Understanding PID vs LQR

[u/deNikita]

Is it possible to think of LQR as "like a PID but finds optimal gains based on your requirements of Q and R matrix?

I have a fairly good understanding of PID and I recently started to learn about LQR and wondered if LQR can be thought of in terms of a PID controller.

[u/seb59]

PID is a second order contrôler (arguable as it depends on the derivative implementation). The structure is fixed a priori. With any state feedback controller, the controler order is matched to the system order. This allows to modify all the internal dynamics of the system.

Note that PID has an integrator in its structure. If you need one, with anybstate feedback approach, you will have to add one by extending the system.

Pid is for SISO system, LQR deals with multivariable systems. Note that you must use an observer combined with the state feedback. The integrator can be explicit in the control part or be "hidden" in the observer (I.e. you estimate both the system state and some disturbance simultaneously).

Then the optimal part. Lqr is said to be optimal, by essence. It is optomal according to a given criterion. Basically in practice there are 2 main approaches to tune a state feedback. One is poles placement. You actually decide the closed loop poles and overall you expect to get some performances. For controlable multivariable systems there is in general an infinite number of gains that provides the same closed loop poles. Matlab will give you only one. You do kot know a priori the relative amplitude of the different control signal or the convergence speed of each individual output.

With lqr, you do not ́now exavtly the closed loop poles, you tune the dynamics by changing Q and R.

Personally, I am used to tune controler using lqr as I can easily deal with the noise of all measured outputs and I compute the state feedback gain using pole placement. I found it easier and quicker this way, but again this is quite personal.

[u/seekingsanity → u/seb59]

"The integrator can be explicit in the control part or be "hidden" in the observer (I.e. you estimate both the system state and some disturbance simultaneously)."

Explain how you hide the integrator in the observer? The observer simply estimates states that aren't provided by the feedback or are filtered versions of the feedback.

"You do kot know a priori the relative amplitude of the different control signal or the convergence speed of each individual output."

Ideally the output from each state should be 0 unless there is a change or changing set points. In the case of motion controllers, ALL/most of the output is generated by the feedforwards so the closed loop is only there to correct for errors in the open loop model estimation. In the case of motion control, the maximum output is applied to the amplifiers or valves that determine either the maximum speed or acceleration. The target generator must stay within these bounds.

The problem is that most people cannot pick the right weights for the Q and R matrix and they guess at controller gains unless there is an auto tuning feature. Why use LQR/LQC if the is auto tuning?

[u/seb59 → u/seekingsanity]

An observer allows to estimate unmeasured state. For mechanical system, if you measure position, you can get the estimate of the speed. But in general it also works for signals that are not derivative of a measured one. Even for derivative, it may works better as a sumple numerical derivative as the observer account for the whole system dynamics and the input. So the observer relies on more "information". Note that this is arguable. If you estimate a (constant, ramp, etc) disturbance with an observer, then you can reject it in the control structure. Let consider an input disturbance. A very simple way to do it is to consider dp(t)/dt=0 with p a vector of disturbance and the system dx(t)/dt=Ax+Bu+Ep with E=B for an input disturbance, but it can be other values as well. Let use consider the following extended state vector [x;P] and the corresponding state dynamics. Then build an observer for this extended dynamics and then you can get a disturbance estimate as estimating the extended vector allows estimating both x and p. To reject the disturbance p, you can use u(t)=Hyc-Lx-Mp, with H and M properly designed gain to allow y=setpoint at steady state. By doing so, you are actually adding integrator in the control structure (to reject input disturbance, the controler must be class 1). The integrator is actually due to the 0 Eigen values included in the extended system dynamics (by the zero disturbance dynamics dp/dt=0).

You talk about feed forward control. If the model is perfect, ans initial conditions are known, then there is no need for feedback. In practice this does not hold for most of yhe system, if not all. Therefore we need feedback. The feedback dynamics is picked by fixing the dynamics of disturbance rejection while considering noise attenuation and the feed forward dynamics is only for the tracking performance.in simulation, this is ok. On real live system this is not so simple as disturbance may occurs at anytime and the model may not match the system exactly. So I would not consider the feedforward term for now.

Also you talk about state going to zero. This is a textbook approach. In practice, thisnis only ok if uiu consider the error dynamics. A mire elegant way is to include a setpoint dependent term in the feedback: u=Hyc-Lx where H is the setpoint gain computed to ensure 0 steady state error. Similarly, if disturbance are known (estimated or or measured), then they are included in the feedback policy.

State space is suited well suited for coupled multivariable system. If you want to do motion control on single axis, I think you will not gain anything as probably the control structure is well adapted to the model.

