Secure communication employing chaotic systems is considered in this chapter. Chaos‐based communication uses chaotic systems as its backbone for information transmission and extraction, and is a field of active research and development and rapid advances in the literature. The theory and methods of synchronizing chaotic systems employing unknown input observers (UIOs) are investigated. New and novel results are presented. The techniques developed can be applied to a wide class of chaotic systems. Applications to the estimation of a variety of information signals, such as speech signal, electrocardiogram, stock price data hidden in chaotic system dynamics, are presented.
- chaotic secure communication
- underwater acoustic communication
- unknown input observers
- nonlinear observers
- reduced‐order observers
With the advances in computing and communication technologies, among others, underwater acoustic communication (UAC) techniques [1–6] have emerged as the predominant mode of underwater communication because of its one key advantage over conventional electromagnetic communication, namely, relatively low attenuation of acoustic waves in water. However, their performance is severely affected by a number of factors, including limited channel bandwidth, time‐varying channel characteristics, complex ambient noise, and multipath distortion that results from multiple reflections of sound waves from top and bottom surfaces of water, especially in a relatively shallow waterbody.
Over the past decade, chaos‐based underwater acoustic communication (CUAC) techniques have attracted a lot of interest from a number of researchers [7–12], because such techniques are potentially more cost‐effective (for example, requiring lesser number of component modules) compared with conventional communication schemes. The CUAC techniques proposed to date can be broadly divided into two categories, namely, coherent detection based CUAC methods , and non‐coherent detection based CUAC techniques. The coherent detection based methods rely on synchronization to reconstruct a copy of the transmitted signal at the receiver end, whereas non‐coherent detection methods [8–12] utilize a variety of data recovery methods without requiring any synchronized reconstruction of the transmitted message.
In this chapter, we focus our attention on the synchronization based CUAC techniques, especially on observer‐based synchronization methods, because the underlying theory is very well understood and has proven to be reliable and robust in many control applications. Also, such methods may potentially turn out to be easier to implement, as compared with many non‐coherent CUAC techniques.
At the outset, we should point out that the main goal of this chapter is to present the fundamental concepts of observer‐based chaotic synchronization and their applications to secure chaotic communication. With this in mind and owing to space limitation, we omit discussion of the robustness issues [13–23] of such techniques here. However, we should point out that the theory of robust observer design in the presence of noise and uncertainties has been well researched in control literature, and these ideas are deemed to be useful for synchronized based CUAC as well [18–23].
The methodologies used for CUAC have many things in common with chaos‐based wired and wireless communication. Research and development in these fields have been advancing rapidly in the literature [7–16]. In contrast to conventional communication systems which use sinusoidal carriers to transmit information, chaos‐based communication uses chaotic systems as its backbone for information transmission and recovery. The advantages of employing chaos‐based systems include, among others, (i) the communication is difficult to detect and decrypt; (ii) the transmission is hidden from unauthorized receivers; (iii) the communication is more resistant to jamming and interferences because of the broadband characteristics of the chaos‐based carriers. The advantages above are due to the following characteristics: (i) a chaotic system is dissipative; (ii) chaotic systems have unstable equilibrium points; (iii) its trajectories are aperiodic and bounded; and (iv) its trajectories have a sensitive dependence on their initial conditions, i.e., trajectories originated from slightly different initial conditions will soon become totally different. We remark that some of these characteristics may, in fact, be undesirable.
The organization of this chapter is as follows. Section 2 introduces three nonlinear chaotic systems that are utilized for designing chaotic communication systems in follow‐up sections. Next a general discussion of unknown input observers is presented in Section 3. Section 4 presents the theory and design of unknown input observers for chaotic secure communication. Finally, the conclusions and plan for future research are provided in Section 5.
2. Nonlinear systems with application in chaotic communication
Consider a general nonlinear system described by
where is the system state vector, the output measurement, an unknown disturbance vector which can be treated as a message vector that carries useful information; is a smooth vector field, a smooth function, and .
The unknown disturbance
where is the message state, is a “picking matrix” that picks the appropriate components of
Eqs. (1) and (2) is widely used for the design of linear and nonlinear observers, unknown input observers (UIO), and unknown input observers for secure communication [24–46]. When applied to the design of unknown input observers (UIOs) for secure communication based on chaotic systems, (1) and (2) can be combined and expressed as
The following three chaotic systems in the form of (3) will be utilized for designing chaotic communication systems in this chapter.
