In control theory, the minimum energy control is the control u ( t ) {\displaystyle 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 ) {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t)}
y ( t ) = C x ( t ) + D u ( t ) {\displaystyle \mathbf {y} (t)=C\mathbf {x} (t)+D\mathbf {u} (t)}
with initial state x ( t 0 ) = x 0 {\displaystyle x(t_{0})=x_{0}} . One seeks an input u ( t ) {\displaystyle u(t)} so that the system will be in the state x 1 {\displaystyle x_{1}} at time t 1 {\displaystyle t_{1}} , and for any other input u ¯ ( t ) {\displaystyle {\bar {u}}(t)} , which also drives the system from x 0 {\displaystyle x_{0}} to x 1 {\displaystyle x_{1}} at time t 1 {\displaystyle t_{1}} , the energy expenditure would be larger, i.e.,
∫ t 0 t 1 u ¯ ∗ ( t ) u ¯ ( t ) d t ≥ ∫ t 0 t 1 u ∗ ( t ) u ( t ) d t . {\displaystyle \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 ) = ∫ t 0 t e A ( t − τ ) B B ∗ e A ∗ ( t − τ ) d τ . {\displaystyle W_{c}(t)=\int _{t_{0}}^{t}e^{A(t-\tau )}BB^{*}e^{A^{*}(t-\tau )}d\tau .}
Assuming W c {\displaystyle W_{c}} 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 ] . {\displaystyle 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
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