Wavelet Differential Neural Network Observer - Semantic Scholar
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 9, SEPTEMBER 2009
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Wavelet Differential Neural Network Observer Isaac Chairez Abstract—State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with a state observation problem when the dynamic model of a plant contains uncertainties or it is completely unknown. Differential neural network (NN) approach is applied in this uninformative situation but with activation functions described by wavelets. A new learning law, containing an adaptive adjustment rate, is suggested to imply the stability condition for the free parameters of the observer. Nominal weights are adjusted during the preliminary training process using the least mean square (LMS) method. Lyapunov theory is used to obtain the upper bounds for the weights dynamics as well as for the mean squared estimation error. Two numeric examples illustrate this approach: first, a nonlinear electric system, governed by the Chua’s equation and second the Lorentz oscillator. Both systems are assumed to be affected by external perturbations and their parameters are unknown. Index Terms—Continuous systems, learning schemes, neural network (NN), sliding-mode observers, state estimation, wavelet approximation.
I. INTRODUCTION ODELING theory (using multiple physical principles, chemical laws, etc.) represents the most usual manner to formalize the systems dynamics knowledge. However, in several real situations, modeling rules may fail to generate acceptable reproductions of reality [1]. In those cases, nonparametric identification (using adaptive methods) can be successfully applied to overcome possible modeling deficiencies. Within the nonparametric identification framework, function approximation techniques play an important role avoiding the necessity of accurate mathematical plant description [2]. Among others, wavelets have been applied [3] (multiscaled signals analysis and synthesis, time-frequency time-series analysis in digital and continuous processing, etc.) to approximate nonlinear function (by numerical reconstruction) and to solve ordinary and partial differential equations [1]–[3]. Wavelets theory is well posed to approximate nonlinear functions with local nonlinear sections and fast variations using two main properties: finite support and self-similarity. Besides, neural networks (NNs) have become an attractive tool for modeling complex nonlinear systems. NNs are particularly powerful for handling large scale problems. However, NN implementation suffers from lack of efficient constructive approaches: choosing network structures and determining the neuron parameters. It has been proven that artificial neural net-
M
Manuscript received October 19, 2007; revised March 01, 2009 and May 16, 2009; accepted May 21, 2009. First published August 11, 2009; current version published September 02, 2009. The author is with the Professional Interdisciplinary Unit of Biotechnology, UPIBI-IPN, México D.F., ZP. 07430 México (e-mail: [email protected]. mx). Digital Object Identifier 10.1109/TNN.2009.2024203
works (ANNs) can approximate a wide range of piecewise nonlinear functions to any desired degree of accuracy [4]. It is generally accepted that NN training algorithm plays an important role in NN applications. For instance, in conventional gradient-descent type, sensitivity of unknown system is required during the online training process [5]. However, it is difficult to acquire good information for unknown or highly nonlinear dynamics. Besides, local minimum of performance index remains to be the main inconvenience in backpropagation algorithm [6]. Radial basis function (RBF) networks are often used in order to improve the NN learning efficiency. Poggio and Girosi [7], [8] analyzed several network architectures to determine their approximation abilities and pointed out that RBF networks possess the best approximation capability. These advantages are further strengthened with the introduction of a wavelet into NN structures [9]. The main advantage of wavelet networks over similar architectures (such as multilayer perceptrons and networks of RBF using sigmoid functions as kernel [6]) is the possibility to optimize wavelet network structure by means of efficient deterministic constructive algorithms [10]. Also, as noted in [11] and [12], NNs have limited ability to characterize local features such as discontinuities in curvature, jumps in objective function, and others. These local features, which are located in time and/or frequency, typically embody important information of the system such as aberrant process modes or faults [11]. Bottou and Vapnik [13] have addressed that improved localized modeling can aid both data reduction (or compression) and subsequent classification tasks that rely on accurate representation of local features. Zhang and Benveniste [9], Zhang [10], and Bakshi and Stephanopoulos [14] improved upon this weakness of the NN by developing wavelet neural networks (WNNs), which are a type of feedforward NN. The WNN is based on global mean square error (MSE) as well. The existence of wavelet-based model selection methods is more important, as addressed in [12] (e.g., [15]) focused on data denoising and the avoidance of excessive number of wavelet coefficients/bases in their approximation models. In this study, the family of basis functions for the RBF network is replaced by an orthogonal basis (i.e., the scaling functions in the theory of wavelets) to form a WNN. Roughly speaking, the WNN is considered as a kind of RBF networks [16] and possesses more advantages than the general networks as was remarked in [10], [17], and [18]. These WNNs have important characteristics such as faster convergence, avoidance of local minimum, easy decision, and adaptation of structure [19], [20]. There are many research results [18], [19], [21], [22] dedicated to developing new adjustment laws for the WNN weights. Here, it should be noted that all WNN analyses were performed over static NNs described by recurrent algorithms. Today, many researchers have developed WNN applications, which combine the NN capability to learn using processed information [23] and wavelet decomposition capability [24], [25].
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These applications include identification and control of uncertain dynamic systems (see [16], [26], and [27] for more details). In [9], the notion of a wavelet network was proposed as an alternative to conventional feedforward NN. Approximate arbitrary nonlinear functions based on the wavelet transformation theory and a backpropagation algorithm was adapted for WNN training. Zhang et al. [16] described a wavelet-based NN for function learning and estimation. The structure of this network was similar to the RBF network except that radial functions were replaced by orthonormal scaling functions. Also, Zhang [10] presented WNN construction algorithms for the purpose of nonparametric regression estimation, using a constructive method and defining a complete basis for the state space. From the point of view of function representation, the traditional RBF networks can approximate any function that is in the space spanned by the family of basis functions. However, basis functions are generally not orthogonal and redundant. That means that RBF network representation for a given function is not unique and is probably not the most efficient. At this point, all WNN applications have been developed for the so-called static schemes (like multilayer structure). However, exploiting the fact of being universal approximations, it may straightforwardly substitute continuous system uncertainties by NNs containing large number of unknown parameters (weights) to be adjusted. In general, this class of NNs is known as differential neural networks (DNNs) and it possesses two important characteristics: its adjustable parameters may appear as linear elements in the NN description and such NN may be modified using differential equations [6], [28]. This approach transforms the original function approximation problem into a nonlinear robust adaptive feedback design. The DNN approach avoids many problems related to global extremum search and converts the learning process into a particular feedback scheme [28]. If the mathematical model of a process is incomplete or partially known, the DNN theory provides an effective instrument to attack a wide spectrum of problems such as nonparametric trajectory identification, state estimation, trajectories tracking, etc., [29]. In view of DNN continuous structure, more detailed techniques should be applied to solve important questions on the new NN proposal (convergence, for example). Lyapunov’s stability theory (especially the so-called controlled Lyapunov theory) has been used within the NN framework [28], [30]. Actually, this is the main tool to prove DNN improvements on the estimation problems or the control action design, even though there exists a general trend to increase the number of nonlinear systems for which the aforementioned works can be applied. As a consequence, there are novel results on stability, convergence to arbitrarily small sets, and robustness to modeling imperfections. In this paper, we have developed the application of the wavelet theory over the DNN structures (state estimation and nonparametric identification). The new approach known as differential wavelet neural network (DWNN) is completely presented, including observer structures, corresponding convergence schemes, and continuous learning algorithms. The effectiveness of the novel DNN topology is shown by two different numerical examples: the Lorentz oscillator and the Chua circuit.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 20, NO. 9, SEPTEMBER 2009
II. DNN OBSERVER DESIGN This section describes the class of uncertain systems to be treated using the DWNNs, their properties, and the type of external perturbations affecting their nonlinear dynamics. Section II-A describes the concept of a wavelet differential network and details its capability to approximate a certain class of nonlinear ordinary differential equation (ODE) dynamics. Section II-C describes the state observer structure, the wavelet approximation, etc. Sections II-D and II-E describe the weights dynamics which permits DWNN reproduction for nonlinear dynamics. Also, the weights bounded behavior is shown by means of a special kind of a controlled Lyapunov-like analysis. A. Uncertain Nonlinear Systems Dynamic uncertain nonlinear system to be analyzed in this paper is described by the following set of ordinary differential equations: (1) describes the system trajectories in System state , that is, a closed set where the system state remains during all system evolution time. Online available output signal is a linear combination (performed by the matrix with constant entries) of state components, and is a bounded control action belonging to the following set:
T T
Admissible set includes state feedback control actions, even linear functions, or discontinuous designs (sliding-mode controllers, for example). Output constant is assumed to be a priori known. real matrix satisfies a local LipVector function schitz condition, i.e., there exist two finite positive conand such that stants is accom, and , . (This plished for any , .) Nonlinear is, in fact, a direct consequence of and system is affected by state/parameter uncertainties measurement disturbances . Both unknown perturbations are bounded in ellipsoidal sense [31], that is, there are two and a pair of positive-definite matrices positive scalars , such that the following inequalities are fulT . Here, it should filled: be admitted that perturbation upper bound keeps the inside the positive nonlinear system dynamics cone defined by the Lipschitz condition for all . Last supposition keeps valid the existence and uniqueness conditions of perturbed nonlinear dynamics (1). Besides, nonlinear dynamics is bounded at the zero point condition given as . Nominal output ) is unisystem (without external perturbations, i.e., formly observable [32].
