System Identification Using Multilayer Differential Neural Networks: A New Result

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 [1–4]. 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 [9–12]. Their success is based on their capability of providing arbitrarily good approximations to any smooth function [13–15] 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,limsup_𝑇→∞1𝑇^𝑇01−𝑓⁰+Υ𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2‖‖𝑃^1/2Δ_𝑡‖‖₊Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡≤𝑓₀+Υ,(1.1) where Δ_𝑡 is the identification error, 𝑄₀ is a positive definite matrix, 𝑓₀ 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]₊=𝑧0𝑧𝑧≥<00,.(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 [26–29] 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 Δ_𝑡 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 𝑓₀ 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 by𝑥̇_𝑡=𝑓𝑥_𝑡,𝑢_𝑡,𝑡+𝜉_𝑡,(2.1) where 𝑥_𝑡∈ℜ^𝑛 is the measurable state vector for 𝑡∈ℜ⁺∶={𝑡∶𝑡≥0},𝑢_𝑡∈ℜ^𝑞 is the control input, 𝑓∶ℜ^𝑛×ℜ^𝑞×ℜ⁺→ℜ^𝑛 is an unknown nonlinear vector function which represents the nominal dynamics of the system, and 𝜉_𝑡∈ℜ^𝑛 represents a deterministic disturbance. 𝑓(𝑥_𝑡,𝑢_𝑡,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(⋅)∶ℜ^𝑞→ℜ^𝑠.

Consider the following parallel structure of multilayer neural network𝑑𝑑𝑡𝑥̂^𝑡=𝐴𝑥̂_𝑡+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡,(2.2) where 𝑥̂_𝑡∈ℜ^𝑛 is the state of the neural network, 𝑢_𝑡∈ℜ^𝑞 is the control input, 𝐴∈ℜ^𝑛×𝑛 is a Hurwitz matrix which can be specified by the designer, the matrices 𝑊_1,𝑡∈ℜ^𝑛×𝑚 and 𝑊_2,𝑡∈ℜ^𝑛×𝑟 are the weights of output layers, the matrices 𝑉_1,𝑡∈ℜ^𝑚×𝑛 and 𝑉_2,𝑡∈ℜ^𝑟×𝑛 are the weights of hidden layers, 𝜎(⋅) is the activation vector-function with sigmoidal components, that is, 𝜎(⋅)∶=[𝜎₁(⋅),…,𝜎_𝑚(⋅)]^𝑇,𝜎_𝑗(𝑣)∶=𝑎^𝜎𝑗1+exp−∑^𝑚𝑖=1𝑐_{𝜎𝑗,𝑖}𝑣_𝑖−𝑑^𝜎𝑗,for𝑗=1,…,𝑚,(2.3) where 𝑎_𝜎𝑗, 𝑐_{𝜎𝑗,𝑖}, and 𝑑_𝜎𝑗 are positive constants which can be specified by the designer, 𝜙(⋅)∶ℜ^𝑟→ℜ^𝑟×𝑠 is also a sigmoidal function, that is,𝜙_𝑖𝑗(𝑧)∶=𝑎^𝜙𝑖𝑗1+exp−∑^𝑟𝑙=1𝑐_{𝜙𝑖𝑗,𝑙}𝑧_𝑙−𝑑^𝜙𝑖𝑗for𝑖=1,…,𝑟,𝑗=1,…,𝑠,(2.4) where 𝑎_𝜙𝑖𝑗, 𝑐_{𝜙𝑖𝑗,𝑙}, and 𝑑_𝜙𝑖𝑗 are positive constants which can be specified by the designer, 𝛾(⋅)∶ℜ^𝑞→ℜ^𝑠 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 𝑥_𝑡 and the input 𝑢_𝑡, adjusting on line the weights 𝑊_1,𝑡,𝑊_2,𝑡,𝑉_1,𝑡, and 𝑉_2,𝑡 by proper learning laws such that the identification error Δ_𝑡∶=𝑥̂_𝑡−𝑥_𝑡 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)‖≤𝐿₁‖𝑥−𝑧‖+𝐿₂‖𝑢−𝑣‖;𝑥,𝑧∈ℜ^𝑛;𝑢,𝑣∈ℜ^𝑞;0≤𝐿₁,𝐿₂<∞.(2.5)

