Compute or plot sector index as function of frequency
the relative sector indices for the dynamic system
a given sector matrix
Q. These indices measure
by how much the sector bound is satisfied (index less than 1) or violated
(index greater than 1) at a given frequency. (See About Sector Bounds and Sector Indices for
more information about the meaning of the sector index.)
chooses the frequency range and number of points based on the dynamics
Let the following be an orthogonal decomposition of the symmetric
Q into its positive and negative parts.
The sector index plot is only meaningful if has a proper stable inverse. In that case, the sector indices are the singular values of:
w is a cell array of the form
sectorplot plots the sector index at frequencies
w is a vector of frequencies,
sectorplot plots the sector index at each
Plot the sector index to visualize the frequencies at which the I/O trajectories of lie within the sector defined by:
In U/Y space, this sector is the shaded region of the following diagram.
The Q matrix for this sector is given by:
a = 0.1; b = 10; Q = [1 -(a+b)/2 ; -(a+b)/2 a*b];
A trajectory lies within the sector S when for all T > 0,
In the frequency domain, this same condition can be expressed as:
To check whether
G satisfies or violates this condition at any frequency, plot the sector index for
H = [G;1].
G = tf([1 2],[1 1]); sectorplot([G;1],Q)
The plot shows that the sector index is less than 1 at all frequencies. Therefore, the trajectories of G(s) fit within in the specified sector Q at all frequencies.
Examine the sector plot of a 2-output, 2-input system for a particular sector.
rng(4,'twister'); H = rss(3,4,2); Q = [-5.12 2.16 -2.04 2.17 2.16 -1.22 -0.28 -1.11 -2.04 -0.28 -3.35 0.00 2.17 -1.11 0.00 0.18]; sectorplot(H,Q)
H is 2-by-2, there are two lines on the sector plot. The largest value of the sector index exceeds 1 below about 0.5 rad/s and in a narrow band around 3 rad/s. Therefore,
H does not satisfy the sector bound represented by
H— Model to analyze
Model to analyze against sector bounds, specified as a dynamic
system model such as a
H can be continuous or discrete. If
H is a generalized model with tunable or uncertain
sectorplot analyzes the current, nominal value
To analyze whether all I/O trajectories (u(t),y(t) of a linear system G lie in a
particular sector, use
H = [G;I], where
nu is the number of inputs of
H is a model array, then
sectorplot plots the sector index of all models in
the array on the same plot. When you use output arguments to get
H must be a single model.
index— Sector indices
Sector indices as a function of frequency, returned as a matrix.
the sector indices computed at the frequencies
you supplied them, or
wout if you did not.
as many columns as there are values in
and as many rows as
H has inputs. Thus the value
the sector indices in descending order at the frequency
For example, suppose that
G is a 3-input,
Q is a suitable sector matrix,
w is a 1-by-30 vector of frequencies, then
the following syntax returns a 3-by-30 matrix
H = [G;eyes(3)] index = sectorplot(H,Q,w);
index(:,k) contains the three sector
H, in descending order, at the frequency
For more information, see About Sector Bounds and Sector Indices.