Design of a robust LMI-based model predictive control method for surge instability in interconnected compressor systems in the presence of uncertainty and disturbance

ABSTRACT Surge is the most significant instability observed in the compressors, and its control requires the exact dynamics of the compressor systems. Uncertainty in compressor characteristics and unknown opening percentage of the throttle and spillback valves, as well as disturbances in compressor’s flow and pressures, are among the major issues to be addressed in controller design for surge instability. Furthermore, in compressor systems that consist of several individual compressors, their reciprocal effects should also be taken into account. This paper presents an LMI-based decentralized robust model predictive control to ensure the stability of the compressor system against surge instability, uncertainty, and disturbance. The proposed scheme benefits from the optimized control signal with minimum computational complexity to overcome the destabilizing effects in a complex compressor system. The considered working class for this compressor system is a continuous-time nonlinear system. Through this method, the optimization problem is designed for a worst-case scenario. The implementation results of the presented robust controller for a compressor system, consisting of three parallel and series compressors, suggest the effectiveness of the presented method.


Introduction
A compressor is a mechanical device that increases the pressure of a gas by reducing its volume. Centrifugal compressors, sometimes called radial compressors, are a subclass of dynamic turbo-machinery and achieve a pressure rise by adding kinetic energy/velocity to a continuous flow of fluid through the rotor or impeller (Brun & Kurz, 2018;Tibrewala et al., 2014). The most important applications of centrifugal compressors are in oil refineries, natural-gas processing, petrochemical, chemical plants, air-conditioning and refrigeration and HVAC, in industry and manufacturing to supply compressed air for all types of pneumatic tools and in air separation plants to manufacture purified and product gases (Bloch & Soares, 1998).
Surge is flow phenomenon instability at low mass flow rate operation for which the impeller cannot add enough energy to overcome the system resistance or backpressure (Semlitsch & Mihăescu, 2016). At low mass flow rate operation, the pressure ratio over the impeller is high. The high back pressure, downstream of the impeller, pushes flow back over the tips of the rotor blades towards the impeller eye (inlet) (Sundström et al., 2018). The rapid CONTACT Hashem Imani Marrani imani.hashem@gmail.com reversal flow exhibits a strong rotational component, which affects the flow angles at the leading edge of the blades. The deterioration of the flow angles causes the impeller to be inefficient and less flow is delivered downstream. Thereby, the plenum downstream of the impeller is emptied and the (back) pressure drops. As a result, less flow reverses over the rotor tips and the impeller gain becomes again efficient. These cyclic events cause large vibrations, increase temperature and change rapidly the axial thrust. These occurrences can damage the rotor seals, rotor bearings, the compressor driver and cycle operation. Most turbo machines are designed to easily withstand occasional surging. However, if the machine is forced to surge repeatedly for a long period of time, or if it is poorly designed, repeated surges can result in a catastrophic failure. Surge control is of greater importance in complex compressor systems consisting of several parallel and series compressors for which a higher fluid pressure increase is expected. A complex system consists of many components that interact with each other, and due to dependencies, competitions, relationships, or other types of interactions between their parts or between a given system and its environment; its modelling and controlling are inherently difficult (Ladyman et al., 2013). Systems that are 'complex' have certain characteristics that result from these relationships, such as nonlinearity, spontaneous emergence and order, compatibility, and feedback loops among others. The type of surge control system specified for a given system is largely determined by the type of compressor being used and the compressor system dynamics. If a system has a single compressor with a very steady behaviour, a simple control system may be appropriate. On the other hand, a complex system with multiple compressors, varying demand, and many types of end uses will require a more sophisticated surge control strategy. Due to the reciprocal effects of the compressors, their behaviour is more sensitive adjacent to the surge line, and thus a more careful control design is imperative. In any case, careful consideration should be given to compressor control system selection because it can be the most important single factor affecting system performance, efficiency and safety.
