Applied Mathematics and Computation 218 (2011) 22–31

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Adaptive control of linear time invariant systems via a wavelet networkand applications to control Lorenz chaos

Yan Wu ⇑, Jonathan S. TannerDepartment of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA

a r t i c l e i n f o

Keywords:Lyapunov stabilityWavelet networkLinear time-invariant systemsLorenz system

0096-3003/$ - see front matter Published by Elsevidoi:10.1016/j.amc.2011.05.037

⇑ Corresponding author.E-mail address: yan@georgiasouthern.edu (Y. W

a b s t r a c t

Compactly supported orthogonal wavelets have certain properties that are useful forcontroller design. In this paper, we explore the mechanism of a wavelet controller byintegrating the controller with linear time-invariant systems (LTI). A necessary conditionfor effective control is that the compact support of the wavelet network covers the statespace where the state trajectories stay. Closed-form bounds on the design parametersof the wavelet controller are derived, which guarantee asymptotic stability of wavelet-controlled LTI systems. The same wavelet controller is then applied to the Lorenzequations. The control objective is to stabilize the Lorenz system well into its normallychaotic region at one of its equilibria.

Published by Elsevier Inc.

1. Introduction

The applications of wavelet theory have been thriving in many branches of the engineering sciences during the pastdecade. Wavelet-based multiresolution analysis is superior to the Fourier analysis because of its ‘‘zoom-in’’ (localization)property in both time (space) and frequency domains, which is powerful in capturing nonlinear behavior in dynamical sys-tems. As such, wavelet theory has drawn a great deal of attention from the control community on the topics of systemidentification and adaptive controller designs.

There are a number of papers dedicated to the application of wavelet analysis in the study of nonlinear dynamicalsystems, mostly in system identification, signal reconstruction, and solution of nonlinear differential/integral equations.Xin and Sano investigated the system identification of an FIR model via wavelet packets filter bank using an adaptive scheme[1]. The wavelet filter bank approach is shown to be effective in dealing with ill-conditioned process for reconstructing thefilter coefficients. Sureshbabu and Farrel [2] developed a system identification algorithm using orthogonal wavelets.Cao et al. [3] applied wavelets to predict chaotic time series from nonlinear dynamical systems. Haar wavelet with variablestepsize is proposed in [4] to solve integral and differential equations. The Haar wavelet based approach is effective in adapt-ing the solution process to irregularities in the solution. Hashish et al. [5] suggested a wavelet-Galerkin method to solvenonhomogeneous 2-D heat equation in finite rectangular domains. The test functions for the two-dimensional Galerkinmethod are scaling functions associated with certain wavelets. The method is adaptive to various boundary conditions withsimplified geometry constraints on the domain of the testing function. After Zhang and Benveniste [6] introduced a newprediction technique, so-called wavelet networks, a number of adaptive control techniques based on the idea of waveletnetworks were developed [7–11]. A common feature of these wavelet-based adaptive control methods is that the controllerconsists of a negative state feedback (proportional controller) and a nonlinear component approximated by a waveletnetwork. The nonlinear component of the controller is used to eliminate the nonlinearity of the original dynamical system

er Inc.

u).

Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31 23

so that the nonlinear system is reduced to either a linear system or a much simplified nonlinear system. Hence, the classicalcontroller design approaches can be applied [10]. There are other variants of wavelet-based adaptive controller designs inthe literature. In [12], a wavelet network modeled only with scale functions maps the input space of the uncertain nonlinearmodel to output space in wavelons, followed by an implementation of nonlinear adaptive predictive control. It is shown in[13] that a wavelet controller is effective in eliminating repetitive errors in disk drives by accurately approximating periodicfunctions so that the disks function smoothly without the presence of disturbances. A wavelet adaptive backstepping controlmethod is proposed in [14], in which the wavelet network serves as the system identifier in the backstepping controller bytaking advantage of the adaptive learning capability of the wavelet network. The output of the wavelet network is used inconstructing the robust controller recursively so that the system achieves the tracking performance at the desired attenua-tion level. The proposed wavelet-based controller design in this paper is different from the existing adaptive controller de-signs. Based on our observations, many controllers, such as a regulator, resemble the waveform of a wavelet. More precisely,many controllers as functions belong to L2ðRnÞ, i.e. the space of square-integrable functions. The wavelet networks are widelyused in system identification and control in which the wavelons are dilated and translated variants of the mother wavelet,which are localized in both space and frequency domains. This architecture has a powerful approximation property inapproximating any L2ðRnÞ functions to arbitrary precision by linear combinations of wavelets [15]. This motivates us to rep-resent a controller in terms of a subset of wavelet basis functions. Wang et al. [16] adopted this idea to control chaos in non-linear systems. However, they merely reported numerical simulation results without providing qualitative analysis on thestability of the wavelet-controlled nonlinear systems. It is the purpose of this paper to present a qualitative analysis onthe mechanism and stability properties of a wavelet approximated regulator when it is applied to LTI systems.

The rationale for our choice of using LTI systems to study the characteristics of a wavelet controller is two fold: Firstly, aLTI system is simple in structure, yet a benchmark system in the area of control. This allows us to study the mechanism of awavelet controller without being sidetracked by the complexity of the dynamical system itself; Secondly, whenever the con-trol objective is to stabilize a nonlinear system in the vicinity of one of its equilibria, such as a regulator, a common approachis to linearize the original system at the specific equilibrium point, which leads to a LTI model.

The rest of this paper is organized as follows. In Section 2, the structure of a wavelet controller is discussed.Stability bounds on the design parameters of a wavelet controller are presented in the main theorem of this paper,followed by a proof. Numerical simulations for wavelet-controlled LTI systems are reported in Section 3. Applicationsof a wavelet controller to control Lorenz chaos are discussed in Section 4, and the main results are summarized in theconclusion section.

2. Wavelet-based control for LTI systems

A wavelet function [15], w 2 L2ðRnÞ, satisfies

ZRn

jWðxÞj2

jðxÞj dx <1; ð1Þ

where W(x) is the Fourier transform of w(x). Both w(x) and W(x) are compactly supported or nearly compactly supported intheir respective domains. With appropriate shifts (tk) and dilations (dl) applied to the (mother) wavelet function w(x), oneobtains a denumerable family of wavelets

U ¼wlkðxÞjwlkðxÞ ¼ detðDlÞ1=2wðDlx� tkÞ; tk 2 Rn;

Dl ¼ diagðdlÞ;dl 2 Rnþ; l; k 2 Z

8<:

9=; ð2Þ

satisfying the frame property: there exist two constants, A > 0 and 0 < B < 1, such that, for all f 2 L2( Rn), the followinginequalities hold:

Akfk26

X/2Uj < /; f > j2 6 Bkfk2

: ð3Þ

As a result, the wavelet family U is dense in L2ðRnÞ.Under the assumption that a wavelet controller u is in L2(Rn), we will represent a controller in terms of a subset of wave-

lets in (2). It is noted that a wavelet controller is calculated from the sampled values of state variables. We will focus on thestability issues of the real-time implementation of a wavelet controller. We first identify the design parameters as the sam-pling interval T and the adaptive gain g from the iterative Eq. (7). We will prove that the wavelet-controlled LTI system isasymptotically stable at the origin. As a result of the proof, we obtain bounds on the design parameters of a wavelet control-ler. The significance of the proof is that we provide sufficient conditions and computable bounds for the design parametersthat guarantee stability of the wavelet-controlled LTI plant. Those bounds can be used in real-time to update the parametersonline so that the dynamical system is stabilized.

