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Journal of Applied Nonlinear Dynamics 13(2) (2024) 223--234 | DOI:10.5890/JAND.2024.06.004

Ravi Kumar Ranjan, B.B. Sharma

Department of Electrical Engineering, National Institute of Technology, Hamirpur, 177005, H.P, India

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Abstract

The present work addresses simplistic approach of synchronization strategy for a class of nonlinear chaotic systems with reduced order observer in master-slave configuration. To establish synchronization objective, the system state dynamics is decoupled into dynamics of measurable and unmeasurable states to obtain the estimation error dynamics. Lyapunov stability criterion based observer stabilizes the estimation error which implies synchronization. Reduced order observer design scheme utilizes only the measurable states of master system for estimation. Even to tackle case of multiple states nonlinearity, only output states of system is required which makes it simpler and economical for practical controllers applications. Moreover, to avoid dependence of reduced order observer on derivative of output state dynamics, suitable coordinate transformation is proposed. For real-time applicability, synchronization effectiveness is utilized for secure chaotic communication using n-cipher encryption-decryption methodology. The derived scheme of synchronization and secure communication is also validated in presence of external random noise but its scope can be extended to class of nonlinear systems as well. For numerical simulations, detailed results for chaotic Lorenz system belonging to addressed class are presented.} [\hfill Chaotic system\par \hfill Reduced order observer\par \hfill Lyapunov stability\par \hfill Chaotic synchronization\par \hfill Secure communication\par][Ricardo Luiz Viana][17 June 2022][2 December 2022][1 April 2024][2024 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle %\thispagestyle{fancy} \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} \label{sec1} In control systems, synchronization of two dynamically identical or non-identical systems is the event when state trajectories of these systems evolve invariantly with time. The aspect of control and synchronization for nonlinear systems was extensively studied in past by many researchers due to their wide applications in different areas of human interest. The design of controller to achieve synchronization for nonlinear systems is not a trivial task. Controller synthesis becomes further tricky for nonlinear chaotic systems. Chaotic systems can be characterized by at-least one positive Lyapunov exponent. These systems show complex dynamical behaviour which may diverge rapidly with small perturbation and are also highly susceptible to their initial conditions. Due to these attributes, design of control and synchronization schemes for these systems has been a tedious task. Over the years, many researchers had contributed a lot to address these challenges \cite{pecora1990synchronization,rega2010controlling,dai2012fuzzy,khettab2018enhanced,noroozi2009observer,fallahi2008application,dongmo2022difference}. The notion of synchronization of nonlinear chaotic systems was introduced by Pecora and Carroll \cite{pecora1990synchronization} in their pioneer work, where the criterion for synchronization was exploited based on the sign of the sub-Lyapunov exponents. Later, many other propositions were introduced by other researchers for achieving synchronism of chaotic systems. In these schemes, the most common one is synchronization of systems in master-slave configuration using appropriate control function, where the key idea is to force slave system states to mimic the states of master system. Control and synchronization schemes for such configurations are extensively applied to wide practical applications such as secure communication \cite{fallahi2008application}, cryptography \cite{shukla2018secure}, system biology\cite{postavaru2021covid}, robotics \cite{Jalaik2016ObserverBN}, etc. In these application domains, linear and nonlinear control tools were developed leading to the possibility of synchronization of two or more chaotic systems. Mostly, these ideas were underlying two basic approaches: first is Poincare map linearization known as