Nonlinear Optimization Solution

Nonlinear Optimization solution uses generateNLJuMPcodes to solve the Problems. The arguments are the same as generateJuMPcodes

OptControl.generateNLJuMPcodes — Function

generateNLJuMPcodes(L, F, state, u, tspan, t0, tf) -> Any
generateNLJuMPcodes(L, F, state, u, tspan, t0, tf, Φ) -> Any
generateNLJuMPcodes(
    L,
    F,
    state,
    u,
    tspan,
    t0,
    tf,
    Φ,
    tf_constraint;
    state_ub,
    state_lb,
    u_ub,
    u_lb,
    N,
    discretization,
    model_name,
    writeFilePath,
    tolerance
) -> Any

generateNLJuMPcodes is used to solve Nonlinear control 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

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

tf_constraint: special requirements for end time, default: nothing. Example:

\[x_{1}+x_{2}=0\]

state_ub: state's upper limit, length of state_ub must be equal to length of state, default: nothing
state_lb: state's lower limit, length of state_lb must be equal to length of state, default: nothing
u_ub: u's upper limit, length of u_ub must be equal to length of u, default: nothing
u_lb: u's lower limit, length of u_lb must be equal to length of u, default: nothing
N: Number of discrete, default: 1000

\[dt = (endTime - startTime) / N\]

discretization: discretization methods, default: "trapezoidal"
model_name: name of JuMP model, default: "model"
writeFilePath: path of generated code, default: nothing
tolerance: acceptable_tol of Ipopt, default: 1.0e-6

Example:

using OptControl,ModelingToolkit
@variables t u x
f = exp(x) + u
L = 0.5 * u^2
t0 = [1.0]
tf = [0.0]
tspan = (0.0, 2.0)
N = 100
sol = generateNLJuMPcodes(L, f, x, u, tspan, t0, tf; N=N)

source

Let's see some examples. If you are not familar with ModelingToolkit.jl, you can see the Symbolics.jl document (which ModelingToolkit.jl based on) and try some tests about symbolic computation

using OptControl,ModelingToolkit
@variables x[1:2] y[1:2]
print(scalarize(rand(1:10,2,2)*x+rand(1:10,2,2)*y))

Num[3x[2] + 4y[1] + 6y[2] + x[1], 2x[1] + 4x[2] + 6y[1] + 6y[2]]

PS: scalarize is from Symbolics.jl

If you need, pass a name to writeFilePath and do some changes in script.

Example 1

To solve:

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

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

using OptControl, ModelingToolkit, Test
@variables t u x
f = exp(x) + u
L = u^2
t0 = [1.0]
tf = [0.0]
tspan = (0.0, 2.0)
N = 100
sol = generateNLJuMPcodes(L, f, x, u, tspan, t0, tf; N=N)

@test isapprox.(0.0, sol[1][end, :], atol=1.0e-5) == [true for i in 1:length(sol[1][end, :])]

Test Passed

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

using OptControl, ModelingToolkit, Test
@variables t u x[1:2]
f = [exp(x[1]) + cos(x[2]), sin(x[1]) + x[2]] + [1, 0] * u
L = u^2
t0 = [1.0, 1.0]
tf = [0.0, 0.0]
tspan = (0.0, 2.0)
N = 100
sol = generateNLJuMPcodes(L, f, x, u, tspan, t0, tf; N=N)
@test isapprox.(0.0, sol[1][end, :], atol=1.0e-5) == [true for i in 1:length(sol[1][end, :])]

Test Passed

Nonlinear Optimization Solution

Example 1

Example 2