# Robust Adaptive Control via Neural Linearization and Compensation

Authors: Roberto Carmona Rodríguez; Wen Yu
Journal: Journal of Control Science and Engineering
 (2012)

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## 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.


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## Body

## 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 as(1)xt.=f(xt,ut,t),xt∈Rn,ut∈Rn,
where f(xt) 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-parallelstructure is used):(2)x̂t.=Ax̂t+W1,tσ(x̂t)+W2,tϕ(x̂t)γ(ut),
where x̂t∈ℜn is the state of the neural network, W1,t,W2,t∈ℜn×n are the weight matrices, A∈ℜn×n is a stable matrix. The vector functionsσ(·)∈ℜn,  ϕ(·)∈ℜn×n is a diagonal matrix. Function γ(·) is selected as ∥γ(ut)∥2≤u¯., for example γ(·) may be linear saturation function,
(3)γ(ut)={ut,if|ut|<b,u¯,if|ut|≥b.
The elements of the weight matrices are selected as monotone increasing functions, a typical presentation is sigmoid function: (4)σi(x̂t)=ai1+e-bix̂t-ci,
where ai,bi,ci>0. In order to avoid ϕ(x̂t)=0, we select(5)ϕi(x̂t)=ai1+e-bix̂t+ci.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{ai},ai:=-1/RiCi,Ri>0 and Ci>0.Ri and Ci are the resistance and capacitance at the ith node of the network, respectively.Let us define identification error as(6)Δt=x̂t-xt.
Generally, dynamic neural network (2) cannot follow the nonlinear system (1) exactly. The nonlinear system may be written as(7)xt.=Axt+W10σ(xt)+W20ϕ(xt)γ(ut)-f̃t,
where W10 and W20 are initial matrices of W1,t and W2,t(8)W10Λ1-1W10T≤W¯1,W20Λ2-1W20T≤W¯2.W¯1 and W¯2 are prior known matrices, vector function f̃t can be regarded as modelling error and disturbances. Because σ(·) and ϕ(·) are chosen as sigmoid functions, clearly they satisfy the following Lipschitzproperty:(9)σ̃TΛ1σ̃≤ΔtTDσΔt,(ϕ̃tγ(ut))TΛ2(ϕ̃tγ(ut))≤u¯ΔtTDϕΔt,
where σ̃=σ(x̂t)-σ(xt),ϕ̃=ϕ(x̂t)-ϕ(xt),  Λ1,  Λ2,  Dσ, and Dϕ are known positive constants matrices. The error dynamic is obtained from (2) and (7):(10)Δ.t=AΔt+W̃1,tσ(x̂t)+W̃2,tϕ(x̂t)γ(ut)+W10σ̃+W20ϕ̃γ(ut)+f̃t,
where W̃1,t=W1,t-W10,W̃2,t=W2,t-W20. As in [4, 5, 9, 10], we assume modeling error is bounded.(A1)
the unmodeled dynamicf̃ satisfies(11)f̃tTΛf-1f̃t≤η¯.Λf is a known positive constants matrix.If we define(12)R=W¯1+W¯2+Λf,Q=Dσ+u¯Dϕ+Q0,
and the matrices A and Q0 are selected to fulfill the following conditions:(1)
the pair (A,R1/2) is controllable, the pair (Q1/2,A) is observable,(2)
local frequency condition [9] satisfies frequency condition:
(13)ATR-1A-Q≥14[ATR-1-R-1A]R[ATR-1-R-1A]T,
then the following assumption can be established.(A2)
There exist a stable matrixA and a strictly positive definite matrix Q0 such that the matrix Riccati equation:(14)ATP+PA+PRP+Q=0
has a positive solution P=PT>0.This condition is easily fulfilled if we selectA as stable diagonal matrix. Next Theorem states the learning procedure of neuroidentifier.Theorem 2.
