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# Multiloop stability criteria

## Multiloop stability criteria

(OP)
Hi,

In many textbooks, the Nyquist criteria is used for analyzing the stability for closed loop systems. But they are mostly single input single output(SISO) systems. But there are many systems which have multiple loops, or in general multiple input multiple output(MIMO) systems. For example: from my electronics background there are various filters like BiQuad filters, delta-sigma loops and other countless examples.

1) How does one come up with a stability criteria for such systems?
2) Is there a single number like phase margin and gain margin which one needs to check? Which loop should be used?

I know there is sequential loop closing method proposed by Bode. But if I have yet to see a practical example of closing loops using that. I would appreciate if someone can refer a tutorial or book on this topic.

Regards
DB

### RE: Multiloop stability criteria

Common problem, there are several approaches depending on the system complexity i.e. the number of simple closed loops.

The relative gain array is probably the most universal, for idealistic models of the system you can look at the system poles and performa a sensitivity analysis (effectively a variation of the root locus approach).

The McAvoy wrote a book on the subject back in the day, published by the ISA. Books by Greg Shinskey are also beneficial, as are Bela Liptak's handbooks.

### RE: Multiloop stability criteria

(OP)
Hi hacksaw,

Thanks for the response. After a quick review of relative gain array(RGA), it seems it looks at the steady state gains, and one needs to use that information to tune the loop. What happens if there are frequency dependent blocks are present, like integrators, differentiators, or any frequency dependent feedback? How does RGA relate to classical phase margin(PM), gain margin(GM) and modulus margin(MM) criterias?

Currently I am closing my MIMO design by following a two step process as below:
1) by breaking one loop at a time while keeping the other loops closed. Check the PM,GM and MM for each individual loop.
2) If all loops are stable, the complete MIMO feedback system is stable.

So for MIMO systems one gets an array of PM,GM and MM corresponding to each one of the loops. But I don't have full confidence in this method. It sounds intuitively ok, but would like to see a mathematical proof for this.

Thanks
DB

### RE: Multiloop stability criteria

The RGA also applies to dynamic systems, it provides the same information as Nyquist, etc. but as with all analysis it is only as good as your model...and the assumption of linearity

As to your statement regarding individual loop stability translating into overall system stability,
there are conditionally stable processes, that can become unstable even if all the loops are stable.... flight stability, control of processes that are inherently untable, or processes where phase changes can occur...

### RE: Multiloop stability criteria

(OP)
Hi hacksaw,

In integrated electronics/circuit work, the model/transfer function is always known. Do you have an example where even though the individual loops are stable but the overall MIMO system is unstable?

Thanks

### RE: Multiloop stability criteria

Examples are contained in the texts mentioned, and actual experience in the field where analytical tools are not always available or funding available to quantify.

A literature search is a good place to start!

### RE: Multiloop stability criteria

The stability issues usually involve system where phase changes occur (thermal, chemical) or where the control action alters the so call process gain functions.

Airframe stability, fractionator columns, steam systems,...

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