Minimum energy control

In control theory, the minimum energy control is the control [math]\displaystyle{ 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]\displaystyle{ \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t) }[/math]

[math]\displaystyle{ \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t) }[/math]

with initial state [math]\displaystyle{ x(t_0)=x_0 }[/math]. One seeks an input [math]\displaystyle{ u(t) }[/math] so that the system will be in the state [math]\displaystyle{ x_1 }[/math] at time [math]\displaystyle{ t_1 }[/math], and for any other input [math]\displaystyle{ \bar{u}(t) }[/math], which also drives the system from [math]\displaystyle{ x_0 }[/math] to [math]\displaystyle{ x_1 }[/math] at time [math]\displaystyle{ t_1 }[/math], the energy expenditure would be larger, i.e.,

[math]\displaystyle{ \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]\displaystyle{ W_c(t)=\int_{t_0}^t e^{A(t-\tau)}BB^*e^{A^*(t-\tau)} d\tau. }[/math]

Assuming [math]\displaystyle{ W_c }[/math] is nonsingular (if and only if the system is controllable), the minimum energy control is then

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ x_1 }[/math] at [math]\displaystyle{ t_1 }[/math].

See also

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