Nyquist stability criteria for MIMO systems

[deba_fire]

Hi All,

The Nyquist criteria which is described in various undergraduate textbooks is for single input and single output. There exists a similar Nyquist criteria for MIMO systems. Atleast a google search presents a lot of results. But I have difficulty in its application.

There are opamp based circuits which have MIMO loops. Example: Two-thomas biquad circuit, delta-sigma loops, and other countless examples. Till now I haven't seen any multiple loop opamp based circuits design using such criteria. Can someone point me to a good textbook/tutorial where multiple loop systems/circuits are designed using the above criteria?

Thanks

 

[deba_fire]

Hi All,

After a some survey in control systems, it seems that relative gain array(RGA) criteria is used for tuning MIMO feedback systems. Has anyone closed MIMO systems using this technique? But this method doesn't look at the classical phase margin(PM), modulus margin(MM) and gain margin(GM) criterias, so I have my doubts on this.

Currently I follow a three step approach to close MIMO loops:
1) Break one loop at a time, while keeping other loops closed.
2) Check PM, GM and MM for each individual loop one at a time.
3) If all loops are stable, complete MIMO feedback is stable.

The above method sound intuitively corrrect, but I donot feel complete confidence. Would appreciate if someone can comment on this.

I am particularly intereseted in stability analysis of the attached circuit. It is a common complex receive filter. There are multiple loops present. How does one go about it?

Thanks

 

[FvM]

In principle, loop gain should be analyzed for each feedback loop separately.

If the stages have identical parameters, stability analysis of the shown circuit can be reduced to one (differential) loop gain measurement by symmetry considerations. Prerequisite is that the amplifier output have sufficient low impedance, otherwise the analysis must at worst case include the full filter chain.

Another point is that loops are linked to some degree. In most designs, instability of one loop will show in all others. This can be expected for a filter, measuring the transfer function will be usually sufficient.

 

[pancho_hideboo]

See https://www.designers-guide.org/Forum/YaBB.pl?num=1243078889

 

[deba_fire]

Hi pancho_hideboo

Thanks for sharing the nice post. It discusses the same query which is troubling me.

In summary, for MIMO systems, the classical control doesn't apply. The physical meaning of GM, PM and MM are not clear for MIMO feedback systems. But then I am sure people in control engineering know how to deal with such systems. As lots of process control, flight control etc. fall under this criteria. But somehow in electronics that knowledge is not used. These are just my thoughts.

Can someone else shed more light on this?

Thanks

 

[LvW]

In summary, for MIMO systems, the classical control doesn't apply. The physical meaning of GM, PM and MM are not clear for MIMO feedback systems.

I rater think, they apply. This is also true for GM, PM and MM - however, for each of the existing loopd separately.
That means: Each loop may have (and will have) other stability margins. Note, stability margins do not belong to a closed-loop system but to a feedback loop.
Normally, in a multi-loop system, we will have one (or more) local loops and one dominant overall loop. This overall loop will mainly determine the closed-loop response.

 

[deba_fire]

Hi LvW,

So if one follows the method mentioned above(repeated for convenience),

1) Breaking one loop at a time, while keeping other loops closed.
2) Check PM, GM and MM for each of the individual loops one at a time.
3) If all loops are stable, complete MIMO feedback is stable.

Is the above rule a sufficient condition for MIMO feedback stability? Can the MIMO loop be unstable even if all loops are stable individually?

Thanks

 

[LvW]

Hi LvW,
So if one follows the method mentioned above(repeated for convenience),
1) Breaking one loop at a time, while keeping other loops closed.
2) Check PM, GM and MM for each of the individual loops one at a time.
3) If all loops are stable, complete MIMO feedback is stable.

Is the above rule a sufficient condition for MIMO feedback stability? Can the MIMO loop be unstable even if all loops are stable individually?

In the following you will find a quote from "Boris J. Lurie, Paul J. Enright: Classical Feedback Control", Marcel Dekker, Inc. New York.

(Begin of quote)
"Bode formulated the Nyquist criterion for multiloop systems as follows:

When a linear system is stable with certain loops disconnected, it is stable when these loops are closed if and only if the total numbers of clockwise and counterclockwise encirclements of the point (-1,0) are equal to each other in a series of Nyquist diagrams drawn for each loop and obtained by beginning with all loops open and closing the loops successively in any order leading to the system normal configuration." (End of quote)

 

[pancho_hideboo]

Is the above rule a sufficient condition for MIMO feedback stability?

You can not understand contents I posted at all.

Modern control theory is surely useful for judging stability of MIMO System.

Your complex filter is formulated by using state space variable equations.
So modern control theory is straight forward to apply.

https://en.wikipedia.org/wiki/State-space_representation

We argue and study stability of Delta-Sigma ADC using Modern Control theory.
Here loop filters have multiple loops and could be complex.

 

[deba_fire]

Hi All,

Thanks for your inputs. Sure state space seems to be the full proof method for studying MIMO. But I have very little intuition on tuning loops using state space.

Just a thought which I want your feedback on. Any differential opamp is a 2x2 MIMO system. But due to the nature/symmetry of the circuits, for most of the analog design, one is concerned only about common mode(CM) and differential mode(DM) loop. And for closing such designs, one makes sure that CM and DM loops are stable. The other two transfer functions are not even looked at due to the symmetry of the circuit. Thus, for closing the differential design, the CM and DM loops should be stable.

Extending the above reasoning, for MIMO also all individual loops should be stable. There is not a single number which characterizes the stability of MIMO. One needs to look at all the loops. Due to symmetry in the circuit, few loops can be ignored just like the differential opamp.

So for the complex filter above, around each and every opamp, if the DM and CM loops are stable, the whole filter is stable.

Thanks