Robust Adaptive Control via Neural Linearization and Compensation

Abstract

We propose a new type of neural adaptive control via dynamic neural networks. For a class of unknown nonlinear systems, a neural identifier-based feedback linearization controller is first used. Dead-zone and projection techniques are applied to assure the stability of neural identification. Then four types of compensator are addressed. The stability of closed-loop system is also proven.

1. Introduction

Feedback control of the nonlinear systems is a big challenge for engineer, especially when we have no complete model information. A reasonable solution is to identify the nonlinear, then a adaptive feedback controller can be designed based on the identifier. Neural network technique seems to be a very effective tool to identify complex nonlinear systems when we have no complete model information or, even, consider controlled plants as “black box”.

Neuroidentifier could be classified as static (feed forward) or as dynamic (recurrent) ones [1]. Most of publications in nonlinear system identification use static networks, for example multilayer perceptrons, which are implemented for the approximation of nonlinear function in the right-side hand of dynamic model equations [2]. The main drawback of these networks is that the weight updating utilize information on the local data structures (local optima) and the function approximation is sensitive to the training dates [3]. Dynamic neural networks can successfully overcome this disadvantage as well as present adequate behavior in presence of unmodeled dynamics because their structure incorporate feedback [4–6].

Neurocontrol seems to be a very useful tool for unknown systems, because it is model-free control, that is, this controller does not depend on the plant. Many kinds of neurocontrol were proposed in recent years, for example, supervised neuro control [7] is able to clone the human actions. The neural network inputs correspond to sensory information perceived by the human, and the outputs correspond to the human control actions. Direct inverse control [1] uses an inverse model of the plant cascaded with the plant, so the composed system results in an identity map between the desired response and the plant one, but the absence of feedback dismisses its robustness; internal model neurocontrol [8] that used forward and inverse model is within the feedback loop. Adaptive neurocontrol has two kinds of structure: indirect and direct adaptive control. Direct neuroadaptive may realize the neurocontrol by neural network directly [1]. The indirect method is the combination of the neural network identifier and adaptive control, the controller is derived from the on-line identification [5].

In this paper we extend our previous results in [9, 10]. In [9], the neurocontrol was derived by gradient principal, so the neural control is local optimal. No any restriction is needed, because the controller did not include the inverse of the weights. In [10], we assume the inverse of the weights exists, so the learning law was normal. The main contributions of this paper are (1) a special weights updating law is proposed to assure the existence of neurocontrol. (2) Four different robust compensators are proposed. By means of a Lyapunov-like analysis, we derive stability conditions for the neuroidentifier and the adaptive controller. We show that the neuroidentifier-based adaptive control is effective for a large classes of unknown nonlinear systems.

2. Neuroidentifier

The controlled nonlinear plant is given as𝑥_𝑡̇=𝑓𝑥_𝑡,𝑢_𝑡,𝑡,𝑥_𝑡∈ℜ^𝑛,𝑢_𝑡∈ℜ^𝑛,(1) where 𝑓(𝑥_𝑡) is unknown vector function. In order to realize indirect neural control, a parallel neural identifier is used as in [9, 10] (in [5] the series-parallel structure is used):𝑥̂_𝑡̇=𝐴𝑥̂_𝑡+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥̂_𝑡𝛾𝑢_𝑡,(2) where 𝑥̂_𝑡∈ℜ^𝑛 is the state of the neural network, 𝑊_1,𝑡,𝑊_2,𝑡∈ℜ^𝑛×𝑛 are the weight matrices, 𝐴∈ℜ^𝑛×𝑛 is a stable matrix. The vector functions𝜎(⋅)∈ℜ^𝑛,  𝜙(⋅)∈ℜ^𝑛×𝑛 is a diagonal matrix. Function 𝛾(⋅) is selected as ‖𝛾(𝑢_𝑡)‖²≤𝑢., for example 𝛾(⋅) may be linear saturation function, 𝛾𝑢_𝑡=𝑢^𝑡,if||𝑢_𝑡||<𝑏,𝑢,if||𝑢_𝑡||≥𝑏.(3) The elements of the weight matrices are selected as monotone increasing functions, a typical presentation is sigmoid function: 𝜎_𝑖𝑥̂_𝑡=𝑎^𝑖1+𝑒^−𝑏𝑖𝑥̂_𝑡−𝑐_𝑖,(4) where 𝑎_𝑖,𝑏_𝑖,𝑐_𝑖>0. In order to avoid 𝜙(𝑥̂_𝑡)=0, we select𝜙_𝑖𝑥̂_𝑡=𝑎^𝑖1+𝑒^−𝑏𝑖𝑥̂_𝑡+𝑐_𝑖.(5)

Remark 1. The dynamic neural network (2) has been discussed by many authors, for example [4, 5, 9, 10]. It can be seen that Hopfield model is the special case of this networks with 𝐴=diag{𝑎_𝑖},𝑎_𝑖∶=−1/𝑅_𝑖𝐶_𝑖,𝑅_𝑖>0 and 𝐶_𝑖>0.𝑅_𝑖 and 𝐶_𝑖 are the resistance and capacitance at the ith node of the network, respectively.

