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State ReconstructionLinear Quadratic Regulator

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,


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.

State ReconstructionLinear Quadratic Regulator