# Optimal Robust Adaptive FuzzyH∞ Tracking Control without Reaching Phase for Nonlinear System

Authors: El Mehdi Mellouli; Siham Massou; Ismail Boumhidi
Journal: Journal of Control Science and Engineering
 (2013)

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## Abstract

An optimalH∞ tracking-based indirect adaptive fuzzy controller for a class of perturbed uncertain affine nonlinear systems without reaching phase is being developed in this paper. First a practical Interval Type-2 (IT2) fuzzy system is used in an adaptive scheme to approximate the system using a nonlinear model and to determine the optimal value of the H∞ gain control. Secondly, to eliminate the trade-off between H∞ tracking performance and high gain at the control input, a modified output tracking error has been used. The stability is ensured through Lyapunov synthesis and the effectiveness of the proposed method is proved and the simulation is also given to illustrate the superiority of the proposed approach.


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## Body

## 1. Introduction

After the approximation-based adaptive fuzzy controller (AFC) of Wang [1] for a class of uncertain affine nonlinear system, many approaches and ideas have been developed in recent years to overcome the difficulty in controller design [2–4].The primary feature that characterizes the fuzzy logic is its high capacity for representing and modelling the nonlinear systems with imprecise uncertainty, as the universal approximation theorem, by Lee and Tomizuka, illustrates [5].Many effective adaptive fuzzy control schemes have been developed to incorporate with human expert knowledge and information in a systematic way, which can also guarantee various stability and performance [6]. The most important issue for Fuzzy Logic Systems (FLSs) is how to get a system design with the guarantee of stability and control performance [7–9].An adaptive fuzzy control system includes uncertainties caused by unmodeled dynamics, Fuzzy Approximation Errors (FAEs), and external disturbance, which cannot be effectively handled by the FLS and may degrade the tracking performance of the closed-loop system [10, 11]. The AFC combined with H∞ control technique is an effective approach for rejecting those uncertainties, ensuring stability and consistent performance [12–14].The research of fuzzy model underH∞ has attracted many attentions in recent years [4, 15], such as the apparent similarities between H∞ and fuzzy control which motivate considerable research efforts in combining the two approaches for achieving more superior performance.Moreover, to the best of our knowledge, the control gain needs to be known in all previousH∞, indirect adaptive fuzzy controller (HIAFC) approaches, such as the arbitrary choice of the gain which does not always give good results, for which we propose in this work a method for extracting automatically the optimum gain from the Lyapunov equation whilst respecting system stability.The convergence of the system in the initial time needs the appearance of high gain at the control input, and the high gain is unavoidable in all previousH∞ tracking-based AFC approaches. The best method to solve the problem of the tradeoffs between H∞ tracking performance and high gain at the control input is to eliminate the reaching phase. During the reaching phase the tracking error cannot be controlled directly and the system response is sensitive to parameter uncertainties. Several methods have been proposed to completely eliminate the reaching phase [16].This paper focuses on a class of Single-Input Single-Output (SISO) perturbed uncertain affine nonlinear systems involving external disturbances without exact knowledge of dynamic functions. Firstly, we use the type-2 fuzzy technique to determine the optimal value of theH∞ gain control. Secondly, a modified output tracking error is used to eliminate the reaching phase [4].The paper is organized as follows: Section2 presents the problem statement. Section 3 gives the control design strategy. An illustration example is described in Section 4. Finally, the simulation results are being used to demonstrate the effectiveness and performance of the proposed approach.