With increasing interest to the robust control appro aches in various applications Shi et al., 2017a;Shi et al., 2017b;Wang et al., 2021;Xiong et al., 2016;Yu et al., 2020;Zhang et al., 2018;Zhu et al., 2020), several methods have been proposed for active surge control in constant speed compressors. In Dominic et al. (2016), the feed-forward control method has been used to control the compressor pressure, which is robust to uncertainty and disturbance. Due to the high performance of adaptive methods Ma & Xu, 2020;Zhang et al., 2020), references (Ghanavati, Salahshoor, Jahed-Motlagh, et al., 2018b;Ghanavati, Salahshoor, Motlagh, et al., 2018a;Ziabari, et al., 2017) have used adaptive and back-stepping approaches to overcome the effects of uncertainty and disturbance. In recent years, model predictive control methods have also been proposed to control surge instability in fixed speed compressors (Imani et al., 2017a;Imani et al., 2017b;Imani, Malekizade, et al., 2018;Marrani et al., 2019). Nonlinear model predictive control method is designed for compressors Geritzer model considering the CCV actuator (Imani et al., 2017a;Imani, Malekizade, et al., 2018). In Marrani et al. (2019), Tube-MPC is designed for discrete time systems where H ∞ method has been used as an auxiliary controller. By taking into account the effects of pipe, robust model predictive control is provided in Imani et al. (2017b).
It is well established that in interconnected systems, the centralized control framework for each subsystem must use the data of the whole system (D' Andrea & Dullerud, 2003). In decentralized control schemes, the controller designed for each subsystem can only use its own local data. Due to the high independence of decentralized controllers, some people have tried to create design approaches that can ensure the stability and performance of the system.
Due to the unique capabilities of the MPC controller in the centralized regulation of large systems with numerous input and output variables, so far little attention has been paid to decentralized MPC algorithms with guaranteed stability. This is due to the following reasons: (1)-the multivariate nature of MPC, which makes it easy to form a centralized regulator.
(2) Problems of ensuring the stability of decentralized algorithms due to the implicit control law obtained from the MPC optimization process (Mayne et al., 2000).
Constructing coordinated decentralized control systems needs dynamic interaction among different units in the design process of control systems. Also, the disturbance, uncertainty and nonlinear properties are reflected in the process of optimization and stability assurance for the designing of decentralized MPC controllers (Alessio et al., 2011;Alessio & Bemporad, 2008, June;Gudi & Rawlings, 2006;Magni & Scattolini, 2006). In Tuan et al. (2015), a new decentralized predictive control scheme has been purposed for a plant made of interconnected systems that provide a method to stabilize a nominal large-scale plant using limited and decentralized controllers.
Despite these efforts, the decentralized controller design faces several major challenges. The first difficulty lies in the stability proof of the general closed-loop system in the presence of the controllers, which only use the local information of their respective subsystems. In other words, it is evident that the stability of each subsystem might not always ensure the stability of the whole system and plant. Another challenge is due to the limitation of the available data and lack of connection between various controllers, which can lead to reduced closed-loop performance under a decentralized control platform (Cui & Jacobsen, 2002). The necessity of addressing the nonlinear dynamics of the subsystems, the disturbances present in the system dynamics, and the parameter uncertainties of the system are other concerns regarding the decentralized robust predictive controller design. To the best knowledge of the authors, there has been no effective control method presented for cooperated decentralized control of a complex compressor system consisting of several parallel and series compressors.
This paper presents a novel LMI-based decentralized robust predictive control scheme for complex compressor systems which covers the followings: (1) The proposed decentralized method is able to address the mutual effects of compressors, and the most important property of this model is its independent communication with various local controllers.
(2) The presented method is based on a predictive model in which the state and control signal constraints are taken into account, and the optimization of the objective function is also performed. (3) In order to reduce the computational time and complexity, the LMI method is employed to solve the optimization problem in each time step. (4) The effects of disturbance on compressor flow and pressure are considered. (5) The effects of uncertainty in the throttle valve, Spillback valve, and also compressor characteristics are covered. (6) The stability is ensured by the Lyapunov method using the proposed controller, and the mutual effects of the compressors, as well as the effects of nonlinearity, uncertainty, and disturbances with unspecified upper bound, are addressed at the same time.
The organization of this paper is as follows. In Section 2, the preliminaries are presented. Section 3 describes the proposed control method in full detail. The implementation of the proposed method for complex compressor system surge control is provided in Section 4. Sections 5 and 6 present the simulation results and conclusions, respectively.