(A2)The differences of functions 𝜎(⋅) and 𝜙(⋅) fulfil the generalized Lipschitz conditions 𝜎^𝑇𝑡Λ₁𝜎_𝑡≤Δ^𝑇𝑡Λ_𝜎Δ_𝑡,𝛾^𝑇𝑢_𝑡𝜙^𝑇𝑡Λ₂𝜙_𝑡𝛾𝑢_𝑡≤Δ^𝑇𝑡Λ_𝜙Δ_𝑡‖‖𝛾𝑢_𝑡‖‖²,(2.6) where 𝜎_𝑡∶=𝜎𝑉₁₀𝑥̂_𝑡−𝜎𝑉₁₀𝑥_𝑡,𝜙_𝑡∶=𝜙𝑉₂₀𝑥̂_𝑡−𝜙𝑉₂₀𝑥_𝑡,(2.7)Λ₁∈ℜ^𝑚×𝑚, Λ₂∈ℜ^𝑟×𝑟, Λ_𝜎∈ℜ^𝑛×𝑛, Λ_𝜙∈ℜ^𝑛×𝑛 are known positive definite matrices, 𝑉⁰¹∈ℜ^𝑚×𝑛 and 𝑉⁰²∈ℜ^𝑟×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: 𝜎_𝑡𝜎^𝑡∶=𝜎𝑉_1,𝑡𝑥̂_𝑡−𝜎𝑉₁₀𝑥̂_𝑡=𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+𝜈_𝜎,𝛾𝑢_𝑡∶==𝜙𝑉_2,𝑡𝑥̂_𝑡−𝜙𝑉⁰²𝑥̂_𝑡𝛾𝑢_𝑡(2.8)_𝑠_𝑖=1𝜙_𝑖𝑉_2,𝑡𝑥̂_𝑡−𝜙_𝑖𝑉⁰²𝑥̂_𝑡𝛾_𝑖𝑢_𝑡=^𝑠_𝑖=1𝐷^𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡+𝑣_𝑖𝜙𝛾^𝑖𝑢_𝑡,(2.9) where 𝐷_𝜎=𝜕𝜎(𝑌)𝜕𝑌|||_𝑌=𝑉⁰¹𝑥̂_𝑡∈ℜ^𝑚×𝑚,𝐷_𝑖𝜙=𝜕𝜙^𝑖(𝑍)𝜕𝑍||||_𝑍=𝑉02𝑥̂_𝑡∈ℜ^𝑟×𝑟,(2.10)𝜈_𝜎∈ℜ^𝑚 and 𝜈_𝑖𝜙∈ℜ^𝑛 are unknown vectors but bounded by ‖𝜈_𝜎‖^2Λ₁≤𝑙₁‖𝑉_1,𝑡𝑥̂_𝑡‖^2Λ₁, ‖𝜈_𝑖𝜙‖^2Λ₂≤𝑙₂‖𝑉_2,𝑡𝑥̂_𝑡‖^2Λ₂, respectively; 𝑉_1,𝑡∶=𝑉_1,𝑡−𝑉⁰¹, 𝑉_2,𝑡∶=𝑉_2,𝑡−𝑉⁰², 𝑙₁ and 𝑙₂ are positive constants which can be defined as 𝑙₁∶=4𝐿^2𝑔,1, 𝑙₂∶=4𝐿^2𝑔,2, where 𝐿_𝑔,1 and 𝐿_𝑔,2 are global Lipschitz constants for 𝜎(⋅) and 𝜙_𝑖(⋅), respectively.(A3)The nonlinear function 𝛾(⋅) is such that ‖𝛾(𝑢_𝑡)‖²≤𝑢 where 𝑢 is a known positive constant.(A4)Unmodeled dynamics 𝑓_𝑡 is bounded by ‖‖𝑓_𝑡‖‖^2Λ₃≤𝑓₀+𝑓₁‖‖𝑥_𝑡‖‖^2Λ₃,(2.11) where 𝑓₀ and 𝑓₁ are known positive constants and Λ₃∈ℜ^𝑛×𝑛 is a known positive definite matrix and 𝑓_𝑡 can be defined as 𝑓_𝑡∶=𝑓(𝑥_𝑡,𝑢_𝑡,𝑡)−𝐴𝑥_𝑡−𝑊⁰¹𝜎(𝑉⁰¹𝑥_𝑡)−𝑊⁰²𝜙(𝑉⁰²𝑥_𝑡)𝛾(𝑢_𝑡);𝑊⁰¹∈ℜ^𝑛×𝑚 and 𝑊⁰²∈ℜ^𝑛×𝑟 are constant matrices which can be selected by the designer.(A5)The deterministic disturbance is bounded, that is, ‖𝜉_𝑡‖^2Λ₄≤Υ, Λ₄ is a known positive definite matrix.(A6)The following matrix Riccati equation has a unique, positive definite solution 𝑃: 𝐴^𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄=0,(2.12) where 𝑅=2𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇+2𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇+Λ³⁻¹+Λ⁴⁻¹,𝑄=Λ_𝜎+𝑢Λ_𝜙+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,𝑅^1/2) is controllable, and the pair (𝑄^1/2,𝐴) is observable.(b)The following matrix inequality is fulfilled: 14𝐴^𝑇𝑅⁻¹−𝑅⁻¹𝐴𝑅𝐴^𝑇𝑅⁻¹−𝑅⁻¹𝐴^𝑇≤𝐴^𝑇𝑅⁻¹𝐴−𝑄.(2.14) Both conditions can relatively easily be fulfilled if 𝐴 is selected as a stable diagonal matrix.

(A7)It exists a bounded control 𝑢_𝑡, 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 𝜕𝑉⁰𝜕𝑥𝑓𝑥^𝑡,𝑢_𝑡,𝑡≤−𝜆‖‖𝑥_𝑡‖‖².(2.15) Additionally, the inequality 𝜆≥𝑓₁‖Λ₃‖ must be satisfied.