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T
B. Wavelet Differential Neural Network Approximation Nonlinear function approximation using wavelets is based on the following properties. Consider the closed space , with the following properties: space sequence is nested 1)
2) ( is the intersection operator); , ( is the direct sum operator); 3) 4) . It is seen that the whole space can be represented by
for some
. Let
be a basic scaling function such that with , such that then there exists a basic function with . is called an orthogonal wavelet if the set of funcFunction is an orthonormal basis of , that is tions
Several kinds of wavelet basis [33] have been developed. For example, the Daubechies, Morlet, Mexican Hat, Meyer families, and many others (in fact, further development of new families of wavelet bases continues to receive considerable attention from in scientific community). Now let us consider a function . In view of previous properties, can be rewritten as [24]
with a weighting constant vector. Remark 1: Appropriate selection of wavelet functions is an important task to construct an adequate approximation of nonlinear functions. Many wavelet functions have been reported in literature [33] that have remarkable results in approximating nonlinear unknown functions. Which one is the most suitable basis in practical application depends on particular design specifications. parameters in wavelet netRemark 2: work design are closely related to the quality approximation , although the wavelet network has been demonstrated to be more effective than sigmoid-based NNs to reproduce uncertain nonlinear functions satisfying the Lipschitz condition, particularly, if the associated Lipschitz constant is large. In view of the WNN approximation capabilities, notice could be always presented (by Stone–Weisstrass that and Kolmogorov theorems [34]) by the composition of and modeling error nominal terms [as is usual when a model-free approximation is applied and the error is given by (2)]. Nonlinear dynamics (1) can be exposed as (3) where the nominal dynamics may be selected according to a desired design (in this case, by means of a combination of wavelet and a special kind of NN with continuous dynamics). Here, the vector of parameters should be adjusted to obtain the best and nonlinear possible matching between the nominal dynamics (in other words, to reduce as much as possible, the presence). In view of the nonlinear dynamics properties (Lipschitz condition and the class of admissible controls), the following upper bound for the error modeling is observed:
Last expression is called a wavelet series expansion of . Based on this series expansion, a wavelet network has the following mathematical structure: T
that can be used to approximate any nonlinear smooth function with the adequate selection of integers , , , and , where T
T T
where the estimation error
T
This wavelet expansion suggests an alternative for an NN representation. Following the Stone–Wiestrass theorem, if is the NN approximation error, then for any arbitrary positive constant , there such that are some constants
(4) is defined by
According to the DNN approach, nominal dynamics is proposed using the Stone–Weisstrass theorem and its corresponding extension for WNN. This nominal dynamics (following the WNN theory) is suggested to be
(2) for all
T
, the
. In case of argument
should be modified to
(5)
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, , . Matrices of activation functions and are usually constructed with components defined by sigmoid functions. In this study, these functions are changed to wavelet functions to build an orthogonal and nonredundant basis for the approximation of the uncertain nonlinear function . It is easy to prove (as a consequence of their bounded structure) that each component in activation functions satisfies sector conditions given by
eter selection, while Section II-D is related with the quality of the suggested state estimation scheme. D. DNN Parameter Adjustment Procedure provide the Usually in NN, the weights function approximation capacity. Nonlinear weight updating (learning) law is described by following matrix differential equations: T T
T
T
T
if if T
(7)
T
Last assumption is a consequence of continuity property for . continuous wavelet functions, which belong to class C. Observer Structure Following the Luenberger-like observers introduced in [28] and the sliding-mode observer proposition given by [35], the DNN estimator structure is suggested as follows [36]:
T is defined by Matrix with a small positive scalar value (typically 0.01). Matrices and represent the distance between to their corresponding best the current values of fitted values and , that is, . Matrices are symmetric positive-definite matrices T . Time-varying function is the output . Variables error defined as are time-varying functions to adjust the learning rates. Matrices T are the positive-definite solutions for the Riccati equations
(6)
T
. Vector is the observer state. and are adaptive parameters that should be adjusted to reproduce (as well as possible) the nominal dynamics given in (5). Constant matrices are, correspondingly, the output-based correction matrices for the linear and variable structure terms. Funcassociated to sliding-mode obtion server is defined as where
T
T T
T
T
T
T
(8) Nonlinear differential (6) has discontinuous right-hand side. Hence, the existence and uniqueness properties for this equation are understood in the sense of differential inclusion theory [37]. DNN state estimator must solve the following two different tasks simultaneously. should converge as close 1) WNN parameters as possible (depending on the values of the external uncer. tainties) to the most adequate but unknown values 2) DNN state should converge asymptotically to a small zone close to (supplied by a ellipsoid upper bound) with size depending on the presence of the external perturba, tions with the ellipsoid bounds given above ( , and ). Both processes (the DNN parameter adjustment and the observer state convergence) can be achieved by means of a suitable Lyapunov function and the adaptive control theory. Section II-C deals with the adjustment process for the adequate DNN param-
and are positive definite as well. In Here fact, they must be selected (over a large set of possible values) just to ensure the existence of a solution for (8). T Remark 3: Special class of Riccati equation has positive solution if and only if the following four conditions are observed: — Condition 1: matrix is stable. is controllable. — Condition 2: pair — Condition 3: Pair is observable. — Condition 4: matrices should be selected to satisfy the following inequality: T
T
T
T
Last condition restricts the largest eigenvalue of avoiding the inexistence of Riccati equation positive solution. State estimation problem for uncertain nonlinear systems analyzed in this study could be stated as follows.
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Considering the nonlinear system conditions (external perturbation presence, uncertain mathematical description, etc.) for , with an adequate any admissible control strategy selection of matrices and with the WNN observer structure supplied with the adjustment law (7) (including the se), and the upper bound for the averaged lection of estimation error defined as (9) T must be obtained, and if it is possible, reduced to its lowest achievable value, using the adjustment of the free parameters included in the WDNN structure. This averaged error is quite similar to the mean squared error that is commonly used in NN theory. State estimation process solution actually means to select correctly the matrices . State estimation result is given in the following. Theorem 1: If there exist positive-definite matrices and positive constant such that the matrix Riccati (8) has positive-definite solutions, then the WDNN observer (6) supplied with the learning law (7) and using any guarantying that the closed-loop matrix is stable, matrix that is
is Hurwitz
Here, it is also necessary to use a numerical interpolator algorithm to manage the sampled data as piecewise continuous signals. This is an important restriction for the observer design but, on the other hand, the same problem appears with any other NN structure, where the training phase is the most crucial stage in this class of approaches. Obviously, the data must be sampled with fixed frequency containing enough information to process the parametric identification [39] and fulfilling the corresponding persistent excitation condition [39]. In view appearing linearly with respect to the activaof tion functions, the application of the matrix least mean square algorithm is suggested (see [40] and [39]) to attain the training stage. To do that, (3) should be represented in its integral form
which, in turn, may be represented in a linear regression form
(10)
and selecting T
(11)
provides the following upper bound for the averaged (or mean squared) norm of the state estimation error:
where
(12) Matrix least square estimate
of
is given by
T
(13) Proof: Proof of this theorem is given in the Appendix . An important remark can be introduced here for the unperturbed case, i.e., when noises and the modeling error are absent. Corollary 1: WDNN observer estimation error norm (12) converges asymptotically to zero when the state and output uncertainties vanish ( and ), the admissible control ellipse , and has a support function that crosses at zero point . no modeled dynamics is absent E. DNN Training Algorithm To complete the learning algorithms (7), the knowledge of the nominal matrices incorporated in is required. Nevertheless, they are actually unknown. The so-called training process consists of the identification for suit. This may be done offline able approximation of (before the beginning of the state estimation) using some avail. able or even fictitious [38] experimental data
T
(14)
or using the differential form T
T T T
(15)
It is possible to show [41] that the training quality is given by T
T
T
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The upper bound for the training process is given by
where and
and
is the upper bound of is T
T
(20)
then the weight time trajectories (7) are bounded, and moreover, for any , it may be concluded that Therefore, the learning laws become T
T
(21) (16) which are numerically realizable, because the weights are now available. Adaptive learning algorithms help to reject perturbation effects, and at the same time, to enforce the estimation error inside the region defined by (12). Considering the external perturbation effect, nominal weights obtained by the training method cannot be substituted in the observer structure. This can be explained better considering that the system is assumed to be affected by different uncertainties (parametric, noises, etc.). As a consequence, adaptive weight laws must be applied to adjust the observer structure depending on the current noises values, modeling error, etc. In fact, the application of (16) provides a certain class of robustness property to the state estimator (6).