Preliminaries
Consider the following continuous-time nonlinear systeṁ where x(t) ∈ R n x shows the system states, u(t) ∈ R n u the control input, w(t, x) : R n x → R n w continuous nonlinear uncertainty function. w(t, x) is considered in the following set The system has the following limitations With x ∈ X, w ∈ W. Then, the system trajectory starting from x(t 0 ) ∈ ⊆ X will remain in the set , where holds for any ε > 0.

Lemma 2.3 (Schur complements (Boyd et al., 1994):
The LMI and S(x) are affine functions of x, and are equivalent to

Robust LMI-based decentralized MPC
Consider a process plant consisting of h interconnected nonlinear systems, each denoted as s i , i = 1, . . . , h in the continuous time state space where x i (t) ∈ R n x i shows the system states, u i (t) ∈ R n u i is the control input, f i (x) : R n x i → R n θ i specifies continuous nonlinear function, θ i (t) ∈ R n θ i indicates uncertainty in the system, d i (t) ∈ R n x i specifies bounded and unknown system disturbances, g i (x) : R n x → R n x i is the interactive (or coupling) continuous nonlinear function, and x(t) ∈ R n x shows the process plant states. The disturbances are considered in the following set The system has the following limitations where X i ⊂ R n x i is bounded and U i ⊂ R n u i is the compact. with will haveẋ The state-feedback control law for system (12) in kT time is chosen as These control signals are true for the following constraint Finally, the chosen infinite horizon quadratic cost function is specified as: (15) where Q i and R i are positive definite weight matrices. In the objective function (15), the uncertain but negative effect term is introduced with weight μ i , where μ i is a positive constant (Tahir & Jaimoukha, 2011).
Theorem 3.1: Consider system (12), x i (kT) is the measure value in sampling time of kT. There is a state-feedback control law (13) that is true in stability condition and in input constraint (14) in every moment. If the optimization problem with LMI constraints can be feasible.
(16) where X i > 0 and Y i are matrixes obtained from the abovementioned optimization problem. By this way, the statefeedback matrix in every moment is obtained as K i = Y i X −1 i .

Proof: Considering a quadratic Lyapunov function, we have
In the sampling time of kT assume that V i (x i (t)) is true in the following condition In order to obtain the robust efficiency, we should have x i (∞, kT) = 0 which results V i (x i (∞, kT)) = 0. By integrating both sides of the equation (19), we have where γ i is a positive scalar (the upper bound of the objective (15)). In order to obtain an MPC robust algorithm, the Lyapunov function should be minimized considering the upper bound (Ghaffari et al., 2013, June). So defining X i = γ i P −1 i and using Schur Complements, we have In the following, according to Lemma 2.1, for system (12) we havė Then, according to (17), we havė According to Lemma 2, we have By substituting (26) in (25), we have Consider where λ imax is the maximum eigenvalue of P i and ε i I is the corresponding upper bound (Poursafar et al., 2010), then By choosing Equation (29) is reduced to Pre-and post-multiplying by X i , Given (28), we have Substituting P i = γ i X −1 i and pre-multiplying by X i , we have Finally, the input constraint is investigated in (Poursafar et al., 2010). According to (14), we have Consistent with (18) and (20), it is known that the states of x i (kT + τ , kT) are determinate and ellipsoid invariant set Therefore, From (36) and (37), the input two-norm constraint in (36) can be rewritten as It is the same as following inequality by applying Schur complement. (39) So, the proof is completed.

Robust model predictive control on surge
Surge is a condition that occurs on compressors when the amount of gas is insufficient to compress and the turbine blades lose their forward thrust, causing a reverse movement in the shaft. It can cause extensive structural damage in the machine because of the violent vibration and high thermal loads that generally accompany the instability. In recent years, several methods have been proposed to control the instability Imani, Jahed-Motlagh, et al., 2018;Taleb Ziabari et al., 2012;Taleb Ziabari et al., 2017). For this reason, compressor system control as one of the most practical systems is considered in this section.

