ssequiv

Equivalence transformation for state-space models

Since R2023b

Syntax

sysT = ssequiv(sys,T1,e1,T2,e2)

Description

ssequiv performs an equivalence transformation on a state-space model.

Input Model	Transformed Model
$$\begin{array}{c}E\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$	$$\begin{array}{c}{T}_{L}E{T}_R\dot{x}=T_{L}A{T}_{R}x+T_{L}Bu\\ y=C{T}_{R}x+Du\end{array}$$

Here:

• The left-transformation matrix is defined as $$T_L=T_1^{e_1}$$.

• The right-transformation matrix is defined as $$T_R=T_2^{e_2}$$.

• e1 and e2 are either –1 or 1. That is, T_L is T1 or its inverse and T_R is T2 or its inverse.

sysT = ssequiv(sys,T1,e1,T2,e2) performs the equivalence transformation for sys.

Examples

Equivalence Transformation for State-Space Model

Perform an equivalence transform for a state space model.

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

rng(0)
sys = rss(5);  
T1 = randn(5);
rng(2)
T2 = randn(5);

Perform the following transformation on sys.

$$\begin{array}{l}\dot{\mathit{x}}={\mathit{T}}_1^{-1}\mathit{A}{\mathit{T}}_2\mathit{x}+{\mathit{T}}_1^{-1}\mathit{B}\\ \mathit{y}={\mathit{CT}}_2\end{array}$$

sysT = ssequiv(sys,T1,-1,T2,1);

Plot the frequency response of both models.

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

The responses of both models match closely.

Equivalence Transformation for Generalized State-Space Models

ssequiv 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.

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

Specify transformation matrices.

T1 = [1 -2;3 5];  
T2 = T1;

Apply the transformation

$$\begin{array}{l}\dot{\mathit{x}}={\mathit{T}}_{\mathit{L}}\mathit{A}{\mathit{T}}_{\mathit{R}}^{-1}\mathit{x}+{\mathit{T}}_{\mathit{L}}\mathit{B}\\ \mathit{y}=\mathit{C}{\mathit{T}}_{\mathit{R}}^{-1}+\mathit{Du}\end{array}$$.

tsys = ssequiv(sys,T1,1,T2,-1);

Compare this transformed model with the model from decomposed tsys.

sigma(sys,tsys,"r--")

The responses of both models match closely.

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 ss or dss models.

• Generalized or uncertain LTI models, such as genss 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 and Internal Structure of Generalized Models.

• Identified state-space idss (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

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

T1 — Left-transformation matrix

matrix | []

Left-transformation matrix, specified as an n-by-n square invertible matrix, where n is the number of states. The left-transformation matrix is defined as $$T_L=T_1^{e_1}$$, that is, T1 or its inverse.

To omit the left transformation, specify T1 and e1 as [].

For example, the following code transforms A, B, C, E of sys to AT^T, B, CT^T, ET^T.

sys = ssequiv(sys,[],[],T',1);

e1 — Left-transformation matrix exponent

–1 | 1 | []

Left-transformation matrix exponent, specified as –1 or 1.

To omit the left transformation, specify T1 and e1 as [].

For example, the following code transforms A, B, C, E of sys to AT^T, B, CT^T, ET^T.

sys = ssequiv(sys,[],[],T',1);

T2 — Right-transformation matrix

matrix | []

Right-transformation matrix, specified as an n-by-n square invertible matrix, where n is the number of states. The right-transformation matrix is defined as $$T_R=T_2^{e_2}$$, that is, T1 or its inverse.

To omit the right transformation, specify T2 and e2 as [].

For example, the following code transforms A, B, C, E of sys to T^-1A, T^-1B, C, T^-1E.

sys = ssequiv(sys,T,-1,[],[]);

e2 — Right-transformation matrix exponent

–1 | 1 | []

Right-transformation matrix exponent, specified as –1 or 1.

To omit the right transformation, specify T2 and e2 as [].

For example, the following code transforms A, B, C, E of sys to T^-1A, T^-1B, C, T^-1E.

sys = ssequiv(sys,T,-1,[],[]);

Output Arguments

sysT — Transformed model

dynamic system model

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

Version History

Introduced in R2023b