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As part of a control systems course I have to implement control for a 3 DOF SISO mass spring damper model of a robotic cutter arm. The arm is to follow a 5Hz triangle wave reference which is positioning it. As you can see from the picture, the robotic arm requires damped and tracking control.

open loop response

I have derived the state equations and successfully simulated the system open loop in simulink to recreate the same waveform as above. I have had success implementing full state feedback with LQR control. I have also used an integrator to achieve tracking of the output waveform. The final step is to design an estimator (since in real life not all states could be accessed).

Although I have achieved some form of control I am not entirely sure if it is correct. Does anyone have examples of 3DOF SISO with estimator, tracked and/or damped control in state space? So far all I have found is examples of active suspension:

Design of LQR controller for active suspension system

Also, here is a link for anyone who is curious about how to derive equations of motion for MBK system:

Multiple Degree-of-Freedom Example

user7083
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Hieronymus Blois
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    Welcome to engineering.SE! I am not entirely sure what your question is, do you want us to verify that your control approach is correct? In that case, you need to provide details about what you did. – OpticalResonator Nov 10 '18 at 12:34
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    Thanks for replying OpticalResonator. Essentially I am going to use an estimator and a full state feedback.However, in MATLAB when I try to do pole placement it tells me that my matrixes are too close to singular and badly scaled. I have the same problem when trying to ascertain controllability and observabillity. The rank function returns 4 for rank(Mc) and then 6 when I do rank(Mc,10e-99). Is my system controllable? Are these discrepancies arising from MATLABS precision? Thanks – Hieronymus Blois Nov 10 '18 at 19:48
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    `Is my system controllable?` Well, what are your states and what is your state matrix? You said you're asking about a SISO system - what is your input? What is your (single!) output? I would bet that it's **not** controllable as you *probably* wrote it, because you can't control x1's position, x2's position, and x3's position, all with a single input. Remember that the controllability check tries to determine if you can uniquely affect each state with the control input. – Chuck Dec 14 '18 at 20:03

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