Design of robust control for uncertain fuzzy quadruple-tank systems with time-varying delays

The robust H∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_\infty$$\end{document} observer-based control design is addressed here for non-linear Takagi-Sugeno (T-S) fuzzy systems with time-varying delays, subject to uncertainties and external disturbances. This is motivated by the quadruple-tank with time delay control problem. The observer design methodology is based on constructing an appropriate Lyapunov–Krasovskii functional (LKF) for an augmented system formed from the original and the delayed states. The bilinear terms are transferred to the linear matrix inequalities, thanks to a change of variables which can be solved in one step. Furthermore, by employing the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_2$$\end{document} performance index, the adverse effects of persistent bounded disturbances is largely avoided. The proposed method has the advantage of relating the controller and Lyapunov function to both the original and delayed states. Then, the controller and observer gains are obtained simultaneously by solving these inequalities with off-the-shelf software (Yalmip/MATLAB toolbox). Finally, an application to a simulated quadruple-tank system with time delay is carried out to demonstrate the benefits of the proposed technique, showing a compromise between controller simplicity and robustness that outperforms previous approaches.


Introduction
Takagi-Sugeno (T-S) systems are a kind of the fuzzy system introduced in Tanaka and Wang (2004) to facilitate the use of fuzzy system tools for some nonlinear systems. Because of the effective representation of a nonlinear system as a set of local linear models that are interpolated by nonlinear functions, T-S fuzzy methods have proved useful in a variety of problems (Ammar et al. 2018;Tuan VLB, and Hajjaji 2018;Chaibi et al. 2019;Naami et al. 2019;Ejegwa 2020;Yang et al. 2020;Saif et al. 2020;Dutta and Doley 2021;Zhang and Huang 2021;Kchaou and Jerbi 2021;Ech-charqy et al. 2020;D'Urso 2017;Najariyan et al. 2017). Stability analysis and controller design can then be handled with this technique (Takagi and Sugeno 1985;Tanaka et al. 1998) for nonlinear systems (Xie et al. (2020)), by an equivalent combination of linear systems. In this context, they have been shown to be useful as universal approximators in (Buckley (1992)) and (Castro (1995)), making possible to extend classical linear model techniques to a wide range of problems, including stabilization, observation, regulation and filtering.
Over the last few decades, many research approaches have studied the observer-based control in the context of disturbances or noises that might create instabilities in nonlinear closed-loop systems ). This control is often obtained under the assumption that the entire state vector can be accessed by output measurement. This is complex in practice when there are disturbances (Wang and Lam (2021)), or when some of the external disturbances to the system are unavailable (Wei and Ma (2021)). In several studies, the problem of constructing observers for systems with noise has been solved by using the H 1 filtering approach (Tuan VLB, and Hajjaji 2018;Xie et al. 2019;Naami et al. 2021). Moreover, (He et al. (2021)) developed some important works on stabilisation for T-S fuzzy descriptor systems. Recently, a straightforward method for observer-based H 1 control for discretetime Takagi-Sugeno (T-S) fuzzy systems has been presented (Chang et al. (2015); Zhi (2021)). An observerbased controller as an alternative to direct static state feedback is discussed in (Chang et al. (2016); Mahmoud et al. (2021)). We also point out ) where dynamic output feedback controller was described for Lipschitz nonlinear systems under input saturation. On the other hand, the H 1 observer design for uncertain one-sided Lipschitz systems with time-varying delay (albeit without taking the controller into account) is proposed in (Yan et al. (2020)).
In such context, the presence of parametric uncertainties, make the stabilization of the system more complicated. As a result, attention is being paid to uncertain processes. For example, the authors in ) have been investigated the problem of robust observer-based control for one-sided Lipschitz nonlinear systems subject to parametric uncertainties and external disturbances. Several other articles have addressed the stable stabilization problem of uncertain Takagi-Sugeno (T-S) fuzzy models (Ahammed and Azeem 2019;Dong et al. 2021;Zhu et al. 2021). In (Islam et al. (2020)), the robust controller design for an uncertain fuzzy system with time-varying time-delay was investigated using a fuzzy functional observer. Several results dealing with observerbased controller design method for Lipschitz nonlinear systems with uncertain parameters and disturbances have also been published (Zemouche et al. 2017;Rastegari et al. 2019;Dinh 2021). Among them we emphasize (Xu et al. (2019)), based on minimizing an index related to the state estimation performance, to optimize the actual value of the uncertainty. Furthermore, using a fuzzy description of the uncertainty bound, the optimal design of the controller is envisaged in ), using a comprehensive fuzzy performance index that involves the performance and the control cost.
