## Abstract

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.

### Keywords

- chaotic secure communication
- underwater acoustic communication
- chaos
- unknown input observers
- nonlinear observers
- reduced‐order observers

## 1. Introduction

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 [7], 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

The unknown disturbance **d** in (1) is assumed to be generated by the exosystem

where **m** to form **d**,

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

where **Ax** is the linear part of **f(x**), *state‐dependent* or *multiplicative* and/or *additive* message signals depending on *additive* messages.

The following three chaotic systems in the form of (3) will be utilized for designing chaotic communication systems in this chapter.

(1) Rossler system [47]

The Rossler system described by

can be modified by chaotic parameter modulation resulting in a system with *state‐dependent (multiplicative)* and *additive* messages as

where

(2) Genesio‐Tesi system [49]

The Genesio‐Tesi system given by

can be modified in the form of (3) with state‐dependent and additive message signals as,

where *a, b* and *c* are the chaotic parameters satisfying

(3) Chua circuit [50]

The Chua circuit

may be modulated in a form with state‐dependent and additive messages as

where *c* = −0.14 are the chaotic parameters. A different modification scheme is given in Ref. [51].

It is noted that, although chaotic systems are sensitive to variations of their chaotic parameters

## 3. General unknown‐input observers (UIOs)

Consider (3), which can be expressed more compactly as,

where

Consider a Luenberger‐like nonlinear observer for (3) given by [27–31, 34],

or more compactly as, with (10),

where

Although (11) and (12) provide a more intuitive form for a Luenberger‐like observer, a linear and nonlinear UIO can be expressed in an alternate form as [31, 52],

where

Three classes of UIOs can be distinguished from the extended state estimate *full‐order UIO* that addresses the estimation of the entire system vector *reduced‐order UIO* that deals with partial‐state and message estimations; and (iii) if *t*, then (13) is an UIO for only message estimation.

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 *synchronization*, which is closely related to the stability of the UIO. Synchronization is a property of the estimation error

where

**Definition 1: Synchronization**

*The drive system* (3) or (10) *and the UIO response system* (11), (12) or (13) *are said to be synchronized if the estimation error* *given by* (14) *satisfies*

*i.e., the UIO is capable of generating an estimate* *that tracks* *asymptotically as*

**Remark 1:** The condition

To proceed, the estimation error (14) satisfies, with (10) and (12),

It follows that **m** and **y**. Further, if a gain

The results above are stated in the following theorem.

**Theorem 1:** *Consider the error Eq.* (15) *with an equilibrium point at* *If a gain matrix* *exists such that* (15) *is asymptotically stable,* then *as* *. *∎

The next task is to determine the gain

### 4.1. Jacobian linearization: full‐order UIO

Linearization of (3) or (10) about the equilibrium point

where

The resulting linearized system is given by

The following assumption is crucial to the construction of UIOs.

**Assumption 1: Observability**

The pair

where

An observer can be constructed for (17) if and only if

Using (17), a linear UIO for full‐state and message estimation can be constructed as

where _{}can be determined, for example, by pole‐placement, such that

Using (17) and (18), the estimation errors

which is exponentially stable, i.e.,

Once a constant

where

which can be linearized about

In summary, we have the following theorem.

**Theorem 2:** *Let* *be an observable pair so that there exists a constant gain* *such that* *in* (19) *is Hurwitz. Then* (20) *is an exponentially stable dynamical system for sufficiently small* *. Further,*

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.

**Example 1**: Rossler system [47]

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

It can be shown that the Rossler system

The stability status of

It follows that **A**, **A**, **A** in the sequel. Therefore, using (18) and (24), it follows that the UIO for full‐state and message estimations has the form

where the gain **Q** and **R**, respectively, where **Q** and **R** matrices can be treated as free design parameters to be chosen and adjusted such that the performance of the UIO is satisfactory. A general method for choosing **Q** and **R** is to set them as diagonal matrices

The overall UIO for full‐state and message estimations can be implemented as (see (22))

where the messages

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 **LQR** command as

where **Ab** and **Cb** denote **P** is the solution of the algebraic Riccati equation (ARE)

where **Q** and **R** that produced a suitable _{}were found to be given by, respectively, (note that the adjustment of **Q** was nontrivial),