The state space representation of a LTI plant with a wavelet controller is given by

_x ¼ Axþ Bu; ð4Þ

24 Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31

where A and B are known constant matrices. The wavelet controller u satisfies

uðtÞ ¼ uðkTÞ if kT 6 t < ðkþ 1ÞT;

where

uðkTÞ ¼XN

i¼1

wiðkTÞwðDiðxðkTÞ � tiÞÞ; k ¼ 1;2; . . . ; ð5Þ

where wi’s are the wavelet coefficients, w is a multidimensional wavelet function, Di is a diagonal matrix known as the scal-ing matrix with scaling factors along its main diagonal, and ti is a shifting vector. Details of constructing Di and ti are dis-cussed in [3]. Since u 2 L2(Rn), a wavelet expansion for the controller u such as (5) exists because of (3). Therefore, thewavelet coefficients wi exist and they will be calculated recursively. It is observed from (5) that a wavelet controller is up-dated at the samples and stays constant between samples. A performance index is defined as E = xTPx/2, where P is a positivedefinite matrix. The matrix P is obtained from solving the associated Lyapunov equation

AcP þ PATc ¼ �BBT ; ð6Þ

where Ac = A � BK, and K is a feedback gain matrix so that Ac is a stability matrix. The existence of such a positive definitematrix P that satisfies (6) is guaranteed if {A,B} is a controllable pair [17].

We adopt the learning algorithm proposed in [6] to recursively minimize the performance of index E, which yields aniterative formula for the wavelet coefficient wi,

wiðkTÞ ¼ wiððk� 1ÞTÞ � gðkTÞBT PxðkTÞ � wðDiðxðkTÞ � tiÞÞ k ¼ 1;2; . . . ; ð7Þ

where T is the sampling interval and g is the adaptive gain. In what follows, we present the main result of this paper.

Theorem 1. Consider the linear time-invariant system (4) with a wavelet controller given by (5), and the wavelet coefficientwisatisfying the adaptive law (7). If the design parameters T and g satisfy

T < min �eðPÞ=kmax;1=lf g; ~g1 < g < g2;

where P is a positive definite matrix satisfying (6), kmax is the largest eigenvalue of the symmetric matrix H from (9), l is the largestsingular value of the state matrix A, �e is given by (15), g2 and ~g1 are given by (18) and (19) respectively. Then the closed-loopsystem (4) is asymptotically stable at the origin.

We prove the theorem by first deriving a bound on g that guarantees _V < 0 at the samples tk = kT, then we derive addi-tional bounds on g and a bound on T that guarantee _V < 0 between samples. All norms used in the proof are 2-norms.

Proof. Consider the Lyapunov function V = xTPx/2, then _V ¼ xT PAxþ xT PBu. Use the adaptive law for wi, let�x ¼ xðkTÞ; �wi ¼ wiððk� 1ÞTÞ; k ¼ 1;2; . . ., let �w

�� ��2 ¼PN

i¼1w2i , where wi = w(Di(x � ti)). We have _VðkTÞ ¼ �xT PA�xþ

PNi¼1�xT

PB �wiwi � g BT P�x��� ���2

�w�� ��2

< 0 which yields

gðkTÞ > �xT PA�xþXN

i¼1

�xT PB �wiwi

!,BT P�x��� ���2

kwk2� �

: ð8Þ

Our next step is to show that _V < 0 between samples, i.e. _VðsÞ < 0 for s 2 (kT, (k + 1)T). It is sufficient to obtain_VðkTÞ þ T €Vmax < 0, where €Vmax is an upper bound for €V over (kT, (k + 1)T). We find €Vmax as follows:

€V ¼x

u

� �T AT PAþ PA2þðAT Þ2P2

12 PABþ AT PB

12 BT AT P þ BT PA BT PB

" #|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

H

x

u

� �: ð9Þ

It can be verified that the largest eigenvalue of the symmetric matrix H satisfies kmax > 0. Therefore, €V 6 kmaxðkuk2 þ kxk2Þ,where u = u(kT) and x = x(s), s 2 (kT, (k + 1)T). We can solve for x from (4) over the interval [ kT, (k + 1)T], i.e.xðsÞ ¼ eAðs�kTÞ�xþ

