Minimum energy control

In control theory, the minimum energy control is the control ${\displaystyle u(t)}$ that will bring a linear time invariant system to a desired state with a minimum expenditure of energy.¹

Let the linear time invariant (LTI) system be

${\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)}$

${\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)}$

with initial state ${\displaystyle x(t_{0})=x_{0}}$. One seeks an input ${\displaystyle u(t)}$ so that the system will be in the state ${\displaystyle x_{1}}$ at time ${\displaystyle t_{1}}$, and for any other input ${\displaystyle {\bar {u}}(t)}$, which also drives the system from ${\displaystyle x_{0}}$ to ${\displaystyle x_{1}}$ at time ${\displaystyle t_{1}}$, the energy expenditure would be larger, i.e.,

${\displaystyle \int _{t_{0}}^{t_{1}}{\bar {u}}^{*}(t){\bar {u}}(t)dt\ \geq \ \int _{t_{0}}^{t_{1}}u^{*}(t)u(t)dt.}$

To choose this input, first compute the controllability Gramian

${\displaystyle W_{c}(t)=\int _{t_{0}}^{t}e^{A(t-\tau )}BB^{*}e^{A^{*}(t-\tau )}d\tau .}$

Assuming ${\displaystyle W_{c}}$ is nonsingular (if and only if the system is controllable), the minimum energy control is then

${\displaystyle u(t)=-B^{*}e^{A^{*}(t_{1}-t)}W_{c}^{-1}(t_{1})[e^{A(t_{1}-t_{0})}x_{0}-x_{1}].}$

Substitution into the solution

${\displaystyle x(t)=e^{A(t-t_{0})}x_{0}+\int _{t_{0}}^{t}e^{A(t-\tau )}Bu(\tau )d\tau }$

verifies the achievement of state ${\displaystyle x_{1}}$ at ${\displaystyle t_{1}}$.