modalreal

Compute modal state-space realization

Since R2023b

Syntax

[msys,blks] = modalreal(sys)

[msys,blks,TL,TR] = modalreal(sys)

[___] = modalreal(sys,Name=Value)

Description

[msys,blks] = modalreal(sys) returns a modal realization msys of an LTI model sys. This is a realization where A or (A,E) are block diagonal and each block corresponds to a real pole, a complex pair, or a cluster of repeated poles. blks is a vector containing block sizes down the diagonal.

[msys,blks,TL,TR] = modalreal(sys) also returns the block-diagonalizing transformations TL and TR.

[___] = modalreal(sys,Name=Value) specifies options for controlling the block size and normalizing 2-by-2 blocks associated with complex pairs.

Examples

Convert State-Space Model to Modal Form

pendulumCartSSModel.mat contains the state-space model of an inverted pendulum on a cart where the outputs are the cart displacement $\mathit{x}$ 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 form and extract the block sizes.

[msys,blks,TL,TR] = modalreal(sys)

msys =
 
  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.

blks = 4×1

     1
     1
     1
     1

TL = 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

TR = 4×4

   16.0000   -4.7631   -0.0037    0.0037
         0    0.2381    0.0203    0.0199
         0    0.0040   -0.0366    0.0369
         0   -0.0002    0.2015    0.2009

msys is the modal realization of sys, blks represents the block sizes down the diagonal, and TL and TR represent the block-diagonalizing transformation matrices.

Modal Realization of Zero-Pole-Gain Model

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 obtain a modal realization using the function modalreal.

sys = zpk([1 -1],[0 -10 -10.0001 1+1i 1-1i 1+1i 1-1i],100);
[msys1,blks1] = modalreal(sys);
blks1

blks1 = 3×1

     1
     4
     2

msys1.A

ans = 7×7

         0         0         0         0         0         0         0
         0    1.0000    2.1220         0         0         0         0
         0   -0.4713    1.0000    1.5296         0         0         0
         0         0         0    1.0000    1.8439         0         0
         0         0         0   -0.5423    1.0000         0         0
         0         0         0         0         0  -10.0000    4.0571
         0         0         0         0         0         0  -10.0001

msys1.B

ans = 7×1

    0.1600
   -0.0052
    0.0201
   -0.0975
    0.2884
         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 msys1 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 condition number of the block-diagonalizing transformation. Set SepTol to 1e-10 for this example.

[msys2,blks2] = modalreal(sys,SepTol=1e-10);
blks2

blks2 = 4×1

     1
     4
     1
     1

msys2.A

ans = 7×7

         0         0         0         0         0         0         0
         0    1.0000    2.1220         0         0         0         0
         0   -0.4713    1.0000    1.5296         0         0         0
         0         0         0    1.0000    1.8439         0         0
         0         0         0   -0.5423    1.0000         0         0
         0         0         0         0         0  -10.0000         0
         0         0         0         0         0         0  -10.0001

msys2.B

ans = 7×1
105 ×

    0.0000
   -0.0000
    0.0000
   -0.0000
    0.0000
    1.6267
    1.6267

The A matrix of msys2 includes separate diagonal elements for the poles near s = -10. Increasing the condition number results in some very large values in the B matrix.

Normalize Diagonal Blocks in Modal Realizations

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

$$sys\left(s\right)=100\frac{\left(s-1\right)\left(s+1\right)}{s\left(s+10\right)\left(s+10.0001\right){\left(s-\left(3+4i\right)\right)}^2}$$

Create a zpk model of this system and obtain a modal realization using the function modalreal.

sys = zpk([1 -1],[0 -10 -10.0001 3+4i 3-4i],100);
[msys1,blks1] = modalreal(sys);
blks1

blks1 = 3×1

     1
     2
     2

msys1.A

ans = 5×5

         0         0         0         0         0
         0    3.0000    8.7637         0         0
         0   -1.8257    3.0000         0         0
         0         0         0  -10.0000    8.8001
         0         0         0         0  -10.0001

msys1 is a state-space model with a block of sizes 2 for the two poles near s = -10, and a pair of complex poles at s = 3+4i and s = 3-4i.

You can normalize the values of 2-by-2 blocks to show the actual pole values using the Normalize option. Additionally, relax the relative accuracy of the block diagonalizing transformation to separate the block near s = -10.

