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Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator

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Abstract

In this paper, a secure chaotic scheme for communications in noisy public channel is proposed. This scheme is based on the concept of carrier encryption in addition to the typical data encryption techniques. At the transmitter end, the unified chaotic system with adaptive parameter and a hyperchaotic Rössler system with uncertain parameters are coupled, constrained and used as a new hyperchaotic system which generates waveforms that are different from those of any known chaotic oscillator. After modulating one of the outputs of the system with the encrypted data signal, the outputs of the system are encrypted using a set of pre-defined encryption rules and transmitted to the receiving end through a noisy public communications channel. At the receiving end, the received outputs are decrypted and the transmitted data are retrieved by reconstructing the constrained hyperchaotic signals using the novel discrete-time iterative decomposed uncertain constrained extended Kalman filter (IDUCEKF). The proposed state estimator, besides being used to handle the estimation problem of uncertain constrained nonlinear dynamical systems, reduces the required processing time and gives good numerical performance. Simulation results are firstly presented to illustrate the applicability of the IDUCEKF in synchronizing the states of the constrained hyperchaotic system. Then, the proposed secure communication scheme is applied to transmit images, and the quality of the transmission process is assessed. The obtained results show the effectiveness of the proposed approach.

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Acknowledgments

This research was supported by Kuwait University under research Grant No. EE 02/14.

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Authors and Affiliations

  1. Electrical Engineering Department, College of Engineering, Kuwait University, P.O. Box 5969, 13060, Safat, Kuwait

    Mohamed F. Hassan

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  1. Mohamed F. Hassan
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Correspondence to Mohamed F. Hassan.

Appendices

Appendix 1

Proof of Lemma 4.1

For the ith subsystem, by linearizing the system dynamics around \(\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}})\) and \(\bar{\varvec{b}}\) while neglecting higher-order terms, we get:

$$\begin{aligned}&{\varvec{f}}_{\varvec{i}} ({\varvec{x}}({\varvec{k}}),{\varvec{b}}({\varvec{k}}))\cong {\varvec{f}}_{\varvec{i}} (\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}),\bar{\varvec{b}})+{\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{x}}({\varvec{k}})\nonumber \\&\quad -\,\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}))+{\hat{{{\varvec{E}}}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{b}}({\varvec{k}})-\bar{\varvec{b}}) \end{aligned}$$
(83)

where \({\hat{\varvec{F}}}_{\varvec{i}}^{\prime } \), \({\hat{\varvec{E}}}_{\varvec{i}}^{\prime } \) are as defined by (18).

Substituting (83) into the ith subsystem model (14), we get:

$$\begin{aligned}&{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)\cong {\varvec{f}}_{\varvec{i}} (\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}),\bar{\varvec{b}})+{\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{x}}({\varvec{k}})-\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}))\nonumber \\&\quad +\,{\hat{\varvec{E}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{b}}({\varvec{k}})-\bar{\varvec{b}})+{\varvec{w}}_{\varvec{i}} ({\varvec{k}}) \end{aligned}$$
(84)

The conditional expectation of (84) given the measurements up to the kth instant of time is given by:

$$\begin{aligned}&\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\quad \cong \,{\varvec{E}}\Big \{ \Big [ {\varvec{f}}_{\varvec{i}} (\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}),\bar{\varvec{b}})+{\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{x}}({\varvec{k}})-\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}))\nonumber \\&\qquad +\,{\hat{\varvec{E}}}_{\varvec{i}}^{\prime } ({\varvec{k}})({\varvec{b}}({\varvec{k}})-\bar{\varvec{b}})+{\varvec{w}}_{\varvec{i}} ({\varvec{k}}) \Big ] \Big |{\varvec{Y}}^{{\varvec{k}}} \Big \} \end{aligned}$$
(85)

where \({\varvec{Y}}^{{\varvec{k}}}=[{\varvec{y}}^{{\varvec{T}}}(1) {\varvec{y}}^{{\varvec{T}}}(2) \cdots {\varvec{y}}^{{\varvec{T}}}({\varvec{k}})]\)

Since \(\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}})\cong {\varvec{E}}\left\{ {{{\varvec{x}}({\varvec{k}})} |{\varvec{Y}}^{{\varvec{k}}}} \right\} \), the variables \({\varvec{Y}}^{{\varvec{k}}}\), \({\varvec{w}}_{\varvec{i}} ({\varvec{k}})\) and \({\varvec{b}}({\varvec{k}})\) are independent, \({\varvec{E}}\left\{ {{\varvec{w}}_{\varvec{i}} ({\varvec{k}})} \right\} =0\) and \({\varvec{E}}\left\{ {{\varvec{b}}({\varvec{k}})} \right\} =\bar{\varvec{b}}\), then the predicted estimate \(\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\) takes the form:

$$\begin{aligned} \hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\cong {\varvec{f}}_{\varvec{i}} (\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}}),\bar{\varvec{b}}) \end{aligned}$$
(86)

let \({\tilde{\varvec{x}}}({\varvec{k}}| {\varvec{k}})={\varvec{x}}({\varvec{k}})-\hat{\varvec{x}}({\varvec{k}}| {\varvec{k}})\), \({\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})={\varvec{x}}_{\varvec{i}} ({\varvec{k}})-\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\), and \(\tilde{{\varvec{b}}}({\varvec{k}})={\varvec{b}}({\varvec{k}})-\bar{\varvec{b}}\), then using (84) and (86), we get:

$$\begin{aligned} {\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\cong {\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}}){\tilde{\varvec{x}}}({\varvec{k}}| {\varvec{k}})+{\hat{\varvec{E}}}_{\varvec{i}}^{\prime } ({\varvec{k}})\tilde{\varvec{b}}({\varvec{k}})+{\varvec{w}}_{\varvec{i}} ({\varvec{k}})\nonumber \\ \end{aligned}$$
(87)

