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Journal of Applied Mathematics
Volume 2012, Article ID 529176, 20 pages
http://dx.doi.org/10.1155/2012/529176
Research Article

System Identification Using Multilayer Differential Neural Networks: A New Result

1Centro Universitario de Ciencias Exactas e Ingenierías, Universidad de Guadalajara, Boulevard Marcelino García Barragán No. 1421, 44430 Guadalajara, JAL, Mexico
2Sección de Estudios de Posgrado e Investigación, ESIME-UA, IPN, Avenida de las Granjas No. 682, 02250 Santa Catarina, NL, Mexico

Received 22 November 2011; Revised 30 January 2012; Accepted 2 February 2012

Academic Editor: Hector Pomares

Copyright © 2012 J. Humberto Pérez-Cruz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In previous works, a learning law with a dead zone function was developed for multilayer differential neural networks. This scheme requires strictly a priori knowledge of an upper bound for the unmodeled dynamics. In this paper, the learning law is modified in such a way that this condition is relaxed. By this modification, the tuning process is simpler and the dead-zone function is not required anymore. On the basis of this modification and by using a Lyapunov-like analysis, a stronger result is here demonstrated: the exponential convergence of the identification error to a bounded zone. Besides, a value for upper bound of such zone is provided. The workability of this approach is tested by a simulation example.

1. Introduction

During the last four decades system identification has emerged as a powerful and effective alternative to the first principles modeling [14]. By using the first approach, a satisfactory mathematical model of a system can be obtained directly from an input and output experimental data set [5]. Ideally no a priori knowledge of the system is necessary since this is considered as a black box. Thus, the time employed to develop such model is reduced significantly with respect to a first principles approach. For the linear case, system identification is a problem well understood and enjoys well-established solutions [6]. However, the nonlinear case is much more challenging. Although some proposals have been presented [7], the class of considered nonlinear systems can result very limited. Due to their capability of handling a more general class of systems and due to advantages such as the fact of not requiring linear in parameters and persistence of excitation assumptions [8], artificial neural networks (ANNs) have been extensively used in identification of nonlinear systems [912]. Their success is based on their capability of providing arbitrarily good approximations to any smooth function [1315] as well as their massive parallelism and very fast adaptability [16, 17].

An artificial neural network can be simply considered as a nonlinear generic mathematical formula whose parameters are adjusted in order to represent the behavior of a static or dynamic system [18]. These parameters are called weights. Generally speaking, ANN can be classified as feedforward (static) ones, based on the back propagation technique [19] or as recurrent (dynamic) ones [17]. In the first network type, system dynamics is approximated by a static mapping. These networks have two major disadvantages: a slow learning rate and a high sensitivity to training data. The second approach (recurrent ANN) incorporates feedback into its structure. Due to this feature, recurrent neural networks can overcome many problems associated with static ANN, such as global extrema search, and consequently have better approximation properties. Depending on their structure, recurrent neural networks can be classified as discrete-time ones or differential ones.

The first deep insight about the identification of dynamic systems based on neural networks was provided by Narendra and Parthasarathy [20]. However, none-stability analyses of their neuroidentifier were presented. Hunt et al. [21] called attention to determine the convergence, stability and robustness of the algorithms based on neural networks for identification and control. This issue was addressed by Polycarpou and Ioannou [16], Rovithakis and Christodoulou [17], Kosmatopoulos et al. [22], and Yu and Poznyak [23]. Given different structures of continuous-time neural networks, the stability of their algorithms could be proven by using Lyapunov-like analysis. All aforementioned works considered only the case of single-layer networks. However, as it is known, this kind of networks does not necessarily satisfy the property of universal function approximation [24]. And although the activation functions of single-layer neural networks are selected as a basis set in such a way that this property can be guaranteed, the approximation error can never be made smaller than a lower bound [24]. This drawback can be overcome by using multilayer neural networks. Due to this better capability of function approximation, the case multilayer was considered in [25] for feedforward networks and for continuous time recurrent neural networks for first time in [26] and subsequently in [27]. By using Lyapunov-like analysis and a dead-zone function, boundedness for the identification error could be guaranteed in [26]. The following upper bound for the “average” identification error was reported,limsup𝑇1𝑇𝑇0𝑓10+Υ𝜆min𝑃1/2𝑄0𝑃1/2𝑃1/2Δ𝑡+Δ𝑇𝑡𝑄0Δ𝑡𝑓𝑑𝑡0+Υ,(1.1) where Δ𝑡 is the identification error, 𝑄0 is a positive definite matrix, 𝑓0 is a upper bound for the modeling error, Υ is an upper bound for a deterministic disturbance, and []+ is a dead-zone function defined as[𝑧]+=𝑧𝑧0,0𝑧<0.(1.2) Although, in [28], open-loop analysis based on the passivity method for a multilayer neural network was carried out and certain simplifications were accomplished, the main result about the aforementioned identification error could not be modified. In [29], the application of the multilayer scheme for control was explored. Since previous works [2629] are based on this “average” identification error, one could wonder about the real utility of this result. Certainly, boundedness for this kind of error does not guarantee that Δ𝑡 belongs to 𝐿2 or 𝐿. Besides, none value for upper bound of identification error norm is provided. Likewise, none information about the speed of the convergence process is presented. Another disadvantage of this approach is that the upper bound for the modeling error 𝑓0 must be strictly known a priori in order to implement the learning laws for the weight matrices. In order to avoid these drawbacks, in this paper, we propose to modify the learning laws employed in [26] in such a way that their implementation does not require anymore the knowledge of an upper bound for the modeling error. Besides, on the basis of these new learning laws, a stronger result is here guaranteed: the exponential convergence of the identification error norm to a bounded zone. The workability of the scheme developed in this paper is tested by simulation.

2. Multilayer Neural Identifier

Consider that the nonlinear system to be identified can be represented bẏ𝑥𝑡𝑥=𝑓𝑡,𝑢𝑡,𝑡+𝜉𝑡,(2.1) where 𝑥𝑡𝑛 is the measurable state vector for 𝑡+={𝑡𝑡0},𝑢𝑡𝑞 is the control input, 𝑓𝑛×𝑞×+𝑛 is an unknown nonlinear vector function which represents the nominal dynamics of the system, and 𝜉𝑡𝑛 represents a deterministic disturbance. 𝑓(𝑥𝑡,𝑢𝑡,𝑡) represents a very ample class of systems including affine and nonaffine-in-control nonlinear systems. However, when the control input appears in a nonlinear fashion in the system state equation (2.1), throughout this paper, such nonlinearity with respect to the input is assumed known and represented by 𝛾()𝑞𝑠.