Now, consider the learning law: 𝑊𝑊̇̇_1,𝑡2,𝑡=−𝑠_𝑡𝐾₁𝑃Δ_𝑡𝜎^𝑇𝑉_1,𝑡𝑥̂_𝑡+𝑠_𝑡𝐾₁𝑃Δ_𝑡𝑥̂^𝑇𝑡𝑉^𝑇1,𝑡𝐷^𝑇𝜎,=−𝑠_𝑡𝐾₂𝑃Δ_𝑡𝛾^𝑇𝑢_𝑡𝜙^𝑇𝑉_2,𝑡𝑥̂_𝑡+𝑠_𝑡𝐾₂𝑃Δ_𝑡𝑥̂^𝑇𝑡𝑉^{𝑇𝑠2,𝑡}𝑉̇_1,𝑡_𝑖=1𝛾^𝑖𝑢_𝑡𝐷^𝑇𝑖𝜙,=−𝑠_𝑡𝐾₃𝐷^𝑇𝜎𝑊^𝑇1,𝑡𝑃Δ_𝑡𝑥̂^𝑇𝑡−𝑠_𝑡𝑙₁2𝐾³𝑉̇_2,𝑡Λ₁𝑉_1,𝑡𝑥̂_𝑡𝑥̂^𝑇𝑡,=−𝑠_𝑡𝐾_4𝑠_𝑖=1𝛾^𝑖𝑢_𝑡𝐷^𝑇𝑖𝜙𝑊^𝑇2,𝑡𝑃Δ_𝑡𝑥̂^𝑇𝑡−𝑠_𝑡𝑠𝑙₂𝑢2𝐾⁴Λ₂𝑉_2,𝑡𝑥̂_𝑡𝑥̂^𝑇𝑡,(2.16) where 𝑠 is the number of columns corresponding to 𝜙(⋅),𝐾₁∈ℜ^𝑛×𝑛,𝐾₂∈ℜ^𝑛×𝑛,𝐾₃∈ℜ^𝑚×𝑚, and 𝐾₄∈ℜ^𝑟×𝑟 are positive definite matrices which are selected by the designer. 𝑠_𝑡 is a dead-zone function which is defined as 𝑠_𝑡∶=1−𝜇‖‖𝑃^1/2Δ_𝑡‖‖₊𝜇=𝑓,₀[𝑧]₊=𝑧0𝑧𝑧≥<00,,+Υ𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2.(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 𝑊_1,𝑡, 𝑊_2,𝑡𝑉_1,𝑡, and 𝑉_2,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: Δ_𝑡,𝑊_1,𝑡,𝑊_2,𝑡,𝑉_1,𝑡,𝑉_2,𝑡∈𝐿_∞,(2.18)(b)the identification error Δ_𝑡 satisfies the following tracking performance:limsup_𝑇→∞1𝑇^𝑇01−𝑓⁰+Υ𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2‖‖𝑃^1/2Δ_𝑡‖‖₊Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡≤𝑓₀+Υ.(2.19)

In order to prove this result, the following nonnegative function was utilized:𝑉_𝑡∶=𝑉+⁰t+r‖‖𝑉𝑃^1/2_𝑇2,𝑡Δ_𝑡‖‖−𝜇²⁺+tr𝑊^𝑇1,𝑡𝐾¹⁻¹𝑊_1,𝑡+tr𝑊^𝑇2,𝑡𝐾²⁻¹𝑊_2,𝑡+tr𝑉^𝑇1,𝑡𝐾³⁻¹𝑉_1,𝑡𝐾⁴⁻¹𝑉_2,𝑡,(2.20) where 𝑊_1,𝑡∶=𝑊_1,𝑡−𝑊⁰¹; 𝑊_2,𝑡∶=𝑊_2,𝑡−𝑊⁰².

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 Ω∈ℜ^𝑛, unmodeled dynamics 𝑓_𝑡 is bounded by ‖𝑓_𝑡‖^2Λ₃≤𝑓₀ where 𝑓₀ 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, 𝑓_𝑡 is given by 𝑓_𝑡∶=𝑓(𝑥_𝑡,𝑢_𝑡,𝑡)−𝐴𝑥_𝑡−𝑊⁰¹𝜎(𝑉⁰¹𝑥_𝑡)−𝑊⁰²𝜙(𝑉⁰²𝑥_𝑡)𝛾(𝑢_𝑡). Note that 𝑊⁰¹𝜎(𝑉⁰¹𝑥_𝑡) and 𝑊⁰²𝜙(𝑉⁰²𝑥_𝑡)𝛾(𝑢_𝑡) are bounded functions because 𝜎(⋅) and 𝜙(⋅) are sigmoidal functions. As 𝑥_𝑡 belongs to Ω, clearly 𝑥_𝑡 is also bounded. Therefore, assumption B4 implies implicitly that 𝑓(𝑥_𝑡,𝑢_𝑡,𝑡) is a bounded function in a compact set Ω∈ℜ^𝑛.

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:𝑊̇_1,𝑡=−2𝑘₁𝑃Δ_𝑡𝜎^𝑇𝑉_1,𝑡𝑥̂_𝑡+2𝑘₁𝑃Δ_𝑡𝑥̂^𝑇𝑡𝑉^𝑇1,𝑡𝐷^𝑇𝜎−𝛼2𝑊^1,𝑡𝑊̇_2,𝑡,=−2𝑘₂𝑃Δ_𝑡𝛾^𝑇𝑢_𝑡𝜙^𝑇𝑉_2,𝑡𝑥̂_𝑡+2𝑘₂𝑃Δ_𝑡𝑥̂^𝑇𝑡𝑉^{𝑇𝑠2,𝑡}_𝑖=1𝛾^𝑖𝑢_𝑡𝐷^𝑇𝑖𝜙−𝛼2𝑊^2,𝑡𝑉̇_1,𝑡,=−2𝑘₃𝐷^𝑇𝜎𝑊^𝑇1,𝑡𝑃Δ_𝑡𝑥̂^𝑇𝑡−𝑘₃𝑙₁Λ₁𝑉_1,𝑡𝑥̂_𝑡𝑥̂^𝑇𝑡−𝛼2𝑉^1,𝑡𝑉̇_2,𝑡,=−2𝑘_4𝑠_𝑖=1𝛾^𝑖𝑢_𝑡𝐷^𝑇𝑖𝜙𝑊^𝑇2,𝑡𝑃Δ_𝑡𝑥̂^𝑇𝑡−𝑘₄𝑠𝑙₂𝑢Λ₂𝑉_2,𝑡𝑥̂_𝑡𝑥̂^𝑇𝑡−𝛼2𝑉^2,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∶=𝜆_min(𝑃^−1/2𝑄₀𝑃^−1/2); 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 𝑊_1,𝑡, 𝑊_2,𝑡, 𝑉_1,𝑡, and 𝑉_2,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: Δ_𝑡,𝑊_1,𝑡,𝑊_2,𝑡,𝑉_1,𝑡,𝑉_2,𝑡∈𝐿_∞,(3.2)(b)the norm of identification error converges exponentially to a region bounded given bylim_𝑡→∞‖‖𝑥_𝑡−𝑥̂_𝑡‖‖≤𝑓₀+Υ𝛼𝜆_min(𝑃).(3.3)