, the suggested Lyapunov function Proof: a) For has the following structure: T
Here Since the function ferentiation) T
T
F. Bounded DNN Weights Behavior Now it is possible to consider the stability analysis of the learning laws (expressed as matrix differential equations). This part of the study may be conducted using new Lyapunov functions and for the weights and , respectively. Here, let us apply the conventional second Lyapunov method to determine the upper bound for the weights dynamics. Here, it is necessary to define the following Lyapunov function:
T
To guarantee the nonincreasing property for of the observer trajectories, inequality (19) must be observed. That condition . The last inequality permits avoids the finite-time escape of the existence of several learning law schemes depending on the structure. b) Considering the Lyapunov function given by (17), a similarly analysis can be done for the weights
(17) and the constant (18) Boundedness property for the weights trajectories is shown in the following. and satisfy the folTheorem 2: If the parameters lowing inequalities: T T
T T
T
T
(19)
is differentiable, one has (by direct dif-
where
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Following a similar procedure for , one gets that if inequality (20) is achieved, then it is implied that , and . Therefore, one may conclude that hence,
This completes the proof. Restriction (19) permits construction of many different learning schemes. In fact, the number of possible adjustment , , just need to be a rules is unbounded because , and fulfills (19) and (20) nonnegative function simultaneously. One example demonstrating the flexibility of the suggested approach for the weights dynamics adjustment is detailed below. Example 1 (Sigmoid Learning Law): Define
(22) It is easy to check that
If
Fig. 1. Chua’s circuit state x DNN, and EKF.
trajectory and three observers results: WDNN,
(the voltage across well-known nonlinear circuit is given by the capacitor and the parallel nonlinear resistance ), (the voltage across capacitor and the inductor ), and (the current through the inductor ). The direct application of Kirchoff’s laws leads to the next set of nonlinear ODEs
satisfies the differential inclusion T
(24)
we get T
(23)
Hence, by the previous theorem, it may be guaranteed that is bounded. In the same way, a similar exercise for can be obtained. It is important to note that learning laws should be adjusted online, i.e., any changes on weight values imply . a corresponding change in the time-varying parameter Then, the adaptive process is carried out in two steps: the first one by the learning law application and the second one by the , designing. learning law rates Here, it must be understood that there is a great number of possible solutions for the inclusion of (23). However, the case where the equality is attained can be taken (for numerical simulations). III. SIMULATION RESULTS This section shows the numerical results for two examples using the observation technique suggested in this paper. For each example, a comparison between the WDDN, DNN, and extended Kalman–Bucy observers is included. A. Chua’s Circuit Chua’s circuit is an interesting electronic system presenting bifurcations and chaotic phenomena such as double scroll and double hook. This special behavior depends on the selection of the electronic components. The mathematical model [42] for the
Parameters of the circuit (24) are , , , , and . Let us consider that is measurable and just the voltage on capacitor of magnitude perturbed with pseudowhite noise ( ). Local observability for with uniform distribution in Chua’s circuit is ensured using [43]. This is necessary before the WDNN observer can be applied. State estimation results are depicted in Figs. 1 and 2. There, it is shown that complete state is successfully reproduced by the as the output signal. Moreover, acWDNN observer using ceptable reproduction of unknown states is reached in a short time (less than 5 s). Figs. 1 and 2 also show the comparison between the NN-based observers using wavelet and sigmoid activation functions. Actually, it is not sufficient to compare one NN-based technique with another NN-based technique. The true test of usefulness and benefit is that NN-based technique does a better job than a good conventional (non-NN-based) technique. Therefore, an additional comparison between WDNN estimator and a standard technique [extended Kalman filter (EKF)] was included in Figs. 1 and 2. As could be seen, the considered observers seem to be identically efficient. Fig. 3 demonstrates the time evolution of the mean squared estimation error (12). Here, it could be noted that better performance is obtained by the observer using wavelet functions, that is, the mean squared error converges faster to zero when the WDNN is used.
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TABLE I COMPARISONS BETWEEN DIFFERENT ESTIMATION ERROR VALUES (DNN, WDNN, AND EKF) FOR THE CHUA CIRCUIT AT DIFFERENT TIMES
WDNN, DNN, and EKF at different times. In Table I, better performance obtained by the WDNN is clarified. Fig. 2. Chua’s circuit state x DNN, and EKF.
B. Lorentz Oscillator trajectory and three observers results: WDNN,
Fig. 3. Time evolution of estimation error (12) for the Chua circuit estimation process. Comparison between WDNN and EKF observers.
Lorentz model is used for the fluid conviction description especially for some feature at atmospheric dynamic [44]. The controlled model is described by the following set of ODEs
where , and represent the measures of fluid velocity and horizontal and vertical temperature variations. These might seem to be very simple equations, but due to the nonlinear terms, they cannot be solved analytically. Parameters , , and are positive parameters that represent the Prandatl and Tayleigh is the geometric factor, correspondingly. If numbers, while , the origin is a stable equilibrium point. If , the system has two stable equilibrium points with components given by and one unstable equilibrium point (the origin). If , three equilibrium points become unstable. Chaotic behavior , , and . is defined with The method suggested in this paper shows an appropriate performance even when parameters , , and vary with time in a small region (this characteristic was proposed to simulate the ). Lorentz oscillator trajectories are completely presence of reproduced after a short time (less than 3 s) as is shown in Figs. 4 and 5. These results confirm the improvement provided by the suggested wavelet observer. Table II shows the time evolution of the mean squared estimation error (12) for the WDNN, DNN, and EKF at different times. In Table II, better performance obtained by the WDNN is clarified. IV. CONCLUSION
Fig. 4. Lorentz oscillator x DNN, and EKF.
trajectory and three observers results: WDNN,
Two main improvements are obtained using the waveletbased observer: less mean squared error and faster convergence in earlier stages of the estimation process. Table I shows the time evolution of the mean squared estimation error (12) for the
This paper presents the application of sliding mode theory, DNN, and wavelet approaches for the nonparametric modeling of unknown models belonging to the given class of nonlinear systems described above. This observer seems to be an interesting tool in the numerical reconstruction for unmeasurable information even if the mathematical model of the nonlinear system is uncertain. Parameters involved in the WDNN-observer structure provide not just the stability for the estimation algorithm but also they are shown (by the formalism supported in the Lyapunov
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APPENDIX Proof: The dynamics of ODE:
is governed by the following
(25) and . To construct the converge algorithm, define the energetic function (Liapunov-like) as where
Fig. 5. Lorentz oscillator x DNN, and EKF.
