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I have heard that LQR and MCP have common similarities. The difference is that MPC is using QP-programming and LQR using Riccati Equations. With QP-programming, constraints can be applied.

If we compare non constrained MPC and LQR, the difference is that LQR is optimal for a infinity time window and MPC is optimal for just a specific time window. I think it called time window. I hope you understand what word I want to use.

But is LQR obsolete compared to non constrained MPC? I mean, in practice, it would be better to have a basic non constrained MPC than LQR because an optimal state space model does not exist in reallity? Basic non constrained MPC uses simple least square to compute the future input signals for the system.

Are there situations there LQR is better in use than MPC?

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Depending on the prediction horizon and model dimensions MPC will be more or much more computationally expensive compared to LQR. This may lead to added delay or require more expensive hardware. Namely (infinite horizon) LQR just needs to compute a gain matrix beforehand and after that the computation cost online is fairly small.

It can also be noted that LQR can be used as a roll-out policy for MPC and there also exists finite horizon LQR, which has a time varying gain matrix (which can still be calculated in advance).

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  • $\begingroup$ It could be added that the MPC optimiziation can fail if no feasible solution is found and that the LQR comes with robustness guarantees (see: Safonov & Athans (1977) Gain and phase margin for multiloop LQG regulators) which cannot be made for the MPC. $\endgroup$
    – MrYouMath
    Feb 19 '19 at 8:19
  • $\begingroup$ @MrYouMath Since MPC and LQR can solve the same problem, then if MPC would fail LQR would as well. The same holds for the margins, however I do not know if one could still talk about margins in the finite horizon problem. $\endgroup$ Feb 19 '19 at 8:44
  • $\begingroup$ Are you sure that they solve the same problem? MPC is used with a moving horizon and LQR is used for infinite horizon. $\endgroup$
    – MrYouMath
    Feb 19 '19 at 9:13
  • $\begingroup$ If we assume that we have a fast modern computer that solve MPC on 0.001 seconds. Like a embedded Microcontroller. $\endgroup$
    – MrYui
    Feb 19 '19 at 9:36
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    $\begingroup$ @MrYouMath receding horizon is indeed different compared to (in)finite LQR. But if you choose a terminal cost in MPC based on the solution to the infinite horizon LQR then they do yield the same results. This is essentially what is meant when using LQR as roll out policy. This can also gives stability guarantees, which you do not have with no roll out policy. $\endgroup$ Feb 19 '19 at 12:54

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