# Robust Adaptive Fuzzy Output Tracking Control for a Class of Twin-Roll Strip Casting Systems

Authors: Yu-Jun Zhang; Li-Bing Wu; Hong-Yang Zhao; Xiao-Dong Hu; Wen-Yu Zhang; Dong-Ying Ju
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

This paper is concerned with the adaptive fuzzy control problem for a class of twin-roll strip casting systems. By using fuzzy logic systems (FLSs) to approximate the compounded nonlinear functions, a novel robust output tracking controller with adaptation laws is designed based on the high gain observer. First, the nonlinear dynamic equations for the roll gap and the molten steel level are constructed, respectively. Then, the mean value theorem is employed to transform the nonaffine nonlinear systems to the corresponding affine nonlinear systems. Moreover, it is also proved that all the closed-loop signals are bounded and the systems output tracking errors can converge to the desired neighborhoods of the origin via the Lyapunov stability analysis. Finally, simulation results, based on semiexperimental system dynamic model and parameters, are worked out to show the effectiveness of the proposed adaptive fuzzy design method.


---

## Body

## 1. Introduction

As is well known, the strip casting combines two processes of continuous casting and hot rolling into a single production; consequently, it brings in a lot of advantages including lower investment cost, energy saving, less space requirements. More specifically, compared with the conventional continuous casting [1], the production line and the production cost of the twin-roll strip casting process are significantly shortened and reduced, respectively. Meanwhile, due to a high cooling rate for the strip casting, the mechanical properties of metallic materials can be increased [2]. Nevertheless, the strip casting process is always with nonlinear uncertainty, external disturbance, and coupled behaviors, and the roll gap and the molten steel level control problems are still important research topics to guarantee steel strip quality.In [3], the model of the continuous casting process with various nonlinearities was proposed, and the corresponding controller was also designed. The authors in [4] developed an adaptive algorithm for the mould level control of a continuous steel slab casting. The modeling and control problem for a class of twin-roll strip casting system was studied in [5]. Correspondingly, some successful adaptive fuzzy or neural network approaches for the molten steel level control have been studied (see, e.g., [6–8] and the references therein) in the casting process. Recently, based on the perturbation method, a decoupling control strategy in [9] was proposed to obtain a uniform sheet thickness and keep a constant roll separating force in the strip casting process. By using twin-roll casting technology (TRC), the optimized process parameters and their effects on TRC of 7050 aluminum alloys strips were studied in [10].It should be pointed out that, in most of the results about nonlinear systems, the considered parameter uncertainties and disturbances satisfy matching condition [11, 12]. Besides, the above-mentioned control approaches require that all the states of the systems are available; thus they cannot be applied to nonaffine nonlinear system with unmeasured states. In particular, for the roll gap and the molten steel level control of twin-roll strip casting process, it is difficult to measure the rates of change of the roll gap and the molten steel level by using proper sensors. In [13], the adaptive fuzzy tracking control problem for a class of uncertain nonaffine nonlinear systems with nonsymmetric dead-zone inputs was investigated; however, the proposed approaches can only handle the SISO nonaffine nonlinear systems rather than MIMO systems with complex coupling terms. To the best knowledge of the authors, it is the first time the adaptive fuzzy tracking control is developed for MIMO nonaffine nonlinear casting systems with immeasurable states, which are very meaningful and more practical.The above considerations motivate our study work. Especially, inspired by [8, 13], by means of fuzzy approximation technique, the adaptive fuzzy output tracking control problem for a class of twin-roll strip casting systems is considered. Moreover, compared with the existing results, the main contributions of this paper are as follows: (1) The novel fuzzy tracking controllers with adaptation laws are designed by using fuzzy logic systems to approximate the compound nonlinear functions. (2) In order to handle the nonaffine coupling terms, the implicit function theorem and the mean value theorem are invoked, respectively. It is thus that the MIMO nonaffine nonlinear system can be transformed into the corresponding affine nonlinear system by this way. (3) By making use of adaptive mechanism driven by the estimation states obtained from the high gain observer, the influence of nonlinear parameter uncertainties and external disturbances is restrained effectively. It is also shown that the output tracking errors of the roll gap and the molten steel level can converge to the desired neighborhoods via the Lyapunov stability analysis.The rest of the paper is organized as follows. In Section2, the mathematical model for the strip casting process is given. The adaptive fuzzy output tracking control problem is addressed in Section 3. Simulations and experimental analysis are then provided in Section 4 to verify the effectiveness of the proposed approach. Finally, Section 5 draws the conclusions.