Subject to assumptionsA1 and A2   being satisfied, if the weights W1,t and W2,t are updated as
(15)W.1,t=st[-K1PΔtσT(x̂t)],W.2,t=stPr[-K2Pϕ(x̂t)γ(ut)ΔtT],
where K1,K2>0,P is the solution of Riccati equation (14), Pri[ω](i=1,2) are projection functions which are defined as ω=K2Pϕ(x̂t)γ(ut)ΔtT(16)Pr[-ω]={-ω,condition,-ω+‖W̃2,t‖2tr(W̃2,tT(K2P)W̃2,t)ωotherwise,
where the “condition” is ∥W̃2,t∥<r or [∥W̃2,t∥=randtr(-ωW̃2,t)≤0],r<∥W20∥ is a positive constant. st is a dead-zone function
(17)st={1,if‖Δt‖2>λmin⁡-1(Q0)η¯,0,otherwise,
then the weight matrices and identification error remain bounded, that is,
(18)Δt∈L∞,W1,t∈L∞,W2,t∈L∞,
for any T>0 the identification error fulfills the following tracking performance:
(19)1T∫0T‖Δt‖Q02dt≤κη¯+Δ0TPΔ0T,
where κ is the condition number of Q0 defined as κ=λmax(Q0)/λmin(Q0).Proof.
Select a Lyapunov function as(20)Vt=ΔtTPΔt+tr⁡{W̃1,tTK1-1W̃1,t}+tr⁡{W̃2,tTK2-1W̃2,t},
where P∈ℜn×n is positive definite matrix. According to (10), the derivative is
(21)V.t=ΔtT(PA+ATP)Δt+2ΔtTPW̃1,tσ(x̂t)+2ΔtTPW̃2,tϕ(x̂t)γ(ut)+2ΔtTPf̃t+2ΔtTP[W1*σ̃+W1*ϕ̃γ(ut)]+2tr⁡{W̃.1,tTK1-1W̃1,t}+2tr⁡{W̃.2,tTK2-1W̃2,t}.
Since ΔtTPW1*σ̃t is scalar, using (9) and matrix inequality
(22)XTY+(XTY)T≤XTΛ-1X+YTΛY,
where X,Y,Λ∈ℜn×k are any matrices, Λ is any positive definite matrix, we obtain
(23)2ΔtTPW1*σ̃t≤ΔtTPW1*Λ1-1W1*TPΔt+σ̃tTΛ1σ̃t≤ΔtT(PW¯1P+Dσ)Δt,2ΔtTPW2*ϕ̃tγ(ut)≤ΔtT(PW¯2P+u¯Dϕ)Δt.
In view of the matrix inequality (22) and (A1),
(24)2ΔtTPf̃t≤ΔtTPΛfPΔt+η¯.
So we have
(25)V.t≤ΔtT[PA+ATP+P(W¯1+W¯2+Λf)P+(Dσ+u¯Dϕ+Q0)]Δt+2tr⁡{W̃.1,tTK1-1W̃1,t}+2ΔtTPW̃1,tσ(x̂t)+η¯-ΔtTQ0Δt+2tr⁡{W̃.2,tTK2-1W̃2,t}+2ΔtTPW̃2,tϕ(x̂t)γ(ut).
Since W̃1,t.=W.1,t and W̃2,t.=W.2,t, if we use (A2), we have
(26)V.t≤2tr⁡{[K1-1W.1,tT+K1PΔtσT(x̂t)]W̃1,t}+η¯-ΔtTQ0Δt+2tr⁡{[K2-1W.2,t+Pϕ(x̂t)γ(ut)ΔtT]W̃2,t}.(I)
if∥Δt∥2>λmin-1(Q0)η¯, using the updating law as (15) we can conclude that
(27)V.t≤2tr⁡{[Pr[Pϕ(x̂t)γ(ut)ΔtT]+Pϕ(x̂t)γ(ut)ΔtT]W̃2,t}-ΔtTQ0Δt+η¯,
(a)
if∥W̃2,t∥<r or [∥W̃2,t∥=randtr(-ωW̃2,t)≤0],V.t≤-λmin(Q0)∥Δt∥2+η¯<0,(b)
if∥W̃2,t∥=r  and  tr(-ωW̃2,t)>0(28)V.t≤2tr⁡{K2P‖W̃2,t‖2tr⁡(W̃2,tT(K2P)W̃2,t)ωW̃2,t}-ΔtTQ0Δt+η¯≤-ΔtTQ0Δt+η¯<0.Vt is bounded. Integrating (27) from 0 up to T yields
(29)VT-V0≤-∫0TΔtTQ0Δtdt+η¯T.
