canon

Canonical state-space realization

Syntax

csys = canon(sys,type)

csys = canon(sys,'modal',condt)

[csys,T]= canon(___)

Description

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

For information on controllable and observable canonical forms, see Canonical State-Space Realizations.

csys = canon(sys,'modal',condt) specifies an upper bound condt on the condition number of the block-diagonalizing transformation. Use condt if you have close lying eigenvalues in csys.

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

Examples

Convert State-Space Model to Companion Canonical Form

aircraftPitchSSModel.mat contains the state-space matrices of an aircraft where the input is elevator deflection angle $\delta$ and the output is the aircraft pitch angle $\theta$.

$$\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{\dot{}}{\alpha }}\\ \underset{}{\overset{\dot{}}{q}}\\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]=\left[\begin{array}{ccc}-0.313& 56.7& 0\\ -0.0139& -0.426& 0\\ 0& 56.7& 0\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[\begin{array}{c}0.232\\ 0.0203\\ 0\end{array}\right]\left[\delta \right]\\ y=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\left[\begin{array}{c}\alpha \\ q\\ \theta \end{array}\right]+\left[0\right]\left[\delta \right]\end{array}$$

Load the model data to the workspace and create the state-space model sys.

load('aircraftPitchSSModel.mat');
sys = ss(A,B,C,D)

sys =
 
  A = 
            x1       x2       x3
   x1   -0.313     56.7        0
   x2  -0.0139   -0.426        0
   x3        0     56.7        0
 
  B = 
           u1
   x1   0.232
   x2  0.0203
   x3       0
 
  C = 
       x1  x2  x3
   y1   0   0   1
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

Convert the resultant state-space model sys to companion canonical form.

csys = canon(sys,'companion')

csys =
 
  A = 
               x1          x2          x3
   x1           0           0  -1.809e-16
   x2           1           0     -0.9215
   x3           0           1      -0.739
 
  B = 
       u1
   x1   1
   x2   0
   x3   0
 
  C = 
            x1       x2       x3
   y1        0    1.151  -0.6732
 
  D = 
       u1
   y1   0
 
Continuous-time state-space model.

csys is the companion canonical form of sys.

Convert State-Space Model to Modal Canonical Form

pendulumCartSSModel.mat contains the state-space model of an inverted pendulum on a cart where the outputs are the cart displacement x and the pendulum angle $\theta$. The control input u is the horizontal force on the cart.

$$\begin{array}{l}\left[\begin{array}{c}\underset{}{\overset{\dot{}}{x}}\\ \underset{}{\overset{¨}{x}}\\ \underset{}{\overset{\dot{}}{\theta }}\\ \underset{}{\overset{¨}{\theta }}\end{array}\right]=\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& -0.1& 3& 0\\ 0& 0& 0& 1\\ 0& -0.5& 30& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{\dot{}}{x}}\\ \theta \\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 2\\ 0\\ 5\end{array}\right]u\\ y=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}x\\ \underset{}{\overset{\dot{}}{x}}\\ \theta \\ \underset{}{\overset{\dot{}}{\theta }}\end{array}\right]+\left[\begin{array}{c}0\\ 0\end{array}\right]u\end{array}$$

First, load the state-space model sys to the workspace.

load('pendulumCartSSModel.mat','sys');

Convert sys to modal canonical form and extract the transformation matrix.

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

csys =
 
  A = 
           x1      x2      x3      x4
   x1       0       0       0       0
   x2       0   -0.05       0       0
   x3       0       0  -5.503       0
   x4       0       0       0   5.453
 
  B = 
          u1
   x1  1.875
   x2  6.298
   x3   12.8
   x4  12.05
 
  C = 
              x1         x2         x3         x4
   y1         16     -4.763  -0.003696   0.003652
   y2          0   0.003969   -0.03663    0.03685
 
  D = 
       u1
   y1   0
   y2   0
 
Continuous-time state-space model.

T = 4×4

    0.0625    1.2500   -0.0000   -0.1250
         0    4.1986    0.0210   -0.4199
         0    0.2285  -13.5873    2.4693
         0   -0.2251   13.6287    2.4995

csys is the modal canonical form of sys, while T represents the transformation between the state vectors of sys and csys.

Convert Zero-Pole-Gain Model to Modal Canonical Form

For this example, consider the following system with doubled poles and clusters of close poles:

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

Create a zpk model of this system and convert it to modal canonical form using the string 'modal'.

sys = zpk([1 -1],[0 -10 -10.0001 1+1i 1-1i 1+1i 1-1i],100);
csys1 = canon(sys,'modal');
csys1.A

ans = 7×7

         0         0         0         0         0         0         0
         0    1.0000    1.0000         0         0         0         0
         0   -1.0000    1.0000    3.2459         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    4.0571
         0         0         0         0         0         0  -10.0001

csys1.B

ans = 7×1

    0.1600
   -0.0052
    0.0427
   -0.0975
    0.5319
         0
    4.0095

sys has a pair of poles at s = -10 and s = -10.0001, and two complex poles of multiplicity 2 at s = 1+i and s = 1-i. As a result, the modal form csys1 is a state-space model with a block of size 2 for the two poles near s = -10, and a block of size 4 for the complex eigenvalues.