The covariance matrix of the prediction error of the ith subsystem is defined by:

$$\begin{aligned} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} } ({\varvec{k}}+1| {\varvec{k}})={\varvec{E}}\left\{ {{\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}}) {\tilde{\varvec{x}}}^{{\varvec{T}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})} \right\} \end{aligned}$$
(88)

Substituting from (87) into (88) and with the use of model properties (13) (i.e., \({\varvec{w}}({\varvec{k}})\), and \(\tilde{\varvec{b}}({\varvec{k}})\) are independent, \({\varvec{x}}_{\varvec{i}} ({\varvec{k}})\) is independent of \({\varvec{w}}({\varvec{k}})\) and \({\varvec{b}}({\varvec{k}})\), and \({\varvec{E}}\left\{ {\tilde{\varvec{b}}({\varvec{k}})} \right\} ={\varvec{E}}\left\{ {{\varvec{b}}({\varvec{k}})-\bar{\varvec{b}}} \right\} =0)\), one gets after simple mathematical manipulations:

$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^0 ({\varvec{k}}+1| {\varvec{k}} +1)={\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} } ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\quad \cong \,{\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}}) {\varvec{P}}_{{\varvec{k}}| {\varvec{k}}} {\hat{\varvec{F}}}_{\varvec{i}}^{{\prime }\varvec{T}} ({\varvec{k}})\nonumber \\&\qquad +\,{\hat{\varvec{E}}}_{\varvec{i}}^{\prime } ({\varvec{k}}) {\varvec{\varGamma }} ({\varvec{k}}){\hat{\varvec{E}}}_{\varvec{i}}^{{\prime }{\varvec{T}}} ({\varvec{k}})+{\varvec{Q}}_{{\varvec{ii}}} ({\varvec{k}}) \end{aligned}$$
(89)

The cross-covariance matrix of the prediction error of the ith and the jth subsystems is defined by:

$$\begin{aligned} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} } ({\varvec{k}}+1| {\varvec{k}})={\varvec{E}}\left\{ {{\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}}) {\tilde{\varvec{x}}}^{{\varvec{T}}}_{\varvec{j}} ({\varvec{k}}+1| {\varvec{k}})} \right\} \end{aligned}$$
(90)

With the assumptions of the model properties and the independency of the noise subvectors \({\varvec{w}}_{\varvec{i}} ({\varvec{k}})\) and \({\varvec{w}}_{\varvec{j}} ({\varvec{k}})\), it is easy to show that

$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}}}^0 ({\varvec{k}}+1| {\varvec{k}} +1)={\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} } ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\quad \cong \,{\hat{\varvec{F}}}_{\varvec{i}}^{\prime } ({\varvec{k}}) {\varvec{P}}_{{\varvec{k}}| {\varvec{k}}} {\hat{\varvec{F}}}_{\varvec{i}}^{{\prime }{\varvec{T}}} ({\varvec{k}})+{\hat{\varvec{E}}}_{\varvec{i}}^{\prime } ({\varvec{k}}) {\varvec{\varGamma }} ({\varvec{k}}){\hat{\varvec{E}}}_{\varvec{i}}^{{\prime }{\varvec{T}}} ({\varvec{k}}) \end{aligned}$$
(91)

Equation (86) gives the predicted estimate of \({\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)\) denoted by \(\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\) for \(i\in \{1,2,{\ldots },N\}\), Eq. (89) gives the covariance matrices of the prediction estimation error of \({\varvec{x}}_{\varvec{i}} ({\varvec{k}})\), whereas Eq. (91) gives the cross-covariance matrices of the estimation errors of \({\varvec{x}}_{\varvec{i}} ({\varvec{k}})\) and \({\varvec{x}}_{\varvec{j}} ({\varvec{k}}) \quad \forall \; i,j \in \left\{ {i+1, 2, \ldots ,N} \right\} \).

Now, assume that we received the measurement vector \({\varvec{y}}({\varvec{k}}+1)\) at the \((k+1)\)th sampling instant. By decomposing \({\varvec{y}}({\varvec{k}}+1)\in {\varvec{R}}^{{\varvec{m}}}\) into M subvectors \({\varvec{y}}_{\varvec{r}} ({\varvec{k}}+1)\in {\varvec{R}}^{{\varvec{m}}_{\varvec{r}}}\) where \(r\in \{1,2,\ldots , M\}\) and \(\sum _{r=1}^M {m_r =m} \), then the predicted estimates of the state subvectors \(\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\) for \(i\in \{1,2,\ldots , N\}\) will be updated successively using the measurement subvectors \({\varvec{y}}_{\varvec{r}} ({\varvec{k}}+1)\) for \(r\in \{1,2,\ldots , M\}\). Therefore, we have

$$\begin{aligned}&\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad \cong {\varvec{E}}\Big \{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\varvec{Y}}^{{\varvec{k}}}, {\varvec{y}}_1 ({\varvec{k}}+1),\nonumber \\&\qquad {\varvec{y}}_2 ({\varvec{k}}+1),\ldots , {\varvec{y}}_{\varvec{M}} ({\varvec{k}}+1) \Big \} \end{aligned}$$
(92)

where

$$\begin{aligned} {\varvec{y}}_{\varvec{r}} ({\varvec{k}}\!+\!1)\!=\!{\varvec{h}}_{\varvec{r}} ({\varvec{x}}({\varvec{k}}\!+\!1),{\varvec{c}}({\varvec{k}}\!+\!1))\!+\!{\varvec{v}}_{\varvec{r}} ({\varvec{k}}\!+\!1) \end{aligned}$$
(93)