## 2. Problem Formulations

Considering the followingnth-order SISO affine nonlinear dynamical system, Chen et al. [14]:
(1)x˙1=x2x˙2=x3⋮x˙n=f(x_,t)+g(x_,t)u(t)+d(t)y=x1.
Or equivalently
(2)x(n)=f(x_,t)+g(x_,t)u(t)+d(t)y=x,
where x_=[x,x˙,…,x(n-1)]T=[x1,x2,…,xn]T∈Rn is the state vector of the systems which is assumed to be available for measurement, u∈R and y∈R are, respectively, the input and the output of the systems. f(x_,t) and g(x_,t) are two functions that are unknown, nonlinear, and continuous; d(t) denotes the external disturbance. For (1) to be controllable, we require that g(x_,t)≠0 for x_ in certain controllability region. Assume that the given reference y_r is bounded and have up to (n-1) bounded derivatives. The reference vector is denoted as y_r=[yr,y˙r,…,yr(n-1)]T. Define the tracking error e=yr-y and the error vector e_=[e1,e2,…,en]T=[e,e˙,…,e(n-1)]T∈Rn.Assumption 1.
For allx_∈D, there exist unknown bounded f¯(x_,t),g¯(x_,t), and d¯(t) such that |f(x_,t)|≤f¯(x_,t), |g(x_,t)|≤g¯(x_,t) and |d(t)|≤d¯(t) hold, where compact set D⊂Rn is a certain controllable region.
During the AFC design, to improve the tracking performance under the external disturbance, an additionalH∞ compensator associated with an attenuation level is usually suggested to apply, Chen et al. [14]. If the prescribed attenuation level is smaller, the tracking performance is better while the control input gain is higher as the output of the H∞ compensator becomes larger.
To avoid high control input gain, we have modified the following output tracking error Yilmaz and Hurmuzlu [16]:
(3)E(t)=e(t)-ψ(t),
where (condition  1) ψ(t) is designed to make E(t) small enough at the onset of the motion t=0, and (condition 2) should rapidly vanish as the motion evolves at t>0.
A suggestedψ(t) is given in the following exponential form:
(4)ψ(t)=γ(t)exp(F(t))γ(t)=∑i=0n-11i!γ(i)(t0)(t-t0)i,γ(i)(t0)=diγdti(t=t0),(5)F(t)=-αt
with γ(i)(t0)(i=0,1,…,n-1) is selected to satisfy condition one and F(t) is selected to satisfy condition two. For the selections of γ(t)(i)(t0)one can follow the methods in Yilmaz and Hurmuzlu [16], on the other side α is selected to satisfy condition two [4].
Now, the objective of this paper is to determine the optimal value of theH∞ gain control, in a way to force y(t) to follow a given bounded reference signal yr(t).
Let us denote the parameter tracking errorΦ_=θ_-θ_* for some parameter estimate θ_ and optimal parameter estimate θ_* of Type-2 Fuzzy Logic System (T2FLS). Let w denote the sum of error due to fuzzy modelling approximations. Then our design objective is to impose an adaptive fuzzy control algorithm so that the following asymptotically stable tracking
(6)E(n)+kn-1E(n-1)+⋯+k0E=0
is achieved while w=0 (i.e., in the case of perfect fuzzy approximation and free of external disturbance). While w appears, the following H∞ tracking performance is requested [17]:
(7)∫0TE_TQE_dt≤2V(0)+ρ2∫0Tw2dtT∈[0∞],
where E_=[E1,E2,…,En]=[E,E˙,…,E(n-1)]T. V is the Lyapunov function, Q=QT>0, ρ∈R+ is the prescribed attenuation level, and w∈L2[0,T].
Q, V, and w will be defined in the next subsection.Remark 2.
(i) The roots of polynomialL^(s)=s(n)+kn-1s(n-1)+⋯+k0s in the characteristic equation of (6) are all in the open left-half plane via an adequate choice of coefficients k0,k1,…,kn-1.
(ii) If the system starts with initial conditionV(0)=0, then the H∞ performance in (7) can be rewritten as
(8)supw∈L2[0T]∥E_∥Q∥w∥≤ρ,
where ∥E_∥Q2=∫0TE_TQE_dt, and ∥w∥2=∫0TwTwdt, that is, the L2-gain form w to the tracking error E_ must be equal to or less than ρ.


## 3. Control Design Strategy

### 3.1. Indirect Adaptive Control Scheme

In this section, we propose a new optimalH∞ tracking-based indirect adaptive output-feedback fuzzy controller that eliminates the reaching phase, with guaranteed stability of the closed loop system. Based on the combination of the H∞ optimal control with fuzzy logic control, using fuzzy identifier and fuzzy logic control, the H∞ control design relies on the solution of an algebraic Riccati equation.If the system (1) is well known and g(x_)≠0 then the control should be designed to have the following idealized control law:
(9)u*=1g(x_,t)(-f(x_,t)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+uh),
where uh=ℜ·E_TPB.However, in practice the functionsf(x_,t) and g(x_,t) are unknown, thus the ideal controller in (9) cannot be realized, and the choice of the H∞ gain control ℜ does not always give good results. In this case the nonlinear functions f(x_,t) and g(x_,t) are approximated using T2-fuzzy systems universal approximation property and by the same technique we determine the optimal H∞ gain control. Hence, the fuzzy adaptive control law is as follows:
(10)u=1g^(x_,θ_g)(-f^(x_,θ_f)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+u^h),
where u^h=ℜ^·E_TPB defined the auxiliary control employed to attenuate the approximation error of the fuzzy model and to eliminate the external disturbance.f
^
(
x
_
,
θ
_
f
), g^(x_,θ_g), and ℜ^(x_,θ_ℜ) are the type-2 fuzzy approximation of f(x_,t),g(x_,t), and ℜ.


