Minimum energy control

From Wikipedia, the free encyclopedia

You have new messages (last change).

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(t0) = x0. One seeks an input u(t) so that the system will be in the state x1 at time t1, and for any other input \bar{u}(t), which also drives the system from x0 to x1 at time t1 the energy expenditure would be larger

\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 Wc 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_1-t_0)}x_0+\int_{t_0}^{t_1}e^{A(t_1-\tau)}Bu(\tau)d\tau

verifies the achievement of state x1 at t1.

[edit] See also

Retrieved from "http://en.wikipedia.org../../../m/i/n/Minimum_energy_control.html"