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Hello fellows,
I am trying to design a controller for the Inverted Pendulum on a cart problem, and I would ask for your help into this subject.
Already tried with a pole placement technique but could not get enough satisfactory results so I am trying to set the optimal Root Loci based on the Letov and Chang symmetrical Root Loci which appears in the Thomas Kailath book.
It seems that whether depends on what amount control energy you have available, then you could set the poles in two different manners. If the cost of the energy control is high then you only have to move the unstable poles to their symmetrical locations from the imaginary axis.
My problem is that my plant have one pole at the origin and another unstable one so although I reflect the unstable one, already appears the pole at the origin in the set of the optimal poles. (Maybe I am getting this wrong?). Actually my plant has two zeros and two poles.(If I consider the system as a SISO with the pendulum angular position as the output, then a cancellation occurs and the TF only has one zero at the origin and three poles) Therefore the system is controllable but not observable here.(I was guessing since this cancellation occurred then I must forget about the pole at the origin and reflect the unstable pole twice). The goal is to apply Ackermann with this locations and try a full feedback controller.
When the cost of control is low, there is a cheap control situation so then you choose two poles to be at the stable zeros of the H(s)*H(-s) and another two poles in a Butterworth configuration.
With this configuration the problem of having a pole at the origin in closed loop does not appear and it seems to generate a workable controller. But my problem here is that this configuration is dependent on a factor "r" and I do not understand quite well how to get the optimal value of "r" from the root loci of H(s)*H(-s).
Could someone shed some light into this topics?
Or lead me towards some lectures, notes, pdfs or even better...books!!
Any kind of help will be very appreciated!!
Best regards!!
I am trying to design a controller for the Inverted Pendulum on a cart problem, and I would ask for your help into this subject.
Already tried with a pole placement technique but could not get enough satisfactory results so I am trying to set the optimal Root Loci based on the Letov and Chang symmetrical Root Loci which appears in the Thomas Kailath book.
It seems that whether depends on what amount control energy you have available, then you could set the poles in two different manners. If the cost of the energy control is high then you only have to move the unstable poles to their symmetrical locations from the imaginary axis.
My problem is that my plant have one pole at the origin and another unstable one so although I reflect the unstable one, already appears the pole at the origin in the set of the optimal poles. (Maybe I am getting this wrong?). Actually my plant has two zeros and two poles.(If I consider the system as a SISO with the pendulum angular position as the output, then a cancellation occurs and the TF only has one zero at the origin and three poles) Therefore the system is controllable but not observable here.(I was guessing since this cancellation occurred then I must forget about the pole at the origin and reflect the unstable pole twice). The goal is to apply Ackermann with this locations and try a full feedback controller.
When the cost of control is low, there is a cheap control situation so then you choose two poles to be at the stable zeros of the H(s)*H(-s) and another two poles in a Butterworth configuration.
With this configuration the problem of having a pole at the origin in closed loop does not appear and it seems to generate a workable controller. But my problem here is that this configuration is dependent on a factor "r" and I do not understand quite well how to get the optimal value of "r" from the root loci of H(s)*H(-s).
Could someone shed some light into this topics?
Or lead me towards some lectures, notes, pdfs or even better...books!!
Any kind of help will be very appreciated!!
Best regards!!
