On detectability and observer design for rectangular linear descriptor systems

Abstract

A new method is proposed to check the detectability for a class of rectangular linear time invariant descriptor systems. The method is based on the properties of restricted system equivalent, derived here from a given descriptor system by means of simple matrix theory. Equivalence between the detectability of a given descriptor system and that of a normal system is established. The proposed result is applied to design full- and reduced-order observers for the same class of descriptor systems. Some illustrative examples are provided.

Appendix: Algorithm to find matrix \(R\)

1.:

Determine

$$\begin{aligned}&p_1:= \hbox {rank of matrix } C\\&n_0 \times n:= \hbox {order of matrix } E. \end{aligned}$$

2.:

Check $\hbox {rank}\begin{bmatrix} E \\ C \\ \end{bmatrix}=n$, then go to steps 3–8.

3.:

Carry out the singular value decomposition (SVD) of matrix $C=U_1\begin{bmatrix} D_1&\quad 0 \\ 0&\quad 0 \\ \end{bmatrix} V_1^T$.

4.:

Calculate $P=V_1\begin{bmatrix} D_1^{-1}&\quad 0 \\ 0&\quad I_{n-p_1} \\ \end{bmatrix}$.

5.:

Calculate $E_{2}=EP\begin{bmatrix} 0_{p_1 \times (n-p_1)} \\ I_{n-p_1} \\ \end{bmatrix}$.

6.:

Carry out the SVD of matrix $E_{2}=U_2\begin{bmatrix} D_2 \\ 0 \\ \end{bmatrix}V_2^T$.

7.:

Calculate $R_0=\begin{bmatrix} 0&\quad I_{n_0+p_1-n} \\ V_2D_2^{-1}&\quad 0 \\ \end{bmatrix}U_2^T$.

8.:

Calculate $R=P\begin{bmatrix} 0_{(n-n_0) \times n_0} \\ R_0\\ \end{bmatrix}$.