Consider the following parallel structure of multilayer neural network𝑑𝑑𝑡̂𝑥𝑡=𝐴̂𝑥𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡,(2.2) where ̂𝑥𝑡𝑛 is the state of the neural network, 𝑢𝑡𝑞 is the control input, 𝐴𝑛×𝑛 is a Hurwitz matrix which can be specified by the designer, the matrices 𝑊1,𝑡𝑛×𝑚 and 𝑊2,𝑡𝑛×𝑟 are the weights of output layers, the matrices 𝑉1,𝑡𝑚×𝑛 and 𝑉2,𝑡𝑟×𝑛 are the weights of hidden layers, 𝜎() is the activation vector-function with sigmoidal components, that is, 𝜎()=[𝜎1(),,𝜎𝑚()]𝑇,𝜎𝑗(𝑎𝑣)=𝜎𝑗1+exp𝑚𝑖=1𝑐𝜎𝑗,𝑖𝑣𝑖𝑑𝜎𝑗,for𝑗=1,,𝑚,(2.3) where 𝑎𝜎𝑗, 𝑐𝜎𝑗,𝑖, and 𝑑𝜎𝑗 are positive constants which can be specified by the designer, 𝜙()𝑟𝑟×𝑠 is also a sigmoidal function, that is,𝜙𝑖𝑗(𝑎𝑧)=𝜙𝑖𝑗1+exp𝑟𝑙=1𝑐𝜙𝑖𝑗,𝑙𝑧𝑙𝑑𝜙𝑖𝑗for𝑖=1,,𝑟,𝑗=1,,𝑠,(2.4) where 𝑎𝜙𝑖𝑗, 𝑐𝜙𝑖𝑗,𝑙, and 𝑑𝜙𝑖𝑗 are positive constants which can be specified by the designer, 𝛾()𝑞𝑠 represents the nonlinearity with respect to the input—if it exists—which is assumed a priori known for the system (2.1). It is important to mention that 𝑚 and 𝑟, that is, the number of neurons for 𝜎() and the number of rows for 𝜙(), respectively, can be selected by the designer.

The problem of identifying system (2.1) based on the multilayer differential neural network (2.2) consists of, given the measurable state 𝑥𝑡 and the input 𝑢𝑡, adjusting on line the weights 𝑊1,𝑡,𝑊2,𝑡,𝑉1,𝑡, and 𝑉2,𝑡 by proper learning laws such that the identification error Δ𝑡=̂𝑥𝑡𝑥𝑡 can be reduced.

Hereafter, it is considered that the following assumptions are valid;(A1)System (2.1) satisfies the (uniform on 𝑡) Lipschitz condition, that is,𝑓(𝑥,𝑢,𝑡)𝑓(𝑧,𝑣,𝑡)𝐿1𝑥𝑧+𝐿2𝑢𝑣;𝑥,𝑧𝑛;𝑢,𝑣𝑞;0𝐿1,𝐿2<.(2.5)

(A2)The differences of functions 𝜎() and 𝜙() fulfil the generalized Lipschitz conditions 𝜎𝑇𝑡Λ1𝜎𝑡Δ𝑇𝑡Λ𝜎Δ𝑡,𝛾𝑇𝑢𝑡𝜙𝑇𝑡Λ2𝜙𝑡𝛾𝑢𝑡Δ𝑇𝑡Λ𝜙Δ𝑡𝛾𝑢𝑡2,(2.6) where 𝜎𝑡𝑉=𝜎10̂𝑥𝑡𝑉𝜎10𝑥𝑡,𝜙𝑡𝑉=𝜙20̂𝑥𝑡𝑉𝜙20𝑥𝑡,(2.7)Λ1𝑚×𝑚, Λ2𝑟×𝑟, Λ𝜎𝑛×𝑛, Λ𝜙𝑛×𝑛 are known positive definite matrices, 𝑉01𝑚×𝑛 and 𝑉02𝑟×𝑛 are constant matrices which can be selected by the designer.As 𝜎() and 𝜙() fulfil the Lipschitz conditions and from Lemma A.1 proven in [26] the following is true: 𝜎𝑡𝑉=𝜎1,𝑡̂𝑥𝑡𝑉𝜎10̂𝑥𝑡=𝐷𝜎𝑉1,𝑡̂𝑥𝑡+𝜈𝜎,𝜎𝑡𝛾𝑢𝑡𝜙𝑉=2,𝑡̂𝑥𝑡𝑉𝜙02̂𝑥𝑡𝛾𝑢𝑡=(2.8)𝑠𝑖=1𝜙𝑖𝑉2,𝑡̂𝑥𝑡𝜙𝑖𝑉02̂𝑥𝑡𝛾𝑖𝑢𝑡=𝑠𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡+𝑣𝑖𝜙𝛾𝑖𝑢𝑡,(2.9) where 𝐷𝜎=𝜕𝜎(𝑌)|||𝜕𝑌𝑌=𝑉01̂𝑥𝑡𝑚×𝑚,𝐷𝑖𝜙=𝜕𝜙𝑖(𝑍)||||𝜕𝑍𝑍=𝑉02̂𝑥𝑡𝑟×𝑟,(2.10)𝜈𝜎𝑚 and 𝜈𝑖𝜙𝑛 are unknown vectors but bounded by 𝜈𝜎2Λ1𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1, 𝜈𝑖𝜙2Λ2𝑙2𝑉2,𝑡̂𝑥𝑡2Λ2, respectively; 𝑉1,𝑡=𝑉1,𝑡𝑉01, 𝑉2,𝑡=𝑉2,𝑡𝑉02, 𝑙1 and 𝑙2 are positive constants which can be defined as 𝑙1=4𝐿2𝑔,1, 𝑙2=4𝐿2𝑔,2, where 𝐿𝑔,1 and 𝐿𝑔,2 are global Lipschitz constants for 𝜎() and 𝜙𝑖(), respectively.(A3)The nonlinear function 𝛾() is such that 𝛾(𝑢𝑡)2𝑢 where 𝑢 is a known positive constant.(A4)Unmodeled dynamics 𝑓𝑡 is bounded by 𝑓𝑡2Λ3𝑓0+𝑓1𝑥𝑡2Λ3,(2.11) where 𝑓0 and 𝑓1 are known positive constants and Λ3𝑛×𝑛 is a known positive definite matrix and 𝑓𝑡 can be defined as 𝑓𝑡=𝑓(𝑥𝑡,𝑢𝑡,𝑡)𝐴𝑥𝑡𝑊01𝜎(𝑉01𝑥𝑡)𝑊02𝜙(𝑉02𝑥𝑡)𝛾(𝑢𝑡);𝑊01𝑛×𝑚 and 𝑊02𝑛×𝑟 are constant matrices which can be selected by the designer.(A5)The deterministic disturbance is bounded, that is, 𝜉𝑡2Λ4Υ, Λ4 is a known positive definite matrix.(A6)The following matrix Riccati equation has a unique, positive definite solution 𝑃: 𝐴𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄=0,(2.12) where 𝑅=2𝑊01Λ11𝑊01𝑇+2𝑊02Λ21𝑊02𝑇+Λ31+Λ41,𝑄=Λ𝜎+𝑢Λ𝜙+𝑄0,(2.13)𝑄0 is a positive definite matrix which can be selected by the designer.

