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{align} \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{align} $$

Making $$\begin{bmatrix}y_1 & y_2\end{bmatrix}^{T}$$ the subject, we then have:

$$

\begin{align} \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{align} $$

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.