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compreal

Compute companion state-space realization

Since R2023b

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    Syntax

    csys = compreal(sys)
    csys = compreal(sys,type)
    [csys,T] = compreal(___)

    Description

    csys = compreal(sys) returns the controllable companion realization of the single-input LTI model sys.

    example

    csys = compreal(sys,type) returns the realization of the model sys depending on type. Specify type as "c" for controllable companion form or "o" for observable companion form.

    • "c" — Computes the controllable companion realization for a single-input LTI model sys. This is the same as the first syntax.

    • "o" — Computes the observable companion realization for a single-output LTI model sys.

    example

    [csys,T] = compreal(___) also returns the transformation matrix T for explicit models with matrices A, B, C such that

    • The controllable companion form is T-1AT, T-1B, CT.

    • The observable companion form is TAT-1, TB, CT-1.

    example

    Examples

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    This example uses:

    Open Live Script

    aircraftPitchSSModel.mat contains the state-space matrices of an aircraft where the input is elevator deflection angle δ and the output is the aircraft pitch angle θ.

    [α˙q˙θ˙]=[-0.31356.70-0.0139-0.4260056.70][αqθ]+[0.2320.02030][δ]y=[001][αqθ]+[0][δ]

    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 controllable companion form.

    csys = compreal(sys)
    csys =
 
  A = 
              x1         x2         x3
   x1          0          0  1.914e-15
   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 controllable companion form of sys.

    Open Live Script

    The file icEngine.mat contains one data set with 1500 input-output samples collected at 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 observable companion form. 

    [osys,T] = compreal(sys,"o");

    Compare frequency response confidence bounds of sys to osys.

    bp = bodeplot(sys,osys,'r.');
showConfidence(bp)

    MATLAB figure

    The frequency response confidence bounds are identical.

    Additionally, compreal returns a transformation matrix T such that the observable companion form is Ao=TAT-1, Bo=TB, Co=CT-1.

    Input Arguments

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    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.

    Companion realization type, specified as "c" (controllable companion form) or "o" (observable companion form).

    For a system with characteristic polynomial

    P(s)=sn+αn−1sn−1+αn−2sn−2+…+α1s+α0,

    the function returns the companion forms as follows.

    • Controllable Companion Form

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

      Accom=[0100⋮00010⋮00001⋮0…………⋱…0000⋮1−α0−α1−α2−α3  ⋮−αn−1], Bccom=[10⋮0].

    • Observable Companion Form

      In the observable companion realization, the characteristic polynomial of the system appears explicitly in the last row of the A matrix.

      Aocom=[000⋮0−α0100⋮0−α1010⋮0−α2001⋮0−α3………⋱……000⋮1−αn−1], Cocom=[10…0].

    For more information, see State-Space Realizations.

    Output Arguments

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    Companion state-space form of the dynamic model, returned as an ss (Control System Toolbox) model object. csys is a state-space realization of sys in the companion form specified by type (controllable or observable).

    Transformation matrix, returned as an n-by-n matrix, where n is the number of states.

    • For explicit state-space models with matrices A, B, C, the function returns T such that

      • The controllable companion form is T-1AT, T-1B, CT.

      • The observable companion form is TAT-1, TB, CT-1.

    • For descriptor state-space models, the function always returns an empty value [].

    Tips

    Computing companion realizations often involves ill-conditioned transformations and loss of accuracy. Use modalreal or balreal as numerically stable alternatives.

    Version History

    Introduced in R2023b

    See Also

    | (Control System Toolbox) | (Control System Toolbox) | (Control System Toolbox) | (Control System Toolbox) | (Control System Toolbox)

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