\u0026#x201c;chaotification\u0026#x201d; of real systems by dynamic state feedback

$: \u0026#x201C;Chaotification\u0026#x201D; of Real Systems by Dynamic State Feedback$
Testing Ourselves

Levent Sevgi D�u, University Electronics and Communication

Eng. Dept.

Zeamet Sokak, No 21, Acibadem - Kadik6y

Istanbul, Turkey

Email: [email protected]. [email protected]

http://www3.dogus.edu.trllsevgi

It has been four years since the "Testing Ourselves" column took off! We started with the first quiz back in February 2007. We

have had a variety of quizzes and tutorials since then. This is the twenty-fourth column. Please go ahead and write down your comments and requests on the topics presented in this column, especially if they are positive and constructive. These mean a lot to us.

We promised to present our last tutorial, in the October 20 I 0 issue, on electromagnetic-precursory-based earthquake prediction. Unfortunately, that hasn't been completed yet. The issue is important and publicly sensitive: it therefore needs extra care and high precision. We hope to make that ready in one of the future issues, either in the February or in April 2011 issues of the Magazine.

We have been preparing some interesting tutorials with much useful MATLAB code. One of them is on microstrip-line filter design. This has been in my mind for some time. I have given some professional two- or three-day EMC courses to the electronics industry. In one of these courses, I was surprised to see that many of the design engineers of a major defense company, who were supposed to design microstrip circuits, had never seen microstrip lines in their undergraduate or graduate programs. This was awkward! However, I then realized that this is the case for many

others. We'll therefore review both the systematic method of microstrip filter design from Le circuits to transmission lines and microstrip lines, and practical design guidelines. We'll include useful computer code for that. Another tutorial will be on path finding and planning for swarm robotics. It seems to be interesting to see how the FDTD model can be used for this purpose. What about reviewing the group and phase velocities in electromagnetics with simple MATLAB code? Dou you want to see a tutorial that explains those things called GTD, PTD, UTD, etc., with nice illustrations? We'll do these, too. Finally, we're planning to prepare a tutorial on academic publishing, together with the Editor-in-Chief of this Magazine, Ross Stone. That will trigger an interesting discussion, I believe, if we can present the tutorial as it is in our minds.

Please note that we're looking for interesting quizzes and tutorials. As always, this as an open call to all our expert members! We shouldn't forget that there are many members and student members out there who do not have easy access to lecturers or trainers like you, so this is our social endeavor to assist those members. Please also note that the keyword of the tutorials is simplicity , so, try to keep it as simple as possible, so that I can understand. If 1 can, so can everybody!

"Chaotification" of Real Systems by Dynamic State Feedback

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Sava$ $ahin1 and Ciineyt Giizeli$2 1 Department of Control and Automation Technology, Ege Vocational School

Ege University, Campus, Bornova, 35100, Izmir, Turkey E-mail: [email protected]

2Department of Electrical and Electronics Engineering, Dokuz EylOI University Tmaztepe Campus, Buca, 35160, Izmir, Turkey

E-mail: [email protected]

IEEE Antennas and Propagation Magazine, Vol. 52, No.6, December 2010

Abstract

Chaos - which, to the best of the knowledge of the authors, is the most complex behavior of deterministic dynamic systems -is observed in many systems modeled by nonlinear ordinary or partial differential and difference equations. The vast amount of chaos literature is mainly devoted to the analysis and implementation of chaotic dynamics and to chaos control, i.e., the design of controllers changing the behavior of an originally chaotic system so that it possesses a stable equilibrium or limit cycle. Against the mainstream chaos studies in control, a new field, called "chaotification" (also called "chaos anti-control" or "chaotization") has emerged, in order to exploit chaotic behavior instead of escaping from it. This is inspired by limited but successful engineering applications of chaotic dynamics in cryptology, secure communication, and mixing of liquids. This paper presents a brief review of chaotic dynamics and chaos control. It also presents a novel chaotification method, which can be applied to any input state of a system that can be linearized, including linear controllable systems as special ca-ses. The chaotification introduced - which is realized by a dynamic state feedback increasing the order of the open-loop system to have the same chaotic dynamics as a reference chaotic system - is used to ·chaotify" a real dc motor to have the celebrated Lorenz chaotic dynamics.

Keywords: Chaos; chaotification; Lorenz chaotic system; dynamic state feedback; feedback linearization; dc motor

1. Introduction

Chaos is a random-like behavior of nonlinear dynamic deterministic system. A chaotic trajectory of a system defined by

nonlinear differential equations is a complicated, bounded solution that does not converge to an equil ibrium state or to a periodic (or also a quasi-periodic) orbit [I]. Chaotic dynamics - more precisely, a chaotic invariant set - has the following three distinguishing features: i) sensitive dependence on initial conditions, i .e., trajectories originating from nearby states diverge as time proceeds; ii) topological transitivity, i .e . , every non-empty open subset of the phase space is traveled by the trajectory originating from an arbitrary open subset; and iii) denseness of the (necessarily unstable) periodic orbits, which yields a broad frequency spectrum [2, 3].