OGY method and the second scheme is based on time delayed feedback (Pyragas method) \cite{rega2010controlling}. For these methods, as a prerequisite, determination of the unstable periodic orbits of the chaotic systems is required in advance for proposing the controlling algorithms. Along with these, various other tools were also developed by the researchers like sliding mode control \cite{dai2012fuzzy}, variable structure control \cite{khettab2018enhanced}, adaptive control \cite{noroozi2009observer}, observer based approach \cite{fallahi2008application,ranjan2021stabilization}, backstepping method \cite{dongmo2022difference} and so on. These synchronization approaches are broadly based on Lyapunov exponent based schemes or state estimation based schemes. In the first approach, synchronization between chaotic systems is established considering them as drive system and driven system (duplicate subsystem). The drive system transmits synchronizing signals in the form of some of its state variables to the driven system \cite{juang2000synchronization}. This synchronizing signal forces state variables of the driven system to synchronize with the other state variables which are not passed from the drive system. The synchronization can be achieved by following underlying necessary and sufficient conditions viz. all the Lyapunov exponents associated with the variational equation of driven system must be negative definite. The competence of synchronization with a well-structured synchronization signal was further utilized in chaotic secure communication \cite{jiang2014observer}. This proposed approach has limited scope as no general criterion was derived to choose the duplicate subsystem which may ensure negative definite Lyapunov exponents of driven system using the linearization principle. In the state estimation based method, the synchronization between systems may be established by considering the system in a master-slave configuration. The state estimation of the slave can be developed to follow the original master system using the observer based scheme. The observer is obtained by stabilizing the estimation error using various control strategies. Above mentioned tools are utilized to address control and synchronization of nonlinear systems under different scenarios and with various nonlinearities. In \cite{hespanha1999certainty}, the associated nonlinearities are addressed by Lipschitz condition to obtain the state estimation for a continuous-time system. The formulated observer, guarantees the state estimation as long as the associated nonlinear part of the original system satisfies the necessary Lipschitz bounds. The technique can be blended with linear matrix inequality (LMI) for a better efficient optimized result, as established in \cite{chen2011intelligent}. To obtain non-fragile observer based robust synchronization with noise disturbance, the LMI scheme can be infused with $H_\infty$ technique \cite{hou2007h} assuming that the disturbances are norm-bounded. Another aspect to handle nonlinearities suggested in the literature is by using Sliding mode control (SMC), in which synchronization is achieved keeping the system trajectories on the sliding surface. SMC observer based synchronization was approached by many authors under several aspects. In \cite{tran2015novel}, the authors used high-order SMC based observer approach to tackle uncertainties of chaotic systems for finite time synchronization. Other scopes for observer based synchronization can also be found in literature to handle nonlinearities, disturbances and uncertainties by using techniques like backstepping approach \cite{Jalaik2016ObserverBN, shukla2018secure}, unknown input observer (UIO) scheme \cite{sharma2018unknown}, its extension for fault detection and isolation (FDI) in \cite{chen2006fault}, observer formulation by contraction theory \cite{sharma2011observer}, etc. In a simple observer based synchronization for the system in master-slave configuration, the observer may use all the states of the master system for slave states estimation. Otherwise, if a set of limited states is used by the observer, then the same may be termed as reduced order observer. For reduced order, if the observer is using only the output states then it is known as output feedback based reduced