Using the multiple projection approach [41, 42], we have:

$$\begin{aligned}&\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad \cong {\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\varvec{Y}}^{{\varvec{k}}}, {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}), \right. \nonumber \\&\qquad \quad {\tilde{\varvec{y}}}^{1}_2 ({\varvec{k}}+1| {{\varvec{k}}+1}), {\tilde{\varvec{y}}}^{2}_3 ({\varvec{k}}+1| {{\varvec{k}}+1}),\nonumber \\&\qquad \qquad \left. \ldots ,{\tilde{\varvec{y}}}_{\varvec{M}}^{{\varvec{M}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1}) \right\} \nonumber \\&\quad \cong {\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\varvec{Y}}^{{\varvec{k}}} \right\} \nonumber \\&\qquad +\,{\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \right\} \nonumber \\&\qquad +\,\sum \limits _{{\varvec{r}}=2}^{{\varvec{M}}} {\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\tilde{\varvec{y}}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1}) \right\} \nonumber \\&\quad \cong {\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\varvec{Y}}^{{\varvec{k}}} \right\} \nonumber \\&\qquad +\,\sum \limits _{{\varvec{r}}=1}^{\varvec{M}} {\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\tilde{\varvec{y}}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1|{{\varvec{k}}+1}) \right\} \end{aligned}$$
(94)

where for \(r=1\) we have:

$$\begin{aligned} {\tilde{\varvec{y}}}^{0}_1 ({\varvec{k}}+1| {{\varvec{k}}+1})= & {} {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}})\nonumber \\\cong & {} {\varvec{y}}_1 ({\varvec{k}}+1)-\hat{{{\varvec{y}}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \nonumber \\ \hat{{{\varvec{y}}}}^{0}_1 ({\varvec{k}}+1| {{\varvec{k}}+1})= & {} \hat{{{\varvec{y}}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \nonumber \\\cong & {} \hat{\varvec{h}}_1 (\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}}),\bar{\varvec{c}}) \end{aligned}$$
(95)

and for \(r \in \{2, 3, \ldots ,M\}\), we have:

$$\begin{aligned}&{\tilde{\varvec{y}}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})={\varvec{y}}_{\varvec{r}} ({\varvec{k}}+1)-\hat{\varvec{y}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1}) \nonumber \\&\hat{\varvec{y}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\cong \hat{\varvec{h}}_{\varvec{r}} (\hat{\varvec{x}}^{{\varvec{r}}-1}({\varvec{k}}+1| {{\varvec{k}}+1}),\bar{\varvec{c}}) \end{aligned}$$
(96)

Using the first submeasurement vector \({\varvec{y}}_1({\varvec{k}}+1)\), the approximated filtered estimate \(\hat{\varvec{x}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) for \(i\in \{1,2,\ldots , N\}\) is such that:

$$\begin{aligned}&\hat{\varvec{x}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad \cong {\varvec{E}}\left\{ {{{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\varvec{Y}}^{{\varvec{k}}}} \right\} +{\varvec{E}}\left\{ {{{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)} | {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}})} \right\} \nonumber \\&\quad \cong \hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})+{\varvec{P}}_{{\varvec{x}}_{\varvec{i}} {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\qquad \times \,{\varvec{P}}^{-1}_{{\tilde{\varvec{y}}}_1 {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}}){\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \\&\quad \cong \hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})+{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1) {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \nonumber \\&\quad \cong \hat{\varvec{x}}^{0}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})+ {\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1){\tilde{\varvec{y}}}^{0}_1 ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \end{aligned}$$
(97)

where

$$\begin{aligned}&{\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}})={\varvec{h}}_1 ({\varvec{x}}({\varvec{k}}+1),{\varvec{c}}({\varvec{k}}+1))\nonumber \\&\quad +\,{\varvec{v}}_1 ({\varvec{k}}+1)-\hat{\varvec{h}}_{ 1} (\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}}),\bar{\varvec{c}})\end{aligned}$$
(98)
$$\begin{aligned}&{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1)\cong {\varvec{P}}_{{\varvec{x}}_{\varvec{i}} {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}}){\varvec{P}}^{-1}_{{\tilde{\varvec{y}}}_1{\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}}) \end{aligned}$$
(99)