Chaos is observed in many real-world systems, including biological, chemical, electrical, and mechanical systems. As will be explained in Section 2, chaos has been shown to exist in a diverse set of nonlinear dynamic systems, including distributed systems defined by partial differential equations, fractional-order continuous-time systems, first-order discrete-time systems, as well as the very-well-known continuous-time systems defined by thirdorder ordinary differential equations, e.g. Lorenz, Rossler and Chua chaotic systems [4-6]. The chaos l iterature is mainly devoted to the analysis, implementation, and control of chaotic systems [\-21]. However, available engineering applications of chaos are quite limited: cryptology, secure communication, and the mixing of liquids are among the rare engineering fields where chaotic systems have been demonstrated to provide successful alternative solutions, as compared to conventional solutions [20-33].

The cryptology applications exploit the random-like sequences produced by the trajectories of a deterministic chaotic system [22-23]. Secure-communication applications exploit the spread nature of the frequency spectrum of the carrier chaotic signals employed, which are produced via so-called chaotic phase synchronization at the transmitter parts of the communication systems [24-27]. The liquid-mixing applications of chaos are inspired by the observations that high impeller speeds, time-varying impeller speeds, and also continuous perturbations of the flow can yield effective mixing for viscous fluids having low or moderate Reynolds numbers. High and time-varying impeller speeds can prevent the formation of segregated regions in the vicinity of impellers [3\, 32]. On the other hand, chaotic vibrations providing time-varying impeller speeds with a broad frequency spectrum


have been observed to be suitable as the aforementioned continuous perturbations [33].

This paper is in the reverse direction to the mainstream of chaos studies. These can essentially be divided into two main groups: i) analysis and implementation of chaotic systems, and ii) chaos control (the so-called control of chaos), to change the asymptotical behavior of a chaotic system into a stable equilibrium or l imit cycle, by using a specific control scheme. Instead, this paper follows the direction of chaos studies relying on the assumption that chaos can be useful for some engineering applications. "Chaotification" (the so-called anti-control of chaos), i .e ., changing the non-chaotic asymptotical behavior of a considered system into a chaotic system, may be useful in some cases.

This paper gives a brief overview of chaos studies. It presents a novel chaotification method, which is based on matching the system dynamics of a given system to the dynamics of a reference chaotic system by dynamic state feedback. The chaotification method introduced can be applied to any linear time-invariant controllable system, and also to any input-state nonlinear system that can be linearized. To demonstrate its useful real-life engineering applications, dynamic state feedback chaotification is applied to a permanent-magnet dc motor, by matching the closed loop dynamics of the motor to the well-known Lorenz' s chaotic circuit.

The method exploits any chaotic system of arbitrary dimension as the reference model, with no need to transform the system into a special form. It thus has the advantage of exploiting the vast amount of literature on the analysis, synthesis, and hardware and software realization of chaotic systems. The universality and implementation efficiency of the proposed chaotification regarding both the system to be "chaotified" and the system to be taken as reference thus provide a tool for developing chaos-based solutions to real-life engineering problems.

Considering its wide-spread use in engineering applications and its generic feature, i.e., the rel iance of its working principles on one of the four fundamental forces of nature, electromagnetic interaction, namely, the dc motor, is chosen in this paper as the case for the real system to be chaotified. This is done by introducing dynamic state feedback: the <;Iosed-Ioop system will possess a behavior such as that produced by the celebrated Lorenz chaotic system.

223

The paper is organized as follows. Section 2 gives a brief overview on chaos, chaos control, and chaotification. In Section 3, the dynamic�state�feedback chaotification of linear time-invariant controllable systems, based on the Lorenz system as the reference chaotic system, is described. The application of the proposed chaotification method to permanent-magnet dc motors is explained in subsection 4.1. The experimental setup and implementation results for a chaotified dc motor speed-control system is described in detail in subsection 4.2. Conclusions and potential engineering applications of the chaotification method presented, and of the chaotified dc motor, are given in Section 5.

2. Chaos, Chaos Control, and Chaotification

2. 1 Chaos

The simplest asymptotical behavior of dynamic systems is equilibrium, i.e., a constant trajectory. Many systems - especially, control systems - are designed to be operated at a desired set-point (that is, to track a constant reference signal). This is usually achieved by reaching an asymptotically stable eqUilibrium point via a stabilizing controller. Periodic and then quasi-periodic asymptotical behaviors, which come after equilibrium in the hierarchy of system dynamics, can also arise naturally in real systems, for instance, as an orbit that a satellite should track. Periodic and quasi-periodic trajectories are, indeed, sums of periodic time waveforms, each of which is frequency written as an integerweighted combination of a finite set of base frequencies. Such trajectories may display time dependencies that are too complex. They can easily be distinguished from a chaotic trajectory by considering their frequency spectrum and their limit sets. The limit set of chaotic dynamics, which is the most-complex deterministic dynamic behavior currently known, is of fractal dimension. This is as opposed to the equilibrium dynamics, which has a zero-dimensional limit set. Periodic and quasi-periodic dynamics have limit sets shaped like a one or higher integer-dimensional k-torus. Chaotic trajectories have broadband frequency spectra, while the periodic and quasi-periodic trajectories have a discrete Fourier spectrum.

