Regularization technique for Stabilization of polynomial by bringing unstable poles into unit circle

I have been trying to stabilize a characteristic polynomial of a model in the z-domain. Therefore I've been trying to bring its poles inside the unit circle without affecting the dynamics of the system. First I thought of the polystab function, which didn't help because for some reason the couple of unstable poles I tried to stabilize ended up to a system with even more unstable poles. Are there any opinions/tips on the use of the polystab function? Or is there any function useful for "pulling" the unstable poles into the unit circle?

[M.]

What exactly do you mean by "bring its poles inside the unit circle without affecting the dynamics of the system" ?

If you go from an unstable model to a stable one, then, by definition, you affect the dynamics of the system.

[Anast]

You are right and let me correct my question by adding that the model is estimated via system identification methods, for example an ARX is estimated. Sometimes the estimated model may have some poles outside the unit circle and some other times no. We may want to keep the specific order of the ARX , so what do we do then?

[Mathieu NOE]

We don't know which method / functions you are using. This can lead to such situations

are you working in time domain or frequency doamin ?

are the input data coming from an always satbe system (ie the problem is only due to the identification method used here ? )

so data and code if you can share , it would help !

[Tianyu]

If you estimated your ARX model using arx function in system identification toolbox, you can enforce the estimated model to be stable using "EnforceStability" in arxOptions

https://www.mathworks.com/help/ident/ref/arxoptions.html#btcxwg1-1_sep_shared-EnforceStability

polystab seems indeed fulfill your purpose, i.e. stabilize the model without altering reponse. Can you please share the code so we can take a further look?

[Mr. Pavl M.]

%Can be any system to stabilize after improved SI:
A = [1 1; -3 -5];
B = [12; 1];
C = [1 -7];
D = 2;

ts = 0.01;
dsys = ss(A,B,C,D,ts)

dsys =
 
  A = 
       x1  x2
   x1   1   1
   x2  -3  -5
 
  B = 
       u1
   x1  12
   x2   1
 
  C = 
       x1  x2
   y1   1  -7
 
  D = 
       u1
   y1   2
 
Sample time: 0.01 seconds
Discrete-time state-space model.

%G = press_max*minreal(dsys)
G = minreal(dsys);
[b3, a3] = ss2tf(G.A,G.B,G.C,G.D)

b3 = 1×3

2.0000 13.0000 316.0000

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a3 = 1×3

1.0000 4.0000 -2.0000

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G3 = tf(b3, a3,ts) %discrete model

G3 =
 
  2 z^2 + 13 z + 316
  ------------------
    z^2 + 4 z - 2
 
Sample time: 0.01 seconds
Discrete-time transfer function.

%Not "very" conventional, but functions fully, entirely in number of cases:
hmin01 = polystab(b3)

hmin01 = 1×3

2.0000 0.0823 0.0127

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hmin1 = hmin01/norm(hmin01)*norm(b3)

hmin1 = 1×3

316.0000 13.0000 2.0000

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hmin02 = polystab(a3)

hmin02 = 1×3

1.0000 -0.2247 -0.1010

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hmin2 = hmin02/norm(hmin02)*norm(a3)

hmin2 = 1×3

4.4495 -1.0000 -0.4495

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G4 = tf(hmin1,hmin2,ts) %discrete model

G4 =
 
    316 z^2 + 13 z + 2
  ----------------------
  4.449 z^2 - z - 0.4495
 
Sample time: 0.01 seconds
Discrete-time transfer function.

figure
pzmap(G3)

figure
pzmap(G4)

Answer to your question is:

polystab TCE built - in function or alike is allright, okay also to find medium value use.

1. There are confidently more accurate and precise id methods, besides ARX. Sorry, PTSD caused to me not from here, which times poles are how deep inside unit circle?There is good rationale here, we don't have catatonic, nor delusions so by ICD11 and DSM5 there were no schizo of us and there is no schizo now. So to stabilize good, fairly well to continue not downwards, which means not to undermine, not to diminish importance of the study and labour of the system responses and systems behaviour.
2. Have you applied [K,prec] = place(A,B,p) ?, what is your desired poles radius,number of poles, ( constraint<r<1)?