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getSectorCrossover

Crossover frequencies for sector bound

Description

example

wc = getSectorCrossover(H,Q) returns the frequencies at which the following matrix M(ω) is singular:

M(ω)=H(jω)HQH(jω).

When a frequency-domain sector plot exists, these frequencies are the frequencies at which the relative sector index (R-index) for H and Q equals 1. See About Sector Bounds and Sector Indices for details.

Examples

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Find the crossover frequencies for the dynamic system G(s)=(s+2)/(s+1) and the sector defined by:

S={(y,u):au2<uy<bu2},

for various values of a and b.

In U/Y space, this sector is the shaded region of the following diagram (for a, b > 0).

The Q matrix for this sector is given by:

Q=[1-(a+b)/2-(a+b)/2ab];a=0.1,b=10.

getSectorCrossover finds the frequencies at which H(s)HQH(s) is singular, for H(s)=[G(s);I]. For instance, find these frequencies for the sector defined by Q with a = 0.1 and b = 10.

G = tf([1 2],[1 1]); 
H = [G;1];

a = 0.1;  
b = 10; 
Q = [1 -(a+b)/2 ; -(a+b)/2 a*b];

w = getSectorCrossover(H,Q)
w =

  0x1 empty double column vector

The empty result means that there are no such frequencies.

Now find the frequencies at which HHQH is singular for a narrower sector, with a = 0.5 and b = 1.5.

a2 = 0.5;  
b2 = 1.5; 
Q2 = [1 -(a2+b2)/2 ; -(a2+b2)/2 a2*b2];

w2 = getSectorCrossover(H,Q2)
w2 = 1.7321

Here the resulting frequency is where the R-index for H and Q2 is equal to 1, as shown in the sector plot.

sectorplot(H,Q2)

Figure contains an axes object. The axes object contains an object of type line. This object represents H.

Thus, when a sector plot exists for a system H and sector Q, getSectorCrossover finds the frequencies at which the R-index is 1.

Input Arguments

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Model to analyze against sector bounds, specified as a dynamic system model such as a tf, ss, or genss model. H can be continuous or discrete. If H is a generalized model with tunable or uncertain blocks, getSectorCrossover analyzes the current, nominal value of H.

To get the frequencies at which the I/O trajectories (u,y) of a linear system G lie in a particular sector, use H = [G;I], where I = eyes(nu), and nu is the number of inputs of G.

Sector geometry, specified as:

  • A matrix, for constant sector geometry. Q is a symmetric square matrix that is ny on a side, where ny is the number of outputs of H.

  • An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)’ = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency.

The matrix Q must be indefinite to describe a well-defined conic sector. An indefinite matrix has both positive and negative eigenvalues.

For more information, see About Sector Bounds and Sector Indices.

Output Arguments

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Sector crossover frequencies, returned as a vector. The frequencies are expressed in rad/TimeUnit, relative to the TimeUnit property of H. If the trajectories of H never cross the boundary, wc = [].

Version History

Introduced in R2016a