observer \cite{tao2018synchronization}. Reduced order based observer reduces the complexity of observer structure. This design was initially developed for linear systems by developing estimates for the unmeasurable states only. Later, the methodology was extended for nonlinear systems by developing necessary and sufficient conditions for ensuring existence of reduced order observer in \cite{nijmeijer1997observer}. Additionally, authors in \cite{zhu2002note} developed the conditions that guarantee the existence of a reduced order based observer for Lipschitz nonlinear systems. This approach was further extended with LMI formulation in \cite{xu2009reduced, boukal2015unknown}. In \cite{pai2012adaptive}, author had presented synchronization for Lipschitz nonlinear system with uncertainties and bounded disturbances based adaptive sliding mode observer. These synchronization approaches were strictly restricted for the class of nonlinear systems with nonlinearities following Lipschitz condition. With the advancement in SMC, an unknown input sliding mode observer with LMI conditions for chaotic communication was proposed by the author in \cite{sharma2017nonlinear}. Without loss of generality, it should be noted that the robustness of conventional SMC schemes is improved by using the discontinuous control terms and these discontinuities introduce undesirable chattering phenomena. In \cite{lin2018chattering}, authors had investigated the methodology to reduce the chattering, however, system external disturbances and system uncertainties were not considered. Parallelly, some other methods like backstepping \cite{laoye2008chaos, singh2019nonlinear,chauhan2021synchronization}, PI control based reduced order observer \cite{castro2015new}, invariant manifold based observer \cite{karagiannis2008invariant}, the algebraic method based observers scheme \cite{boutat2009new}, etc were also explored. Some authors also used alternative contraction theory based approach to design the reduced order observer \cite{yi2021reduced} with the coordinate transformation scheme. In \cite{sharma2017nonlinear}, contraction theory based approach was exploited for unknown input observers, which was further extended for discrete-time systems in \cite{sharma2016reduced}. The major drawback for observers derived in the contraction theory framework is that it is mainly applicable to systems with differentiable nonlinearities. Keeping in view above literature, it can be highlighted that reduced order observer formulation and its application in synchronization for nonlinear systems is a challenging issue, especially for nonlinear chaotic systems and very limited results are available related to synchronization of such systems. Especially generalization of the available methodology is limited, or the derived reduced observer structures are difficult for practical implementations. In the present manuscript, output feedback based synchronization scheme is explored by formulating a systematic reduced order observer based design methodology. The synchronization scheme is developed by considering the systems in master-slave configuration. Unknown states of slave systems are estimated to make them follow the corresponding states of master system. The observer formulation is developed for a class of nonlinear chaotic systems satisfying certain conditions on nonlinear part. Many well-known chaotic systems fit in the proposed class of nonlinear systems. The assumptions on nonlinearities were relaxed further to fit the scheme for some extended class of nonlinear systems. The design scheme was further exploited for addressing challenge of secure communication. In the secure communication scheme, master and slave systems are considered as transmitter and receiver, respectively. The message signal is added to one of the state of transmitter to achieve chaotic masking which is encrypted using n-cypher encryption. The transmitter system states are estimated using reduced order-based observer. Afterwards, at receiver end, the chaotic encrypted signal is received and decrypted to reconstruct original message after demodulation. The proposed design scheme for synchronization and secure communication uses only measurable states for observer synthesis. The design methodology is justified with Lorenz