Since \({\varvec{x}}({\varvec{k}}+1)=\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}})+{\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})\) and \({\varvec{c}}({\varvec{k}}+1)=\bar{\varvec{c}}+\tilde{{\varvec{c}}}({\varvec{k}}+1)\), and assuming that \({\varvec{x}}({\varvec{k}}+1)\) is close to \(\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}})\) and \({\varvec{c}}({\varvec{k}}+1)\) is close to \(\bar{\varvec{c}}\) such that \({\varvec{h}}_1 ({\varvec{x}}({\varvec{k}}+1),{\varvec{c}}({\varvec{k}}+1))\) can be replaced by its first-order approximation, i.e., we assume \({\varvec{h}}_1 ({\varvec{x}}({\varvec{k}}+1),{\varvec{c}}({\varvec{k}}+1))\) is affine on a neighborhood of \({\varvec{x}}({\varvec{k}}+1),\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}})\) and \({\varvec{c}}({\varvec{k}}+1), \bar{\varvec{c}}\). Then, we have:

$$\begin{aligned}&{\varvec{h}}_1 ({\varvec{x}}({\varvec{k}}+1),{\varvec{c}}({\varvec{k}}+1))\cong \hat{\varvec{h}}_1 (\hat{\varvec{x}}({\varvec{k}}+1| {\varvec{k}}),\bar{\varvec{c}})\nonumber \\&\quad +\,\hat{\varvec{H}}_1 ({\varvec{k}}+1){\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})+\hat{\varvec{D}}_1 ({\varvec{k}}+1)\tilde{{\varvec{c}}}({\varvec{k}}+1)\nonumber \\ \end{aligned}$$
(100)

where \(\hat{\varvec{H}}_1 ({\varvec{k}}+1)=\hat{\varvec{H}}^{0}_1 ({\varvec{k}}+1), \hat{\varvec{D}}_1 ({\varvec{k}}+1)=\hat{\varvec{D}}_1^0 ({\varvec{k}}+1)\) are defined by (26) for \(r=1\).

Substituting from (100) into (98) while using (95), we get:

$$\begin{aligned}&{\tilde{\varvec{y}}}^{0}_1 ({\varvec{k}}+1| {{\varvec{k}}+1})={\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\quad \cong \hat{\varvec{H}}_1 ({\varvec{k}}+1){\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})+\hat{\varvec{D}}_1 ({\varvec{k}}+1)\tilde{{\varvec{c}}}({\varvec{k}}+1)\nonumber \\&\quad +\,{\varvec{v}}_1 ({\varvec{k}}+1) \end{aligned}$$
(101)

The cross-covariance matrix \({\varvec{P}}_{{\varvec{x}}_{\varvec{i}} {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}})\) is defined by:

$$\begin{aligned}&{\varvec{P}}_{{\varvec{x}}_{\varvec{i}} {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}})={\varvec{E}}\left\{ {{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1) {\tilde{\varvec{y}}}^{{\varvec{T}}}_1 ({\varvec{k}}+1| {\varvec{k}})} \right\} \nonumber \\&\quad = {\varvec{E}}\left\{ {\left( {\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})+{\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})} \right) {\tilde{\varvec{y}}}^{{\varvec{T}}}_1 ({\varvec{k}}+1| {\varvec{k}})} \right\} \end{aligned}$$

From the Gaussian conditional estimator [48], since \({\varvec{E}}\left\{ {\hat{\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}}) {\tilde{\varvec{y}}}^{{\varvec{T}}}_1 ({\varvec{k}}+1| {\varvec{k}})} \right\} \cong 0\), then we have:

$$\begin{aligned}&{\varvec{P}}_{{\varvec{x}}_{\varvec{i}} {\tilde{\varvec{y}}}_1} ({\varvec{k}}+1| {\varvec{k}})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{y}}}_1 } ({\varvec{k}}+1| {\varvec{k}}) \\&\quad \cong {\varvec{E}}\left\{ {\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})\left( \hat{\varvec{H}}_1 ({\varvec{k}}+1){\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})\right. \right. \nonumber \\&\qquad \left. \left. +\,\hat{\varvec{D}}_1 ({\varvec{k}}+1)\tilde{{\varvec{c}}}({\varvec{k}}+1)+{\varvec{v}}_1 ({\varvec{k}}+1) \right) ^{{\varvec{T}}} \right\} \end{aligned}$$

Since \({\varvec{E}}\left\{ {\tilde{{\varvec{c}}}({\varvec{k}}+1)} \right\} =0\), \({\varvec{E}}\left\{ {{\varvec{v}}_1 ({\varvec{k}}+1)} \right\} =0\) and \({\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})\) is independent of both \(\tilde{{\varvec{c}}}({\varvec{k}}+1)\) and \({\varvec{v}}_1 ({\varvec{k}}+1)\), we get:

$$\begin{aligned} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{y}}}_1 } ({\varvec{k}}+1| {\varvec{k}})={\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}} ({\varvec{k}}+1| {\varvec{k}})\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}+1) \end{aligned}$$
(102)

Using (101) and recalling that \({\tilde{\varvec{x}}}({\varvec{k}}+1| {\varvec{k}})\), \(\tilde{{\varvec{c}}}({\varvec{k}}+1)\), and \({\varvec{v}}_1 ({\varvec{k}}+1)\) are mutually independent, we get:

$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{y}}}_1 {\tilde{\varvec{y}}}_1} ({\varvec{k}}\!+\!1| {\varvec{k}})\!=\!\hat{\varvec{H}}_1 ({\varvec{k}}\!+\!1){\varvec{P}}({\varvec{k}}\!+\!1| {\varvec{k}})\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}\!+\!1)\nonumber \\&\quad +\,\hat{\varvec{D}}_1 ({\varvec{k}}+1){\varvec{U}}({\varvec{k}}+1)\hat{\varvec{D}}^{{\varvec{T}}}_1 ({\varvec{k}}+1)+{\varvec{R}}_{11} ({\varvec{k}}+1)\nonumber \\ \end{aligned}$$
(103)

where \({\varvec{U}}({\varvec{k}}+1)\) is the covariance matrix of the output model parameter \({\varvec{c}}({\varvec{k}}+1)\) and \({\varvec{R}}_{11} ({\varvec{k}}+1)\) is the covariance matrix of the output noise subvector \({\varvec{v}}_1 ({\varvec{k}}+1)\).

