State-space canonical realization
csys = canon(sys,type)
csys = canon(sys,'modal',condt)
Any linear dynamic system model, except for frd models.
String specifying the type of canonical form of csys. type can take one of the two following values:
Increase condt to reduce the size of the eigenvalue clusters in the A matrix of csys. Setting condt = Inf diagonalizes A.
xc = Tx
This argument is available only when sys is state-space model.
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:
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:
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 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.
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
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.
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 .