LMIs in Control/pages/Small Gain Theorem

LMIs in Control/Matrix and LMI Properties and Tools/Small Gain Theorem

The Small Gain Theorem provides a sufficient condition for the stability of a feedback connection.

Theorem

Suppose $B$ is a Banach Algebra and $Q\in B$. If $\Vert Q\Vert <1$, then $(I-Q{)}^{-1}$ exists, and furthermore,

                   $(I-Q{)}^{-1}=\sum _{k=0}^{\mathrm{\infty }}{Q}^k$ 

Proof

Assuming we have an interconnected system $(G,K)$:

$y_1=G(u_1-y_2)$ and, $y_2=K(u_2-y_1)$

The above equations can be represented in matrix form as

$\begin{array}{cc}\left[\begin{array}{cc}I& 0\\ 0& I\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]& =\left[\begin{array}{cc}0& -G\\ -K& 0\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]+\left[\begin{array}{cc}G& 0\\ 0& K\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]\end{array}$

Making ${\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^T$ the subject, we then have:

$\begin{array}{ccc}\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]& ={\left[\begin{array}{cc}I& G\\ K& I\end{array}\right]}^{-1}\left[\begin{array}{cc}G& 0\\ 0& K\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]& =\left[\begin{array}{cc}(I-GK{)}^{-1}G& -G(I-KG{)}^{-1}K\\ -K(I-GK{)}^{-1}G& (I-KG{)}^{-1}K\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]\end{array}$

If $(I-GK{)}^{-1}$ is well-behaved, then the interconnection is stable. For $(I-GK{)}^{-1}$ to be well-behaved, $\Vert (I-GK{)}^{-1}\Vert$ must be finite.

Hence, we have $\Vert (I-GK{)}^{-1}\Vert <\mathrm{\infty }$

$\Vert G\Vert \Vert K\Vert =\Vert Q\Vert$ and $\Vert Q\Vert <I$ for the higher exponents of $\Vert Q\Vert$ to converge to $0.$

Conclusion

If $\Vert Q\Vert <1$, then this implies stability, since the higher exponents of $Q$ in the summation of $\sum _{k=0}^{\mathrm{\infty }}{Q}^k$ will converge to $0$, instead of blowing up to infinity.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.