## 2. System Model for the Strip Casting Process

### 2.1. Molten Metal Level Equation

In this subsection, a diagram of the strip casting process is shown in Figure1, and the corresponding mathematical model for the molten steel leveling dynamics is described as in [8]. Concretely, the following dynamic equation can be derived from (1)–(4) in [8] (1)dxhdt=F-1xg,xhL-1Qin-Qout-Lxhdxgdt,where R is the roll radius, L is the length of the roll cylinders, Qin and Qout are the input flow and output flow of the pool between the two rolls, respectively, and F(xg,xh)=xg+2R-2R2-xh2.Figure 1
Schematic diagram of the twin-roll casting process.In addition, for convenience, the input flowQin is taken as Qin=k(t)u, where u is the control input, and the gain k(t) is determined empirically. The output flow Qout can be derived from the product of roll surface tangential velocity v, roll gap xg, and the length of the roll cylinder L; that is, Qout=Lxgv. So the molten metal level equation (1) is rewritten as (2)dxhdt=F-1xg,xhL-1ku-Lxgv-Lxhdxgdt.By introducing the coordinate transformationsx11=xg, x12=dxg/dt, x21=xh, and x22=dxh/dt, it follows from (2) that the following MIMO nonaffine nonlinear system can be obtained: (3)x˙i1=xi2x˙i2=fix,ui+diyi=xi1,i=1,2,where x=[x1,x2]T and xi=[xi1,xi2]T stand for the state variables, di is the unknown disturbance, yi and ui denote the system output and the control input of the ith subsystem, respectively, and fi(x,ui) is the unknown and smooth nonaffine nonlinear function with ui being electric servomotor control.Remark 1.
Applying Newton’s Second Law, it is easy to obtain thatf1(x,u1)=(1/M)(Ku1-fM(x)) with M being the mass of roll, fM(x) being viscous resistance, and u1 being servomotor control. At the same time, the nonaffine nonlinear function f2(x,u2) can be derived by taking derivative of (2).The objective of this paper is to design an adaptive fuzzy output feedback controllerui such that all the closed-loop error signals are uniformly ultimately bounded, and the system output yi tracks a reference trajectory ydi within a desired compact set in the presence of unknown nonaffine nonlinear coupling term. Then, to ensure the feasibility of the considered problem, the necessary assumptions are required for the nonaffine nonlinear system (3).Assumption 2.
The desired reference trajectoryydi is known and smooth, and its derivative is also continuous. That is, there exist unknown positive constants d-i, d˙-i, and d¨-i such that ydit≤d-i, y˙dit≤d˙-i, and y¨dit≤d¨-i, respectively.Assumption 3 (see [13]).
For allx∈R4 and ui∈R in the ith subsystem (3), there always exist positive constants fi1 and fi2 such that the following inequality holds: (4)0<fi1≤∂fix,ui∂ui≤fi2,i=1,2.Assumption 4.
The external disturbancedi is bounded; that is, there exists an unknown positive constant di∗ such that dit≤di∗.Remark 5.
It can be seen that Assumptions2 and 4 are quite standard in most of the references for nonlinear tracking control, which means that the external disturbances, the reference signals, and their derivatives are bounded. Assumption 3 is used to decouple the nonaffine nonlinear term for the ith subsystem, which implies that the change rate of the control input gain is bounded.


### 2.2. Fuzzy Logic Systems (FLSs)

Generally speaking, for an FLS, it consists of four parts: the knowledge base, the singleton fuzzifier, product inference, and center average defuzzifier, respectively. First, construct the knowledge base for FLS with the following IF-THEN rules:⁡Ri: If X1 is F1i and … and Xn is Fni.ThenY is Bi, i=1,2,…,N.Next, the FLS with the singleton fuzzifier, product inference, and center average defuzzifier can be expressed as(5)YX=∑i=1Nθi∏j=1nμFjiXj∑i=1N∏j=1nμFjiXj,where X=[X1,X2,…,Xn]T∈Rn, μFji is the membership of Fji, and θi=maxY∈R[μBi(Y)]. Let (6)ϕiX=∏j=1nμFjiXj∑i=1N∏j=1nμFjiXj,i=1,2,…,N,ϕ(X)=[ϕ1(X),ϕ2(X),…,ϕN(X)]T, and θ=[θ1,θ2,…,θN]T. Hence, the FLS can be rewritten in the following form: (7)YX=θTϕX.Lemma 6 (see [14]).
LetF(X) be a continuous function defined on a compact set ΩX. Then, for any given constant ϵ>0, there exists a FLS Y(X) in the form of (7) such that (8)supX∈ΩX⁡FX-YX=FX-θTϕX<ϵ.Similar to [13], the optimal parameter vectors θ∗ of FLS are defined as (9)θ∗=arg⁡minθ∈Ωθ⁡supX∈ΩX⁡FX-θTϕX,where Ωθ and ΩX are compact regions for θ and X, respectively. Furthermore, from Lemma 6, the fuzzy approximation error δ∗(X) is defined as (10)FX=θ∗TϕX+δ∗X,∀X∈ΩX⊂Rn.


