \end{array}\right]\right) \\ = 1-\left[\Delta_{1} \cdots \Delta_{n}\right]\left[\begin{array}{c} b_{1} a_{1} \\ b_{2} a_{2} \\ \vdots \\ b_{n} a_{n} \end{array}\right] \end{array}\nonumber$

and $$\sigma_{\max }(\Delta)=\max _{i}\left|\Delta_{i}\right|$$. Hence,

$\frac{1}{\mu(M)}=\inf _{\Delta_{1}, \ldots, \Delta_{n}}\left\{\max _{i}\left|\Delta_{i}\right|\left[\left[\Delta_{1} \cdots \Delta_{n}\right]\left[\begin{array}{c} \overline{b_{1}} a_{1} \\ b_{2} a_{2} \\ \vdots \\ b_{n} a_{n} \end{array}\right]=1\right\}\right.\nonumber$

Optimizing the RHS, it follows that (verify)

$\frac{1}{\mu(M)}=\frac{1}{\sum_{i=1}^{n}\left|\overline{b_{i}} a_{i}\right|} \leftrightarrow \mu(M)=\sum_{i=1}^{n}\left|\overline{b_{i}} a_{i}\right|\nonumber$

Notice that the SISO robust disturbance rejection problem is a rank-one problem. This follows since

$M=\left[\begin{array}{c} -W_{1} K \\ W_{2} \end{array}\right]\left[\frac{P_{0}}{1+P_{0} K} \quad \frac{1}{1+P_{0} K}\right]\nonumber$

Then

$\mu(M(j \omega))=\left|\frac{W_{1} P_{0} K}{1+P_{0} K}(j \omega)\right|+\left|\frac{W_{2}}{1+P_{0} K}(j \omega)\right|\nonumber$

which is the condition we derived before.

#### Coprime Factor Perturbations

Consider the class of SISO systems

$\Omega=\left\{\frac{N(s)}{D(s)} \mid N=N_{0}+\Delta_{1} W_{1}, D=D_{0}+\Delta_{2} W_{2},\left\|\Delta_{i}\right\|<1\right\}\nonumber$

where the nominal plant is $$N_{0}/D_{0}$$ with the property that both $$N_{0}$$ and $$D_{0}$$ are stable with no common zeros in the RHP. Assume that $$K$$ stabilizes $$N_{0}/D_{0}$$. This block diagram is shown in Figure 21.7.

Figure 21.7: Coprime Factor Perturbation Model

The closed loop block diagram can be mapped to the $$M- \Delta$$ diagram where

\begin{aligned} M &=\left[\begin{array}{cc} -\frac{W_{1} K}{D_{0}+N_{0} K} & -\frac{W_{1} K}{D_{0}+N_{0} K} \\ \frac{W_{2}}{D_{0}+N_{0} K} & \frac{W_{2}}{D_{0}+N_{0} K} \end{array}\right] \\ &=\left[\begin{array}{c} -\frac{W_{1} K}{D_{0}+N_{0} K} \\ \frac{W_{2}}{D_{0}+N_{0} K} \end{array}\right] [1 \ 1] \end{aligned}\nonumber

Hence, $$M$$ has rank 1 and

$\mu(M(j \omega))=\left|\frac{W_{1} K}{D_{0}+N_{0} K}\right|+\left|\frac{W_{2}}{D_{0}+N_{0} K}\right|\nonumber$

#### Robust Hurwitz Stability of Polynomials with Complex Perturbations

Another application of the structured singular value with rank one matrices is the robust stability of a family of polynomials with complex perturbations of the coeffcients. In this case let $$\delta =\left[\begin{array}{cccc} \delta_{n-1} & \delta_{n-2} & \ldots & \delta_{0} \end{array}\right]^{T}$$ and consider the polynomial family

$P(s, \delta)=s^{n}+\left(a_{n-1}+\gamma_{n-1} \delta_{n-1}\right) s^{n-1}+\ldots+\left(a_{0}+\gamma_{0} \delta_{0}\right)\nonumber$

where $$a_{i}$$, $$\gamma_{i}$$, and $$\delta_{i} \in \mathbb{C}$$ and $$\left|\delta_{i}\right| \leq 1$$. We want to obtain a condition that is both necessary and sufficient for the Hurwitz stability of the entire family of polynomials $$P(s, \delta)$$. We can write the polynomials in this family as

