Hi Alireza,
you can use Sylvester matrix equation to solve the Jordan form assignment (pole placement). If you want to assign Jordan form L to state space matrices (A, B), you require A+BF ~ L, e.g. then there exists regular matrix X, that A+BF = X L inv(X), so reordering this means AX - XL + BFX = 0. And then you can assign H = FX and you get Sylvesters matrix equation AX - XL + BH = 0. When you compute X, then F = H inv(X). For more you can see our article : M.Schlegel, J.Königsmarková Parametric Jordan Form Assignment Revisited, where moreover the minimal parametrization of all state feedback is explained.
Hello, Alireza,
Beside the very nice response posted by Jana, I add the following note:
The choice of a suitable pair (W,H), with A+BF = XWX^-1, and H = FX, can lead to several parametric explicit formulae derived from the solution of the Sylvester Equation. See my paper:
Article A unified point of view for pole placement formulas in singl...
And references therein for more details. Also, it can be pointed that any of these solutions are numerically sensible to the condition number of X. More on this issue is given in details in the authoritative book
NUMERICAL LINEAR ALGEBRA AND APPLICATIONS, by B.N. Datta:
http://bookstore.siam.org/ot116/
I hope it can help you.
Hi,
the contribution from Alireza is also interesting. I would add that when we have one input (B is only one column), than there exists only one state feedback F assigning the Jordan form L, or this set is empty. In this case when only one solution exists, you can generate randomly the matrix H, then compute X(H) as the solution of Sylvesters matrix equation (AX - XL + BH = 0) and then compute F = H inv(X) and for arbitrary H you get another X(H), but same matrix F. According to our previous article, you can also use direct the parametric matrix (Q) (where also no parameter is). In case of more inputs, the situation is more complex and more matices F exist, the set of all state feedbacks can be parameterized.
Just to add and remind.
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