Let us define identification error asΔ_𝑡=𝑥̂_𝑡−𝑥_𝑡.(6) Generally, dynamic neural network (2) cannot follow the nonlinear system (1) exactly. The nonlinear system may be written as𝑥_𝑡̇=𝐴𝑥_𝑡+𝑊⁰¹𝜎𝑥_𝑡+𝑊⁰²𝜙𝑥_𝑡𝛾𝑢_𝑡−𝑓_𝑡,(7) where 𝑊⁰¹ and 𝑊⁰² are initial matrices of 𝑊_1,𝑡 and 𝑊_2,𝑡𝑊⁰¹Λ¹⁻¹𝑊^10𝑇≤𝑊₁,𝑊⁰²Λ²⁻¹𝑊^20𝑇≤𝑊₂.(8)𝑊₁ and 𝑊₂ are prior known matrices, vector function 𝑓_𝑡 can be regarded as modelling error and disturbances. Because 𝜎(⋅) and 𝜙(⋅) are chosen as sigmoid functions, clearly they satisfy the following Lipschitz property:𝜙_𝑡𝜎^𝑇Λ₁𝜎≤Δ^𝑇𝑡𝐷_𝜎Δ_𝑡,𝛾(𝑢_𝑡)^𝑇Λ₂𝜙^𝑡𝛾𝑢_𝑡≤𝑢Δ^𝑇𝑡𝐷_𝜙Δ_𝑡,(9) where 𝜎=𝜎(𝑥̂_𝑡)−𝜎(𝑥_𝑡),𝜙=𝜙(𝑥̂_𝑡)−𝜙(𝑥_𝑡),  Λ₁,  Λ₂,  𝐷_𝜎, and 𝐷_𝜙 are known positive constants matrices. The error dynamic is obtained from (2) and (7):Δ̇_𝑡=𝐴Δ_𝑡+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥̂_𝑡𝛾𝑢_𝑡+𝑊⁰¹𝜎+𝑊⁰²𝜙𝛾𝑢_𝑡+𝑓_𝑡,(10) where 𝑊_1,𝑡=𝑊_1,𝑡−𝑊⁰¹,𝑊_2,𝑡=𝑊_2,𝑡−𝑊⁰². As in [4, 5, 9, 10], we assume modeling error is bounded.(A1) the unmodeled dynamic 𝑓 satisfies𝑓^𝑇𝑡Λ^𝑓−1𝑓_𝑡≤𝜂.(11)Λ_𝑓 is a known positive constants matrix.

If we define𝑅=𝑊₁+𝑊₂+Λ_𝑓,𝑄=𝐷_𝜎+𝑢𝐷_𝜙+𝑄₀,(12) and the matrices 𝐴 and 𝑄₀ are selected to fulfill the following conditions:(1)the pair (𝐴,𝑅^1/2) is controllable, the pair (𝑄^1/2,𝐴) is observable,(2) local frequency condition [9] satisfies frequency condition: 𝐴^𝑇𝑅⁻¹𝐴−𝑄≥14𝐴^𝑇𝑅⁻¹−𝑅⁻¹𝐴𝑅𝐴^𝑇𝑅⁻¹−𝑅⁻¹𝐴^𝑇,(13) then the following assumption can be established. (A2) There exist a stable matrix 𝐴 and a strictly positive definite matrix 𝑄₀ such that the matrix Riccati equation:𝐴^𝑇𝑃+𝑃𝐴+𝑃𝑅𝑃+𝑄=0(14) has a positive solution 𝑃=𝑃^𝑇>0.

This condition is easily fulfilled if we select 𝐴 as stable diagonal matrix. Next Theorem states the learning procedure of neuroidentifier.

Theorem 2. Subject to assumptions A1 and A2   being satisfied, if the weights 𝑊_1,𝑡 and 𝑊_2,𝑡 are updated as 𝑊̇𝑊̇_1,𝑡=𝑠_𝑡−𝐾₁𝑃Δ_𝑡𝜎^𝑇𝑥̂_𝑡_2,𝑡,=𝑠_𝑡Pr−𝐾₂𝑃𝜙𝑥̂_𝑡𝛾𝑢_𝑡Δ^𝑇𝑡,(15) where 𝐾₁,𝐾₂>0,𝑃 is the solution of Riccati equation (14), Pr_𝑖[𝜔](𝑖=1,2) are projection functions which are defined as 𝜔=𝐾₂𝑃𝜙(𝑥̂_𝑡)𝛾(𝑢_𝑡)Δ^𝑇𝑡Pr[−𝜔]=⎧⎪⎨⎪⎩−−𝜔𝜔,+‖‖𝑊^2,𝑡condition,‖‖²𝑡𝑟𝑊^𝑇2,𝑡𝐾₂𝑃𝑊_2,𝑡𝜔otherwise,(16) where the “condition” is ‖𝑊_2,𝑡‖<𝑟 or [‖𝑊_2,𝑡‖=𝑟andtr(−𝜔𝑊_2,𝑡)≤0],𝑟<‖𝑊⁰²‖ is a positive constant. 𝑠_𝑡 is a dead-zone function 𝑠_𝑡=1,if‖‖Δ^𝑡0,otherw‖‖is²e>,𝜆^−1min𝑄₀𝜂,(17) then the weight matrices and identification error remain bounded, that is, Δ_𝑡∈𝐿_∞,𝑊_1,𝑡∈𝐿_∞,𝑊_2,𝑡∈𝐿_∞,(18) for any 𝑇>0 the identification error fulfills the following tracking performance: 1𝑇^𝑇0‖‖Δ_𝑡‖‖^2𝑄₀𝑑𝑡≤𝜅𝜂+Δ^𝑇0𝑃Δ₀𝑇,(19) where 𝜅 is the condition number of 𝑄₀ defined as 𝜅=𝜆_max(𝑄₀)/𝜆_min(𝑄₀).