We obtain

Note that the eigenvalues

The performance of the UIO is displayed in Figures 2 and 3. The initial conditions used in the simulations were: *t =* 100, and (b) the electrocardiogram (ECG)

### 4.2. Reduced‐order UIO for message estimation for completely known x(*t*)

The objective here is to estimate the unknown message signal vector *derived measurement* derived from

Before launching into the design of UIOs for message estimation, let us consider a general disturbance estimation problem described by

where **y** the measured or known output vector; **d** is assumed to be generated by

where

The objective is to estimate the unknown disturbance **d** using the output **d** can be estimated based on a *derived measurement* instead of **y**.

**Lemma 1:** Estimation of **d** based on derived measurement

*Consider the systems described by* (27) *and* (28)*. A Luenberger‐like nonlinear observer can be constructed for disturbance estimation as*

*where* *is an estimate of* *, and* *is the observer gain to be determined such that* *asymptotically.*

**Proof:** Define a *derived measurement* equation derived from the output

Since **x** is known,

which constitutes a standard form or “pattern” for constructing a nonlinear observer. Hence, a candidate Luenberger‐like observer can be constructed for estimating **d** based on

Substituting (30) into (32) yields (29). ∎

**Remark 2:** When *disturbance observer‐based control (DOBC)* technique applicable to both linear and nonlinear systems. However, it is not clear how and why the derivative term *derived measurement* provides a clear insight, specifically, it clearly shows how **x** for measurement, the derived measurement formulation will pave the way for the design of reduced‐order observers for joint partial‐state and disturbance estimations (see Section 4.3) using only those state variables that are available by direct measurement, thereby extending the DOBC technique and applications. ∎

**Remark 3:** The presence or origin of

To continue further, the derivative

where

where

which is identical to the enhanced observer presented in Ref. [61] (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

where

is the *derived‐measurement* in the form of (30). Most importantly,

where

To proceed, it follows from (37) and (39) that the estimation error defined by

which shows that if

The results above are summarized in the following theorem.

**Theorem 3:** *Consider (36)–(39). If there exists a gain* *such that (40) is asymptotically stable for all x, then* *asymptotically. *∎

Note that since **x**, it complicates the determination of _{} is a constant matrix, then

Using (36), it follows that (35) takes on the form,

A main task in applying (41) is the determination of **p**(**x**). If we set _{}as a linear function of **x**, i.e.,

where

which shows that

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

We remark that the UIO governed by the third equation in (45) is a nonlinear observer with its gain

**Example 2:** Genesio‐Tesi system [49]

Consider the Genesio‐Tesi system described by (7) with additive messages and output

Using (37) with

where

An observer for (47) can be constructed as

which is obtainable from (39). Since

To determine the gain

we obtain, for

The final result for implementation can be obtained by combing Eqs. (46) and (49) as

Since

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

### 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

Let

where **y** is the output. Using (52), we assume that (51) can be partitioned as

which can be rearranged to give

where

Eq. (54) constitutes a standard form that can be used to construct an observer for estimating the inaccessible partial‐state **m** based on the derived measurement

where

**Remark 4:** In (39), the reduced‐order UIO was derived using the output

The estimation error

The preceding error equation is a version of (15). Hence from Theorem 1, the origin

The next task is to eliminate

Choosing

where

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

Using (51) and (60), the main results for the construction of UIO for partial‐state and message estimations are stated in the following theorem.

**Theorem 4:** *Consider the augmented system (54), where* *is the derived measurement. A candidate UIO for partial‐state and message estimations is given by*

If *the gains* *and* *exist such that* (61) *is asymptotically stable, then* *and* *as*

**Example 3:** Chua’s circuit [50]

Consider the Chua circuit described by (8), modified here with an additive message *m* as,

where

Using (52), let the output be chosen as

where

Using (62) and (63), the combined partial‐state and message system has the form

where

Using (61) or (65) and (62), a UIO for partial‐state and message estimations can be constructed based on

The gain

The message

## 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.