Proof of Theorem 3.2. Before beginning analysis, the dynamics of the identification error Δ_𝑡 must be determined. The first derivative of Δ_𝑡 is 𝑑Δ_𝑡𝑑𝑡=𝑑𝑑𝑡𝑥̂^𝑡−𝑥_𝑡.(3.4) Note that an alternative representation for (2.1) could be calculated as follows: 𝑥̇_𝑡=𝐴𝑥_𝑡+𝑊⁰¹𝜎𝑉⁰¹𝑥_𝑡+𝑊⁰²𝜙𝑉⁰²𝑥_𝑡𝛾𝑢_𝑡+𝑓_𝑡+𝜉_𝑡.(3.5) Substituting (2.2) and (3.5) into (3.4) yields Δ̇_𝑡=−𝐴𝑊𝑥̂_𝑡02+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡−𝐴𝑥_𝑡−𝑊⁰¹𝜎𝑉⁰¹𝑥_𝑡=𝐴Δ_𝑡𝜙𝑉⁰²𝑥_𝑡𝛾𝑢_𝑡−𝑓_𝑡−𝜉_𝑡,−𝑊₀₂+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡−𝑊⁰¹𝜎𝑉⁰¹𝑥_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡𝜙𝑉⁰²𝑥_𝑡𝛾𝑢_𝑡−𝑓_𝑡−𝜉_𝑡.(3.6) Subtracting and adding the terms 𝑊⁰¹𝜎(𝑉_1,𝑡𝑥̂_𝑡), 𝑊⁰¹𝜎(𝑉⁰¹𝑥̂_𝑡), 𝑊⁰²𝜙(𝑉_2,𝑡𝑥̂_𝑡)𝛾(𝑢_𝑡), and 𝑊⁰²𝜙(𝑉⁰²𝑥̂_𝑡)𝛾(𝑢_𝑡) and considering that 𝑊_1,𝑡∶=𝑊_1,𝑡−𝑊⁰¹, 𝑊_2,𝑡∶=𝑊_2,𝑡−𝑊⁰², 𝜎_𝑡∶=𝜎(𝑉₁₀𝑥̂_𝑡)−𝜎(𝑉₁₀𝑥_𝑡), 𝜙_𝑡∶=𝜙(𝑉₂₀𝑥̂_𝑡)−𝜙(𝑉₂₀𝑥_𝑡), 𝜎_𝑡∶=𝜎(𝑉_1,𝑡𝑥̂_𝑡)−𝜎(𝑉₁₀𝑥̂_𝑡), 𝜙′_𝑡𝛾(𝑢_𝑡)∶=(𝜙(𝑉_2,𝑡𝑥̂_𝑡)−𝜙(𝑉₂₀𝑥̂_𝑡))𝛾(𝑢_𝑡), (3.6) can be expressed as Δ̇_𝑡=−𝐴𝑊Δ_𝑡01+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡−𝑊⁰¹𝜎𝑉_1,𝑡𝑥̂_𝑡+𝑊⁰¹𝜎𝑉_1,𝑡𝑥̂_𝑡−𝑊⁰¹𝜎𝑉⁰¹𝑥̂_𝑡+𝑊⁰¹𝜎𝑉⁰¹𝑥̂_𝑡−𝑊₀₂𝜎𝑉⁰¹𝑥_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡−𝑊⁰²𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡+𝑊⁰²𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡𝜙𝑉⁰²𝑥̂_𝑡𝛾𝑢_𝑡+𝑊⁰²𝜙𝑉⁰²𝑥̂_𝑡𝛾𝑢_𝑡−𝑊⁰²𝜙𝑉⁰²𝑥_𝑡𝛾𝑢_𝑡−𝑓_𝑡−𝜉_𝑡=+𝐴𝑊Δ_𝑡02+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+𝑊⁰¹𝜎′_𝑡+𝑊⁰¹𝜎_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡+𝑊⁰²𝜙′_𝑡𝛾𝑢_𝑡Δ̇_𝑡𝜙_𝑡𝛾𝑢_𝑡−𝑓_𝑡−𝜉_𝑡,=𝐴Δ_𝑡+𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡+𝑊⁰¹𝜎_𝑡+𝑊⁰²𝜙_𝑡𝛾𝑢_𝑡+𝑊⁰¹𝜎′_𝑡+𝑊⁰²𝜙′_𝑡𝛾𝑢_𝑡−𝑓_𝑡−𝜉_𝑡.(3.7) In order to begin analysis, the following nonnegative function is selected: 𝑉_𝑡=Δ^𝑇𝑡𝑃Δ_𝑡+12𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡+12𝑘₂+1tr𝑊^𝑇2,𝑡𝑊_2,𝑡2𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡+12𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡,(3.8) where 𝑃 is a positive solution for the Riccati matrix equation given by (2.12). The first derivative of 𝑉_𝑡 is 𝑉_𝑡̇=𝑑𝑑𝑡Δ^𝑇𝑡𝑃Δ_𝑡+𝑑𝑑𝑡21𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡+𝑑𝑑𝑡21𝑘₂+𝑑tr𝑊^𝑇2,𝑡𝑊_2,𝑡𝑑𝑡21𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡+𝑑𝑑𝑡21𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡.(3.9) Each term of (3.9) will be calculated separately. For (𝑑/𝑑𝑡)(Δ^𝑇𝑡𝑃Δ_𝑡), 𝑑𝑑𝑡Δ^𝑇𝑡𝑃Δ_𝑡=2Δ^𝑇𝑡𝑃Δ̇_𝑡.(3.10) substituting (3.7) into (3.10) yields 𝑑𝑑𝑡Δ^𝑇𝑡𝑃Δ_𝑡=+2Δ2Δ^𝑇𝑡𝑃_𝑇𝑡𝐴Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡−2Δ_𝑇𝑡𝑃𝑊⁰¹𝜎_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰²𝜙_𝑡𝛾𝑢_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎^𝑡+2Δ^𝑇𝑡𝑃𝑊⁰²𝜙^𝑡𝛾𝑢_𝑡𝑃𝑓_𝑡−2Δ^𝑇𝑡𝑃𝜉_𝑡.(3.11) The terms 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎_𝑡, 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙_𝑡𝛾(𝑢_𝑡), −2Δ^𝑇𝑡𝑃𝑓_𝑡, and −2Δ^𝑇𝑡𝑃𝜉_𝑡 in (3.11) can be bounded using the following matrix inequality proven in [26]: 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝑋^𝑇Γ⁻¹𝑋+𝑌^𝑇Γ𝑌,(3.12) which is valid for any 𝑋,𝑌∈ℜ^𝑛×𝑘 and for any positive definite matrix 0<Γ=Γ^𝑇∈ℜ^𝑛×𝑛. Thus, for 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎_𝑡 and considering assumption (A2), 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎_𝑡=Δ^𝑇𝑡𝑃𝑊⁰¹𝜎_𝑡+𝜎^𝑇𝑡𝑊⁰¹^𝑇𝑃Δ_𝑡≤Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝜎^𝑇𝑡Λ₁𝜎_𝑡≤Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+Δ^𝑇𝑡Λ_𝜎Δ_𝑡.(3.13) For 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙_𝑡𝛾(𝑢_𝑡),and considering assumptions (A2) and (A3) 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙_𝑡𝛾𝑢_𝑡=Δ^𝑇𝑡𝑃𝑊⁰²𝜙_𝑡𝛾𝑢_𝑡+𝛾^𝑇𝑢_𝑡𝜙^𝑇𝑡𝑊⁰²^𝑇𝑃Δ_𝑡≤≤ΔΔ^𝑇𝑡_𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝛾^𝑇𝑢_𝑡𝜙^𝑇𝑡Λ₂𝜙_𝑡𝛾𝑢_𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝑢Δ^𝑇𝑡Λ_𝜙Δ_𝑡.