The Perturbations Estimation in Two Gas Plants

The perturbations are the unwanted and unknown inlets in nonlinear plants which can affect the outlets. In this article, an estimator is studied for the variables and perturbations estimation in nonlinear plants. The saturation map is used in our estimator instead of the signum map to decrease the chattering, and we ensure the estimator convergence by the Lyapunov analysis. The conditions required by our estimator gains are found to reach the variables error convergence, and these gains are used for the perturbations estimation. An algorithm is proposed to choose the gains for achieving a satisfactory performance in our estimator. The studied estimator is applied for the variables and perturbations estimation in the gas turbine and gasification plants.


I. INTRODUCTION
The perturbations are the unwanted and unknown inlets in nonlinear plants which can affect the outlets. This issue has occurred in many nonlinear plants. Since perturbations can affect the sensors, actuators, or plants yielding additional costs, and since most nonlinear plants regulators require the knowledge of perturbations; an approach for the perturbations estimation is welcome.
There are some studies about regulators for perturbed plants. In [1] and [2], the active strategy for the perturbations attenuation is mentioned. In [3] and [4], the singular perturbations approach for the perturbations attenuation is considered. The variables and perturbations estimation in plants is focused in [5]- [7], and [8]. In [9]- [12], and [13], the robust analysis for the perturbed plants stabilization is focused. The perturbations estimation with fuzzy regulators is mentioned in [14] and [15]. In [16]- [18], and [19], the authors use the neuro-fuzzy approximations for the perturbed plants regulation. The adaptive laws for the perturbed plants regulation are focused in [20], [21], and [22]. In [23] and [24], the structure theory for the perturbations attenuation is mentioned. From the above studies, in [1], [2], [5], [6], [7]- [11], The associate editor coordinating the review of this manuscript and approving it for publication was Ning Sun . [12], [13], [17], and [19], the authors use approaches for the variables or perturbations estimation in nonlinear plants; then it would be welcome to be focused on this issue. The novelty of this article is that each nonlinear plant has a different structure; consequently, a special estimator with the structure of the gas plants must be discussed.
There are various estimators who use the plant outlets for the variables estimation [1], [2], [5], [6], but there are not many estimators who use the plant outlets for the perturbations estimation. In [7], [8], [11], [13], [17], [19], previous studies of estimators are focused for the perturbations estimation, but with the two below differences: in the previous studies estimators with adaptive or feedback outlets are used, while in this article an estimator with sliding modes outlets is used, in the plants of previous studies the noise is not used, while in the plants of this article the noise is used.
In this article, the sliding mode approach is utilized in our estimator for the variables and perturbations estimation in nonlinear plants. Since the sliding mode approach uses the signum map, it can yield the unwanted chattering [3], [12], [14], [15].
The first contribution of this article is that an estimator is designed, it is described by the following characteristics: 1) the saturation map is used in our estimator instead of the signum map to decrease the chattering; and later, we ensure VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ the estimator convergence by the Lyapunov analysis, 2) find the conditions required by our estimator gains that allow it to reach the variables error convergence; it yields an acceptable performance in the variables and perturbations estimation. The second contribution of this article is that an algorithm is proposed to choose the gains for achieving a satisfactory performance in our estimator, it is described as follows: 1) we choose a value for the gain 1, 2) we obtain the estimator matrix, 3) we obtain the eigenvalues of the estimator matrix, if the real parts of all the eigenvalues of the estimator matrix are negative, then the gain 1 is correctly chosen and we can go to the step 4, otherwise, we must return to the step 1, 4) we choose the matrix 1 of the Lyapunov equation, 5) we substitute matrix 1 and estimator matrix into the Lyapunov equation, and we find the matrix 2 of the Lyapunov equation, if matrix 1 and matrix 2 of the Lyapunov equation are positive definite, then matrix 1 is correctly chosen and we can go to the step 6, otherwise, we must return to step 1, 6) we choose the gain 2, gain 3, and gain 4 to solve the estimator, if the estimator reaches an acceptable exactness for the variables and perturbations estimation in the nonlinear plant, then the algorithm finishes, otherwise, we must return to step 6.
Our estimator is applied for the variables and perturbations estimation in the gas turbine and gasification plants. The gas turbine plant is used for the electrical energy generation from the gas [25], while the gasification plant is used for the gas generation from biomass [26].
The rest of the article is described below. Section II presents the estimator design containing the variables error convergence, and the perturbations estimation in nonlinear plants, later, an algorithm is proposed to choose the gains for achieving a satisfactory performance in our estimator. Sections III and IV estimators are studied for the variables and perturbations estimation in the gas turbine and gasification plants. Section V express the conclusion and future work.