Many different definitions of chaos have been proposed in the literature. The following definition is given by Wiggins [2] for continuous-time autonomous systems, and agrees with Devaney's definition [3] proposed for discrete-time systems. The definition is based on the features of sensitive dependence on initial conditions, topological transitivity, and denseness of periodic trajectories. The definition of Wiggins requires the concepts of flow and an invariant set to be introduced, beforehand.

Let :i::::: f ( x ), with :i::= dx , be an autonomous system with a dl

continuously differentiable function f(o): RR � RR . Let its solu-

tion due to initial state Xo = x (/0 ) ERR be denoted as x (I; xo, 10 ) at a specific time, t, where the initial time, to, is assumed to be

zero, with no loss of generality. Assume that the solution

x (t; xo, to) exists for alI initial states Xo ERR , and let it be defined

for all t � to . A solution due to a specific initial state, xo, is indeed

a function mapping time, t, to the state, x (t ) , at that time t, given

as x ( 0; xo, 10 ) : R+ � R n . On the other hand, each initial state pro-

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duces a possibly different solution, namely, a function

x ( 0; 0, (0) : R+xR n � R n , which is called flow in the topological

dynamics literature, and a state-transition function in the systemtheory literature. This solution can be defined for representing the

mapping from time t and initial state Xo to the state x (I) at that

time t. Now, let A E RR be a closed and bounded subset of the

state space R R that is invariant under the flow x(o;o,to):R+xRn �Rn, i.e., X(/;Xo,/o)EA for all Xo EA and

for all 1 � to .

2. 1. 1 Definition: Chaotic Invariant Set [2]

The invariant set, A E RR, of the autonomous system

:i: = f (x) is said to be chaotic if:

1. The flow x( o;o,to) is sensitive, depending on the initial

conditions in A E RR , i.e., 3& � 0 such that VXo E A and for some U neighborhood of xo, 3xo E U and

3t > 0 such that Ilx(t;xo,fo)-X(t;xo,to)11 > & .

2. The flow x ( 0; 0, to) is topologically transitive on A, i.e., for each pair of open sets U, V c A, 3t E R such

that x (I; U, to ) n V :;t: rp, where rp stands for the empty set.

3. Periodical trajectories of the flow x ( 0;0,/0 ) are dense

in A, i.e., in each open neighborhood of a periodic trajectory, there must exist another periodic trajectory.

Sensitive dependence on initial conditions is usually measured by calculating the largest Lyapunov exponent, A., defined by Equation (1) [34], where x, x denote two trajectories due to the initial states xo, xo:

(1)

The largest Lyapunov exponent, A., is less than zero, equal to zero, and greater than zero when the associated (bounded solution) system dynamics has stable equilibrium, stable periodic (limit cycle) or quasi-periodic, and chaotic behavior, respectively. In order to get insight into the above definitions, one can verify that the largest Lyapunov exponent for a scalar linear time-invariant system

x = ax is equal to a , as shown below, where the solution due to

initial state Xo has the form x(t;xo,to):::: eat Xo at a specific time t,

so the flow is x( o;o,to):::: ea(o) (0) :

[ (at 1 . I)] 1 e xo-xo

= lim sup -log 1 -

' 1 t-+oo 1 Xo Xo

=a.

IEEE Antennas and Propagation Magazine, Vol. 52, No. 6, December 2010

This means that an unstable scalar system with a > 0 has the property of sensitive dependence on initial conditions. This can also be observed by the application of the definition as

Ilx(t;xo,to ) -x(t;io,to )11 = leat Xo - eat io)1 = eat IXo - io l > & for

I & t = -log --- > 0 values. However, the solutions of such a Ixo -io l

unstable systems are not bounded, and they also do not possess the other two features intrinsic to chaotic systems.

The largest Lyapunov exponent is the most widely used tool for determining the existence of chaotic dynamics. Indeed, a large class of chaos studies that is restricted to the numerical analysis only due to the difficulty of checking the conditions in definition 2.Ll in a theoretical way is based on the calculation of the Lyapunov exponents, and also on constructing bifurcation diagrams demonstrating the change of qualitative limit behavior of the systems with respect to the parameters. A set of dynamic systems the dynamics of which are shown to be chaotic in the l iterature by theoretical and/or numerical methods is presented below.

As a consequence ofthe fact that any bounded solution of an autonomous planar vector field tends to either an equilibrium point or a periodic orbit, as stated by the Poincare-Bendixson Theorem [35], chaos can be created at least in a three-dimensional autonomous vector field defined by a third-order ordinary differential equation system in the state-equation form (as wil l be seen below, this fact is true for integer-dimensional vector fields.) . For the Lorenz system given by the third-order autonomous state equation in Equation (2), where the states correspond to the convective fluid motion, the horizontal temperature variation and the vertical temperature variation are such examples. These were derived by the meteorologist E. N. Lorenz in 1963, for modeling the random-like behavior of weather [4]:

x = a(y-x),

y = rx- y-20xz, (2)

z = 5xy-bz.