Because κ≥1, we have
(30)∫0TΔtTQ0Δtdt≤V0-VT+∫0TΔtTQ0Δtdt≤V0+η¯T,≤V0+κη¯T,
where κ is condition number of Q0(II)
If∥Δt∥2≤λmin-1(Q0)η¯, the weights become constants, Vt remains bounded. And(31)∫0TΔtTQ0Δtdt≤∫0Tλmax⁡(Q0)‖Δt‖2dt≤λmax⁡(Q0)λmin⁡(Q0)η¯T≤V0+κη¯T.
From (I) and (II),Vt is bounded, (18) is realized. From (20) and W̃1,t=W1,t-W10,W̃2,t=W2,t-W20 we know V0=Δ0TPΔ0. Using (30) and (31), (19) is obtained. The theorem is proved.Remark 3.
The weight update law (15) uses two techniques. The dead-zone st is applied to overcome the robust problem caused by unmodeled dynamic f̃t. 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 W2,t inside the ball of center W20 and radius r. If ∥W̃2,t∥<r, we use the normal gradient algorithm. When W2,t-W20 is on the ball, and the vector W2,t points either inside or along the ball, that is, (d/dt)∥W̃2,t∥2=2tr(-ωW̃2,t)≤0,we also keep this algorithm. If tr(-ωW̃2,t)>0,tr[(-ω+(∥W̃2,t∥2/tr(W̃2,tT(K2P)W̃2,t))ω)W̃2,t]<0, so (d/dt)∥W̃2,t∥2<0,W2,t are directed toward the inside or the ball, that is, W2,t will never leave the ball. Since r<∥W20∥,W2,t≠0.Figure 1
Projection algorithm.Remark 4.
Figure1 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, W10 and W20, after T0, the identification error should become smaller, that is, W1,T0and W2,T0 are better than W10 and W20. We use following steps to find the initial weights.(1)
Start from any initial value forW10=W1,0,W20=W2,0.(2)
Do identification until training time arrivesT0.(3)
If the∥Δ(T0)∥<∥Δ(0)∥, let W1,T0,W2,T0 as a new W10 and W20, go to 2 to repeat the identification process.(4)
If the∥Δ(T0)∥≥∥Δ(0)∥, stop this offline identification, now W1,T0,W2,T0 are the final initial weights.Remark 5.
Since the updating rate isKiP (i=1,2), and Ki 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: f̃=0) we obtain η¯=0 and, hence,
(32)limsup⁡T→∞1T∫0T‖Δt‖Q02dt=0
from which, for this special situation, the asymptotic stability property (∥Δt∥→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 as(33)xt.=Axt+W1*σ(xt)+W2*ϕ(xt)γ(ut)+f̃=Axt+W1,tσ(x̂t)+W2,tϕ(xt)γ(ut)+f̃+W̃1,tσ(x̂t)+W̃2,tϕ(xt)γ(ut)+W1,t*σ̃t+W1*ϕ̃γ(ut).Equation (33) can be rewritten as(34)xt.=Axt+W1,tσ(x̂t)+W2,tϕ(xt)γ(ut)+dt,
where(35)dt=f̃+W̃1,tσ(x̂t)+W̃2,tϕ(xt)γ(ut)+W1,t*σ̃t+W1*ϕ̃γ(ut).