Remark 2.1. Based on [30, 31], it can be established that the matrix Riccati equation (2.12) has a unique positive definite solution 𝑃 if the following conditions are satisfied;(a)The pair (𝐴,𝑅1/2) is controllable, and the pair (𝑄1/2,𝐴) is observable.(b)The following matrix inequality is fulfilled: 14𝐴𝑇𝑅1𝑅1𝐴𝑅𝐴𝑇𝑅1𝑅1𝐴𝑇𝐴𝑇𝑅1𝐴𝑄.(2.14) Both conditions can relatively easily be fulfilled if 𝐴 is selected as a stable diagonal matrix.

(A7)It exists a bounded control 𝑢𝑡, such that the closed-loop system is quadratic stable, that is, it exists a Lyapunov function 𝑉0>0 and a positive constant 𝜆 such that 𝜕𝑉0𝑓𝑥𝜕𝑥𝑡,𝑢𝑡𝑥,𝑡𝜆𝑡2.(2.15) Additionally, the inequality 𝑓𝜆1Λ3 must be satisfied.

Now, consider the learning law: ̇𝑊1,𝑡=𝑠𝑡𝐾1𝑃Δ𝑡𝜎𝑇𝑉1,𝑡̂𝑥𝑡+𝑠𝑡𝐾1𝑃Δ𝑡̂𝑥𝑇𝑡𝑉𝑇1,𝑡𝐷𝑇𝜎,̇𝑊2,𝑡=𝑠𝑡𝐾2𝑃Δ𝑡𝛾𝑇𝑢𝑡𝜙𝑇𝑉2,𝑡̂𝑥𝑡+𝑠𝑡𝐾2𝑃Δ𝑡̂𝑥𝑇𝑡𝑉𝑇𝑠2,𝑡𝑖=1𝛾𝑖𝑢𝑡𝐷𝑇𝑖𝜙,̇𝑉1,𝑡=𝑠𝑡𝐾3𝐷𝑇𝜎𝑊𝑇1,𝑡𝑃Δ𝑡̂𝑥𝑇𝑡𝑠𝑡𝑙12𝐾3Λ1𝑉1,𝑡̂𝑥𝑡̂𝑥𝑇𝑡,̇𝑉2,𝑡=𝑠𝑡𝐾4𝑠𝑖=1𝛾𝑖𝑢𝑡𝐷𝑇𝑖𝜙𝑊𝑇2,𝑡𝑃Δ𝑡̂𝑥𝑇𝑡𝑠𝑡𝑠𝑙2𝑢2𝐾4Λ2𝑉2,𝑡̂𝑥𝑡̂𝑥𝑇𝑡,(2.16) where 𝑠 is the number of columns corresponding to 𝜙(),𝐾1𝑛×𝑛,𝐾2𝑛×𝑛,𝐾3𝑚×𝑚, and 𝐾4𝑟×𝑟 are positive definite matrices which are selected by the designer. 𝑠𝑡 is a dead-zone function which is defined as 𝑠𝑡𝜇=1𝑃1/2Δ𝑡+,[𝑧]+=𝑓𝑧𝑧0,0𝑧<0,𝜇=0+Υ𝜆min𝑃1/2𝑄0𝑃1/2.(2.17) Based on this learning law, the following result was demonstrated in [26].

Theorem 2.2. If the assumptions (A1)–(A7) are satisfied and the weight matrices 𝑊1,𝑡, 𝑊2,𝑡𝑉1,𝑡, and 𝑉2,𝑡 of the neural network (2.2) are adjusted by the learning law (2.16), then(a)the identification error and the weights are bounded: Δ𝑡,𝑊1,𝑡,𝑊2,𝑡,𝑉1,𝑡,𝑉2,𝑡𝐿,(2.18)(b)the identification error Δ𝑡 satisfies the following tracking performance:limsup𝑇1𝑇𝑇0𝑓10+Υ𝜆min𝑃1/2𝑄0𝑃1/2𝑃1/2Δ𝑡+Δ𝑇𝑡𝑄0Δ𝑡𝑓𝑑𝑡0+Υ.(2.19)

In order to prove this result, the following nonnegative function was utilized:𝑉𝑡=𝑉0+𝑃1/2Δ𝑡𝜇2+𝑊+tr𝑇1,𝑡𝐾11𝑊1,𝑡𝑊+tr𝑇2,𝑡𝐾21𝑊2,𝑡𝑉+tr𝑇1,𝑡𝐾31𝑉1,𝑡𝑉+tr𝑇2,𝑡𝐾41𝑉2,𝑡,(2.20) where 𝑊1,𝑡=𝑊1,𝑡𝑊01; 𝑊2,𝑡=𝑊2,𝑡𝑊02.

3. Exponential Convergence of the Identification Process

Consider that the assumptions (A1)–(A3) and (A5)-(A6) are still valid but the assumption (A4) is slightly modified as follows.(B4) In a compact set Ω𝑛, unmodeled dynamics 𝑓𝑡 is bounded by 𝑓𝑡2Λ3𝑓0 where 𝑓0 is a constant not necessarily a priori known.

