Differential Equations Solution

Differential Equations Solution uses DifferentialEquations.jl to solve the Problems. Build the problem and call generateDEcodes, work done!

OptControl.generateDEcodes — Function

generateDEcodes(L, F, state, u, tspan, t0, tf) -> Any
generateDEcodes(L, F, state, u, tspan, t0, tf, Φ) -> Any
generateDEcodes(
    L,
    F,
    state,
    u,
    tspan,
    t0,
    tf,
    Φ,
    dt;
    guess,
    writeFilePath
) -> Any

generateDEcodes will generate the solution code that using DifferentialEquations.jl to solve problem

A general form of optimal control problem:

\[min (\Phi(\boldsymbol{x}(t_f),t_f)+\int_{t_0}^{t_f} L[\boldsymbol{x}(t),\boldsymbol{u}(t),t]dt) \\ s.t. \hspace{0.4cm} \dot{\boldsymbol{x}} =f[\boldsymbol{x}(t),\boldsymbol{u}(t),t]\]

args:

L: above L

\[L[\boldsymbol{x}(t),\boldsymbol{u}(t),t]\]

F: above f

\[f[\boldsymbol{x}(t),\boldsymbol{u}(t),t]\]

state: states(variable) in F
u: control variable u in F
tspan: time field
t0: initial value at tspan[1], length of t0 must be equal to length of state
tf: final(end) value at tspan[2], length of tf must be equal to length of state
Φ: above Φ, default: nothing

\[\Phi(\boldsymbol{x}(t_f),t_f)\]

dt: dt of BVProblem, default: 0.1
guess: initial value of BVProblem, default: 1.0. length of guess must be equal to (2*length(state)+length(u))
writeFilePath: path of generated code, default: nothing

Example:

using OptControl
@variables t u x[1:2]
f = [0 1; 0 0] * x + [0, 1] * u
L = 0.5 * u^2
t0 = [1.0, 1.0]
tf = [0.0, 0.0]
tspan = (0.0, 2.0)
sol = generateDEcodes(L, f, x, u, tspan, t0, tf)

source

Overview of generateDEcodes and generateJuMPcodes

generateDEcodes and generateJuMPcodes using the same way to build problem, and they both can solve Fixed Value(Example1) and Free End(Example2). But generateDEcodes cannot deal with End constraint and Add variable limit. Meanwhile, generateDEcodeshave lower accuracy than generateJuMPcodes. Maybe accuracy can be improved by pass different solver parameters. You can have a try.

Example	generateJuMPcodes	generateDEcodes
1. Fixed Value	✅	✅
2. Free Value	✅	✅
3. Constraint	✅	❌
4. Multiple `x` and `u`	✅	✅
5. variable limit	✅	❌

✅ = supported

❌ = not supported

Example 1: Fixed value

To solve:

\[min \int_{0}^{2} u^2dt \newline s.t. ~~~~~ \dot{\boldsymbol{x}} =\begin{bmatrix}0&1 \newline 0&0\end{bmatrix}\boldsymbol{x}+ \begin{bmatrix}0 \newline 1 \end{bmatrix}u \newline \boldsymbol{x}(0) = \begin{bmatrix} 1 \newline 1 \end{bmatrix}, \boldsymbol{x}(2)=\begin{bmatrix} 0 \newline 0 \end{bmatrix}\]

Just define variables and build functions. Call generateDEcodes and get the results.

The analytical solution of $x_1$ is

\[x_1(t) = 0.5*t^3-1.75*t^2+t+1\]

and we can campare the difference between them by using Mean Square Error(MSE).

using OptControl, Statistics, ModelingToolkit
@variables t u x[1:2]
f = [0 1; 0 0] * x + [0, 1] * u
L = 0.5 * u^2
t0 = [1.0, 1.0]
tf = [0.0, 0.0]
tspan = (0.0, 2.0)
sol = generateDEcodes(L, f, x, u, tspan, t0, tf)
nu = [sol.u[i][1] for i in 1:length(sol.u)]
xs = collect(range(tspan[1], tspan[2], length=length(sol.u)))
an = @.(0.5 * xs^3 - 1.75 * xs^2 + xs + 1)
mean((an - nu).^2)

7.279902475761595e13

Example 2: Free End

To solve:

If variable is free, use nothing.

The analytical solution of $x_1$ is

\[x_1(t) = t^3-3.0*t^2+t+1\]

and get MSE.

using OptControl, Statistics, ModelingToolkit
@variables t u x[1:2]
f = [0 1; 0 0] * x + [0, 1] * u
L = 0.5 * u^2
t0 = [1.0, 1.0]
tf = [0.0, nothing]
tspan = (0.0, 1.0)
N = 100
sol = generateDEcodes(L, f, x, u, tspan, t0, tf)
nu = [sol.u[i][1] for i in 1:length(sol.u)]
xs = collect(range(tspan[1], tspan[2], length=length(sol.u)))
an = @.(xs^3 - 3.0 * xs^2 + xs + 1)
mean((an - nu).^2)

0.020118421612599108

Example 3: Multiple `x` and `u`

To solve:

\[min \int_{0}^{2} u^2dt \newline s.t. ~~~~~ \dot{\boldsymbol{x}} =\begin{bmatrix}0&1 \newline 0&0\end{bmatrix}\boldsymbol{x}+ \begin{bmatrix}1&0 \newline 0&1 \end{bmatrix}\boldsymbol{u} \newline \boldsymbol{x}(0) = \begin{bmatrix} 1 \newline 1 \end{bmatrix}, \boldsymbol{x}(2)=\begin{bmatrix} 0.0 \newline free \end{bmatrix} \]

The analytical solution of $u_1,u_2$ is

\[\begin{matrix} u_1=-\frac{9}{14}\\u_2=\frac{9}{14}*t-\frac{9}{7} \end{matrix} \]

and get MSE.

using OptControl, Statistics, ModelingToolkit
@variables t u[1:2] x[1:2]
f = [0 1; 0 0] * x + [1 0; 0 1] * u
L = 0.5 * (u[1]^2 + u[2]^2)
t0 = [1.0, 1.0]
tf = [0.0, nothing]
tspan = (0.0, 2.0)
sol = generateDEcodes(L, f, x, u, tspan, t0, tf)
nu1 = [sol.u[i][5] for i in 1:length(sol.u)]
nu2 = [sol.u[i][6] for i in 1:length(sol.u)]
xs = collect(range(tspan[1], tspan[2], length=length(sol.u)))
an_u2 = @.(9 / 14 * xs - 9 / 7)
res1 = mean((-9 / 14 .- nu1).^2)
res2 = mean((an_u2 - nu2).^2)
(res1,res2)

(1.4141364774585914e13, 1.3718426377342753e14)

Differential Equations Solution

Example 1: Fixed value

Example 2: Free End

Example 3: Multiple x and u

Example 3: Multiple `x` and `u`