Proof. Select a Lyapunov function as 𝑉_𝑡=Δ^𝑇𝑡𝑃Δ_𝑡+tr𝑊^𝑇1,𝑡𝐾¹⁻¹𝑊_1,𝑡+tr𝑊^𝑇2,𝑡𝐾²⁻¹𝑊_2,𝑡,(20) where 𝑃∈ℜ^𝑛×𝑛 is positive definite matrix. According to (10), the derivative is 𝑉_𝑡̇=+Δ2^𝑇𝑡Δ𝑃_𝑇𝑡𝐴+𝐴^𝑇𝑃Δ_𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑥̂_𝑡𝑃𝑊_2,𝑡𝜙𝑥̂_𝑡𝛾𝑢_𝑡+2Δ^𝑇𝑡𝑃𝑓_𝑡+2Δ^𝑇𝑡𝑃𝑊^∗1𝜎+𝑊^∗1𝜙𝛾𝑢_𝑡+2tr𝑊̇^𝑇1,𝑡𝐾¹⁻¹𝑊_1,𝑡+2tr𝑊̇_𝑇2,𝑡𝐾²⁻¹𝑊_2,𝑡.(21) Since Δ^𝑇𝑡𝑃𝑊^∗1𝜎_𝑡 is scalar, using (9) and matrix inequality 𝑋^𝑇𝑌+𝑋^𝑇𝑌^𝑇≤𝑋^𝑇Λ⁻¹𝑋+𝑌^𝑇Λ𝑌,(22) where 𝑋,𝑌,Λ∈ℜ^𝑛×𝑘 are any matrices, Λ is any positive definite matrix, we obtain 2Δ^𝑇𝑡𝑃𝑊^∗1𝜎_𝑡≤Δ^𝑇𝑡𝑃𝑊^∗1Λ¹⁻¹𝑊^1∗𝑇𝑃Δ_𝑡+𝜎^𝑇𝑡Λ₁𝜎_𝑡2Δ_𝑇𝑡≤Δ^𝑇𝑡𝑃𝑊₁𝑃+𝐷_𝜎Δ^𝑡,𝑃𝑊^∗2𝜙_𝑡𝛾𝑢_𝑡≤Δ^𝑇𝑡𝑃𝑊₂𝑃+𝑢𝐷_𝜙Δ^𝑡.(23) In view of the matrix inequality (22) and (A1), 2Δ^𝑇𝑡𝑃𝑓_𝑡≤Δ^𝑇𝑡𝑃Λ_𝑓𝑃Δ_𝑡+𝜂.(24) So we have 𝑉_𝑡̇≤Δ^𝑇𝑡+𝑃𝐷𝐴_𝜎+𝐴^𝑇𝑃+𝑃𝑊₁+𝑊₂+Λ_𝑓𝑃+𝑢𝐷_𝜙+𝑄₀Δ_𝑡+2tr𝑊̇^𝑇1,𝑡𝐾¹⁻¹𝑊_1,𝑡+2Δ^𝑇𝑡𝑃𝑊_1,𝑡𝜎𝑥̂_𝑡+𝜂−Δ^𝑇𝑡𝑄₀Δ_𝑡+2tr𝑊̇^𝑇2,𝑡𝐾²⁻¹𝑊_2,𝑡+2Δ^𝑇𝑡𝑃𝑊_2,𝑡𝜙𝑥̂_𝑡𝛾𝑢_𝑡.(25) Since 𝑊̇_1,𝑡=𝑊̇_1,𝑡 and 𝑊̇_2,𝑡=𝑊̇_2,𝑡, if we use (A2), we have 𝑉_𝑡̇≤2tr𝐾¹⁻¹𝑊̇^𝑇1,𝑡+𝐾₁𝑃Δ_𝑡𝜎^𝑇𝑥̂_𝑡𝑊_1,𝑡+𝜂−Δ^𝑇𝑡𝑄₀Δ_𝑡+2tr𝐾²⁻¹𝑊̇_2,𝑡+𝑃𝜙𝑥̂_𝑡𝛾𝑢_𝑡Δ^𝑇𝑡𝑊_2,𝑡.(26)(I)if ‖Δ_𝑡‖²>𝜆^−1min(𝑄₀)𝜂, using the updating law as (15) we can conclude that 𝑉_𝑡̇≤2trPr𝑃𝜙𝑥̂_𝑡𝛾𝑢_𝑡Δ^𝑇𝑡+𝑃𝜙𝑥̂_𝑡𝛾𝑢_𝑡Δ^𝑇𝑡𝑊_2,𝑡−Δ_𝑇𝑡𝑄₀Δ_𝑡+𝜂,(27)(a)if ‖𝑊_2,𝑡‖<𝑟 or [‖𝑊_2,𝑡‖=𝑟andtr(−𝜔𝑊_2,𝑡)≤0],𝑉_𝑡̇≤−𝜆_min(𝑄₀)‖Δ_𝑡‖²+𝜂<0,(b)if ‖𝑊_2,𝑡‖=𝑟 and tr(−𝜔𝑊_2,𝑡)>0𝑉_𝑡̇≤2tr⎧⎪⎨⎪⎩𝐾₂𝑃‖‖𝑊^2,𝑡‖‖²tr𝑊^𝑇2,𝑡𝐾₂𝑃𝑊_2,𝑡𝜔𝑊^2,𝑡⎫⎪⎬⎪⎭−Δ_𝑇𝑡≤−Δ_𝑇𝑡𝑄₀Δ_𝑡+𝜂𝑄₀Δ_𝑡+𝜂<0.(28)𝑉_𝑡 is bounded. Integrating (27) from 0 up to 𝑇 yields 𝑉_𝑇−𝑉₀≤−^𝑇0Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡+𝜂𝑇.(29) Because 𝜅≥1, we have ^𝑇0Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡≤≤𝑉𝑉₀₀−𝑉_𝑇+^𝑇0Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡≤𝑉₀+𝜂𝑇,+𝜅𝜂𝑇,(30) where 𝜅 is condition number of 𝑄₀(II)If ‖Δ_𝑡‖²≤𝜆^−1min(𝑄₀)𝜂, the weights become constants, 𝑉_𝑡 remains bounded. And^𝑇0Δ^𝑇𝑡𝑄₀Δ_𝑡𝑑𝑡≤≤𝜆^𝑇0𝜆_max𝑄₀‖‖Δ_𝑡‖‖²𝑑𝑡_max𝑄₀𝜆_min𝑄₀𝜂𝑇≤𝑉₀+𝜅𝜂𝑇.(31)
From (I) and (II), 𝑉_𝑡 is bounded, (18) is realized. From (20) and 𝑊_1,𝑡=𝑊_1,𝑡−𝑊⁰¹,𝑊_2,𝑡=𝑊_2,𝑡−𝑊⁰² we know 𝑉₀=Δ^𝑇0𝑃Δ₀. Using (30) and (31), (19) is obtained. The theorem is proved.