### 3.2. Interval Type-2 Fuzzy Logic System (IT2FLS)

For an Interval Type-2 Fuzzy Logic System IT2FLS withM, total number of IF-THEN rules in the rule base, the jth rule can be written as follows:
(11)Rj:ifx1isμy1jand…andxnisμynj,thenyisθj.
Equation (11) represents a T2 fuzzy relation between the input and the output spaces of the FLS, where μF1j’s are antecedent type-2 sets, y is the output, and θj’s are the consequent T2 fuzzy singleton.Since fuzzy sets are type-2, we need to perform the reduction operation type. This operation will give each function estimatedy^ two vector of the fuzzy basis functions [18]
(12)Y^cos=(θy1,…,θyM,Wy1,…,WyM)=∫θy1…∫θyM∫Wy1…∫WyM1∑i=1Mwyiθyi/∑i=1MwyiY^cos is the interval set determined by two end points y^r and y^l, and wyi∈Wyi=[wyli,wyri] is the firing interval.Accordingly, the firing interval bounds for theith rule of an IT2FLS with n inputs, wyli and wyri, can be rewritten as follows:
(13)wylj=μ_y1j(x1)×μ_y2j(x2)×⋯×μ_ynj(xn)=∏i=1nμ_yij(xi)wyri=μ¯y1j(x1)×μ¯y2j(x2)×⋯×μ¯ynj(xn)=∏i=1nμ¯yij(xi).
Using the centre of gravity, the defuzzified crisp output for each output is given by Liang and Mendel [19]:
(14)y^=y^l+y^r2y^l can be represented as a vector of fuzzy basis functions (FBFs) expansion as follows:
(15)y^l=∑i=1Mwyliθyi∑i=1Mwyli=∑i=1Mθyiξyli=θ_yTξ_yl(x_)ξ_ylT(x_) is the FBF vector of y^l such that ξ_ylT(x_)=[ξyl1,…,ξylM] whose components are given by
(16)ξyli=wyli∑i=1Mwyliθ_yT=[θy1,…,θyM] represents the conclusion of T2FLS.Similar to the foregoing we have(17)y^r=∑i=1Mwyriθyi∑i=1Mwyri=∑i=1Mθyiξyri=θ_yTξ_yr(x_).
Substituting (17) and (15) in (14) then the output of the T2FLS can be given as follows:
(18)y^=θ_yTξ_fl+θ_yTξ_fr2=θ_yTξ_y(x_),
where ξ_y(x_)=(ξ_yl+ξ_yr)/2The previous equation (18) will be used, in an indirect adaptive control, to approximate the unknown system dynamics and to determine the optimal H∞ gain control.Therefore the expression (18) can be expressed as:
(19)y^f=f^(x_θ_f)=θ_fTξf(x_)y^g=g^(x_θ_g)=θ_gTξg(x_)y^ℜ=ℜ^(x_θ_ℜ)=θ_ℜTξℜ(x_).
Define the compact sets D={x_:∥x_∥≤Mx},
(20)Ωf={θ_f∈Rn∥θ_f∥≤Mf},Ωg={θ_g∈Rn∥θ_g∥≤Mg},Ωℜ={θ_ℜ∈Rn∥θ_ℜ∥≤Mℜ},
where Mf,Mg, and Mℜ are given constants.The minimum approximation error is defined by(21)w=f^(x_,θ_f*)-f(x_,t)+(g^(x_,θ_g*)-g(x_,t))u-(ℜ^(x_,θ_ℜ*)-ℜ)(ETPB)2-d(t),
where θ_f*, θ_g*, and θ_ℜ* are an optimal parameter vector defined as
(22)θ_f*=argminθf∈Ωf(supx∈D|f^(x_,θ_f)-f(x_,t)|)θ_g*=argminθg∈Ωg(supx∈D|g^(x_,θ_g)-g(x_,t)|)θ_ℜ*=argminθℜ∈Ωℜ(supx∈D|ℜ^(x_,θ_ℜ)-ℜ|).(23)Suchthat:ℜ>1r,
where r is a positive constant used below.


