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

1Software School, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 2School of Science, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 3School of Materials and Metallurgy, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China 4College of International Finance and Bank, University of Science and Technology Liaoning, Anshan, Liaoning 114051, China


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][7][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 Section 2, 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.

Molten Metal Level Equation.
In this subsection, a diagram of the strip casting process is shown in Figure 1, 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] where  is the roll radius,  is the length of the roll cylinders,  in and  out are the input flow and output flow of the pool between the two rolls, respectively, and (  ,  ℎ ) =   + 2 − 2√ 2 −  2 ℎ .In addition, for convenience, the input flow  in is taken as  in = (), where  is the control input, and the gain () is determined empirically.The output flow  out can be derived from the product of roll surface tangential velocity V, roll gap   , and the length of the roll cylinder ; that is,  out =   V.So the molten metal level equation ( 1) is rewritten as By introducing the coordinate transformations  11 =   ,  12 =   /,  21 =  ℎ , and  22 =  ℎ /, it follows from (2) that the following MIMO nonaffine nonlinear system can be obtained: where  = [ 1 ,  2 ]  and   = [ 1 ,  2 ]  stand for the state variables,   is the unknown disturbance,   and   denote the system output and the control input of the th subsystem, respectively, and   (,   ) is the unknown and smooth nonaffine nonlinear function with   being electric servomotor control.
The objective of this paper is to design an adaptive fuzzy output feedback controller   such that all the closed-loop error signals are uniformly ultimately bounded, and the system output   tracks a reference trajectory   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 trajectory   is known and smooth, and its derivative is also continuous.That is, there exist unknown positive constants   , ḋ  , and Assumption 3 (see [13]).For all  ∈ R 4 and   ∈ R in the th subsystem (3), there always exist positive constants  1 and  2 such that the following inequality holds: Assumption 4. The external disturbance   is bounded; that is, there exists an unknown positive constant  *  such that Remark 5.It can be seen that Assumptions 2 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 th subsystem, which implies that the change rate of the control input gain is bounded.

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: Next, the FLS with the singleton fuzzifier, product inference, and center average defuzzifier can be expressed as where () = [ 1 (),  2 (), . . .,   ()]  , and  = [ 1 ,  2 , . . .,   ]  .Hence, the FLS can be rewritten in the following form: Lemma 6 (see [14]).Let () be a continuous function defined on a compact set Ω  .Then, for any given constant  > 0, there exists a FLS () in the form of ( 7) such that sup Similar to [13], the optimal parameter vectors  * of FLS are defined as where Ω  and Ω  are compact regions for  and , respectively.Furthermore, from Lemma 6, the fuzzy approximation error  * () is defined as

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: where   > 0 is a sufficiently small constant and the parameter  1 is appropriately chosen such that where   fl  2 +  1  1 and  ()  represents the th time derivative of   .Moreover, if all the observer states satisfy that According to Lemma 6, the estimation of unmeasurable state vector is defined as Next, to facilitate control system design from nonaffine form to affine form, the tracking error and the filtered tracking error are defined as ê = x −   = [ê 1 , ê2 ]  ∈ R 2 and ê = [  , 1]ê  , respectively, where   = [  , ẏ  ]  is the reference state vector and   is appropriately chosen coefficient such that +  is an Hurwitz polynomial; that is, ê → 0 as ê → 0.
Denote the nonlinear functions   (,   ) = (  (,  Remark 8.For the th 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 the th subsystem as follows: with adaptation laws where θ and D are the estimate values of  *  and  *  , respectively, and Γ  = Γ   > 0,   ,   ,   ,   ,  = 1, 2, are positive design parameters.

Stability Analysis.
In this subsection, the stability of the resulting closed-loop system is given in the following theorem.
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   and the molten steel level  ℎ can track to the empirical reference signals.

Simulations and Experimental Analysis
In this section, in order to verify the effectiveness of 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  = 150 mm,  = 200 mm,  = 300 kg, and V  = 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.The height y 1 of molten metal The desired height y d1 of molten metal Furthermore, the fuzzy membership functions are chosen as follows: ] , ] , ] , ] , ] ,  = 1, 2, . . ., .
Define fuzzy basis functions as where  = [ 1 ,  2 , . . .,   ]  .The tracking error of molten steel level The observer state x11 The observer state x12   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 and D , as well as the designed control   , 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.

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 estimate of 휃 11 The estimate of 휃 12 The estimate of 휃 13 The estimate of 휃 14 The estimate of 휃 15 The estimate of 휃 16 The estimate of 휃 17 The estimates of The estimate of 휃 21 The estimate of 휃 22 The estimate of 휃 23 The estimate of 휃 24 The estimate of 휃 25 The estimate of 휃 26 The estimate of 휃 27

Figure 1 :
Figure 1: Schematic diagram of the twin-roll casting process.

Figure 2 :
Figure 2: Trajectories of the system output  1 and the desired reference signal  1 .

Figure 3 :
Figure 3: Trajectories of the system output  2 and the desired reference signal  2 .

Figure 4 : 1 Figure 5 :
Figure 4: Trajectory of the system tracking error of molten steel level.