Minimum energy control

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 : \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t) : \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{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 \bar{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.,

: \int_{t_0}^{t_1} \bar{u}^(t) \bar{u}(t) dt \ \geq \ \int_{t_0}^{t_1} u^(t) u(t) dt.

To choose this input, first compute the controllability Gramian

: W_c(t)=\int_{t_0}^t e^{A(t-\tau)}BB^e^{A^(t-\tau)} d\tau.

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+\int_{t_0}^{t}e^{A(t-\tau)}Bu(\tau)d\tau

verifies the achievement of state x_1 at t_1.