### 3.3.H∞ Tracking Performance Design in Indirect Adaptive Fuzzy System

Choose theH∞ compensator uh as
(24)uh=ℜ^E_TPB,
where P is the solution of the following Riccati equation:
(25)ATP+PA+Q-PB(2r-1ρ2)BTP=0,
where Q>0,ρ is prescribed attenuation level and r is positive constant verified r<2ρ2.Theorem 3.
If we select the following adaptive fuzzy control law in the nonlinear system (1)
(26)u=1θ_gTξg(x_)(-θ_fTξf(x_)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+θ_ℜTξℜ(x_)E_TPB∑i=0n-1kiE(i)),
where
(27)θ˙f={-γ1E_TPBξf(x)if(∥θ_f∥<Mfor∥θ_f∥=Mf,E_TPBξf(x)>0∥θ_f∥)-γ1E_TPBξf(x)+γ1E_TPBξf(x)×θ_fTθ_fξf(x)∥θf∥2otherwise(28)θ˙g={-γ2E_TPBξg(x)uif(∥θ_g∥<Mgor∥θ_g∥=Mg,E_TPBξg(x)u>0∥θ_g∥)-γ2E_TPBξg(x)u+γ2E_TPBξg(x)u×θ_gTθ_gξg(x)∥θg∥2otherwise(29)θ˙ℜ={γ3(E_TPB)2ξ(x)ℜif(E_TPBξ(x)>0ℜ∥θ_ℜ∥<Mℜor∥θ_ℜ∥=Mℜ,E_TPBξ(x)>0ℜ)γ3(E_TPB)2ξ(x)ℜ-γ3(E_TPB)2ξ(x)ℜ×θ_ℜTθ_ℜξ(x)ℜ∥θℜ∥2otherwise.
With P=PT≥0 is the solution of the Riccati equation (25), then the H∞ tracking performance in (7) is achieved for a prescribed attenuation level ρ.Proof.
We have(30)x(n)=f(x_,t)+g(x_,t)u+d(t).
And the equation of the control already proposed
(31)u=1g^(x_,θ_g)(-f^(x_,θ_f)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+u^h],
where
(32)u^h=ℜ^E_TPB.
And ℜ^=θ_ℜTξℜ.
Utilising (3) and substituting (30) into (31), the output error dynamics may be expressed as
(33)E(n)=-∑i=0n-1kiE(i)+f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t).
The error dynamics can be represented by(34)E˙_=AE_+B[f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t)(g^(x_,θ_g)-g(x_,t))],
where
(35)A=[010⋯0001⋯⋮⋮⋱⋮⋮1-k0-k1⋯⋯-kn],B=[0⋮01].
Consider the following Lyapunov function:(36)V=12E_TPE_+12γ1ΦfTΦf+12γ2ΦgTΦg+12γ3ΦℜTΦℜ,
where
(37)Φf=θ_f-θ_f*,Φg=θ_g-θ_g*,Φℜ=θ_ℜ-θ_ℜ*.(38)V˙=12E_T(ATP+PA)E_+E_TPB×[f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t)(g^(x_,θ_g)-g(x_,t))]+1γ1ΦfTΦ˙f+1γ2ΦgTΦ˙g+1γ3ΦℜTΦ˙ℜ.
Utilizing (25) and (32) into (38)
(39)V˙=-12E_TQE_+(E_TPB)2r-(E_TPB)22ρ2+[w-ℜE_TPB]E_TPB+1γ1ΦfT(Φ˙f+γ1ξf(x)E_TPB)+1γ2ΦgT(Φ˙g+γ2ξg(x)E_TPBu)+1γ3ΦℜT(Φ˙ℜ-γ3ξℜ(x)(E_TPB)2)-(E_TPB)2ℜ,
where w is defined in (21), the V˙ can be written as
(40)V˙=-12E_TQE_-12((E_TPB)ρ-ρw)2+12(ρw)2+(1r-ℜ)(E_TPB)2+1γ1ΦfT(Φ˙f+γ1E_TPBξf(x))+1γ2ΦgT(Φ˙g+γ2E_TPBξg(x)u)+1γ3ΦℜT(Φ˙ℜ-γ3(E_TPB)2ξℜ(x)).
By consideration of the update law (27), (28), (23), and (29), V˙ can be written as
(41)V˙≤-12E_TQE_+12(ρw)2.
Integrating the above equality from t=0 to T yields (0≤T≤∞):
(42)V(T)-V(0)≤-12∫0TE_TQE_dt+12ρ2∫0Tw2dt.
Since V(T)≥0 the above inequality implies the following inequality:
(43)∫0TE_TQE_dt≤2VL(0)+ρ2∫0Tw2dt,T∈[0∞].
Hence, the inequality (7) holds. This completes the proof of theorem. So the system is stable and the error will asymptotically converge to zero; that is, H∞ a performance is achieved.