Remark 3.1. (B4) is a common assumption in the neural network literature [17, 22]. As mentioned in Section 2, 𝑓𝑡 is given by 𝑓𝑡=𝑓(𝑥𝑡,𝑢𝑡,𝑡)𝐴𝑥𝑡𝑊01𝜎(𝑉01𝑥𝑡)𝑊02𝜙(𝑉02𝑥𝑡)𝛾(𝑢𝑡). Note that 𝑊01𝜎(𝑉01𝑥𝑡) and 𝑊02𝜙(𝑉02𝑥𝑡)𝛾(𝑢𝑡) are bounded functions because 𝜎() and 𝜙() are sigmoidal functions. As 𝑥𝑡 belongs to Ω, clearly 𝑥𝑡 is also bounded. Therefore, assumption B4 implies implicitly that 𝑓(𝑥𝑡,𝑢𝑡,𝑡) is a bounded function in a compact set Ω𝑛.

Although certainly assumption (B4) is more restrictive than assumption (A4), from now on, assumption (A7) is not needed anymore.

In this paper, the following modification to the learning law (2.16) is proposed:̇𝑊1,𝑡=2𝑘1𝑃Δ𝑡𝜎𝑇𝑉1,𝑡̂𝑥𝑡+2𝑘1𝑃Δ𝑡̂𝑥𝑇𝑡𝑉𝑇1,𝑡𝐷𝑇𝜎𝛼2𝑊1,𝑡,̇𝑊2,𝑡=2𝑘2𝑃Δ𝑡𝛾𝑇𝑢𝑡𝜙𝑇𝑉2,𝑡̂𝑥𝑡+2𝑘2𝑃Δ𝑡̂𝑥𝑇𝑡𝑉𝑇𝑠2,𝑡𝑖=1𝛾𝑖𝑢𝑡𝐷𝑇𝑖𝜙𝛼2𝑊2,𝑡,̇𝑉1,𝑡=2𝑘3𝐷𝑇𝜎𝑊𝑇1,𝑡𝑃Δ𝑡̂𝑥𝑇𝑡𝑘3𝑙1Λ1𝑉1,𝑡̂𝑥𝑡̂𝑥𝑇𝑡𝛼2𝑉1,𝑡,̇𝑉2,𝑡=2𝑘4𝑠𝑖=1𝛾𝑖𝑢𝑡𝐷𝑇𝑖𝜙𝑊𝑇2,𝑡𝑃Δ𝑡̂𝑥𝑇𝑡𝑘4𝑠𝑙2𝑢Λ2𝑉2,𝑡̂𝑥𝑡̂𝑥𝑇𝑡𝛼2𝑉2,𝑡,(3.1) where 𝑘1,𝑘2,𝑘3, and 𝑘4 are positive constants which are selected by the designer; 𝑃 is the solution of the Riccati equation given by (2.12); 𝛼=𝜆min(𝑃1/2𝑄0𝑃1/2); 𝑠 is the number of columns corresponding to 𝜙(). By using the constants 𝑘1,𝑘2,𝑘3, and 𝑘4 in (3.1) instead of the matrices 𝐾1,𝐾2,𝐾3, and 𝐾4 in (2.16), the tuning process of the neural network (2.2) is simplified. Besides, none dead-zone function is now required. Based on the learning law (3.1), the following result is here established.

Theorem 3.2. If the assumptions (A1)–(A3), (B4), (A5)-(A6) are satisfied and the weight matrices 𝑊1,𝑡, 𝑊2,𝑡, 𝑉1,𝑡, and 𝑉2,𝑡 of the neural network (2.2) are adjusted by the learning law (3.1), then(a)the identification error and the weights are bounded: Δ𝑡,𝑊1,𝑡,𝑊2,𝑡,𝑉1,𝑡,𝑉2,𝑡𝐿,(3.2)(b)the norm of identification error converges exponentially to a region bounded given bylim𝑡𝑥𝑡̂𝑥𝑡𝑓0+Υ𝛼𝜆min(𝑃).(3.3)

