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
: $\dot(t) = A \mathbf(t) + B \mathbf(t)$
: $\mathbf(t) = C \mathbf(t) + D \mathbf(t)$
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 $\bar(t)$, which also drives the system from $x_0$ to $x_1$ at time $t_1$, the energy expenditure would be larger, i.e.,

:$\int_^ \bar^*(t) \bar(t) dt \ \geq \ \int_^ u^*(t) u(t) dt.$

To choose this input, first compute the controllability Gramian

:$W_c(t)=\int_^t e^BB^*e^ d\tau.$

Assuming $W_c$ is nonsingular (if and only if the system is controllable), the minimum energy control is then

: u(t) = -B^*e^W_c^(t_1) ^x_0-x_1

Substitution into the solution

:$x(t)=e^x_0+\int_^e^{A(t-\tau)}Bu(\tau)d\tau$

verifies the achievement of state $x_1$ at $t_1$.

See also

* LTI system theory
* Control engineering
* State space (controls)
* Variational Calculus
Control theory