system belonging to the proposed class of nonlinear systems. This class of nonlinear chaotic systems have multiple states nonlinearities even though only measurable states are required for synchronization and secure communication. Use of limited state in proposed observer scheme makes the controller more efficient for practical applications and synchronization robust against external noise disturbances. In nutshell, methodology proposed here presents a simplistic synchronization approach for a wide class of nonlinear chaotic systems. Unlike the previous approaches in literature, knowledge of bounds of associated nonlinearities is not required as in case of sector bounds or Lipschitz nonlinearity \cite{lan2016full, sharma2017nonlinear, zhu2008full, alain2019robust}. Overall, the main contribution of present work can be highlighted as follows:\begin{itemize} \item To develop the reduced order observer based synchronization scheme using output feedback for a given class of nonlinear systems in master-slave configuration \item To extend the derived results of reduced order based observer for other nonlinear systems closely linked to proposed class \item To extend the proposed results to secure communication scheme by using encryption-decryption based on n-shift ciphering approach \end{itemize} The paper is structured as follows: Problem formulation for the proposed class of nonlinear system is given in Section \ref{sec:2}. The main results are enumerated in Section \ref{sec:3} for reduced-order based synchronization scheme. Further, the proposed scheme is extended to state estimation based secure chaotic communication and message recovery in Section \ref{sec:4}. The developed results are justified with numerical simulations in Section \ref{sec:5}. Finally, the concluding remarks about the established scheme are pointed out in Section \ref{sec:6}. %%%%%%%%%%%%%%% \section{Problem formulation}\label{sec:2} The objective of work presented here is to develop reduced order observer based synchronizing strategy for a class of nonlinear systems. To design estimation scheme framework, the problem formulation is considered in master-salve configuration and output of master system is considered measurable which is further utilized for reduced order observer based synchronization scheme. The designed observer will estimate unknown states of the slave system for synchronization with the reference master system. The mathematical description of the master system belonging to proposed class of nonlinear systems is represented as \begin{equation} \begin{split} &\dot{\mathbf{x}}(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{f}\left(\mathbf{x}\left(t\right)\right) \\ &{\mathbf{y}}(t)=\mathbf{C}{\mathbf{x}}(t) \end{split} \end{equation} where ${\mathbf{x}}(t)\in R^n$, ${\mathbf{y}}(t)\in R^p$ are the state and output vectors of the system, respectively, and $n\geq p$, for all practical situations. $\mathbf{A}\in R^{n\times n}$ is the matrix of system parameters associated with the linear part of the system description in $(1)$ and ${\mathbf{C}\in R}^{p\times n}$ is a constant output matrix. The structure of $\mathbf{C}$ plays a critical role in reduced order observer design, as it denotes the states available in output and has a special form of $[\begin{matrix}\mathbf{I}_P&\mathbf{0}\\\end{matrix}]$. Further, it is assumed that $\mathbf{A}$ and $\mathbf{C}$ matrices represent observable pair. The nonlinearities associated with the proposed class is represented by the vector function $\mathbf{f}\left(\mathbf{x}\left(t\right)\right)$ and it satisfies the following assumptions as given below: \begin{assumption} For system $(1)$, the nonlinear function $\mathbf{f}\left(\mathbf{x}\left(t\right)\right):R^n\rightarrow R^n$; can also be expressed as \[ \mathbf{f}\left(\mathbf{x}\left(t\right)\right)=\mathbf{F}\left(\mathbf{y}\left(t\right)\right)\mathbf{x}\left(t\right) ; \text{ where } \mathbf{F}\left(\mathbf{y}\left(t\right)\right)=\left[\begin{matrix}\mathbf{F}_{11} \left(\mathbf{y}(t)\right)&\mathbf{F}_{12}\left(\mathbf{y}\right(t))\\ \mathbf{F}_{21}\left(\mathbf{y}(t)\right)&\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\\\end{matrix}\right], \] is a $(n\times n)$ matrix function of output states. \end{assumption} To achieve the proposed observer based