T
trajectory and three observers results: WDNN,
T
T
T
. Its time derivative is T
T
T
T
T
T
Fig. 6. Time evolution of estimation error (12) for the Lorentz oscillator estimation process. Comparison between WDNN and EKF observers. T
TABLE II COMPARISONS BETWEEN DIFFERENT NORMS OF ESTIMATION ERROR (DNN, WDNN, AND EKF) FOR THE LORENTZ OSCILLATOR AT DIFFERENT TIMES
T T
T
T
T
T T
T
T T T
(26)
Using (25) implies T T T
method) that they remain bounded with the specific selection of some parameters related to the learning rate. Moreover, strength of the proposed method is demonstrated by two computer simulations of well-known complex nonlinear models: the Chua circuit and the Lorentz oscillator. Considering the class of admissible control described in this paper, the WDNN observer proposed here could be used in the output-based control algorithms.
T
(27) T T Notice that T . Let us esT timate the rest of the terms using the matrix inequality T T T valid for any and
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T
T
any T
T
T
T
T
T T
T T T
T
T
T
T
T
T
T
T
T T
T
T
T
T
T T
T
T
T
T
T
T
T
T
T T
T
T
T
T
T
T
T T
(28) On the other hand, the term
T
T
T
T
T
T T
T T
could be analyzed by
T
T T T T
T T
T
Besides, if the following parameters are chosen as , then
T
T T
T T
T T
T T
T T
By means of the following inequality and the
selection:
T
Finally, using the adaptive laws given for each DNN weight maT . After the integration of trices, we have the previous inequality, , dividing over and taking the upper limit, last inequality becomes
T
and using the Jensen’s inequality, we have
T
So, the function
If we solve the previous inequality, the theorem statement is proved.
is bounded now as T
T T
T T
REFERENCES T T
[1] Y. Meyer, Wavelets and Applications. Berlin, Germany: SpringerVerlag, 1992. [2] D. Hartmut and H. J. Hartmut, Fundamentals of Wavelets. New York: Wiley, 1997, pp. 101–146.
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[3] C. Jaideva, G. Andrew, and K. Chan, Fundamentals of Wavelets. New York: Wiley, 2001, pp. 345–390. [4] O. Omidvar and D. L. Elliott, Neural Systems for Control. New York: Academic, 1997. [5] F. J. Lin, W. J. Hwang, and R. J. Wai, “Ultrasonic motor servo drive with on-line trained neural network model-following controller,” Proc. Inst. Electr. Eng. Electr. Power Appl., vol. 145, no. 2, pp. 105–110, 1998. [6] S. Haykin, Neural Networks. A comprehensive Foundation. New York: IEEE Press, 1994. [7] T. Poggio and F. Girosi, “Networks for approximation and learning,” Proc. IEEE Conf. Decision Control, vol. 78, no. 9, pp. 1481–1497, 1990. [8] T. Poggio and F. Girosi, “Regularization algorithms for learning that are equivalent to multilayer networks,” Science, vol. 247, pp. 978–982, 1990. [9] Q. Zhang and A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 889–898, Nov. 1992. [10] Q. Zhang, “Using wavelet network in nonparameters estimation,” IEEE Trans. Neural Netw., vol. 8, no. 2, pp. 227–236, Mar. 1997. [11] J. Jin and J. Shi, “Feature-preserving data compression of stamping tonnage information using wavelets,” Technometrics, vol. 41, no. 4, pp. 327–339, 1999. [12] L. Martell, “Wavelet-based data reduction and de-noising procedures,” Ph.D. dissertation, Dept. Electr. Eng., North Carolina State Univ., Raleigh, NC, 2000. [13] L. Bottou and V. Vapnik, “Local learning algorithms,” Neural Comput., vol. 4, pp. 888–900, 1992. [14] B. Bakshi and G. Stephanopoulos, “Wave-net: A multiresolution, hierarchical neural network with localized learning,” AIChE J., vol. 39, no. 1, pp. 57–81, 1993. [15] H. Saito, “Wavelets in geophysics,” in Simultaneous Noise Suppression and Signal Compression Using a Library of Orthonormal Bases and the Minimum Description Length Criterion. New York: Academic, 1994, pp. 299–324. [16] J. Zhang, G. Walter, and Y. Miao, “Wavelet neural networks for function learning,” IEEE Trans. Signal Process., vol. 43, no. 6, pp. 1485–1496, Jun. 1995. [17] L. Jiao, J. Pan, and Y. Fang, “Multiwavelet neural network and its approximation properties,” IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1060–1066, Sep. 2001. [18] J. Xu and Y. Tan, “Nonlinear adaptive wavelet control using constructive wavelet networks,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 25–27. [19] C. P. Bernard and J. J. E. Slotine, “Adaptive control with multiresolution bases,” in Proc. 36th IEEE Conf. Decision Control, San Diego, CA, 1997, pp. 3884–3889. [20] A. Ikonomopoulos and A. Endou, “Wavelet decomposition and radial basis function networks for system monitoring,” IEEE Trans. Nuclear Sci., vol. 45, no. 5, pp. 2293–2301, Oct. 1998. [21] N. Sureshbabu and J. A. Farrell, “Wavelet-based system identification for nonlinear control,” IEEE Trans. Autom. Control, vol. 44, no. 2, pp. 412–417, Feb. 1999. [22] R. M. Sanner and J. J. E. Slotine, “Structurally dynamic wavelet networks for the adaptive control of uncertain robotic systems,” in Proc. IEEE Conf. Decision Control, New Orleans, LA, 1995, pp. 2460–2467. [23] Y. C. Pati and P. S. Krishnaprasad, “Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations,” IEEE Trans. Neural Netw., vol. 4, no. 1, pp. 73–85, Jan. 1993. [24] B. Delyon, A. Juditsky, and A. Benveniste, “Accuracy analysis for wavelet approximations,” IEEE Trans. Neural Netw., vol. 6, no. 2, pp. 332–348, Mar. 1995. [25] T. Lindblad and J. M. Kinser, “Inherent features of wavelets and pulse coupled networks,” IEEE Trans. Neural Netw., vol. 10, no. 3, pp. 607–614, May 1999.
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[26] Y. Qussar, I. I. Rivals, L. Personnaz, and G. Dreyfus, “Training wavelet networks for nonlinear dynamic input-output modeling,” Neurocomputing, vol. 20, pp. 173–188, 1998. [27] R. J. Wai and J. M. Chang, “Intelligent control of induction servo motor drive via wavelet neural network,” Electric Power Syst. Res., vol. 61, no. 1, pp. 67–76, 2002. [28] A. Poznyak, E. Sanchez, and W. Yu, Differential Neural Networks for Robust Nonlinear Control (Identification, State Estimation and Trajectory Tracking). Singapore: World Scientific, 2001. [29] F. L. Lewis, A. Yesildirek, and K. Liu, “Multilayer neural-net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Netw., vol. 7, no. 2, pp. 1–11, Mar. 1996. [30] G. Rovithakis and M. Christodoulou, “Adaptive control of unknown plants using dynamical neural networks,” IEEE Trans. Systs. Man Cybern., vol. 24, no. 3, pp. 400–412, Mar. 1994. [31] A. B. Kurshanski and P. Varaiya, “Ellipsoidal techniques for reachability analysis,” in Lecture Notes in Computer Science (Hybrid Systems: Computation and Control). Berlin, Germany: Springer-Verlag, 2000, vol. 1790, pp. 202–214. [32] A. Krener and A. Isidori, “Linearization by output injection and nonlinear observers,” Syst. Control Lett., vol. 3, pp. 47–52, 1983. [33] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM, 1992. [34] N. E. Cotter, “The Stone-Weierstrass theorem and its application to neural networks,” IEEE Trans. Neural Netw., vol. 1, no. 4, pp. 290–295, Dec. 1990. [35] V. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992. [36] T. Poznyak, I. Chairez, and A. Poznyak, “Application of a neural observer to phenols ozonation in water: Simulation and kinetic parameters identification,” Water Res., vol. 39, pp. 2611–2620, 2005. [37] A. F. Fillipov, Differential Equations With Discontinuous Right-Hand Side. Boston, MA: Kluwer, 1998. [38] V. Stepanyan and N. Hovakimyan, “Robust adaptive observer design for uncertain systems with bounded disturbances,” IEEE Trans. Neural Netw., vol. 18, no. 5, pp. 1392–1403, Sep. 2007. [39] L. Lung, System Identification, Theory for the User. New York: Springer-Verlag, 1979. [40] C. Kunusch, “Identificacion De Sistemas Dinamicos,” Universidad Nacional de la Plata, Buenos Aires, Argentina, 2003. [41] A. Poznyak, J. Escobar, and Y. Shtessel, “Stochastic sliding modes identification,” in Proc. Workshop Variable Structure Syst., Alghero, Italy, Italy, Jun. 2006, pp. 5–9. [42] T. Vincent and J. Yu, “Control of chaotic systems,” Dyn. Control, vol. 1, pp. 35–52, 1991. [43] J. P. Gauthier and G. Bornard, “Observability for any u(t) of a class of nonlinear systems,” IEEE Trans. Autom. Control, vol. AC-26, no. 4, pp. 922–926, Aug. 1981. [44] Y. Zeng and S. N. Singh, “Adaptive control of chaos in Lorenz system,” Dyn. Control, vol. 7, pp. 143–154, 1997.
