18.4: Analytic Constraints- The "Waterbed" Effect

One performance limitation of LTI SISO Feedback systems (these systems have rational sensitivity transfer functions), is known as the waterbed effect. Loosely speaking, when one designs a controller to "push" the sensitivity function in a particular direction, another part of the sensitivity function necessarily "pulls" back in the opposite direction. This effect is due to a property of analytic functions \(f (s)\) as stated by Cauchy's theorem. In words, this theorem says that the line integral of an analytic function around any simple closed contour \(C\) in a region R is zero, i.e.,

\[\int_{C} f(s) d s=0\nonumber\]

for every contour \(C\) in R.

A proof of this theorem will not be shown here but can be found in standard complex analysis textbooks. One consequence of this theorem is the following integral constraint (known as Bode's Integral) on the rational sensitivity transfer function \(S(jw)\):

\[\int_{0}^{\infty} \ln |S(j w \mid d w=\sum_{i} \pi \operatorname{Re}\left(p_{i}\right).\nonumber\]

where \(\sum_{i} \pi \operatorname{Re}\left(p_{i}\right)\) is the sum over the unstable open-loop poles (poles of \(P (jw)K(jw)\)). This result holds for all closed-loop systems as long as the product \(P K\) has relative degree two. The result implies that making \(S(jw)\) small at almost all frequencies (a common performance objective) is impossible since the integrated value of \(\ln |S(j w)|\) over all frequencies must be constant. This constant is zero for open-loop stable systems (\(P K\) stable) and positive otherwise. Therefore, lowering the sensitivity function in one range of frequencies, increases the same function in another range-hence the name "waterbed effect." Figure 18.5 below illustrates this phenomenon.

Constraints on Singular Value Plots

From what we have seen already, it is clear that singular value plots over all frequencies are the MIMO system analogs of Bode plots. The following fact establishes some simple bounds involving singular values of \(S\) and \(T\) :

Fact 18.5.1

If \(S=(I+P K)^{-1}\) and \(T=(I+P K)^{-1}P K\) then the following hold

\[\left|1-\sigma_{\max }(S)\right| \leq \sigma_{\max }(T) \leq 1+\sigma_{\max }\tag{S}\]

and

\[\left|1-\sigma_{\max }(T)\right| \leq \sigma_{\max }(S) \leq 1+\sigma_{\max }\tag{T}\]

Proof

Since \(S + T = I\) then clearly

\[\sigma_{\max }(T)=\sigma_{\max }(I-S) \leq \sigma_{\max }(I)+\sigma_{\max }\tag{S}\]

and therefore \(\sigma_{\max }(T) \leq 1+\sigma_{\max }(S)\nonumber\). For any element \(x \in \mathbb{C}^{n}\) with \(\|x\|_{2}=1\) we have

\[\begin{aligned}
x-S x &=T x \\
\left|\|x\|_{2}-\|S x\|_{2}\right| & \leq\|x-S x\|_{2}=\|T x\|_{2} \\
\left|1-\|S x\|_{2}\right| & \leq \sigma_{\max }(T) \\
\left|1-\sigma_{\max }(S)\right| & \leq \sigma_{\max }(T)
\end{aligned}\nonumber\]

Combining this relation with \(\sigma_{\max }(T) \leq 1+\sigma_{\max }(S)\nonumber\), we obtain

\[\left|1-\sigma_{\max }(S)\right| \leq \sigma_{\max }(T) \leq 1+\sigma_{\max }\tag{S}\]

The other relation follows in exactly the same manner.