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  to  at time , 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  at . 

See also
 LTI system theory
 Control engineering
 State space (controls)

Category:Control theory