Isaac Chairez received the M.S. degree in automatic control and the Ph.D. degree in control theory from the CINVESTAV-IPN, México, in 2002 and 2005, respectively. Since 2003, he has been with the Bioelectronics Department, Professional Interdisciplinary Unit of Biotechnology, UPIBI-IPN, México. His professional activities have been concentrated on nonlinear control theory, stability theory, neural networks, image processing, sliding mode theory, and some practical research projects in these and other fields. His current research interests are in high-order sliding modes, differential neural networks to identify partial differential equations and their applications to control and observation, and nonlinear robust output-feedback control.
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Wavelet Differential Neural Network Observer Isaac Chairez Abstract—State estimation for uncertain systems affected by external noises is an important problem in control theory. This paper deals with a state observation problem when the dynamic model of a plant contains uncertainties or it is completely unknown. Differential neural network (NN) approach is applied in this uninformative situation but with activation functions described by wavelets. A new learning law, containing an adaptive adjustment rate, is suggested to imply the stability condition for the free parameters of the observer. Nominal weights are adjusted during the preliminary training process using the least mean square (LMS) method. Lyapunov theory is used to obtain the upper bounds for the weights dynamics as well as for the mean squared estimation error. Two numeric examples illustrate this approach: first, a nonlinear electric system, governed by the Chua’s equation and second the Lorentz oscillator. Both systems are assumed to be affected by external perturbations and their parameters are unknown. Index Terms—Continuous systems, learning schemes, neural network (NN), sliding-mode observers, state estimation, wavelet approximation.
I. INTRODUCTION ODELING theory (using multiple physical principles, chemical laws, etc.) represents the most usual manner to formalize the systems dynamics knowledge. However, in several real situations, modeling rules may fail to generate acceptable reproductions of reality [1]. In those cases, nonparametric identification (using adaptive methods) can be successfully applied to overcome possible modeling deficiencies. Within the nonparametric identification framework, function approximation techniques play an important role avoiding the necessity of accurate mathematical plant description [2]. Among others, wavelets have been applied [3] (multiscaled signals analysis and synthesis, time-frequency time-series analysis in digital and continuous processing, etc.) to approximate nonlinear function (by numerical reconstruction) and to solve ordinary and partial differential equations [1]–[3]. Wavelets theory is well posed to approximate nonlinear functions with local nonlinear sections and fast variations using two main properties: finite support and self-similarity. Besides, neural networks (NNs) have become an attractive tool for modeling complex nonlinear systems. NNs are particularly powerful for handling large scale problems. However, NN implementation suffers from lack of efficient constructive approaches: choosing network structures and determining the neuron parameters. It has been proven that artificial neural net-
M
Manuscript received October 19, 2007; revised March 01, 2009 and May 16, 2009; accepted May 21, 2009. First published August 11, 2009; current version published September 02, 2009. The author is with the Professional Interdisciplinary Unit of Biotechnology, UPIBI-IPN, México D.F., ZP. 07430 México (e-mail: [email protected]. mx). Digital Object Identifier 10.1109/TNN.2009.2024203
works (ANNs) can approximate a wide range of piecewise nonlinear functions to any desired degree of accuracy [4]. It is generally accepted that NN training algorithm plays an important role in NN applications. For instance, in conventional gradient-descent type, sensitivity of unknown system is required during the online training process [5]. However, it is difficult to acquire good information for unknown or highly nonlinear dynamics. Besides, local minimum of performance index remains to be the main inconvenience in backpropagation algorithm [6]. Radial basis function (RBF) networks are often used in order to improve the NN learning efficiency. Poggio and Girosi [7], [8] analyzed several network architectures to determine their approximation abilities and pointed out that RBF networks possess the best approximation capability. These advantages are further strengthened with the introduction of a wavelet into NN structures [9]. The main advantage of wavelet networks over similar architectures (such as multilayer perceptrons and networks of RBF using sigmoid functions as kernel [6]) is the possibility to optimize wavelet network structure by means of efficient deterministic constructive algorithms [10]. Also, as noted in [11] and [12], NNs have limited ability to characterize local features such as discontinuities in curvature, jumps in objective function, and others. These local features, which are located in time and/or frequency, typically embody important information of the system such as aberrant process modes or faults [11]. Bottou and Vapnik [13] have addressed that improved localized modeling can aid both data reduction (or compression) and subsequent classification tasks that rely on accurate representation of local features. Zhang and Benveniste [9], Zhang [10], and Bakshi and Stephanopoulos [14] improved upon this weakness of the NN by developing wavelet neural networks (WNNs), which are a type of feedforward NN. The WNN is based on global mean square error (MSE) as well. The existence of wavelet-based model selection methods is more important, as addressed in [12] (e.g., [15]) focused on data denoising and the avoidance of excessive number of wavelet coefficients/bases in their approximation models. In this study, the family of basis functions for the RBF network is replaced by an orthogonal basis (i.e., the scaling functions in the theory of wavelets) to form a WNN. Roughly speaking, the WNN is considered as a kind of RBF networks [16] and possesses more advantages than the general networks as was remarked in [10], [17], and [18]. These WNNs have important characteristics such as faster convergence, avoidance of local minimum, easy decision, and adaptation of structure [19], [20]. There are many research results [18], [19], [21], [22] dedicated to developing new adjustment laws for the WNN weights. Here, it should be noted that all WNN analyses were performed over static NNs described by recurrent algorithms. Today, many researchers have developed WNN applications, which combine the NN capability to learn using processed information [23] and wavelet decomposition capability [24], [25].
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These applications include identification and control of uncertain dynamic systems (see [16], [26], and [27] for more details). In [9], the notion of a wavelet network was proposed as an alternative to conventional feedforward NN. Approximate arbitrary nonlinear functions based on the wavelet transformation theory and a backpropagation algorithm was adapted for WNN training. Zhang et al. [16] described a wavelet-based NN for function learning and estimation. The structure of this network was similar to the RBF network except that radial functions were replaced by orthonormal scaling functions. Also, Zhang [10] presented WNN construction algorithms for the purpose of nonparametric regression estimation, using a constructive method and defining a complete basis for the state space. From the point of view of function representation, the traditional RBF networks can approximate any function that is in the space spanned by the family of basis functions. However, basis functions are generally not orthogonal and redundant. That means that RBF network representation for a given function is not unique and is probably not the most efficient. At this point, all WNN applications have been developed for the so-called static schemes (like multilayer structure). However, exploiting the fact of being universal approximations, it may straightforwardly substitute continuous system uncertainties by NNs containing large number of unknown parameters (weights) to be adjusted. In general, this class of NNs is known as differential neural networks (DNNs) and it possesses two important characteristics: its adjustable parameters may appear as linear elements in the NN description and such NN may be modified using differential equations [6], [28]. This approach transforms the original function approximation problem into a nonlinear robust adaptive feedback design. The DNN approach avoids many problems related to global extremum search and converts the learning process into a particular feedback scheme [28]. If the mathematical model of a process is incomplete or partially known, the DNN theory provides an effective instrument to attack a wide spectrum of problems such as nonparametric trajectory identification, state estimation, trajectories tracking, etc., [29]. In view of DNN continuous structure, more detailed techniques should be applied to solve important questions on the new NN proposal (convergence, for example). Lyapunov’s stability theory (especially the so-called controlled Lyapunov theory) has been used within the NN framework [28], [30]. Actually, this is the main tool to prove DNN improvements on the estimation problems or the control action design, even though there exists a general trend to increase the number of nonlinear systems for which the aforementioned works can be applied. As a consequence, there are novel results on stability, convergence to arbitrarily small sets, and robustness to modeling imperfections. In this paper, we have developed the application of the wavelet theory over the DNN structures (state estimation and nonparametric identification). The new approach known as differential wavelet neural network (DWNN) is completely presented, including observer structures, corresponding convergence schemes, and continuous learning algorithms. The effectiveness of the novel DNN topology is shown by two different numerical examples: the Lorentz oscillator and the Chua circuit.
