compreal

Compute companion state-space realization

自 R2023b 起

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语法

csys = compreal(sys)

csys = compreal(sys,type)

[csys,T] = compreal(___)

说明

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

示例

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.

示例

[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.

示例

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Convert State-Space Model to Companion Canonical Form

此示例使用:

Control System Toolbox Control System Toolbox

打开实时脚本

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 $θ$ .

$\begin{array}{l} [\begin{array}{c} α_{}^{˙} \\ q_{}^{˙} \\ θ_{}^{˙} \end{array}] = [\begin{array}{ccc} - 0.313 & 56.7 & 0 \\ - 0.0139 & - 0.426 & 0 \\ 0 & 56.7 & 0 \end{array}] [\begin{array}{c} α \\ q \\ θ \end{array}] + [\begin{array}{c} 0.232 \\ 0.0203 \\ 0 \end{array}] [δ] \\ y = [\begin{array}{ccc} 0 & 0 & 1 \end{array}] [\begin{array}{c} α \\ q \\ θ \end{array}] + [0] [δ] \end{array}$

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.
Model Properties

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.
Model Properties

csys is the controllable companion form of sys.

Convert System to Observable Companion Form

打开实时脚本

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 $A_{o} = {TAT}^{- 1}$ , $B_{o} = TB$ , $C_{o} = {CT}^{- 1}$ .

输入参数

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`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.

`type` — Companion realization type
`"c"` (默认) | `"o"`

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

For a system with characteristic polynomial

$P (s) = s^{n} + α_{n - 1} s^{n - 1} + α_{n - 2} s^{n - 2} + \dots + α_{1} s + α_{0},$

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.
$A_{c c o m} = [\begin{matrix} \begin{array}{l} 0 \\ 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{array} & \begin{array}{l} 0 \\ 0 \\ 1 \\ 0 \\ ⋮ \\ 0 \end{array} & \begin{matrix} \begin{matrix} \begin{array}{l} 0 \\ 0 \\ 0 \\ 1 \\ ⋮ \\ 0 \end{array} & \begin{array}{l} \dots \\ \dots \\ \dots \\ \dots \\ ⋱ \\ \dots \end{array} \end{matrix} & \begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \\ ⋮ \\ 1 \end{array} & \begin{array}{l} - α_{0} \\ - α_{1} \\ - α_{2} \\ - α_{3} \\ ⋮ \\ - α_{n - 1} \end{array} \end{matrix} \end{matrix}], B_{c c o m} = [\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}] .$
Observable Companion Form
In the observable companion realization, the characteristic polynomial of the system appears explicitly in the last row of the A matrix.
$A_{o c o m} = [\begin{matrix} \begin{array}{l} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ - α_{0} \end{array} & \begin{array}{l} 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ - α_{1} \end{array} & \begin{matrix} \begin{matrix} \begin{array}{l} 0 \\ 1 \\ 0 \\ ⋮ \\ 0 \\ - α_{2} \end{array} & \begin{array}{l} 0 \\ 0 \\ 1 \\ ⋮ \\ 0 \\ - α_{3} \end{array} \end{matrix} & \begin{array}{l} \dots \\ \dots \\ \dots \\ ⋱ \\ \dots \\ \dots \end{array} & \begin{array}{l} 0 \\ 0 \\ 0 \\ ⋮ \\ 1 \\ - α_{n - 1} \end{array} \end{matrix} \end{matrix}], C_{o c o m} = [\begin{matrix} 1 & 0 & \dots & 0 \end{matrix}] .$

For more information, see State-Space Realizations.

Output Arguments

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`csys` — Companion state-space form of the dynamic model
`ss` model object

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).

`T` — Transformation matrix
matrix | `[]`

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 [].

提示

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

版本历史记录

在 R2023b 中推出

另请参阅

主题

State-Space Realizations

compreal

语法

说明

示例

Convert State-Space Model to Companion Canonical Form

Convert System to Observable Companion Form

输入参数

sys — Dynamic system dynamic system model

type — Companion realization type "c" (默认) | "o"

Output Arguments

csys — Companion state-space form of the dynamic model ss model object

T — Transformation matrix matrix | []

提示

版本历史记录

另请参阅

主题

`sys` — Dynamic system
dynamic system model

`type` — Companion realization type
`"c"` (默认) | `"o"`

`csys` — Companion state-space form of the dynamic model
`ss` model object

`T` — Transformation matrix
matrix | `[]`