If updated law of W1,t and W2,t is (15), W1,t and W2,t are bounded. Using the assumption (A1), dt is bounded as d¯=supt∥dt∥.The object of adaptive control is to force the nonlinear system (1) following a optimal trajectory xt*∈ℜr which is assumed to be smooth enough. This trajectory is regarded as a solution of a nonlinear reference model:(36)ẋt*=φ(xt*,t),
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(xt*,t)=0, x*(0)=c, c is constant. Let us define the sate trajectory error as(37)Δt*=xt-xt*.
From (34) and (36) we have(38)Δ.t*=Axt+W1,tσ(x̂t)+W2,tϕ(xt)γ(ut)+dt-φ(xt*,t).
Let us select the control action γ(ut) as linear form(39)γ(ut)=U1,t+[W2,tϕ(x̂t)]-1U2,t,
where U1,t∈ℜn is direct control part and U2,t∈ℜn is a compensation of unmodeled dynamic dt. As φ(xt*,t),xt*, W1,tσ(x̂t) and W2,tϕ(x̂t) are available, we can select U1,t as(40)U1,t=[W2,tϕ(x̂t)]-1[φ(xt*,t)-Axt*-W1,tσ(x̂t)].
Because ϕ(x̂t) in (5) is different from zero, and W2,t≠0 by the projection approach in Theorem 2. Substitute (39) and (40) into (38), we have So the error equation is(41)Δ.t*=AΔt*+U2,t+dt.
Four robust algorithms may be applied to compensate dt.(A) Exactly Compensation
From (7) and (2) we have
(42)dt=(xt.-x̂.t)-A(xt-x̂t).
If xt. is available, we can select U2,t as U2,ta=-dt, that is,
(43)U2,ta=A(xt-x̂t)-(xt.-x̂.t).
So, the ODE which describes the state trajectory error is
(44)Δ.t*=AΔt*.
Because A is stable, Δt* is globally asymptotically stable.
(45)lim⁡t→∞Δt*=0.(B) An Approximate Method
Ifxt. is not available, an approximate method may be used as
(46)x.t=xt-xt-ττ+δt,
where δt>0, is the differential approximation error. Let us select the compensator as
(47)U2,tb=A(xt-x̂t)-(xt-xt-ττ-x̂.t).
So U2,tb=U2,ta+δt, (44) become
(48)Δ.t*=AΔt*+δt.
Define Lyapunov-like function as
(49)Vt=Δt*TP2Δt*,P2=P2T>0.
The time derivative of (49) is
(50)V.t=Δt*(ATP2+P2A)Δt*+2Δt*TP2δt,2ΔtTP2δt can be estimated as
(51)2Δt*TP2δt≤Δt*TP2ΛP2Δt*+δtTΛ-1δt
where Λ is any positive define matrix. So (50) becomes
(52)V.t≤Δt*(ATP2+P2A+P2ΛP2+Q2)Δt*+δtTΛ-1δt-Δt*TQ2Δt*,
where Q is any positive define matrix. Because A is stable, there exit Λ and Q2 such that the matrix Riccati equation:
(53)ATP2+P2A+P2ΛP2+Q2=0
has positive solution P2=P2T>0. Defining the following seminorms:
(54)‖Δt*‖Q22=lim⁡T→∞¯1T∫0TΔt*Q2Δt*dt,
where Q2=Q2>0 is the given weighting matrix, the state trajectory tracking can be formulated as the following optimization problem:
(55)Jmin⁡=min⁡utJ,J=‖xt-xt*‖Q22.
Note that
(56)lim⁡T→∞¯1T(Δ0*TP2Δ0*)=0
based on the dynamic neural network (2), the control law (47) can make the trajectory tracking error satisfies the following property:
(57)‖Δt*‖Q22≤‖δt‖Λ-12.
A suitable selection of Λ and Q2 can make the Riccati equation (53) has positive solution and make ∥Δt*∥Q22 small enough if τ is small enough.(C) Sliding Mode Compensation
Ifxt. is not available, the sliding mode technique may be applied. Let us define Lyapunov-like function as
(58)Vt=Δt*TP3Δt*,
where P3 is a solution of the Lyapunov equation:
(59)ATP3+P3A=-I.