(3.14) By using (3.12) and given assumptions (B4) and (A5), −2Δ^𝑇𝑡𝑃𝑓_𝑡 and −2Δ^𝑇𝑡𝑃𝜉_𝑡 can be bounded, respectively, by −2Δ^𝑇𝑡𝑃𝑓_𝑡=−Δ^𝑇𝑡𝑃𝑓_𝑡−𝑓^𝑇𝑡𝑃Δ_𝑡≤Δ^𝑇𝑡𝑃Λ³⁻¹𝑃Δ_𝑡+𝑓^𝑇𝑡Λ₃𝑓_𝑡−2Δ_𝑇𝑡≤Δ^𝑇𝑡𝑃Λ³⁻¹𝑃Δ_𝑡+𝑓₀,𝑃𝜉_𝑡=−Δ^𝑇𝑡𝑃𝜉_𝑡−𝜉^𝑇𝑡𝑃Δ_𝑡≤Δ^𝑇𝑡𝑃Λ⁴⁻¹𝑃Δ_𝑡+𝜉^𝑇𝑡Λ₄𝜉_𝑡≤Δ^𝑇𝑡𝑃Λ⁴⁻¹𝑃Δ_𝑡+Υ.(3.15) Considering (2.8), 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎′_𝑡 can be developed as 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎′_𝑡=2Δ^𝑇𝑡𝑃𝑊⁰¹𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰¹𝜈_𝜎.(3.16) By simultaneously adding and subtracting the term 2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡 into the right-hand side of (3.16), 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎′_𝑡=2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰¹𝜈_𝜎.(3.17) By using (3.12) and considering assumption (A2), the term Δ^𝑇𝑡𝑃𝑊⁰¹𝜈_𝜎 can be bounded as 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜈_𝜎=Δ^𝑇𝑡𝑃𝑊⁰¹𝜈_𝜎+𝜈^𝑇𝜎𝑊⁰¹^𝑇𝑃Δ_𝑡≤Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝜈^𝑇𝜎Λ₁𝜈_𝜎≤Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁.(3.18) And consequently, 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎′_𝑡 is bounded by 2Δ^𝑇𝑡𝑃𝑊⁰¹𝜎′_𝑡≤2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁.(3.19) For 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙′_𝑡𝛾(𝑢_𝑡) and considering (2.9), 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙′_𝑡𝛾𝑢_𝑡=2Δ^𝑇𝑡𝑃𝑊^02𝑠_𝑖=1𝐷^𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡+𝜈_𝑖𝜙=2Δ_𝑇𝑡𝛾^𝑖𝑢_𝑡𝑃𝑊^02𝑠_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡+2Δ^𝑇𝑡𝑃𝑊^02𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡.(3.20) Adding and subtracting the term 2Δ^𝑇𝑡𝑃𝑊_2,𝑡∑^𝑠𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖(𝑢_𝑡) into the right-hand side of (3.20), 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙′_𝑡𝛾𝑢_𝑡=2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡−2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1+2Δ_𝑇𝑡𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡𝑃𝑊^02𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡.(3.21) By using (3.12), 2Δ^𝑇𝑡𝑃𝑊⁰²∑^𝑠𝑖=1𝜈_𝑖𝜙𝛾_𝑖(𝑢_𝑡) can be bounded by 2Δ^𝑇𝑡𝑃𝑊^02𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡=Δ^𝑇𝑡𝑃𝑊^02𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡+^𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖(𝑢_𝑡)^𝑇𝑊⁰²^𝑇𝑃Δ_𝑡≤Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+^𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖(𝑢_𝑡)^𝑇Λ_2𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡,(3.22) but considering that ^𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖(𝑢_𝑡)^𝑇Λ_2𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡≤𝑠^𝑠_𝑖=1𝛾^2𝑖𝑢_𝑡𝜈^𝑇𝑖𝜙Λ₂𝜈_𝑖𝜙(3.23) and from assumptions (A2) and (A3), the following can be concluded: 2Δ^𝑇𝑡𝑃𝑊^02𝑠_𝑖=1𝜈_𝑖𝜙𝛾_𝑖𝑢_𝑡≤Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝑠𝑙₂𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂.(3.24) Thus, 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙′_𝑡𝛾(𝑢_𝑡) is bounded by 2Δ^𝑇𝑡𝑃𝑊⁰²𝜙′_𝑡𝛾𝑢_𝑡≤2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡−2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1+Δ_𝑇𝑡𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝑠𝑙₂𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂.