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| {{Orphan|date=December 2009}}
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| 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.
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| Let the linear time invariant (LTI) system be
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| : <math>\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
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| : <math>\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>
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| 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
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| :<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> | |
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| To choose this input, first compute the controllability gramian
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| :<math> W_c(t)=\int_{t_0}^t e^{A(t-\tau)}BB^*e^{A^*(t-\tau)} d\tau.</math> | |
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| Assuming <math>W_c</math> is nonsingular (if and only if the system is controllable), the minimum energy control is then
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| :<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>
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| Substitution into the solution
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| :<math>x(t)=e^{A(t_1-t_0)}x_0+\int_{t_0}^{t_1}e^{A(t_1-\tau)}Bu(\tau)d\tau</math>
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| verifies the achievement of state <math>x_1</math> at <math>t_1</math>.
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| ==See also== | |
| * [[LTI system theory]]
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| * [[Control engineering]]
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| * [[State space (controls)]]
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| [[Category:Control theory]]
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