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II. DNN OBSERVER DESIGN This section describes the class of uncertain systems to be treated using the DWNNs, their properties, and the type of external perturbations affecting their nonlinear dynamics. Section II-A describes the concept of a wavelet differential network and details its capability to approximate a certain class of nonlinear ordinary differential equation (ODE) dynamics. Section II-C describes the state observer structure, the wavelet approximation, etc. Sections II-D and II-E describe the weights dynamics which permits DWNN reproduction for nonlinear dynamics. Also, the weights bounded behavior is shown by means of a special kind of a controlled Lyapunov-like analysis. A. Uncertain Nonlinear Systems Dynamic uncertain nonlinear system to be analyzed in this paper is described by the following set of ordinary differential equations: (1) describes the system trajectories in System state , that is, a closed set where the system state remains during all system evolution time. Online available output signal is a linear combination (performed by the matrix with constant entries) of state components, and is a bounded control action belonging to the following set:
T T
Admissible set includes state feedback control actions, even linear functions, or discontinuous designs (sliding-mode controllers, for example). Output constant is assumed to be a priori known. real matrix satisfies a local LipVector function schitz condition, i.e., there exist two finite positive conand such that stants is accom, and , . (This plished for any , .) Nonlinear is, in fact, a direct consequence of and system is affected by state/parameter uncertainties measurement disturbances . Both unknown perturbations are bounded in ellipsoidal sense [31], that is, there are two and a pair of positive-definite matrices positive scalars , such that the following inequalities are fulT . Here, it should filled: be admitted that perturbation upper bound keeps the inside the positive nonlinear system dynamics cone defined by the Lipschitz condition for all . Last supposition keeps valid the existence and uniqueness conditions of perturbed nonlinear dynamics (1). Besides, nonlinear dynamics is bounded at the zero point condition given as . Nominal output ) is unisystem (without external perturbations, i.e., formly observable [32].
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T
B. Wavelet Differential Neural Network Approximation Nonlinear function approximation using wavelets is based on the following properties. Consider the closed space , with the following properties: space sequence is nested 1)
2) ( is the intersection operator); , ( is the direct sum operator); 3) 4) . It is seen that the whole space can be represented by
for some
. Let
be a basic scaling function such that with , such that then there exists a basic function with . is called an orthogonal wavelet if the set of funcFunction is an orthonormal basis of , that is tions
Several kinds of wavelet basis [33] have been developed. For example, the Daubechies, Morlet, Mexican Hat, Meyer families, and many others (in fact, further development of new families of wavelet bases continues to receive considerable attention from in scientific community). Now let us consider a function . In view of previous properties, can be rewritten as [24]
with a weighting constant vector. Remark 1: Appropriate selection of wavelet functions is an important task to construct an adequate approximation of nonlinear functions. Many wavelet functions have been reported in literature [33] that have remarkable results in approximating nonlinear unknown functions. Which one is the most suitable basis in practical application depends on particular design specifications. parameters in wavelet netRemark 2: work design are closely related to the quality approximation , although the wavelet network has been demonstrated to be more effective than sigmoid-based NNs to reproduce uncertain nonlinear functions satisfying the Lipschitz condition, particularly, if the associated Lipschitz constant is large. In view of the WNN approximation capabilities, notice could be always presented (by Stone–Weisstrass that and Kolmogorov theorems [34]) by the composition of and modeling error nominal terms [as is usual when a model-free approximation is applied and the error is given by (2)]. Nonlinear dynamics (1) can be exposed as (3) where the nominal dynamics may be selected according to a desired design (in this case, by means of a combination of wavelet and a special kind of NN with continuous dynamics). Here, the vector of parameters should be adjusted to obtain the best and nonlinear possible matching between the nominal dynamics (in other words, to reduce as much as possible, the presence). In view of the nonlinear dynamics properties (Lipschitz condition and the class of admissible controls), the following upper bound for the error modeling is observed:
Last expression is called a wavelet series expansion of . Based on this series expansion, a wavelet network has the following mathematical structure: T
that can be used to approximate any nonlinear smooth function with the adequate selection of integers , , , and , where T
T T
where the estimation error
T
This wavelet expansion suggests an alternative for an NN representation. Following the Stone–Wiestrass theorem, if is the NN approximation error, then for any arbitrary positive constant , there such that are some constants
(4) is defined by
According to the DNN approach, nominal dynamics is proposed using the Stone–Weisstrass theorem and its corresponding extension for WNN. This nominal dynamics (following the WNN theory) is suggested to be
(2) for all
T
, the
. In case of argument
should be modified to
(5)
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, , . Matrices of activation functions and are usually constructed with components defined by sigmoid functions. In this study, these functions are changed to wavelet functions to build an orthogonal and nonredundant basis for the approximation of the uncertain nonlinear function . It is easy to prove (as a consequence of their bounded structure) that each component in activation functions satisfies sector conditions given by
eter selection, while Section II-D is related with the quality of the suggested state estimation scheme. D. DNN Parameter Adjustment Procedure provide the Usually in NN, the weights function approximation capacity. Nonlinear weight updating (learning) law is described by following matrix differential equations: T T
T
T
T
if if T
(7)
T
Last assumption is a consequence of continuity property for . continuous wavelet functions, which belong to class C. Observer Structure Following the Luenberger-like observers introduced in [28] and the sliding-mode observer proposition given by [35], the DNN estimator structure is suggested as follows [36]:
T is defined by Matrix with a small positive scalar value (typically 0.01). Matrices and represent the distance between to their corresponding best the current values of fitted values and , that is, . Matrices are symmetric positive-definite matrices T . Time-varying function is the output . Variables error defined as are time-varying functions to adjust the learning rates. Matrices T are the positive-definite solutions for the Riccati equations
(6)
T
. Vector is the observer state. and are adaptive parameters that should be adjusted to reproduce (as well as possible) the nominal dynamics given in (5). Constant matrices are, correspondingly, the output-based correction matrices for the linear and variable structure terms. Funcassociated to sliding-mode obtion server is defined as where
T
T T
T
T
T
T
(8) Nonlinear differential (6) has discontinuous right-hand side. Hence, the existence and uniqueness properties for this equation are understood in the sense of differential inclusion theory [37]. DNN state estimator must solve the following two different tasks simultaneously. should converge as close 1) WNN parameters as possible (depending on the values of the external uncer. tainties) to the most adequate but unknown values 2) DNN state should converge asymptotically to a small zone close to (supplied by a ellipsoid upper bound) with size depending on the presence of the external perturba, tions with the ellipsoid bounds given above ( , and ). Both processes (the DNN parameter adjustment and the observer state convergence) can be achieved by means of a suitable Lyapunov function and the adaptive control theory. Section II-C deals with the adjustment process for the adequate DNN param-
and are positive definite as well. In Here fact, they must be selected (over a large set of possible values) just to ensure the existence of a solution for (8). T Remark 3: Special class of Riccati equation has positive solution if and only if the following four conditions are observed: — Condition 1: matrix is stable. is controllable. — Condition 2: pair — Condition 3: Pair is observable. — Condition 4: matrices should be selected to satisfy the following inequality: T
T
T
T
Last condition restricts the largest eigenvalue of avoiding the inexistence of Riccati equation positive solution. State estimation problem for uncertain nonlinear systems analyzed in this study could be stated as follows.