(3.25) Consequently, given (3.13), (3.14), (3.15), (3.19), and (3.25), (𝑑/𝑑𝑡)(Δ^𝑇𝑡𝑃Δ_𝑡) can be bounded as 𝑑𝑑𝑡Δ^𝑇𝑡𝑃Δ_𝑡≤+ΔΔ^𝑇𝑡_𝑇𝑡𝐴^𝑇𝑃+𝑃𝐴Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+Δ^𝑇𝑡Λ_𝜎Δ_𝑡+Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡++𝑢2ΔΔ^𝑇𝑡_𝑇𝑡Λ_𝜙Δ_𝑡+Δ^𝑇𝑡𝑃Λ³⁻¹𝑃Δ_𝑡+𝑓₀+Δ^𝑇𝑡𝑃Λ⁴⁻¹𝑃Δ_𝑡+Υ𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁+2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1−2Δ_𝑇𝑡𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡+Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝑠𝑙₂𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂.(3.26) With respect to (𝑑/𝑑𝑡)((1/2𝑘₁)tr{𝑊^𝑇1,𝑡𝑊_1,𝑡}), using several properties of the trace of a matrix, 𝑑𝑑𝑡21𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡=12𝑘₁tr𝑑𝑑𝑡𝑊^𝑇1,𝑡𝑊_1,𝑡=21𝑘₁tr^⋅𝑊^𝑇1,𝑡𝑊_1,𝑡+𝑊^{𝑇⋅1,𝑡}𝑊_1,𝑡=12𝑘₁tr^⋅𝑊^𝑇1,𝑡𝑊_1,𝑡+tr𝑊^{𝑇⋅1,𝑡}𝑊_1,𝑡=𝑘1₁tr𝑊^{⋅𝑇1,𝑡}𝑊_1,𝑡.(3.27) As 𝑊_1,𝑡∶=𝑊_1,𝑡−𝑊⁰¹, the derivative of 𝑊_1,𝑡 is clearly 𝑊^⋅_1,𝑡=𝑊^⋅_1,𝑡. However, 𝑊^⋅_1,𝑡 is given by the learning law (3.1). Therefore, by substituting (3.1) into 𝑊^⋅_1,𝑡=𝑊^⋅_1,𝑡 and the corresponding expression into the right-hand side of (3.27), it is possible to obtain 𝑑𝑑𝑡21𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡=1𝑘₁tr−2𝑘₁𝜎𝑉_1,𝑡𝑥̂_𝑡Δ^𝑇𝑡𝑃𝑊_1,𝑡+2𝑘₁𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡Δ^𝑇𝑡𝑃𝑊_1,𝑡−𝛼2𝑊^𝑇1,𝑡𝑊_1,𝑡=−2tr𝜎𝑉_1,𝑡𝑥̂_𝑡Δ^𝑇𝑡𝑃𝑊_1,𝑡+2tr𝐷^𝜎𝑉_1,𝑡𝑥̂_𝑡Δ^𝑇𝑡𝑃𝑊_1,𝑡−𝛼2𝑘₁=−2trΔ_𝑇𝑡tr𝑊^𝑇1,𝑡𝑊_1,𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+2trΔ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−𝛼2𝑘₁=−2Δ_𝑇𝑡tr𝑊^𝑇1,𝑡𝑊_1,𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−𝛼2𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡.(3.28) Proceeding in a similar way for 𝑑𝑑𝑡21𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡,𝑑𝑑𝑡21𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡,𝑑𝑑𝑡21𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡,(3.29) it is possible to obtain 𝑑𝑑𝑡21𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡=−2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡+2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1−𝛼𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡2𝑘₂𝑑𝑡𝑟𝑊^𝑇2,𝑡𝑊_2,𝑡,𝑑𝑡21𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡=−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁−𝛼2𝑘₃𝑑tr𝑉^𝑇1,𝑡𝑉_1,𝑡,𝑑𝑡21𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡=−2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡∑_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡−𝑠𝑙₂𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂−𝛼2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡.(3.30) By substituting (3.26), (3.28), and (3.30) into (3.9), the following bound for 𝑉^⋅_𝑡 can be determined: 𝑉_𝑡̇≤+ΔΔ^𝑇𝑡_𝑇𝑡𝐴^𝑇𝑃+𝑃𝐴Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡+2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+Δ^𝑇𝑡Λ_𝜎Δ_𝑡+Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡++2𝑢ΔΔ^𝑇𝑡_𝑇𝑡Λ_𝜙Δ_𝑡+Δ^𝑇𝑡𝑃Λ³⁻¹𝑃Δ_𝑡+𝑓₀+𝑓₁𝑥^𝑇𝑡Λ₃𝑥_𝑡+Δ^𝑇𝑡𝑃Λ⁴⁻¹𝑃Δ_𝑡+Υ𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡+Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁+2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡−2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1+Δ_𝑇𝑡𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡+2Δ_𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+𝑠𝑙₂𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑉_1,𝑡𝑥̂_𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−𝛼2𝑘₁+2Δ_𝑇𝑡tr𝑊^𝑇1,𝑡𝑊_1,𝑡−2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾𝑢_𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡−𝛼2𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡−2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝐷_𝜎𝑉_1,𝑡𝑥̂_𝑡−𝑙₁‖‖𝑉_1,𝑡𝑥̂_𝑡‖‖^2Λ₁−𝛼2𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡−2Δ^𝑇𝑡𝑃𝑊_𝑠2,𝑡_𝑖=1−𝑠𝑙₂𝐷_𝑖𝜙𝑉_2,𝑡𝑥̂_𝑡𝛾_𝑖𝑢_𝑡𝑢‖‖𝑉_2,𝑡𝑥̂_𝑡‖‖^2Λ₂−𝛼2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡.(3.31) Simplifying like terms 𝑉_𝑡̇≤Δ^𝑇𝑡𝐴^𝑇𝑃+𝑃𝐴Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰¹Λ¹⁻¹𝑊⁰¹^𝑇𝑃Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊⁰²Λ²⁻¹𝑊⁰²^𝑇𝑃Δ_𝑡+−Δ𝛼^𝑇𝑡𝑃Λ³⁻¹𝑃Δ_𝑡+Δ^𝑇𝑡𝑃Λ⁴⁻¹𝑃Δ_𝑡+Δ^𝑇𝑡Λ_𝜎Δ_𝑡+𝑢Δ^𝑇𝑡Λ_𝜙Δ_𝑡+𝑓₀+Υ2𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡−𝛼2𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡−𝛼2𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡−𝛼2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡.(3.32) Adding and subtracting Δ^𝑇𝑡𝑄₀Δ_𝑡 into the right-hand side of the last inequality yields the expression 𝐴^𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄. That is 𝑉_𝑡̇≤−Δ^𝑇𝑡𝑄₀Δ_𝑡+Δ^𝑇𝑡𝐴^𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄Δ_𝑡+𝑓₀+Υ−𝛼2𝑘₁−𝛼tr𝑊^𝑇1,𝑡𝑊_1,𝑡2𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡−𝛼2𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡−𝛼2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡.(3.33) However, the expression 𝐴^𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄 is, in accordance with the assumption (A6), equal to zero. Therefore, 𝑉_𝑡̇≤−Δ^𝑇𝑡𝑄₀Δ_𝑡−𝛼2𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡−𝛼2𝑘₂tr𝑊^𝑇2,𝑡𝑊_2,𝑡−𝛼2𝑘₃−𝛼tr𝑉^𝑇1,𝑡𝑉_1,𝑡2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡+𝑓⁰+Υ.(3.34) Now, considering that Δ^𝑇𝑡𝑄₀Δ_𝑡=Δ^𝑇𝑡𝑃^1/2𝑃^−1/2𝑄₀𝑃^−1/2𝑃^1/2Δ_𝑡 and using Rayleigh’s inequality, the following can be obtained: Δ^𝑇𝑡𝑄₀Δ_𝑡≥𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2Δ^𝑇𝑡𝑃Δ_𝑡(3.35) or alternatively −Δ^𝑇𝑡𝑄₀Δ_𝑡≤−𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2Δ^𝑇𝑡𝑃Δ_𝑡.(3.36) In view of (3.36), it is possible to establish that 𝑉_𝑡̇≤−𝜆_min𝑃^−1/2𝑄₀𝑃^−1/2Δ^𝑇𝑡𝑃Δ_𝑡−𝛼2𝑘₁tr𝑊^𝑇1,𝑡𝑊_1,𝑡−𝛼2𝑘₂−𝛼tr𝑊^𝑇2,𝑡𝑊_2,𝑡2𝑘₃tr𝑉^𝑇1,𝑡𝑉_1,𝑡−𝛼2𝑘₄tr𝑉^𝑇2,𝑡𝑉_2,𝑡+𝑓⁰+Υ.(3.37) As 𝛼∶=𝜆_min(𝑃^−1/2𝑄₀𝑃^−1/2), finally the following bound for 𝑉_𝑡̇ can be concluded: 𝑉_𝑡̇≤−𝛼𝑉_𝑡+𝑓₀+Υ.(3.38) Equation (3.38) can be rewritten in the following form 𝑉_𝑡̇+𝛼𝑉_𝑡≤𝑓₀+Υ.(3.39) Multiplying both sides of the last inequality by exp(𝛼𝑡), it is possible to obtain exp(𝛼𝑡)𝑉_𝑡̇+𝛼exp(𝛼𝑡)𝑉_𝑡≤𝑓₀+Υexp(𝛼𝑡).(3.40) The left-hand side of (3.40) can be rewritten as 𝑑𝑑𝑡exp(𝛼𝑡)𝑉^𝑡≤exp(𝛼𝑡)𝑓₀+Υ(3.41) or equivalently as 𝑑exp(𝛼𝑡)𝑉_𝑡≤exp(𝛼𝑡)𝑓₀+Υ𝑑𝑡.(3.42) Integrating both sides of the last inequality yields exp(𝛼𝑡)𝑉_𝑡−𝑉₀≤^𝑡0𝑓⁰+Υexp(𝛼𝜏)𝑑𝜏.(3.43) Adding 𝑉₀ to both sides of the inequality, exp(𝛼𝑡)𝑉_𝑡≤𝑉₀+^𝑡0𝑓⁰+Υexp(𝛼𝜏)𝑑𝜏.(3.44) Multiplying both sides of the inequality (3.44) by exp(−𝛼𝑡), the following can be obtained: 𝑉_𝑡≤exp(−𝛼𝑡)𝑉₀+exp(−𝛼𝑡)^𝑡0𝑓⁰+Υexp(𝛼𝜏)𝑑𝜏.(3.45) and, consequently, 𝑉_𝑡≤𝑉₀exp(−𝛼𝑡)+𝑓⁰+Υ𝛼(1−exp(−𝛼𝑡)).(3.46) As 𝑃 and 𝑄₀ are positive definite matrices, then 𝛼 is always a positive scalar and therefore 𝑉_𝑡 is an upperly bounded function. However, in reference to (3.8), 𝑉_𝑡 is also a nonnegative function. Consequently, Δ_𝑡,𝑊_1,𝑡,𝑊_2,𝑡,𝑉_1,𝑡,𝑉_2,𝑡∈𝐿_∞, 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 Δ^𝑇𝑡𝑃Δ_𝑡≤𝑉_𝑡. Besides, from Rayleigh’s inequality, 𝜆_min(𝑃)Δ^𝑇𝑡Δ_𝑡≤Δ^𝑇𝑡𝑃Δ_𝑡. Consequently, 𝜆_min(𝑃)Δ^𝑇𝑡Δ_𝑡≤𝑉_𝑡. Nonetheless, in accordance with (3.46), 𝑉_𝑡 is bounded by 𝑉₀exp(−𝛼𝑡)+((𝑓₀+Υ)/𝛼)(1−exp(−𝛼𝑡)). This means that ‖‖Δ_𝑡‖‖≤⎷𝑉₀exp(−𝛼𝑡)+𝑓₀+Υ/𝛼(1−exp(−𝛼𝑡))𝜆_min(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 Λ₁∈ℜ^𝑚×𝑚, Λ₂∈ℜ^𝑟×𝑟, Λ_𝜎∈ℜ^𝑛×𝑛, Λ_𝜙∈ℜ^𝑛×𝑛, Λ₃∈ℜ^𝑛×𝑛, and Λ₄∈ℜ^𝑛×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 𝑥_𝑡. 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 𝑓_𝑡 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 𝑊_1,𝑡,𝑊_2,𝑡,𝑉_1,𝑡, and 𝑉_2,𝑡, some authors, for example [26], simply select 𝑊_1,0=𝑊₁₀,𝑊_2,0=𝑊₂₀,𝑉_1,0=𝑉₁₀, and 𝑉_2,0=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 [26–29].