(1) Rossler system 
The Rossler system described by
can be modified by chaotic parameter modulation resulting in a system with
(2) Genesio‐Tesi system 
The Genesio‐Tesi system given by
can be modified in the form of (3) with state‐dependent and additive message signals as,
(3) Chua circuit 
The Chua circuit
may be modulated in a form with state‐dependent and additive messages as
where , and
It is noted that, although chaotic systems are sensitive to variations of their chaotic parameters , most systems do accommodate suitable modifications of some of these parameters. This property has precisely been exploited for the designs of UIOs for secure communication and many control‐based synchronization schemes in the literature.
3. General unknown‐input observers (UIOs)
Consider (3), which can be expressed more compactly as,
, , and .
or more compactly as, with (10),
where is an estimate of , , and is the observer gain matrix to be determined such that the observer has desirable properties, such as generating an estimate that can track (or converge to) asymptotically in the face of unknown disturbances.
where , is a smooth vector field, a smooth function, and .
Three classes of UIOs can be distinguished from the extended state estimate , namely, (i) if , then (13) is a
The design of all the three classes of UIOs discussed above for secure communication will be addressed in Section 4.
4. Unknown‐input observers (UIOs) for chaotic secure communication
The analysis and design of UIOs for secure communication using a drive‐response scheme in this section will be based on (10)–(13). Hence, (3) or (10) will serve as the drive system, while (11), (12) or (13) as the response system.
In the drive‐response chaotic communication theory and applications, one of the most important issues is
where and .
To proceed, the estimation error (14) satisfies, with (10) and (12),
It follows that is an equilibrium point of (15), i.e., for all ,
The results above are stated in the following theorem.
The next task is to determine the gain so that the candidate observer (12) or (13) becomes an asymptotically or exponentially stable observer. The matrix can take on various forms depending on the type of systems being considered and/or the design techniques. For a general nonlinear system, can be determined as a function of the estimate , i.e., [27, 28]; for nonlinear systems under Jacobian linearization, can be obtained as a constant matrix [29, 30]; for extended Kalman‐Bucy filtering using Jacobian linearization, the filter gain matrix can be approximated by its steady‐state value . We shall focus on Jacobian linearization in Section 4.1 below with applications to full‐order UIOs for state and message estimations using constant gain . Section 4.2 addresses the design of reduced‐order UIOs for message estimation, while the design of reduced‐order UIOs for partial‐state and message estimations is considered in Section 4.3.
4.1. Jacobian linearization: full‐order UIO
Linearization of (3) or (10) about the equilibrium point yields
, , , , , .
The resulting linearized system is given by
The following assumption is crucial to the construction of UIOs.
The pair in (17) is observable, i.e.,
where is the observability matrix
An observer can be constructed for (17) if and only if is an observable pair. Hence when the Jacobian linearization method yields a pair that is not observable, then the Jacobian linearization method is not applicable to the system under consideration. However, other methods may work, such as feedback linearization [53, 54].
Using (17), a linear UIO for full‐state and message estimation can be constructed as
where is the constant UIO gain matrix to be determined. Note that and , and (18) is simply a Luenberger observer . Since is an observable pair by Assumption 1, then can be determined, for example, by pole‐placement, such that is Hurwitz, i.e., all the eigenvalues of are located in the open left half‐complex plane.
Using (17) and (18), the estimation errors and satisfy
which is exponentially stable, i.e., and exponentially for all and . It follows that and exponentially.
Once a constant has been determined, it can then be substituted into (12), whereby the resulting nonlinear UIO has the form
where is an arbitrary initial condition. Further, (15) becomes
which can be linearized about to give (19). Hence the dynamics of (21) close to the origin are well described by (19) for sufficiently small .
In summary, we have the following theorem.
Using (10) and (20), the overall chaotic system‐based UIO for full‐state and message estimations under the Jacobin linearization scheme can be implemented as
A block diagram for (22) is shown in Figure 1.
Consider the Rossler system with state‐dependent and additive messages described by (5), with the output arranged as ,
The preceding equation can be expressed as
where for simplicity. Note that is an observable pair for all .
It can be shown that the Rossler system given by (4) has two equilibrium points, for ,
The stability status of and can be determined by checking the eigenvalues of the Jacobian matrices and . We obtain,
It follows that
where the gain is to be determined such that is Hurwitz. The next task is then to find , which may be obtained by using the pole‐placement or Kalman‐Bucy filter design method. We shall use the Kalman‐Bucy filter technique. We note that in the design of a Kalman‐Bucy filter [55, 56], the known covariance matrices of the system noise and measurement noise are given by
The overall UIO for full‐state and message estimations can be implemented as (see (22))
where the messages and are injected into the Rossler system directly (see Figure 1 also), thereby the message model is omitted in (26); however, the model matrix is needed in the message observer equation (third equation).