R skT eAðs�tÞBudt. As such, an upper bound on kxk can be obtained in terms of the know quantities via the

triangle inequality and properties of matrix norms,

kxk 6 ek�xk þ ekBk � kuk=l;

where l is the largest singular value of the state matrix A. The above upper bound on kxk is obtained under the condition

T <1l : ð10Þ

Meanwhile, an upper bound on kuk is found to be kuk 6 kPN

i¼1 �wiwik þ gkBT P�xkk�wk2. Therefore,

€V 6 kmax e2k�xk2 þ 2a2kuk þ r2kuk2�

¼D €Vmax;

Fig. 1. Suitable quadratic curves for g.

Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31 25

where a2 ¼ e2kBkk�xk=l and r2 = e2kBk2/l2 + 1. Recall that we need _VðkTÞ þ T €Vmax < 0, which guarantees _V < 0 between sam-ples. We proceed as follows

_VðkTÞ þ T €Vmax < ðM þ eS2Þ � kBT P�xk2k�wk2 � 2a2kBT P�xkk�wk2 þ 2r2k�wk2kXN

i¼1

wiwikkBT P�xk !

e

!g

þ r2kBT P�xk2k�wk4eg2; ð11Þ

where e = kmaxT, M ¼ �xT PA�xþPN

i¼1�xT PB �wiwi, and S2 ¼ e2k�xk2 þ 2a2kPN

i¼1 �wiwik þ r2kPN

i¼1 �wiwik2. Therefore, _VðkTÞ þ T €Vmax < 0

implies an inequality for g,

r2ek�wk2g2 � ð1� 2d2eÞgþ M þ eS2

BT P�x��� ���2

�w�� ��2

< 0; ð12Þ

where d2 ¼ a2 þ r2kPN

i¼1 �wiwik�

=kBT P�xk. It shows in Fig. 1 the suitable quadratic curves for g as g has to be positive,which implies the following inequalities on e

e <1

2d2 ð13Þ

and

ð1� 2d2eÞ2 � 4r2eðM þ eS2Þ= BT P�x��� ���2

> 0

which can further be written as

4e2k�xk2e2=kBT P�xk2 þ 4Xe� 1 < 0; ð14Þ

where X ¼ d2 þ r2M=kBT P�xk2. An illustrative plot for the quadratic function in (14) is given in Fig. 2.

Fig. 2. Quadratic curve for e.

26 Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31

The positive zero in Fig. 2 is found to be e2 ¼ 1= 2Xþ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2 þ e2k�xk2=kBT P�xk2

q� �. Therefore, the condition e < e2

guarantees (14). Note that e2 is expressed in terms of known quantities so that it can be calculated explicitly. Consider (13),we choose

�eðPÞ ¼min 1=ð2d2Þ;1= 2Xþ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2 þ e2 �xk k2

BT P�x��� ���2

vuuut0BB@

1CCA

8>><>>:

9>>=>>;: ð15Þ

Therefore, e < �eðPÞ guarantees that (12) has a solution satisfying the condition illustrated by Fig. 1. We are ready to obtain anupper bound on the sampling interval T. From e < �eðPÞ or kmaxT < �eðPÞ, one has

T <�eðPÞkmax

: ð16Þ

Now, we solve for g from (12). The solution for g is

g1 < g < g2; ð17Þ

where

g1;2 ¼1� 2d2e�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2d2e� 2

� 4r2eðM þ eS2Þ= BT P�x��� ���2

r2r2e �w

�� ��2 : ð18Þ

Hence, condition (17) leads to _VðkTÞ þ T €Vmax < 0, therefore _VðsÞ < 0, for s 2 (kT, (k + 1)T). Recall that, in order to have_VðkTÞ < 0 (at the samples), g must satisfy (8), which can be written as

g > ð�xT PA�xþXN

i¼1

�xT PB �wiwiÞ=ðkBT P�xk2k�wk2Þ ¼ M=ðkBT P�xk2k�wk2Þ:

