I am studying control systems, and I want to solve following problem. Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e. For a given full rank AinRn*n, with lambda_i(A)>1, solve the following

reinzogoq 2022-09-11 Answered
I am studying control systems, and I want to solve following problem.
Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e.
For a given full rank A R n × n , with λ i ( A ) > 1, solve the following
minimize X R n × n , B R n × m t r ( B X B ) subject to X = A X ( I + B B X ) 1 A ( A , B )  is stabilizable
From my understanding, since all eigenvalues of A are outside of unit circle (discrete-time system), we can change condition (A,B) is stabilizable with (A,B) is controllable, which is equivalent to rank([ r a n k ( [ B A B A 2 B A n 1 B ] ) = n
The problem is for sure feasible, since for any full rank A, there is B such that rank condition is satisfied and we can solve DARE.
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Answers (1)

Jaylen Mcmahon
Answered 2022-09-12 Author has 16 answers
I tried to use dual problem, maybe someone can help me to finish it. By creating random B, usually we get stabilizable pair (A,B), so lets ignore the second constraint for now.
trace ( B X B ) = trace ( B B X ) = trace ( A X 1 A X ) trace ( I ), so we can minimize trace trace ( A X 1 A X ) instead of trace trace ( B X B ).
Rewrite X = A X ( I + B B X ) 1 A to B B A X 1 A + X 1 = 0, then Lagrangian function:
Λ ( B , X , V ) = trace ( A X 1 A X ) + trace ( V B B ) trace ( V A X 1 A ) + trace ( V X 1 ) , Λ ( B , X , V ) B = ( V + V ) B = 0 , Λ ( B , X , V ) B = ( A X 1 A X 1 A X A X 1 ) + ( X 1 A V A X 1 ) ( X 1 V X 1 ) = 0.
then we define g ( B , X , V ) = inf B , X Λ ( B , X , V ) and dual problem becomes: maxVg(B,X,V).
First we need to find what B and X minimize g(B,X,V). Assuming that V + V and B are both nonzero, I found that either V + V or B must be singular to satisfy ( V + V ) B = 0

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