Therefore, substituting from (102) and (103) into (99) while using (22), the gain matrix \({\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1)\) is given by:

$$\begin{aligned} {\varvec{K}}_{1{\varvec{i}},{\varvec{k}}+1}^1= & {} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}} ({\varvec{k}}+1| {\varvec{k}})\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}+1)\nonumber \\&\times \,\Big [ {\hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}({\varvec{k}}+1| {\varvec{k}})\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}+1)} \nonumber \\&+\,\hat{\varvec{D}}_1 ({\varvec{k}}+1){\varvec{U}}({\varvec{k}}+1)\hat{\varvec{D}}^{{\varvec{T}}}_1 ({\varvec{k}}+1)\nonumber \\&+\,{\varvec{R}}_{11} ({\varvec{k}}+1) \Big ]^{-1} \nonumber \\= & {} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}}^0 ({\varvec{k}}+1| {\varvec{k}} +1)\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}+1)\nonumber \\&\times \,\Big [ \hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}({\varvec{k}}+1| {\varvec{k}})\hat{\varvec{H}}^{\varvec{T}}_1 ({\varvec{k}}+1) \nonumber \\&+\,\hat{\varvec{D}}_1 ({\varvec{k}}+1){\varvec{U}}({\varvec{k}}+1)\hat{\varvec{D}}^{{\varvec{T}}}_1 ({\varvec{k}}+1)\nonumber \\&+\,{\varvec{R}}_{11} ({\varvec{k}}+1) \Big ]^{-1} \end{aligned}$$
(104)

The covariance matrix of the estimation error of the filtered estimate \(\hat{\varvec{x}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) denoted by \({\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\) is given by:

$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad ={\varvec{E}}\left\{ {{\tilde{\varvec{x}}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1}){\tilde{\varvec{x}}}^{1^{{\varvec{T}}}}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})} \right\} \end{aligned}$$
(105)

where the error \({\tilde{\varvec{x}}}^{1}_i ({\varvec{k}}+1| {{\varvec{k}}+1})\) is defined as:

$$\begin{aligned}&{\tilde{\varvec{x}}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad ={\varvec{x}}_{\varvec{i}} ({\varvec{k}}+1)-\hat{\varvec{x}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1}) \nonumber \\&\quad ={\tilde{\varvec{x}}}_{\varvec{i}} ({\varvec{k}}+1| {\varvec{k}})-{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1) {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \end{aligned}$$
(106)

Substituting from (106) into (105) while using (99) and after simple mathematical manipulation, it is easy to get:

$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} } ({\varvec{k}}+1| {\varvec{k}})-{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1)\nonumber \\&\quad \hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}{\tilde{\varvec{x}}}_{\varvec{i}}} ({\varvec{k}}+1| {\varvec{k}}) \nonumber \\&{\varvec{Or}}\,{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\!\cong \!{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^0 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\!-\!{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}\!+\!1)\nonumber \\&\quad \hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}{\tilde{\varvec{x}}}_{\varvec{i}} }^0 ({\varvec{k}}+1| {{\varvec{k}}+1}) \end{aligned}$$
(107)

The cross-covariance matrix of the error of the filtered estimates \(\hat{\varvec{x}}^{1}_{\varvec{i}} ({\varvec{k}}\!+\!1| {{\varvec{k}}+1})\) and \(\hat{\varvec{x}}^{1}_{\varvec{j}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) denoted by \({\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\) is given by:

$$\begin{aligned} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})={\varvec{E}}\!\left\{ {{\tilde{\varvec{x}}}^{1}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1}){\tilde{\varvec{x}}}^{1^{{\varvec{T}}}}_{\varvec{j}} ({\varvec{k}}+1| {{\varvec{k}}+1})} \!\right\} \end{aligned}$$

where \({\tilde{\varvec{x}}}^{1}_i ({\varvec{k}}+1| {{\varvec{k}}+1})\) is defined by (106) and \({\tilde{\varvec{x}}}^{1}_{\varvec{j}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) is given by:

$$\begin{aligned}&{\tilde{\varvec{x}}}^{1}_{\varvec{j}} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad \cong {\varvec{x}}_{\varvec{j}} ({\varvec{k}}+1)-\hat{\varvec{x}}^{1}_{\varvec{j}} ({\varvec{k}}+1| {{\varvec{k}}+1}) \nonumber \\&\quad \cong {\tilde{\varvec{x}}}_{\varvec{j}} ({\varvec{k}}+1| {\varvec{k}})-{\varvec{K}}_{1{\varvec{j}}}^1 ({\varvec{k}}+1) {\tilde{\varvec{y}}}_1 ({\varvec{k}}+1| {\varvec{k}}) \end{aligned}$$
(108)

Again, using (106) and (108), it is easy to show that the cross-covariance matrices for \(j\in \{i+1,\ldots ,N\} \,\, {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}}}^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\) are such that:

$$\begin{aligned} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^1 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\cong & {} {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} } ({\varvec{k}}\!+\!1| {\varvec{k}})-{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}\!+\!1)\nonumber \\&\hat{\varvec{H}}_1 ({\varvec{k}}\!+\!1){\varvec{P}}_{{\tilde{\varvec{x}}}{\tilde{\varvec{x}}}_{\varvec{j}} } ({\varvec{k}}\!+\!1| {\varvec{k}}) \nonumber \\ {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^1 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\!\cong & {} \! {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^0 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\!-\!{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}\!+\!1)\nonumber \\&\hat{\varvec{H}}_1 ({\varvec{k}}\!+\!1){\varvec{P}}_{{\tilde{\varvec{x}}}{\tilde{\varvec{x}}}_{\varvec{j}} }^0 ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\nonumber \\ \end{aligned}$$
(109)