## 2.1. Molten Metal Level Equation

In this subsection, a diagram of the strip casting process is shown in Figure1, and the corresponding mathematical model for the molten steel leveling dynamics is described as in [8]. Concretely, the following dynamic equation can be derived from (1)–(4) in [8] (1)dxhdt=F-1xg,xhL-1Qin-Qout-Lxhdxgdt,where R is the roll radius, L is the length of the roll cylinders, Qin and Qout are the input flow and output flow of the pool between the two rolls, respectively, and F(xg,xh)=xg+2R-2R2-xh2.Figure 1
Schematic diagram of the twin-roll casting process.In addition, for convenience, the input flowQin is taken as Qin=k(t)u, where u is the control input, and the gain k(t) is determined empirically. The output flow Qout can be derived from the product of roll surface tangential velocity v, roll gap xg, and the length of the roll cylinder L; that is, Qout=Lxgv. So the molten metal level equation (1) is rewritten as (2)dxhdt=F-1xg,xhL-1ku-Lxgv-Lxhdxgdt.By introducing the coordinate transformationsx11=xg, x12=dxg/dt, x21=xh, and x22=dxh/dt, it follows from (2) that the following MIMO nonaffine nonlinear system can be obtained: (3)x˙i1=xi2x˙i2=fix,ui+diyi=xi1,i=1,2,where x=[x1,x2]T and xi=[xi1,xi2]T stand for the state variables, di is the unknown disturbance, yi and ui denote the system output and the control input of the ith subsystem, respectively, and fi(x,ui) is the unknown and smooth nonaffine nonlinear function with ui being electric servomotor control.Remark 1.
Applying Newton’s Second Law, it is easy to obtain thatf1(x,u1)=(1/M)(Ku1-fM(x)) with M being the mass of roll, fM(x) being viscous resistance, and u1 being servomotor control. At the same time, the nonaffine nonlinear function f2(x,u2) can be derived by taking derivative of (2).The objective of this paper is to design an adaptive fuzzy output feedback controllerui such that all the closed-loop error signals are uniformly ultimately bounded, and the system output yi tracks a reference trajectory ydi within a desired compact set in the presence of unknown nonaffine nonlinear coupling term. Then, to ensure the feasibility of the considered problem, the necessary assumptions are required for the nonaffine nonlinear system (3).Assumption 2.
The desired reference trajectoryydi is known and smooth, and its derivative is also continuous. That is, there exist unknown positive constants d-i, d˙-i, and d¨-i such that ydit≤d-i, y˙dit≤d˙-i, and y¨dit≤d¨-i, respectively.Assumption 3 (see [13]).
For allx∈R4 and ui∈R in the ith subsystem (3), there always exist positive constants fi1 and fi2 such that the following inequality holds: (4)0<fi1≤∂fix,ui∂ui≤fi2,i=1,2.Assumption 4.
The external disturbancedi is bounded; that is, there exists an unknown positive constant di∗ such that dit≤di∗.Remark 5.
It can be seen that Assumptions2 and 4 are quite standard in most of the references for nonlinear tracking control, which means that the external disturbances, the reference signals, and their derivatives are bounded. Assumption 3 is used to decouple the nonaffine nonlinear term for the ith subsystem, which implies that the change rate of the control input gain is bounded.


## 2.2. Fuzzy Logic Systems (FLSs)

Generally speaking, for an FLS, it consists of four parts: the knowledge base, the singleton fuzzifier, product inference, and center average defuzzifier, respectively. First, construct the knowledge base for FLS with the following IF-THEN rules:⁡Ri: If X1 is F1i and … and Xn is Fni.ThenY is Bi, i=1,2,…,N.Next, the FLS with the singleton fuzzifier, product inference, and center average defuzzifier can be expressed as(5)YX=∑i=1Nθi∏j=1nμFjiXj∑i=1N∏j=1nμFjiXj,where X=[X1,X2,…,Xn]T∈Rn, μFji is the membership of Fji, and θi=maxY∈R[μBi(Y)]. Let (6)ϕiX=∏j=1nμFjiXj∑i=1N∏j=1nμFjiXj,i=1,2,…,N,ϕ(X)=[ϕ1(X),ϕ2(X),…,ϕN(X)]T, and θ=[θ1,θ2,…,θN]T. Hence, the FLS can be rewritten in the following form: (7)YX=θTϕX.Lemma 6 (see [14]).
LetF(X) be a continuous function defined on a compact set ΩX. Then, for any given constant ϵ>0, there exists a FLS Y(X) in the form of (7) such that (8)supX∈ΩX⁡FX-YX=FX-θTϕX<ϵ.Similar to [13], the optimal parameter vectors θ∗ of FLS are defined as (9)θ∗=arg⁡minθ∈Ωθ⁡supX∈ΩX⁡FX-θTϕX,where Ωθ and ΩX are compact regions for θ and X, respectively. Furthermore, from Lemma 6, the fuzzy approximation error δ∗(X) is defined as (10)FX=θ∗TϕX+δ∗X,∀X∈ΩX⊂Rn.