Therefore, we need

~g1 ¼maxM

BT P�x��� ���2

�w�� ��2

;g1

8><>:

9>=>;: ð19Þ

Then, condition (17) is updated to be

~g1 < g < g2: ð20Þ

However, condition (20) causes no conflict only if g2 > M= kBT P�xk2k�wk2�

which is shown as follows

g2 ¼1� 2d2e�

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2d2e� 2

� 4r2e M þ eS2�

= BT P�x��� ���2

r2r2e �w

�� ��2

¼2M þ 2eS2�

= BT P�x��� ���2

�w�� ��2

� �

1� 2d2e�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� 2d2e� 2

� 4r2e M þ eS2�

= BT P�x��� ���2

r :

Note that 1 � 2d2e 2 (0,1). We discuss the relation between g2 and M= kBT P�xk2k�wk2�

in two cases

(i) If M < 0, one immediately has M= kBT P�xk2k�wk2�

< 0 < g2.(ii) If M > 0, it is easy to see that

0 < ð1� 2d2eÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 2d2eÞ2 � 4r2eðM þ eS2Þ= BT P�x

��� ���2r

< 1;

therefore,

g2 > ð2M þ 2eS2Þ= kBT P�xk2k�wk2�

> M=ðkBT P�xk2k�wk2Þ:

Therefore, g2 > M= kBT P�xk2k�wk2�

is established. Finally, the conditions on the design parameters, g and T, coupled with a

condition on T from(10), are given by T < min �eðPÞ=kmax;1=lf g and ~g1 < g < g2, which guarantees _V < 0 for all t. h

Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31 27

Remark 1. The online feature of a wavelet-based controller design is readily seen from the proof of Theorem 1. The upperbound on T indicates the duration of time for the system to be stable with the amount of control calculated at the samples.Meanwhile, the bounds on g can be estimated via state samples according to (18)–(20) so that the adaptive gain g stayswithin the bounds to guarantee the stability of the wavelet-controlled LTI system.

Remark 2. The dimensional parameters of a wavelet network, i.e. the scaling matrices Di and the shifting vectors ti in (5), arecalculated to guarantee that the operative region (union of the compact supports of the wavelets) of the wavelet networkencompasses the state trajectories of the dynamical system. This requirement is crucial for an effective wavelet-based con-trol. A wavelet controller will be futile to control the state trajectories if the trajectories veer off the effective region of thewavelet network because the controller is zero outside this region. Di and ti can be calculated off-line if the dynamical systemis known a priori. Otherwise, a gradient-type approach may be used for an online calculation of the dimensional parameters,such as the algorithm proposed in [6]. Calculating Di and ti only requires the samples of the state trajectories right before thecontroller is activated. Therefore, online construction of a wavelet network can be done instantly without causing seriousdelays.

Remark 3. It is well-known that a wavelet network is capable of identifying disturbances or sudden changes present in thestate trajectories [2]. The disturbances are recorded in the wavelet coefficients (7), which are then used to recalculate thewavelet controller to eliminate the disturbances and stabilize the state trajectories. We will further demonstrate this obser-vation with numerical simulations.

Remark 4. The quantity k�wk2 in the denominator of the bounds (18) and (19) for the design parameters is never zero. This isguaranteed by the requirement that the compact support of the wavelet network covers the region where the state trajec-tories stay, see Remark 2.

Remark 5. Further effort can be made to find a global bound on T, i.e. an upper bound on T (theoretical bound) that is inde-pendent of �x(measurement of state samples). However, the local bounds on T given by (10) and (16) are more practical asthey can be computed explicitly and applied to adjust T online.