methodology, system dynamics of the considered class is decoupled into two parts: the measurable state variable dynamics $\dot{\mathbf{y}}(t)$ and the unmeasurable state dynamics $\dot{\mathbf{\phi}}(t)$. Thus, the partitioned dynamics of system $(1)$ can be expressed as: \[ \left[\begin{matrix}\dot{\mathbf{y}}(t)\\\dot{\mathbf{\phi}}(t)\\\end{matrix}\right]= \left[\begin{matrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\\end{matrix}\right] \left[\begin{matrix}{\mathbf{y}}(t)\\\mathbf{\phi}(t)\\\end{matrix}\right]+ \left[\begin{matrix}\mathbf{F}_{11}\left({\mathbf{y}}(t)\right)& \mathbf{F}_{12}\left(\mathbf{y}(t)\right)\\\mathbf{F}_{21}\left(\mathbf{y}(t)\right)&\mathbf{F}_{22} \left(\mathbf{y}(t)\right)\\\end{matrix}\right]\left[\begin{matrix}{\mathbf{y}}(t)\\{\mathbf{\phi}}(t)\\\end{matrix}\right] ; \] where ${\mathbf{\phi}}(t)\in R^{n-p}$ are the leftover states which are not available in output, The different sub-matrices here are defined as $\mathbf{A}_{11}\in R^{p\times p}$, $\mathbf{A}_{12}\in R^{p\times\left(n-p\right)}$, $\mathbf{A}_{21}\in R^{\left(n-p\right)\times p}$ and $\mathbf{A}_{22}\in R^{\left(n-p\right)\times\left(n-p\right)}$. The nonlinear submatrices are defined as $\mathbf{F}_{11}\left(\mathbf{y}(t)\right)\in R^{p\times p}$, $\mathbf{F}_{12}\left(\mathbf{y}(t)\right)\in R^{p\times\left(n-p\right)}$, $\mathbf{F}_{21}\left(\mathbf{y}(t)\right)\in R^{\left(n-p\right)\times p}$ and $\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\in R^{\left(n-p\right)\times\left(n-p\right)}$, respectively. \begin{assumption} It is assumed that the nonlinearities are associated with system states that are not present in output state dynamics i.e., $\mathbf{F}_{11}\left(\mathbf{y}(t)\right)$ $\&$ $\mathbf{F}_{12}\left(\mathbf{y}(t)\right)$ are zero and the submatrix $\mathbf{F}_{22}\left(\mathbf{y}(t)\right)$ is skew-symmetric. \end{assumption} Thus, the original master system dynamics expressed in $(1)$, can be further restructured as \begin{equation} \left[\begin{matrix}\dot{\mathbf{y}}(t)\\\dot{\mathbf{\phi}}(t)\\\end{matrix}\right]= \left[\begin{matrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\\mathbf{A}_{21}&\mathbf{A}_{22}\\\end{matrix}\right]\left[\begin{matrix}{\mathbf{y}}(t)\\{\mathbf{\phi}}(t)\\\end{matrix}\right]+\left[\begin{matrix}0&0\\\mathbf{F}_{21}\left(\mathbf{y}(t)\right) &\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\\\end{matrix}\right] \left[\begin{matrix}{\mathbf{y}}(t)\\{\mathbf{\phi}}(t)\\\end{matrix}\right]. \end{equation} %%%%%%%%%%%%%% \begin{remark} The above-mentioned assumptions are realistic as several nonlinear systems including chaotic systems like Lorenz system, Lue system, Lorenz-Stenflo system, Chen system etc.~are expressible in the form given in $(2)$.\end{remark} %%%%%%%%%%%%%%%%%% The measurable state dynamics $\dot{\mathbf{y}}(t)$ in Eq.~$(2)$ will be used to design the reduced order observer. However, it consists of an unmeasurable state vector ${\mathbf{\phi}}(t)$, so a necessary coordinate transformation will be required to eliminate unmeasurable state vector ${\mathbf{\phi}}(t)$ in the description of measurable state vector dynamics ${\dot{\mathbf{y}}}(t)$.~Afterwards, the formulated observer is applied to the unmeasurable state dynamics $\dot{\mathbf{\phi}}(t)$. The scheme is discussed in detail in next section. %%%%%%%%%%%%%%%%%% \section{Main results} \label{sec:3} In the previous section, mathematical structure of the master system was presented in partitioned form in terms of measurable state vector ${\mathbf{y}}(t)$ and unmeasurable state vector ${\mathbf{\phi}}(t)$.~Here, reduced order observer framework based synchronization for the class of the systems given in $(1)$ is elaborated. Reduced order observer gains can be derived by using the Lyapunov stability criterion. The observer design scheme is presented in the form of theorem followed by its proof. From Eq.~$(2)$, the output dependent state vector dynamics is rewritten as \begin{equation}\dot{\mathbf{y}}(t)-\mathbf{A}_{11}\mathbf{y}(t)=\mathbf{A}_{12}\mathbf{\phi}(t). \end{equation} As the left-hand part of Eq.