Rossler, Chua, Sprott and Lii-Chen systems [5-8] are other examples of third-order autonomous continuous-time chaotic systems, which have been analyzed theoretically and also experimental ly with a vast amount of research.

Chaos is a phenomenon intrinsic to nonlinear dynamic systems, i .e. , being nonlinear and being dynamic are both necessary in generating chaotic behavior. However, chaos is not a phenomenon unique to continuous-time integer-dimensional systems defined by ordinary differential equations. The discrete-time dynamic model of population growth in Equation (3), called a logistic map, which exhibits chaotic behavior for most of the c values beyond 3.57, shows the possibility of creating chaos even in one dimension for the discrete-time case [36]:

(3)

In the Kuramoto-Sivashinsky equation, Equation (4), the real function u = u (x, t) of space and time satisfies periodic boundary

conditions. This constitutes an example from the domain of partial differential equations, shown to be chaotic for R = 2 by calculating its Lyapunov exponents [37]:


au au I a2u a4u -=-u------al ax R ax2 ax4' (4)

Chaos is also observed in fractional-order state equations [38]. The fractional-order Chua circuit in Equation (5) shows that continuous-time dynamic systems with a fractional order less than three can exhibits chaos. Here, the Riemann-Liouville fractional order (n -1:5: q < n e Z) derivative of a function f (I) is defined

in Equation (6) by means of an integer-order derivative (n E Z ) of a convolutional integral and the Gamma function, 1(0):

(5)

(6)

2.2 Chaos Control

The above examples show that chaos is a natural phenomenon, arising with a diverse set of nonlinear dynamic systems. The observed chaos might be a desired behavior, as in biological systems such as healthy heart and brain [39, 40]. However, chaos is considered to be an undesired behavior in most control systems. A control law is needed to drive the system exhibiting chaotic behavior to a l imit cycle or to an asymptotically stable equilibrium. Starting with the work by Ott et al . [9], a number of methods have been developed in this direction, constituting a new field of research called chaos control . Three widely accepted chaos-control methods developed in the l iterature are given below.

2.2.1 The Feed-Forward Control Method

The feed-forward control method is an open-loop control scheme for chaos control . For a desired periodic x* (t) trajectory,

the behavior of a chaotic system, i = f ( x) + Bu, is changed by

using a suitable control signal , u(t) =B+ {i* (/ ) -f [x* (t)]} ,

where B+ is the generalized inverse. This method provides the desired trajectory in an asymptotic sense for uniformly Lyapunov stable Jacobian matrices of f(-) , evaluated along the desired trajectory. It is widely used in the l iterature because of its simplicity [13-15, 21].

2.2.2 The Ott-Grebogi-Yorke (OGY) Method

The Ott-Grebogi-Yorke (OGY) method [9, 20-21] is based on the construction of the Poincare map x(k+I)= p[ x(k),u(k)]

225

for a given system i = f (x, u ) , by introducing a Poincare section

transverse to the desired (goal) trajectory, x'" (I ) . The OGY

method applies a state-feedback control u(k)=Kx(k) to the lin

earized Poincare map x(k+ I) = Ax(k)+ Bu(k), with x =: x-xo. This is valid around the point Xo = x'" (0) taking place on the Poincare section. Indeed, this control input defines accessible control parameters of the system along which small perturbations are appl ied to the system whenever I lxll � s for some smalls > O.

2.2.3 The Time-Delayed Feedback Method

The time-delayed feedback method, also known as the Pyragas method, is based on applying a delayed state feedback control u = K[x(/) - X (I -,)] to stabilize a periodic solution with

period, of the system i = f( x,u ) [ 1 8-2 1 ].

2.3 Chaotification

Chaotification, also called anti-control or chaotization, aims at establishing a desired chaotic behavior for an originally nonchaotic system. This is done in order to benefit from the distinguishing features of chaos, i.e., random-like determinism, a broad frequency spectrum, etc. Three of the chaotification methods developed in the l iterature are given below.

2.3.1 The Vanecek-Celikovsky Method

The Vanecek-Celikovsky method [4 1 ], which is based on the Shilnikov Theorem, considers a continuous-time control system in Lure' s form. For such systems, an odd monotonic nonlinearity exists in the feedback path, and a l inear time-invariant system is in the feed-forward path. The linear feedback gain is chosen to insure that i) some of the closed-loop poles are in the open left-hand side, ii) some of the closed-loop poles are in the right-hand side, ii i) some of the closed-loop poles are negative real, iv) some of the closed-loop poles have a nonzero imaginary part, and v) there is no pole on the jOJ axis.

2.3.2 The Chen Method

The Chen method [42, 43] was proposed to chaotify discretetime systems x(k+ 1) =f[x(k )]+u(k) by using feedback control

in the form u(k)=Kg{O'[x(k)]}, where K,O'>O and g(o) is a

sine or piecewise-linear sawtooth function. The amplitude, K, and frequency, 0', are chosen based on the function considered, f(e), in a manner suitable to changing the Lyapunov exponents of the closed-loop system to be positive.

