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

$\dot{𝐱}(t)=A𝐱(t)+B𝐮(t)$

$𝐲(t)=C𝐱(t)+D𝐮(t)$

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 $\overline{u}(t)$, which also drives the system from $x_0$ to $x_1$ at time $t_1$, the energy expenditure would be larger, i.e.,

$\underset{t_0}{\overset{t_1}{\int }}{\overline{u}}^{*}(t)\overline{u}(t)dt\ge \underset{t_0}{\overset{t_1}{\int }}{u}^{*}(t)u(t)dt.$

To choose this input, first compute the controllability Gramian

$W_c(t)=\underset{t_0}{\overset{t}{\int }}{e}^{A(t-\tau )}B{B}^{*}{e}^{A^{*}(t-\tau )}d\tau .$

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

$x(t)=e^{A(t-t_0)}{x}_0+\underset{t_0}{\overset{t}{\int }}{e}^{A(t-\tau )}Bu(\tau )d\tau$

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

See also

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