LMIs in Control/pages/Insensitive Strip Region Design

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Insensitive Strip Region Design

Suppose if one were interested in robust stabilization 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. This would be accomplished with the help of 2 design problems: the insensitive strip region design and insensitive disk region design (see link below for the latter).

The System

Suppose we consider the following continuous-time linear system that needs to be controlled:

{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}&=Ax+Bu,\\y&=Cx\\\end{cases}}\end{aligned}}}

where ${\displaystyle x\in \mathbb {R} ^{n}}$, ${\displaystyle y\in \mathbb {R} ^{m}}$, and ${\displaystyle u\in \mathbb {R} ^{r}}$ are the state, output and input vectors respectively. 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: ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$.

The Optimization Problem

Consider the above linear system as well as 2 scalars ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$. Then the output feedback control law ${\displaystyle u=Ky}$ would be such that ${\displaystyle {\gamma _{1}}<{\lambda _{i}}(A_{c}^{s})<{\gamma _{1}}}$, where:

{\displaystyle {\begin{aligned}{A_{c}^{s}}&\triangleq {\frac {1}{2}}{\langle {A_{c}}\rangle _{s}}={\frac {{(A+BKC)^{T}}+(A+BKC)}{2}}\\\end{aligned}}}

Letting ${\displaystyle K}$ being the solution to the above problem, then

{\displaystyle {\begin{aligned}{\gamma _{1}}&<{\alpha _{1}}\leq {Re({\lambda _{i}}(A+BKC))\leq {\alpha _{2}}<{\gamma _{2}}},&i=1,2,...,n\\\end{aligned}}}

where

{\displaystyle {\begin{aligned}{\begin{cases}{\alpha _{1}}&={\lambda _{min}}({A_{c}^{s}})\\{\alpha _{2}}&={\lambda _{max}}({A_{c}^{s}})\\\end{cases}}\end{aligned}}}

The LMI: Insensitive Strip Region Design

Using the above info, we can simplify the problem by setting ${\displaystyle \gamma _{1}}$ to ${\displaystyle -\infty }$ for all practical applications. This then simplifies our problem and results in the following LMI:

{\displaystyle {\begin{aligned}{\begin{cases}{\text{min }}\gamma \\{\text{s.t. }}{(A+BKC)^{T}}+(A+BKC)<{\gamma }I\\\end{cases}}\end{aligned}}}

Conclusion:

If the resulting solution from the LMI above produces a negative ${\displaystyle \gamma }$, then the output feedback controller ${\displaystyle K}$ is Hurwitz-stable. Hoewever, if ${\displaystyle \gamma }$ is a really small positive number, then ${\displaystyle {\alpha _{2}}=\lambda _{max}(A_{c}^{s})}$ must be negative for the controller to be Hurwitz-stable.

Implementation

• Example Code - A GitHub link that contains code that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

Links to other closely-related LMIs