II. THE ESTIMATOR FOR THE VARIABLES AND PERTURBATIONS ESTIMATION
In this section, a variables estimator, and a perturbations estimator, which are termed as estimator will be studied for the variables and perturbations estimation in nonlinear plants.
In this article, a special nonlinear plant will be used in which the outlets have a linear combination with the variables, the variables have a nonlinear combination with the variables, and the perturbations are entered additively. The nonlinear plant is [25], [26]: h ∈ n as plant variables, v ∈ m as the plant inlets, y ∈ as the plant outlets, u ∈ as the perturbations, f (h, v) ∈ n as a nonlinear map, ∈ as the noise, A ∈ n×n , B ∈ n×1 , C ∈ p×n as matrices.

A. THE VARIABLES ESTIMATOR
The goal of the variables estimator is that using the inlets and outlets, the variables of the variables estimator should estimate the nonlinear plant variables.
The estimator error y is: y as the variables estimator outlet, h = h − h as the variables error, h as the estimator variables. The variables estimator is: h as the estimator variables, y as the estimator outlets, sat(·) as the saturation map, M as a matrix where MC ∈ n×n is a positive semi-definite constant, K ∈ n×1 and E ∈ n×1 .

B. THE CONVERGENCE ANALYSIS OF THE VARIABLES ESTIMATOR
In this sub-section, the convergence of the variables estimator applied to nonlinear plants is analyzed based on the Lyapunov approach [3], [4].
The closed loop model of the variables estimator is the subtraction of (3) to (1) and using the estimator error (2) as: The nonlinear map f is bounded as f ≤ f , |·| as the absolute value. The below theorem analyzes the variables estimator convergence.
Theorem 1: The variables error of the variables estimator (2)-(3) applied to estimate the nonlinear plant variables h (1) is convergent, γ = λ min (Q s P −1 s ), f + Bu + B ≤ u, u ≤ E, · as the Euclidean norm in n , |·| as the absolute value, P s ∈ n×n and Q s ∈ n×n are positive definite matrices which meet: A s as is expressed in (4). Proof: The Lyapunov candidate map is: The derivative of (4) is: 83082 VOLUME 8, 2020 Using the second term of (7), sat(MC h) = sat( h), and f + Bu + B ≤ u, is: Using (8) in (7) is: Using A T s P s + P s A s + µP s = −Q s of (5), is: The equation of (9) can be expressed as: , , we notice that there are three cases of the saturation map. 1) If h > 1, then sat( h) = 1 and h = h , we replace in (10) as: 2) If h ≤ 1, then sat( h) = h and h T h = h T h , we replace in (10) as: (10) as: since (11), (12), (13), the three cases we have the same inequality expressed as: Using γ = λ min (Q s P −1 s ) (14) becomes to: Since (15), it concludes that the variables error of the variables estimator applied to estimate the nonlinear plants variables is convergent.

C. THE PERTURBATIONS ESTIMATOR
The goal of the perturbations estimator is that using the outlets, the perturbations of the perturbations estimator should estimate the nonlinear plant perturbations.
Since the Theorem 1, it is: Using, lim in the first equality of (4) is: Since all the terms of (17) are bounded independently of T , it becomes to:  (2), (18), u only allows to estimate perturbations u, and since in this article the map sat(·) is used, it can reduce the unwanted chattering. The possibility of changing the non-continuous map sat(·) for a softer one also could reduce the chattering.
Remark 3: Note it is not required that the nonlinear plant (1) must be convergent to achieve a satisfactory performance in our estimator.
Remark 4: For the satisfactory operation of the estimator (2), (3), (18), the theory and application conditions mentioned below must be met: a) propose a gain K that meets the theory condition (5) such that the variables h of the variables estimator (2), (3) must reach as soon as possible the nonlinear plant variables h of (1), b) propose the gain   (2), (18) that meets the application condition such that the estimated perturbations u of (2), (18) should reach as soon as possible the nonlinear plant perturbations u of (1). In case that the proposed gains K , E do not work, you have to start over.

