I agree with your comments in the last paragraph. But, if you ensure the stability of the each loop independently. Then all other (global) intervention can be  classified as

1. disturbances/input signals. These signals should not create any instability in the loops. Though the performance of the loop in terms of tracking may vary. This behavior can be considered as a design parameter.

2. The global intervention in terms in controller parameters and structure. This can cause change in the dynamic behavior of the loop. The adaptive control theory can be helpful in the stability and performance analysis of such systems. 

Thanks for both comments however they seem to me quite difficult to be followed. Can you for example clarify what you mean? Or can you introduce me some related papers with almost similar technique on such problems?

Do you have an actuator at each p location?

I propose to influence the diagonal entries of the error covariance matrix to refelect this amount of uncertainty in the unmeasurable states for these instances and to maintain them on a suitable value.

Muhammad,

Yeap. Assume that we have actuator at each location.

Raphael,

Error covariance matrix of the ALL state variables? And this error must be known? Note that there is no noise or any kind of stochasticity in the system. You mean I can model the potential uncertainties, caused by different controllers, by noises? Then, I can use stochastic control?

Well, not being able to measure may be considered an uncertainty, so a state estimator may help. You can reduce the trust of the KF estimate of state i by increasing only the i,i-th diagonal entry.

Seyed,

I would start saying that no one can guarantee that your problem has a safe solution, and this is in case you do have a control solution for each one of the cases.

Moreover, as far as I can see, the problem seems to be very complicated, too general and not clear at all and I think that first you must try to make it clear, first of all for yourself.

I don't know what you mean by local state feedback when you do not measure the entire state. If you want to use constant feedback, you mean constant output feedback.

 For simplicity, let's start with a SISO system, namely, where you can only measure one output signal. No one knows if stabilization with constant output feedback is possible at all. That's why people moved to using dynamic controllers. If you say that you only happen to be able to measure different outputs, and you do not aim to do something special with these outputs, I would try to maybe design a dynamic controller for the few outputs that are always measured and ignore the rest. If this does not seem at all right to you, it is again because the problem is not clear to me.

Now, if you have to stick to constant feedback, if the system is known and you can have some frequency analysis, etc., for each case, then you might be able to see if a constant feedback can stabilize your system.

But then, just assume that same feedback controller is used, yet just the gain varies during operation, either because of environmental conditions or because, as in your case, you decided to let it change. While the system is stable for each one of these values, no one can guarantee that it remains stable when you let the gain vary within the "admissible" domain (and doesn't matter whether the gain varies continuously or is switched). In spite of the above, people do use gain scheduling and they show that it works, yet I don't know of any guaranteed proof of stability.

Switching controllers is much less safe.

I will not make this a dissertation. I just happen to come from the Adaptive Control world, where we have been burnt by assuming that we could let the adaptive gains vary within the "admissible" domain and then were forced to find out that it is quite a job to guarantee stability of the adaptive system for the specific algorithm that changes the adaptive gains.

Dear Itzhak and Rafael,

First, Thanks a lot for your very informative responses. They are very kind of you.

Second, once again, I asked here about static output feedback and found out that there is no guaranteed solution for it since it is a bilinear matrix inequality problem...

But, look, here I avoided the term "Static Output Feedback" since I didn't want to end up with the same conclusions. I do not know how it is called but in fact it is not a regular static output feedback problem. Since the whole states are available globally but not locally! In fact at each location, we have a static output feedback but since we are going to control from all locations, globally all state variables are playing a role in the overall controls (An augmented control vector from all locations).  Do you see my point? So it is not globally a static output feedback problem.

I do not know who gain scheduling can help here. I may need to study some material more carefully. Could you please give me some suggestions for it?

Best regards,

Mehran 

 "Since the whole states are available globally but not locally! In fact at each location, we have a static output feedback but since we are going to control from all locations, globally all state variables are playing a role in the overall controls (An augmented control vector from all locations).  Do you see my point?"

No, I am afraid I don't. Writing "globally" and "locally" may have different meanings for different problems and people and I am afraid your description still looks too much like the "general problem." My mind is now very much into stability, so "globally stable" means that all trajectories are stable from any position you may start, while "locally stable" means only from some positions, usually within some restricted domain. 

In your case, same words pretty clearly mean something else, yet not clear what else. Do you have an LTI system that at different times measure different state variables? Then at each time you have a limited number of state variables to use. What does it mean that "globally all state variables..."  ???