The present work focuses on the design of H 1 observerbased controllers for delayed continuous-time Takagi-Sugeno fuzzy systems, in the presence of parameter uncertainties and external disturbances. The research is motivated by a process composed of four interconnected tanks; the model of this process is also used to demonstrate the approach. There are some previous studies on observerbased controllers for fuzzy systems: for instance, in ), these controllers were applied to a similar quadruple-tank system; in (Naami et al. (2021)), the robust H 1 control problem was studied with parameter uncertainties. However, these previous studies did not take into account the time delay, motivating this work. The main contribution of this paper is then the proposal of a direct approach for designing H 1 observer-based controllers for T-S fuzzy systems with uncertainties, external disturbances and time-varying delays. The approach is based on proposing a Lyapunov-Krasovskii functional with time-delay information . Based on it, a controller is proposed based on using both the original state and the time-delay state. Using this approach allows to develop stability conditions expressed as LMIs, so the design conditions are also expressed as LMIs. The solution of these LMIs makes it possible to obtain the observer and controller gains.
The paper is organised as follows. Section 2 describes the mathematical modelling of the fuzzy quadruple-tank systems with time delay, and some previous results. Section 3 presents the main contributions , describing the observer-based controller and the design methodology. Section 4 provides the application to the quadruple-tank system with time delay. Finally, the paper draws some conclusions. Notation The following standard notation are used in the paper. The superscript ð:Þ T represent the matrix transpose. P [ 0 means that P is a symmetric positive definite matrix, and H e ¼ P þ P T f g. The symbol Ã denotes a symmetric block. I denotes the identity matrix with appropriate dimensions.
2 Model description and preliminaries 2.1 Fuzzy quadruple-tank systems Figure 1 shows the schematic diagram of the quadrupletank model with disturbances (Johansson and Nunes (1998)). The process is composed of four identical cylindrical tanks numbered 1-4, and a reservoir 5. The two pumps transfer the water from the reservoir to tank-3 and tank-4. The objective is to control the liquid levels in tank-3 and tank-4, by the observation of the dynamic states of tank-1 and tank-2. A delay in the inputs of the system is introduced to make the control problem more challenging (Shah and Patel (2019)). The continuous-time nonlinear system is represented by the following equations: where x k ðtÞ is the liquid level in the tank number k,Ã pk is the cross sectional area of tank k,ã pk is the cross-sectional area of the outlet of tank k, g is the gravity constant, g j ðtÞ 2 0; 1 ½ is the valve flow proportion, and # j ðtÞ is the control signal of pump j, with the gaink #j . The parameters of the process are condensed in Table 1. The measured output signals yðtÞ 2 R 2 , correspond in this study to the levels of tanks 1 and 2. d(t) is the time it takes for the liquid to move to tank-1 from tank-3, tank-2 from tank-4, tank-1 from pump-1, tank-4 from pump-1, tank-2 from pump-2, tank-3 from pump-2, and tank-3 and tank-4. This delay satisfies that: The nonlinear termsF k ðtÞ ¼ ffiffiffiffiffiffiffiffiffi ffi x k ðtÞ p of the level in the tank-k 8k ¼ 1; . . .; 4, the following sector rules apply: If x k ðtÞ is N 1 ThenF k ðtÞ ¼C Fz1 x k ðtÞ If x k ðtÞ is N 2 ThenF k ðtÞ ¼C Fz2 x k ðtÞ with N 1 and N 2 the fuzzy sets. As a result, the following is the T-S Fuzzy model: It should be noted that the membership functions (MFs) are with the following properties: where Remark 1 The rules (3) are generalized to calculate only u j ðx 1 Þ, j ¼ 1; 2, by using the same fuzzy form and nonlinear function characteristics of the model. Thanks to this, the number of membership functions is reduced from 8 to 2. Table 2 details the fuzzy parameters.
The quadruple-tank system can be expressed by the uncertain T-S Fuzzy model in (3) by: where l i ðuðtÞÞ is the grade of membership of uðtÞ, xðtÞ 2 R n is the state vector, yðtÞ 2 R p is the output vector, uðtÞ 2 R r is the input vector, wðtÞ 2 l q 2 is the unknown exogenous disturbance, A i 2 R nÂn , A d i 2 R nÂn , B i 2 R nÂr and C i 2 R pÂn ði ¼ 1; . . .; mÞ, are known constant matrices. Finally, DA i ðtÞ, DA d i ðtÞ ði ¼ 1; . . .; mÞ and DCðtÞ are unknown matrices that represent model uncertainty.