Remark 3. The weight update law (15) uses two techniques. The dead-zone 𝑠_𝑡 is applied to overcome the robust problem caused by unmodeled dynamic 𝑓_𝑡. In presence of disturbance or unmodeled dynamics, adaptive procedures may easily go unstable. The lack of robustness of parameters identification was demonstrated in [11] and became a hot issue in 1980s. Dead-zone method is one of simple and effective tool. The second technique is projection approach which may guarantee that the parameters remain within a constrained region and do not alter the properties of the adaptive law established without projection [12]. The projection approach proposed in this paper is explained in Figure 1. We hope to force 𝑊_2,𝑡 inside the ball of center 𝑊⁰² and radius 𝑟. If ‖𝑊_2,𝑡‖<𝑟, we use the normal gradient algorithm. When 𝑊_2,𝑡−𝑊⁰² is on the ball, and the vector 𝑊_2,𝑡 points either inside or along the ball, that is, (𝑑/𝑑𝑡)‖𝑊_2,𝑡‖²=2tr(−𝜔𝑊_2,𝑡)≤0, we also keep this algorithm. If tr(−𝜔𝑊_2,𝑡)>0,tr[(−𝜔+(‖𝑊_2,𝑡‖²/tr(𝑊^𝑇2,𝑡(𝐾₂𝑃)𝑊_2,𝑡))𝜔)𝑊_2,𝑡]<0, so (𝑑/𝑑𝑡)‖𝑊_2,𝑡‖²<0,𝑊_2,𝑡 are directed toward the inside or the ball, that is, 𝑊_2,𝑡 will never leave the ball. Since 𝑟<‖𝑊⁰²‖,𝑊_2,𝑡≠0.

Figure 1 

Projection algorithm.

Remark 4. Figure 1 and (7) show that the initial conditions of the weights influence identification accuracy. In order to find good initial weights, we design an offline method. From above theorem, we know the weights will convergence to a zone. We use any initial weights, W⁰¹ and W⁰², after 𝑇₀, the identification error should become smaller, that is, 𝑊_1,𝑇0and 𝑊_2,𝑇0 are better than 𝑊⁰¹ and 𝑊⁰². We use following steps to find the initial weights.(1)Start from any initial value for 𝑊⁰¹=𝑊_1,0,𝑊⁰²=𝑊_2,0.(2)Do identification until training time arrives 𝑇₀.(3)If the ‖Δ(𝑇₀)‖<‖Δ(0)‖, let 𝑊_1,𝑇0,𝑊_2,𝑇0 as a new 𝑊⁰¹ and 𝑊⁰², go to 2 to repeat the identification process.(4)If the ‖Δ(𝑇₀)‖≥‖Δ(0)‖, stop this offline identification, now 𝑊_1,𝑇0,𝑊_2,𝑇0 are the final initial weights.

Remark 5. Since the updating rate is 𝐾_𝑖𝑃 (𝑖=1,2), and 𝐾_𝑖 can be selected as any positive matrix, the learning process of the dynamic neural network (15) is free of the solution of Riccati equation (14).

Remark 6. Let us notice that the upper bound (19) turns out to be ‘‘sharp’’, that is, in the case of not having any uncertainties (exactly matching case: 𝑓=0) we obtain 𝜂=0 and, hence, limsup_𝑇→∞1𝑇^𝑇0‖‖Δ_𝑡‖‖^2𝑄₀𝑑𝑡=0(32) from which, for this special situation, the asymptotic stability property (‖Δ_𝑡‖→_𝑡→∞0) follows. In general, only the asymptotic stability ‘‘in average’’ is guaranteed, because the dead-zone parameter 𝜂 can be never set zero.

3. Robust Adaptive Controller Based on Neuro Identifier

From (7) we know that the nonlinear system (1) may be modeled as𝑥_𝑡̇==𝐴𝐴𝑥𝑥_𝑡𝑡+𝑊^∗1𝜎𝑥_𝑡+𝑊^∗2𝜙𝑥_𝑡𝛾𝑢_𝑡+𝑓+𝑓++𝑊𝑊_1,𝑡1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥_𝑡𝛾𝑢_𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥_𝑡𝛾𝑢_𝑡+𝑊^∗1,𝑡𝜎_𝑡+𝑊^∗1𝜙𝛾𝑢_𝑡.(33)

Equation (33) can be rewritten as𝑥_𝑡̇=𝐴𝑥_𝑡+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥_𝑡𝛾𝑢_𝑡+𝑑_𝑡,(34) where𝑑_𝑡=𝑓+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥_𝑡𝛾𝑢_𝑡+𝑊^∗1,𝑡𝜎_𝑡+𝑊^∗1𝜙𝛾𝑢_𝑡.(35) If updated law of 𝑊_1,𝑡 and 𝑊_2,𝑡 is (15), 𝑊_1,𝑡 and 𝑊_2,𝑡 are bounded. Using the assumption (A1), 𝑑_𝑡 is bounded as 𝑑=sup_𝑡‖𝑑_𝑡‖.