## 3.1. Indirect Adaptive Control Scheme

In this section, we propose a new optimalH∞ tracking-based indirect adaptive output-feedback fuzzy controller that eliminates the reaching phase, with guaranteed stability of the closed loop system. Based on the combination of the H∞ optimal control with fuzzy logic control, using fuzzy identifier and fuzzy logic control, the H∞ control design relies on the solution of an algebraic Riccati equation.If the system (1) is well known and g(x_)≠0 then the control should be designed to have the following idealized control law:
(9)u*=1g(x_,t)(-f(x_,t)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+uh),
where uh=ℜ·E_TPB.However, in practice the functionsf(x_,t) and g(x_,t) are unknown, thus the ideal controller in (9) cannot be realized, and the choice of the H∞ gain control ℜ does not always give good results. In this case the nonlinear functions f(x_,t) and g(x_,t) are approximated using T2-fuzzy systems universal approximation property and by the same technique we determine the optimal H∞ gain control. Hence, the fuzzy adaptive control law is as follows:
(10)u=1g^(x_,θ_g)(-f^(x_,θ_f)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+u^h),
where u^h=ℜ^·E_TPB defined the auxiliary control employed to attenuate the approximation error of the fuzzy model and to eliminate the external disturbance.f
^
(
x
_
,
θ
_
f
), g^(x_,θ_g), and ℜ^(x_,θ_ℜ) are the type-2 fuzzy approximation of f(x_,t),g(x_,t), and ℜ.


## 3.2. Interval Type-2 Fuzzy Logic System (IT2FLS)

For an Interval Type-2 Fuzzy Logic System IT2FLS withM, total number of IF-THEN rules in the rule base, the jth rule can be written as follows:
(11)Rj:ifx1isμy1jand…andxnisμynj,thenyisθj.
Equation (11) represents a T2 fuzzy relation between the input and the output spaces of the FLS, where μF1j’s are antecedent type-2 sets, y is the output, and θj’s are the consequent T2 fuzzy singleton.Since fuzzy sets are type-2, we need to perform the reduction operation type. This operation will give each function estimatedy^ two vector of the fuzzy basis functions [18]
(12)Y^cos=(θy1,…,θyM,Wy1,…,WyM)=∫θy1…∫θyM∫Wy1…∫WyM1∑i=1Mwyiθyi/∑i=1MwyiY^cos is the interval set determined by two end points y^r and y^l, and wyi∈Wyi=[wyli,wyri] is the firing interval.Accordingly, the firing interval bounds for theith rule of an IT2FLS with n inputs, wyli and wyri, can be rewritten as follows:
(13)wylj=μ_y1j(x1)×μ_y2j(x2)×⋯×μ_ynj(xn)=∏i=1nμ_yij(xi)wyri=μ¯y1j(x1)×μ¯y2j(x2)×⋯×μ¯ynj(xn)=∏i=1nμ¯yij(xi).
Using the centre of gravity, the defuzzified crisp output for each output is given by Liang and Mendel [19]:
(14)y^=y^l+y^r2y^l can be represented as a vector of fuzzy basis functions (FBFs) expansion as follows:
(15)y^l=∑i=1Mwyliθyi∑i=1Mwyli=∑i=1Mθyiξyli=θ_yTξ_yl(x_)ξ_ylT(x_) is the FBF vector of y^l such that ξ_ylT(x_)=[ξyl1,…,ξylM] whose components are given by
(16)ξyli=wyli∑i=1Mwyliθ_yT=[θy1,…,θyM] represents the conclusion of T2FLS.Similar to the foregoing we have(17)y^r=∑i=1Mwyriθyi∑i=1Mwyri=∑i=1Mθyiξyri=θ_yTξ_yr(x_).
Substituting (17) and (15) in (14) then the output of the T2FLS can be given as follows:
(18)y^=θ_yTξ_fl+θ_yTξ_fr2=θ_yTξ_y(x_),
where ξ_y(x_)=(ξ_yl+ξ_yr)/2The previous equation (18) will be used, in an indirect adaptive control, to approximate the unknown system dynamics and to determine the optimal H∞ gain control.Therefore the expression (18) can be expressed as:
(19)y^f=f^(x_θ_f)=θ_fTξf(x_)y^g=g^(x_θ_g)=θ_gTξg(x_)y^ℜ=ℜ^(x_θ_ℜ)=θ_ℜTξℜ(x_).
Define the compact sets D={x_:∥x_∥≤Mx},
(20)Ωf={θ_f∈Rn∥θ_f∥≤Mf},Ωg={θ_g∈Rn∥θ_g∥≤Mg},Ωℜ={θ_ℜ∈Rn∥θ_ℜ∥≤Mℜ},
where Mf,Mg, and Mℜ are given constants.The minimum approximation error is defined by(21)w=f^(x_,θ_f*)-f(x_,t)+(g^(x_,θ_g*)-g(x_,t))u-(ℜ^(x_,θ_ℜ*)-ℜ)(ETPB)2-d(t),
where θ_f*, θ_g*, and θ_ℜ* are an optimal parameter vector defined as
(22)θ_f*=argminθf∈Ωf(supx∈D|f^(x_,θ_f)-f(x_,t)|)θ_g*=argminθg∈Ωg(supx∈D|g^(x_,θ_g)-g(x_,t)|)θ_ℜ*=argminθℜ∈Ωℜ(supx∈D|ℜ^(x_,θ_ℜ)-ℜ|).(23)Suchthat:ℜ>1r,
where r is a positive constant used below.


