Dear friends and colleagues! I have designed a Simulink model of an inverted pendulum control system. In order to study the LQR I have also linearised the model and obtained the K gain matrix for the controller, which works well for the linear system, now I am hoping to apply it to the non-linear system as well. However, I do not quite understand the reference for the control system.
The system has 4 states: position of the cart, pendulum angle and their derivatives. I am modeling a data acquisition system, so I assume I can only have encoders measuring the position and angle, and then I model software derivatives I could perform on a computer. Then I have a block to get u=K1*x1+ K2*x2+K3*x3+K4*x4. That seems to be my input in the system. I, however, assume there must be some comparison block, but I can not seem to wrap my head around it. In the diagram I have attached to this question you can see the reference is set to zero, but I did it merely to have some refernece in general. But what is this zero? What if my desired states are not zero? What if i want to move my reference to some other point during execution pf the program? Which one do I move? If I still want to stabilise the pendulum but not at x=0, but somewhere at x=ref?
Hi, so far, it seems you are on the right path. However, I have some remarks:
1. what's your control objective? I assume the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. Then that desired position and angle (with respect to your model) giving you the upright position are your controller reference (could be 180° or 0°, it depends which angle you took as the reference to develop your model).
2. Because you are thinking in implementation and as you said the only measurements available will be position and angle, then you would need an observer to estimate the other two states (derivatives). I recommend a Kalman Filter.
3. In order to better understand how to use your linear approximation to control your non-linear model, the following paper could help (which explain how to translate your linear model to the operating point of the nonlinear system):
Article Linearization: Students Forget the Operating Point
Regards.
Jorge Humberto Urrea-Quintero
Dear Jorge, thank you for your remarks!
1. Yes, the control objective is to stabilise the pendulum at the desired cart position. And ultimately to have two reference points: cart position and pendulum angle. While pendulum angle is the main reference in this case, I choose to formulate the initial problem as follows in order to understand the principles of state-space control in Simulink: what if I don't care, where the cart ends up? I just need one reference then - pendulum angle = 0. The question is where should this zero be in terms of Simulink among the rest of references in the future, e.g. when I need two reference points?
2. Agreed, but I could just obtain derivatives from the position and angle signals numerically. Observer is in my opinion needed in case I need a state I can't directly or indirectly obtain from the model. Maybe I am wrong, please, correct me in this case.
3. Thank you very much for the paper, I will definitely look into it!
Best regards, Alex
Hi Alexander Babin
what if I don't care, where the cart ends up? I just need one reference then - pendulum angle = 0. The question is where should this zero be in terms of Simulink among the rest of references in the future, e.g. when I need two reference points?
Yes, you only need the reference for the angle.
I think the confusing part for you now is that you have full feedback state, but what you need is only the feedback of your controlled variable and then be sure the other state variables remain stable. Then if you want to control another variable in the future, you add the feedback law for that variable too.
In summary, you only need the reference of your angle and leave the other variables free.
I could just obtain derivatives from the position and angle signals numerically. Observer is in my opinion needed in case I need a state I can't directly or indirectly obtain from the model. Maybe I am wrong, please, correct me in this case.
Yes, you could use your model as the system states estimator, but using an observer would be more reliable because you somehow mix the information of your model with the measurements you have.
Regards.
Dear Jorge Humberto Urrea-Quintero ,
thanks again for your detailed answers! That means that even thout the u=Kx control signal is based on the full state of the system, I need to compare it with only one state reference? Maybe I have confused myself even more no, but I have sketched a few options, please, point me to the correct one. ref signal is my one constant theta. The other signal entering the summation block with a minus sign is the state of the system, which is a vector. In the first case I cant compare a vector to a scalar, however, the second option does not feel right, because in this case I am comparing the output signal of a controller with a scalar reference, so the input of the system is the error between a reference and a control signal, which makes no sense! I have a feeling the first schematic must be right, but how do I compare a vector to a scalar?
So, again, from my viewpoint, you don't need the full state feedback, you only need the feedback of your controlled variable (angle in this case). Then, because you have only one variable in your feedback loop, you compare a scalar with a scalar in the summation block. By the way, you could solve your problem using a PI/PID controller because even if you have several states, you only have one output making your problem a SISO problem.
Just take in mind that if you use the full feedback approach, then you need 4 references and you need to control all your variables at the same time, no just your angle. In fact, I'm not sure your system would be controllable in that case because of the dofs you have.
Hope my answer helps!
I'm guessing the answer is in the fact that other states must be constant zeros, then essentially I will be comparing the changing state of pendulum angle with its reference, but that can't be true, because at least one more state is non zero and not constant
Dear Jorge Humberto Urrea-Quintero ,
no, it must be LQR :) I'm setting this task before myself. So full state feedback it is.
So, if my state vector is X=[x, theta, x dot, theta dot], my reference vector must be ref=[0, 0, 0, 0], not a constant, mustnt it? In case the first schematic is correct.
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