226

2.3.3 Model-Based Static Feedback C haotification

Model-based static feedback chaotification [44, 45] was realized as first transforming the reference chaotic system into the Brunovsky canonical form via a transformation (in general, nonlinear), and then matching a considered system of the same order to the transformed reference chaotic system by a suitable static feedback. The dynamic state feedback chaotification method presented in this paper is indeed an extension of the model-based static chaotification method developed by MorgUl [44, 45]. This method does not need to transform the reference chaotic system into the Brunovsky form while matching, thus providing the superiority of exploiting efficient hardware realizations, and also benefiting from the extensive analysis results for the reference chaotic systems already available in the l iterature.

3. Dynamic State Feedback Chaotification of Linear Time-Invariant Systems

Consider an nth order linear, time-invariant, controllable, and observable system, which is to be chaotified. With no loss of generality, assume that the system has a single input, and is defined in the following controllable canonical form. Furthermore, assume that all state variables can be measured directly.

XI 0 0 0 XI 0

x2 0 0 0 x2 0

0 + u, (7) xn-I 0 0 0 0 xn-I 0

xn -aI

-a2 -an xn

where the Xj stand for state variables, and u stands for the scalar input. Now, consider the Lorenz system defined in Equation (8) [4, 25,46] as the reference chaotic system:

x = O'(y-x),

y = rx-y-20xz , (8)

:i = 5xy-bz.

Redefining the above Lorenz states x, y, and z as

Xn+I ';'y, (9)

and choosing the control input, u, as

( 1 0)


the l inear system given in Equation (7) is augmented to the following nonlinear system of order (n + 2) :

XI 0 0 0 0 xI x2 0 0 0 0 0 x2

xn_1 0 0 0 0 xn_1 (II)

xn 0 0 -(J" (J" 0 Xn xn+1 0 0 r -I -20xn Xn+1 xn+2 0 0 0 5xn -b Xn+2

With the above choice of the control input, u, the last equation of the system in Equation (7) becomes identical to the first Lorenz equation in Equation (8) under the change of variables in Equation (9). The state feedback defined by Equation (lO) can be interpreted as a nonlinear dynamic state feedback, since the control input, u, brings in an extra state variable, xn+I ' which is nonline-

arly and also dynamically dependent on the nth state variable, xn' as can be observed from Equation (8) and (9). In an indirect way, the control input, u , in Equation ( 1 0) brings in another extra state variable, xn+2, yielding the last three equations of the augmented system, to match the Lorenz state equations in Equation (8) but with the new variables in Equation (9). Thus, the resulting nonlinear system in Equation (II) has exactly the same chaotic dynamics as the Lorenz system, in such a way that the dynamics of the last three state variables match the dynamics of the states in the Lorenz system, and the first (n -I) state variables are simply the integral,

at different level, of the state xn. This means that the new highdimensional system of Equation ( 1 1 ) has a low-order chaotic dynamics.

The proposed chaotification scheme is depicted in Figure I, where a part of the Lorenz system constitutes a dynamic control ler, together with the state feedbacks.

It should be noted that as shown in [30], the dynamic state feedback chaotification method presented can also be applied to any nonlinear system that can have its input state linearized.

=�-�-2�2 2=5x�-�2

4. Chaotification of Permanent-Magnet dc Motor by Dynamic State Feedback

To demonstrate the applicability and real-life engineeringapplication potentials of the simple dynamic state feedback method that has been introduced, a permanent-magnet dc motor is considered as the system to be chaotified. This is due to its wide-spread engineering usage, and due to its prominent status as having the features of the reliance of its working principles on one of the four fundamental forces of nature, i .e . , electromagnetic interaction [29, 30].

4.1 Chaotification of dc Motor Based on a Simplified First-Order Model

In the experiments, an RF-3 1 OT A series dc motor, which is easy to use and can be acquired easily with very low cost, was used. Thus, anyone who is interested in the appl ications of the proposed chaotification method to a real dc motor may attempt to reproduce the experiments, and observe how a chaotified dc motor runs. Just for simplicity, a first-order simplified model was chosen for the permanent-magnet dc motor: X = -amx + bmu, where x stands for the angular velocity of the motor, i .e. , its rpm, and u stands for the armature input voltage. The first-order system parameters of the RF -3 lOT A series dc motor used were identified as am = 2 and bm = 5600, by measuring its response to a step input.

Applying the proposed chaotification method to the simpl ified model of the dc motor, and using the Lorenz chaotic system reference, the augmented system became a third-order nonlinear system. This is identical to the Lorenz system, since the original system, i .e., the dc motor, had a first-order dynamic model. The resulting system is given in Equation ( 1 2), where the original symbols of the Lorenz state variables are preserved, and the first one i s taken equal to the dc motor's state:

y = rx -y-20xz, (12)

Figure 1. A block diagram of the proposed chaotifying control scheme.

IEEE Antennas and Propagation Magazine, Vol. 52, No.6, December 2010 227

z = 5xy -bz .