## 3.3.H∞ Tracking Performance Design in Indirect Adaptive Fuzzy System

Choose theH∞ compensator uh as
(24)uh=ℜ^E_TPB,
where P is the solution of the following Riccati equation:
(25)ATP+PA+Q-PB(2r-1ρ2)BTP=0,
where Q>0,ρ is prescribed attenuation level and r is positive constant verified r<2ρ2.Theorem 3.
If we select the following adaptive fuzzy control law in the nonlinear system (1)
(26)u=1θ_gTξg(x_)(-θ_fTξf(x_)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+θ_ℜTξℜ(x_)E_TPB∑i=0n-1kiE(i)),
where
(27)θ˙f={-γ1E_TPBξf(x)if(∥θ_f∥<Mfor∥θ_f∥=Mf,E_TPBξf(x)>0∥θ_f∥)-γ1E_TPBξf(x)+γ1E_TPBξf(x)×θ_fTθ_fξf(x)∥θf∥2otherwise(28)θ˙g={-γ2E_TPBξg(x)uif(∥θ_g∥<Mgor∥θ_g∥=Mg,E_TPBξg(x)u>0∥θ_g∥)-γ2E_TPBξg(x)u+γ2E_TPBξg(x)u×θ_gTθ_gξg(x)∥θg∥2otherwise(29)θ˙ℜ={γ3(E_TPB)2ξ(x)ℜif(E_TPBξ(x)>0ℜ∥θ_ℜ∥<Mℜor∥θ_ℜ∥=Mℜ,E_TPBξ(x)>0ℜ)γ3(E_TPB)2ξ(x)ℜ-γ3(E_TPB)2ξ(x)ℜ×θ_ℜTθ_ℜξ(x)ℜ∥θℜ∥2otherwise.
With P=PT≥0 is the solution of the Riccati equation (25), then the H∞ tracking performance in (7) is achieved for a prescribed attenuation level ρ.Proof.
We have(30)x(n)=f(x_,t)+g(x_,t)u+d(t).
And the equation of the control already proposed
(31)u=1g^(x_,θ_g)(-f^(x_,θ_f)+yr(n)-ψ(n)+∑i=0n-1kiE(i)+u^h],
where
(32)u^h=ℜ^E_TPB.
And ℜ^=θ_ℜTξℜ.
Utilising (3) and substituting (30) into (31), the output error dynamics may be expressed as
(33)E(n)=-∑i=0n-1kiE(i)+f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t).
The error dynamics can be represented by(34)E˙_=AE_+B[f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t)(g^(x_,θ_g)-g(x_,t))],
where
(35)A=[010⋯0001⋯⋮⋮⋱⋮⋮1-k0-k1⋯⋯-kn],B=[0⋮01].
Consider the following Lyapunov function:(36)V=12E_TPE_+12γ1ΦfTΦf+12γ2ΦgTΦg+12γ3ΦℜTΦℜ,
where
(37)Φf=θ_f-θ_f*,Φg=θ_g-θ_g*,Φℜ=θ_ℜ-θ_ℜ*.(38)V˙=12E_T(ATP+PA)E_+E_TPB×[f^(x_,θ_f)-f(x_,t)+(g^(x_,θ_g)-g(x_,t))u-u^h-d(t)(g^(x_,θ_g)-g(x_,t))]+1γ1ΦfTΦ˙f+1γ2ΦgTΦ˙g+1γ3ΦℜTΦ˙ℜ.
Utilizing (25) and (32) into (38)
(39)V˙=-12E_TQE_+(E_TPB)2r-(E_TPB)22ρ2+[w-ℜE_TPB]E_TPB+1γ1ΦfT(Φ˙f+γ1ξf(x)E_TPB)+1γ2ΦgT(Φ˙g+γ2ξg(x)E_TPBu)+1γ3ΦℜT(Φ˙ℜ-γ3ξℜ(x)(E_TPB)2)-(E_TPB)2ℜ,
where w is defined in (21), the V˙ can be written as
(40)V˙=-12E_TQE_-12((E_TPB)ρ-ρw)2+12(ρw)2+(1r-ℜ)(E_TPB)2+1γ1ΦfT(Φ˙f+γ1E_TPBξf(x))+1γ2ΦgT(Φ˙g+γ2E_TPBξg(x)u)+1γ3ΦℜT(Φ˙ℜ-γ3(E_TPB)2ξℜ(x)).
By consideration of the update law (27), (28), (23), and (29), V˙ can be written as
(41)V˙≤-12E_TQE_+12(ρw)2.
Integrating the above equality from t=0 to T yields (0≤T≤∞):
(42)V(T)-V(0)≤-12∫0TE_TQE_dt+12ρ2∫0Tw2dt.
Since V(T)≥0 the above inequality implies the following inequality:
(43)∫0TE_TQE_dt≤2VL(0)+ρ2∫0Tw2dt,T∈[0∞].
Hence, the inequality (7) holds. This completes the proof of theorem. So the system is stable and the error will asymptotically converge to zero; that is, H∞ a performance is achieved.


