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 $\|Q\|<1$, then $(I-Q)^{-1}$ exists, and furthermore,

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

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

${\begin{aligned}{\begin{bmatrix}I&0\\0&I\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}&={\begin{bmatrix}\ 0&-G\\\ -K&0\\\end{bmatrix}}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}+{\begin{bmatrix}G&0\\0&K\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}\end{aligned}}$

Making ${\begin{bmatrix}y_{1}&y_{2}\end{bmatrix}}^{T}$ the subject, we then have:

${\begin{aligned}{\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}}&={\begin{bmatrix}\ I&G\\\ K&I\\\end{bmatrix}}^{-1}{\begin{bmatrix}G&0\\0&K\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}&={\begin{bmatrix}\ (I-GK)^{-1}G&-G(I-KG)^{-1}K\\\ -K(I-GK)^{-1}G&(I-KG)^{-1}K\\\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}}\end{aligned}}$

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

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

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

Conclusion

If $\|Q\|<1$, then this implies stability, since the higher exponents of $Q$ in the summation of $\sum _{k=0}^{\infty }Q^{k}$ 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.