## 3. Adaptive Tracking Controller Design and Stability Analysis

### 3.1. Adaptive Tracking Controller Design

In this subsection, we shall present an adaptive fuzzy control scheme only based on output variable. Therefore, the high gain observer is introduced to design adaptive output fuzzy tracking controller, and the corresponding lemma is given as follows.Lemma 7 (see [15]).
Consider the following linear system:(11)ϵiz˙i1=zi2,ϵiz˙i2=-μi1zi2-zi1+yi,i=1,2,where ϵi>0 is a sufficiently small constant and the parameter μi1 is appropriately chosen such that s2+μi1s+1 is an Hurwitz polynomial. If the output function yi and its kth time derivatives yi(k) are bounded, that is, there exist positive constants Yi0, Yik satisfying yi≤Yi0, yik≤Yik, then it is obtained that (12)ηik≔zikϵik-1-yik-1=-ϵiϕik,k=1,2,where ϕi≔zi2+μi1zi1 and ϕi(k) represents the kth time derivative of ϕi. Moreover, if all the observer states satisfy that zik≤z-ik with z-ik>0, then there exist η-ik>0 such that ηik≤ϵiη-ik.According to Lemma6, the estimation of unmeasurable state vector is defined as (13)x^i=xi1,zi2ϵiT=x^i1,x^i2T,i=1,2.Next, to facilitate control system design from nonaffine form to affine form, the tracking error and the filtered tracking error are defined as e^i=x^i-xid=[e^i1,e^i2]T∈R2 and e^is=[λi,1]e^i, respectively, where xid=[yid,y˙id]T is the reference state vector and λi is appropriately chosen coefficient such that s+λi is an Hurwitz polynomial; that is, e^i→0 as e^is→0.Then, taking the derivative ofe^is gives (14)e^˙is=fix,ui-y¨id+x~˙i2+0,λie^i+di,where x^i2=xi2+x~i2. By using Assumption 3 and the implicit function theorem [16], there exists a unique and continuous ideal control ui∗=Ui(x)∈Ωui such that fi(x,ui∗)=fix,Uix=0 for all x∈Ωx, where Ωx and Ωui are two compact sets. By adding and subtracting γie^is in (14) and applying it to the mean value theorem, we have (15)e^˙is=fix,ui∗+∂fix,ui0∂uiui-ui∗+γie^is-y¨id+x~˙i2+0,λie^i+di-γie^is=μiui-∂fix,ui0∂uiui∗+∂fix,ui0∂ui-μiui+γieis+0,λiei-y¨id+x~˙i2+di+γie~is+0,λie~i-γie^is,where ui0 is some point between zero and ui, γi>0, μi>0, e^i=ei+e~i, and e^is=eis+e~is.Denote the nonlinear functionsHi(x,ei)=(∂fix,ui0/∂ui)ui∗-(∂fi(x,ui0)/∂ui-μi)ui-γieis-[0,λi]ei; it follows from (10) that Hi(x,ei) can be approximated by the following form: (16)Hix,ei=θi∗Tϕix,ei+δi∗x,eifor all (x,ei)∈Ωx×Ωei with Ωei being a compact set. Consequently, substituting (16) into (15) yields (17)e^˙is=μiui-θi∗Tϕix,ei+Dit-γie^is,where Di(t)=-δi∗(x,ei)-y¨id+x~˙i2+di+γie~is+[0,λi]e~i. From Lemma 7 and using Assumptions 2 and 4, it can be concluded that there exists an unknown upper bound Di∗ such that Dit≤Di∗.Remark 8.
For theith nonaffine nonlinear subsystem (3), Assumption 3 plays an important role in the controller design. The reason is that the implicit function theorem is employed to transform the nonaffine nonlinear coupling term into the corresponding affine term based on this assumption. In addition, the similar decoupling method in [13] has been developed; however, it is required that less adjustable parameters are used for the controller design in this paper.Moreover, the adaptive fuzzy tracking controller is designed for theith subsystem as follows: (18)ui=μi-1-αie^is+θ^iTϕix^,e^i-e^isD^i2e^isD^itanh⁡e^is/δi+δiwith adaptation laws (19)θ^˙i=-Γiϕix^,e^ie^is+σiθ^i,D^˙i=-βiσiD^i+βie^is,where θ^i and D^i are the estimate values of θi∗ and Di∗, respectively, and Γi=ΓiT>0, αi,δi,βi,σi, i=1,2, are positive design parameters.