The object of adaptive control is to force the nonlinear system (1) following a optimal trajectory 𝑥^∗𝑡∈ℜ^𝑟 which is assumed to be smooth enough. This trajectory is regarded as a solution of a nonlinear reference model:𝑥̇^∗𝑡=𝜑𝑥^∗𝑡,𝑡,(36) with a fixed initial condition. If the trajectory has points of discontinuity in some fixed moments, we can use any approximating trajectory which is smooth. In the case of regulation problem 𝜑(𝑥^∗𝑡,𝑡)=0, 𝑥^∗(0)=𝑐, 𝑐 is constant. Let us define the sate trajectory error asΔ^∗𝑡=𝑥_𝑡−𝑥^∗𝑡.(37) From (34) and (36) we haveΔ̇^∗𝑡=𝐴𝑥_𝑡+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙𝑥_𝑡𝛾𝑢_𝑡+𝑑_𝑡−𝜑𝑥^∗𝑡,𝑡.(38) Let us select the control action 𝛾(𝑢_𝑡) as linear form𝛾𝑢_𝑡=𝑈_1,𝑡+𝑊_2,𝑡𝜙𝑥̂_𝑡⁻¹𝑈_2,𝑡,(39) where 𝑈_1,𝑡∈ℜ^𝑛 is direct control part and 𝑈_2,𝑡∈ℜ^𝑛 is a compensation of unmodeled dynamic 𝑑_𝑡. As 𝜑(𝑥^∗𝑡,𝑡),𝑥^∗𝑡, 𝑊_1,𝑡𝜎(𝑥̂_𝑡) and 𝑊_2,𝑡𝜙(𝑥̂_𝑡) are available, we can select 𝑈_1,𝑡 as𝑈_1,𝑡=𝑊_2,𝑡𝜙𝑥̂_𝑡⁻¹𝜑𝑥^∗𝑡,𝑡−𝐴𝑥^∗𝑡−𝑊_1,𝑡𝜎𝑥̂_𝑡.(40) Because 𝜙(𝑥̂_𝑡) in (5) is different from zero, and 𝑊_2,𝑡≠0 by the projection approach in Theorem 2. Substitute (39) and (40) into (38), we have So the error equation isΔ̇^∗𝑡=𝐴Δ^∗𝑡+𝑈_2,𝑡+𝑑_𝑡.(41) Four robust algorithms may be applied to compensate 𝑑_𝑡.

(A) Exactly Compensation
From (7) and (2) we have 𝑑_𝑡=𝑥_𝑡̇−𝑥̇̂_𝑡−𝐴𝑥_𝑡−𝑥̂_𝑡.(42) If 𝑥_𝑡̇ is available, we can select 𝑈_2,𝑡 as 𝑈^𝑎2,𝑡=−𝑑_𝑡, that is, 𝑈^𝑎2,𝑡=𝐴𝑥_𝑡−𝑥̂_𝑡−𝑥_𝑡̇−𝑥̇̂_𝑡.(43) So, the ODE which describes the state trajectory error is Δ̇^∗𝑡=𝐴Δ^∗𝑡.(44) Because 𝐴 is stable, Δ^∗𝑡 is globally asymptotically stable. lim_𝑡→∞Δ^∗𝑡=0.(45)

(B) An Approximate Method
If 𝑥_𝑡̇ is not available, an approximate method may be used as 𝑥̇_𝑡=𝑥^𝑡−𝑥_𝑡−𝜏𝜏+𝛿^𝑡,(46) where 𝛿_𝑡>0, is the differential approximation error. Let us select the compensator as 𝑈^𝑏2,𝑡=𝐴𝑥_𝑡−𝑥̂_𝑡−𝑥^𝑡−𝑥_𝑡−𝜏𝜏−𝑥̇̂^𝑡.(47) So 𝑈^𝑏2,𝑡=𝑈^𝑎2,𝑡+𝛿_𝑡, (44) become Δ̇^∗𝑡=𝐴Δ^∗𝑡+𝛿_𝑡.(48) Define Lyapunov-like function as 𝑉_𝑡=Δ^𝑡∗𝑇𝑃₂Δ^∗𝑡,𝑃₂=𝑃^𝑇2>0.(49) The time derivative of (49) is 𝑉_𝑡̇=Δ^∗𝑡𝐴^𝑇𝑃₂+𝑃₂𝐴Δ^∗𝑡+2Δ^𝑡∗𝑇𝑃₂𝛿_𝑡,(50)2Δ^𝑇𝑡𝑃₂𝛿_𝑡 can be estimated as 2Δ^𝑡∗𝑇𝑃₂𝛿_𝑡≤Δ^𝑡∗𝑇𝑃₂Λ𝑃₂Δ^∗𝑡+𝛿^𝑇𝑡Λ⁻¹𝛿_𝑡(51) where Λ is any positive define matrix. So (50) becomes 𝑉_𝑡̇≤Δ^∗𝑡𝐴^𝑇𝑃₂+𝑃₂𝐴+𝑃₂Λ𝑃₂+𝑄₂Δ^∗𝑡+𝛿^𝑇𝑡Λ⁻¹𝛿_𝑡−Δ^𝑡∗𝑇𝑄₂Δ^∗𝑡,(52) where 𝑄 is any positive define matrix. Because 𝐴 is stable, there exit Λ and 𝑄₂ such that the matrix Riccati equation: 𝐴^𝑇𝑃₂+𝑃₂𝐴+𝑃₂Λ𝑃₂+𝑄₂=0(53) has positive solution 𝑃₂=𝑃^𝑇2>0. Defining the following seminorms: ‖‖Δ^∗𝑡‖‖^2𝑄₂=lim_𝑇→∞1𝑇^𝑇0Δ^∗𝑡𝑄₂Δ^∗𝑡𝑑𝑡,(54) where 𝑄₂=𝑄₂>0 is the given weighting matrix, the state trajectory tracking can be formulated as the following optimization problem: 𝐽_min=min_𝑢𝑡𝐽,𝐽=‖‖𝑥_𝑡−𝑥^∗𝑡‖‖^2𝑄₂.(55) Note that lim_𝑇→∞1𝑇Δ^0∗𝑇𝑃₂Δ^∗0=0(56) based on the dynamic neural network (2), the control law (47) can make the trajectory tracking error satisfies the following property: ‖‖Δ^∗𝑡‖‖^2𝑄₂≤‖‖𝛿_𝑡‖‖^2Λ₋₁.(57) A suitable selection of Λ and 𝑄₂ can make the Riccati equation (53) has positive solution and make ‖Δ^∗𝑡‖^2𝑄₂ small enough if 𝜏 is small enough.