Using (41) whose time derivative is
(60)V.t=Δt*(ATP3+P3A)Δt*+2Δt*TP3U2,t+2Δt*TP3dt.
According to sliding mode technique, we may select u2,t as
(61)U2,tc=-kP3-1sgn⁡(Δt*),k>0,
where k is positive constant,
(62)sgn⁡(Δt*)={1Δt*>00Δt*=0-1Δt*<0sgn⁡(Δt*)=[sgn⁡(Δ1,t*),…sgn⁡(Δn,t*)]T∈Rn.
Substitute (59) and (61) into (60)
(63)V.t=-‖Δt*‖2-2k‖Δt*‖+2Δt*TPdt≤-‖Δt*‖2-2k‖Δt*‖+2λmax⁡(P)‖Δt*‖‖dt‖=-‖Δt*‖2-2‖Δt*‖(k-λmax⁡(P)‖dt‖).
If we select
(64)k>λmax⁡(P3)d¯,
where d¯ is define as (35), then V.t<0. So,
(65)lim⁡t→∞Δt*=0.(D) Local Optimal Control
Ifxt. is not available and xt. is not approximated as (B). In order to analyze the tracking error stability, we introduce the following Lyapunov function:
(66)Vt(Δt*)=Δt*P4Δt*,P4=P4T>0.
Using (41), whose time derivative is
(67)V̇t=Δt*(ATP4+P4A)Δt*+2Δt*TP4U2,t+2Δt*TP4dt,2Δt*TP4dt can be estimated as
(68)2Δt*TP4dt≤Δt*P4Λ4-1P4Δt*+dtTΛ4dt.
Substituting (68) in (67), adding and subtracting the term Δt*TQ4Δt* and U2,tdTR4U2,td with Q4=Q4T>0 and R4=R4T>0, we formulate
(69)V̇t≤Δt*(ATP4+P4A+P4Λ4P4+Q4)Δt*+2Δt*TP4U2,td+U2,tdTR4U2,td+dtTΛ4-1dt-Δt*QΔt*-U2,tdTR4U2,td.
Because A is stable, there exit Λ4 and Q4 such that the matrix Riccati equation:
(70)ATP4+P4A+P4Λ4P4+Q4=0.
So (69) is
(71)V̇t≤-(‖Δt*‖Q42+‖U2,td‖R42)+Ψ(U2,td)+dtTΛ4-1dt,
where
(72)Ψ(U2,td)=2Δt*TP4U2,td+U2,tdTR4U2,td.
We reformulate (71) as
(73)‖Δt*‖Q42+‖U2,td‖R42≤Ψ(U2,td)+dtTΛ4-1dt-V̇t.
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
(74)lim⁡T→∞¯1T∫0TΔt*TQ4Δt*dt+lim⁡T→∞¯1T∫0TU2,tdTR4U2,tddt≤lim⁡T→∞¯1T∫0TdtTΛ4-1dtdt+lim⁡T→∞¯1T∫0TΨ(U2,td)dt+lim⁡T→∞¯1T∫0T[-V̇t]dt.
In the view of definitions of the seminorms (55), we have
(75)‖Δt*‖Q42+‖U2,td‖R42≤‖dt‖Λ4-12+lim⁡T→∞¯1T∫0TΨ(U2,td)dt.
It fixes a tolerance level for the trajectory-tracking error. So, the control goal now is to minimize Ψ(U2,td) and ∥dt∥Λ4-12. To minimize ∥dt∥Λ4-12, we should minimize Λ4-1. From (13), if select Q4 to make (70) have solution, we can choose the minimal Λ4-1 as
(76)Λ4-1=A-TQ4A-1.
To minimizing Ψ(U2,td), we assume that, at the given t (positive), x*(t) and x̂(t) are already realized and do not depend on U2,td. We name the U2,td*(t) as the locally optimalcontrol, because it is calculated based only on “local” information. The solution of this optimization problem is given by
(77)min⁡Ψ(u2,td)=2Δt*TP4u2,td+U2,tdTR4U2,td.subject:A0(U1,t+U2,td)≤B0.