### 3.2. Stability Analysis

In this subsection, the stability of the resulting closed-loop system is given in the following theorem.Theorem 9.
Consider the MIMO nonaffine nonlinear system (3) with unmeasurable states, under the condition that Assumptions 2–4 hold and the estimation states can be obtained from the high gain observer (11). On the compact set Ωx×Ωei, the adaptive fuzzy tracking controller (18) and the parameter updated laws (19) are constructed; then all the closed-loop system error signals are uniformly ultimately bounded. Moreover, the parameter estimation errors D~i, θ~i and tracking error eis remain as the compact sets ΩD~i, Ωθ~i, and Ωeis in the sense that (20)Ωeis=eis∈R∣eis≤ΩiΩD~i=D~i∈R∣D~i≤βiΩiΩθ~i=θ~i∈Rn∣θ~i≤ΩiλminΓi-1,i=1,2,where Ωi=2(Vit=0+ρi2/ρi1) with ρi1 and ρi2 defined as (21)ρi1≔min2αi+γi-12,σi-1/ψiλmaxΓi-1,σi-2κτi∗βi,ρi2≔αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2.Proof.
Define the Lyapunov functionVi(eis,θ~i,D~i) as (22)Vieis,θ~i,D~i=12eis2+12βi-1D~i2+12θ~iTΓi-1θ~i,where D~i=D^i-Di∗ and θ~i=θ^i-θi∗, i=1,2, are the parameter estimation errors. Taking the derivative of Vi yields (23)V˙i=eise˙is+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i=eisμiui-θi∗Tϕix,ei+Dit-γie^is+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i.By substituting (18) into (23), one can obtain that (24)V˙i=-αi+γieise^is+θ^iTϕix^,e^ieis-eise^isD^i2e^isD^itanh⁡e^is/δi+δi-θi∗Tϕix,eieis+eisDi+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis-e^is2D^i2e^isD^itanh⁡e^is/δi+δi+e~ise^isD^i2e^isD^itanh⁡e^is/δi+δi+e^isDi-e~isDi+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i,where e~is=e^is-eis. Using the adaptive control laws (19), (24) becomes (25)V˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis-e^is2D^i2e^isD^itanh⁡e^is/δi+δi+e~ise^isD^i2e^isD^itanh⁡e^is/δi+δi+e^isDi∗+e~isDi∗-σiD~iD^i+e^isD~i-θ~iTϕix^,e^ie^is-σiθ~iTθ^i.
Invoking inequalities0≤btanh⁡(b/a)≤b, ∀b∈R, a>0 and b≤btanh⁡(b/a)+κa, ∀b∈R, a>0, κ=0.2785 given in [17], it follows from (25) that (26)V˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis+e^isD^iδie^isD^i+δi+e~isD^i+κe~isD^i2+e~isDi∗-σiD~i2-σiD~iDi∗-θ~iTϕix^,e^ie^is-σiθ~iTθ^i.For the FLS error term on the right side of (26), adding and subtracting θi∗Tϕi(x^,e^i) yield (27)θ^iTϕix^,e^i-θi∗Tϕix^,e^i+θi∗Tϕix^,e^i-θi∗Tϕix,ei=θ~iTϕix^,e^i+θi∗Tϕix^,e^i-ϕix,ei.Using the fact that ϕiTϕi≤1 and the triangle inequality, we have (28)θ^iTϕix^,e^i-θi∗Tϕix,ei≤θ~iTϕx^,e^i+2θi∗2D~iDi∗=D^i2-Di∗2-D~i2≥-Di∗2-D~i22θ~iTθ^i=θ~i2+θ^i2-θi∗2≥θ~i2-θi∗2.
Invoking (28), (26) becomes (29)V˙i≤-αi+γieis2-αi+γieise~is-θ~iTϕix^,e^ie~is+2θi∗+e^isD^iδie^isD^i+δi+e~isD~i+κe~isD^i2+2e~isDi∗-σi2D~i2+σi2Di∗2-σi2θ~i2+σi2θi∗2.From the definition in (12) and the fact that e~is=[λi,1]e~i, it can be obtained that e~is is bounded; that is, there exists a positive constant τi∗ subject to e~is≤τi∗. Moreover, by utilizing the inequalities 2ab≤(1/c)a2+cb2, ∀c>0, a,b∈R and 0≤a/(a+b)<1, ∀a≥0, b>0, we have (30)V˙i≤-αi+γieis2+12eis2+αi+γi2τi∗22+12ψiθ~i2+τi∗2ψi2+2θi∗+δi+12φiτi∗2+D~i2φi2+2κτi∗D~i2+2κτi∗Di∗2+2τi∗Di∗-σi2D~i2+σi2Di∗2-σi2θ~i2+σi2θi∗2≤-αi+γi-12eis2-12σi-1ψiθ~i2-12σi-2κτi∗D~i2+αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2≤-ρi1Vi+ρi2,where ρi1 and ρi2 are given by the following form: (31)ρi1≔min⁡2αi+γi-12,σi-1/ψiλmaxΓi-1,σi-2κτi∗βiρi2≔αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2.
By appropriately adjusting the design parametersμi, αi, βi, γi, σi, δi, and ψi such that αi+γi-1/2>0, σi-1/ψi>0, and σi-2κτi∗>0, respectively, the closed-loop system stability can be guaranteed. Finally, by multiplying eρi1t and integrating over [0,t] on both sides of (30), we have (32)Vi≤Vit=0-ρi2ρi1e-ρi1t+ρi2ρi1≤Vit=0+ρi2ρi1.It follows from (22) and (32) that (33)eis≤2Vit=0+ρi2ρi1D~i≤2βiVit=0+ρi2ρi1θ~i≤2Vit=0+ρi2/ρi1λminΓi-1.Consequently, the parameter estimation errors D~i, θ~i and tracking error es are bounded from (31) and remain as the compact sets ΩD~i, Ωθ~i, and Ωeis in the sense that (34)Ωeis=eis∈R∣eis≤ΩiΩD~i=D~i∈R∣D~i≤βiΩiΩθ~i=θ~i∈Rn∣θ~i≤ΩiλminΓi-1,i=1,2,where Ωi=2(Vit=0+ρi2/ρi1) with ρi1 and ρi2 defined as (31). The proof is completed.Remark 10.
It is worth mentioning that the authors in [13] considered the adaptive fuzzy tracking control problem for a class of SISO nonaffine nonlinear systems. However, the approach proposed in [13] cannot be applied to MIMO nonaffine nonlinear casting systems with immeasurable coupling states. In this paper, the mean value theorem and the fuzzy approximation method are employed to transform the nonaffine nonlinear systems to the corresponding affine nonlinear systems. Also, based on Lyapunov stability analysis, it is shown that the proposed adaptive fuzzy output tracking control scheme can guarantee that the roll gap xg and the molten steel level xh can track to the empirical reference signals.


