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 ?
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, I have heard about LQR.
Basically, we place poles according a criteria that we want to minimize.
I will learn more about that.
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.
'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.
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.
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 ?
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.