Proof of Theorem 3.2. Before beginning analysis, the dynamics of the identification error Δ𝑡 must be determined. The first derivative of Δ𝑡 is 𝑑Δ𝑡=𝑑𝑑𝑡𝑑𝑡̂𝑥𝑡𝑥𝑡.(3.4) Note that an alternative representation for (2.1) could be calculated as follows: ̇𝑥𝑡=𝐴𝑥𝑡+𝑊01𝜎𝑉01𝑥𝑡+𝑊02𝜙𝑉02𝑥𝑡𝛾𝑢𝑡+𝑓𝑡+𝜉𝑡.(3.5) Substituting (2.2) and (3.5) into (3.4) yields ̇Δ𝑡=𝐴̂𝑥𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡𝐴𝑥𝑡𝑊01𝜎𝑉01𝑥𝑡𝑊02𝜙𝑉02𝑥𝑡𝛾𝑢𝑡𝑓𝑡𝜉𝑡,=𝐴Δ𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡𝑊01𝜎𝑉01𝑥𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡𝑊02𝜙𝑉02𝑥𝑡𝛾𝑢𝑡𝑓𝑡𝜉𝑡.(3.6) Subtracting and adding the terms 𝑊01𝜎(𝑉1,𝑡̂𝑥𝑡), 𝑊01𝜎(𝑉01̂𝑥𝑡), 𝑊02𝜙(𝑉2,𝑡̂𝑥𝑡)𝛾(𝑢𝑡), and 𝑊02𝜙(𝑉02̂𝑥𝑡)𝛾(𝑢𝑡) and considering that 𝑊1,𝑡=𝑊1,𝑡𝑊01, 𝑊2,𝑡=𝑊2,𝑡𝑊02, 𝜎𝑡=𝜎(𝑉10̂𝑥𝑡)𝜎(𝑉10𝑥𝑡), 𝜙𝑡=𝜙(𝑉20̂𝑥𝑡)𝜙(𝑉20𝑥𝑡), 𝜎𝑡=𝜎(𝑉1,𝑡̂𝑥𝑡)𝜎(𝑉10̂𝑥𝑡), 𝜙𝑡𝛾(𝑢𝑡)=(𝜙(𝑉2,𝑡̂𝑥𝑡)𝜙(𝑉20̂𝑥𝑡))𝛾(𝑢𝑡), (3.6) can be expressed as ̇Δ𝑡=𝐴Δ𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡𝑊01𝜎𝑉1,𝑡̂𝑥𝑡+𝑊01𝜎𝑉1,𝑡̂𝑥𝑡𝑊01𝜎𝑉01̂𝑥𝑡+𝑊01𝜎𝑉01̂𝑥𝑡𝑊01𝜎𝑉01𝑥𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡𝑊02𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+𝑊02𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡𝑊02𝜙𝑉02̂𝑥𝑡𝛾𝑢𝑡+𝑊02𝜙𝑉02̂𝑥𝑡𝛾𝑢𝑡𝑊02𝜙𝑉02𝑥𝑡𝛾𝑢𝑡𝑓𝑡𝜉𝑡=𝐴Δ𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+𝑊01𝜎𝑡+𝑊01𝜎𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+𝑊02𝜙𝑡𝛾𝑢𝑡+𝑊02𝜙𝑡𝛾𝑢𝑡𝑓𝑡𝜉𝑡,̇Δ𝑡=𝐴Δ𝑡+𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+𝑊01𝜎𝑡+𝑊02𝜙𝑡𝛾𝑢𝑡+𝑊01𝜎𝑡+𝑊02𝜙𝑡𝛾𝑢𝑡𝑓𝑡𝜉𝑡.(3.7) In order to begin analysis, the following nonnegative function is selected: 𝑉𝑡=Δ𝑇𝑡𝑃Δ𝑡+12𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡+12𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡+12𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡+12𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡,(3.8) where 𝑃 is a positive solution for the Riccati matrix equation given by (2.12). The first derivative of 𝑉𝑡 is ̇𝑉𝑡=𝑑Δ𝑑𝑡𝑇𝑡𝑃Δ𝑡+𝑑1𝑑𝑡2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡+𝑑1𝑑𝑡2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡+𝑑1𝑑𝑡2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡+𝑑1𝑑𝑡2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡.(3.9) Each term of (3.9) will be calculated separately. For (𝑑/𝑑𝑡)(Δ𝑇𝑡𝑃Δ𝑡), 𝑑Δ𝑑𝑡𝑇𝑡𝑃Δ𝑡=2Δ𝑇𝑡𝑃̇Δ𝑡.(3.10) substituting (3.7) into (3.10) yields 𝑑Δ𝑑𝑡𝑇𝑡𝑃Δ𝑡=2Δ𝑇𝑡𝑃𝐴Δ𝑡+2Δ𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+2Δ𝑇𝑡𝑃𝑊01𝜎𝑡+2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡+2Δ𝑇𝑡𝑃𝑊01𝜎𝑡+2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡2Δ𝑇𝑡𝑃𝑓𝑡2Δ𝑇𝑡𝑃𝜉𝑡.(3.11) The terms 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡, 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾(𝑢𝑡), 2Δ𝑇𝑡𝑃𝑓𝑡, and 2Δ𝑇𝑡𝑃𝜉𝑡 in (3.11) can be bounded using the following matrix inequality proven in [26]: 𝑋𝑇𝑌+𝑌𝑇𝑋𝑋𝑇Γ1𝑋+𝑌𝑇Γ𝑌,(3.12) which is valid for any 𝑋,𝑌𝑛×𝑘 and for any positive definite matrix 0<Γ=Γ𝑇𝑛×𝑛. Thus, for 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡 and considering assumption (A2), 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡=Δ𝑇𝑡𝑃𝑊01𝜎𝑡+𝜎𝑇𝑡𝑊01𝑇𝑃Δ𝑡Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝜎𝑇𝑡Λ1𝜎𝑡Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+Δ𝑇𝑡Λ𝜎Δ𝑡.(3.13) For 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾(𝑢𝑡),and considering assumptions (A2) and (A3) 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡=Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡+𝛾𝑇𝑢𝑡𝜙𝑇𝑡𝑊02𝑇𝑃Δ𝑡Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝛾𝑇𝑢𝑡𝜙𝑇𝑡Λ2𝜙𝑡𝛾𝑢𝑡Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑢Δ𝑇𝑡Λ𝜙Δ𝑡.(3.14) By using (3.12) and given assumptions (B4) and (A5), 2Δ𝑇𝑡𝑃𝑓𝑡 and 2Δ𝑇𝑡𝑃𝜉𝑡 can be bounded, respectively, by 2Δ𝑇𝑡𝑃𝑓𝑡=Δ𝑇𝑡𝑃𝑓𝑡𝑓𝑇𝑡𝑃Δ𝑡Δ𝑇𝑡𝑃Λ31𝑃Δ𝑡+𝑓𝑇𝑡Λ3𝑓𝑡Δ𝑇𝑡𝑃Λ31𝑃Δ𝑡+𝑓0,2Δ𝑇𝑡𝑃𝜉𝑡=Δ𝑇𝑡𝑃𝜉𝑡𝜉𝑇𝑡𝑃Δ𝑡Δ𝑇𝑡𝑃Λ41𝑃Δ𝑡+𝜉𝑇𝑡Λ4𝜉𝑡Δ𝑇𝑡𝑃Λ41𝑃Δ𝑡+Υ.(3.15) Considering (2.8), 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡 can be developed as 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡=2Δ𝑇𝑡𝑃𝑊01𝐷𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊01𝜈𝜎.(3.16) By simultaneously adding and subtracting the term 2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡 into the right-hand side of (3.16), 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡=2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊01𝜈𝜎.(3.17) By using (3.12) and considering assumption (A2), the term Δ𝑇𝑡𝑃𝑊01𝜈𝜎 can be bounded as 2Δ𝑇𝑡𝑃𝑊01𝜈𝜎=Δ𝑇𝑡𝑃𝑊01𝜈𝜎+𝜈𝑇𝜎𝑊01𝑇𝑃Δ𝑡Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝜈𝑇𝜎Λ1𝜈𝜎Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1.(3.18) And consequently, 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡 is bounded by 2Δ𝑇𝑡𝑃𝑊01𝜎𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡+Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1.(3.19) For 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾(𝑢𝑡) and considering (2.9), 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡=2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡+𝜈𝑖𝜙𝛾𝑖𝑢𝑡=2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡+2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡.(3.20) Adding and subtracting the term 2Δ𝑇𝑡𝑃𝑊2,𝑡𝑠𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖(𝑢𝑡) into the right-hand side of (3.20), 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡=2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡+2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡.(3.21) By using (3.12), 2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖(𝑢𝑡) can be bounded by 2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡=Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡+𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖(𝑢𝑡)𝑇𝑊02𝑇𝑃Δ𝑡Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖(𝑢𝑡)𝑇Λ2𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡,(3.22) but considering that 𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖(𝑢𝑡)𝑇Λ2𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡𝑠𝑠𝑖=1𝛾2𝑖𝑢𝑡𝜈𝑇𝑖𝜙Λ2𝜈𝑖𝜙(3.23) and from assumptions (A2) and (A3), the following can be concluded: 2Δ𝑇𝑡𝑃𝑊02𝑠𝑖=1𝜈𝑖𝜙𝛾𝑖𝑢𝑡Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ2.