Add a toy 2D system with nonlinear dynamics.

[hongkai-dai]

We can certify the CLF/CBF are compatible for this toy system.

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This change is 

[hongkai-dai]

+@Chuanruijiang for review please, thanks!

Reviewable status: 0 of 5 files reviewed, all discussions resolved (waiting on @Chuanruijiang)

[Chuanruijiang]

Reviewed 2 of 5 files at r1, all commit messages.
Reviewable status: 2 of 5 files reviewed, 2 unresolved discussions (waiting on @hongkai-dai)

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compatible_clf_cbf/clf_cbf.py line 575 at r1 (raw file):

        Note that this function only adds the constraint
        -(1+ϕᵢ,₀(x))*bᵢ(x) +∑ⱼϕᵢ,ⱼ(x)pⱼ(x) is sos
        It doesn't add the constraint ϕᵢ,₀(x), ϕᵢ,ⱼ(x) are sos.

Why we don't need these \phi(x) polynomials to be SOS?

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examples/nonlinear_toy/toy_system.py line 24 at r1 (raw file):

                    {
                        sym.Monomial(x[0]): sym.Expression(-1),
                        sym.Monomial(x[0], 3): sym.Expression(-1.0 / 6.0),

Maybe sym.Expression(1.0/6.0)? I saw that the system dynamics above is dot(x1) = -x0 + 1/6(x0)^3

[hongkai-dai]

Reviewable status: 1 of 5 files reviewed, 2 unresolved discussions (waiting on @Chuanruijiang)

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compatible_clf_cbf/clf_cbf.py line 575 at r1 (raw file):

Previously, Chuanruijiang wrote…

Why we don't need these \phi(x) polynomials to be SOS?

Because we will use this function in two cases (during bilinear alternation)

1. Fix the Lagrangian multiplier \phi(x), search for the CLF/CBF.
2. Fix the CLF/CBF, search for the Lagrangian multiplier \phi.

In the second case, we will need the condition that \phi(x) is sos; while in the first case we do not.

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examples/nonlinear_toy/toy_system.py line 24 at r1 (raw file):

Previously, Chuanruijiang wrote…

Maybe sym.Expression(1.0/6.0)? I saw that the system dynamics above is dot(x1) = -x0 + 1/6(x0)^3

Done. Good catch

[Chuanruijiang]

Reviewable status: 1 of 5 files reviewed, 2 unresolved discussions (waiting on @hongkai-dai)

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compatible_clf_cbf/clf_cbf.py line 575 at r1 (raw file):

Previously, hongkai-dai (Hongkai Dai) wrote…

Because we will use this function in two cases (during bilinear alternation)

1. Fix the Lagrangian multiplier \phi(x), search for the CLF/CBF.
2. Fix the CLF/CBF, search for the Lagrangian multiplier \phi.

In the second case, we will need the condition that \phi(x) is sos; while in the first case we do not.

Ah, I see. So in this file, "class compatible_cbfclf " should be used when we fix the lagrangian and find b(x),V(x). And the first three classes are used when we fix the b(x), V(x) and compute the lagrangian. Is that correct?

[hongkai-dai]

Reviewable status: 1 of 5 files reviewed, 1 unresolved discussion (waiting on @Chuanruijiang)

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compatible_clf_cbf/clf_cbf.py line 575 at r1 (raw file):

Previously, Chuanruijiang wrote…

Ah, I see. So in this file, "class compatible_cbfclf " should be used when we fix the lagrangian and find b(x),V(x). And the first three classes are used when we fix the b(x), V(x) and compute the lagrangian. Is that correct?

class CompatibleClfCbf will be used in both cases: 1) when we fix the Lagrangian and 2) when we fix the CLF/CBF. In both cases we will call this function _add_barrier_safe_constraint. But when we search for the Lagrangian, we will additionally add the constraint that the Lagrangian is sos.

[Chuanruijiang]

Reviewed 2 of 5 files at r1, 2 of 2 files at r2, all commit messages.
Reviewable status: complete! all files reviewed, all discussions resolved (waiting on @hongkai-dai)