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canon

State-space canonical realization

Syntax

csys = canon(sys,type)
[csys,T]= canon(sys,type)
csys = canon(sys,'modal',condt)

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

sys

Any linear dynamic system model, except for frd models.

type

Canonical form of csys, specified as one of the following values:

condt

Positive scalar value specifying an upper bound on the condition number of the block-diagonalizing transformation that converts sys to csys. This argument is available only when type is 'modal'.

Increase condt to reduce the size of the eigenvalue clusters in the A matrix of csys. Setting condt = Inf diagonalizes A.

Default: 1e8

Output Arguments

csys

State-space (ss) model. csys is a state-space realization of sys in the canonical form specified by type.

T

Matrix specifying the transformation between the state vector x of the state-space model sys and the state vector xc of csys:

xc = Tx

.

This argument is available only when sys is state-space model.

Examples

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Consider a system with doubled poles and clusters of close poles:

$$G\left( s \right) = 100\frac{{\left( {s - 1} \right)\left( {s + 1}
\right)}}{{s\left( {s + 10} \right)\left( {s + 10.0001} \right){{\left(
{s - \left( {1 + i} \right)} \right)}^2}{{\left( {s - \left( {1 - i} \right)}
\right)}^2}}}$$

Create a linear model of this system, and convert it to modal canonical form.

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.

Gc.A
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

Separate the two poles near $s = -10$ by increasing the value of the condition number of the block-diagonalizing transformation. The default value of the condition number is 1e8.

Gc2 = canon(G,'modal',1e10);
Gc2.A
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 condition number of A is that the B matrix includes some large values.

format shortE
Gc2.B
ans =

   3.2000e-01
  -6.5691e-03
   5.4046e-02
  -1.9502e-01
   1.0637e+00
   3.2533e+05
   3.2533e+05

Estimate a state-space model that is freely parameterized.

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

Convert the estimated model to companion canonical form.

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,'r.');
showConfidence(h)

The frequency response confidence bounds are identical.

More About

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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 (λ1,σ±jω,λ2), the modal A matrix is of the form

[λ10000σω00ωσ0000λ2]

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

p(s)=sn+α1sn1++αn1s+αn

the corresponding companion A matrix is

A=[00....0αn100..0αn1010.:::0..::0..10α20....01α1]

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.

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

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Introduced before R2006a

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