Eigenvalues of a matrix sum

Question

I have a control system problem, which ends up in that the eigenvalues of the system matrix should have a negative real part, then the system is stable.

The system matrix is real but not symmetric. The values depend on the system parameters, but zeros will remain zeros and equal value will be equal.

M  =

 -126.49961    41.887902    1120.2778    13417.173  - 175.32356  
 -41.887902  - 126.49961  - 13417.173    1120.2778  - 28.050364  
  0.1282520    0.         - 3.4974646    0.           0.5473535  
  0.           0.1282520    0.         - 3.4974646    0.0875722  
  5.63488      0.9015356  - 153.66458  - 24.585101    0.  

This one is fine, all eigenvalues have negative real part.

However, there is a disturbance that adds up:

 Me  =

  - 308.54625  - 49.364927    0.    0.    0.  
  - 49.364927  - 7.8979926    0.    0.    0.  
    0.           0.           0.    0.    0.  
    0.           0.           0.    0.    0.  
    0.           0.           0.    0.    0.  

One pair of eigenvalues of (M + Me) has positive real part.

Now, I can add a compensation like

 Mc  =

  - r  - x    0.    0.    0.  
    x  - r    0.    0.    0.  
    0.   0.   0.    0.    0.  
    0.   0.   0.    0.    0.  
    0.   0.   0.    0.    0.  

here with r = 85.41 and x = 41.88

and (M + Me + Mc) again has eigenvalues with negative real part if r and x are high enough.

The question is: How can I find sufficient values for r and x without numerically calculating the matrix eigenvalues? Is there some approximation?

Thanks for any hint Alexander

Answer 1

You could use the Gershgorin circle theorem. For an $n\times n$ matrix $M$, consider $n$ circles $C_i$, $i=1,2,\ldots n$ in the complex plane centered at $M_{ii}$, with radius $\sum_{j\neq i}|M_{ij}|$. Each eigenvalue of $M$ lies in at least one of these circles. So you want to add to $M$ a correction that shifts all circles such that they lie in the left-half of the complex plane.

Answer 2

This is a constrained nearest stable matrix problem. You can try using the technique in Guglielmi, Lubich Matrix Stabilization Using Differential Equations https://doi.org/10.1137/16M1105840 .