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10. Optimal Control Systems Design
Given the constraints  on control functions  that form the set of admissible controls  for all  and the constraints  on the state trajectories  that form the set of admissible trajectories  for all  , the optimal control problem is to find an admissible control function  that forces the continuous-time system
 to follow an admissible trajectory  while minimizing the performance criterion
 If the solution to the optimal control problem can be found in the form
 then the control is said to exist in the closed-loop form, and Eq. (10.3) is referred to as the optimal control law.
In Eq. (10.2), the function  is the cost associated with error in the terminal state at time  , and  penalizes for transient state errors and control effort. In the particular case of quadratic cost functions,
 and
 or (in the form that includes the cross-term P)
 where the desired state is assumed to be  .
Matrices  ,  , and  must be square;  and  must have a length equal to the number of states; and  must correspond in dimension to the number of inputs. Additionally, to ensure that the solution is unique and finite, matrices  and  must be positive semidefinite and matrix  must be positive definite. The cross-term problem in Eq. (10.6) is reducible to the one in Eq. (10.5), and so matrix  must be of a form that brings about suitable  and  .
The components of the matrices reflect the emphasis the designer places on corresponding errors. For instance, if  is a diagonal matrix, a relatively larger value of  means that more control effort will be allotted to regulate input  . In a sense, the art of choosing the elements of  ,  , and  is similar to the art of selecting the proper pole locations in the feedback design via pole assignment.
The optimal control problem can be restated similarly for discrete-time control systems
 and the performance criterion is
Control System Professional addresses both continuous- and discrete-time problems.
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