Minimum energy control

{{Unreferenced|date=December 2009}}

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

: $\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; A\; \backslash mathbf\{x\}(t)\; +\; B\; \backslash mathbf\{u\}(t)$

: $\backslash mathbf\{y\}(t)\; =\; C\; \backslash mathbf\{x\}(t)\; +\; D\; \backslash mathbf\{u\}(t)$

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 $\backslash bar\{u\}(t)$, which also drives the system from $x\_0$ to $x\_1$ at time $t\_1$, the energy expenditure would be larger, i.e.,

:$\backslash int\_\{t\_0\}^\{t\_1\}\; \backslash bar\{u\}^*(t)\; \backslash bar\{u\}(t)\; dt\; \backslash \; \backslash geq\; \backslash \; \backslash int\_\{t\_0\}^\{t\_1\}\; u^*(t)\; u(t)\; dt.$

To choose this input, first compute the controllability Gramian

:$W\_c(t)=\backslash int\_\{t\_0\}^t\; e^\{A(t-\backslash tau)\}BB^*e^\{A^*(t-\backslash tau)\}\; d\backslash tau.$

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

:$x(t)=e^\{A(t-t\_0)\}x\_0+\backslash int\_\{t\_0\}^\{t\}e^\{A(t-\backslash tau)\}Bu(\backslash tau)d\backslash tau$

verifies the achievement of state $x\_1$ at $t\_1$.