Minimum energy control

In control theory, the minimum energy control is the control

that will bring a linear time invariant system to a desired state with a minimum expenditure of energy.

Let the linear time invariant (LTI) system be with initial state

One seeks an input

so that the system will be in the state

at time

, and for any other input

, which also drives the system from

, the energy expenditure would be larger, i.e., To choose this input, first compute the controllability Gramian Assuming

is nonsingular (if and only if the system is controllable), the minimum energy control is then Substitution into the solution verifies the achievement of state