[msys2,blks2] = modalreal(sys,Normalize=true,SepTol=1e-10);
blks2

blks2 = 4×1

     1
     2
     1
     1

msys2.A

ans = 5×5

         0         0         0         0         0
         0    3.0000    4.0000         0         0
         0   -4.0000    3.0000         0         0
         0         0         0  -10.0000         0
         0         0         0         0  -10.0001

For complex poles, this option normalizes the 2-by-2 block of complex poles $\mathit{a}\pm \mathit{bi}$ to $\left[\begin{array}{cc}\mathit{a}& \mathit{b}\\ -\mathit{b}& \mathit{a}\end{array}\right]$.

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 (Control System Toolbox), zpk (Control System Toolbox), ss (Control System Toolbox), or pid (Control System Toolbox) models.

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

• Identified LTI models, such as idtf, idss, idproc, idpoly, and idgrey models.

You cannot use frequency-response data models such as frd (Control System Toolbox) models.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: [msys,blks] = modalreal(sys,Normalize=true)

SepTol — Relative accuracy of block diagonalization

1e-12 (default) | scalar between 0 and 1

Relative accuracy of block diagonalization, specified as a scalar between 0 and 1.

This option limits the condition number of the block diagonalizing transformation to roughly SepTol/eps. Increasing SepTol helps yield smaller blocks at the expense of accuracy.

Normalize — Normalize diagonal blocks

false or 0 (default) | true or 1

Normalize 1-by-1 and 2-by-2 diagonal blocks, specified as a logical 0 (false) or 1 (true).

When Normalize is true, the function normalizes the block to show the pole values.

• For explicit models, the function normalizes 2-by-2 blocks associated with complex pairs a±jb to $$\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$$.

• For descriptor models, the function normalizes

  • 1-by-1 blocks to A_j = a, E_j = 1.

  • 2-by-2 blocks to A_j = $$\left[\begin{array}{cc}a& b\\ -b& a\end{array}\right]$$, E_j = I.

Output Arguments

msys — Modal state-space realization

ss model object

Modal state-space realization of the dynamic model, returned as an ss (Control System Toolbox) model object. msys is a realization where A or (A,E) are block diagonal and each block corresponds to a real pole, a complex pair, or a cluster of repeated poles.

blks — Block sizes

vector

Block sizes in the block-diagonal realization, returned as a vector.

TL — Left-side matrix of block-diagonalizing transformation

matrix | []

Left-side matrix of the block-diagonalizing transformation, returned as a matrix with dimensions Nx-by-Nx, where Nx is the number of states in the model sys.

The algorithm transforms the state-space realization (A, B, C, D, E) of a model to block diagonal matrices (A_m, B_m, C_m, D_m, E_m) given by:

• For explicit state-space models

  $$\begin{array}{l}\begin{array}{cc}{A}_m=T_{L}A{T}_R,& B_m=T_{L}B,\end{array}\\ \begin{array}{ccc}{C}_m=C{T}_R,& D_m=D,& E=T_L{T}_R=I\end{array}\end{array}$$

• For descriptor state-space models

  $$\begin{array}{l}\begin{array}{cc}{A}_m=T_{L}A{T}_R,& B_m=T_{L}B,\end{array}\\ \begin{array}{ccc}{C}_m=C{T}_R,& D_m=D,& E_m=T_{L}E{T}_R\end{array}\end{array}$$

The function returns an empty value [] for this argument when the input model sys is not a state-space model.

TR — Right-side matrix of block-diagonalizing transformation

matrix | []

Right-side matrix of the block-diagonalizing transformation, returned as a matrix with dimensions Nx-by-Nx, where Nx is the number of states in the model sys.The algorithm transforms the state-space realization (A, B, C, D, E) of a model to block diagonal matrices (A_m, B_m, C_m, D_m, E_m) given by:

• For explicit state-space models

  $$\begin{array}{l}\begin{array}{cc}{A}_m=T_{L}A{T}_R,& B_m=T_{L}B,\end{array}\\ \begin{array}{ccc}{C}_m=C{T}_R,& D_m=D,& E=T_L{T}_R=I\end{array}\end{array}$$

• For descriptor state-space models

  $$\begin{array}{l}\begin{array}{cc}{A}_m=T_{L}A{T}_R,& B_m=T_{L}B,\end{array}\\ \begin{array}{ccc}{C}_m=C{T}_R,& D_m=D,& E_m=T_{L}E{T}_R\end{array}\end{array}$$

The function returns an empty value [] for this argument when the input model sys is not a state-space model.

Version History

Introduced in R2023b