The key task now is the determination of a suitable UIO gain based on (24) that yields acceptable performance. The design can be accomplished by using Matlab’s
where in (see (24)). The parameter matrices
Note that the eigenvalues are spread apart widely and have two complex conjugate poles.
The performance of the UIO is displayed in Figures 2 and 3. The initial conditions used in the simulations were: and . The signals to be estimated are: (a) a voice message injected into the drive system at
4.2. Reduced‐order UIO for message estimation for completely known x(
The objective here is to estimate the unknown message signal vector by assuming that the entire state vector is accessible by direct measurement, i.e., full‐state measurement, and does not have to be estimated. Hence, without loss of generality, the output can be assumed to be given by . This leads to the construction of a reduced‐order UIO for message (disturbance) estimation. In general, a reduced‐order observer based on full‐state or partial‐state measurement has an interesting structure and is an active area of research in the literature for system controls and disturbance estimation, see for example [24–26, 57–60]. The reduced‐order UIO designed in this section for message estimation will be based on a
Before launching into the design of UIOs for message estimation, let us consider a general disturbance estimation problem described by
where is the state vector, a known control input vector, an unknown disturbance vector, and
The objective is to estimate the unknown disturbance
which constitutes a standard form or “pattern” for constructing a nonlinear observer. Hence, a candidate Luenberger‐like observer can be constructed for estimating
Substituting (30) into (32) yields (29). ∎
To continue further, the derivative in (29) can be eliminated by moving the term to the left side of the equation to yield
where . Defining ,
where is to be determined. If , then (33) can be expressed as
which is identical to the enhanced observer presented in Ref.  (see for example, Eq. (3.5), p. 44).
We now return to message estimation in chaotic systems. We can start with (3), which can be expressed as, with full‐state measurement given by ,
Since the entire state vector is known for all , (36) can be rearranged as
where is the observer gain matrix to be determined.
To proceed, it follows from (37) and (39) that the estimation error defined by satisfies
which shows that if is a suitable stabilizing gain, then can be made to converge to zero asymptotically for arbitrary , thereby .
The results above are summarized in the following theorem.
Note that since is a function of
Using (36), it follows that (35) takes on the form,
A main task in applying (41) is the determination of
where is a constant matrix, then we obtain from (34), . Further, if and is an observable pair, then can be determined readily by, for example, the pole‐placement method, such that is Hurwitz. Moreover, (40) becomes,
which shows that , thereby exponentially for arbitrary . In addition, in this case, the enhanced UIO (41) reduces to
It can be shown that the preceding equation can be obtained by using the linearized system (17) and setting .
Once a suitable gain has been determined, such as , it can then be substituted into (41), and the overall chaotic system‐based UIO for message estimation can be implemented as, with (36),
We remark that the UIO governed by the third equation in (45) is a nonlinear observer with its gain replaced by a constant . Other methods may be used to determine a suitable , such as linear matrix inequality (LMI), see for example Ref. .
Consider the Genesio‐Tesi system described by (7) with additive messages and output
Using (37) with , the preceding equation can be arranged in the form of an LTI system as
where is the derived measurement, and it can be shown that is an observable pair for all , i.e., .
An observer for (47) can be constructed as
which is obtainable from (39). Since is an observable pair, a constant gain can be determined such that is Hurwitz. Further, eliminating the derivative term in (48) yields
To determine the gain , let the poles of be selected as . Using Matlab’s pole‐placement command,
we obtain, for ,
Since and are injected directly into the Genesio‐Tesi system (46), the message model is not needed and is omitted in (50); however, the model matrix is required in the estimation equation (second equation in (50)). The signal is the nine‐term Fourier series of a square wave, and is a mix signal consisting of a trapezoid, sine wave, and ramp and exponential functions. It would be difficult to generate these rather complicated signals, in particular , by using the simple model , and/or a more general model proposed in Refs. [24–26], where the elements of are unknown sequences of random delta functions. For simulation studies, the mix signal can easily be generated by the following Matlab codes and injected into (50):
Mix signal : m2=0.05*t*((t>0)&(t<10))+0.5*((t>=10)&(t<=20)) ‐0.05*(t‐30)*((t>20)&(t<=30))+0.25*sin(t‐30)*((t>=30)&(t<58.27)) +(1/20)*(t‐58.27)*((t>=58.27)&(t<78.27)) +1*exp(‐0.2*(t‐78.27))*((t>=78.27)&(t<200)).
