20.6: The Small-Gain Theorem

Since every term in \(G_{wz}\) other than \((I-M \Delta)^{-1}\) is known to be stable, we shall have stability of \(G_{wz}\), and hence guaranteed stability of the actual closed-loop system, if \((I-M \Delta)^{-1}\) is stable for all allowed \(\Delta\). In what follows, we will arrive at a condition - the small-gain condition - that guarantees the stability of \((I-M \Delta)^{-1}\). It can also be shown (see Appendix) that if this condition is violated, then there is a stable \(\Delta\) with \(\|\Delta\|_{\infty} \leq 1\) such that \((I-M \Delta)^{-1}\) and \(\Delta (I-M \Delta)^{-1}\) are unstable, and \(G_{wz}\) is unstable for some choice of \(z\) and \(w\).

Theorem 20.1 ("Unstructured" Small- Gain Theorem)

Define the set of stable perturbation matrices \(\Delta \triangleq\left\{\Delta \mid\|\Delta\|_{\infty} \leq 1\right\}\). If \(M\) is stable, then \((I-M \Delta)^{-1}\) and \(\Delta (I-M \Delta)^{-1}\) are stable for each \(\Delta\) in \(Delta\) if and only if \(\|M\|_{\infty}<1\).

Proof

The proof of necessity (see Appendix) is by construction of an allowed \(\Delta\) that causes \((I-M \Delta)^{-1}\) and \(\Delta(I-M \Delta)^{-1}\) to be unstable if \(\|M\|_{\infty} \geq 1\), and ensures that \(G_{wz}\) is unstable. For here, we focus on the proof of sufficiency. We need to show that if \(\|M\|_{\infty}<1\) then \((I-M \Delta)^{-1}\) has no poles in the closed right half-plane for any \(\Delta \in \Delta\), or equivalently that \(I-M \Delta\) has no zeros there. For arbitrary \(x \neq 0\) and any \(s_{+}\) in the closed right half-plane (CRHP), and using the fact that both \(M\) and \(\Delta\) are well-defined throughout the CRHP, we can deduce that

\[\begin{aligned}
\left\|\left[I-M\left(s_{+}\right) \Delta\left(s_{+}\right)\right] x\right\|_{2} & \geq\|x\|_{2}-\left\|M\left(s_{+}\right) \Delta\left(s_{+}\right) x\right\|_{2} \\
& \geq\|x\|_{2}-\sigma_{\max }\left[M\left(s_{+}\right) \Delta\left(s_{+}\right)\right]\|x\|_{2} \\
& \geq\|x\|_{2}-\|M\|_{\infty}\|\Delta\|_{\infty}\|x\|_{2} \\
&>0
\end{aligned} \ \tag{20.10}\]

The first inequality above is a simple application of the triangle inequality. The third inequality above results from the Maximum Modulus Theorem of complex analysis, which says that the largest magnitude of a complex function over a region of the complex plane is found on the boundary of the region, if the function is analytic inside and on the boundary of the region. In our case, both \(q^{\prime} M^{\prime} M q\) and \(q^{\prime} \Delta^{\prime} \Delta q\) are stable, and therefore analytic, in the CRHP, for unit vectors \(q\); hence their largest values over the CRHP are found on the imaginary axis. The final inequality in the above set is a consequence of the hypotheses of the theorem, and establishes that \(I-M \Delta\) is nonsingular - and therefore has no zeros - in the CRHP.