I want to design a controller for a model that has an integrator.For this reason, i would to design a PD controller instead of PID and so on. almost of books and refs discuss about tuning rules of PI and PID controllers. Is there any rules about tuning of PD Controller?
There are a lot of different approaches to design a controller, if the transfer function of the system is known. You can use bode, root-locus, Nyquist, etc to design your controller. In your case, by using a PD controller, you basically add a zero to your system Kp+Kds -> (s+Kp/Kd). As you know, zeros add positive angle to the system (move the root-locus to the left) or in frequency-domain they introduce phase-lead to the system (increase the phase margin). For a second order system for instance, if you want to use root-locus, first obtain the desired location of your closed-loop poles from the design requirements such as the settling time, percent overshoot, etc, and calculate the total angle of deficiency from poles and zeros of your system and additional positive angle required from the PD controller in order to get +- 180 degrees to place the closed-loop poles at the desired location. Another approach is to combine the PD controller with the transfer function and obtain the closed-loop transfer function G_cl=GK/(1+GK) and compare it with standard second order system Wn^2/(s^2+2*zeta*Wn*s+Wn^2) and it will tell you what Kp and Kd gains will result in the desired second order transfer function (again Wn and zeta can be found from the design requirements). Keep in mind that PD controller by itself is not realizable for real-time implementation (because it is not proper), therefore you need to add a low-pass filter to make the PD controller realizable and attenuate high frequency measurement noise. You can have a look at the following link for further explanations.
http://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/pph02-ch13.pdf
You may use numerous approaches that give you some results. But, if you wish to get a higher performance in a reliable way, you hove to define your loop and control requirements more precisely. In general, the necessity to use an integral action does not depend just on the plant model, but also on the disturbances acting. In the simplest case, the P control might be satisfactory. Use of PD control requires to pay higher attention to the measurement and quantization noise, see e.g.
https://www.researchgate.net/publication/269808455_Tuning_of_a_filtered_pole_assignment_controller_for_an_integral_plant
when you may achieve significant improvements for n>2. For a 2nd order dominant dynamics and relatively short additional delays, the control constraints may become on importance
https://www.researchgate.net/publication/269811203_Constrained_pole_assignment_control_for_a_2nd_order_integral_system
Conference Paper Tuning of a filtered pole assignment controller for an integral plant
Conference Paper Constrained pole assignment control for a 2nd order integral system
Probably the richest collection of different PI and PID tuning rules and structures (including also PD) you may find in several publications by O'Dvyer, as e.g.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.7556
A. O’Dwyer, Handbook of PI and PID controller tuning rules. 2nd Ed.
London: Springer, 2006
Handbook of PI and PID controller tuning rules. 3rd Ed., Imperial College Press,
2009
“A summary of PI and PID controller tuning rules for processes
with time delay. Part 1, PI controller tuning rules,” Preprints IFAC
Workshop on Digital Control, Past, present and future of PID Control,
Terassa, Spain, pp. 175–180, 2000.
“An Overview of Tuning Rules for the PI and PID Continuous-
Time Control of Time-Delayed Single-Input, Single-Output (SISO)
Processes,” in PID Control in the Third Millennium. Lessons Learned
and New Approaches, R. Vilanova and A. Visioli, Eds. Springer, 2012.
But, these may only be used as a phone directory
A PD controller often offers faster setpoint tracking than the PID controller - adding the integral mode somehow slows down the closed-loop response. But, it should be noted that, a PD controller does not remove the steady-state offset in the presence of an input disturbance - it only removes offset for servo control and output disturbance type. Since, a PD controller might suffer from serious impulsive spike in its control output (I.e., this can damage the equipment involved) due to a step change in the setpoint, as well as being vulnerable to measurement noise, for a practical implementation the PD controller is often augmented with a first-order lag filter - which is equivalent to a Proportional controller augmented with lead-lag filter. A relatively simple and generalized approach to design or tune this type of controller is using the latest control design approach - the multi-scale control.
A simple PD controller design is the method of Cohen Coon (G. H. Cohen and G. A. Coon. Theoretical consideration of retarded control.Trans. ASME, 75:827–834, 1953.)
I enclose the implementation of this method in Matlab
https://www.erpublication.org/admin/vol_issue1/upload%20Image/IJETR022181.pdf
http://ukacc.group.shef.ac.uk/proceedings/control2006/papers/f165.pdf
this wil be usefu lfor u
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