Choosing the control input as

( 1 3)

the resulting chaotified dc motor system is depicted in Figure 2 .

The dc motor model chaotified by the above dynamic state feedback with the Lorenz reference chaotic system has the phase portrait shown in Figure 3a, and the time waveforms in Figure 3b. The MATLAB m-fiIe codes for the phase portrait and time waveforms used in the computer simulations are given in Tables 1 -3.

The Lyapunov exponents of the dc motor model chaotified by the Lorenz chaotic system were calculated using the TISEAN package program [47,48] to be 4 .35 , 0.1, and -7.10. The time series used in the calculation was obtained from the x state generated by the MATLAB m-fiIe for solving the chaotified dc motor equation with a sampling frequency of 1 17 Hz, and for a data length of 10000 points. As expected, the largest Lyapunov exponent was found to be positive, indicating that the resulting system was chaotic. The sum of the Lyapunov exponents was found to be negative, indicating that the chaotic trajectory was a bounded trajectory. It should be noted that different sampling frequencies and data lengths, and also different methods and techniques used for solving differential equations, may yield different results for Lyapunov exponents. Table 4 compares the Lyapunov exponents found by the TISEAN program as described above to the two other different sets of Lyapunov exponents found in [49, 50], all of which belonging to Lorenz systems with different parameters but in the chaotic mode.

Figure 2. A diagram of the proposed chaotification of the dc motor.

5

4

N 3

2

O L-------�--------�------�------� -4 -2 o

x 2 4

Figure 3a. The x-z projection of the phase portrait obtained from the simulation of the chaotified dc motor.

228

Figure 3b. Time wavef'orms of the x and z states or the simulated chaotified dc motor.

Table 1. The MATLAB code for the chaotified dc motor.

% Chaotified DC motor Equation and the Plots: % -----------------------------------------------------

am=2; bm=5600; sigma=lO; r=56.6; b=5 .02; xm=[ 1 .0 1 , 0.0 1 , 1.02]; tempxm=xm; delta_t=0.0085 ; for k= 1 :5000 xm(2 )=tempxm(2 )+( r*tempxm( I )-tempxm(2)-20*tempxm( I )*tempxm(3»*delta _ t; xm(3)=tempxm(3)+(5*tempxm( I )*tempxm(2)b*tempxm(3»*delta _ t; u=(llbm)*(-(sigma-am)*tempxm(I)+sigma*(tempxm(2»); xm(l )=tempxm( I )+( -am*tempxm( 1 )+ bm*u )*delta _ t; tempxm=xm; x(k)=xm( I); y(k)=xm(2); z(k)=xm(3); end; figure; plot(x,z); xlabel('x'); ylabel(,x');figure; plot(x,'-'); hold on; plot(z,'-.'); xlabel('samples'); ylabel('x and z');

Table 2. The MATLAB code for the bifurcation diagram (r).

% The bifurcation diagram (r): % -----------------------------------------------------am=2; bm=5600; sigma=IO; r=0.02; b=5.02; x=zeros(25 0, I); xm=[ 1 .0 1 , 0.0 1 , 1.02]; tempxm=xm; delta_t=0.0085 ; hold on; while r<IOO

for k= 1 :250, u=(llbm)*(-(sigma-am)*tempxm(l) + sigma*(tempxm(2»); xm(1 )=tempxm( I )+( -am*tempxm( 1)+ bm*u)*delta _t; xm(2)=tempxm(2)+(r*tempxm( I )-tempxm(2)-20*tempxm( 1 )*

tempxm(3»*delta_t; xm(3)=tempxm(3)+( 5*tempxm( I )*tempxm(2)

b*tempxm(3»*delta _ t; tempxm=xm; x(k)=xm(I); y(k)=xm(2); z(k)=xm(3); plot(r,x( 190:250),'b.'); xlabeICr'); ylabel('x');

end; r=r+l

end;


4.2 Experimental Setup for the Proposed dc Motor Chaotification System

The whole dc motor chaotification system depicted in Figure 2 was implemented as a fully analog system in a modular fashion. The plant was the RF-3 1 OTA series dc motor, which indeed implemented the first state equation in Equation ( 1 2). The second and third state equations of the Lorenz system of Equation ( 1 2), which constitute the nonlinear dynamic controller part, were realized with the circuit configuration in Figure 5. There, two AD633 analog multipliers realized nonlinear terms. Two LF353 opamps, together with eleven resistors ( RI = R2 = R5 = 1 00 kO, R3 = R4 = I kQ, R6 = R8 = 1 0 kO, R7 = 1 MO, RVI = RV3 = 1 00 kO, RV2 = 470 kO) and two capacitors ( CI = C2 = 1 0 nF), realized linear weighted summations and integrations. The motor speed, which was the state variable to be fed back, was measured by a tachometer-generator. This was again an RF -3 IOTA series dc motor-generator, connected to the shaft of

. the dc motor. The tachometer-generator output voltage, considered to be proportional to the motor speed, constituted the feedback signal , which was again analog. The other part of the hardware realization, comprised of two LF353 opamps, realized the summers and amplifications in Figure 2. This provided the control input driving the dc motor. The experimental setup of the proposed chaotification scheme is given in Figure 6.

