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 $$t_1$$.