𝐻_∞ Control Theory Using in the Air Pollution Control System

Abstract

In recent years, air pollution control has caused great concern. This paper focuses on the primary pollutant SO₂ in the atmosphere for analysis and control. Two indicators are introduced, which are the concentration of SO₂ in the emissions (PSO₂) and the concentration of SO₂ in the atmosphere (ASO₂). If the ASO₂ is higher than the certain threshold, then this shows that the air is polluted. According to the uncertainty of the air pollution control systems model, 𝐻_∞ control theory for the air pollution control systems is used in this paper, which can change the PSO₂ with the method of improving the level of pollution processing or decreasing the emissions, so that air pollution system can maintain robust stability and the indicators ASO₂ are always operated within the desired target.

1. Introduction

The main feature of the 𝐻_∞ control theory is based on the frequency design method of using the state-space model, and this theory presents an effective method to solve the uncertainty problem of external disturbance to the system. In order to overcome the drawbacks of the classical control theory and the modern control theory, 𝐻_∞ control theory established technology and method of the loop shaping in the frequency domain, which combines the classic frequency-domain and the modern state-space method. The design problem of the control system is converted to the 𝐻_∞ control problem, which made the system closer to the actual situation and meet the actual needs. So it gives the robust control system design method, which obtains 𝐻_∞ controller by solving two Riccati equations. This method fully considered the impact of system uncertainty, which not only can ensure the robust stability of the control system, but also can optimize some performance indices. It is the optimal control theory in frequency domain, and the parameters design of 𝐻_∞ controller is more effective than optimal regulator [1].

So the 𝐻_∞ control theory for the air pollution control systems can solve the uncertainty of the air pollution control systems model, which can get the control strategy to change the PSO₂, so that air pollution system can maintain robust stability and the indicators ASO₂ are always operated within the desired target.

2. Standard 𝐻_∞ Control Problem

2.1. Problem Description

The standard 𝐻_∞ control problem is shown in Figure 1, which consists of 𝐺 and 𝐾. 𝐺 is a generalized control target which is a given part of the system. 𝐾 is 𝐻_∞ controller, and it needs to be designed.

Figure 1 

The diagram of the standard 𝐻∞ control problem.

It is supposed that 𝐺 and 𝐾 are described as the transfer function matrix of the linear time invariant system [2]. Then, 𝐺(𝑠) and 𝐾(𝑠) are proper rational matrices, Decomposing 𝐺(𝑠) as 𝐺𝐺=¹¹𝐺₁₂𝐺₂₁𝐺₂₂. (1) Its state-space realization is 𝑥̇=𝐴𝑥+𝐵₁𝜔+𝐵₂𝑢,𝑧=𝐶₁𝑥+𝐷₁₁𝜔+𝐷₁₂𝑢,𝑦=𝐶₂𝑥+𝐷₂₁𝜔+𝐷₂₂𝑢. (2) It is denoted as: 𝐺=𝐴𝐵₁𝐵₂𝐶₁𝐷₁₁𝐷₁₂𝐶₂𝐷₂₁𝐷₂₂, (3) where 𝑥∈𝑅^𝑛 is the state vector, 𝜔 is the external input, 𝑢 is the control input, 𝑧 is the controlled output, and 𝑦 is the measured output. They are all vector signals. To compare (1) and (3), we can obtain 𝐺_𝑖𝑗=𝐶_𝑖(𝑠𝐼−𝐴)⁻¹𝐵_𝑗+𝐷_𝑖𝑗,𝑖,𝑗=1,2. (4) Obviously, the closed-loop transfer function matrix from 𝜔 to 𝑧 can be expressed as 𝑇_𝑧𝜔(𝑠)=𝐺₁₁+𝐺₁₂𝐾𝐼−𝐺₂₂𝐾⁻¹𝐺₂₁=𝐹_𝑙(𝐺,𝐾). (5)

The problem of 𝐻_∞ optimal control theory is to find a proper rational controller 𝐾 for closed-loop control system, and make the closed-loop control system internally stabile, and then minimize the 𝐻_∞ norm of closed-loop transfer function matrix 𝑇_𝑧𝜔(𝑠) [3]: min_{𝐾stability𝐺}𝐹_𝑙(𝐺,𝐾)_∞<1. (6)