I am not trying to claim anything here, just to express my confusion. If it is not secret, or at least trying to not reveal secrets, I think you must present some more of your actual problem and its aims.

"I do not know who gain scheduling can help here. I may need to study some material more carefully. Could you please give me some suggestions for it?"

I mentioned gain scheduling only as a simpler example of a system that may just need different gain values for different situations. Each gain is perfectly fit for the specific situation. Now, when situations change dynamically and you let the gains vary correspondingly, things may "work" yet may also stop working.

I still think we must better understand your specific problem first.

I understand, problem in the question as follows.

1. n plants  at different location are given. Sk linear plants are represented  by Ak,  Bk and Ck   matrices of the linear state space systems,  Where, k=1,2, ...

2. Further, A1=A2=..., B1=B2=.. and C1=C2=...

3. All the states of Sk are are not available locally. For example some states of S1 are measurable at location 1. Then some states of S2 are measurable, and so on.

4. union of the states has all the states of Ak but not from the same location.

5. using these states, it is desired to stabilize all the plants.

6.  Do all the eigen values of Ak are in left hand side?

If it is correct, then I may have few more queries about the problem.

Dear Muhammad,

The second condition is a little bit different:

C1 and C2 and C3 and... are not equal. But A1 and A2 and A3 and .. are the same, Let's call them A. The Bs are also same and let's call them simply B. In fact, because of this, we say the available state variables at each location are different.

A is stable. 

I guessed so, yet i waited for you to say so.

Before everything else: is your system known? Are A, B, C1,C2...Ck,... known?

Now,we reach the other issue: I asked what you want to do.

If A is stable and you only want to stabilize the system, then the "solution" is do nothing.

OK, I guess this is not so. However, at no point I saw anything that may tell what the purpose of your work is.  In regular Control works, a position, velocity, concentration of fluid, paper thickness, etc. must be permanently monitored and "controlled," or, in other words, made to behave in some desired way. What is yours?

Dear Itzhak,

Stabilization is "one" of the purposes of control designs. Sometimes we are interested in obtain a "desired performance" from the system which in a very low level controllers is just considering its stability. It can be for example reducing the overshoots or chatterings.

WOW! How did I know? :-)

Sir, at the time when I started doing these things and "squeezing" the best performance from real systems, I am not sure your parents were born. However, this has never been the general solution for the general problem, like the case seems to be, or at least seems to be presented, here.. I was just trying to guess what YOU might be willing to do with a system where, the way you put it,, it seems that you may have 15 (or 50) state variables and at one "location" you may measure a position and a current, at another location a velocity and a concentration, etc. Then how  can one guess what YOU want to "control?" 

Even in a system with ONE single output, if it is stable with no feedback,then  first you try some constant feedback and  try to make it as fast as you can. This is a P controller, yet it may lead to overshoot, because it may need some damping and so you add a D and tune it to get the damping you need. Then, you may not like the steady-state error and you ad an I, ending with a PID controller. If you do not want dynamic controllers, not even PID, then you may try to get the best you can with the proper P.

However, this is when you have a given output that you want to control. Sorry, but the way I read here, I am afraid I can only wish you luck, as you seem to keep having different outputs and I can have no idea what you want to do with which.

Thanks for your supports so far. I will write the problem in details later.

BTW, my parents are older than you for sure (just joking :)). 

Joking is always good. (I hope my :-) was clear)

Now, I think that, with or without me, it is good for you to have a clear picture of what you call measurements and what you call "controlled output."

1. You do not need gain scheduling for this problem.

2. Assume that all the states are available because the problem statement says that union of the  measured variable is the state space of the individual plants.

3. The plant is stable, design a static controller so that the closed loop remain stable as per the performance described in the problem statement.

4. In this case the stability of the global system is reduced to the stability of a local closed-loop. In this case all the states are bounded, they will not produce divergence in the system. 

5. As some of the states are obtained from the other plants any deviation in the state can be treated as disturbance.

6. Analyse the effect of this disturbance. The problem under discussion will remain bounded from input to states and from input to output and states to the output. The only remaining issue to be discussed is the affect of the these disturbed states on the output performance.

7. By defining some deviation bound on the states can be considered at the design stage to minimize the undesired behavior on the plant output.

