When considering the convergence of the error dynamics
\(e=x-\hat{x}\), the resulting dynamics of the candidate Lyapunov function are usually formed in a quadratic formula involving
e and
\(\Delta f=f(x,t)-f(\hat{x},t)\), with the following general form
$$\begin{aligned} \begin{pmatrix} e^T&\Delta f^T \end{pmatrix}\mathcal {M}\begin{pmatrix} e\\ \Delta f \end{pmatrix}. \end{aligned}$$
(7)
To ensure the negative definiteness of (
7),
\(\mathcal {M}<0 \) must be satisfied to guarantee convergence. Here
\(\mathcal {M}\) is a block matrix that generally involves the system matrices along with some other unknown matrices in its blocks. As a result,
\(\mathcal {M}<0 \) is resulted in the form of an LMI.
\(\mathcal {M}\) also takes contribution from the nonlinearity
f using (
5) as follows
$$\begin{aligned} ||f(x,t)-f(\hat{x},t)||\le & {} \lambda _f ||(x-\hat{x})|| \nonumber \\ \Rightarrow \Delta f^T \Delta f\le & {} \lambda _f^2 e^Te \nonumber \\ \Rightarrow 0\le & {} \begin{pmatrix} e^T&\Delta f^T \end{pmatrix}\!\!\begin{pmatrix} \lambda _f^2 I &{}0\\ 0&{} -I \end{pmatrix}\!\!\begin{pmatrix} e\\ \Delta f \end{pmatrix}. \end{aligned}$$
(8)
So, the contributing term in
\(\mathcal {M}\) is the matrix
$$\begin{aligned} \begin{pmatrix} \lambda _f^2 I&{}0\\ 0&{} -I \end{pmatrix} \end{aligned}$$
(9)
In some works, this matrix is scaled by an additional parameter
\(\epsilon >0\) to improve the solvability of the LMI as
$$\begin{aligned} \begin{pmatrix} \epsilon \lambda _f^2 I &{}0\\ 0&{} -\epsilon I \end{pmatrix}, \end{aligned}$$
(10)
becomes the contributing term. Moreover, in some works, the nonlinearity is considered to be acting on a linear combination of the states and is written as
f(
Hx,
t), as in (
1). This way, considering (
5), the resulting matrix is
$$\begin{aligned} \begin{pmatrix} \epsilon \lambda _f^2 H^T H &{} 0\\ 0&{} -\epsilon I \end{pmatrix}. \end{aligned}$$
(11)
In some earlier works [
4,
14,
66,
67], the problem of secure communications was considered on state space systems. Input information signal was considered as a system state that augmented the system state as described in Sect.
2.2. This converts the system into a rectangular descriptor form, and then an observer is designed. The nonlinearity
f(
x,
s) is assumed to satisfy Lipschitz constraint with respect to
x(
t) and
s(
t), where
s(
t) is the information signal. The feasibility of the observer is formulated under the solvability of an LMI and a set of algebraic constraints on the system matrices. Boutayeb et al. [
4] use an important assumption, also present in other works, that the information signal is also transmitted through the output, that is
\(R\ne 0\) in system (
3). The results are illustrated using the Lorenz system [
37] to transmit a sinusoidal signal. For this, two cases are considered. In the first, the information is linearly injected in the system. In the second case, the information is also injected through the nonlinearity
f(
x,
s,
y), which depends on the state, information signal, and output, and is assumed to be Lipschitz with respect to its first two arguments. In [
14], the design is more complex, as the transmission system consists of two chaotic systems, one is used for signal encryption, and another for the synchronization process. The observer is adaptive, where the adaptive term is used to compensate for the unknown Lipschitz constant of the nonlinear term. In the numerical examples, the Rössler [
58] and Lorenz systems are first considered to encrypt and transmit two sinusoidal signals simultaneously. In a second example, a modified Chua circuit [
75] is taken to encrypt and transmit a binary image. In [
66], the augmentation is done by adding a duplicate of the system of differential equations to make the resulting system square.
Square descriptor systems were considered to design full and reduced order observers [
38]. In this article, full-order observers are given in the descriptor form. These observers were further converted to reduced order observers in the state space form. In the numerical example, a one-dimensional observer is designed for a two-dimensional system. Gupta et al. [
21] designed an observer for square descriptor systems. The authors show that the detectability of the linear part is necessary for the solvability of the LMI that is required for observer convergence. Using systems with sinusoidal and cosinusoidal nonlinearities, two numerical examples are then presented. The authors extended these results for rectangular descriptor systems with UIs in [
22]. The UIs were removed from the output first and then a reduced observer is appropriately designed to nullify their effect from the dynamic equation. An application to secure communications was described on Lorenz system, with a sinusoidal information signal that is estimated, and an unknown input sinusoidal signal acting on the system. Lorenz system is used for secure communications and the nonlinearity is multiplied by a full column rank matrix converting it to
Rf(
x), which helps simplifying the error dynamics [
7]. The reference [
22] also discusses the benefits and drawbacks of augmentation technique presented in Sect.
2.2 while considering UIs.