## 4. An Illustrative Example

### 4.1. The Dynamic Model

In this section consider a single-link robot manipulator governed by the following dynamic model [20]:
(44)x˙1=x2x˙2=-dfx2ml2-gvcos(x1)l+(1ml2)u+d(t)y=x1,
where x1: Position, x2: Velocity, f(x_,t): Nonlinear term depending on x_, m, df: Mass and damping, l: Length of the manipulator, gv: is the gravitational acceleration and d(t)=0.1*randn is the disturbance.We assume that the positionx1 and the velocity x2 are available for measurements, where m=2kg, l=1m, gv=9.8m/s2, and df=1.0kgm2/s.


### 4.2. Controller Parameters Design

Step 1.
In the first step, we need to define the type-2 fuzzy sets for modelling the unknown functions entering into the creation of the control law and to determine the optimal value of theH∞ gain control. The choice of the number of fuzzy sets and constant Mf, Mg, and Mℜ are related to knowledge of expert on the system.The fuzzy membership functions are chosen as(45)μ¯Fi1(xi)=exp(-(xi-cni2*sigi)2),μ¯Fi2(xi)=exp(-(xi-cpi2*sigi)2),μ¯Fi3(xi)=1-μFi1(xi)-μFi2(xi)μ_Fi1(xi)=0.85*exp(-(xi-cni2*sigi)2),μ_Fi2(xi)=0.85*exp(-(xi-cpi2*sigi)2),μ_Fi3(xi)=0.85*(1-μFi1(xi)-μFi2(xi))(i=1,2),
such as cp1=-cn1=π/3, cp2=-cn2=π, sig1=5, and sig2=1.53 with Mf=6, Mg=1, and Mℜ=10.Step 2.
Determine parameters of the modified errorE in (3).
Chooseα=5 in (5).
To determineγ(t) in (4) one can follow the method in Yilmaz and Hurmuzlu, and one can make
(46)ψ(0)=γ(0)=yr(0)-x1(0)ψ˙(0)=-αγ(0)+γ˙(0)=y˙r(0)-x2(0)
with
(47)yr(0)=0,y˙r(0)=π23,x1(0)=π6,x2(0)=π.
Thus, one gets
(48)γ(0)=π6,γ˙(0)=(-π6)α+π23-π.Step 3.
Design parameters of the control law.
The control parameters for simulation are chosen as follows:k0=1, k1=2, γ1=0.002, γ2=0.0001, γ3=0.001, Q=5 eye (2), and ρ=0.1.
The solution to Riccati equation forQ is
(49)P=[5.05030.05030.05030.0513].