8. You may incorporate adaptive tracking control to penalize the state measurement and the state deviation.

I think the problem is clear.

You may use the following references to further investigate the solution.

1. Linear systems  by Thomas Kailath

2. Linear System Theory, CT Chen

3. Control Systems Design Goodwin

4. Essentials of Robust Control Kemin Zhuo

I read the above discussions. I agree with Itzhak Barkana that a dynamic controller is needed to assure closed loop stability and performance of (A,B, Ci). I assume that (A, B, Ci) are linear time-invariant systems. The parametrization of all stabilizing controllers can help (see the book of Vidyasagar) and optimal or robust control design, if instead static output feedback is needed it could be "locally" more complicated. My concern is about the "global problem", It is well known that even if the individual closed loop systems are stable the interconnection of them could be unstable. A "global" Lyapunov function can help or if you can make a problem statement of the "global" system as a linear parametric varying (LPV) system into a convex set then there are results about stability such as Horisberger Theorem or the works of Apkarian.

If the local plants are strictly proper and stable, and the local loops are stable, the global system will be stable.

In general I disagree with Muhammad Shafiq, see for instance the works of,

Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

Hai Lin, and Panos J. Antsaklis, 

Stability Criteria for Switched and Hybrid Systems

Robert Shorten, Fabian Wirth, Oliver Mason, Kai Wulff and Christopher King

Maybe for this particular problem stability of the whole system can be assured when the local closed loop systems are stable, but I can not see the problem statement in this concern.

Dear Dr. Rene Galifo: Can you give a simple example for strictly proper plants and closed looo stable, become unstable in this scenario?

Dear Muhammad Shafiq,

Let consider two linear time invariant systems, each describing the motion of two particles in "small" oscillations. Their state space realizations are,

A1=[-1   0; 0  -w1*w1], B1=0 and

A2=[-1 0; 0 -w2*w2], B2=0

So, the state trajectories in the phase plane (x2 vs x1) of both systems are ellissi centered at (0,0) where the potencial energy is less than the total energy and the size of this ellissi depends on the initial energy.

Suppose that the ellisi of the first system is larger in the horizontal axes than in the vertical axis and the second system is larger in the vertical axes than in the horizontal axes. Hence, starting in positive horizontal axis of the second system and commuting to the first system in the vertical axis, then commuting to the second system in the horizontal axis, and so on, will generate state trajectories of the overall system that are increasing or unstable.

But A=A1=A2 in the given problem. Futher, A is stable.

This takes us back to the old conjectures of Aizerman and Kalman.

First, assume that your closed-loop system is stable for the gain K1 and for also for the gain K2. The question is: can you switch arbitrarily between K1 and K2? As we would see no problem, then the intuitive answer is yes and this is what those great names thought. 

OK, unlike in a perfect LTI system, the answer is: don't know. It may work for some and may not work at all for other "stable" K1 and K2. 

Can try:

A. Dewey and E. Jury, "A note on Aizerman's conjecture," IEEE Transactions on Automatic Control, vol. AC-19, pp. 482-483, 1963.

Then, one may ask why should one switch? If the system is stable for any fixed gain between 1 and 10, then let the gain vary arbitrarily yet smoothly between the admissible bounds. Again, for some variation may "work," and even nicely, yet for some small change may totally diverge. 

At some point in time I also thought that I had the proof that things must work. However it was only an "almost" proof. It then forced me to learn so much about passivity (positive realness in LTI systems) and stability of nonlinear systems that in retrospect I call this my best and happiest mistake ever. 

I also like the reference,

Two Counterexamples to Aizerman’s Conjecture, R. E. FITTS, IEEE Trans. on Automatic Control. 

Yes, of course Fitts. I gave Dewy&Jury as a first example.

Also, get

Jan Willems: The Analysis of Feedback Systems

at

http://homes.esat.kuleuven.be/~jwillems/Books/MITMonograph.pdf

pp. 177: Counterexample to Aizerman...

(and I did not want to mention the example in my book 

Kaufman, Barkana, Sobel: Direct Adaptive Control Algorithms, Springer)

Still, Seyed, my question remains what the objective of control is here. If A is stable and all you want is stability, you just let it work and you don't need any feedback.

Are you telling us that you have different C1, C2, ... because first you want to control y=C1 x and then y =C2 x, etc. ?