4.3 Experimental Results from the Proposed dc Motor Chaotification System

The dc motor chaotified by the fully analog hardware realization in Figure 5 was also analyzed experimentally with the setup of

Table 3. The MATLAB code for the bifurcation diagram (b).

% The bifurcation diagram (b) % -----------------------------------------------------

am=2; bm=5600; sigma= 1 0; r=56.6; b=0.02; x=zeros(250, I); xm=[ 1. 0 1 , 0.01, 1 .02]; tempxm=xm; delta_t=0.0085 ; hold on; while b<25

for k= 1 :250, u=( Ilbm)*( -(sigma-am )*tempxm( I) + sigma *(tempxm(2))); xm(I)=tempxm(I)+(-am*tempxm(I)+ bm*u)*delta_t; xm(2)=tempxm(2)+(r*tempxm( I )-tempxm(2)-20*tempxm( 1 )*

tempxm(3»*delta_t; xm(3)=tempxm(3)+( 5 *tempxm( 1 )*tempxm(2)

b*tempxm(3»*delta _t; tempxm=xm; x(k)=xm(I); y(k)=xm(2); z(k)=xm(3); plot(r,x( 1 90:250),'b.'); xlabel('b'); ylabel('x');

end; b=b+0.4

end;

Table 4. The Lyapunov exponents for the Lorenz chaotic system.

Lyapunov Exponents

From + 0 -

Sprott [49] 1 .5 0 -22.37

Peterson [50] 0.09 0 - 1 4.57

Chaotified dc motor* 4.35 0. 1 -7 .10


..

3

2

w 0

-1

-2

·3

� L-____________ � ____ �� ____ � ____ �

Figure 4a. A bifurcation diagram obtained from the simulation of the chaotified dc motor, in terms of the r parameter.

5

.50�----�5------�1�O------1�5�----�20�----�25 b

Figure 4b. A bifurcation diagram obtained from the simulation ofthe chaotified dc motor, in terms of the b parameter.

Figure 6. Snapshots of the x-z projections of the phase portrait of the chaotified dc motor are given in Figures 7a-7c and 8a-8c, obtained for different choices of resistor values. These RVI and RV2 resistors, which were parts of the integrator opamp circuits (U2A and U2B), respectively tuned the rand b parameters of the Lorenz chaotic system given in Equation ( 1 2). Thus, different dynamic modes, ranging from an asymptotically stable equilibrium point to the Lorenz-type chaotic attractor, were observed, as demonstrated in Figures 7a-7c and 8a-8c .

5. Conclusion and Future Directions

A chaotic-reference-model-based dynamic state feedback chaotification method was presented in this paper. This was presented together with an overview of chaotic-system examples from diverse fields, and an overview of chaos control and chaotification methods. Besides being easily implemented, the chaotification method does not necessitate sophisticated knowledge of chaotic dynamics. The method is somewhat universal , since it is applicable to any kind of controllable l inear time-invariant system, and also to any nonlinear system with an input state that can be linearized. It keeps the original form of the reference chaotic system when

229

R3 Rl

R4 I·

y

U2 \ 1 '.':'t C3

Figure 5. The analog realization of the chaotifled dc motor system, based on the Lorenz chaotic system as the reference model.

230

Figure 6. The experimental setup for chaotiflcation of the dc motor.


II1II x - v

0'':'-

v" 1:� 2: 2.S6U

v.c

. . . . . . ..

.. . . .. . · · ·t:Jf' 1: 1.38feU 2: :S.2&iIV

. F ...... ., . . . ·.t ·· ·· ·· . . ··•···· l' 4 854Hz - . i. 2: 697Hz

CHI'" ICH2'" 11ees/s I l�V IV 2�s xY

DwtTCJSIe 1: 7.23% 2: 87.27% RIle .. 1:86.46iq 2: 336. 2IiIs � 9. 9900Hz ITRIC"'I EDGE I ACQ 1 USB CHI/" AUTO SAMPLE

Figure 7a. A snapshot on the x-y mode of the oscilloscope, where the signals in the x and z states of the Lorenz circuits were fed to the x and y channels, respectively. While RV2 and RV3 were both equal to 50 kO, the RVI value was chosen to be 10 kO.

.. x - v Run Aut.o v" � 1:� 2: 2.32V V ..

..•.. . . 1: -2.ee.u ._, .,' .,� 2: 3S.e.u .. ' :;i��. ':" . . ""'--r

•• • ••• • • • + • • • •• • •

. �.i�...;�.;. ·

·:··"'t�i.:- ···· · t

. . ..•.. . . 1: 19 .Hz I �. : AJ.': !! 1.489Hz Ii

DwtTCJSIe 1: 52.61% 2: 96.92% RIle .. I 1: 26.ee.-s •

� ________ � ________ � __ � ____ � __ �2:�685��.p�� 1. z CHI '" ICH2'" I 1905h I XV I ee...v I V 2:50to1s ITRIC"'I EDGE I ACQ I USB CHI/" AUTO SAMPLE S.nd • •

Figure 7b. A snapshot as in Figure 7a, but with RVI set to 30kO.

x - v Run

·· · ·t····.· · . ·.· .· ••• . •

CHI'" ICH2'" 11eesh 1 I�V IV 2S�s

v" 1: 16."'" 2: 12..u V ..

