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What? If you have a good model of your system then you can (relatively easily) turn it into an optimization problem and use a LQR to choose gains for you. It’s

What? If you have a good model of your system then you can (relatively easily) turn it into an optimization problem and use a LQR to choose gains for you. It’s true, this method still gives you knobs you might have to tune (Q and R), but these are easily conceptualized because they slide the cost of state excursions and control effort along a Pareto boundary. That means that you can get optimal PID gains just by saying how much you penalize a state excursion vs. how much you penalize control effort (e.g. distance from your reference point v.s. fuel use, or something like that.) It’s also certainly true that a LQR won’t help you if you have no a priori knowledge of your system, but for many of the mechanisms we need to control this isn’t a problem.

sgillen 7y ago
If you have a good linear model for your system you can pull out an LQR. It does not account for how bad your linearization of the true dynamics might be. Also from my experience you still need to tune the Q and R the same amount that you would P I and D. and I'm not convinced that it is strictly better at all. In industry at least PID absolutely dominates. It's also important to realize that just because the LQR is derived by solving an optimization problem, that doesn't mean it gives you the best possible gains for whatever you want. You got the optimal controller to minimize a quadratic cost you made up for a linearized approximation of your system. Iterative design (guess and check) is absolutely still the state of the art for control design, and using an LQR does not escape that.
patrick5415 7y ago
Your whole argument is based around “if you have good model...”. This where most all control theory falls on its face. Getting a good is hard. For lqr that model better be mostly linear. Oh, you system model isn’t first order? Now you need an estimator/Kalman filter. Or more sensors. That’s just more complexity for questionable benifit, which is why lqr is beloved by academics [0]. For anything that would be adaquetly controlled with pid, stick with that. After about 2 hours fiddling with the knobs, it’ll be close enough. This whole idea of optimality is based on bad intuition. Which states do you care about? Why? Is that more valuable than control effort? Why? Who is doing the economic analysis to determine what ultimately costs us more money? In the end, this thing are tuned just like pid: you stop when the step response looks nice. Besides all that, for a lot of systems, and in particular flexible structures with a lot of states, “penalizing state excursions” isn’t really useful intuition to begin with for almost all state space represtations. You are better off with a pid and notch filter. [0] I decided to complete my PhD in controls so could make statements like that with at least marginal credibility.

sgillen 7y ago

If you have a good linear model for your system you can pull out an LQR. It does not account for how bad your linearization of the true dynamics might be. Also from my experience you still need to tune the Q and R the same amount that you would P I and D. and I'm not convinced that it is strictly better at all. In industry at least PID absolutely dominates. It's also important to realize that just because the LQR is derived by solving an optimization problem, that doesn't mean it gives you the best possible gains for whatever you want. You got the optimal controller to minimize a quadratic cost you made up for a linearized approximation of your system. Iterative design (guess and check) is absolutely still the state of the art for control design, and using an LQR does not escape that.

patrick5415 7y ago

Your whole argument is based around “if you have good model...”. This where most all control theory falls on its face. Getting a good is hard. For lqr that model better be mostly linear. Oh, you system model isn’t first order? Now you need an estimator/Kalman filter. Or more sensors. That’s just more complexity for questionable benifit, which is why lqr is beloved by academics [0]. For anything that would be adaquetly controlled with pid, stick with that. After about 2 hours fiddling with the knobs, it’ll be close enough. This whole idea of optimality is based on bad intuition. Which states do you care about? Why? Is that more valuable than control effort? Why? Who is doing the economic analysis to determine what ultimately costs us more money? In the end, this thing are tuned just like pid: you stop when the step response looks nice. Besides all that, for a lot of systems, and in particular flexible structures with a lot of states, “penalizing state excursions” isn’t really useful intuition to begin with for almost all state space represtations. You are better off with a pid and notch filter. [0] I decided to complete my PhD in controls so could make statements like that with at least marginal credibility.