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Considering the nonlinear system conditions (external perturbation presence, uncertain mathematical description, etc.) for , with an adequate any admissible control strategy selection of matrices and with the WNN observer structure supplied with the adjustment law (7) (including the se), and the upper bound for the averaged lection of estimation error defined as (9) T must be obtained, and if it is possible, reduced to its lowest achievable value, using the adjustment of the free parameters included in the WDNN structure. This averaged error is quite similar to the mean squared error that is commonly used in NN theory. State estimation process solution actually means to select correctly the matrices . State estimation result is given in the following. Theorem 1: If there exist positive-definite matrices and positive constant such that the matrix Riccati (8) has positive-definite solutions, then the WDNN observer (6) supplied with the learning law (7) and using any guarantying that the closed-loop matrix is stable, matrix that is
is Hurwitz
Here, it is also necessary to use a numerical interpolator algorithm to manage the sampled data as piecewise continuous signals. This is an important restriction for the observer design but, on the other hand, the same problem appears with any other NN structure, where the training phase is the most crucial stage in this class of approaches. Obviously, the data must be sampled with fixed frequency containing enough information to process the parametric identification [39] and fulfilling the corresponding persistent excitation condition [39]. In view appearing linearly with respect to the activaof tion functions, the application of the matrix least mean square algorithm is suggested (see [40] and [39]) to attain the training stage. To do that, (3) should be represented in its integral form
which, in turn, may be represented in a linear regression form
(10)
and selecting T
(11)
provides the following upper bound for the averaged (or mean squared) norm of the state estimation error:
where
(12) Matrix least square estimate
of
is given by
T
(13) Proof: Proof of this theorem is given in the Appendix . An important remark can be introduced here for the unperturbed case, i.e., when noises and the modeling error are absent. Corollary 1: WDNN observer estimation error norm (12) converges asymptotically to zero when the state and output uncertainties vanish ( and ), the admissible control ellipse , and has a support function that crosses at zero point . no modeled dynamics is absent E. DNN Training Algorithm To complete the learning algorithms (7), the knowledge of the nominal matrices incorporated in is required. Nevertheless, they are actually unknown. The so-called training process consists of the identification for suit. This may be done offline able approximation of (before the beginning of the state estimation) using some avail. able or even fictitious [38] experimental data
T
(14)
or using the differential form T
T T T
(15)
It is possible to show [41] that the training quality is given by T
T
T
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The upper bound for the training process is given by
where and
and
is the upper bound of is T
T
(20)
then the weight time trajectories (7) are bounded, and moreover, for any , it may be concluded that Therefore, the learning laws become T
T
(21) (16) which are numerically realizable, because the weights are now available. Adaptive learning algorithms help to reject perturbation effects, and at the same time, to enforce the estimation error inside the region defined by (12). Considering the external perturbation effect, nominal weights obtained by the training method cannot be substituted in the observer structure. This can be explained better considering that the system is assumed to be affected by different uncertainties (parametric, noises, etc.). As a consequence, adaptive weight laws must be applied to adjust the observer structure depending on the current noises values, modeling error, etc. In fact, the application of (16) provides a certain class of robustness property to the state estimator (6).
, the suggested Lyapunov function Proof: a) For has the following structure: T
Here Since the function ferentiation) T
T
F. Bounded DNN Weights Behavior Now it is possible to consider the stability analysis of the learning laws (expressed as matrix differential equations). This part of the study may be conducted using new Lyapunov functions and for the weights and , respectively. Here, let us apply the conventional second Lyapunov method to determine the upper bound for the weights dynamics. Here, it is necessary to define the following Lyapunov function:
T
To guarantee the nonincreasing property for of the observer trajectories, inequality (19) must be observed. That condition . The last inequality permits avoids the finite-time escape of the existence of several learning law schemes depending on the structure. b) Considering the Lyapunov function given by (17), a similarly analysis can be done for the weights
(17) and the constant (18) Boundedness property for the weights trajectories is shown in the following. and satisfy the folTheorem 2: If the parameters lowing inequalities: T T
T T
T
T
(19)
is differentiable, one has (by direct dif-
where
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Following a similar procedure for , one gets that if inequality (20) is achieved, then it is implied that , and . Therefore, one may conclude that hence,
This completes the proof. Restriction (19) permits construction of many different learning schemes. In fact, the number of possible adjustment , , just need to be a rules is unbounded because , and fulfills (19) and (20) nonnegative function simultaneously. One example demonstrating the flexibility of the suggested approach for the weights dynamics adjustment is detailed below. Example 1 (Sigmoid Learning Law): Define
(22) It is easy to check that
If
Fig. 1. Chua’s circuit state x DNN, and EKF.
trajectory and three observers results: WDNN,
(the voltage across well-known nonlinear circuit is given by the capacitor and the parallel nonlinear resistance ), (the voltage across capacitor and the inductor ), and (the current through the inductor ). The direct application of Kirchoff’s laws leads to the next set of nonlinear ODEs
satisfies the differential inclusion T
(24)
we get T
(23)
Hence, by the previous theorem, it may be guaranteed that is bounded. In the same way, a similar exercise for can be obtained. It is important to note that learning laws should be adjusted online, i.e., any changes on weight values imply . a corresponding change in the time-varying parameter Then, the adaptive process is carried out in two steps: the first one by the learning law application and the second one by the , designing. learning law rates Here, it must be understood that there is a great number of possible solutions for the inclusion of (23). However, the case where the equality is attained can be taken (for numerical simulations). III. SIMULATION RESULTS This section shows the numerical results for two examples using the observation technique suggested in this paper. For each example, a comparison between the WDDN, DNN, and extended Kalman–Bucy observers is included. A. Chua’s Circuit Chua’s circuit is an interesting electronic system presenting bifurcations and chaotic phenomena such as double scroll and double hook. This special behavior depends on the selection of the electronic components. The mathematical model [42] for the
Parameters of the circuit (24) are , , , , and . Let us consider that is measurable and just the voltage on capacitor of magnitude perturbed with pseudowhite noise ( ). Local observability for with uniform distribution in Chua’s circuit is ensured using [43]. This is necessary before the WDNN observer can be applied. State estimation results are depicted in Figs. 1 and 2. There, it is shown that complete state is successfully reproduced by the as the output signal. Moreover, acWDNN observer using ceptable reproduction of unknown states is reached in a short time (less than 5 s). Figs. 1 and 2 also show the comparison between the NN-based observers using wavelet and sigmoid activation functions. Actually, it is not sufficient to compare one NN-based technique with another NN-based technique. The true test of usefulness and benefit is that NN-based technique does a better job than a good conventional (non-NN-based) technique. Therefore, an additional comparison between WDNN estimator and a standard technique [extended Kalman filter (EKF)] was included in Figs. 1 and 2. As could be seen, the considered observers seem to be identically efficient. Fig. 3 demonstrates the time evolution of the mean squared estimation error (12). Here, it could be noted that better performance is obtained by the observer using wavelet functions, that is, the mean squared error converges faster to zero when the WDNN is used.
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TABLE I COMPARISONS BETWEEN DIFFERENT ESTIMATION ERROR VALUES (DNN, WDNN, AND EKF) FOR THE CHUA CIRCUIT AT DIFFERENT TIMES
WDNN, DNN, and EKF at different times. In Table I, better performance obtained by the WDNN is clarified. Fig. 2. Chua’s circuit state x DNN, and EKF.
B. Lorentz Oscillator trajectory and three observers results: WDNN,
Fig. 3. Time evolution of estimation error (12) for the Chua circuit estimation process. Comparison between WDNN and EKF observers.
