Questions in Control Theory (pole placement, stability, Kalman's Criterion)

Discussion in 'Robotics & Mechatronics' started by Mike.B, Nov 24, 2014.

1. Mike.BNew Member

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Hi everyone !

Currently, I am studying Control Theory and I believe it is a fascinating field ! I still have a lot to learn !
Anyway, I have a question to ask because I am not very sure.
I have just learned about Pole Placement. I know the key is to find a gain K in order to meet my specifications and change system's behavior by what I desire. In order to do that, it is necessary to check a criterion: Kalman's Criterion (to know if my system is controllable and if I can place my pole as I want).

My first question: Imagine a system that is initially no stable (with positive eigenvalues for example) .
Can I assume it does not check Kalman's Criterion ?
Can I do a Pole Placement to make it stable ?
Are there other methods to make a system stable ?

My second question: I am working with linear system. However, I think in life, real systems are not necessarily linear.
Can I do a pole placement in a nonlinear system ?

Thank you for your help !
Have a nice day !

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All good questions. I labeled each of your questions (there are actually more than two there) so that I can reference them and try to answer each one.

Q1: Correct me if I'm wrong, but isn't Kalman's criterion related to controllability/observability? Stability can be checked with pole locations. If all poles are in the left half plane, then the system is stable. Can you quote which Kalman Criterion you are referring to please? I'd like to be sure.

Q2: If the system is controllable then yes you can stabilize an unstable system. A rank check on the controllability matrix tells you if it is controllable. That is the beauty of pole placement. It allows you to place the poles anywhere you want, in principle, if the system is controllable. However, this assumes the system is completely linear, but if the system has input limits (as all systems do) then you may not be able to drive the system hard enough to make the solution practical.

Q3: Yes, many other methods are available. LQR design is probably the next one to consider. LQR design tends to make very stable and robust systems if the system is controllable.

Q4: No system is truly linear. Some systems appear very linear over a particular range of signals and if you confine yourself to that range, you are all set. Other system are not linear at all at any point. If the nonlinearity has certain characteristics, such as being smooth and not to severe, and if it meets certain other criteria you can linearize the system and do pole placement design at various operating points. Then you can allow the gains to be dependent on operating point, or you can use constant gains and check to make sure that they don't move too much (or particularly make sure they don't go to the right half plane and become unstable) from your design point based on one operating point.

3. Mike.BNew Member

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Hello SteveB !

Thank you for your help, I appreciate a lot !
I will try to reply at best and explain my point of view.

Q1 :
Yes, Kalman's criterion is related to controllability:
'A system is controllable if the controllability matrix R has full row rank'
and R=[B AB A²B ...]

I agree with what you said.
If we don't check Kalman's criterion, does it implies anything about stability ? Why a system won't check Kalman's criterion ? Is there a reason ?

Q2:
Actually, you have just solved my problem. I wasn't sure we can stabilize a system with pole placement (I knew we can change its behavior and how its answer)
It is really powerful !
"However, this assumes the system is completely linear".
Aright, that's what I thought.

Q3: Yes, I have heard about LQR.
Basically, we place poles according a criteria that we want to minimize.

Q4: 'No system is truly linear.' ==> It is a pity. It would be much more simple !
If we cannot linearize the system, what kind of controller can we apply directly (assuming system is stable) ? Can you name them ?
I am sure PID is not suitable.

I add a question because you seem to be very comfortable with those notions and I would like to get your point of view.

Q5:
I am currently a student.
According to you, are there lots of job opportunies in control theory in laboratory/company ?
I like as well embedded systems. Is it common to embed a controller on a FPGA or a microcontroller ?

Thank for your help and have a nice evening !

4. DaveNew Member

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Generally we use the Kalman criteria if we want to make a feedback system or an observer. The two problems are "duals" of each other, which is a general property that falls out of the math. Every control feedback problem can be recast into an observer estimator problem. For example. the LQR feedback design problem is the dual of the Kalman filter/observer problem.

Hence, the controllability matrix used for feedback control design and the observability matrix used for observer design are dual versions of each other and the check for "observability" or "controllability" are duals of each other. Basically, we check these things because the criteria tell us whether we can actually fully control, or fully observe a system. If a system is not controllable, you may need more outputs to use for feedback control and if a system is not observable, you may need more measurements to estimate the variable that you are not measuring directly.

Yes, we do place the poles using minimization concepts, but we do so indirectly. We actually calculate the gains that produce the minimization. But of course, those gains result in particular pole locations. In pole-placement design, we calculate the gains that give us those poles we want. So. typically we get very different designs using these two approaches. Some problems are better solved with one technique over the other.

There are various types of controllers. - too many to mention them all. Certainly, you can try PID control, but you might need the gains to change based on operating point, depending on how severe the nonlinearity is. You can try pole-placement via full state feedback, or LQR design, but again, the gains may need to be a function of operating point. There is H-infinity design, and sliding mode control, simple bang-bang control and various nonlinear approaches, included nonlinear PID.

It is very common to embed a controller into a FPGA or microcontroller. As time goes by, I see less and less analog control and more digital control.

As far as jobs, I don't have a good idea about that. I always viewed feedback design and microcontroller design as two of the many tools that an electrical engineer needs to know well. I've worked in many fields, never specializing in control work or microprocessor design, but just about all work I've done required some aspects of these things. If you want to actually target that field as your specialization, I see nothing wrong with that, as it is a fun and rewarding type of work, in my view. But, as to how to actually find the places that need a specialist in that area, I don't have a good idea about that.