Consider the following first-order nonlinear system:𝑥̇_𝑡=−𝑥_𝑡+1.9sin𝑥_𝑡+2.2cos𝑥_𝑡−3sin𝑥_𝑡cos𝑥_𝑡+𝑥_𝑡𝑢_𝑡,(5.1) with the initial condition 𝑥₀=0.7 and the input given by 𝑢_𝑡=sin(𝑡).

For simplicity, 𝜉_𝑡 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 as𝜎(𝑧)=21+𝑒^−2𝑧−1,𝜙(𝑦)=11+𝑒^−0.5𝑦𝐷−0.5,_𝜎=𝜕𝜎(𝑧)𝜕𝑧|||_𝑧=𝑉⁰¹𝑥̂_𝑡=4𝑒^−2𝑉01𝑥̂_𝑡1+𝑒^−2𝑉01𝑥̂_𝑡²,𝐷_𝑖𝜙=𝜕𝜙^𝑖(𝑦)𝜕𝑦||||_𝑦=𝑉02𝑥̂_𝑡=0.5𝑒^−0.5𝑉02𝑥̂_𝑡1+𝑒^−0.5𝑉02𝑥̂_𝑡²𝐴=−5,Λ₁,(5.2)Λ=1,Λ₂=1,Λ₃=1,Λ_𝜎=1,_𝜙𝑊=1,𝑢=1,𝑄₀=2,𝑊⁰¹=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=2𝑊⁰¹Λ¹⁻¹(𝑊⁰¹)^𝑇+2𝑊⁰²Λ²⁻¹(𝑊⁰²)^𝑇+Λ³⁻¹+Λ⁴⁻¹,𝑄=Λ_𝜎+𝑢Λ_𝜙+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 𝑥̂₀=0.1.

The results of the identification process are displayed in Figures 1 and 2. In Figure 1, the state 𝑥_𝑡 of the nonlinear system (5.1) is represented by solid line whereas the state 𝑥̂_𝑡 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 Δ_𝑡∶=𝑥̂_𝑡−𝑥_𝑡 is showed in Figure 2. Clearly, the new learning laws proposed in this paper exhibit a satisfactory behavior.

Figure 1 

Identification process.

Figure 2 

Identification error evolution.

Now, which is the practical advantage of this method with respect to previous works [26–29]? The determination of 𝑓₀ and 𝑓₁ more still (parameters associated with assumption A.4) can result difficult. Besides, assuming that 𝑓₁ is equal to zero, 𝑓₀ can result excessively large inclusive for simple systems. For example, for system (5.1) and the values selected for the parameters of the identifier, 𝑓₀ can approximately be estimated as 140. This implies that the learning laws (2.16) are activated only when |Δ_𝑡|≥70 due to the dead-zone function 𝑠_𝑡. Thus, although the results presented in works [26–29] are technically right, on these conditions, that is, 𝑓₀≈70, 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 𝑓_𝑡. 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 𝑓₀ 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).