E of the perturbations estimator
The Figure 2 shows the proposed algorithm to choose the gains K , E, M , h for achieving a satisfactory performance in our estimator, the request Re (eigen (A ss )) < 0 represents if the real part in the eigenvalues of A ss are negative, the request Q s > 0 represents if Q s is positive definite, and the request h ∼ = h, u ∼ = u represents of h, u achieve a satisfactory performance in the estimation of h, u. The proposed algorithm of Figure 2 detailed as follows: 1) we choose a value for K , 2) we obtain A s = A − KC, 3) we obtain the eigenvalues of A ss = sI − A s , if the real parts af all the eigenvalues of A ss are negative, then the gain K is correctly chosen and we can go to the step 4, otherwise, we must return to the step 1, 4) we choose P s , 5) we substitute P s and A s into the Lyapunov equation A T s P s + P s A s = −Q s , and we find Q s , if P s and Q s are positive definite, then P s is correctly chosen, the Theorem 1 is met, and we can go to the step 6, otherwise, we must return to step 1, 6) we choose the gains E, M , and h to solve the proposed estimator as

if the proposed estimator reaches an acceptable exactness for the variables and perturbations estimation in the nonlinear plant
then the algorithm finishes, otherwise, we must return to step 6.
In the below sections, the root mean squared error (MSE) is used for comparisons, it is:

III. THE GAS TURBINE PLANT
The Figure 3 shows the gas turbine plant. The gas turbine plant consists of a compressor, a combustion chamber, a turbine, and a power turbine.  6 , h 9 = P 3 , h 10 = P 4 , h 11 = P 5 , h 12 = P 6 , and the outlet is y = P 6 . Table 1 shows the gas turbine plant constants.
In the Figures 4 and 5, since the Estimator 1 reaches better variables and perturbations estimation of the plant than the Estimator 2, it is seen that the Estimator 1 reaches better performance. In addition, the Figures 4 and 5 show that in the variables and perturbations estimation of Estimator 1, 83086 VOLUME 8, 2020 the unwanted chattering is not presented. In the Table 2, since the MSE is smaller for the Estimator 1 than for the Estimator 2, it is seen that the Estimator 1 reaches better exactness for the variables and perturbations estimation.
The gains K ∈ 11 , E ∈ 11 , M ∈ 11 , and h ∈ for the Estimator 1 are chosen using the proposed algorithm of Figure 2 detailed as follows: 1) we choose a value for   Table 4 shows the MSE of (19).
From Figures 7 and 8, since the Estimator 1 reaches better variables and perturbations estimation of the plant than the Estimator 2, it is seen that the Estimator 1 reaches better performance. In addition, the Figures 7 and 8 show that in the variables and perturbations estimation of Estimator 1, the unwanted chattering is not presented. In the Table 4, since the MSE is smaller for the Estimator 1 than for the Estimator 2, it is seen that the Estimator 1 reaches better exactness for the variables and perturbations estimation.
Remark 5: In the past two sections, it would be very tedious and expensive to have the sensors to measure all plant variables. Consequently, it highlights one of the major contributions of this article, it is that with the outlets measurement, our estimator can roughly estimate the variables and perturbations.

V. CONCLUSION
In this article, an estimator was studied for the variables and perturbations estimation in nonlinear plants. The variables error convergence was analyzed with the Lyapunov approach. Our estimator was compared to a previous estimator in the gas turbine and gasification plants concluding that our estimator reached a better performance than the previous estimator for the variables and perturbations estimation. In addition, in our estimator, the unwanted chattering is not presented. Our estimator can be applied to many types of nonlinear plants such as electric, mechanical, hydraulic or thermal. In the future work, we will seek to use another alternative map that allows us to reduce the unwanted chattering for this type of estimators, we will explore other types of strategies for the perturbations estimation, for the perturbations attenuation, or for trajectories reaching in perturbed plants. JOSÉ  VICTOR GARCIA is currently pursuing the Ph.D. degree with the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional. He has published two articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems.
GUADALUPE JULIANA GUTIERREZ is currently a Full Time Professor with the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional. She has published ten articles in international journals. Her fields of interest are robotic systems, modeling, and intelligent systems.
TOMAS MIGUEL VARGAS is currently pursuing the Ph.D. degree with the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional. He has published two articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems.
GENARO OCHOA received the Ph.D. degree from the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional, in 2017. He has published nine articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems.
RICARDO BALCAZAR received the Ph.D. degree from the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional, in 2017. He has published eight articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems. JAIME PACHECO is currently a Full Time Professor with the Sección de Estudios de Posgrado e Investigación, ESIME Azcapotzalco, Instituto Politécnico Nacional. He has published 25 articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems.
JESUS ALBERTO MEDA-CAMPAÑA (Member, IEEE) is currently a Full Time Professor with the Sección de Estudios de Posgrado e Investigación, ESIME Zacatenco, Instituto Politécnico Nacional. He has published 30 articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems.
DANTE MUJICA-VARGAS is currently a Full Time Professor with the Tecnológico Nacional de México/CENIDET. He has published 15 articles in international journals. His fields of interest are robotic systems, modeling, and intelligent systems. VOLUME 8, 2020