### 4.3. Simulation Result

Simulation result is presented to validate performance and robustness of the proposed approaching that we have been usingT-S fuzzy logic to determine automatically the gain of the H∞ control and modifying the output tracking error to eliminate the reaching phase.Three fuzzy sets for each input have been found sufficient for an efficient system design. Fuzzy sets for inputsx1 and x2 are defined according to the membership functions presented forward in Step 1.The sampling time is defined as10ms and the running time as 15s.Figure1 present a responses of the output y(t) versus the desired output yr(t).Figure 1
Reponses of they(t) and the yr(t).Figures2 and 3 present successively the tracking error and the control signal that we apply the proposed method.Figure 2
The tracking error.Figure 3
The control signalu(t).


## 4.1. The Dynamic Model

In this section consider a single-link robot manipulator governed by the following dynamic model [20]:
(44)x˙1=x2x˙2=-dfx2ml2-gvcos(x1)l+(1ml2)u+d(t)y=x1,
where x1: Position, x2: Velocity, f(x_,t): Nonlinear term depending on x_, m, df: Mass and damping, l: Length of the manipulator, gv: is the gravitational acceleration and d(t)=0.1*randn is the disturbance.We assume that the positionx1 and the velocity x2 are available for measurements, where m=2kg, l=1m, gv=9.8m/s2, and df=1.0kgm2/s.


## 4.2. Controller Parameters Design

Step 1.
In the first step, we need to define the type-2 fuzzy sets for modelling the unknown functions entering into the creation of the control law and to determine the optimal value of theH∞ gain control. The choice of the number of fuzzy sets and constant Mf, Mg, and Mℜ are related to knowledge of expert on the system.The fuzzy membership functions are chosen as(45)μ¯Fi1(xi)=exp(-(xi-cni2*sigi)2),μ¯Fi2(xi)=exp(-(xi-cpi2*sigi)2),μ¯Fi3(xi)=1-μFi1(xi)-μFi2(xi)μ_Fi1(xi)=0.85*exp(-(xi-cni2*sigi)2),μ_Fi2(xi)=0.85*exp(-(xi-cpi2*sigi)2),μ_Fi3(xi)=0.85*(1-μFi1(xi)-μFi2(xi))(i=1,2),
such as cp1=-cn1=π/3, cp2=-cn2=π, sig1=5, and sig2=1.53 with Mf=6, Mg=1, and Mℜ=10.Step 2.
Determine parameters of the modified errorE in (3).
Chooseα=5 in (5).
To determineγ(t) in (4) one can follow the method in Yilmaz and Hurmuzlu, and one can make
(46)ψ(0)=γ(0)=yr(0)-x1(0)ψ˙(0)=-αγ(0)+γ˙(0)=y˙r(0)-x2(0)
with
(47)yr(0)=0,y˙r(0)=π23,x1(0)=π6,x2(0)=π.
Thus, one gets
(48)γ(0)=π6,γ˙(0)=(-π6)α+π23-π.Step 3.
Design parameters of the control law.
The control parameters for simulation are chosen as follows:k0=1, k1=2, γ1=0.002, γ2=0.0001, γ3=0.001, Q=5 eye (2), and ρ=0.1.
The solution to Riccati equation forQ is
(49)P=[5.05030.05030.05030.0513].


## 4.3. Simulation Result

Simulation result is presented to validate performance and robustness of the proposed approaching that we have been usingT-S fuzzy logic to determine automatically the gain of the H∞ control and modifying the output tracking error to eliminate the reaching phase.Three fuzzy sets for each input have been found sufficient for an efficient system design. Fuzzy sets for inputsx1 and x2 are defined according to the membership functions presented forward in Step 1.The sampling time is defined as10ms and the running time as 15s.Figure1 present a responses of the output y(t) versus the desired output yr(t).Figure 1
Reponses of they(t) and the yr(t).Figures2 and 3 present successively the tracking error and the control signal that we apply the proposed method.Figure 2
The tracking error.Figure 3
The control signalu(t).


## 5. Conclusions

In this paper, we have proposed a new method to determine the optimal value of theH∞ gain control based on type-2 fuzzy logic systems, and to eliminate the trade-off between H∞ tracking performance and high gain at the control input, we have used the modification in the output tracking error.The parameters of the dynamics systems are estimated by using the fuzzy model. Furthermore, the parameters can be tuned on-line by the adaptive law based on Lyapunov synthesis.



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*Source: 498461-2013-04-22.xml*