3. Numerical simulations

As a result of Theorem 1, we obtain theoretical bounds on the design parameters of a wavelet controller. In this section,we will demonstrate with numerical simulations that those bounds are achievable in practice. Single-input all-state feed-back and multiple-input all-state feedback LTI systems are considered in the numerical tests. Morlet wavelet is used inthe computation of a wavelet controller.

The first example is a marginally stable single-input system,

A ¼�4 6 1�6 6 26 �2 �3

264

375; B ¼

111

264

375:

In response to Remark 3 from the previous section, perturbations are added to the state trajectories after the system is sta-bilized, see Figs. 3 and 4. The wavelet network automatically identified the disturbances. As a result, the wavelet controller isconstructed to eliminate the disturbances and stabilize the state trajectories.

The sampling interval and the adaptive gain are chosen at T = 0.01 and g = 0.6 at the beginning of the simulation. The sam-pling interval T always satisfies the bound during simulation, and the adaptive gain g is updated to a value between 0.2 and0.4 in order to satisfy the bounds stated in Theorem 1, see Fig. 5. The simulation shows that the wavelet controller is effectivein stabilizing the LTI system whenever instability presents in the state trajectories. The state variables of the linear systemstay close to zero with the average error below 1e�16 after the controller is activated.

We also apply a wavelet controller to an unstable multiple-input system,

A ¼9 4 �7�4 1 22 2 �1

264

375; B ¼

1 00 10 0

264

375:

The controller for this LTI system is set to be u = [u1,u2]T, which has two control inputs. The controllers are activated at thebeginning of the simulation with T = 0.01 and g = 0.4. The adaptive gain g is updated occasionally. It stays at the chosen valuemost of the time during simulation; see Figs. 6 and 7.

0 5 10 15 20 25 30 35-50

0

50

Out

put x

0 5 10 15 20 25 30 35-50

0

50

Out

put y

0 5 10 15 20 25 30 35-100

0

100

Out

put z

Time

Fig. 3. Wavelet controlled marginally stable LTI system. The controller is activated at t = 20 s; a perturbation at the size of 10 is added to each state variableat t = 30 s.

15 20 25 30 35

-15

-10

-5

0

5

10

Con

trol I

nput

Time

Fig. 4. Wavelet controller.

20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time

boun

ds

Upper BoundLower Bound

Fig. 5. Upper and lower bounds for the adaptive gain g.

28 Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31

0 2 4 6 8 10 12 14 16 18 20-20

-10

0

10

Out

put x

0 2 4 6 8 10 12 14 16 18 20-5

0

5

10

Out

put y

0 2 4 6 8 10 12 14 16 18 20-5

0

5

Out

put z

Time

Fig. 6. Wavelet controlled unstable LTI system. The controller is activated at t = 0 s.

0 2 4 6 8 10 12-10

0

10

20

Con

trol I

nput

u1

0 2 4 6 8 10 12-10

-5

0

5

10

Con

trol I

nput

u2

Time

Fig. 7. Wavelet controllers.

Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31 29

4. Application to control Lorenz chaos

In this section, we apply the wavelet-based controller design to control Lorenz chaos. We choose to linearize the Lorenzequations at one of its equilibria, which results in a LTI model. A wavelet controller is then designed based on the structure ofthe LTI plant. After obtaining the design Eqs. (5) and (7) of the wavelet controller as well as the bounds for the design param-eters, we incorporate the wavelet controller back into the original Lorenz equations to eliminate chaos, i.e. the Lorenz systemis stabilized at an equilibrium point well into its usually chaotic regime.