(C) Sliding Mode Compensation
If 𝑥_𝑡̇ is not available, the sliding mode technique may be applied. Let us define Lyapunov-like function as 𝑉_𝑡=Δ^𝑡∗𝑇𝑃₃Δ^∗𝑡,(58) where 𝑃₃ is a solution of the Lyapunov equation: 𝐴^𝑇𝑃₃+𝑃₃𝐴=−𝐼.(59) Using (41) whose time derivative is 𝑉_𝑡̇=Δ^∗𝑡𝐴^𝑇𝑃₃+𝑃₃𝐴Δ^∗𝑡+2Δ^𝑡∗𝑇𝑃₃𝑈_2,𝑡+2Δ^𝑡∗𝑇𝑃₃𝑑_𝑡.(60) According to sliding mode technique, we may select 𝑢_2,𝑡 as 𝑈^𝑐2,𝑡=−𝑘𝑃³⁻¹sgnΔ^∗𝑡,𝑘>0,(61) where 𝑘 is positive constant, sgnΔ^∗𝑡=⎧⎪⎨⎪⎩1Δ^∗𝑡0Δ_∗𝑡>0−1Δ_∗𝑡=0sgnΔ_∗𝑡<0=sgnΔ^∗1,𝑡,…sgnΔ^∗𝑛,𝑡^𝑇∈ℜ^𝑛.(62) Substitute (59) and (61) into (60) 𝑉_𝑡̇=−‖‖Δ^∗𝑡‖‖²−2𝑘‖‖Δ^∗𝑡‖‖+2Δ^𝑡∗𝑇𝑃𝑑_𝑡=≤−−‖‖‖‖ΔΔ^∗𝑡_∗𝑡‖‖²−2𝑘‖‖Δ^∗𝑡‖‖+2𝜆_max(𝑃)‖‖Δ^∗𝑡‖‖‖‖𝑑_𝑡‖‖‖‖²−2‖‖Δ^∗𝑡‖‖𝑘−𝜆_max(𝑃)‖‖𝑑_𝑡‖‖.(63) If we select 𝑘>𝜆_max𝑃₃𝑑,(64) where 𝑑 is define as (35), then 𝑉_𝑡̇<0. So, lim_𝑡→∞Δ^∗𝑡=0.(65)