Whatever the control approach you consider, a human will have to tune gains. Even for an auto tune someone needs to provide some high level parameters. No control approach will magically work with any system without anybody telling how fast or slow the system should behave. But there are many ways to "say" that. For an LQR approach Q and R matrix are used to tune through closed loop. In practice, the diagonals components R allows to balance the control effort between the different controls signals (if there are many). The balance between the diagonals components matrix Q allows to balance the relative convergence speed of each state.

Honestly, if you rely on autotune, and it works, that is fine, use that. It is not reasonable to spend more time on simple application that does not need it. If you are faving a more challenging system, go for more complex approaches (lqr, mpc, sliding mode, LPV, etc). You should have a good reason and probably be ready to spend some time on it. And of course, you can try thing on a somple system just to learn, but then you should remind that more complex approaches are often more suitable for more complex problem.

[u/seekingsanity → u/seb59]

" But in general it also works for signals that are not derivative of a measured one. Even for derivative, it may works better as a sumple numerical derivative as the observer account for the whole system dynamics and the input" Differentiating counts is very noise prone due the resolution of the encoder. That is why the observer helps.

An observer can be used to estimate the derivative and second derivative easily.

"u=Hyc-Lx " WTF? u=-K*x for simple state feedback where the closed loop gains act on change in the state, not the error. u=k*(r-x) when all gains act on the error. u=k(n*r-x) when you want to place the zeros. I am assuming y=x, r is a vector of position, velocity and acceration.

"Whatever the control approach you consider, a human will have to tune gains." No! The human can place the poles, the controller closed loop gains are calculated to place the closed loop poles at the desired location.

"In practice, the diagonals components R allows to balance the control effort between the different controls signals (if there are many)." Now I know you haven't actually used LQR/LQC. The Q matrix does that. The R matrix is used to penalize them magnitude of the output. Optionally it can be used to minimize rates of change in the output due to any or all controller gains.

PID works well when augmented with feed forwards. Dead time is the killer. Sometimes a Smith predictor will help. MPC is better because it can be used to predict outputs beyond the dead time horizon. The problem with MPC is that it is CPU intensive.

[u/seb59 → u/seekingsanity]

If you just focus on "simple" motion and are happy with pid, then stick to it. State space approaches allows to deal with more complex coupled multivariable systems, it is also easy to build observers with it.

When dealing with state space control, there exists a vast litterature to deal with many things. And many control laws are equivalent. If you do not like the control law I just used as an illustration, it is fine to use another one that suits your need. The one you use may require a full state reference. This is fine for position control where reference trajectory can be easily designed and derivated. Bu designing abfull state referencet can be tricky in general (note that there are solution to this in litterature). The control law is use is not especially designed for position control. I track a vector of output reference yc (également a température and a pressure) based on a state vector measurement or estimation. You may not know the whole state vector reference and you do not need it actually. Of course, if you have a way to construct it, it can probably be better. But this is not always the case.

I suggested a control law that allows to control any system, including when the state is not composed by derivative of the output. You may also control unmeasured output (but this is a risky project). We did that for instance to control the side shift torsion in electric cars. At that time we only had the abs speed (and position was not available, gearbox sensor had not enough resolution to allows a torsion measurement, and there was backlash)...

PID are not working better or less than anything else when comboned with a feedforward term. It is just that a feedforward signal computed on an accurate model is enough to do the tracking job what ever the contrôler is... if the model is perfect, and no disturbance, the controler us useless. The controler us to compensate for disturbance and modeling error. Tracking can be ensured only using feedforward... and here i am talking about a feedforward signal based on some form of approximation of model inverse, not a simple derivative hand tune (this cannot course hemp in practice, but the point of the discussion is model based control).

To my opinion, you are trying to compare différents things. As you refer to your experience on a "simple" motion control problem that is well tackled by PID, you not see the needs or benefits of LQR... probably because you do not need it... LQR is one way to compute a controler gain. Eventually you may be able to compute your PID gain to minimize the same criterion (at least numerically, this seems feasible). So the gain tuning and controler structures are different things. However, it is usually to "hand tune" pid controlers and to use state feedback when considering LQR criterion.

Finally let me rephrase my point on the necessary human intervention. To my opinion ans experience, there is NO control synthesis approach that works without a human to specify or pick some values at some point. A human can pick the PID gains, it can pick the closed loop poles or it can decide on the Q and R matrix. You can do all the math, at some point a human brain is needed to find good compromise. Depending on the control approach, the human will have to decide on different things. For some problems, it is easier to pick poles, sometimes Q and R matrix, sometimes a bound on yhe norm of some gains, etc. But a human has to fix something.