In control theory, the minimum energy control is the control u ( t ) 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
x ˙ ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) with initial state x ( t 0 ) = x 0 . One seeks an input u ( t ) so that the system will be in the state x 1 at time t 1 , and for any other input u ¯ ( t ) , which also drives the system from x 0 to x 1 at time t 1 , the energy expenditure would be larger, i.e.,
∫ t 0 t 1 u ¯ ∗ ( t ) u ¯ ( t ) d t ≥ ∫ t 0 t 1 u ∗ ( t ) u ( t ) d t . To choose this input, first compute the controllability gramian
W c ( t ) = ∫ t 0 t e A ( t − τ ) B B ∗ e A ∗ ( t − τ ) d τ . Assuming W c is nonsingular (if and only if the system is controllable), the minimum energy control is then
u ( t ) = − B ∗ e A ∗ ( t 1 − t ) W c − 1 ( t 1 ) [ e A ( t 1 − t 0 ) x 0 − x 1 ] . Substitution into the solution
x ( t ) = e A ( t − t 0 ) x 0 + ∫ t 0 t e A ( t − τ ) B u ( τ ) d τ verifies the achievement of state x 1 at t 1 .