~$(3)$ is not required to be estimated and can be considered as a new output variable, defined as \begin{equation} \mathbf{\nu}(t)=\dot{\mathbf{y}}(t)-\mathbf{A}_{11}\mathbf{y}(t) \end{equation} where, $\mathbf{\nu}(t)\in R^p$ is the coordinate transformation of output which will be utilized to establish the reduced order observer based estimation of the unmeasured states. Using Eqs.~$(3)$ and $(4)$, coordinate transformation can also be represented as \begin{equation} \mathbf{\nu}(t)=\mathbf{A}_{12}\mathbf{\phi}(t). \end{equation} Further, the unmeasured state dynamics of master system, using expression $(4)$ is given as \begin{equation} \dot{\mathbf{\phi}}(t)=\mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22}\mathbf{\phi}(t)+ \mathbf{F}_{21}\left(\mathbf{y}(t)\right)\mathbf{y}(t) +\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\mathbf{\phi}(t). \end{equation} Similarly, for slave system, the estimated coordinate transformation can be given as \begin{equation} \begin{split} \hat{\mathbf{\nu}}(t)=\dot{\mathbf{y}}(t)-\mathbf{A}_{11}\mathbf{y}(t) =\mathbf{A}_{12}\hat{\mathbf{\phi}}(t) \end{split} \end{equation} where $\hat{\mathbf{\nu}}(t)$ and $\hat{\mathbf{\phi}}(t)$ represent the estimates of $\mathbf{\nu}(t)$ and $\mathbf{\phi}(t)$, respectively. The dynamics of the unknown state vector $\mathbf{\phi}(t)$ can be estimated with the new coordinate transformation of output.~So, for the unknown states, observer dynamics is considered as \begin{equation} \dot{\hat{\mathbf{\phi}}}(t)=\mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22}\hat{\mathbf{\phi}}(t)+\mathbf{F}_{21}\left(\mathbf{y}(t)\right)\mathbf{y}(t)+\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\hat{\mathbf{\phi}}(t)+\mathbf{L}\left(\mathbf{\nu}(t)-\hat{\mathbf{\nu}}(t)\right) \end{equation} where $\mathbf{L}\in R^{(n-p)\times p}$ is the observer gain to be derived from the reduced order observer based scheme. The estimated state of slave system given in $(8)$ can be rewritten using $(5)$ and $(7)$ as \begin{equation} \dot{\hat{\mathbf{\phi}}}(t)=\mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22} \hat{\mathbf{\phi}}(t)+\mathbf{F}_{21}(\mathbf{y}(t))\mathbf{y}(t) +\mathbf{F}_{22}(\mathbf{y}(t))\hat{\mathbf{\phi}}(t)+\mathbf{L} \mathbf{A}_{12}(\mathbf{\phi}(t)-\hat{\mathbf{\phi}}(t)). \end{equation} For master system in $(8)$ and the proposed observer in $(9)$, the estimation error $\mathbf{e}_{\phi}(t) \in \Re^{n-p}$, can be defined as $\mathbf{e}_{\phi}(t)=\mathbf{\phi}(t)-\hat{\mathbf{\phi}}(t)$. By replacing $\dot{\mathbf{\phi}}(t)$ and $\dot{\hat{\mathbf{\phi}}}(t)$ from Eqs.~$(6)$ and $(9)$, respectively, estimation error dynamics for the unknown state vector of the system can be computed as \begin{equation} {\dot{\mathbf{e}}}_{\phi}(t)=\mathbf{A}_{22}\mathbf{e}_{\phi}(t)+ \mathbf{F}_{22}\left(\mathbf{y}(t)\right)\mathbf{e}_{\phi}(t)-\mathbf{L} \mathbf{A}_{12}\mathbf{e}_{\phi}(t). \end{equation} The main goal is to identify reduced order observer gain $\mathbf{L}$, which ensures estimation error $\mathbf{e}_{\phi}(t)\in R^{(n-p)}$ convergence to zero as time $t\rightarrow\infty$, i.e. $ \lim_{t\to{\infty}}||{{\mathbf{e}}}_{\phi}\left(t\right)||=\lim_{t\to{\infty}}||{\mathbf{\phi}}\left(t,t_0\right)|-|{\hat{\mathbf{\phi}}}\left(t,t_0\right)||=0.$ \begin{theorem} For the system represented in $(2)$, with estimation error dynamics corresponding to unknown states given in $(10)$, if the reduced order based observer gain matrix $\mathbf{L}$ is selected such that, matrix $\mathbf{P}=\left(\mathbf{A}_{22}-\mathbf{L}\mathbf{A}_{12}\right)< \mathbf{0}$, i.e $(n-p)\times(n-p)$ matrix $\mathbf{P}$ is negative definite, then the estimates of unknown states $\hat{\mathbf{\phi}}(t)$ converge to the true value $\mathbf{\phi}(t)$ which further ensures synchronism of master system with slave system modeled as reduced order observer. \end{theorem} \begin{proof} Let the quadratic Lyapunov candidate function in terms of estimation errors be $\mathbf{V}(t)=\frac{1}{2}\mathbf{e}_\mathrm{\phi}^T(t)\mathbf{e}_{\phi}(t)$. Using Eq.~$(10)$, the time derivative $\dot{\mathbf{V}}(t)$ can be evaluated as \begin{equation} \dot{\mathbf{V}}(t)=\mathbf{e}_\mathrm{\phi}^T(t)\left(\mathbf{A}_{22}- \mathbf{L}\mathbf{A}_{12}\right)\mathbf{e}_{\phi}(t)+ \mathbf{e}_{\phi}^T(t)\left(\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\right)\mathbf{e}_{\phi}(t). \end{equation} Using the skew-symmetric nature of $\mathbf{F}_{22}\left(\mathbf{y}(t)\right)$, as per $Assumption. 