(3.24) Thus, 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾(𝑢𝑡) is bounded by 2Δ𝑇𝑡𝑃𝑊02𝜙𝑡𝛾𝑢𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡+Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ2.(3.25) Consequently, given (3.13), (3.14), (3.15), (3.19), and (3.25), (𝑑/𝑑𝑡)(Δ𝑇𝑡𝑃Δ𝑡) can be bounded as 𝑑Δ𝑑𝑡𝑇𝑡𝑃Δ𝑡Δ𝑇𝑡𝐴𝑇Δ𝑃+𝑃𝐴𝑡+2Δ𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+Δ𝑇𝑡Λ𝜎Δ𝑡+Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑢Δ𝑇𝑡Λ𝜙Δ𝑡+Δ𝑇𝑡𝑃Λ31𝑃Δ𝑡+𝑓0+Δ𝑇𝑡𝑃Λ41𝑃Δ𝑡+Υ+2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡+Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1+2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡+Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ2.(3.26) With respect to (𝑑/𝑑𝑡)((1/2𝑘1𝑊)tr{𝑇1,𝑡𝑊1,𝑡}), using several properties of the trace of a matrix, 𝑑1𝑑𝑡2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡=12𝑘1𝑑tr𝑊𝑑𝑡𝑇1,𝑡𝑊1,𝑡=12𝑘1tr𝑊𝑇1,𝑡𝑊1,𝑡+𝑊𝑇1,𝑡𝑊1,𝑡=12𝑘1tr𝑊𝑇1,𝑡𝑊1,𝑡𝑊+tr𝑇1,𝑡𝑊1,𝑡=1𝑘1tr𝑊𝑇1,𝑡𝑊1,𝑡.(3.27) As 𝑊1,𝑡=𝑊1,𝑡𝑊01, the derivative of 𝑊1,𝑡 is clearly 𝑊1,𝑡=𝑊1,𝑡. However, 𝑊1,𝑡 is given by the learning law (3.1). Therefore, by substituting (3.1) into 𝑊1,𝑡=𝑊1,𝑡 and the corresponding expression into the right-hand side of (3.27), it is possible to obtain 𝑑1𝑑𝑡2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡=1𝑘1tr2𝑘1𝜎𝑉1,𝑡̂𝑥𝑡Δ𝑇𝑡𝑃𝑊1,𝑡+2𝑘1𝐷𝜎𝑉1,𝑡̂𝑥𝑡Δ𝑇𝑡𝑃𝑊1,𝑡𝛼2𝑊𝑇1,𝑡𝑊1,𝑡𝜎𝑉=2tr1,𝑡̂𝑥𝑡Δ𝑇𝑡𝑃𝑊1,𝑡𝐷+2tr𝜎𝑉1,𝑡̂𝑥𝑡Δ𝑇𝑡𝑃𝑊1,𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡Δ=2tr𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡Δ+2tr𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡=2Δ𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡.(3.28) Proceeding in a similar way for 𝑑1𝑑𝑡2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡,𝑑1𝑑𝑡2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡,𝑑1𝑑𝑡2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡,(3.29) it is possible to obtain 𝑑1𝑑𝑡2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡=2Δ𝑇𝑡𝑃𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡𝛼2𝑘2𝑊𝑡𝑟𝑇2,𝑡𝑊2,𝑡,𝑑1𝑑𝑡2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡=2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡,𝑑1𝑑𝑡2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡=2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ2𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡.(3.30) By substituting (3.26), (3.28), and (3.30) into (3.9), the following bound for 𝑉𝑡 can be determined: ̇𝑉𝑡Δ𝑇𝑡𝐴𝑇Δ𝑃+𝑃𝐴𝑡+2Δ𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+Δ𝑇𝑡Λ𝜎Δ𝑡+Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑢Δ𝑇𝑡Λ𝜙Δ𝑡+Δ𝑇𝑡𝑃Λ31𝑃Δ𝑡+𝑓0+𝑓1𝑥𝑇𝑡Λ3𝑥𝑡+Δ𝑇𝑡𝑃Λ41𝑃Δ𝑡+Υ+2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡+Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1+2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡+Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ22Δ𝑇𝑡𝑃𝑊1,𝑡𝜎𝑉1,𝑡̂𝑥𝑡+2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡2Δ𝑇𝑡𝑃𝑊2,𝑡𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑢𝑡+2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡𝛼2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡2Δ𝑇𝑡𝑃𝑊1,𝑡𝐷𝜎𝑉1,𝑡̂𝑥𝑡𝑙1𝑉1,𝑡̂𝑥𝑡2Λ1𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡2Δ𝑇𝑡𝑃𝑊𝑠2,𝑡𝑖=1𝐷𝑖𝜙𝑉2,𝑡̂𝑥𝑡𝛾𝑖𝑢𝑡𝑠𝑙2𝑢𝑉2,𝑡̂𝑥𝑡2Λ2𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡.(3.31) Simplifying like terms ̇𝑉𝑡Δ𝑇𝑡𝐴𝑇Δ𝑃+𝑃𝐴𝑡+2Δ𝑇𝑡𝑃𝑊01Λ11𝑊01𝑇𝑃Δ𝑡+2Δ𝑇𝑡𝑃𝑊02Λ21𝑊02𝑇𝑃Δ𝑡+Δ𝑇𝑡𝑃Λ31𝑃Δ𝑡+Δ𝑇𝑡𝑃Λ41𝑃Δ𝑡+Δ𝑇𝑡Λ𝜎Δ𝑡+𝑢Δ𝑇𝑡Λ𝜙Δ𝑡+𝑓0𝛼+Υ2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡𝛼2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡.(3.32) Adding and subtracting Δ𝑇𝑡𝑄0Δ𝑡 into the right-hand side of the last inequality yields the expression 𝐴𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄. That is ̇𝑉𝑡Δ𝑇𝑡𝑄0Δ𝑡+Δ𝑇𝑡𝐴𝑇Δ𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄𝑡+𝑓0𝛼+Υ2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡𝛼2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡.(3.33) However, the expression 𝐴𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄 is, in accordance with the assumption (A6), equal to zero. Therefore, ̇𝑉𝑡Δ𝑇𝑡𝑄0Δ𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡𝛼2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡+𝑓0+Υ.(3.34) Now, considering that Δ𝑇𝑡𝑄0Δ𝑡=Δ𝑇𝑡𝑃1/2𝑃1/2𝑄0𝑃1/2𝑃1/2Δ𝑡 and using Rayleigh’s inequality, the following can be obtained: Δ𝑇𝑡𝑄0Δ𝑡𝜆min𝑃1/2𝑄0𝑃1/2Δ𝑇𝑡𝑃Δ𝑡(3.35) or alternatively Δ𝑇𝑡𝑄0Δ𝑡𝜆min𝑃1/2𝑄0𝑃1/2Δ𝑇𝑡𝑃Δ𝑡.(3.36) In view of (3.36), it is possible to establish that ̇𝑉𝑡𝜆min𝑃1/2𝑄0𝑃1/2Δ𝑇𝑡𝑃Δ𝑡𝛼2𝑘1𝑊tr𝑇1,𝑡𝑊1,𝑡𝛼2𝑘2𝑊tr𝑇2,𝑡𝑊2,𝑡𝛼2𝑘3𝑉tr𝑇1,𝑡𝑉1,𝑡𝛼2𝑘4𝑉tr𝑇2,𝑡𝑉2,𝑡+𝑓0+Υ.(3.37) As 𝛼=𝜆min(𝑃1/2𝑄0𝑃1/2), finally the following bound for ̇𝑉𝑡 can be concluded: ̇𝑉𝑡𝛼𝑉𝑡+𝑓0+Υ.(3.38) Equation (3.38) can be rewritten in the following form ̇𝑉𝑡+𝛼𝑉𝑡𝑓0+Υ.(3.39) Multiplying both sides of the last inequality by exp(𝛼𝑡), it is possible to obtain ̇𝑉exp(𝛼𝑡)𝑡+𝛼exp(𝛼𝑡)𝑉𝑡𝑓0+Υexp(𝛼𝑡).(3.40) The left-hand side of (3.40) can be rewritten as 𝑑𝑑𝑡exp(𝛼𝑡)𝑉𝑡𝑓exp(𝛼𝑡)0+Υ(3.41) or equivalently as 𝑑exp(𝛼𝑡)𝑉𝑡𝑓exp(𝛼𝑡)0+Υ𝑑𝑡.(3.42) Integrating both sides of the last inequality yields exp(𝛼𝑡)𝑉𝑡𝑉0𝑡0𝑓0+Υexp(𝛼𝜏)𝑑𝜏.(3.43) Adding 𝑉0 to both sides of the inequality, exp(𝛼𝑡)𝑉𝑡𝑉0+𝑡0𝑓0+Υexp(𝛼𝜏)𝑑𝜏.(3.44) Multiplying both sides of the inequality (3.44) by exp(𝛼𝑡), the following can be obtained: 𝑉𝑡exp(𝛼𝑡)𝑉0+exp(𝛼𝑡)𝑡0𝑓0+Υexp(𝛼𝜏)𝑑𝜏.(3.45) and, consequently, 𝑉𝑡𝑉0𝑓exp(𝛼𝑡)+0+Υ𝛼(1exp(𝛼𝑡)).(3.46) As 𝑃 and 𝑄0 are positive definite matrices, then 𝛼 is always a positive scalar and therefore 𝑉𝑡 is an upperly bounded function. However, in reference to (3.8), 𝑉𝑡 is also a nonnegative function. Consequently, Δ𝑡,𝑊1,𝑡,𝑊2,𝑡,𝑉1,𝑡,𝑉2,𝑡𝐿, and, thus, the first part of the Theorem 3.2 has been proven. With respect to the final part of this theorem, from (3.8), it is evident that Δ𝑇𝑡𝑃Δ𝑡𝑉𝑡. Besides, from Rayleigh’s inequality, 𝜆min(𝑃)Δ𝑇𝑡Δ𝑡Δ𝑇𝑡𝑃Δ𝑡. Consequently, 𝜆min(𝑃)Δ𝑇𝑡Δ𝑡𝑉𝑡. Nonetheless, in accordance with (3.46), 𝑉𝑡 is bounded by 𝑉0𝑓exp(𝛼𝑡)+((0+Υ)/𝛼)(1exp(𝛼𝑡)). This means that Δ𝑡𝑉0𝑓exp(𝛼𝑡)+0+Υ/𝛼(1exp(𝛼𝑡))𝜆min(𝑃).(3.47) Finally, taking the limit as 𝑡 of the last inequality, the last part of Theorem 3.2 has been proven.