Following the same procedure, it can be shown that the filtered estimate \(\hat{\varvec{x}}^{{\varvec{r}}}_{\varvec{i}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) corresponding to the measurements \({\varvec{Y}}^{{\varvec{k}}}, {\varvec{y}}_1 ({\varvec{k}}+1), {\varvec{y}}_2 ({\varvec{k}}+1),\ldots , {\varvec{y}}_{\varvec{r}} ({\varvec{k}}+1)\), its covariance matrix \({\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^{\varvec{r}} ({\varvec{k}}+1| {{\varvec{k}}+1})\), and the cross-covariance matrices \({\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^{\varvec{r}} ({\varvec{k}}+1| {{\varvec{k}}+1})\) are given by:

$$\begin{aligned} \hat{\varvec{x}}^{{\varvec{r}}}_{\varvec{i}} ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\!= & {} \!\hat{\varvec{x}}_{\varvec{i}}^{{\varvec{r}}\!-\!1} ({\varvec{k}}\!+\!1| {{\varvec{k}}\!+\!1})\\&+\,{\varvec{K}}_{1{\varvec{i}}}^{\varvec{r}} ({\varvec{k}}+1){\tilde{\varvec{y}}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \end{aligned}$$
(110)

where

$$\begin{aligned}&{\varvec{K}}_{1{\varvec{i}}}^{\varvec{r}} ({\varvec{k}}+1)\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\hat{\varvec{H}}_{\varvec{r}}^{{\varvec{r}}-1^{{\varvec{T}}}} ({\varvec{k}}+1)\nonumber \\&\quad \times \, \Big [ {\hat{\varvec{H}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}}^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\hat{\varvec{H}}_{\varvec{r}}^{{\varvec{r}}-1^{{\varvec{T}}}} ({\varvec{k}}+1)} \nonumber \\&\quad +\,\hat{\varvec{D}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1){\varvec{U}}({\varvec{k}}+1)\hat{\varvec{D}}_{\varvec{r}}^{{\varvec{r}}-1^{{\varvec{T}}}} ({\varvec{k}}+1)\nonumber \\&\quad +\,{\varvec{R}}_{{\varvec{rr}}} ({\varvec{k}}+1) \Big ]^{-1} \end{aligned}$$
(111)
$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {{\tilde{\varvec{x}}}_{\varvec{i}}}}^{\varvec{r}} ({\varvec{k}}+1| {{\varvec{k}}+1})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad -\,{\varvec{K}}_{1{\varvec{i}}}^{\varvec{r}} ({\varvec{k}}+1)\hat{\varvec{H}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}\, {\tilde{\varvec{x}}}_{\varvec{i}} }^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1}) \end{aligned}$$
(112)
$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^{\varvec{r}} ({\varvec{k}}+1| {{\varvec{k}}+1})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{j}} }^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad -\,{\varvec{K}}_{1{\varvec{i}}}^{\varvec{r}} ({\varvec{k}}+1)\hat{\varvec{H}}_{\varvec{r}}^{{\varvec{r}}-1} ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}\, {\tilde{\varvec{x}}}_{\varvec{j}} }^{{\varvec{r}}-1} ({\varvec{k}}+1| {{\varvec{k}}+1}) \end{aligned}$$
(113)
$$\begin{aligned}&{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} } ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&\quad -\,{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1)\hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}\, {\tilde{\varvec{x}}}_{\varvec{i}} } ({\varvec{k}}+1| {\varvec{k}})\nonumber \\&{\varvec{Or}}\,\,{\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^1 ({\varvec{k}}+1| {{\varvec{k}}+1})\cong {\varvec{P}}_{{\tilde{\varvec{x}}}_{\varvec{i}} {\tilde{\varvec{x}}}_{\varvec{i}} }^0 ({\varvec{k}}+1| {{\varvec{k}}+1})\nonumber \\&\quad -\,{\varvec{K}}_{1{\varvec{i}}}^1 ({\varvec{k}}+1)\hat{\varvec{H}}_1 ({\varvec{k}}+1){\varvec{P}}_{{\tilde{\varvec{x}}}\, {\tilde{\varvec{x}}}_{\varvec{i}} }^0 ({\varvec{k}}+1| {{\varvec{k}}+1}) \end{aligned}$$
(114)

\(\square \)

Appendix 2

Proof of Theorem 6.1

  1. 1a.

