LA OPCIÓN MIDDLE STEP CONTROL NO TIENE SENTIDO, NO LA HE ESCRITO
SIN MIDDLE STEP CONTROL NO HAY PASO ADAPTATIVO NO?
HE ESCRITO ESO DE LA APROXIMACIÓN DE LA DERIVADA Y DEL ADJUNTO, AUNQUE NO LO HE ENCONTRADO EN EL LIBRO NI ENTIENDO BIEN PORQUÉ ÉS UNA APROXIMACIÓN (EL P)
This method is used within the GradientMethod method. GradientMethod executes iteratively this rutine in order to get one update of the control in each iteration. In the case of choosing AdaptativeDescent this function updates the control of the following way:
$$u_{old}=u_{new}-\alpha_k dJ$$
where dJ is an approximation of the gradient of J that has been obtained considering the adjoint state of the optimality condition of Pontryagin principle. The optimal control problem is defined by
$$\min J=\min\Psi (t,Y(T))+\int^T_0 L(t,Y,U)dt$$
subject to:
$$\frac{d}{dt}Y=f(t,Y,U).$$
The gradient of $$J$$ is:
$$dJ=\partial_u H=\partial_uL+p\partial_uf$$
An approximation $$p$$ is computed using:
$$-\frac{d}{dt}p = f_Y (t,Y,U)p+L_Y(Y,U)$$
$$ p(T)=\psi_Y(Y(T))$$
Since one the expression of the gradient, we can start with an initial control, solve the adjoint problem and evaluate the gradient. Then one updates the initial control in the direction of the approximate gradient with a step size $$\alpha_k$$. $$\alpha_k$$ is determined by a small variation of the Armijo stepsize rule. In each iteration the algorithm multiplies the stepsize by to $$\alpha_k=2\alpha_{k-1}$$ and checks if $$J(y_k,u_k)<J(y_{k-1},u_{k-1})$$ in case to be true continues to the next iteration, in case to be false it devides by two the stepsize until the condition is fulfilled or the minimum stepsize is reached.
In this routine the user has to choose the minimum step size.
This routine will tell to GradientMethod to stop when the minimum tolerance of the derivative (or the relative error, user's choice) is reached. Moreover there is a maximum of iterations allowed.
MANDATORY INPUTS:
NAME: iCP
DESCRIPTION: Control problem object, it carries all the information about the dynamics, the functional to be minimized and moreover the updates of the current best control find so far.
CLASS: ControlProblem
NAME: tol
DESCRIPTION: the tolerance desired.
CLASS: double
OPTIONAL INPUT PARAMETERS
NAME: InitialLengthStep
DESCRIPTION: This paramter is the length step of the gradient method that is going to be used at the begining of the process. By default, this is 0.1.
CLASS: double
NAME: MinLengthStep
DESCRIPTION: This paramter is the lower bound on the length step of the gradient method if the algorithm needs to have a step size lower than this size it will make the GradientMethod stop.
CLASS: double
OUTPUT PARAMETERS:
All the updates will be carried inside the iCP control problem object.
Name: Unew
Description: Update of the Control
class: a vector valued function in a form of a double matrix
Name: Ynew
Description: Update of State Vector
class: a vector valued function in a form of a double matrix
Name: Jnew
Description: New Value of functional
class: double
Name: dJnew
Description: New Value of gradient
class: a vector valued function in a form of a double matrix
Name: error
Description: Nthe error |dJ|/|U| or |dJ| depending on the choice of the user.
class: double
Name: stop
Description: if this parameter is true, the routine will tell to GradientMethod to stop
class: logical
Citations:
[1] Cohen, William C.,Optimal control theory—an introduction, Donald E. Kirk, Prentice Hall, Inc., New York (1971), 452 poges. $13.50,https://onlinelibrary.wiley.com/doi/abs/10.1002/aic.690170452
LA OPCIÓN MIDDLE STEP CONTROL NO TIENE SENTIDO, NO LA HE ESCRITO
SIN MIDDLE STEP CONTROL NO HAY PASO ADAPTATIVO NO?
HE ESCRITO ESO DE LA APROXIMACIÓN DE LA DERIVADA Y DEL ADJUNTO, AUNQUE NO LO HE ENCONTRADO EN EL LIBRO NI ENTIENDO BIEN PORQUÉ ÉS UNA APROXIMACIÓN (EL P)
This method is used within the GradientMethod method. GradientMethod executes iteratively this rutine in order to get one update of the control in each iteration. In the case of choosing AdaptativeDescent this function updates the control of the following way:
where dJ is an approximation of the gradient of J that has been obtained considering the adjoint state of the optimality condition of Pontryagin principle. The optimal control problem is defined by
$$\min J=\min\Psi (t,Y(T))+\int^T_0 L(t,Y,U)dt$$
subject to:
The gradient of$$J$$ is:
An approximation$$p$$ is computed using:
Since one the expression of the gradient, we can start with an initial control, solve the adjoint problem and evaluate the gradient. Then one updates the initial control in the direction of the approximate gradient with a step size$$\alpha_k$$ . $$\alpha_k$$ is determined by a small variation of the Armijo stepsize rule. In each iteration the algorithm multiplies the stepsize by to $$\alpha_k=2\alpha_{k-1}$$ and checks if $$J(y_k,u_k)<J(y_{k-1},u_{k-1})$$ in case to be true continues to the next iteration, in case to be false it devides by two the stepsize until the condition is fulfilled or the minimum stepsize is reached.
In this routine the user has to choose the minimum step size.
This routine will tell to GradientMethod to stop when the minimum tolerance of the derivative (or the relative error, user's choice) is reached. Moreover there is a maximum of iterations allowed.
MANDATORY INPUTS:
NAME: iCP
DESCRIPTION: Control problem object, it carries all the information about the dynamics, the functional to be minimized and moreover the updates of the current best control find so far.
CLASS: ControlProblem
NAME: tol
DESCRIPTION: the tolerance desired.
CLASS: double
OPTIONAL INPUT PARAMETERS
NAME: InitialLengthStep
DESCRIPTION: This paramter is the length step of the gradient method that is going to be used at the begining of the process. By default, this is 0.1.
CLASS: double
NAME: MinLengthStep
DESCRIPTION: This paramter is the lower bound on the length step of the gradient method if the algorithm needs to have a step size lower than this size it will make the GradientMethod stop.
CLASS: double
OUTPUT PARAMETERS:
All the updates will be carried inside the iCP control problem object.
Name: Unew
Description: Update of the Control
class: a vector valued function in a form of a double matrix
Name: Ynew
Description: Update of State Vector
class: a vector valued function in a form of a double matrix
Name: Jnew
Description: New Value of functional
class: double
Name: dJnew
Description: New Value of gradient
class: a vector valued function in a form of a double matrix
Name: error
Description: Nthe error |dJ|/|U| or |dJ| depending on the choice of the user.
class: double
Name: stop
Description: if this parameter is true, the routine will tell to GradientMethod to stop
class: logical
Citations:
[1] Cohen, William C.,Optimal control theory—an introduction, Donald E. Kirk, Prentice Hall, Inc., New York (1971), 452 poges. $13.50,https://onlinelibrary.wiley.com/doi/abs/10.1002/aic.690170452