Things are connected!
Thank you so much!
Hello, everyone, I'm happy to share that function works very well in my current one-dimensional case, and I will test two-dimensional case recently. Also, I am wondering could you help me with two quick questions.
- can we plot a rectangle with different patterns rather than colors only, e.g., north west lines as follows
- the result of LazySets.Interval(0, 7) ∪ LazySets.Interval(5, 7) has two elements, (0, 7) and (5, 7). Actually, it should be only one (0, 7). The disadvantage of two elements here is that if we want to plot this result with
alpha=0.5, darker color will appear at region (5,7) due to the overlap.
Hello everyone, I guess I found one case that function doesn't work but it is in the set operation table list in the document. Could you please help me to check it?
P = VPolygon([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0]]), Q = UnionSetArray([Hyperrectangle([0.0, 0.0],[3.0, 3.0])]), P ⊆ Q
which document do you mean?
yes, the current implementation only applies to hyperrectangles, but it is actually straightforward to generalize it to polyhedra. i added the code in JuliaReach/LazySets.jl#3012 (you can just define it in your script)
yes, the current implementation only applies to hyperrectangles, but it is actually straightforward to generalize it to polyhedra. i added the code in JuliaReach/LazySets.jl#3012 (you can just define it in your script)
Thanks for your great help. It works now. I want to share that it does not support input -
After changing input
UnionSetArray of hyperrectangle. The reason is that isdisjoint of VPolygon and Hyperrectangle doesn't work. I also found the same problem in intersection function several days ago.After changing input
hyperrectangle to HPolyhedron by writing a small function, then they all work very well now.
I have another small question. I guess I didn't find the correct method.
I want to find the reachable set at next time with initial state X0 and input U under discrete function f as follows.
I want to find the reachable set at next time with initial state X0 and input U under discrete function f as follows.
@taylorize function f(x⁺, x, u)
x⁺[1] = u[1]
x⁺[2] = u[2]
x⁺[3] = x[1]*x[2]
return x⁺
end
X0 = Hyperrectangle([0,0,0], [1,2,1])
U = Hyperrectangle([0,0],[1,1])
I am trying to construct a
ivp as follows and then how to solve it? It seems simple solve doesn't work. (I'm not sure). Or, is there a better way?@taylorize function f(x⁺, x, p, t)
x⁺[1] = x[4]
x⁺[2] = x[5]
x⁺[3] = x[1]*x[2]
x⁺[4] = zero(x[1])
x⁺[5] = zero(x[1])
return x⁺
end
X0 = Hyperrectangle([0,0,0, 0,0], [1,2,1,1,1])
prob = @ivp(x⁺ = f(x), dim:5, x(0) ∈ X0)
unfortunately, we do not support
solve for discrete systems, only for continuous systems (i guess the reason is that we never needed it). technically discrete systems are easier to handle than continuous systems, but it is still a bit of work
i think you could use https://juliareach.github.io/LazySets.jl/dev/lib/lazy_operations/QuadraticMap/#LazySets.QuadraticMap with
overapproximate(qm::QuadraticMap{N, <:AbstractZonotope}, ::Type{<:Zonotope}) where {N}
(see this page) to propagate the set under f
you could also use sparse polynomial zonotopes..
Thanks for your help again @mforets . It works for above simple example, but for one item with both x[1] and x[1]*x[2] as follows, it seems that quadraticmap doesn't work.
x_next[1] = x[1] + x[1]*x[2]
x_next[2] = x[2] + x[1]^2I also tried to solve it by SimpleSparsePolynomialZonotope. It can deal with x[1] + x[1]*x[2], but the default interval of x[1] and x[2] are both [-1,1] and can not change. If I want to consider [25, 30] x [-5,0], it seems too complex to adjust the parameters in SimpleSparsePolynomialZonotope.
Is there a way to solve this case?
This is the third case study in our work (maybe also the last one). The first two is well demonstrated by JuliaReach and LazySets and they are 1-d and 2-d simple systems respectively. I can say JuliaReach is the most complete tool in this field I have ever seen, although it is still under developing. In the third case, we want to use a more complex system. For discrete system, it seems hard to compute the reachable set one time instant later, as I said above.
We have an alternative, using continuous systems. Then the question becomes: After solving
@ivp, sol = solve(prob, T=10.0), how to get the solution at a specific time from sol, e.g., the solution at instant t=3? I didn't find the answer since the document of Flowpipe is still empty now.
the interest about the sparse polynomial zonotope (for this purpose you can try
SimpleSparsePolynomialZonotope) is that the quadratic map is computed without approximation
the linear part is the identity, and the quadratic map is given by
Q1 = [0 1; 0 0] and Q2 = [1 0; 0 0]
- approximated using
SingleSparsePolynomialZonotope(since it uses minkowski sum) - computed exactly using the
exact_sumSparsePolynomialZonotope, which will be added in an upcoming PR
It can deal with x[1] + x[1]*x[2], but the default interval of x[1] and x[2] are both [-1,1] and can not change
Ah, I think I understood what you meant by that. If you want to handle the linear terms as generators, that is also possible. I was proposing to add the linear terms with the quadratic part using set operations. This is cleaner, since you don't have to manually adjust the weights of the generators.
Yeah, I can write it as
I * X + QuadraticMap([Q1, Q2], X). The advantage of SingleSparsePolynomialZonotope is that It can also deal with higher-order terms, not just quadratic terms. However, if I want to consider the X domain which is not [-1,1], it is too complex to adjust the weights of generators, especially for high order term.
If you want to handle the linear terms as generators, that is also possible. I was proposing to add the linear terms with the quadratic part using set operations.
Glad to hear that. If we can add the linear terms with the quadratic part, it seems we can compute all the reachable set of quadratic discrete-time system. I am wondering when will you (or your group) be able to finish it?
Despite all of this, it seems there is still a long way to achieve reachable set of any polynomial discrete-time system. But I totally understand it, since it is still under developing. Actually, I also tried a tool called CORA in matlab, in which nonlieanr discrete-time system reachable set computation has been well developed. However, except from reachable set computation, I also need some set operations like
intersection, union, issubset, in. I can say JuliaReach is the most suitable tool for me to finish our work, but unfortunately I was stuck in this reachable set computation of discrete-time system now.
@Xinyi-Yu if you used the
for the linear algorithms i think there is no such convenience functionality, but it can be done with some tricks
TMJets algorithm (standard for nonlinear problems), you can just use evaluate(sol, 3.0) as in the example herefor the linear algorithms i think there is no such convenience functionality, but it can be done with some tricks
So @Xinyi-Yu I hear that you've tried with
SimpleSparsePolynomialZonotope, but I'm not sure I understand the issue you mention about adjusting weights.
Besides, we have a now a new set type called
SparsePolynomialZonotope which can handle set sums in an exact fashion, so it should give more precise results than the SimpleSparsePolynomialZonotope.