    Equation (67) can be written in the following alternative form:

    $$\begin{aligned} {\varvec{Z}}({\varvec{k}}+1)={\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} {\varvec{Z}}(0) \end{aligned}$$

    Then, the norm of the expected value of the above equation is given by:

    $$\begin{aligned} \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le \Vert {{\varvec{\varPhi }} _{{\varvec{ss}}}^{{\varvec{k}}+1} } \Vert \bar{{\bar{{{\varvec{Z}}}}}}(0) \end{aligned}$$
    (115)

    Since for all the solution of (67) \(\bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\) remains bounded as \({{k}}\rightarrow \infty \), then there exists a constant \({{c}}_1\) such that for \({{k}}+1\ge 0\):

    $$\begin{aligned} \Vert {{\varvec{\varPhi }} _{{\varvec{ss}}}^{{\varvec{k}}+1} } \Vert \le {\varvec{c}}_1 \end{aligned}$$
    (116)

    Equation (68) can be written in the form:

    $$\begin{aligned} {\varvec{Z}}({\varvec{k}}\!+\!\!1)\!=\!{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} {\varvec{Z}}(0)\!+\!\!\sum \limits _{{\varvec{j}}=1}^{{\varvec{k}}-1} {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}-{\varvec{j}}} {\varvec{\varPsi }} ({\varvec{j}}\!+\!1,{\varvec{j}}){\varvec{Z}}({\varvec{j}})} \end{aligned}$$

    Taking the norm of the expected value of this equation while using (70), we have:

    $$\begin{aligned}&\bar{{\bar{{{\varvec{Z}}}}}}_ ({\varvec{k}}+1)\le \Vert {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} } \Vert \bar{{\bar{{{\varvec{Z}}}}}}(0)\nonumber \\&\quad +\,\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {\Vert {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}-{\varvec{j}}} } \Vert } \sqrt{{\varvec{\varTheta }}^{{\varvec{T}}}{\varvec{\varTheta }}}\le {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)\nonumber \\&\quad +\,\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{c}}_1 } {\varvec{\lambda }} ({\varvec{j}})\bar{{\bar{{{\varvec{Z}}}}}}({\varvec{j}}) \end{aligned}$$
    (117)

    Let \(g(j)=c_1 \lambda (j)\),

    $$\begin{aligned} \therefore \,\,\bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)+\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{g}}({\varvec{j}})} \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{j}}) \end{aligned}$$

    From the special Gronwall lemma [55], we have:

    $$\begin{aligned} \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le & {} {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)\prod _{{\varvec{j}}=0}^{{\varvec{k}}-1} (1+{\varvec{g}}({\varvec{j}}))\nonumber \\\le & {} {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)\exp \left( \sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{g}}({\varvec{j}})} \right) \end{aligned}$$
    (118)

    By the hypothesis (69), there exists a constant \(c_2\) such that \(\sum \limits _{k=0}^\infty {\lambda (k)} \le c_2 \). Therefore, we have:

    $$\begin{aligned} \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)\exp ({\varvec{c}}_1 {\varvec{c}}_2 ) \end{aligned}$$
    (119)

    which shows that the expected value of the norm of the homogeneous system is bounded. Since \({\varvec{E}}\{{\varvec{w}}({\varvec{k}})\}={\varvec{E}}\{\tilde{\varvec{b}}({\varvec{k}})\}={\varvec{E}}\{ \tilde{{\varvec{c}}}({\varvec{k}}+1)\}={\varvec{E}}\{{\varvec{v}}({\varvec{k}}+1)\}=0\), then \({\varvec{E}}\{{\varvec{\mu }} ({\varvec{k}})\}=0\) in (64) or (66). Therefore, the norm of the expected value of (66) as given by (117) is bounded according to (119).

  2. 1b.

    Since the expected value of the constrained equilibrium \({\varvec{x}}_{{\varvec{ss}}} \)is detectable, and the pair \(\left( {\hat{{\varvec{F}}}^{\prime }\!({\varvec{k}}), \hat{\varvec{H}}({\varvec{k}}+1)} \right) \) is detectable for the entire domain \({\varvec{\varOmega }}\) except for a finite number of points, then the matrix \({\varvec{\varPhi }}_{{\varvec{ss}}}\) is stable. Therefore, the origin is the zero equilibrium of the expected value of the estimation error. As a result, there exist \(0\le {\varvec{\delta }}({\varvec{k}}+1) <1\) such that \(\Vert {{\varvec{\varPhi }} _{{\varvec{ss}}}^{{\varvec{k}}+1} } \Vert <[{\varvec{\delta }} ({\varvec{k}}+1)]^{{\varvec{k}}+1}\) for \({{k}}+1\ge {{K}}\).

    In this case, i.e., \(k+1\ge K\), and from the Gronwall lemma while using (69), the inequality (117) takes the form:

    $$\begin{aligned} \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le & {} \left\| {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} } \right\| \bar{{\bar{{{\varvec{Z}}}}}}(0)+\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {\left\| {{\varvec{\varPhi }}_{ss}^{k-j} } \right\| } \sqrt{{\varvec{\varTheta }} ^{{\varvec{T}}}{\varvec{\varTheta }} }\nonumber \\\le & {} {\varvec{c}}_1 \bar{{\bar{{{\varvec{Z}}}}}}(0)+\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{c}}_1 } {\varvec{\lambda }} ({\varvec{j}})\bar{{\bar{{{\varvec{Z}}}}}}({\varvec{j}})\nonumber \\ \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le & {} \left\| {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} } \right\| \bar{{\bar{{{\varvec{Z}}}}}}(0)+\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{g}}({\varvec{j}})} \bar{{\bar{{{\varvec{Z}}}}}}(j)\nonumber \\ \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le & {} \left\| {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} } \right\| \bar{{\bar{{{\varvec{Z}}}}}}(0)\prod _{{\varvec{j}}=0}^{{\varvec{k}}-1} {(1+{\varvec{g}}({\varvec{j}})) } \nonumber \\\le & {} \left\| {{\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} } \right\| \bar{{\bar{{{\varvec{Z}}}}}}(0)\exp \left( \sum \limits _{{\varvec{j}}=0}^{\varvec{k}} {{\varvec{g}}({\varvec{j}})} \right) \nonumber \\ \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)\le & {} [{\varvec{\delta }} ({\varvec{k}}+1)]^{{\varvec{k}}+1}\bar{{\bar{{{\varvec{Z}}}}}}(0)\exp ({\varvec{c}}_1 {\varvec{c}}_2 )\nonumber \\\le & {} \bar{{{\varvec{\delta }}}}^{{\varvec{k}}+1}\bar{{\bar{{{\varvec{Z}}}}}}(0)\exp ({\varvec{c}}_1 {\varvec{c}}_2 ) \end{aligned}$$
    (120)

