I have a 4th order strictly proper transfer function. I want it to track a desire trajectory. I have chosen feedforward structure and I have put the feedforward path as the inverse of my system in order to have zero steady state error. How can I design a controller for this system to keep the poles of the closed loop system on the right side of the imaginary coordinate?
Your question is very vague. Usually feedback control is used to alter the dynamic behavior and move the closed-loop poles of the system to the desired locations and feedforward part is designed for tracking performance. As far as I know, you can design a 2DOF controller, in which the feedback part is for performance improvement and disturbance rejection and the feedforward part is for tracking performance. By the way, I think by saying the poles are on the "right-side" you mean the left hand side if you are referring to the root-locus of the system.
If you design your feedforward input properly then the output should match the desired output as long as there is not plant uncertainty. With a real system there will always be some error. You can often handle this error with a feedback controller but the design is not standard unless you have some type of model for the error. Many times something as simple as a PI controller can do the trick. If you want something more advanced then take a look at these papers:
http://faculty.washington.edu/devasia/Research/Papers/Nonlinear_Inverse_IEEE96.pdf
http://faculty.washington.edu/devasia/Research/Papers/robust_inversion_IEEEAC02.pdf
Mr Noshadi thanks for your help. you are right I was mistaken I meant to keep the poles of the closed loop system on the left side of imaginary coordinate.but for the designing the controller my system has two paires of complex poles and the root locus of this system has some paths on the right side of the imaginary coordinate and my aim is to keep the poles of the closed system on the left side for all gains.since my system is strictly proper ad it has no zeros I have to add at least 3 real zeros in order to reach this aim thus my controller is an improper system with 3 zeros and it is not implementable and also simulating an improper transfer function in Simulink has difficulمity.it usually gives error with derivative parts and I don’t know what to do with simulation can you help me with this problem??by feedforward path I do the tracking very well but with a PI control I cant keep the poles on the left side . in my opinion by designing a controller for a feedforward structure we try to keep the poles of the closed loop system on the left side for all frequencies.we don’t have any problem with performance since our tracking is completث???!!!!!!!!!
My first question is why do you want to keep the root-locus on the left for all gains? There is no point in doing that. Because in real-time you can't increase the gain to infinity (because of saturation in the actuators). Therefore, you need to design a controller which gives you the specifications you require in terms of performance and robustness. If you derive the closed-loop sensitivity function (S=1/(1+GK)) and (T=GK/(1+GK)), you need S=0 for good performance and disturbance rejection, and T=0 for robustness and high frequency noise attenuation. This is not possible at the same time (regardless of the system, if you study the Bode's sensitivity integral theorem (or Waterbed effect). So, the idea is to design a controller to give you a good trade-off. Of course you can't realize improper controllers in real-time even if you manage to do it in simulation. unless you add three other high-frequency poles (as low-pass filter) to make the controller realizable. However, you make the controller order non-necessarily high.
Designing a feedforward controller via inversion of the process model has a drawback, which can lead to physically unrealizable controller. In many cases, it is advantageous to combine both feedback and feedforward controllers - while the feedback {could be PID controller) controller serves the purpose of stabilizing the system and ensures zero steady-state offset, the feedforward controller enhances the control performance, either disturbance rejection or setpoint tracking response. You could try the latest design technique based on the multi-scale control scheme for designing a unified feedback-feedforward control system. This new scheme guarantees that the feedforward controller is physically realizable.
Cite
Wameedh Abdul-Adheem
Al-Ma'moun University College
Please refer to these articles:
https://www.researchgate.net/publication/313576646_From_PID_to_Nonlinear_State_Error_Feedback_Controller
https://www.researchgate.net/publication/312046377_Improved_Sliding_Mode_Nonlinear_Extended_State_Observer_based_Active_Disturbance_Rejection_Control_for_Uncertain_Systems_with_Unknown_Total_Disturbance
https://www.researchgate.net/publication/309655805_On_the_Improved_Nonlinear_Tracking_Differentiator_based_Nonlinear_PID_Controller_Design
Article From PID to Nonlinear State Error Feedback Controller
Article Improved Sliding Mode Nonlinear Extended State Observer base...
Article On the Improved Nonlinear Tracking Differentiator based Nonl...
Cite
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