dear Avinash,

Penalty matrices including Q(state x) and R(control u) are positive semidefinite to satisfy the solution of the Algebraic Riccati Equation (ARE). We must note that the Riccati equation can have more than one solution, but there is only one solution that is positive semidefinite.

For more details and information about thi subject i suggest you to see links and attached file in topic.

-Development of a State Dependent Riccati Equation based ... - MAiA

www.maia.ub.es/~gerard/AstroNet-II/.../PrachTekinalp2013.pdf

-linear quadratic optimal control - Systems Control Group

https://www.control.utoronto.ca/people/profs/.../chap6-08.pdf

Best regards

Dear Avinash,

The matrices Q and R imply weightings on the state variables or on the control signals. A simple Q is Sum(qii*xi) for all i's, which would put weightings on all state-variables and result in a diagonal matrix Q. Sometimes, you (or your adviser :-) are only interested in a few state variables, say x1 and x2 out of n=20 variables.

Using only a few weightings in Q, makes it singular. As the computations do not imply any inversion of Q, this is allowed, yet Q must still be POSITIVE semidefinite in order to get a solution.

As R is inverted, it can only be positive DEFINITE.

Hope this helps,

Itzhak

Thanks a lot for the responses/