## 3.1. Adaptive Tracking Controller Design

In this subsection, we shall present an adaptive fuzzy control scheme only based on output variable. Therefore, the high gain observer is introduced to design adaptive output fuzzy tracking controller, and the corresponding lemma is given as follows.Lemma 7 (see [15]).
Consider the following linear system:(11)ϵiz˙i1=zi2,ϵiz˙i2=-μi1zi2-zi1+yi,i=1,2,where ϵi>0 is a sufficiently small constant and the parameter μi1 is appropriately chosen such that s2+μi1s+1 is an Hurwitz polynomial. If the output function yi and its kth time derivatives yi(k) are bounded, that is, there exist positive constants Yi0, Yik satisfying yi≤Yi0, yik≤Yik, then it is obtained that (12)ηik≔zikϵik-1-yik-1=-ϵiϕik,k=1,2,where ϕi≔zi2+μi1zi1 and ϕi(k) represents the kth time derivative of ϕi. Moreover, if all the observer states satisfy that zik≤z-ik with z-ik>0, then there exist η-ik>0 such that ηik≤ϵiη-ik.According to Lemma6, the estimation of unmeasurable state vector is defined as (13)x^i=xi1,zi2ϵiT=x^i1,x^i2T,i=1,2.Next, to facilitate control system design from nonaffine form to affine form, the tracking error and the filtered tracking error are defined as e^i=x^i-xid=[e^i1,e^i2]T∈R2 and e^is=[λi,1]e^i, respectively, where xid=[yid,y˙id]T is the reference state vector and λi is appropriately chosen coefficient such that s+λi is an Hurwitz polynomial; that is, e^i→0 as e^is→0.Then, taking the derivative ofe^is gives (14)e^˙is=fix,ui-y¨id+x~˙i2+0,λie^i+di,where x^i2=xi2+x~i2. By using Assumption 3 and the implicit function theorem [16], there exists a unique and continuous ideal control ui∗=Ui(x)∈Ωui such that fi(x,ui∗)=fix,Uix=0 for all x∈Ωx, where Ωx and Ωui are two compact sets. By adding and subtracting γie^is in (14) and applying it to the mean value theorem, we have (15)e^˙is=fix,ui∗+∂fix,ui0∂uiui-ui∗+γie^is-y¨id+x~˙i2+0,λie^i+di-γie^is=μiui-∂fix,ui0∂uiui∗+∂fix,ui0∂ui-μiui+γieis+0,λiei-y¨id+x~˙i2+di+γie~is+0,λie~i-γie^is,where ui0 is some point between zero and ui, γi>0, μi>0, e^i=ei+e~i, and e^is=eis+e~is.Denote the nonlinear functionsHi(x,ei)=(∂fix,ui0/∂ui)ui∗-(∂fi(x,ui0)/∂ui-μi)ui-γieis-[0,λi]ei; it follows from (10) that Hi(x,ei) can be approximated by the following form: (16)Hix,ei=θi∗Tϕix,ei+δi∗x,eifor all (x,ei)∈Ωx×Ωei with Ωei being a compact set. Consequently, substituting (16) into (15) yields (17)e^˙is=μiui-θi∗Tϕix,ei+Dit-γie^is,where Di(t)=-δi∗(x,ei)-y¨id+x~˙i2+di+γie~is+[0,λi]e~i. From Lemma 7 and using Assumptions 2 and 4, it can be concluded that there exists an unknown upper bound Di∗ such that Dit≤Di∗.Remark 8.
For theith nonaffine nonlinear subsystem (3), Assumption 3 plays an important role in the controller design. The reason is that the implicit function theorem is employed to transform the nonaffine nonlinear coupling term into the corresponding affine term based on this assumption. In addition, the similar decoupling method in [13] has been developed; however, it is required that less adjustable parameters are used for the controller design in this paper.Moreover, the adaptive fuzzy tracking controller is designed for theith subsystem as follows: (18)ui=μi-1-αie^is+θ^iTϕix^,e^i-e^isD^i2e^isD^itanh⁡e^is/δi+δiwith adaptation laws (19)θ^˙i=-Γiϕix^,e^ie^is+σiθ^i,D^˙i=-βiσiD^i+βie^is,where θ^i and D^i are the estimate values of θi∗ and Di∗, respectively, and Γi=ΓiT>0, αi,δi,βi,σi, i=1,2, are positive design parameters.


