# 20.6: The Small-Gain Theorem

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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.

This page titled 20.6: The Small-Gain Theorem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.