Optimization and pole placement for a single input controllable system

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Abstract
Given the system [xdot]=Ax+bu and the cost function J=dt, relations are to be determined among the open-loop characteristic polynomial, the closed-loop characteristic polynomial and the matrices A and Q. Those relations take a simple form if the system is in the standard controllable form. In this case the optimal control law can be found easily without solving the matrix Riccati equation while the minimum value of the cost function, if it is required, can be determined by solving a matrix equation of the form C T. X+XC= −D

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