User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Modified Minimum Gain Lemma

Modified Minimum Gain Lemma

The System
Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.

With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $$ G $$ corresponds to the following matrices $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

Essentially the same concept as the Minimum Gain Lemma; however, this modification ensure that the system being optimized is also Lyapunov stable. This Essentially means that the variable $$ Q $$ must ensure that the system is stable to this degree, otherwise the system itself is infeasible.

The Data
A system of the plant $$ G $$ will be required with the following matrices: $$ A \in \mathbb{R}^{m \times n} $$, $$ B \in \mathbb{R}^{n \times m} $$, $$ C \in \mathbb{R}^{p \times n} $$, and $$ D \in \mathbb{R}^{p \times m} $$.

The LMI: Modified Minimum Gain Lemma
The following two LMI's are equivalent, with the same variable:

Suppose there exists $$ Q \in \mathbb{S}^n $$, and $$ \upsilon \in \mathbb{R}_{\geq 0} $$, where $$ Q > 0 $$ such that:



\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D \\ (B - Q C^T D)^T & \upsilon^2 I - D^T D \end{bmatrix}&\leq 0

\end{align}$$

Or (Obtained using Schur compliment):



\begin{align}

\begin{bmatrix} A Q + Q A^T & B - Q C^T D & 0 \\ (B - Q C^T D)^T & -D^T D & \upsilon I \\ 0 & \upsilon I & -I \end{bmatrix}&\leq 0

\end{align}$$

Conclusion:
Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $$ \upsilon $$ obtained from the optimization. This system will also prove that the plant is Lyapunov stable.

Implementation

 * Minimum Gain Lemma Example