ss2ss

State coordinate transformation for state-space model

语法

sysT = ss2ss(sys,T)

说明

ss2ss performs the similarity transformation z = Tx on the state vector x of a state-space model. For more information, see Algorithms.

sysT = ss2ss(sys,T) performs the state-coordinate transformation of sys using the specified transformation matrix T. The matrix T must be invertible.

示例

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Similarity Transformation for State-Space Model

此示例使用:

Control System Toolbox Control System Toolbox

打开实时脚本

Perform a similarity transform for a state space model.

Generate a random state-space model and a transformation matrix.

rng(0)
sys = rss(5);  
t = randn(5);

Perform the transformation and plot the frequency response of both models.

tsys = ss2ss(sys,t);
bode(sys,'b',tsys,'r--')
legend

MATLAB figure

The responses of both models match closely.

Similarity Transformation for Generalized State-Space Models

此示例使用:

Control System Toolbox Control System Toolbox

打开实时脚本

ss2ss applies state transformation only to the state vectors of the numeric portion of the generalized model.

Create a genss model.

sys = rss(2,2,2) * tunableSS('a',2,2,3) + tunableGain('b',2,3)

Generalized continuous-time state-space model with 2 outputs, 3 inputs, 4 states, and the following blocks:
  a: Tunable 2x3 state-space model, 2 states, 1 occurrences.
  b: Tunable 2x3 gain, 1 occurrences.
Model Properties

Type "ss(sys)" to see the current value and "sys.Blocks" to interact with the blocks.

Specify a transformation matrix and obtain the transformation.

T = [1 -2;3 5];
tsys = ss2ss(sys,T)

Generalized continuous-time state-space model with 2 outputs, 3 inputs, 4 states, and the following blocks:
  a: Tunable 2x3 state-space model, 2 states, 1 occurrences.
  b: Tunable 2x3 gain, 1 occurrences.
Model Properties

Type "ss(tsys)" to see the current value and "tsys.Blocks" to interact with the blocks.

Decompose both models.

[H,B,~,~] = getLFTModel(sys);
[H1,B1,~,~] = getLFTModel(tsys);

Obtain the transformation separately on the model from decomposed sys.

H2 = ss2ss(H,T);

Compare this transformed model with the model from decomposed tsys.

isequal(H1,H2)

ans = logical
   1

Both models are equal.

Similarity Transformation for Identified State-Space Models

打开实时脚本

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);

Specify a random transformation matrix.

T = randn(4);

Obtain the transformation.

sysT = ss2ss(sys,T);

Compare the frequency responses.

bode(sys,'b',sysT,'r--')
legend

MATLAB figure

The responses match closely.

Transformation for Models with Complex Coefficients

此示例使用:

Control System Toolbox Control System Toolbox

打开实时脚本

ss2ss also lets you perform similarity transformation for models with complex coefficients.

For this example, generate a random state-space model with complex coefficients.

rng(0)
sys = ss(randn(5)+1i*randn(5),randn(5,3),randn(2,5)+1i*randn(2,5),0,.1);

Specify a transformation matrix containing complex data.

T = randn(5)+1i*randn(5);

Obtain the transformation.

sysT = ss2ss(sys,T);

Compare the singular values of the frequency response.

sigma(sys,'b',sysT,'r--')
legend

MATLAB figure

The responses match closely for both branches.

输入参数

全部折叠

`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 ss (Control System Toolbox) or dss (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.)
For such models, the state transformation is applied only to the state vectors of the numeric portion of the model. For more information about decomposition of these models, see getLFTModel (Control System Toolbox) and Internal Structure of Generalized Models (Control System Toolbox).
Identified state-space idss models.

If sys is an array of state-space models, ss2ss applies the transformation T to each individual model in the array.

`T` — Transformation matrix
matrix

Transformation matrix, specified as an n-by-n matrix, where n is the number of states. T is the transformation between the state vector of the state-space model sys and the state vector of the transformed model sysT. (See Algorithms.)

Output Arguments

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`sysT` — Transformed model
dynamic system model

Transformed state-space model, returned as a dynamic system model of the same type as sys.

算法

ss2ss performs the similarity transformation $\bar{x} = T x$ on the state vector x of a state-space model.

This table summarizes the transformations returned by ss2ss for each model form.

Input Model	Transformed Model
Explicit state-space models of the form: $\begin{array}{l} \dot{x} = A x + B u \\ y = C x + D u \end{array}$	$\begin{array}{l} \dot{\bar{x}} = T A T^{- 1} \bar{x} + T B u \\ y = C T^{- 1} \bar{x} + D u \end{array}$
Descriptor (implicit) state-space models for the form: $\begin{matrix} E \dot{x} = A x + B u \\ y = C x + D u \end{matrix}$	$\begin{matrix} E T^{- 1} \dot{\bar{x}} = A T^{- 1} \bar{x} + B u \\ y = C T^{- 1} \bar{x} + D u \end{matrix}$
Identified state-space (`idss`) models of the form: $\begin{matrix} \frac{d x}{d t} = A x + B u + K e \\ y = C x + D u + e \end{matrix}$	$\begin{array}{l} \dot{\bar{x}} = T A T^{-}^{1} \bar{x} + T B u + T K e \\ y = C T^{-}^{1} \bar{x} + D u + e \end{array}$

版本历史记录

在 R2006a 之前推出

全部展开

R2021b: `ss2ss` returns different transformation results for descriptor state-space models

For a descriptor state-space model

$\begin{matrix} E \dot{x} = A x + B u \\ y = C x + D u, \end{matrix}$

ss2ss now returns

$\begin{matrix} E T^{- 1} \dot{\bar{x}} = A T^{- 1} \bar{x} + B u \\ y = C T^{- 1} \bar{x} + D u . \end{matrix}$

Previously, the function returned the following transformation.

$\begin{matrix} T E T^{- 1} \dot{\bar{x}} = T A T^{- 1} \bar{x} + T B u \\ y = C T^{- 1} \bar{x} + D u \end{matrix}$

R2021b: Similarity transformation is no longer supported for `mechss` models

ss2ss no longer supports sparse second-order (mechss) models. Performing similarity transformations on mechss models destroys symmetry and has no obvious general form.

另请参阅

balreal (Control System Toolbox) | canon (Control System Toolbox) | balance

主题

Scaling State-Space Models to Maximize Accuracy (Control System Toolbox)

ss2ss

语法

说明

示例

Similarity Transformation for State-Space Model

Similarity Transformation for Generalized State-Space Models

Similarity Transformation for Identified State-Space Models

Transformation for Models with Complex Coefficients

输入参数

sys — Dynamic system dynamic system model

T — Transformation matrix matrix

Output Arguments

sysT — Transformed model dynamic system model

算法

版本历史记录

R2021b: ss2ss returns different transformation results for descriptor state-space models

R2021b: Similarity transformation is no longer supported for mechss models

另请参阅

主题

`sys` — Dynamic system
dynamic system model

`T` — Transformation matrix
matrix

`sysT` — Transformed model
dynamic system model

R2021b: `ss2ss` returns different transformation results for descriptor state-space models

R2021b: Similarity transformation is no longer supported for `mechss` models