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.

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: The novel fuzzy tracking controllers with adaptation laws are designed by using fuzzy logic systems to approximate the compound nonlinear functions. 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. 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.

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 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 in [8] where is the roll radius, is the length of the roll cylinders, and are the input flow and output flow of the pool between the two rolls, respectively, and .

Figure 1 

Schematic diagram of the twin-roll casting process.

In addition, for convenience, the input flow is taken as , where is the control input, and the gain is determined empirically. The output flow can be derived from the product of roll surface tangential velocity , roll gap , and the length of the roll cylinder ; that is, . So the molten metal level equation (1) is rewritten as

By introducing the coordinate transformations , , , and , it follows from (2) that the following MIMO nonaffine nonlinear system can be obtained: where and 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.

Remark 1. Applying Newton’s Second Law, it is easy to obtain that with being the mass of roll, being viscous resistance, and being servomotor control. At the same time, the nonaffine nonlinear function can be derived by taking derivative of (2).

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 , , and such that , , and , respectively.

Assumption 3 (see [13]). For all and in the th subsystem (3), there always exist positive constants and 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.

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: ⁡: If is and … and is . Then is , .

Next, the FLS with the singleton fuzzifier, product inference, and center average defuzzifier can be expressed aswhere , is the membership of , and . Let , and . Hence, the FLS can be rewritten in the following form:

Lemma 6 (see [14]). Let be a continuous function defined on a compact set . Then, for any given constant , there exists a FLS in the form of (7) such that

Similar to [13], the optimal parameter vectors of FLS are defined as where and are compact regions for and , respectively. Furthermore, from Lemma 6, the fuzzy approximation error is defined as

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: where is a sufficiently small constant and the parameter is appropriately chosen such that is an Hurwitz polynomial. If the output function and its th time derivatives are bounded, that is, there exist positive constants , satisfying , , then it is obtained that where and represents the th time derivative of . Moreover, if all the observer states satisfy that with , then there exist such 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 and , respectively, where is the reference state vector and is appropriately chosen coefficient such that is an Hurwitz polynomial; that is, as .

Then, taking the derivative of gives where . By using Assumption 3 and the implicit function theorem [16], there exists a unique and continuous ideal control such that for all , where and are two compact sets. By adding and subtracting in (14) and applying it to the mean value theorem, we have where is some point between zero and , , , , and .

Denote the nonlinear functions ; it follows from (10) that can be approximated by the following form: for all with being a compact set. Consequently, substituting (16) into (15) yields where . From Lemma 7 and using Assumptions 2 and 4, it can be concluded that there exists an unknown upper bound such that .

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 and are the estimate values of and , respectively, and , , , 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 , 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 , and tracking error remain as the compact sets , , and in the sense that where with and defined as

Proof. Define the Lyapunov function as where and , , are the parameter estimation errors. Taking the derivative of yields By substituting (18) into (23), one can obtain that where . Using the adaptive control laws (19), (24) becomes Invoking inequalities , , and , , , given in [17], it follows from (25) that For the FLS error term on the right side of (26), adding and subtracting yield Using the fact that and the triangle inequality, we have Invoking (28), (26) becomes From the definition in (12) and the fact that , it can be obtained that is bounded; that is, there exists a positive constant subject to . Moreover, by utilizing the inequalities , , and , , , we have where and are given by the following form: By appropriately adjusting the design parameters , , , , , , and such that , , and , respectively, the closed-loop system stability can be guaranteed. Finally, by multiplying and integrating over on both sides of (30), we have It follows from (22) and (32) that Consequently, the parameter estimation errors , and tracking error are bounded from (31) and remain as the compact sets , , and in the sense that where with and defined as (31). The proof is completed.