2.2. State-Space 𝐻_∞ Output-Feedback Control

We assume that the state-space representation of the generalized controlled object 𝐺 state-space representation is (2), where 𝑥∈𝑅^𝑛, 𝜔∈𝑅^𝑚1, 𝑢∈𝑅^𝑚2, 𝑧∈𝑅^𝑝1, 𝑦∈𝑅^𝑃2, 𝐴∈𝑅^𝑛×𝑛, 𝐵₁∈𝑅^𝑛×𝑚1, 𝐵₂∈𝑅^𝑛×𝑚2, 𝐶₁∈𝑅^𝑝1×𝑛, 𝐶₂∈𝑅^𝑝2×𝑛, 𝐷₂₁ and 𝐷₂₂ are the corresponding dimension of the real matrix, and controller 𝐾 is dynamic output feedback compensator [4].

Consider that 𝐺 have a special form: 𝐺=𝐴𝐵₁𝐵₂𝐶₁0𝐷₁₂𝐼00𝐼00, (7)

which satisfies the following conditions:(1)(𝐴,𝐵₁) is stabilizable, and (𝐶₁,𝐴) is detectable;(2)(𝐴,𝐵₂) is stabilizable;(3)𝐷^𝑇₁₂𝐶₁𝐷₁₂=0𝐼.

Obviously, 𝑦=[^𝑥_𝜔]; that is, the state 𝑥 of generalized control object and the external input signal 𝜔 could be measured; then, it can be directly used to constitute control law.

As the result, we can obtain Theorem 1.

Theorem 1. If and only if 𝐻_∞∈dom(Ric), 𝑋_∞=Ric(𝐻_∞)≥0, there exists 𝐻_∞ controller 𝑢=−𝐵^𝑇₂𝑋_∞𝑥+𝑄(𝑠)𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥. (8) That is, 𝐾(𝑠)=−𝐵^𝑇₂𝑋_∞−𝑄(𝑠)𝛾⁻²𝐵^𝑇₁𝑋_∞𝑄(𝑠), (9) where 𝑄(𝑠)∈𝑅𝐻_∞, ‖𝑄(𝑠)‖_∞<𝛾, which is made to stabilize the closed-loop control system, and the closed-loop transfer function matrix 𝑇_𝑧𝜔(𝑠) from 𝜔 to 𝑧 is satisfied as ‖𝑇_𝑧𝜔(𝑠)‖_∞<𝛾.

Proof. If Hamilton matrix 𝐻_∞ was considered, there exist 𝑋_∞=Ric(𝐻_∞), and then the derivative of 𝑥^𝑇(𝑡)𝑋_∞𝑥(𝑡) can be achieved, such that 𝑑𝑥𝑑𝑡^𝑇𝑋_∞𝑥=𝑥̇^𝑇𝑋_∞𝑥+𝑥^𝑇𝑋_∞𝑥̇. (10)
Substituting (2) into (10) and considering Riccati equation (11) and orthogonality condition 𝐴^𝑇𝑋_∞+𝑋_∞𝐴+𝛾⁻²𝑋_∞𝐵₁𝐵^𝑇₁𝑋_∞−𝑋_∞𝐵₂𝐵^𝑇₂𝑋_∞+𝐶^𝑇₁𝐶₁=0. (11) So the following formula can be achieved: 𝑑𝑥𝑑𝑡^𝑇𝑋_∞𝑥=𝑥^𝑇𝐴^𝑇𝑋_∞+𝑋_∞𝐴𝑥+𝜔^𝑇𝐵^𝑇₁𝑋_∞𝑥+𝐵^𝑇₁𝑋_∞𝑥^𝑇𝜔+𝑢^𝑇𝐵^𝑇₂𝑋_∞𝑥+𝐵^𝑇₂𝑋_∞𝑥^𝑇𝑢=−𝐶₁𝑥²−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥²+𝐵^𝑇₂𝑋_∞𝑥²+𝜔^𝑇𝐵^𝑇₁𝑋_∞𝑥+𝐵^𝑇₁𝑋_∞𝑥^𝑇𝜔+𝑢^𝑇𝐵^𝑇₂𝑋_∞𝑥+𝐵^𝑇₂𝑋_∞𝑥^𝑇𝑢=−‖𝑧‖²+𝛾²‖𝜔‖²−𝛾²𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥²+𝑢+𝐵^𝑇₂𝑋_∞𝑥². (12) If 𝑥(0)=𝑥(∞)=0, 𝜔∈𝐿₂[0,+∞), to integral above equation from 𝑡=0 to 𝑡=∞, then ‖𝑧‖²₂−𝛾²‖𝜔‖²₂=𝑢+𝐵^𝑇₂𝑋_∞𝑥²₂−𝛾²𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥²₂. (13) According to the equivalence relation, 𝑇_𝑧𝜔(𝑠)_∞<𝛾⟺‖𝑧‖²₂<𝛾²‖𝜔‖²₂. (14)
Form (12) and (13), we can get that 𝑇_𝑧𝜔(𝑠)_∞<𝛾⟺𝑢+𝐵^𝑇₂𝑋_∞𝑥²₂<𝛾²𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥²₂⟺𝑢+𝐵^𝑇₂𝑋_∞𝑥=𝑄(𝑠)𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥, (15) where ‖•‖ represent Euclidean norm.
Hence, 𝑢=−𝐵^𝑇₂𝑋_∞𝑥+𝑄(𝑠)(𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥); the proof is completed.