1: -� 2: 28."'"

�., • ••• . . .. ... . . .... ... ... •• • • 1:

? DwtTC� 1: ? 2: ? RIle ..

1: ? 2: ?

• I kHz xv ITRIC"'1 EbCE I ACQ I USB CHI/" AUTO SAMPLE Send • •

Figure 7c. A snapshot as in Figure 7a, but with RVI set to 50-100 kO.


.. x - v Run Aut.o v"

1: 32."'" 2: .... ...., V ..

1: -31.9uU 2: 6.aa..u

·· · ·.· · ··t· ·· ·.····.··· . �

• • ••• • • •••••• , • •• •••••• 1: ?

xv

2: ? DwtTCJSIe 1: ?

2: ? RIle .. 1: ? 2: ? 269. Hz ITRiC"'1 EDGE 1 ACQ I USB CHI/" AUTO SAMPLE

Figure 8a. A snapshot produced as described in Figure 7a. While R VI was 20 k 0 and R V3 was 50 k 0, the value of RV2 was chosen to be 1 kO.

l1li x - v

:. v,, � 1: 34e.u I 2: 1.88U V ..

1: -2.98MI : .,'" ". 2: -68. 3IIIU : . . . . '

.

Fnt .. ., . •••••••••• . • •••• • • '" �� •• • : •• . • •••••••. •• . 1'1 467Hz

· .. ·iI : · . •

'. : 2: 4. 385Hz DwtTCJSIe 1: :se.�

12: :S.27% RIle ..

1:48:s.� , �2: 63:5. :s-S9.900Hz CHI", ICH2'" 1 10eSIs I 199MV IV 2�s xv ITRiC"'1 EDGE I ACQ 1 US8 CHI /" AUTO SAMPLE

Figure 8b. A snapshot as in Figure 8a, but with R V2 set to 50kO.

x - v

• ,,-f .. .

Run.-v"

1: 348M.' 2: 8.88U

':',;:'" ,. V .. f 1: 28.""'" _ 2: 28. hAJ

��; .: . ' - �., .... . .......... . ...... � .:. •• � ...... : • • •••• • • t •• • • I 12 sa-tz.1 y : � :: 2� 3.m.a

. .: �; ' DwtT C�

xv ITRIC"'I EDCE CHI /" AUTO

1: 26.74% 2: 8:5.32:c

...... 1:41.37 .. 2: 26:5 .... 50. 0900Hz

Figure 8c. A snapshot as in Figure 8a, but with R V2 set to 350 kO.

231

matChing the system to be chaotified to the reference chaotic system. As a consequence, the possibility of using efficient hardware realizations and analysis results for the reference chaotic systems already available in the literature makes it superior to the existing chaotification methods.

The chaotification method presented, with its universal applicabil ity and implementation efficiencies, is promising not only in all engineering areas. It is also promising in applications in physics, chemistry, and biology, where broad-spectrum, random-like determinism and other distinguishing features of chaotic dynamics are considered to be beneficial .

The success of the chaotification method presented depends on the accuracy of the system models desired to be chaotified. Further research is needed in order to enhance the robustness of the proposed chaotification method against unavoidable systemmodeling errors, which is crucial in obtaining a persistent chaotic behavior.

6. Acknowledgement

The authors would like to thank the reviewers for the constructive and helpful comments.

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Introducing the Authors

. Cuneyt GiizeJi� received the BSc, MSc, and PhD degrees in Electrical Engineering from Istanbul Technical University (ITU), in 1 98 1 , 1 984, and 1 988, respectively. He worked in the ITU from 1 982 to 2000, where he became a full Professor. Between 1 989 and 1 99 1 , he was a visiting professor in the Electrical Engineering and Computer Science Department of Berkeley, California. He is currently a Professor in the Department of Electrical and Electronics Engineering at Dokuz Eyliil University, izmir, Turkey. His research interests include artificial neural networks, biomedical signal and image processing, nonlinear circuits, systems, control , and educational systems.

Sava� $ahin received the BSc degree in Electronics and Communication Engineering from Kocaeli University, Kocaeli, Turkey, in 1 996; the MSc degree from the Department of Electrical and EI\!ctronics Engineering, Ege University, izmir, Turkey, in 2003 ; and the PhD degree from the Department of Electrical and Electronics Engineering at Dokuz Eyliil University, izmir, Turkey, in 20 I O. He has been working as an Instructor with the Department of Control and Automation, Ege Vocational School, Ege University, since 2000. His main research interests are in the fields of control systems, industrial automation, chaotic systems, and artificial neural networks. He i s involved in several national projects on control ler design and engineering education. <®

233

\u0026#x201c;chaotification\u0026#x201d; of real systems by dynamic state feedback

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