User:Margav06/sandbox/Click here to continue/Controller synthesis/Generic Insensitive Disk Region Design

Insensitive Disk Region Design

Similar to the insensitive strip region design problem, insensitive disk region design is another way with which robust stabilization can be achieved where closed-loop eigenvalues are placed in particular regions of the complex plane where the said regions has an inner boundary that is insensitive to perturbations of the system parameter matrices.

The System
Suppose we consider the following linear system that needs to be controlled:

\begin{align} \begin{cases} \rho x&=Ax+Bu,\\ y&=Cx\\ \end{cases} \end{align}$$

where $$x\in\mathbb{R}^{n}$$, $$y\in\mathbb{R}^{m}$$, and $$u\in\mathbb{R}^{r}$$ are the state, output and input vectors respectively, and $$\rho$$ represents the differential operator (in the continuous-time case) or one-step shift forward operator (i.e., $$\rho x(k)=x(k+1)$$) (in the discrete-time case). Then the steps to obtain the LMI for insensitive strip region design would be obtained as follows.

The Data
Prior to obtaining the LMI, we need the following matrices: $$A$$, $$B$$, and $$C$$.

The Optimization Problem
Consider the above linear system as well as 2 positive scalars $$\gamma$$ and $$q$$. Then the output feedback control law $$u=Ky$$ would be designed such that:



\begin{align} {\eta}&=||A+BKC+qI||<{\gamma}\\ \end{align}$$

Recalling the definition, we have:



\begin{align} {\mathbb{D}_{q,\eta}}&=\{ {s|s\in\mathbb{C},|s+q|<{\eta}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\eta^2}\}\\ \end{align}$$

and



\begin{align} {\mathbb{D}_{q,r}}&=\{ {s|s\in\mathbb{C},|s+q|<{\gamma}}\}\\ &=\{ x+jy|x,y\in\mathbb{R},{(x+q)^2}+{y^2}<{\gamma^2}\}\\ \end{align}$$

Letting $$K$$ being the solution to the above problem, then

\begin{align} {\lambda_i}(A+BKC)&\in{\mathbb{D}_{q,\eta}}\subset{\mathbb{D}_{q,r}}, &i=1,2,...,n\\ \end{align}$$

The LMI: Insensitive Strip Region Design
Using the above info, we can convert the given problem into an LMI, which - after using Schur compliment Lemma - results in the following:

\begin{align} \begin{cases} \text{min }\gamma\\ \text{s.t. }\begin{bmatrix}-{\gamma}I&&(A+BKC+qI)\\{(A+BKC+qI)^T}&&-{\gamma}I\end{bmatrix}&<0\\ \end{cases} \end{align}$$

Conclusion:
For Schur stabilization, we can choose to solve the problem with $$q=0$$. Schur stability is achieved when $${\gamma}{\leq}1$$. Alternately, if $${\gamma}$$ is greater than (but very close to) 1, then Schur stability is also achieved when $${\eta}=||A+BKC+qI||_2{\leq}1$$.

Implementation

 * Example Code - A GitHub link that contains code (titled "InsensitiveDiskRegion.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * $H_2$ Disk Region Design - LMI for Disk Region Design with minimal $$H_2$$ gain
 * Insensitive Strip Region Design - Equivalent LMI for Insensitive Strip Region Design