Compressor model
In this section, given the proposed decentralized predictive controller, a surge controller is designed for serial and parallel compressors. First, a compressor model is investigated and then a decentralized controller is designed for these combinations: two parallel compressors and one serial compressor. Moore and Greitzer's surge model of the centrifugal compressor is as followṡ where ψ is the coefficient of compressor pressure, φ is the coefficient of compressor's mass flow, d φ (t) and d ψ (t) are the disturbances of flow and pressure, respectively. Also, φ T (ψ) is the characteristic of throttle valve and ψ c (ϕ) is the characteristic of the compressor. B is the Greitzer's parameter and l c shows the length of ducts. Moore and Greitzer's (1986) compressor characteristic is defined as (42) where ψ c0 is the value of characteristic curve in zero dB, H is the half of the height of the characteristic curve, and W is the half of the width of the characteristic curve. The equation for throttle valve characteristic is also derived from  and is as follows where γ T is also the valve's yield. Values of compressor parameters are used in simulation according to (Greitzer, 1976).  The system model equations, considering a CCV, arė Consider ψ V (φ) as the input for system control.
According to Figure 2, the equations for the two parallel compressors and one serial compressor arė where ϑ j , j = 1, . . . , 6 are uncertain parameters and The characteristic of the spill back valve is where γ SB is also the valve's yield. In designing a surge controller in the compressor systems (49), it is assumed that the value of throttle valve, as well as the compressor characteristic, is not known.

Controller design
According to equations (8) and (47), each subsystem is rewritten as Next, the existing constraints in the compressor systems should be incorporated in the optimization problem (15). The first established constraint is on the control input. Since the control signal is a CCV output, so we havē The next constraint and limitation is that the flow has some maximum and minimum values. This constraint should also be considered.
The LMI parameters are selected

Simulation
In this section, simulation is performed in Matlab to demonstrate the performance of the proposed control method. The applied scenario is similar to the scenario presented in the article , and the obtained results for the proposed controller are compared to the robust decentralized tube MPC adaptive control method. The considered scenario is given as follows: In t = 40s the first compressor throttle valve value reduces from γ T 1 = 0.65 to γ T 1 = 0.6, which leads to surge in the first compressor. As such, in t = 50s, the second compressor throttle valve value reduces from γ T 2 = 0.7 to γ T 2 = 0.6, and the second compressor also experiences surge. Also, in t = 60s, the third compressor throttle valve value reduces from γ T 3 = 0.75 to γ T 3 = 0.6, and the third compressor also leads to surge. Finally, In t = 20s the spill back valve value changes from γ SB = 0.2 to γ SB = 0. After applying the presented predictive controller and simulating the compressors behaviour, the following results are obtained.
Figures 3-6 display the pressure, flow, control signal, and trajectory of the first compressor system, respectively. Figure 3 shows a higher pressure increase using the proposed method compared to the reference . Figure 5 also exhibits the lower fluctuations of the compressor flow with the proposed controller. The obtained control signal also has smaller amplitude in the range of 0 and 1, indicative of its practicality. The traversed path on the compressor's characteristic curve shown in Figure 6 ensures compressor operation near the surge line without entering the surge region.
The pressure, flow, the control signal, and trajectory associated with the second compressor are shown in Figures 7-10, respectively. Similar to the first case, the proposed method achieved a higher pressure increase, smaller fluctuations in flow, smaller amplitude for the control signal of the compressor two, and ensured working close to the surge line without entering it.
Figures 10-14 also show the obtained results for compressor 3 in the compressor system. By employing the presented controller, the obtained results for compressor 3 are in agreement with two first compressors.
The obtained results from these figures indicate the effectiveness of the presented method for the compressor system stabilization, surge avoidance, control signal optimization, higher pressure increase, and reduced flow fluctuations.

Conclusions
This paper presents an LMI-based decentralized robust predictive control scheme for a particular class of complex compressor systems, which also includes the multistage compressor system model. The presented control method was implemented to avoid surge instability as the most significant compressor instability. The effects of the nonlinear factors, disturbance, and uncertainty on the compressor characteristic curve and throttle valve and spillback valve are also addressed. Furthermore, the objective function is optimized in each step using the LMI method to obtain the control signal. The simulation results suggest that our proposed approach, compared to the robust adaptive tube MPC control method, was superior in terms of the pressure increase, flow fluctuation, and control signal. Optimal compressor operation near the surge line with the ability to increase the pressure more and the ability to work with less flow are among the results obtained from the simulation section which ensures minimal fluctuations on flow and pressure during disturbance and uncertainty in the complex compressor system, despite the control signal with the lowest amplitude.

Disclosure statement
No potential conflict of interest was reported by the author(s).