Minimum energy control
# 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 $t_{1}$.