It is typical quadratic programming problem. Without restriction U* is selected according to the linear squares optimal control law:
(78)u2,td=-2R4-1P4Δt*.Remark 7.
Approaches (A) and (C) are exactly compensations ofdt, Approach (A) needs the information of x.t. Because Approach (C) uses the sliding mode control  U2,tc 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 W1,t and W2,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]:(79)M(θ)θ..+V(θ,θ.)θ.+G(θ)+Fd(θ.)=τ,
where θ∈ℜ2 consists of the joint variables, θ.∈ℜ2 denotes the links velocity, τ is the generalized forces, M(θ) is the intertie matrix, V(θ,θ.) is centripetal-Coriolis matrix, and G(θ) is gravity vector, Fd(θ.) is the friction vector. M(θ) represents the positive defined inertia matrix. If we define x1=θ=[θ1,θ2] is joint position, x2=θ. is joint velocity of the link, xt=[x1,x2]T, (79) can be rewritten as state space form [15]:(80)x.1=x2,x.2=H(xt,ut),
where ut=τ is control input,(81)H(xt,ut)=-M(x1)-1[C(x1,x2)x.1+G(x1)+Fx.1+ut].
Equation (80) can also be rewritten as(82)x.1=∫0tH(xτ,uτ)dτ+H(x0,u0).
So the dynamic of the two-link robot (79) is in form of (1) with(83)f(xt,ut,t)=∫0tH(xτ,uτ)dτ+H(x0,u0).
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 as(84)x̂.=Ax̂+W1,tσ(x̂t)+W2,tϕ(x̂)u.
We select A=[-1.500-1],ϕ(x̂t)=diag(ϕ1(x̂1),ϕ2(x̂2)),σ(x̂t)=[σ2(x̂2),σ2(x̂2)]T(85)σi(x̂i)=2(1+e-2x̂i)-12,ϕi(x̂i)=2(1+e-2x̂i)+12,
where i=1,2. We used Remark 4  to obtain a suitable W10 and W20, start from random values, T0=100. After 2 loops, ∥Δ(T0)∥ does not decrease, we let the W1,300 and W2,300 as the new W10=[0.513.8-2.31.51] and W20=[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 as(86)τ1=7sin⁡t,τ2=0.Now we check the neurocontrol. We assume the robot is changed att=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 as(87)τPD=-10(θ-θ*)-5(θ.-θ.*),
where θ1*=3;θ2* is square wave. So φ(θ*)=θ.*=0.The neurocontrol is  (39)(88)τneuro=[W2,tϕ(x̂)]+[φ(xt*,t)-Axt*-W1,tσ(x̂)]+[W2,tϕ(x̂)]+U2,t.U2,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 θ. is measurable, as in (43),(89)U2,t=A(θ-θ̂)-(θ.-θ̂.).
The results are shown in Figures 2 and 3.Figure 2
Tracking control ofθ1 (method B).Figure 3
Tracking control ofθ2 (method B).(C)D=0.3.θ. is not available, the sliding mode technique may be applied. we select u2,t as   (61).(90)u2,t=-10×sgn⁡(θ-θ*).
The results are shown in Figures 4 and 5.Figure 4
Tracking control ofθ1 (method C).Figure 5
Tracking control ofθ2 (method C).(D)D=3. We select Q=1/2,  R=1/20,  Λ=4.5, the solution of following Riccati equation:(91)ATP+PA+PΛPt+Q=-P.
is P=[0.33000.33]. If without restriction τ, the linear squares optimal control law:(92)u2,t=-2R-1P(θ-θ*)=[-2000-20](θ-θ*).
The results of local optimal compensation are shown in Figures 6 and 7.Figure 6
Tracking control ofθ1 (method D).Figure 7
Tracking control ofθ2 (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.



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*Source: 867178-2012-04-08.xml*