(D) Local Optimal Control
If 𝑥_𝑡̇ is not available and 𝑥_𝑡̇ is not approximated as (B). In order to analyze the tracking error stability, we introduce the following Lyapunov function: 𝑉_𝑡Δ^∗𝑡=Δ^∗𝑡𝑃₄Δ^∗𝑡,𝑃₄=𝑃^𝑇4>0.(66) Using (41), whose time derivative is 𝑉_𝑡̇=Δ^∗𝑡𝐴^𝑇𝑃₄+𝑃₄𝐴Δ^∗𝑡+2Δ^𝑡∗𝑇𝑃₄𝑈_2,𝑡+2Δ^𝑡∗𝑇𝑃₄𝑑_𝑡,(67)2Δ^𝑡∗𝑇𝑃₄𝑑_𝑡 can be estimated as 2Δ^𝑡∗𝑇𝑃₄𝑑_𝑡≤Δ^∗𝑡𝑃₄Λ⁴⁻¹𝑃₄Δ^∗𝑡+𝑑^𝑇𝑡Λ₄𝑑_𝑡.(68) Substituting (68) in (67), adding and subtracting the term Δ^𝑡∗𝑇𝑄₄Δ^∗𝑡 and 𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡 with 𝑄₄=𝑄^𝑇4>0 and 𝑅₄=𝑅^𝑇4>0, we formulate 𝑉_𝑡̇≤Δ^∗𝑡𝐴^𝑇𝑃₄+𝑃₄𝐴+𝑃₄Λ₄𝑃₄+𝑄₄Δ^∗𝑡+2Δ^𝑡∗𝑇𝑃₄𝑈^𝑑2,𝑡+𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡+𝑑^𝑇𝑡Λ⁴⁻¹𝑑_𝑡−Δ^∗𝑡𝑄Δ^∗𝑡−𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡.(69) Because 𝐴 is stable, there exit Λ₄ and 𝑄₄ such that the matrix Riccati equation: 𝐴^𝑇𝑃₄+𝑃₄𝐴+𝑃₄Λ₄𝑃₄+𝑄₄=0.(70) So (69) is 𝑉_𝑡̇≤−‖‖Δ^∗𝑡‖‖^2𝑄₄+‖‖𝑈^𝑑2,𝑡‖‖^2𝑅₄+Ψ𝑈^𝑑2,𝑡+𝑑^𝑇𝑡Λ⁴⁻¹𝑑_𝑡,(71) where Ψ𝑈^𝑑2,𝑡=2Δ^𝑡∗𝑇𝑃₄𝑈^𝑑2,𝑡+𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡.(72) We reformulate (71) as ‖‖Δ^∗𝑡‖‖^2𝑄₄+‖‖𝑈^𝑑2,𝑡‖‖^2𝑅₄≤Ψ𝑈^𝑑2,𝑡+𝑑^𝑇𝑡Λ⁴⁻¹𝑑_𝑡−𝑉_𝑡̇.(73) Then, integrating each term from 0 to 𝜏, dividing each term by 𝜏, and taking the limit, for 𝜏→∞ of these integrals’ supreme, we obtain lim_𝑇→∞1𝑇^𝑇0Δ^𝑡∗𝑇𝑄₄Δ^∗𝑡𝑑𝑡+lim_𝑇→∞1𝑇^𝑇0≤𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡𝑑𝑡lim_𝑇→∞1𝑇^𝑇0𝑑^𝑇𝑡Λ⁴⁻¹𝑑_𝑡𝑑𝑡+lim_𝑇→∞1𝑇^𝑇0+Ψ𝑈^𝑑2,𝑡𝑑𝑡lim_𝑇→∞1𝑇^𝑇0−𝑉_𝑡̇𝑑𝑡.(74) In the view of definitions of the seminorms (55), we have ‖‖Δ^∗𝑡‖‖^2𝑄₄+‖‖𝑈^𝑑2,𝑡‖‖^2𝑅₄≤‖‖𝑑_𝑡‖‖^2Λ₄₋₁+lim_𝑇→∞1𝑇^𝑇0Ψ𝑈^𝑑2,𝑡𝑑𝑡.(75) It fixes a tolerance level for the trajectory-tracking error. So, the control goal now is to minimize Ψ(𝑈^𝑑2,𝑡) and ‖𝑑_𝑡‖^2Λ₄₋₁. To minimize ‖𝑑_𝑡‖^2Λ₄₋₁, we should minimize Λ⁴⁻¹. From (13), if select 𝑄₄ to make (70) have solution, we can choose the minimal Λ⁴⁻¹ as Λ⁴⁻¹=𝐴^−𝑇𝑄₄𝐴⁻¹.(76) To minimizing Ψ(𝑈^𝑑2,𝑡), we assume that, at the given 𝑡 (positive), 𝑥^∗(𝑡) and 𝑥̂(𝑡) are already realized and do not depend on 𝑈^𝑑2,𝑡. We name the 𝑈^{𝑑∗2,𝑡}(𝑡) as the locally optimal control, because it is calculated based only on “local” information. The solution of this optimization problem is given by subjmecint:𝐴Ψ₀𝑢^𝑑2,𝑡=2Δ^𝑡∗𝑇𝑃₄𝑢^𝑑2,𝑡+𝑈^{𝑑𝑇2,𝑡}𝑅₄𝑈^𝑑2,𝑡.𝑈^1,𝑡+𝑈^𝑑2,𝑡≤𝐵⁰.(77) It is typical quadratic programming problem. Without restriction 𝑈^∗ is selected according to the linear squares optimal control law: 𝑢^𝑑2,𝑡=−2𝑅⁴⁻¹𝑃₄Δ^∗𝑡.(78)

Remark 7. Approaches (A) and (C) are exactly compensations of 𝑑_𝑡, Approach (A) needs the information of 𝑥̇_𝑡. Because Approach (C) uses the sliding mode control  𝑈^𝑐2,𝑡 that is inserted in the closed-loop system, chattering occurs in the control input which may excite unmodeled high-frequency dynamics. To eliminate chattering, the boundary layer compensator can be used, it offers a continuous approximation to the discontinuous sliding mode control law inside the boundary layer and guarantees the output tracking error within any neighborhood of the origin [13].
Finally, we give following design steps for the robust neurocontrollers proposed in this paper.(1)According to the dimension of the plant (1), design a neural networks identifier (2) which has the same dimension as the plant. In (2), 𝐴 can be selected a stable matrix. 𝐴 will influence the dynamic response of the neural network. The bigger eigenvalues of 𝐴 will make the neural network slower. The initial conditions for 𝑊_1,𝑡 and 𝑊_2,𝑡 are obtained as in Remark 4.(2)Do online identification. The learning algorithm is (15) with the dead zone in Theorem 2. We assume we know the upper bound of modeling error, we can give a value for 𝜂.  𝑄₀ is chosen such that Riccati equation (14) has positive defined solution, 𝑅 can be selected as any positive defined matrix because Λ¹⁻¹ is arbitrary positive defined matrix. The updating rate in the learning algorithm (15) is 𝐾₁𝑃, and 𝐾₁ can be selected as any positive defined matrix, so the learning process is free of the solution 𝑃 of the Riccati equations (14). The larger 𝐾₁𝑃 is selected, the faster convergence the neuroidentifier has.(3)Use robust control (39) and one of compensation of (43), (47), (61), and (78).