Remark 3.3. Based on the results presented in [32, 33], and, from the inequality (3.38), uniform stability for the identification error can be guaranteed.

Remark 3.4. Although, in [34], the asymptotic convergence of the identification error to zero is proven for multilayer neural networks, the considered class of nonlinear systems is much more restrictive than in this work.

Remark 3.5. In [35], the main idea behind Theorem 3.2 was utilized but only for the single layer case. In this paper, the generalization for the multilayer case is presented for first time.

4. Tuning of the Multilayer Identifier

In this section, some details about the selection of the parameters for the neural identifier are presented. In first place, it is important to mention that the positive definite matrices Λ1𝑚×𝑚, Λ2𝑟×𝑟, Λ𝜎𝑛×𝑛, Λ𝜙𝑛×𝑛, Λ3𝑛×𝑛, and Λ4𝑛×𝑛 presented throughout assumptions (A2)–(A5) are known a priori. In fact, their selection can be very free. Although, in many cases, identity matrices can be enough, the corresponding freedom of selection can be used to satisfy the conditions specified in Remark 2.1.

Other important design decision is related to the proper number of elements 𝑚 or neurons for 𝜎(). A good point of departure is to select 𝑚=𝑛 where 𝑛 is the dimension of the state vector 𝑥𝑡. Normally, this selection is enough in order to produce adequate results. In other case, 𝑚 should be selected such as 𝑚>𝑛. With respect to 𝜙(), for simplicity, a first attempt could be to set the elements of this matrix as zeroes except for the main diagonal.