The Lorenz equations are usually written as

_x ¼ Pðy� xÞ;_y ¼ Rx� y� xz;_z ¼ xy� bz;

ð21Þ

where P is analogous to the Prandtl number, b is a spatial constant, and R is analogous to the Rayleigh number. As in moststudies, we choose P = 10 and b = 8/3 in what follows. It should be noted that the control input is incorporated with the

40 41 42 43 44 45 46 47 480

0.1

0.2

0.3

0.4

0.5

0.60.7

0.8

0.9

Time

boun

ds

Upper BoundLower Bound

Fig. 8. Upper and lower bounds for the adaptive gain g.

0 5 10 15 20 25 30 35 40 45 50-40

-200

2040

X

0 5 10 15 20 25 30 35 40 45 50-40-20

02040

Y

0 5 10 15 20 25 30 35 40 45 500

20

40

60

Z

Fig. 9. Wavelet-controlled Lorenz system. The wavelet controller is activated at t = 40 s. A perturbation vector, dx = [16,12], is added to the states at t = 44 s.

39.5 40 40.5 41 41.5 42 42.5

-5

0

5

10

cont

rol u

43.5 44 44.5 45 45.5 46 46.5-10

-5

0

5

10

cont

rol u

Time

Fig. 10. The wavelet controller for the Lorenz system. (a) Activated at t = 40 s to stabilize the flow; (b) automatically re-activated at t = 44 s in response tothe perturbations.

30 Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31

Y. Wu, J.S. Tanner / Applied Mathematics and Computation 218 (2011) 22–31 31

Rayleigh number R, which represents the heat source [10]. We first linearize the Lorenz system at one of its equilibriumpoints, (xss,yss,zss), which can be found explicitly (we only consider the two nontrivial steady states). After linearization,the system Jacobian matrix and the input Jacobian matrix of the Lorenz system are found to be

A ¼�10 10 0R� zss �1 �xss

yss xss � 83

264

375; B ¼

0xss

0

264

375: ð22Þ

We choose R = 40 so that the Lorenz system (21) is known to be chaotic. Correspondingly, two of the eigenvalues of matrix Aare in the right half of the complex plane. The Wavelet controller is calculated from (5) and (7) with matrices A and B from(22) as well as the sampled values of the states. We also compute the bounds for the wavelet controller from Theorem 1,which are used to adjust the adaptive gain g and sampling interval T during the control process. Then, the wavelet controlleris inserted in the Lorenz system (21), i.e.

_x ¼ Pðy� xÞ;_y ¼ ðRþ uÞx� y� xz;_z ¼ xy� bz:

We choose T = 0.01. The adaptive gain g is updated during the simulation. It varies from 0.25 to 0.45, see Fig. 8. Numericalresults show that the wavelet-based controller design for LTI systems is effective in controlling chaos and eliminating dis-turbances, Figs. 9 and 10. We also observe that, although the wavelet controller is calculated from the linearized system (22)at a stationary point of (21), it is effective for controlling chaos even if the wavelet controller is activated outside a smallneighborhood of the stationary point, as long as the compact support of the wavelet network covers the state space wherethe state trajectories stay. This result provides evidence for a wavelet controller capable of improving the basin of attractionfor the steady states of the Lorenz system.

5. Conclusion

In this paper, we investigated the stability properties of wavelet-controlled LTI systems. The objective of this work is tostudy the mechanism of a wavelet network for the approximation of a regulator. We choose the LTI systems for the studybecause these systems do not add complexity to the already complicated wavelet networks. It helps us understand the func-tionality of a wavelet network when it is interacting with a dynamical system. We obtained stability bounds for the designparameters of a wavelet controller. Numerical results further showed that those bounds for the design parameters wereachievable. Unstable and marginally stable LTI systems were tested with proposed wavelet controller design. We concludethat a necessary condition for an effective wavelet-based control is that the compact support of the wavelet network con-tains the state space where state trajectories evolve. The effectiveness of a wavelet controller was further demonstratedthrough controlling Lorenz chaos.

Acknowledgments

This work is supported in part by the Georgia Southern University Faculty Research Grant. The authors are grateful for thereviewers’ invaluable suggestions.

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