Mixing
Time-Discretize a Linear Quadratic OCP (Bolza Function)
I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function:
$$J(t_f)=frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + frac{1}{2}int_{t_0}^{t_f}(1+[x^t(t)Qx(t)+u^T(t)Ru(t)])dt$$
$$ text{ where } x=begin{bmatrix}V\gamma\h\rend{bmatrix}P_finmathbb{R}^{4x4}, Qinmathbb{R}^{4x4}, Q_xinmathbb{R}^{4x4}, R=begin{bmatrix}1end{bmatrix}text{ and } u=theta=text{Pitch angle}$$
I would like to minimize the time for a small RC aircraft to go from takeoff at time $t_0$ to cruise in the shortest amount of time $t_f$. I think that there are basically 2 approaches to solving this.
- I can use the ACADO Matlab toolkit to solve the function in this form. I've tried doing this and I cannot seem to understand ACADO well enough to solve it.
- First time-discretize the cost function to put the function in a form which can be solved using another Matlab toolkit such as YALMIP.
Seeing as I've failed on approach #1, I would like to try approach #2. However, I do not understand how to time-discretize this function. I've tried looking through textbooks for how to solve these problems, but I cannot find any text that gives a method to discretize a function like this into a form which can be solved with a solver.
I have initial and final constraints, and have the state space model, but do not really know how to proceed from here
$$dot{x}=begin{bmatrix}dot{V}=frac{1}{m}(Tcos(u-gamma)-D-mgsin(gamma)\dot{gamma}=frac{1}{mV}(Tsin(u-gamma)+L-mgcos(gamma)\dot{h}=Vsin(gamma)\dot{r}=Vcos(gamma)end{bmatrix}$$
I've read that I may need to solve for the Hamiltonian, but I don't understand why, and still would not understand how to proceed solving the problem in matlab once I would solve for it.
control-theory optimal-control
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
|
show 4 more comments
I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function:
$$J(t_f)=frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + frac{1}{2}int_{t_0}^{t_f}(1+[x^t(t)Qx(t)+u^T(t)Ru(t)])dt$$
$$ text{ where } x=begin{bmatrix}V\gamma\h\rend{bmatrix}P_finmathbb{R}^{4x4}, Qinmathbb{R}^{4x4}, Q_xinmathbb{R}^{4x4}, R=begin{bmatrix}1end{bmatrix}text{ and } u=theta=text{Pitch angle}$$
I would like to minimize the time for a small RC aircraft to go from takeoff at time $t_0$ to cruise in the shortest amount of time $t_f$. I think that there are basically 2 approaches to solving this.
- I can use the ACADO Matlab toolkit to solve the function in this form. I've tried doing this and I cannot seem to understand ACADO well enough to solve it.
- First time-discretize the cost function to put the function in a form which can be solved using another Matlab toolkit such as YALMIP.
Seeing as I've failed on approach #1, I would like to try approach #2. However, I do not understand how to time-discretize this function. I've tried looking through textbooks for how to solve these problems, but I cannot find any text that gives a method to discretize a function like this into a form which can be solved with a solver.
I have initial and final constraints, and have the state space model, but do not really know how to proceed from here
$$dot{x}=begin{bmatrix}dot{V}=frac{1}{m}(Tcos(u-gamma)-D-mgsin(gamma)\dot{gamma}=frac{1}{mV}(Tsin(u-gamma)+L-mgcos(gamma)\dot{h}=Vsin(gamma)\dot{r}=Vcos(gamma)end{bmatrix}$$
I've read that I may need to solve for the Hamiltonian, but I don't understand why, and still would not understand how to proceed solving the problem in matlab once I would solve for it.
control-theory optimal-control
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32
|
show 4 more comments
I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function:
$$J(t_f)=frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + frac{1}{2}int_{t_0}^{t_f}(1+[x^t(t)Qx(t)+u^T(t)Ru(t)])dt$$
$$ text{ where } x=begin{bmatrix}V\gamma\h\rend{bmatrix}P_finmathbb{R}^{4x4}, Qinmathbb{R}^{4x4}, Q_xinmathbb{R}^{4x4}, R=begin{bmatrix}1end{bmatrix}text{ and } u=theta=text{Pitch angle}$$
I would like to minimize the time for a small RC aircraft to go from takeoff at time $t_0$ to cruise in the shortest amount of time $t_f$. I think that there are basically 2 approaches to solving this.
