How to give a vector as input in Simulink, to solve a system of differential equations ?

I would like to solve the following system of differential equations. It mathematically describes a 2-DOF car model and it's a very common problem in scientific literature. I managed to solve it using DotNumerics library from C# but I would like to solve it using Matlab, as well.

From which:

So, I have tried to do it using Simulink and I have build the attached block diagram to solve the system.

The 'y' from the system represents the road, which I generated as an array of values using ISO 8608 standard, in a Matlab script.

The prolem is that I don't know how to give this array I generated as input to the attached Simulink model, below. I have to do this because the road is a function of time, so that in each moment of the time the excitation of the road is different.

I used a constant block with a constant value, temporarly, and I also tried to give the name of the array to this block, but certainly, these are not solutions to my problem.

Could you please help me to figure this out ? I have seen a few vague indications in some papers but nothing practical to solve my issue.

Maybe, I could consider to tackle the problem from a different angle by using some Matlab script libraries, instead. Nevertheless, any solution or example would be appreciated.

Thanks in advance.

[Paul]

Are the values in the y-array samples of a coninuous function of a time y(t)? If t is not the independent variable, what is?

How do you want to approximate y() between the samples in the y-array?

[Patrausanu Andrei]

The values from the y-array are generated using a PSD (power spectral density) function, including a cosine function. I attached the script I use to generated the y-array, here. The y-array is called 'road_front' in the script.

I should have mentioned that Ks, Cs, Ku, Cu, Ms, Mu are constants in the system. They have fixed values. The y-array is generated before starting to solve the system. I want to get xs and xs_dot_dot from the system.

Regarding the approximation, I didn't think of anyting so far. In my C# implementation there was no need to do something like this. I just provide the y-array to the Ode45 library. I guess it might be the same case here, as well. But, I if I am wrong, I am opened to any suggestion you may have.

[Paul]

I'm afraid I have to repeat my questions.

Are the values in the y-array samples of a function of time? If not, what is the independent variable?

I don't know anything about the C# implementation and so don't know what it assumed to solve the problem. In Simulink, solving a continuous-time system requires the solver to evaluate the inputs to the integrators at any instant in time. In this case, you have a discrete set of samples in the y-array of what is fundamentally a continous function, the road (I assume that's the height of the road above some baseline level). So, you'll need to decide how Simulink should compuate "the road" in between the samples provided in the y-array. You could assume that the road is constant in between samples, but that would lead to discontinuities in the road height, which sounds like a very bad road. You could assume the road is linearly interpolated between the samples, but then you'd have a discontinuous first derivative, which might not be realisitic either.

So, you'll need to specify how to model the road between samples of its independent variable and what that independent variable is before you can any further.

[Patrausanu Andrei]

I see what you mean. The values in the y-array are generated by a continuous function of time. The time is clearly the independent variable, as you also said. Regarding the interpolation, I would preffer to use the spline interpolation. Maybe, in the future, I may change it to linear, even though it has the drawbacks you talked about. Considering these two theoretical aspects clarified, how shall I proceed ? How can apply it in Simulink ? So far, as you can see in the atached Simulink file, I have been able to give Simulink only one constant value from the y-array.

Answer 1

[Paul]

Use a 1-D Lookup Table block with a Clock connected to its input.

Use the y-array and its corresponding times to specify the table data.

Use the other block parameters to specify the interpolation method and how you want the block to behave if it has to extrapolate off the end of the y-array.

If the y-array data are at equally spaced points in time, there are some other options to use in that block to make the lookup more efficient.

[Patrausanu Andrei]

That seems to be exactly what I have been looking for. Thank you! I did what you said but I have one final question. I am not 100% sure I did all the 'syncronization' settings. In the block diagram attached, I displayed the output of the 1-D Lookup Table in a Scope block and surprinsgly for me, I don't get the road signal as it is in LookUp. I guess there is something related to timing/sampling in the simulations that I am missing. Could you please take a quick look or suggest me what I am missing ? It's the first time I do this.

[Paul]

Sorry, I don't download and open Simulink files.

Not sure what 'syncronization' settings refers to.

If i understand what you have, you should only need change three parameters in th 1-D Lookup Table block, at least to get started. I'm going to assume you have yval and tval defined in the Matlab base workspace, where yval is the name of the 1-dimensional array that contains the road samples and tval is the name of the corresponding time array. Use whatever variable names you have

1. Under the Table and Breakpoints tab, put yval, i.e., the name of the variable, in the Value field of Table data
2. Under the Table and Breakpoints tab, put tval in the Value field of Breakpoints
3. Under the Algorithm tab, use the drop down menu next to Interpolation Method to pick what you want.

If that doesn't work, you can post some screen captures (use the icon next to the paper clip on the Insert menu) showing the block parameters or anything else that illustrates problems.

[Patrausanu Andrei]

In the LookUpTable block, I get my road vector. It is displayed correctly so I guess it should be ok. I was wondering about Breakpoints section. I gave it the time as [1:1000]. I always generate my time aray from 1 to 1000. My road array always have 1000 points. Is it correct ? Moreover, I was wondering about something else. I would have expected to see some other results, but it could be something wrong with my block diagram. If the road is properly given as input to it, and I guess it is, then it must be a problem with what I did. Thank you!

[Paul]

Simulation time starts at T=0. If that table lookup is called before T = 1, make sure to set the Extrapolation method parameters appropriately. If you really have 1000 evenly spaced points starting at t = 0, enter the breakpoints as 0:999. Or, set the Breakpoint Specification to 'even spacing' and set First Point to 0 and Spacing to 1.