# Homework Help: LQR Design: Choosing Q & R matrices for specific eigenvalues

1. Mar 23, 2012

### Expirobo

Assuming we have a closed loop system (A-BK), with stable eigenvalues, how would one choose matrices Q and R such that the eigenvalues of (A-BK) are exactly [-1,-2]?

LTI System:
$\dot{x}=\left[ \begin{array}{cc} 0& 1 \\ 0 & 0 \\ \end{array} \right]x+\left[ \begin{array}{c} 0 \\ 1 \\ \end{array} \right]u$

The performance measure is given by:

$V=\int_0^\infty \left( x'\left[ \begin{array}{cc} 1 & 0 \\ 0 & 2 \\ \end{array} \right]x+u^2\right)\mathrm{d}t.$

So the initial Q is :
$Q=\left[ \begin{array}{cc} 1 & 0 \\ 0 & 2 \\ \end{array} \right]$

and R = 1

However, we want to choose a new matrix for Q and R so that we have stable eigenvalues:
$\lambda=-1 \\ \lambda=-2 \\$

I have gone round and round with this, but I cannot see how to work back through the Hamiltonian Matrix to achieve a desired value for Q & R.

The Hamiltonian is:
$H=\left[ \begin{array}{cc} A & -BR^{-1}B' \\ -Q & -A' \\ \end{array} \right]$

and the stable Eigenvalues of H (ones with negative real parts) are exactly the eigenvalues of the closed loop system A-BK.

$Q=\left[ \begin{array}{cc} 4 & 0 \\ 0 & 5 \\ \end{array} \right]$