Lorentz model is used for the fluid conviction description especially for some feature at atmospheric dynamic [44]. The controlled model is described by the following set of ODEs
where , and represent the measures of fluid velocity and horizontal and vertical temperature variations. These might seem to be very simple equations, but due to the nonlinear terms, they cannot be solved analytically. Parameters , , and are positive parameters that represent the Prandatl and Tayleigh is the geometric factor, correspondingly. If numbers, while , the origin is a stable equilibrium point. If , the system has two stable equilibrium points with components given by and one unstable equilibrium point (the origin). If , three equilibrium points become unstable. Chaotic behavior , , and . is defined with The method suggested in this paper shows an appropriate performance even when parameters , , and vary with time in a small region (this characteristic was proposed to simulate the ). Lorentz oscillator trajectories are completely presence of reproduced after a short time (less than 3 s) as is shown in Figs. 4 and 5. These results confirm the improvement provided by the suggested wavelet observer. Table II shows the time evolution of the mean squared estimation error (12) for the WDNN, DNN, and EKF at different times. In Table II, better performance obtained by the WDNN is clarified. IV. CONCLUSION
Fig. 4. Lorentz oscillator x DNN, and EKF.
trajectory and three observers results: WDNN,
Two main improvements are obtained using the waveletbased observer: less mean squared error and faster convergence in earlier stages of the estimation process. Table I shows the time evolution of the mean squared estimation error (12) for the
This paper presents the application of sliding mode theory, DNN, and wavelet approaches for the nonparametric modeling of unknown models belonging to the given class of nonlinear systems described above. This observer seems to be an interesting tool in the numerical reconstruction for unmeasurable information even if the mathematical model of the nonlinear system is uncertain. Parameters involved in the WDNN-observer structure provide not just the stability for the estimation algorithm but also they are shown (by the formalism supported in the Lyapunov
CHAIREZ: WAVELET DIFFERENTIAL NEURAL NETWORK OBSERVER
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APPENDIX Proof: The dynamics of ODE:
is governed by the following
(25) and . To construct the converge algorithm, define the energetic function (Liapunov-like) as where
Fig. 5. Lorentz oscillator x DNN, and EKF.
T
trajectory and three observers results: WDNN,
T
T
T
. Its time derivative is T
T
T
T
T
T
Fig. 6. Time evolution of estimation error (12) for the Lorentz oscillator estimation process. Comparison between WDNN and EKF observers. T
TABLE II COMPARISONS BETWEEN DIFFERENT NORMS OF ESTIMATION ERROR (DNN, WDNN, AND EKF) FOR THE LORENTZ OSCILLATOR AT DIFFERENT TIMES
T T
T
T
T
T T
T
T T T
(26)
Using (25) implies T T T
method) that they remain bounded with the specific selection of some parameters related to the learning rate. Moreover, strength of the proposed method is demonstrated by two computer simulations of well-known complex nonlinear models: the Chua circuit and the Lorentz oscillator. Considering the class of admissible control described in this paper, the WDNN observer proposed here could be used in the output-based control algorithms.
T
(27) T T Notice that T . Let us esT timate the rest of the terms using the matrix inequality T T T valid for any and
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T
T
any T
T
T
T
T
T T
T T T
T
T
T
T
T
T
T
T
T T
T
T
T
T
T T
T
T
T
T
T
T
T
T
T T
T
T
T
T
T
T
T T
(28) On the other hand, the term
T
T
T
T
T
T T
T T
could be analyzed by
T
T T T T
T T
T
Besides, if the following parameters are chosen as , then
T
T T
T T
T T
T T
T T
By means of the following inequality and the
selection:
T
Finally, using the adaptive laws given for each DNN weight maT . After the integration of trices, we have the previous inequality, , dividing over and taking the upper limit, last inequality becomes
T
and using the Jensen’s inequality, we have
T
So, the function
If we solve the previous inequality, the theorem statement is proved.
is bounded now as T
T T
T T
REFERENCES T T
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CHAIREZ: WAVELET DIFFERENTIAL NEURAL NETWORK OBSERVER
[3] C. Jaideva, G. Andrew, and K. Chan, Fundamentals of Wavelets. New York: Wiley, 2001, pp. 345–390. [4] O. Omidvar and D. L. Elliott, Neural Systems for Control. New York: Academic, 1997. [5] F. J. Lin, W. J. Hwang, and R. J. Wai, “Ultrasonic motor servo drive with on-line trained neural network model-following controller,” Proc. Inst. Electr. Eng. Electr. Power Appl., vol. 145, no. 2, pp. 105–110, 1998. [6] S. Haykin, Neural Networks. A comprehensive Foundation. New York: IEEE Press, 1994. [7] T. Poggio and F. Girosi, “Networks for approximation and learning,” Proc. IEEE Conf. Decision Control, vol. 78, no. 9, pp. 1481–1497, 1990. [8] T. Poggio and F. Girosi, “Regularization algorithms for learning that are equivalent to multilayer networks,” Science, vol. 247, pp. 978–982, 1990. [9] Q. Zhang and A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 889–898, Nov. 1992. [10] Q. Zhang, “Using wavelet network in nonparameters estimation,” IEEE Trans. Neural Netw., vol. 8, no. 2, pp. 227–236, Mar. 1997. [11] J. Jin and J. Shi, “Feature-preserving data compression of stamping tonnage information using wavelets,” Technometrics, vol. 41, no. 4, pp. 327–339, 1999. [12] L. Martell, “Wavelet-based data reduction and de-noising procedures,” Ph.D. dissertation, Dept. Electr. Eng., North Carolina State Univ., Raleigh, NC, 2000. [13] L. Bottou and V. Vapnik, “Local learning algorithms,” Neural Comput., vol. 4, pp. 888–900, 1992. [14] B. Bakshi and G. Stephanopoulos, “Wave-net: A multiresolution, hierarchical neural network with localized learning,” AIChE J., vol. 39, no. 1, pp. 57–81, 1993. [15] H. Saito, “Wavelets in geophysics,” in Simultaneous Noise Suppression and Signal Compression Using a Library of Orthonormal Bases and the Minimum Description Length Criterion. New York: Academic, 1994, pp. 299–324. [16] J. Zhang, G. Walter, and Y. Miao, “Wavelet neural networks for function learning,” IEEE Trans. Signal Process., vol. 43, no. 6, pp. 1485–1496, Jun. 1995. [17] L. Jiao, J. Pan, and Y. Fang, “Multiwavelet neural network and its approximation properties,” IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1060–1066, Sep. 2001. [18] J. Xu and Y. Tan, “Nonlinear adaptive wavelet control using constructive wavelet networks,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 25–27. [19] C. P. Bernard and J. J. E. Slotine, “Adaptive control with multiresolution bases,” in Proc. 36th IEEE Conf. Decision Control, San Diego, CA, 1997, pp. 3884–3889. [20] A. Ikonomopoulos and A. Endou, “Wavelet decomposition and radial basis function networks for system monitoring,” IEEE Trans. Nuclear Sci., vol. 45, no. 5, pp. 2293–2301, Oct. 1998. [21] N. Sureshbabu and J. A. Farrell, “Wavelet-based system identification for nonlinear control,” IEEE Trans. Autom. Control, vol. 44, no. 2, pp. 412–417, Feb. 1999. [22] R. M. Sanner and J. J. E. Slotine, “Structurally dynamic wavelet networks for the adaptive control of uncertain robotic systems,” in Proc. IEEE Conf. Decision Control, New Orleans, LA, 1995, pp. 2460–2467. [23] Y. C. Pati and P. S. Krishnaprasad, “Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations,” IEEE Trans. Neural Netw., vol. 4, no. 1, pp. 73–85, Jan. 1993. [24] B. Delyon, A. Juditsky, and A. Benveniste, “Accuracy analysis for wavelet approximations,” IEEE Trans. Neural Netw., vol. 6, no. 2, pp. 332–348, Mar. 1995. [25] T. Lindblad and J. M. Kinser, “Inherent features of wavelets and pulse coupled networks,” IEEE Trans. Neural Netw., vol. 10, no. 3, pp. 607–614, May 1999.
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Isaac Chairez received the M.S. degree in automatic control and the Ph.D. degree in control theory from the CINVESTAV-IPN, México, in 2002 and 2005, respectively. Since 2003, he has been with the Bioelectronics Department, Professional Interdisciplinary Unit of Biotechnology, UPIBI-IPN, México. His professional activities have been concentrated on nonlinear control theory, stability theory, neural networks, image processing, sliding mode theory, and some practical research projects in these and other fields. His current research interests are in high-order sliding modes, differential neural networks to identify partial differential equations and their applications to control and observation, and nonlinear robust output-feedback control.