𝑢 is the input to ensure ‖𝑇_𝑧𝜔(𝑠)‖_∞<𝛾, where 𝑄(𝑠)(𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥) is the output of the transfer system 𝑄(𝑠) by the input (𝜔−𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥). From (12), the decay of 𝑥^𝑇(𝑡)𝑋_∞𝑥(𝑡)≥0 is the slowest when 𝜔=𝛾⁻²𝐵^𝑇₁𝑋_∞𝑥. Then for this kind of disturbance, we can stabilize (12) by using control strategy 𝑢.

3. Analysis and Synthesis of Air Pollution Control System

Atmospheric quality management is essentially the process of the analysis and synthesis to the air pollution control system. So-called atmospheric system analysis is qualitative and quantitative research for an atmospheric area system or facilities system, which include that to evaluate the current situation, to find out the main environment problems, to put a series of optional target projects, and to establish the quantitative relation between emission and the quality of the permissive atmosphere [5]. The systems synthesis means the plan and design of an air pollution control system on the basis of system analysis to determine the target and to determine a management method of a system, in other words, in order to achieve certain environmental goal to select the optimal planning scheme, to optimal design, to find the optimal management method, and so forth. So the process of the synthesis should include three main steps: determine the target, form better feasibility scheme of system, and optimization decision [6].

Usually, the atmosphere has certain self-purification ability; namely, the atmospheric environment has a certain capacity. It refers to the permissible pollutant emissions within the natural purification capacity, which reach the limiting quantity in order not to destruct the nature material circulation [7]. We can make the quantity of pollutant discharged that meets a certain environmental goal to be permissible total emission. Only when the pollutant emissions beyond atmospheric self-purification ability, namely, exceeds the environmental capacity, there may be air pollution [8].

In this paper, the objective of applying the 𝐻_∞ control theory in air pollution control system is to find out regularity, to make full use of atmospheric environmental self-purification ability; we cannot only develop production but also protect the environment.

4. 𝐻_∞ Control of Air Pollution

In recent years, air pollution control has caused great concern. This paper focuses on primary pollutant SO₂ in the atmosphere for analysis and control, which is mainly pollutant of the PM2.5. We introduced two indicators, which are the concentration of SO₂ in the emissions (PSO₂) and the concentration of SO₂ in the atmosphere (ASO₂). Meanwhile, we can change the content of SO₂ in the emissions (with the method of improving the methods of processing), with the aim of returning atmospheric quality back to the desired value.

4.1. The Mathematical Model of Air Pollution

A certain volume of air mainly accepted the controlled pollutants which discharged from certain purification equipments. In addition, considering that the atmospheric self-purification ability is mainly affected by wind and other meteorological factors, a certain volume of air can be defined as atmospheric section. Thus, we can put forward a second order state-space equation; it describes the relationship between PSO₂ and ASO₂ on an average point of the atmospheric section [9]. The basic idea of modeling is to consider each section as ideal stirred reactor, as shown in Figure 2. So, the parameters and variables of the whole section are consistent, and the output concentration of PSO₂ and ASO₂ is equal to the counterpart concentration in this section. Hence, from the point of view of the mass balance, we can get the following equation.

Figure 2 

The stirred reactor model of an ideal atmosphere section.