4. Simulation

In this section, a two-link robot manipulator is used to illustrate the proposed approach. Its dynamics of can be expressed as follows [14]:𝑀(𝜃)𝜃^..+𝑉𝜃,𝜃̇𝜃̇+𝐺(𝜃)+𝐹_𝑑𝜃̇=𝜏,(79) where 𝜃∈ℜ² consists of the joint variables, 𝜃̇∈ℜ² denotes the links velocity, 𝜏 is the generalized forces, 𝑀(𝜃) is the intertie matrix, 𝑉(𝜃,𝜃̇) is centripetal-Coriolis matrix, and 𝐺(𝜃) is gravity vector, 𝐹_𝑑(𝜃̇) is the friction vector. 𝑀(𝜃) represents the positive defined inertia matrix. If we define 𝑥₁=𝜃=[𝜃₁,𝜃₂] is joint position, 𝑥₂=𝜃̇ is joint velocity of the link, 𝑥_𝑡=[𝑥₁,𝑥₂]^𝑇, (79) can be rewritten as state space form [15]:𝑥̇₂𝑥̇₁=𝑥₂,=𝐻𝑥_𝑡,𝑢_𝑡,(80) where 𝑢_𝑡=𝜏 is control input,𝐻𝑥_𝑡,𝑢_𝑡=−𝑀𝑥₁⁻¹𝐶𝑥₁,𝑥₂𝑥̇₁+𝐺𝑥₁+𝐹𝑥̇₁+𝑢_𝑡.(81) Equation (80) can also be rewritten as𝑥̇₁=^𝑡0𝐻𝑥_𝜏,𝑢_𝜏𝑑𝜏+𝐻𝑥₀,𝑢₀.(82) So the dynamic of the two-link robot (79) is in form of (1) with𝑓𝑥_𝑡,𝑢_𝑡,𝑡=^𝑡0𝐻𝑥_𝜏,𝑢_𝜏𝑑𝜏+𝐻𝑥₀,𝑢₀.(83) The values of the parameters are listed below: 𝑚₁=𝑚₂=1.53kg,  𝑙₁=𝑙₂=0.365m,  𝑟1=𝑟2=0.1,  𝑣1=𝑣2=0.4,  𝑘1=𝑘2=0.8. Let define 𝑥̂=[𝜃̂₁,𝜃̂₂]^𝑇, and 𝑢=[𝜏₁,𝜏₂]^𝑇, the neural network for control is represented as𝑥̇̂=𝐴𝑥̂+𝑊_1,𝑡𝜎𝑥̂_𝑡+𝑊_2,𝑡𝜙(𝑥̂)𝑢.(84) We select 𝐴=^−1.500−1,𝜙(𝑥̂_𝑡)=diag(𝜙₁(𝑥̂₁),𝜙₂(𝑥̂₂)),𝜎(𝑥̂_𝑡)=[𝜎₂(𝑥̂₂),𝜎₂(𝑥̂₂)]^𝑇𝜎_𝑖𝑥̂_𝑖=21+𝑒⁻²𝑥̂_𝑖−1𝜙_𝑖2,𝑥̂_𝑖=21+𝑒⁻²𝑥̂_𝑖+12,(85) where 𝑖=1,2. We used Remark 4  to obtain a suitable 𝑊⁰¹ and 𝑊⁰², start from random values, 𝑇₀=100. After 2 loops, ‖Δ(𝑇₀)‖ does not decrease, we let the 𝑊_1,300 and 𝑊_2,300 as the new 𝑊⁰¹=^{0.513.8−2.31.51} and 𝑊⁰²=^{3.12−2.785.52−4.021}. For the update laws (15), we select 𝜂=0.1,  𝑟=5,  𝐾₁𝑃=𝐾₁𝑃=⁵⁰⁰². If we select the generalized forces as𝜏₁=7sin𝑡,𝜏₂=0.(86)

Now we check the neurocontrol. We assume the robot is changed at 𝑡=480, after that 𝑚₁=𝑚₂=3.5kg,  𝑙₁=𝑙₂=0.5m, and the friction becomes disturbance as 𝐷sin((𝜋/3)𝑡),𝐷 is a positive constant. We compare neurocontrol with a PD control as𝜏_PD=−10𝜃−𝜃^∗−5𝜃̇−𝜃̇^∗,(87) where 𝜃^∗1=3;𝜃^∗2 is square wave. So 𝜑(𝜃^∗)=𝜃̇^∗=0.

The neurocontrol is  (39)𝜏_neuro=+𝑊𝑊_2,𝑡𝜙(𝑥̂)⁺𝜑𝑥^∗𝑡,𝑡−𝐴𝑥^∗𝑡−𝑊_1,𝑡𝜎(𝑥̂)_2,𝑡𝜙(𝑥̂)⁺𝑈_2,𝑡.(88)𝑈_2,𝑡 is selected to compensate the unmodeled dynamics. Sine 𝑓 is unknown method. (A) exactly compensation, cannot be used.

(B) 𝐷=1. The link velocity 𝜃̇ is measurable, as in (43),𝑈_2,𝑡=𝐴𝜃−𝜃̂−𝜃̇−𝜃̇̂.(89) The results are shown in Figures 2 and 3.

Figure 2 

Tracking control of 𝜃1 (method B).

Figure 3 

Tracking control of 𝜃2 (method B).

(C) 𝐷=0.3.𝜃̇ is not available, the sliding mode technique may be applied. we select 𝑢_2,𝑡 as   (61).𝑢_2,𝑡=−10×sgn𝜃−𝜃^∗.(90) The results are shown in Figures 4 and 5.

Figure 4 

Tracking control of 𝜃1 (method C).

Figure 5 

Tracking control of 𝜃2 (method C).

(D) 𝐷=3. We select 𝑄=1/2,  𝑅=1/20,  Λ=4.5, the solution of following Riccati equation:𝐴^𝑇𝑃+𝑃𝐴+𝑃Λ𝑃_𝑡+𝑄=−𝑃̇(91) is 𝑃=^0.33000.33. If without restriction 𝜏, the linear squares optimal control law:𝑢_2,𝑡=−2𝑅⁻¹𝑃𝜃−𝜃^∗=⎡⎢⎢⎣−020−020⎤⎥⎥⎦𝜃−𝜃^∗.(92) The results of local optimal compensation are shown in Figures 6 and 7.

Figure 6 

Tracking control of 𝜃1 (method D).

Figure 7 

Tracking control of 𝜃2 (method D).

We may find that the neurocontrol is robust and effective when the robot is changed.

5. Conclusion

By means of Lyapunov analysis, we establish bounds for both the identifier and adaptive controller. The main contributions of our paper is that we give four different compensation methods and prove the stability of the neural controllers.