View source for Minimum energy control

{{Unreferenced|date=December 2009}}


In [[control theory]], the '''minimum energy control''' is the control <math>u(t)</math> 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
: <math>\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>
with initial state <math> x(t_0)=x_0 </math>.  One seeks an input <math> u(t) </math> so that the system will be in the state <math>x_1</math> at time <math>t_1</math>, and for any other input <math>\bar{u}(t)</math>, which also drives the system from <math>x_0</math> to <math>x_1</math> at time <math>t_1</math>, the energy expenditure would be larger, i.e., 

:<math> \int_{t_0}^{t_1} \bar{u}^*(t) \bar{u}(t) dt \ \geq  \ \int_{t_0}^{t_1} u^*(t) u(t) dt. </math>

To choose this input, first compute the [[controllability Gramian]]

:<math> W_c(t)=\int_{t_0}^t e^{A(t-\tau)}BB^*e^{A^*(t-\tau)} d\tau.</math>

Assuming  <math>W_c</math> is nonsingular (if and only if the system is controllable), the minimum energy control is then

:<math> u(t) = -B^*e^{A^*(t_1-t)}W_c^{-1}(t_1)[e^{A(t_1-t_0)}x_0-x_1].</math>

Substitution into the solution

:<math>x(t)=e^{A(t-t_0)}x_0+\int_{t_0}^{t}e^{A(t-\tau)}Bu(\tau)d\tau</math>

verifies the achievement of state <math>x_1</math> at <math>t_1</math>.

==See also==
* [[LTI system theory]]
* [[Control engineering]]
* [[State space (controls)]]
* [[Variational Calculus]]
[[Category:Control theory]]