C Fz2
Coefficient of fuzzy set in region N 2 0.8149 and the following controller is used for the system (8): where L i 2 R nÂp , K j 2 R xÂn and K d j 2 R xÂn ði; j ¼ 1; . . .; mÞ are the observer and the controller gains, respectively, to be calculated, andxðtÞ 2 R n is the estimation of x(t) provided by the observer. Let us define eðtÞ ¼ xðtÞ ÀxðtÞ, the dynamics of e(t) which can be obtained as follows: By using the controller (12), then the system (8) in closedloop is: Following this, the augmented system can be written as follows:

Problem statement
The main objective of the paper is to design a robust observer-based control design in the presence of process delays and disturbances, such that augmented system (15) is asymptotically stable and satisfies that kfðtÞk 2 kwðtÞk 2 \c ð16Þ

Preliminaries
The following lemmas are required to provide the main results in the following section.
Lemma 1 (Lien (2004)) 8 r, positive constant, and real matrices M, N, and F 2 R of appropriate dimensions, such that F T ðtÞFðtÞ 6 I, the following holds: Lemma 2 (Chang et al. (2015)) For matrices T, Q, U, and W, with appropriate dimensions, and scalarf the inequality, is satisfied if the following condition holds: Lemma 3 (Dong et al. (2017)) For any constant matrix Z ¼ Z T [ 0 and scalar h ! 0 such that the following integrations are well defined, then  3 Main results

Robust observer-based control design
This section aims to investigate the stability problem of system (8) in the presence of uncertainties and disturbances.

Simulation results and discussions
To compensate for the inherent differences between system states and measurement outputs, an uncertain model is used. The uncertainty matrices for the fuzzy quadrupletank system (1) are then the following:  Figure 3 shows the evolution of the measured levels ðx 1 ; x 2 Þ, although Fig. 4 illustrates the evolution of the unmeasured states ðx 3 ; x 4 Þ. Despite the existence of model uncertainties and disturbances, the observer estimates adequately the level of the four tanks, as seen in these figures. Now, by using the YALMIP toolbox (Löfberg (2004)) in Matlab (Higham and Higham (2005)), the LMI (20) in Theorem 1 can be solved, obtaining the following controller gains: The flow distribution rate in the tanks is considered, based on the position of the control valves g 1 ðtÞ and g 2 ðtÞ, which will be modified according to the following rules: If t s ðpÞ ¼ 0; . . .; 20 Then g 1 ðtÞ þ g 2 ðtÞ 2 1; 2 ½ ; MP If t s ðpÞ ¼ 20; . . .; 120 Then g 1 ðtÞ þ g 2 ðtÞ 2 0; 1 ½ ; NMP If t s ðpÞ ¼ 120; . . .; 220 Then g 1 ðtÞ þ g 2 ðtÞ 2 1; 2 ½ ; MP If t s ðpÞ ¼ 220; . . .; 300 Then g 1 ðtÞ þ g 2 ðtÞ 2 0; 1 ½ ; NMP where MP is Minimum Phase setting (Johansson (1997)), and NMP is No-Minimum Phase setting. Using the setting rules, the estimation error of each tank level is presented in Fig. 5. Figure 6 shows the control signals of the observer-based controller designed using the approach in this paper. The results demonstrate the closed-loop stability and the reduction of tracking errors in various situations, including in the presence of parameter uncertainties and perturbations.

Conclusions
This paper has proposed a direct approach to design robust observer-based controllers for T-S fuzzy systems with time-varying delays in the presence of parameter uncertainties and admissible external disturbances. Lyapunov- Krasovskii functional have been used to ensure the robust asymptotic stability, making possible to then select both the controller and observer gains in a single step, by solving a set of LMIs. This methodology ensures that the closed-loop system is robust asymptotically stable with a certain performance c. The results reported in this paper are also relevant from the perspective of discrete-time state estimation. The approach was demonstrated using a simulated quadruple tank laboratory process: an observerbased controller was developed, and tested in simulation, to show the effectiveness of the proposed approach. It must be pointed out that the functional used in this article does not include the integral of the derivative, but this issue appears in the application, contributing to non-linear terms that are more complex. This merits further research.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Data availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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