Another very important question which must be taken into account is the following: how should the weights 𝑊10,𝑊20,𝑉10,𝑉20 be selected? Ideally, these weights should be chosen in such a way that the modelling error or unmodeled dynamics 𝑓𝑡 can be minimized. Likewise, the design process must consider the solution of the Riccati equation (2.12). In order to guarantee the existence of a unique positive definite solution 𝑃 for (2.12), the conditions specified in Remark 2.1 must be satisfied. However, these conditions could not be fulfilled for the optimal weights. Consequently, different values for 𝑊10 and 𝑊20 could be tested until a solution for (2.12) can be found. At the same time, the designer should be aware of that as 𝑊10,𝑊20,𝑉10,𝑉20 take values increasingly different from the optimal ones, the upper bound for unmodeled dynamics in assumption B4 becomes greater. With respect to the initial values for 𝑊1,𝑡,𝑊2,𝑡,𝑉1,𝑡, and 𝑉2,𝑡, some authors, for example [26], simply select 𝑊1,0=𝑊10,𝑊2,0=𝑊20,𝑉1,0=𝑉10, and 𝑉2,0=𝑉20.

Finally, a last recommendation, in order to achieve a proper performance of the neural network, the variables in the identification process should be normalized. In this context, normalization means to divide each variable by its corresponding maximum value.

5. Numerical Example

In this section, a very simple but illustrative example is presented in order to clarify the tuning process of the neural identifier and compare the advantages of the scheme developed in this paper with respect to the results of previous works [2629].

Consider the following first-order nonlinear system:̇𝑥𝑡=𝑥𝑡𝑥+1.9sin𝑡𝑥+2.2cos𝑡𝑥3sin𝑡𝑥cos𝑡+𝑥𝑡𝑢𝑡,(5.1) with the initial condition 𝑥0=0.7 and the input given by 𝑢𝑡=sin(𝑡).

For simplicity, 𝜉𝑡 is assumed equal to zero. It is very important to note that (5.1) is only used as a data generator since apart from the assumptions (A1)–(A3), (B4), (A5)-(A6), none previous knowledge about the unknown system (5.1) is required to satisfactorily carry out the identification process.

The parameters of the neural network (2.2) and the learning laws (3.1) are selected as2𝜎(𝑧)=1+𝑒2𝑧11,𝜙(𝑦)=1+𝑒0.5𝑦𝐷0.5,𝜎=𝜕𝜎(𝑧)|||𝜕𝑧𝑧=𝑉01̂𝑥𝑡=4𝑒2𝑉01̂𝑥𝑡1+𝑒2𝑉01̂𝑥𝑡2,𝐷𝑖𝜙=𝜕𝜙𝑖(𝑦)||||𝜕𝑦𝑦=𝑉02̂𝑥𝑡=0.5𝑒0.5𝑉02̂𝑥𝑡1+𝑒0.5𝑉02̂𝑥𝑡2,(5.2)𝐴=5,Λ1=1,Λ2=1,Λ3=1,Λ𝜎Λ=1,𝜙=1,𝑢=1,𝑄0=2,𝑊01𝑊=1.5,02=0.5,𝑉01=0.9,𝑉02=1.1.(5.3) Note that Riccati equation (2.12) becomes a simple second-order algebraic equation for this case:𝑅𝑃2+2𝐴𝑃+𝑄=0.(5.4) As 𝑅=2𝑊01Λ11(𝑊01)𝑇+2𝑊02Λ21(𝑊02)𝑇+Λ31+Λ41,𝑄=Λ𝜎+𝑢Λ𝜙+𝑄0, and given the previous values for these parameters, (5.4) has the following solution: 𝑃=1. The rest of the parameters for the neural identifier are selected as 𝛼=2,𝑙1=4,𝑙2=0.0625,𝑘1=500,𝑘2=400,𝑘3=600,𝑘4=800. The initial condition for the neural identifier is selected as ̂𝑥0=0.1.

The results of the identification process are displayed in Figures 1 and 2. In Figure 1, the state 𝑥𝑡 of the nonlinear system (5.1) is represented by solid line whereas the state ̂𝑥𝑡 is represented by dashed line. Both states were obtained by using Simulink with the numerical method ode23s. In order to appreciate better the quality of the identification process, the absolute value of the identification error Δ𝑡=̂𝑥𝑡𝑥𝑡 is showed in Figure 2. Clearly, the new learning laws proposed in this paper exhibit a satisfactory behavior.

529176.fig.001
Figure 1: Identification process.
529176.fig.002
Figure 2: Identification error evolution.

Now, which is the practical advantage of this method with respect to previous works [2629]? The determination of 𝑓0 and 𝑓1 more still (parameters associated with assumption A.4) can result difficult. Besides, assuming that 𝑓1 is equal to zero, 𝑓0 can result excessively large inclusive for simple systems. For example, for system (5.1) and the values selected for the parameters of the identifier, 𝑓0 can approximately be estimated as 140. This implies that the learning laws (2.16) are activated only when |Δ𝑡|70 due to the dead-zone function 𝑠𝑡. Thus, although the results presented in works [2629] are technically right, on these conditions, that is, 𝑓070, the performance of the identifier results completely unsatisfactory from a practical point of view since the corresponding identification error is very high. To avoid this situation, it is necessary to be very careful with the selection of weights 𝑊01,𝑊02,𝑉01, and 𝑉02 in order to minimize the unmodeled dynamics 𝑓𝑡. However, with these optimal weights, the matrix Riccati equation could have no solution. This dilemma is overcome by means of the learning laws (3.1) developed in this paper. In fact, as can be appreciated, a priory knowledge of 𝑓0 is not required anymore for the proper implementation of (3.1).

6. Conclusions

In this paper, a modification of a learning law for multilayer differential neural networks is proposed. By this modification, the dead-zone function is not required anymore and a stronger result is here guaranteed: the exponential convergence of the identification error norm to a bounded zone. This result is thoroughly proven. First, the dynamics of the identification error is determined. Next, a proper nonnegative function is proposed. A bound for the first derivative of such function is established. This bound is formed by the negative of the original nonnegative function multiplied by a constant parameter 𝛼 plus a constant term. Thus, the convergence of the identification error to a bounded zone can be guaranteed. Apart from the theoretical importance of this result, from a practical point of view, the learning law here proposed is easier to implement and tune. A numerical example confirms the efficiency of this approach.

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful comments and advice which contributed to improve this paper. First author would like to thank the financial support through a postdoctoral fellowship from Mexican National Council for Science and Technology (CONACYT).

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