Why does XY Graph affect algebraic loop solver?

I have a model involving an algebraic loop that mysteriously refuses to solve ONLY when an XY Graph is attached. I don't think the model's exact purpose is important, but you can read more about it here. I'm using Matlab & Simulink 2011a. The observations I think are important are:

• This model works fine as-is
• When you change Constant1 from 4.8 to 5, the model will no longer solve -- you'll get the error message "Algebraic state in algebraic loop containing 'algebraicscope/Sum1' computed at time 0.0 is Inf or NaN"
• When you then delete the XY Graph block from the model (and leave Constant1 as 5), the model will again solve just fine (the normal scope can be used to verify results; there should be a constant line of value 0 and a constant line of value 4)

It is the simple presence of the XY Graph that prevents Simulink from solving the model under certain conditions! As a beginner in Simulink, it is extremely disturbing to me that the presence of a read-only debugging tool can affect the solution of a model. Am I thinking about something wrong, or is this actually a bug?

This is a graphic of the model in question:

Answer 1

[Doug Eastman]

The problem is that adding blocks to your model (such as the XY Graph) causes the sorted order of all the blocks to change. Since there are no clear data dependencies in an algebraic system like this (which Simulink normally can use to provide a consistent sorted order), adding blocks can cause seemingly random changes. There is nothing special about the XY Block in particular, you will get the same issue if you add a display block to both the X and Y signals for example.

I believe the problem with changing the sorted order is that depending on which blocks are executing first, Simulink will automatically select an initial guess for that block's output. If that guess happens to be poor, the solution will diverge and you get no answer. If that guess is reasonable then it will converge on a solution.

When you want to force Simulink to solve an algebraic equation, a good approach is to use the Algebraic Constraint block. This allows you to explicitly specify an initial guess for one of the parameters and have a more consistent result. For example, here is a revision of your model using the Algebraic Constraint block with X as the algebraic state:

[Ben]

Thanks, this looks extremely promising and makes total sense. I'm not sure exactly what the Algebraic Constraint is supposed to do, but when I add it into the loop, it ensures the error rather than eliminating it -- I'll guess I'll have to read up on its proper use.

BUT, I noticed that the Algebraic Constraint model contained an "Initial Guess" block. I'm not sure where that comes from because I can't find it in my Simulink Library search, but when I copied it out of the Algebraic Constraint model and pasted it directly into the Algebraic Loop, that solved all my problems.

Thanks again for your explanation! Do you have any suggestions where I should read about these kinds of considerations outside of the basic Simulink documentation?

[Doug Eastman]

The Algebraic Constraint block is in the Math Operations library. The way it works is that you send in a signal that you want to constrain to zero. Basically you need to rewrite your equation as a function of one variable that equals zero. In your case I subtracted the calculated Y of the left rod from the Y of the right rod. Since these should be the same when they are connected, the difference should be zero (which I send as the input to the block). The output of the block is the algebraic variable, in your case the X location of the connection.

Honestly you pretty much always want to avoid algebraic loops in your Simulink models. If you have algebraic equations like in this case, it's better to solve them analytically by hand or possibly with the Symbolic Math toolbox and then use the closed-form solution in Simulink. Because of that I don't know of any good resources outside of the documentation that describes it in much detail. But there is a lengthy section of the documentation here that should give you more than enough information.

[Azzi Abdelmalek]

I 'm not really convinced by that the sorted order will be changed by the scope, because that scope is not in the loop. Also in the algebraic loop there is no any sorted order, if there is any, I don't think it will be called an algebraic loop.

[Doug Eastman]

You can very easily prove to yourself that the sorted order changes by displaying the block sorted order before and after deleting the XY Scope (or other blocks for that matter). You'll notice the order shifts dramatically.

Here is the sorted order with the scope:

---- Sorted list for 'algebraicscope' [9 nonvirtual blocks, directFeed=0]

0:0    'algebraicscope/Constant2' (Constant, tid=1)
0:1    'algebraicscope/Constant1' (Constant, tid=1)
0:2    'algebraicscope/rl^2' (Product, tid=1)
0:3    'algebraicscope/Constant3' (Constant, tid=1)
0:4    'algebraicscope/rr^2' (Product, tid=1)
0:5    'algebraicscope/Constant' (Constant, tid=1)
0:6    'algebraicscope/<< Synthesized_Atomic_Subsystem_For_Alg_Loop_1 >>' (SubSystem, tid=0)
0:7    'algebraicscope/XY Graph' (SubSystem, tid=1)
0:8    'algebraicscope/Scope' (SignalViewerScope, tid=0)

---- Sorted list for 'algebraicscope/<< Synthesized_Atomic_Subsystem_For_Alg_Loop_1 >>' [8 nonvirtual blocks, directFeed=1]

1:0    'algebraicscope/dx^2' (Product, tid=0) [algId=1#1]
1:1    'algebraicscope/Sum' (Sum, tid=0) [algId=1#1]
1:2    'algebraicscope/Sqrt' (Sqrt, tid=0) [algId=1#1]
1:3    'algebraicscope/y^2' (Product, tid=0) [algId=1#1]
1:4    'algebraicscope/Sum2' (Sum, tid=0) [algId=1#1]
1:5    'algebraicscope/Sqrt1' (Sqrt, tid=0) [algId=1#1]
1:6    'algebraicscope/Sum3' (Sum, tid=0) [algId=1#1]
1:7    'algebraicscope/Sum1' (Sum, tid=0) [algId=1#1, algVar]