- I can use the ACADO Matlab toolkit to solve the function in this form. I've tried doing this and I cannot seem to understand ACADO well enough to solve it.
- First time-discretize the cost function to put the function in a form which can be solved using another Matlab toolkit such as YALMIP.
Seeing as I've failed on approach #1, I would like to try approach #2. However, I do not understand how to time-discretize this function. I've tried looking through textbooks for how to solve these problems, but I cannot find any text that gives a method to discretize a function like this into a form which can be solved with a solver.
I have initial and final constraints, and have the state space model, but do not really know how to proceed from here
$$dot{x}=begin{bmatrix}dot{V}=frac{1}{m}(Tcos(u-gamma)-D-mgsin(gamma)\dot{gamma}=frac{1}{mV}(Tsin(u-gamma)+L-mgcos(gamma)\dot{h}=Vsin(gamma)\dot{r}=Vcos(gamma)end{bmatrix}$$
I've read that I may need to solve for the Hamiltonian, but I don't understand why, and still would not understand how to proceed solving the problem in matlab once I would solve for it.
control-theory optimal-control
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function:
$$J(t_f)=frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + frac{1}{2}int_{t_0}^{t_f}(1+[x^t(t)Qx(t)+u^T(t)Ru(t)])dt$$
$$ text{ where } x=begin{bmatrix}V\gamma\h\rend{bmatrix}P_finmathbb{R}^{4x4}, Qinmathbb{R}^{4x4}, Q_xinmathbb{R}^{4x4}, R=begin{bmatrix}1end{bmatrix}text{ and } u=theta=text{Pitch angle}$$
I would like to minimize the time for a small RC aircraft to go from takeoff at time $t_0$ to cruise in the shortest amount of time $t_f$. I think that there are basically 2 approaches to solving this.
- I can use the ACADO Matlab toolkit to solve the function in this form. I've tried doing this and I cannot seem to understand ACADO well enough to solve it.
- First time-discretize the cost function to put the function in a form which can be solved using another Matlab toolkit such as YALMIP.
Seeing as I've failed on approach #1, I would like to try approach #2. However, I do not understand how to time-discretize this function. I've tried looking through textbooks for how to solve these problems, but I cannot find any text that gives a method to discretize a function like this into a form which can be solved with a solver.
I have initial and final constraints, and have the state space model, but do not really know how to proceed from here
$$dot{x}=begin{bmatrix}dot{V}=frac{1}{m}(Tcos(u-gamma)-D-mgsin(gamma)\dot{gamma}=frac{1}{mV}(Tsin(u-gamma)+L-mgcos(gamma)\dot{h}=Vsin(gamma)\dot{r}=Vcos(gamma)end{bmatrix}$$
I've read that I may need to solve for the Hamiltonian, but I don't understand why, and still would not understand how to proceed solving the problem in matlab once I would solve for it.
control-theory optimal-control
control-theory optimal-control
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
edited Nov 21 '18 at 19:33
RocketSocks22
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
asked Nov 21 '18 at 18:22
RocketSocks22RocketSocks22
184
184
I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32
|
show 4 more comments
I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32
I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32
|
show 4 more comments
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I assume $u=T$?
– Kwin van der Veen
Nov 21 '18 at 18:59
u is the pitch angle $theta$
– RocketSocks22
Nov 21 '18 at 19:06
So $dot{x}$ is not a function of $u$?
– Kwin van der Veen
Nov 21 '18 at 19:12
I suppose it could be in that $theta = gamma + alpha$ where it can reasonably be assumed that $alpha$ will be constant for this problem. Doing so would simply replace my state variable $gamma$ though, so perhaps this is not the best way to go about this problem? I currently have my thrust modeled as $T=0.5*rho V^2SC_D$ where $C_D$ is a function of $alpha$
– RocketSocks22
Nov 21 '18 at 19:22
^Actually disregard that comment. I updated my problem statement, replacing $alpha$ with $theta-gamma$ to better see the relation
– RocketSocks22
Nov 21 '18 at 19:32