PSO₂ balance equation: 𝑥̇_𝑖=−𝑘_𝑖𝑥_𝑖+𝑄_𝑖−1𝑣_𝑖𝑥_𝑖−1−𝑄_𝑖+𝑄_𝐸𝑣_𝑖𝑥_𝑖. (16)

ASO₂ balance equation: 𝑦̇_𝑖=ℎ_𝑖𝑦^𝑠_𝑖−𝑦_𝑖+𝑄_𝑖−1𝑣_𝑖𝑦_𝑖−1−𝑄_𝑖+𝑄_𝐸𝑣_𝑖𝑦_𝑖−𝑘_𝑖𝑥_𝑖, (17) where 𝑥_𝑖,𝑥_𝑖−1 are the PSO₂ of Section 𝑖 and Section 𝑖−1 (mg/m³), 𝑣_𝑖 is the atmospheric capacity of Section 𝑖 (m³); 𝑄_𝐸 is PSO₂ gas flow rate of Section 𝑖 (m³/d), 𝑦_𝑖,𝑦_𝑖−1 are ASO₂ of Section 𝑖 and Section 𝑖−1 (mg/m³), 𝑘_𝑖,ℎ_𝑖 are daily decay rate of PSO₂ of Section 𝑖 and supply rate of ASO₂ of Section 𝑖, 𝑄_𝑖,𝑄_𝑖−1 are atmosphere gas flow rates of Section 𝑖 and Section 𝑖−1 (m³/d), and 𝑦^𝑠_𝑖 is saturating capacity of SO₂ of Section 𝑖 (mg/m³).

From [5], we can conclude that it is advisable to take the following values as the coefficients in above equations: 𝑘_𝑖=0.32/day,ℎ_𝑖=0.2/day,𝑦^𝑠_𝑖=0.36mg/m³,𝑄_𝐸𝑣_𝑖𝑄=0.1,^𝑖𝑣_𝑖,𝑄_𝑖−1𝑣_𝑖=0.9. (18)

Thus, the mathematical model of Section 𝑖 air pollution is 𝑥̇_𝑖𝑦̇_𝑖=−1.320𝑥−0.32−1.2𝑦^𝑖_𝑖+00.1𝑢+0.9𝑥_𝑖−10.9𝑦_𝑖−1,+1.9 (19) where 𝑢 is control strategy.

4.2. Simulation of Air Pollution 𝐻_∞ Control

𝐻_∞ control problem of air pollution is basically how to control emissions of pollutants in the best way, in order to properly handle the cost of cleaning and the price we pay for too much atmospheric pollution. Using 𝐻_∞ control theory can better achieve this goal, which not only saves investment but also is easy to be realized [10].

In this paper, the simulation object is two sections of the 𝐻_∞ control of atmospheric pollution, the state-space equation is described as 𝑥̇₁𝑦̇₁𝑥̇₂𝑦̇₂=−1.32000𝑥−0.32−1.2000.90−1.32000.9−0.32−1.2¹𝑦₁𝑥₂𝑦₂+00.0.10.1𝑢+5.351.094.191.9 (20)

Using the MATLAB to simulate [11], we can conclude the step response curve of the air pollution control system. We have 𝑢(𝑠)=570613.9602(𝑠+7.702)(𝑠+1.32)²(𝑠+1.2)²(𝑠+4012)(𝑠+1.77)𝑠²+2.5𝑠+1.563𝑠²,+2.4𝑠+1.44𝐾(𝑠)=𝑄(𝑠),−0.67−1.77−2.32−1.52−0.40.10.10000000.10000000.10000000.1008.634.368.246.922.180.14 (21)(22) which is the transfer function and the state-space expression of robust 𝐻_∞ control strategy.

It could be seen from Figure 3, that the response time of the original system is 3.6 s, while the response time of closed loop system is 0.026 s by using 𝐻_∞ control strategy. It means that 𝐻_∞ control strategy makes the step response of the atmosphere system with an improvement; the response time is greatly reduced. Therefore, 𝐻_∞ control strategy is a practical control strategy, which can ensure that air pollution control system is operating steadily within the desired target [12].

(a)   The original closed-loop system

(b) The closed-loop system with 𝐻∞ control strategy

Figure 3 

The system step response.

5. Conclusion

In this paper, the 𝐻_∞ control theory and methods have a great application value on air pollution control system. It can help the environmental protection departments at various levels to analyze the air pollution system, which can ensure the atmosphere quality steady work within the desired target value. Of course, the analysis and control of a large-scale air pollution system that introduced other influencing factors will be very complicated, which will be the focus of the study in this field.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grants nos. 61104062, 61374085, and 61174077), Jiangsu Qing Lan Project, and PAPD.