$P(s, \delta)=P(s, 0)+\tilde{P}(s, \delta)\label{21.20}$

$=\left(s^{n}+a_{n-1} s^{n-1}+\ldots+a_{0}\right)+\left(\gamma_{n-1} \delta_{n-1} s^{n-1}+\ldots+\gamma_{0} \delta_{0}\right)\label{21.21}\) which can also be rewritten as \[P(s, \delta)=P(s, 0)+\left[\begin{array}{ccc} 1 & 1 & \ldots 1 \end{array}\right]\left[\begin{array}{ccccc} \delta_{n-1} & 0 & 0 & \ldots & 0 \\ 0 & \delta_{n-2} & 0 & \ldots & 0 \\ \vdots & & \ddots & & \vdots \\ & & & \delta_{1} & 0 \\ 0 & 0 & \ldots & 0 & \delta_{0} \end{array}\right]\left[\begin{array}{c} \gamma_{n-1} s^{n-1} \\ \gamma_{n-2} s^{n-2} \\ \vdots \\ \gamma_{1} s \\ \gamma_{0} \end{array}\right]\nonumber$

We assume that the center polynomial $$P(s, 0)$$ is Hurwitz stable. This implies that the stability of the entire family $$P(s, \delta)$$ is equivalent to the condition that

$1+\frac{1}{P(j \omega, 0)}\left[\begin{array}{ccc} 1 & 1 & \ldots 1 \end{array}\right]\left[\begin{array}{ccccc} \delta_{n-1} & 0 & 0 & \ldots & 0 \\ 0 & \delta_{n-2} & 0 & \ldots & 0 \\ \vdots & & \ddots & & \vdots \\ & & & \delta_{1} & 0 \\ 0 & 0 & \ldots & 0 & \delta_{0} \end{array}\right]\left[\begin{array}{c} \gamma_{n-1}(j \omega)^{n-1} \\ \gamma_{n-2}(j \omega)^{n-2} \\ \vdots \\ \gamma_{1}(j \omega) \\ \gamma_{0} \end{array}\right] \neq 0\nonumber$

for all $$\omega \in \mathbb{R}$$ and $$\left|\delta_{i}\right| \leq 1$$. This is equivalent to the condition that

$\operatorname{det}\left(I+\frac{1}{P(j \omega, 0)}\left[\begin{array}{c} \gamma_{n-1}(j \omega)^{n-1} \\ \gamma_{n-2}(j \omega)^{n-2} \\ \vdots \\ \gamma_{1}(j \omega) \\ \gamma_{0} \end{array}\right]\left[\begin{array}{lll} 1 & 1 & \ldots 1 \end{array}\right] \Delta\right) \neq 0\nonumber$

for all $$\omega \in \mathbb{R}$$ and $$\Delta \in \Delta$$ with $$\|\Delta\|_{\infty} \leq 1$$. Now using the concept of the structured singular value we arrive at the following condition which is both necessary and sufficient for the Hurwitz stability of the entire family

$\mu(M(j \omega))<1\nonumber$

for all $$\omega \in \mathbb{R}$$, where

$M(j \omega)=\frac{1}{P(j \omega, 0)}\left[\begin{array}{c} \gamma_{n-1}(j \omega)^{n-1} \\ \gamma_{n-2}(j \omega)^{n-2} \\ \vdots \\ \gamma_{1}(j \omega) \\ \gamma_{0} \end{array}\right]\left[\begin{array}{lll} 1 & 1 & \ldots 1 \end{array}\right]\nonumber$

Clearly this is a rank one matrix and by our previous discussion the structured singular value can be computed analytically resulting in the following test

$\frac{1}{|P(j \omega, 0)|} \sum_{i=1}^{n}\left|\gamma_{n-i}\right||\omega|^{n-i}<1\nonumber$

for all $$\omega \in \mathbb{R}$$.