LQR Design: Choosing Q & R matrices for specific eigenvalues

[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.

Please help if you can. Even something to point me in the right direction will help.

BTW, just iterating by hand the matrix Q, I came up with eig(H) = -1,-2 using:

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

Yet, I must show how to achieve something like this mathematically.

Thanks!

 

[Valkarie]

The approach to this problem is to use the eigenvalues of the Hamiltonian matrix H to find the appropriate Q and R. To do this, we can first solve for the eigenvalues of the Hamiltonian matrix:\lambda^2-(tr(A))\lambda+(det(A)+tr(QR^{-1}B'B))=0We know that we want the eigenvalues to be -1 and -2, so we can substitute these values into the equation above. This will give us two equations which we can solve for the terms tr(A) and det(A)+tr(QR^{-1}B'B). We can then use these equations to compute the appropriate values for Q and R.For example, if tr(A)=-3 and det(A)+tr(QR^{-1}B'B)=7, then we can solve for Q and R using the following equations:Q=(-3+\sqrt{10}\,)^2/4 \\R=(7+\sqrt{10}\,)^2/4Therefore, the matrices Q and R that will give us the desired eigenvalues of [-1,-2] are:Q=\left[ \begin{array}{cc}(3+\sqrt{10}\,)^2/4 & 0 \\0 & (3+\sqrt{10}\,)^2/4 \\\end{array} \right]R=\left[ \begin{array}{cc}(7+\sqrt{10}\,)^2/4 & 0 \\0 & (7+\sqrt{10}\,)^2/4 \\\end{array} \right]