    where \(\bar{{{\varvec{\delta }} }}=\sup \{{\varvec{\delta }} ({\varvec{j}}), {\varvec{j}}\in ({\varvec{K}},{\varvec{K}}+1,\ldots ,\infty )\}\) and \(0\le \bar{{{{\varvec{\delta }}}}}<1\)

    Since \(\mathop {\lim }\limits _{{\varvec{k}}\rightarrow \infty } \bar{{{\varvec{\delta }} }}^{{\varvec{k}}+1}=0 \), then \( \mathop {\lim }\limits _{{\varvec{k}}\rightarrow \infty } \bar{{\bar{{{\varvec{Z}}}}}}({\varvec{k}}+1)=0\). Therefore, the norm of the expected value of the estimation error is locally exponentially stable.

  3. 2.

    Equation (71) can be written as:

    $$\begin{aligned} {\varvec{Z}}({\varvec{k}}+1)= & {} {\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}+1} {\varvec{Z}}(0)+\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {\varvec{\varPhi }}_{{\varvec{ss}}}^{{\varvec{k}}-{\varvec{j}}}\nonumber \\&\times \, \left[ {{\varvec{\varPsi }} ({\varvec{j}}+1,{\varvec{j}}){\varvec{Z}}({\varvec{j}})+{\varvec{\mu }} ({\varvec{j}})} \right] \end{aligned}$$
    (121)

    Using (72), (73) and (116), the expected value of the norm of (121) is given by:

    $$\begin{aligned}&\left\| {{\varvec{Z}}({\varvec{k}}+1)} \right\| _{{\varvec{av}}} \le {\varvec{c}}_1 \left\| {{\varvec{Z}}(0)} \right\| _{{\varvec{av}}}\nonumber \\&\quad +\,\sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{c}}_1 } \left\{ {{\varvec{\lambda }} ({\varvec{j}})+{\varvec{\varepsilon }} ({\varvec{j}})} \right\} \left\| {{\varvec{Z}}({\varvec{j}})} \right\| _{{\varvec{av}}} \end{aligned}$$
    (122)

    Let \(g^{\prime } (j)=c_1 \left\{ {\lambda (j)+\varepsilon (j)} \right\} \),

    $$\begin{aligned}&\therefore \,\!\left\| {{\varvec{Z}}({\varvec{k}}\!+\!1)} \right\| _{{\varvec{av}}} \!\le \!{\varvec{c}}_1 \left\| {{\varvec{Z}}(0)} \right\| _{{\varvec{av}}}\nonumber \\&\quad +\sum \limits _{{\varvec{j}}\!=\!0}^{{\varvec{k}}-1} {{\varvec{g}}^{\prime } ({\varvec{j}})} \left\| {{\varvec{Z}}({\varvec{j}})} \right\| _{{\varvec{av}}} \end{aligned}$$
    (123)

    Using Gronwall lemma, we have:

    $$\begin{aligned} \left\| {{\varvec{Z}}({\varvec{k}}+1)} \right\| _{{\varvec{av}}}\le & {} {\varvec{c}}_1 \left\| {{\varvec{Z}}(0)} \right\| _{{\varvec{av}}} \prod _{{\varvec{j}}=0}^{{\varvec{k}}-1} (1+{\varvec{g}}^{\prime } ({\varvec{j}}))\nonumber \\\le & {} {\varvec{c}}_1 \left\| {{\varvec{Z}}(0)} \right\| _{{\varvec{av}}} \exp \left( \sum \limits _{{\varvec{j}}=0}^{{\varvec{k}}-1} {{\varvec{g}}^{\prime } ({\varvec{j}})} \right) \nonumber \\ \end{aligned}$$
    (124)

    By the hypothesis (72), there exists a constant \(c_3\) such that \(\sum \limits _{k=0}^\infty {\left\{ {\lambda (k)+\varepsilon (k)} \right\} } \le c_3 \). Hence, (124) is such that:

    $$\begin{aligned} \left\| {{\varvec{Z}}({\varvec{k}}+1)} \right\| _{{\varvec{av}}} \le {\varvec{c}}_1 \left\| {{\varvec{Z}}(0)} \right\| _{{\varvec{av}}} \exp ({\varvec{c}}_1 {\varvec{c}}_3 ) \end{aligned}$$
    (125)

    which proves the boundedness of the expected value of the norm of the non-homogeneous system and hence the root-mean-square of the estimation error.\(\square \)

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Hassan, M.F. Synchronization of uncertain constrained hyperchaotic systems and chaos-based secure communications via a novel decomposed nonlinear stochastic estimator. Nonlinear Dyn 83, 2183–2211 (2016). https://doi.org/10.1007/s11071-015-2474-6

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