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According to my understanding, if the poles of a system are in RHP, LQR controller can be designed to place the pole in LHP with optimal gain 'K' found by minimizing the cost function (Q, R). I am confused that why is LQR designed for a system whose poles are already in the LHP. For example a system poles are at -10, -6+4i and -6-4i and the system step response final value is reaching say 0.2 in 2 seconds. What is the point of designing an LQR for an already stable system when a simple PID or reference scaling can do the work of reaching the desired value.

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  • $\begingroup$ When compensating a dynamic system, two main aspects are taken into account. First stability and then performance, involving aspects such as overshoot, response time, bandwidth etc, This second item is also essential ... $\endgroup$ – Cesareo May 2 '19 at 8:06
  • $\begingroup$ Thank you for your response. If I design lqr for the system with poles mentioned above and get the desired performance. Let suppose poles move to -5, -2+6i and -2-6i and the final value of graph reaches one. Will the lqr be better in terms of energy expenditure as compared to PID? where lqr will move three poles. lqr will not be optimal in this case right? $\endgroup$ – Papa Devil May 2 '19 at 8:57
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You effectively ask why should I use method $A$ instead of method $B$, in a field where there are many methods to solve the same problem.

If a PID controller gives you the performance you want and you are used to that design strategy, great, no reason to complicate matters.

LQ has some benefits sometimes when it comes to intuitive tuning (in particular for more complicated systems), is completely identical in SISO and MIMO case, and have theoretically appealing properties (infinite gain margin, large phase margin) etc.

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  • $\begingroup$ Thank you for your response. If I design lqr for the system with poles mentioned above and get the desired performance. Let suppose poles move to -5, -2+6i and -2-6i and the final value of graph reaches one. Will the lqr be better in terms of energy expenditure as compared to PID? where lqr will move three poles. lqr will not be optimal in this case right? $\endgroup$ – Papa Devil May 2 '19 at 8:57
  • $\begingroup$ The LQ controller will be optimal w.r.t to the performance measure it minimizes, which is a trade-off between control effort and state error. $\endgroup$ – Johan Löfberg May 2 '19 at 10:11
  • $\begingroup$ thank you very much for your response. $\endgroup$ – Papa Devil May 2 '19 at 11:47

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