The performance of the UIO is displayed in Figures 4 and 5. The initial condition of the Genesio‐Tesi system used in the simulation was , while was calculated by using , which yields where . Figure 4(a) shows and its estimate , and Figure 4(b) exhibits and . The estimation errors were negligible, as can be seen from Figure 4(c) and (d), where the plots of vs. , and vs. are displayed. Note also the Gibb’s phenomenon (the “twin‐towers”) in Figure 4(a). The Genesio‐Tesi attractor is shown in Figure 5. We conclude that the performance of the reduced‐order UIO for message estimation was satisfactory.
4.3. Reduced‐order UIO for partial‐state and message estimations
The objective in this section is to extend the design of reduced‐order UIO for message estimation to the design of UIO for joint partial‐state and message estimations. The results obtained are believed to be new and novel.
Consider a general nonlinear system described by (3) which is expressed here without the output as (see also (1) and (2))
where . The design will be based on a derived measurement formulation.
where and are, respectively, accessible and inaccessible for direct measurement, and
which can be rearranged to give
where denotes the derived measurement.
Eq. (54) constitutes a standard form that can be used to construct an observer for estimating the inaccessible partial‐state and the unknown message
where and are the gain matrices to be determined such that the observer has desirable performance characteristics, in particular, and as .
The estimation error , where , satisfies, with (54) and (55),
The preceding error equation is a version of (15). Hence from Theorem 1, the origin is an equilibrium point of the unforced equation in (56) for all . Further, and if and are stabilizing gains.
The next task is to eliminate in in (55) by moving to the left side of the equation and defining
where , and are to be determined. It follows that
Using (55), (57), (58) and (59), can be expressed as
which can further be reduced to a form given by, for example (44), once the specific structure of the chaotic system under consideration is known and has been determined (see Example 3 for more details).
Using (51) and (60), the main results for the construction of UIO for partial‐state and message estimations are stated in the following theorem.
Consider the Chua circuit described by (8), modified here with an additive message
Using (52), let the output be chosen as
where constitutes the unknown partial state, and the derived measurement can be obtained as
Using (62) and (63), the combined partial‐state and message system has the form
where if , , and is an observable pair for all , i.e., .
Using (61) or (65) and (62), a UIO for partial‐state and message estimations can be constructed based on given by (64) and implemented as
The gain used for the simulations was obtained by choosing the UIO poles as . Using Matlab’s pole‐placement command , we obtain, with ,
The message in (66) is a stock price data consisting of 50 data points where the value of is . To minimize the effect of on the chaotic nature of the Chua circuit, it is scaled down to a small signal as ; this yields . The scaled down signal was then injected into (62) directly. To enhance the estimate , its initial value was calculated by using , which gave , where and . We remark that, since the initial condition of the Chua circuit is known, we can always set , while in the event that the value of is not known, then it can be set as resulting in small mismatches between and during the transient period. The performance of the reduced‐order UIO is shown in Figures 6 and 7. Figure 6(a) shows and its estimate , while Figure 6(c) shows the plot of vs. , which indicates an excellent match. Figure 6(b) displays the message and its estimate , while the plot of vs. in Figure 6(d) shows a clean 45‐degree line indicating an almost perfect match. The plots of vs. are depicted in Figure 7(a)–(c), showing that the small signal has little effect on the chaotic nature of the Chua circuit. We conclude that the performance of the reduced‐order UIO for partial‐state and message estimations was satisfactory. Further, it is emphasized that no Jacobian linearization was employed in this example.
5. Conclusions and plan for future research
In this paper, we showed that secure communication employing chaotic systems can be achieved by synchronizing the dynamics of the drive and response systems. The results are obtained by using unknown‐input observers (UIOs), which serve as the response systems. Three classes of UIOs have been designed, namely, (i) full‐order UIO for estimating all the state variables (full state) and messages in the drive system; (ii) reduced‐order UIO for message estimation based on a derived measurement technique, where the formulation is based on the disturbance observer‐based control (DOBC) theory (recall that the DOBC technique is only applicable to disturbance estimation based on the assumption that all the state variables (full state) in a system are known; and (iii) reduced‐order UIO for partial‐state and message estimations based on partial‐state measurement using the derived‐measurement technique. The reduced‐order UIO for partial‐state and message estimations is novel, and is an extension of the DOBC theory, thereby expanding the technique and applications of DOBC. Our future research and development will be focused on wireless secure communication, robust synchronization in the presence of channel noise and various channel induced distortions, and the designs and applications of disturbance cancellation nonlinear control systems using the well‐known disturbance accommodation control (DAC) theory, thereby unifying the DAC and DOBC approaches and techniques.