## 3.2. Stability Analysis

In this subsection, the stability of the resulting closed-loop system is given in the following theorem.Theorem 9.
Consider the MIMO nonaffine nonlinear system (3) with unmeasurable states, under the condition that Assumptions 2–4 hold and the estimation states can be obtained from the high gain observer (11). On the compact set Ωx×Ωei, the adaptive fuzzy tracking controller (18) and the parameter updated laws (19) are constructed; then all the closed-loop system error signals are uniformly ultimately bounded. Moreover, the parameter estimation errors D~i, θ~i and tracking error eis remain as the compact sets ΩD~i, Ωθ~i, and Ωeis in the sense that (20)Ωeis=eis∈R∣eis≤ΩiΩD~i=D~i∈R∣D~i≤βiΩiΩθ~i=θ~i∈Rn∣θ~i≤ΩiλminΓi-1,i=1,2,where Ωi=2(Vit=0+ρi2/ρi1) with ρi1 and ρi2 defined as (21)ρi1≔min2αi+γi-12,σi-1/ψiλmaxΓi-1,σi-2κτi∗βi,ρi2≔αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2.Proof.
Define the Lyapunov functionVi(eis,θ~i,D~i) as (22)Vieis,θ~i,D~i=12eis2+12βi-1D~i2+12θ~iTΓi-1θ~i,where D~i=D^i-Di∗ and θ~i=θ^i-θi∗, i=1,2, are the parameter estimation errors. Taking the derivative of Vi yields (23)V˙i=eise˙is+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i=eisμiui-θi∗Tϕix,ei+Dit-γie^is+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i.By substituting (18) into (23), one can obtain that (24)V˙i=-αi+γieise^is+θ^iTϕix^,e^ieis-eise^isD^i2e^isD^itanh⁡e^is/δi+δi-θi∗Tϕix,eieis+eisDi+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis-e^is2D^i2e^isD^itanh⁡e^is/δi+δi+e~ise^isD^i2e^isD^itanh⁡e^is/δi+δi+e^isDi-e~isDi+βi-1D~iD~˙i+θ~iTΓi-1θ~˙i,where e~is=e^is-eis. Using the adaptive control laws (19), (24) becomes (25)V˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis-e^is2D^i2e^isD^itanh⁡e^is/δi+δi+e~ise^isD^i2e^isD^itanh⁡e^is/δi+δi+e^isDi∗+e~isDi∗-σiD~iD^i+e^isD~i-θ~iTϕix^,e^ie^is-σiθ~iTθ^i.
Invoking inequalities0≤btanh⁡(b/a)≤b, ∀b∈R, a>0 and b≤btanh⁡(b/a)+κa, ∀b∈R, a>0, κ=0.2785 given in [17], it follows from (25) that (26)V˙i≤-αi+γieis2-αi+γieise~is+θ^iTϕix^,e^i-θi∗Tϕix,eieis+e^isD^iδie^isD^i+δi+e~isD^i+κe~isD^i2+e~isDi∗-σiD~i2-σiD~iDi∗-θ~iTϕix^,e^ie^is-σiθ~iTθ^i.For the FLS error term on the right side of (26), adding and subtracting θi∗Tϕi(x^,e^i) yield (27)θ^iTϕix^,e^i-θi∗Tϕix^,e^i+θi∗Tϕix^,e^i-θi∗Tϕix,ei=θ~iTϕix^,e^i+θi∗Tϕix^,e^i-ϕix,ei.Using the fact that ϕiTϕi≤1 and the triangle inequality, we have (28)θ^iTϕix^,e^i-θi∗Tϕix,ei≤θ~iTϕx^,e^i+2θi∗2D~iDi∗=D^i2-Di∗2-D~i2≥-Di∗2-D~i22θ~iTθ^i=θ~i2+θ^i2-θi∗2≥θ~i2-θi∗2.
Invoking (28), (26) becomes (29)V˙i≤-αi+γieis2-αi+γieise~is-θ~iTϕix^,e^ie~is+2θi∗+e^isD^iδie^isD^i+δi+e~isD~i+κe~isD^i2+2e~isDi∗-σi2D~i2+σi2Di∗2-σi2θ~i2+σi2θi∗2.From the definition in (12) and the fact that e~is=[λi,1]e~i, it can be obtained that e~is is bounded; that is, there exists a positive constant τi∗ subject to e~is≤τi∗. Moreover, by utilizing the inequalities 2ab≤(1/c)a2+cb2, ∀c>0, a,b∈R and 0≤a/(a+b)<1, ∀a≥0, b>0, we have (30)V˙i≤-αi+γieis2+12eis2+αi+γi2τi∗22+12ψiθ~i2+τi∗2ψi2+2θi∗+δi+12φiτi∗2+D~i2φi2+2κτi∗D~i2+2κτi∗Di∗2+2τi∗Di∗-σi2D~i2+σi2Di∗2-σi2θ~i2+σi2θi∗2≤-αi+γi-12eis2-12σi-1ψiθ~i2-12σi-2κτi∗D~i2+αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2≤-ρi1Vi+ρi2,where ρi1 and ρi2 are given by the following form: (31)ρi1≔min⁡2αi+γi-12,σi-1/ψiλmaxΓi-1,σi-2κτi∗βiρi2≔αi+γi2τi∗22+τi∗2ψi2+2θi∗+δi+2τi∗Di∗+σi2θi∗2+2κτi∗+σi2Di∗2.
By appropriately adjusting the design parametersμi, αi, βi, γi, σi, δi, and ψi such that αi+γi-1/2>0, σi-1/ψi>0, and σi-2κτi∗>0, respectively, the closed-loop system stability can be guaranteed. Finally, by multiplying eρi1t and integrating over [0,t] on both sides of (30), we have (32)Vi≤Vit=0-ρi2ρi1e-ρi1t+ρi2ρi1≤Vit=0+ρi2ρi1.It follows from (22) and (32) that (33)eis≤2Vit=0+ρi2ρi1D~i≤2βiVit=0+ρi2ρi1θ~i≤2Vit=0+ρi2/ρi1λminΓi-1.Consequently, the parameter estimation errors D~i, θ~i and tracking error es are bounded from (31) and remain as the compact sets ΩD~i, Ωθ~i, and Ωeis in the sense that (34)Ωeis=eis∈R∣eis≤ΩiΩD~i=D~i∈R∣D~i≤βiΩiΩθ~i=θ~i∈Rn∣θ~i≤ΩiλminΓi-1,i=1,2,where Ωi=2(Vit=0+ρi2/ρi1) with ρi1 and ρi2 defined as (31). The proof is completed.Remark 10.
It is worth mentioning that the authors in [13] considered the adaptive fuzzy tracking control problem for a class of SISO nonaffine nonlinear systems. However, the approach proposed in [13] cannot be applied to MIMO nonaffine nonlinear casting systems with immeasurable coupling states. In this paper, the mean value theorem and the fuzzy approximation method are employed to transform the nonaffine nonlinear systems to the corresponding affine nonlinear systems. Also, based on Lyapunov stability analysis, it is shown that the proposed adaptive fuzzy output tracking control scheme can guarantee that the roll gap xg and the molten steel level xh can track to the empirical reference signals.


