- #1
synMehdi
- 38
- 0
Hi, I am trying to solve a control problem where I have to minimize the fuel consumption of a vehicle:
$$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$
##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the value of the last state - in my case ##x(T)=x(0)##
Subject to some dynamic constraints on the state:
$$\dot{x}(t)=f(x(t),u(t))$$
and some inequality constraints on the control variable (which is what I'm having trouble with)
$$U_{min}<u(t)<U_{max}$$
So far I've been able to solve the unconstrained problem by minimizing the Hamiltonian
$$H(x(t),u(t),p(t))=L(x(t), u(t),t)+p(t) \dot{x(t)}$$
The co-state ##p(t)=p_0## is constant and calculated so that ##x(T)=x(0)##. Also ##u(t)## is function of ##p_0## and some of the quadratic parameters that come from derivative of ##L(x(t),u(t),t)## wrt to ##u(t)## (##\frac{dH}{du}=0##)
This works fine and i have the results I want. My question is how can I incorporate the inequality constrains on the control input: ##u(t)## without losing the ##p_0## that makes ##x(T)=x(0)##
$$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$
##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the value of the last state - in my case ##x(T)=x(0)##
Subject to some dynamic constraints on the state:
$$\dot{x}(t)=f(x(t),u(t))$$
and some inequality constraints on the control variable (which is what I'm having trouble with)
$$U_{min}<u(t)<U_{max}$$
So far I've been able to solve the unconstrained problem by minimizing the Hamiltonian
$$H(x(t),u(t),p(t))=L(x(t), u(t),t)+p(t) \dot{x(t)}$$
The co-state ##p(t)=p_0## is constant and calculated so that ##x(T)=x(0)##. Also ##u(t)## is function of ##p_0## and some of the quadratic parameters that come from derivative of ##L(x(t),u(t),t)## wrt to ##u(t)## (##\frac{dH}{du}=0##)
This works fine and i have the results I want. My question is how can I incorporate the inequality constrains on the control input: ##u(t)## without losing the ##p_0## that makes ##x(T)=x(0)##