dss2ss

Convert descriptor state-space model to explicit form

Since R2024a

Syntax

sys = dss2ss(dsys)

sys = dss2ss(dsys,"consistent")

Description

sys = dss2ss(dsys) eliminates the E matrix in the descriptor state-space model dsys of the form

$$\begin{array}{l}E\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

and returns an explicit state-space model sys of the form

$$\begin{array}{l}{\dot{x}}_e=A_e{x}_e+B_{e}u\\ y=C_e{x}_e+D_{e}u\end{array}$$

Here, dsys must be proper and the number of states x_e in sys is smaller than number of states x in dsys when E is singular (the explicit form removes the algebraic variables). Use findop to compute matching initial conditions when the state is reduced.

sys = dss2ss(dsys,"consistent") takes an array of descriptor state-space models dsys and eliminates E while preserving state consistency, that is, all models in sys share the same state vector x. This requires all E matrices to be invertible.

Examples

Convert Descriptor State-Space Model to Explicit Form

This example shows how to use dss2ss to convert a state-space model in descriptor form to explicit form.

For this example, consider a cube rotating about its corner with inertia tensor J and a damping force F of 0.2 magnitude. The input to the system is the driving torque while the angular velocities are the outputs. The equation is given by:

$$\mathit{J}\frac{\mathit{d}\omega }{\mathit{dt}}+\mathit{F}\omega =\mathit{T}$$

$$\mathit{y}=\omega$$

The state-space matrices for the cube are:

$$\begin{array}{l}A=-F,B=I,C=I,D=0,E=J,\\ where,J=\left[\begin{array}{ccc}8& -3& -3\\ -3& 8& -3\\ -3& -3& 8\end{array}\right]andF=\left[\begin{array}{ccc}0.2& 0& 0\\ 0& 0.2& 0\\ 0& 0& 0.2\end{array}\right]\end{array}$$

Specify the A, B, C and D matrices, and create the continuous-time descriptor state-space model.

J = [8 -3 -3; -3 8 -3; -3 -3 8];
F = 0.2*eye(3);
A = -F;
B = eye(3);
C = eye(3);
D = 0;
E = J;
sysd = dss(A,B,C,D,E);

To convert this model to explicit form, use dss2ss.

syse = dss2ss(sysd);
syse.E

ans =

     []

As you can see, the $\mathit{E}$ matrix is empty. The dss2s function provides an equivalent explicit transformation.

sigma(sysd,syse,'r--')

Input Arguments

dsys — Descriptor state-space model

ss object | ss object array

Descriptor state-space model, specified as a state-space model or an array of state-space models.

Output Arguments

sys — Explicit state-space model

ss object | ss object array

Explicit state-space model, returned as a state-space model or an array of state-space models.

Version History

Introduced in R2024a