And the order without:

---- Sorted list for 'algebraicscope' [8 nonvirtual blocks, directFeed=0]

0:0    'algebraicscope/Constant' (Constant, tid=1)
0:1    'algebraicscope/Constant1' (Constant, tid=1)
0:2    'algebraicscope/Constant2' (Constant, tid=1)
0:3    'algebraicscope/Constant3' (Constant, tid=1)
0:4    'algebraicscope/rr^2' (Product, tid=1)
0:5    'algebraicscope/rl^2' (Product, tid=1)
0:6    'algebraicscope/<< Synthesized_Atomic_Subsystem_For_Alg_Loop_1 >>' (SubSystem, tid=1)
0:7    'algebraicscope/Scope' (SignalViewerScope, tid=1)

---- Sorted list for 'algebraicscope/<< Synthesized_Atomic_Subsystem_For_Alg_Loop_1 >>' [8 nonvirtual blocks, directFeed=1]

1:0    'algebraicscope/Sum2' (Sum, tid=1) [algId=1#1]
1:1    'algebraicscope/Sqrt1' (Sqrt, tid=1) [algId=1#1]
1:2    'algebraicscope/Sum3' (Sum, tid=1) [algId=1#1]
1:3    'algebraicscope/Sum1' (Sum, tid=1) [algId=1#1]
1:4    'algebraicscope/dx^2' (Product, tid=1) [algId=1#1]
1:5    'algebraicscope/Sum' (Sum, tid=1) [algId=1#1]
1:6    'algebraicscope/Sqrt' (Sqrt, tid=1) [algId=1#1]
1:7    'algebraicscope/y^2' (Product, tid=1) [algId=1#1, algVar]

You can see that with the scope, the algebraic variable is chosen as the output of Sum1. Without the scope the algebraic variable is chosen as the output of y^2.

Answer 2

[Azzi Abdelmalek]

Ben, In the previous question http://www.mathworks.com/matlabcentral/answers/58271-why-does-algebraic-loop-not-solve I've simulated the model without XY graph, it works, then added an XY graph, it works again. I've also simulated the second model which did'nt work for reasons that I ignore, but have nothing to do with xy graph

[Ben]

Thanks for testing -- could you check this model without any changes to make sure we're talking about the same thing? That model breaks for me in 2011a. If it works for you, what version of Matlab & Simulink are you using? Thanks!

[Azzi Abdelmalek]

2011b, this file did'nt work

[Ben]

Just to be clear, here's what I experience:

1. Link in this question (algebraicscope.mdl): Works
2. After then changing Constant1 from 4.8 to 5: Doesn't work
3. After then deleting the XY Graph: Works
4. Link in this comment (algebraicscope_broken.mdl): Doesn't work (should be identical to step 2 above)
5. Link in previous question: Doesn't work

It sounds like you have tried #4 above and it was consistent with my experience, and two configurations not listed above but based on #5. If that's correct, it sounds like we're observing the same results.

[Azzi Abdelmalek]

Ben, in fact there is just one thing that allows: it works or it does'nt work, If the algebraic loop has solution it works, otherwise it does'nt. In your example when you add xy-graph it don't work, replace this xy graph with any display block(scope, to workspace,...), because there is no solution to display, simulink shows error, when you remove those display blocks, the simulation does'nt work, and because you are not asking for displaying result, simulink does not show errors. In fact it's not an error, it's just that the solution is [], which can't be displayed.

[Ben]

I don't think you understand what I'm asking and/or my observations. The model I posted for this question has two display mechanisms; the X and Y signals are each connected both to the XY Graph block, and to a scope not shown (you can tell they are connected to that scope by observing the two blue sunglasses icons in the model). There is a single, correct solution in all scenarios I've mentioned in this question. When the model doesn't crash, the off-screen scope always displays the correct answer. However, the model crashes when the XY Graph is present, and then doesn't crash when the XY Graph is removed.

If the algebraic loop has solution it works, otherwise it does'nt.

Yes. The question is why simply removing the XY Graph changes whether the algebraic loop has a solution.

when you add xy-graph

I never add an XY Graph; the original model starts with one, and then I remove it.

replace this xy graph with any display block(scope, to workspace,...)

No replacement is necessary; there is already a second display mechanism present in the original model.

when you remove those display blocks, the simulation does'nt work

No, that's reversed; the simulation works when the XY Graph display block is removed and doesn't work when it's present.

because you are not asking for displaying result, simulink does not show errors.

I am asking for a result to be displayed in all scenarios via the off-screen scope indicated by the two blue glasses icons in the model.

In fact it's not an error, it's just that the solution is [], which can't be displayed.

No. This is the error:

This shows the exact same model that produced that message above working properly. The only change was to delete the XY Graph block.

[Azzi Abdelmalek]

Replace your xy-graph with scopes, you will have the same message. I know that with viewers scope it works.