2$, the time derivative of Lyapunov function in Eq.~$(11)$ reduces to \begin{equation*} \dot{\mathbf{V}}(t)=\mathbf{e}_{\phi}^T(t)\left(\mathbf{A}_{22}-\mathbf{L}\mathbf{A}_{12}\right)\mathbf{e}_{\phi}(t) \end{equation*} The observer gain is selected such that $\mathbf{P}=\left(\mathbf{A}_{22}-\mathbf{L}\mathbf{A}_{12}\right)< \mathbf{0}$, i.e., time derivative of Lyapunov function becomes negative definite. This completes the proof. \end{proof} \begin{remark} The implementation of the reduced observer is derived using the coordinate transformation considered in $(5)$, which contains the derivative term of output state ${\mathbf{y}}(t)$.~The derivative term with measurement vector may pose a problem, as it will amplify the noise associated with measurement. To tackle this issue, for the selected output of system given in Eq.~$(5)$, a linear coordinate transformation is needed in terms of actual output state dynamics $\dot{\mathbf{y}}(t)$ of the system $(2)$. \end{remark} Let us consider such linear coordinate transformation as \begin{equation} \mathbf{\psi}(t)=\hat{\mathbf{\phi}}(t)-\mathbf{L}\mathbf{y}(t). \end{equation} Using Eq.~$(8)$, the linear coordinate transformation dynamics $\dot{\mathbf{\psi}}(t)$ becomes, \[ \dot{\mathbf{\psi}}(t)=\mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22}\hat{{\phi}} +\mathbf{F}_{21}(t)\left(\mathbf{y}\right)\mathbf{y}(t)+\mathbf{F}_{22}\left(\mathbf{y}(t) \right)\hat{\mathbf{\phi}}(t)+\mathbf{L}\left(\mathbf{\nu}(t)-\hat{\mathbf{\nu}}(t)\right) -\mathbf{L}\dot{\mathbf{y}}(t). \] The considered coordinate transformation dynamics $\dot{\mathbf{\psi}}(t)$ can be further reduced by reverse substitution from expression $(5)$ and $(7)$ and the dynamics is modified as \begin{equation} \dot{\mathbf{\psi}}(t)=\mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22}\hat{\mathbf{\phi}}(t)+\mathbf{F}_{21}\left(\mathbf{y}(t)\right)\mathbf{y}(t)+\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\hat{\mathbf{\phi}}(t) +\mathbf{L}(-\mathbf{A}_{11}\mathbf{y}(t)-\mathbf{A}_{12}\hat{\mathbf{\phi}}(t)). \end{equation} The expression $(14)$ can be modified as the function of known states using Eq.~$(13)$ as follows; \begin{equation} \begin{aligned} \dot{\mathbf{\psi}}(t)= & \mathbf{A}_{21}\mathbf{y}(t)+\mathbf{A}_{22}\left(\mathbf{\psi}(t)+\mathbf{L}\mathbf{y}(t) \right)+\mathbf{F}_{21}\left(\mathbf{y}(t)\right)\mathbf{y}(t)+\mathbf{F}_{22}\left(\mathbf{y}(t)\right)\left(\mathbf{\psi}(t)+\mathbf{L}\mathbf{y}(t) \right)\\ & + \mathbf{L}\left(-\mathbf{A}_{11}\mathbf{y}(t)-\mathbf{A}_{12}\left(\mathbf{\psi}(t)+ \mathbf{L}\mathbf{y}(t)\right)\right). % \end{split} \end{aligned} \end{equation} So, rather than realizing the estimates of unmeasurable states from Eq.~$(8)$, the estimates $\hat{\mathbf{\phi}}(t)$ can be obtained as $ \hat{\mathbf{\phi}}(t)=\mathbf{\psi}(t)+\mathbf{L}\mathbf{y}(t) $, with transformed vector $\mathbf{\psi}(t)$ realized through $(13)$. \begin{remark} The essence of proposed methodology for reduced order-based state estimation is that it is not restricted to structural dynamics represented in $(2)$.~The central idea for the implementation of the strategy is to deftly tackle the nonlinearities associated with the system. The approach can be easily extended to the class of nonlinear systems where the associated nonlinearities can be expressed as $\mathbf{f}\left(\mathbf{x}\left(t\right)\right)=\mathbf{F}\left(\mathbf{y}\left(t\right)\right)\mathbf{x}\left(t\right)$, with the sub matrix $\mathbf{F}_{12}\left(\mathbf{y}(t)\right)$ is zero and $\mathbf{F}_{22}\left(\mathbf{y}(t)\right)$ as skew-symmetric. For this, the output state dependent matrix $\mathbf{F}\left(\mathbf{y}\left(t\right)\right)$ of the nonlinearity can be represented as \[ \mathbf{F}\left(\mathbf{y}\left(t\right)\right)= \left[\begin{matrix}\mathbf{F}_{11}\left(\mathbf{y}(t) \right)&0\\\mathbf{F}_{21}\left(\mathbf{y}(t)\right)& \mathbf{F}_{22}\left(\mathbf{y}(t)\right)\\\end{matrix}\right]. \] For such case, Eqs.~$(3)$ to $(5)$ are required to be modified appropriately. Moreover, the methodology is also applicable to the cases where $\mathbf{F}_{11}\left(\mathbf{y}(t)\right)$ and/or $\mathbf{F}_{21}\left(\mathbf{y}(t)\right)$ are zero. \end{remark} \begin{figure*

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