Minimum energy control explained

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

• x (t) = A x (t) + B u (t) y (t) = C x (t) + D 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 \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., \int ^{t _{1}} _{t _{0}} \bar{u} ^{*(t)} \bar{u}(t) dt \geq \int ^{t _{1}} _{t _{0}} u ^{*(t)} u(t) dt.

To choose this input, first compute the controllability Gramian

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

Assuming

W _{c} is nonsingular (if and only if the system is controllable), the minimum energy control is thenu(t) = - B ^{*e} ^{A ^{*(t} _{1- t)}} W ^{-1} _{c} (t ^{A(t _{1-t} _{0)}} _{1 )[e} x _{0-x} _{1].}

Substitution into the solution

x(t) = e ^{A(t - t _{0)}} x _{0+\int} ^{t} ^{t _{0}} 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