popov

Popov robust stability test

collapse all in page

Syntax

[t,P,S,N] = popov(usys,flag)

Description

[t,P,S,N] = popov(usys,flag) uses the Popov criterion to test if the uncertain model usys is robustly stable. Stability is established when t < 0, in which case the function returns a Lyapunov matrix P and multipliers S and N proving stability.

example

Examples

collapse all

Open Live Script

Create a model of a dynamic system some uncertain parameters.

p1 = ureal('p1',10,'Percentage',50);
p2 = ureal('p2',3,'PlusMinus',[-.5 1.2]);
p3 = ureal('p3',0);
A = [-p1 p2; 0 -p1];
B = [-p2; p2+p3];
C = [1 0; 1 1-p3];
D = [0; 0];
usys = ss(A,B,C,D);

Test the stability of the system using popov.

[t,P,S,N] = popov(usys);
 Solver for LMI feasibility problems L(x) < R(x)
    This solver minimizes  t  subject to  L(x) < R(x) + t*I
    The best value of t should be negative for feasibility

 Iteration   :    Best value of t so far 
 
     1                        0.122129
     2                    6.169090e-04
     3                       -5.692904

 Result:  best value of t:    -5.692904
          f-radius saturation:  0.002% of R =  1.00e+07
 
 Robustly stable for the prescribed uncertainty

The function displays information about the optimization. For this system, the result is t < 0, meaning that the system as described is stable.

popov also returns a Lyapunov matrix P and Popov multipliers S and N.

P
P = 2×2

   29.0383    4.7367
    4.7367   38.2164


S
S = 6×6

   19.0273   -1.2865         0         0         0         0
    0.6128   23.7903         0         0         0         0
         0         0   20.3213   12.7462         0         0
         0         0  -13.6064   20.9684         0         0
         0         0         0         0   17.9928  -37.3131
         0         0         0         0   37.3131   18.6171


N
N = 6×6

    1.1233    0.2933         0         0         0         0
    0.2933    0.1214         0         0         0         0
         0         0         0         0         0         0
         0         0         0         0         0         0
         0         0         0         0         0   -6.6889
         0         0         0         0   -6.6889    0.0000

Input Arguments

collapse all

Uncertain dynamic system, specified as a uss model.

Flag to reduce conservatism of computation specified as 0 or 1. When usys contains real parameter uncertainty, you can set flag to 1 to reduce conservatism at the expense of more intensive computations.

Output Arguments

collapse all

Optimal largest eigenvalue of the feasibility problem corresponding to the system stability criterion, returned as a scalar. If t < 0, then the Popov LMIs are feasible and usys is robustly stable.

Lyapunov matrix, returned as a matrix. P determines the quadratic part of the Lyapunov function, xTPx.

Popov multipliers proving stability, returned as matrices.

References

[1] Haddad, W.M., and D.S. Bernstein, “Parameter-Dependent Lyapunov Functions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis: Part 1 and 2,” Proc. Conf. Dec. Contr., 1991, pp. 2274-2279 and 2632-2633.

Version History

Introduced before R2006a

See Also