Dear Anshuman,

The ball and beam system has a dynamic relative to a nonlinear model for testing different control algorithms, which is why it poses a problem with the PID controller. In order to improve the performance of this controller it is used in combination with the LQR controller. Here are examples in attached form that presents quite interesting solutions.

Best regards

I'm thinking there may be a problem with the way you have formulated your problem. I can't see how your computer/controller can output an angle. It may measure the angle of the beam and the control loop may receive an angle reference input from the user. So this gives an angle error as input to the controller. But I expect the output of your computer/controller (i.e. the input to your electro/mechanical system, or plant) will be a voltage or similar, not an angle.

Anyway... I don't think the ' rtu_ActualBeamAngle ' term is necessary.

The term of actual beam angle was not there when I performed the simulation, I read the same equation in a couple of research papers. That's why included this term but it never improved the controller.

Also in the plant model there is no use of voltage though, we can only modify the control angle to stabilize the ball.

My goal is to minimize the cumulative energy of the system.

Can you suggest me how to write the LQR algorithm in C++ in a generalised way? Like the structure of the algorithm . I've the parameters listed in the question , so how to start with them?

Mohamed-Mourad Lafifi

If you could specify an angle as in input to your process (i.e. plant), then get that same angle as an output, then there would be nothing to control!

This is going to make modeling your plant properly very difficult. Thus any principled approach to controller design (e.g. LQR) will most likely fail.

Here is a possible way forward (I don't suppose it matters what we call the plant input, let's just call it x(n) and the output y(n)) ...

Create a simple model of your plant by applying a pulse input x(n), with an amplitude equal to some 'sensible' value.

Log the output y(n) and plot it. There will most likely be a delay term (of many samples) plus something that can be modelled reasonably well by a second-order transfer function.

Derive the delay parameters, gain and pole positions by trial and error if you like, or look at Ogata's text book for a nice least-squares method.

With an approximate (discrete) transfer function defined, you can then think about deriving a controller.

I wouldn't bother deriving the dynamics of the beam and the ball from first principles, becasue there will be all sorts of terms that you are not aware of, like the behaviour of your servo, etc. And it will be the delays and non-linearities throughout the loop that will probably end up confounding you.

Hugh Lachlan Kennedy , the plant modeling is perfect as it is given to me by an organisation. I only have to change the controller design.

Here is the problem statement:

Consider a system consisting of a 1-D beam balancing a ball. The beam is pivoted

at its center and can be rotated about this point to tilt it in both directions.

The ball is in a stable equilibrium on the beam as long as the beam remains

horizontal. Your task is to design a controller for such a system to control the

angle of the beam so as to prevent the ball from rolling off the beam.

The model in the repository is exactly the same as described above with one

change in its initial condition. Instead of the beam being horizontal at the

beginning, it is held at an angle with the ball being at exactly the center of

the beam. It is thus obvious that without any change in the beam angle, the ball

will roll off the beam and fall down.

The system is configured to run for 10 seconds once the simulation begins. Design a controller for the system to control the beam angle such that the ball stays on the beam for the entire duration of simulation.

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My code above is performing the simulation , but it is stuck on the specific amount of energy usage. I want to reduce the amount of energy consumed but unable to do so after a certain extent.

Have a look at the files.

OK. I get it now. Beam angle is the plant input and ball position is the plant output.

Any specific method to reduce the control effort in PID?

My main target is to reduce the amount of energy spent.

Hugh Lachlan Kennedy

Do you need to use integral control? It results in a phase lag (not good for stability) and excessive gain at low-frequencies.

I expect the derivative term will be important though, for a phase lead - because it effectively anticipates where the ball will be in the future.

Well I agree to your point. Hugh Lachlan Kennedy

But on adding the integral term the control effort gets reduced more,than on using the PD.I don't know the exact reason for this though, trying to figure it out.

Also the values remain in the first quadrant only on using PD.