## 4. Simulations and Experimental Analysis

In this section, in order to verify the effectiveness of the proposed adaptive fuzzy control method, the following numerical simulation is performed for MIMO nonaffine nonlinear system (3), and the corresponding system parameters are selected as R=150 mm, L=200 mm, M=300 kg, and vr=10 mpm. These values are chosen from [9]. In addition, the initial roll gap and the desired roll gap are set to be 0 mm and 3 mm, and the initial molten steel level and desired molten steel level are set to be 20 mm and 70 mm, respectively. Furthermore, the fuzzy membership functions are chosen as follows: (35)μFj1=exp⁡-Xj+1.522,μFj2=exp⁡-Xj+122,μFj3=exp⁡-Xj+0.522,μFj4=exp⁡-Xj22,μFj5=exp⁡-Xj-0.522,μFj6=exp⁡-Xj-122,μFj7=exp⁡-Xj-1.522,j=1,2,…,n.Define fuzzy basis functions as (36)ϕiX=∏j=1nμFjiXj∑i=17∏j=1nμFjiXj,i=1,2,…,7,where X=[X1,X2,…,Xn]T.Besides, the simulation parameters are chosen asμi=5, αi=5, βi=0.2, Γi=10, σi=10, and δi=0.1, i=1,2, and the initial values are selected as x1(0)=[0.02,0]T, x2(0)=[0,0.1]T, x^1(0)=[2,-3]T, x^2(0)=[0,-1]T, θ^1(0)=θ^2(0)=[0,0.1,0,0.1,0,0,0.1]T, and D^1(0)=D^2(0)=5. The simulation results are obtained in Figures 2–11. From Figures 2 and 3, the system output signals of yi and the reference signals ydi, respectively, can be seen. Figures 4 and 5 show that the tracking performance is satisfactory and the output tracking errors of the roll gap and the molten steel level can converge to the desired neighborhoods of the origin. The curves of the estimations states are given in Figures 6 and 7. The boundedness of parameter estimations θ^i and D^i, as well as the designed control ui, is demonstrated in Figures 8–11. Besides, the casting 7075 aluminum alloy strip is shown in Figure 12. It can also be seen that the casting strip is flat and there are no obvious cracks on the surface.Figure 2
Trajectories of the system outputy1 and the desired reference signal yd1.Figure 3
Trajectories of the system outputy2 and the desired reference signal yd2.Figure 4
Trajectory of the system tracking error of molten steel level.Figure 5
Trajectory of the system tracking error of roll gap.Figure 6
The observer statesx^11 and x^12.Figure 7
The observer statesx^21 and x^22.Figure 8
The response curves ofθ^1.Figure 9
The response curves ofθ^2.Figure 10
The response curves of the adaptive laws.Figure 11
The respond curves of the control signals.Figure 12
The casting 7075 aluminum alloy strip image showing the smooth surface in twin-roll strip process.


## 5. Conclusion

This paper studies the robust adaptive fuzzy tracking control problem for a class of twin-roll strip casting systems. Based on fuzzy logic systems (FLSs) to approximate the compounded nonlinear functions, a novel adaptive fuzzy output tracking control scheme is developed by using the high gain observer. The mean value theorem is employed to decouple the nonaffine nonlinear systems, and thus it is proved that all the closed-loop signals are bounded and the systems output tracking errors of the roll gap and the molten steel level can converge to the desired neighborhoods of the origin via the Lyapunov stability analysis. Simulation results show the effectiveness of the proposed adaptive control approach.



---

*Source: 6742630-2017-06-05.xml*