Yes - I know that this is a fundamental problem.

However, I think the first question is: What is the term "stability" referred to? To the whole system (with n feedback loops) or to each individual loop?

For my opinion, the stability margin applies to one single feedback loop only because its definition is based on the LOOP GAIN, which - of course- can be defined for one loop only. From this it follows that for a system with N loops we can find N loop gains and, consequently, also N different stability margins. And the logical consequence is:

The worst margin determines the stability properties of the whole system.  

Interesting problem - any other opinions?

I think if the loops are cascaded you could think about the all over stability margin and of course all loops must be stable. But if n loops are not cascade connected, the loops not effected the others like reference,we can not define an all over stability margin i suppose. In second situation the answer above is quite important.

Hope this helps.

thank you  for your response

you can measure by criterien an all over stability margin and see Lyaponov condition.

Calculate the MIMO H-infinity norm for the closed-loop system.  

Quote R. Hanuma Naik's question "how to measure the degree of robust stability of multivariable system?" There are a couple of approaches to achieve this goal.

(1) "Calculate the MIMO H-infinity norm for the closed-loop system." suggested by Zigang Pan.

A good reference book is

S. Skogestad, I. Postlethwaite, "Multivariable feedback control: analysis and design", Wiely, 2005.

(2) "measure by criterien an all over stability margin and see Lyaponov condition" suggested by Mohamed-Mourad Lafifi.

(3) "the stability margin of feedback loop" suggested by Lutz von Wangenheim and Osman Çakıroğlu.

Quote R. Hanuma Naik's question "Is it measured individually to each loop or over all system".

Each loop should be stable in order to make the whole system stable.

In addition to Lutz von Wangenheim's comments "the stability margin applies to one single feedback loop only. The worst margin determines the stability properties of the whole system."

the interaction with other loops will reduce the stability margin of each loop. In other words, the interaction makes the worst margin even worse.

Let us look at the simplest multivariable control system with only 2 loops (the simulation example in [HY05a]).

Assume that each loop has no interaction,

the phase margin of loop #1 is \phi_1,   

the phase margin of loop #2 is \phi_2,

both \phi_1 and \phi_2 should be positive in order to make the whole system stable. That is, \phi_1>0 and \phi_2>0.

However, taking into account the interaction between loop #1 and loop #2,

the real phase margins for both loops are {\phi_1}^' and {\phi_2}^' instead

Direct Nyquist Array (DNA) can be used to analyze the interaction of different loops over a system.

According to DNA stability theorem, {\phi_1}^'

I totally agree with Lutz von Wangenheim, the worst stability margin would impose its characteristics to the entire system. However, those margins are commonly preferred above a certain boundary due to plant's nature. In this way, if the plant (or the portion of the "cascaded" plant) is unlikely to vary, low margins can be used at low risk; otherwise, high variability should be accompanied by conservative margins.

It all comes to the expected variability to me. It would be nonsense to speak about robustness in fully known non-variable plants. If the margin (at any level of the cascade loop) can not attain stability for the worst case plant, then problems appear.

For decentralized PI control stability analysis, as a suggestion one could use Nyquist array analysis as proposed in (D CHEN & D EBORG, "Design of decentralized PI control systems based on Nyquist stability analysis", J Process Control, 13 (2003) 27-39.) The method is fairly easy to understand and apply.

For a more rigorous approach of multivariable control design, the balanced sensitivity analysis proposed in (JD. BLIGHT, RL DAILEY & D GANGSAAS, "Practical control law design for aircraft using multivariable techniques", Int J Control, 59 (1994) 93-137) is quite effective. The method extends the familiar sensitivity function analysis of SISO system to MIMO, and the method can be found in Matlab Robust Control Toolbox. Personally, I have used this method to design decentralized PID control for up to 5x5 MIMO system via a simultaneous MSC tuning approach (not yet published); the method gives good measure of robustness/performance robustness - can provide stability margin.

There are number of methods to measure robustness with perturbations. While Lutz von presented a general solution f robustness, I think you are looking for mathematical prove.  A common measure of robustness for a system matrix in the face of perturbations is the condition number of its eigen vector matrix. Further information can be find in the following paper and reference within.

James Lam , H. K. TSO & N. K. Tsing (1997) Robust

deadbeat regulation, International Journal of Control, 67:4, 587-602, DOI:

10.1080/002071797224090

You can use a artifical block  GM x exp( -Td S ) in loop and change ualue of Gain GM or Time lag Td to find root loci of the closed-loop system by toolbox of Matlab. The roots crossing the imaginary axis give the gain magin(GM) or phase magin(PM) (exp(-TdS)==>exp(-jTdxWc)==>exp(jPM)). This method can be used also in Hardware In Loop(HIL).

In general, the connecting point between controller output and plant input is the point for checking the robustness.The artifical bock GM x exp(-Td S) can be used loop by loop to get the robustness measure(Gain/Phase Margins)  for each channel or uses m blocks for m x m multivariable system and changing GM or Td simultaneously to find an extreme measure of robustness of the overall system.