canon - State-space canonical realization

Syntax

Description

csys = canon(sys,type) transforms the linear model sys into a canonical state-space model csys. The argument type specifies whether csys is in modal or companion form.

[csys,T]= canon(sys,type) also returns the state-coordinate transformation T that relates the states of the state-space model sys to the states of csys.

csys = canon(sys,'modal',condt) specifies an upper bound condt on the condition number of the block-diagonalizing transformation.

Input Arguments

Output Arguments

Definitions

Modal Form

In modal form, A is a block-diagonal matrix. The block size is typically 1-by-1 for real eigenvalues and 2-by-2 for complex eigenvalues. However, if there are repeated eigenvalues or clusters of nearby eigenvalues, the block size can be larger.

For example, for a system with eigenvalues , the modal A matrix is of the form

Companion Form

In the companion realization, the characteristic polynomial of the system appears explicitly in the rightmost column of the A matrix. For a system with characteristic polynomial

the corresponding companion A matrix is

The companion transformation requires that the system be controllable from the first input. The companion form is poorly conditioned for most state-space computations; avoid using it when possible.

Examples

This example uses canon to convert a system having doubled poles and clusters of close poles to modal canonical form.

Consider the system G having the following transfer function:

To create a linear model of this system and convert it to modal canonical form, enter:

G = zpk([1 -1],[0 -10 -10.0001 1+1i 1-1i 1+1i 1-1i],100);
Gc = canon(G,'modal');

The system G has a pair of nearby poles at s = –10 and s = –10.0001. G also has two complex poles of multiplicity 2 at s = 1 + i and s = 1 – i. As a result, the modal form, has a block of size 2 for the two poles near s = –10, and a block of size 4 for the complex eigenvalues. To see this, enter the following command:

Gc.A

This command returns the result:

ans =

         0         0         0         0         0         0         0
         0    1.0000    1.0000         0         0         0         0
         0   -1.0000    1.0000    2.0548         0         0         0
         0         0         0    1.0000    1.0000         0         0
         0         0         0   -1.0000    1.0000         0         0
         0         0         0         0         0  -10.0000    8.0573
         0         0         0         0         0         0  -10.0001

To separate the two poles near s = –10, you can increase the value of condt. For example, entering the commands:

Gc2 = canon(G,'modal',1e10);
Gc2.A

returns the result:

ans =

         0         0         0         0         0         0         0
         0    1.0000    1.0000         0         0         0         0
         0   -1.0000    1.0000    2.0548         0         0         0
         0         0         0    1.0000    1.0000         0         0
         0         0         0   -1.0000    1.0000         0         0
         0         0         0         0         0  -10.0000         0
         0         0         0         0         0         0  -10.0001

The A matrix of Gc2 includes separate diagonal elements for the poles near s = –10. The cost of increasing the maximum condition number of A is that the B matrix includes some large values.

format shortE
Gc2.B

ans =

  3.2000e-001
 -6.5691e-003
  5.4046e-002
 -1.9502e-001
  1.0637e+000
  3.2533e+005
  3.2533e+005

This example estimates a state-space model that is freely parameterized and convert to companion form after estimation.

load icEngine.mat
z = iddata(y,u,0.04);
FreeModel = n4sid(z,4,'InputDelay',2); 
CanonicalModel = canon(FreeModel, 'companion')

Obtain the covariance of the resulting form by running a zero-iteration update to model parameters.

opt = ssestOptions; opt.SearchOption.MaxIter = 0;
CanonicalModel  = ssest(z, CanonicalModel, opt)

Compare frequency response confidence bounds of FreeModel to CanonicalModel.

h = bodeplot(FreeModel, CanonicalModel)

the bounds are identical.

Algorithms

The canon command uses the bdschur command to convert sys into modal form and to compute the transformation T. If sys is not a state-space model, the algorithm first converts it to state space using ss.

The reduction to companion form uses a state similarity transformation based on the controllability matrix [1].

References

[1] Kailath, T. Linear Systems, Prentice-Hall, 1980.

See Also

ctrb | ctrbf | ss2ss

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