Now, separate the two poles near s = -10 by increasing the value of the condition number of the block-diagonalizing transformation. Use a value of 1e10 for this example.

csys2 = canon(sys,'modal',1e10);
csys2.A

ans = 7×7

         0         0         0         0         0         0         0
         0    1.0000    1.0000         0         0         0         0
         0   -1.0000    1.0000    3.2459         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

format shortE
csys2.B

ans = 7×1

   1.6000e-01
  -5.1885e-03
   4.2687e-02
  -9.7508e-02
   5.3187e-01
   1.6267e+05
   1.6267e+05

The A matrix of csys2 includes separate diagonal elements for the poles near s = -10. Increasing the condition number results in some very large values in the B matrix.

Convert System to Companion Canonical Form

The file icEngine.mat contains one data set with 1500 input-output samples collected at the a sampling rate of 0.04 seconds. The input u(t) is the voltage (V) controlling the By-Pass Idle Air Valve (BPAV), and the output y(t) is the engine speed (RPM/100).

Use the data in icEngine.mat to create a state-space model with identifiable parameters.

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

Convert the identified state-space model sys to companion canonical form.

csys = canon(sys,'companion');

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

opt = ssestOptions;
opt.SearchOptions.MaxIterations = 0;
csys = ssest(z,csys,opt);

Compare frequency response confidence bounds of sys to csys.

h = bodeplot(sys,csys,'r.');
showConfidence(h)

The frequency response confidence bounds are identical.

Input Arguments

sys — Dynamic system

dynamic system model

Dynamic system, specified as a SISO, or MIMO dynamic system model. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, ss, or pid models.

• Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

  The resulting canonical state-space model assumes

  • current values of the tunable components for tunable control design blocks.

  • nominal model values for uncertain control design blocks.

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), idproc (System Identification Toolbox), idpoly (System Identification Toolbox), and idgrey (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

You cannot use frequency-response data models such as frd models.

type — Transformation type

'modal' (default) | 'companion'

Transformation type, specified as either 'modal' or 'companion'. If type is unspecified, then canon converts the specified dynamic system model to modal canonical form by default.

The companion canonical form is the same as the observable canonical form. For information on controllable and observable canonical forms, see Canonical State-Space Realizations.

• 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 $$({\lambda }_1,\sigma \pm j\omega ,{\lambda }_2)$$, the modal A matrix is of the form

  $$\left[\begin{array}{cccc}{\lambda }_1& 0& 0& 0\\ 0& \sigma & \omega & 0\\ 0& -\omega & \sigma & 0\\ 0& 0& 0& {\lambda }_2\end{array}\right]$$

• 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

  $$P(s)=s^n+{\alpha }_1{s}^{n-1}+\dots +{\alpha }_{n-1}s+{\alpha }_n$$

  the corresponding companion A matrix is

  $$A=\left[\begin{array}{ccc}\begin{array}{l}0\\ 1\\ 0\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{l}0\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}& \begin{array}{ccc}\begin{array}{cc}\begin{array}{l}0\\ 0\\ 0\\ 1\\ ⋮\\ 0\end{array}& \begin{array}{l}\dots \\ \dots \\ \dots \\ \dots \\ \ddots \\ \dots \end{array}\end{array}& \begin{array}{l}0\\ 0\\ 0\\ 0\\ ⋮\\ 1\end{array}& \begin{array}{l}-{\alpha }_n\\ -{\alpha }_{n-1}\\ -{\alpha }_{n-2}\\ -{\alpha }_{n-3}\\ ⋮\\ -{\alpha }_1\end{array}\end{array}\end{array}\right]$$

  The companion transformation requires that the system is controllable from the first input. The transformation to companion form is based on the controllability matrix which is almost always numerically singular for mid-range orders. Hence, avoid using it when possible.

  The companion canonical form is the same as the observable canonical form. For more information on observable and controllable canonical forms, see Canonical State-Space Realizations.

condt — Upper bound on the condition number of the block-diagonalizing transformation

1e8 (default) | positive scalar

Upper bound on the condition number of the block-diagonalizing transformation, specified as a positive scalar. This argument is available only when type is set to 'modal'.

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

Output Arguments

csys — Canonical state-space form of the dynamic model

ss model object

Canonical state-space form of the dynamic model, returned as an ss model object. csys is a state-space realization of sys in the canonical form specified by type.

T — Transformation matrix

matrix

Transformation matrix, returned as an n-by-n matrix, where n is the number of states. T is the transformation between the state vector x of the state-space model sys and the state vector x_c of csys:

x_c = Tx

.

This argument is available only when sys is an ss model object.

Limitations

• You cannot use frequency-response data models to convert to canonical state-space form.

• The companion form is poorly conditioned for most state-space computations, that is, the transformation to companion form is based on the controllability matrix which is almost always numerically singular for mid-range orders. Hence, 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, canon 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.