Elsevier

Automatica

Volume 38, Issue 8, August 2002, Pages 1373-1380
Automatica

Brief Paper
Adaptive control of compressor surge instability

https://doi.org/10.1016/S0005-1098(02)00031-6Get rights and content

Abstract

In this paper we propose a simple high-gain type adaptive control scheme for surge stabilization in a compression system. We consider a typical configuration in which the plenum throttle position is the control input, while the measured output is given by the total pressure at the compressor inlet. We prove convergence of the control scheme using a simplified, but widely accepted, second-order model. The system does not satisfy strict minimum-phase assumptions, usually introduced in the high-gain type adaptive control context, and therefore the proof has been tailored to the specific problem. We finally validate the proposed controller, by means of both numerical simulation and experiments on a real plant.

Introduction

Surge instability drastically limits the operating region of compression systems. As it is known, surge occurs at low values of the compressor flow rate, causing highly undesirable oscillations in the system. By means of a control system it is possible to attenuate or eliminate the phenomenon, so allowing the plant to operate in naturally unstable points.

Several previous references have considered the problem of surge elimination, and most of them are based on the work of Greitzer 1976a, Greitzer 1976b and Greitzer and Moore (1986a) and Moore and Greitzer (1986b) who proposed dynamical models that have been deeply exploited for the analysis and the design of control systems for compression plant stabilization. Based on a linearized approach, Epstein, Ffowcs William, and Greitzer (1989) introduced the active control for surge suppression, while early successful implementations were reported by Pinsley, Guenette, Epstein, and Greitzer (1991) and Ffowcs Williams and Huang (1989). In a subsequent work (Simon, Valavani, Epstein, & Greitzer, 1993), several actuator/sensor configurations were considered together with a proportional compensator, and a local stability analysis based on a linearized model was carried out.

After these pioneering contributions, the compressor surge control has attracted many researchers as it is evident from several recent contributions which suggested more sophisticated techniques to face the problem. A Lyapunov approach has been proposed in Behnken and Murray (1997) and Gravdal and Egeland (1999a). A non-linear approach based on backstepping has been proposed in Banaszuk and Krener (1999) and Gravdal and Egeland (1997). The problem of bifurcation control is addressed in Kang, Gu, Sparks, and Banda (1997) and McCaughan (1990) based on the Moore–Greitzer model. A feedback linearization method is presented in Badmus, Chowdhury, and Nett (1996). Extensive surveys are provided in Gravdal and Egeland (1999b) and Gu, Sparks, and Banda (1999) and Willems and de Jager (1999).

In the above contributions the system model is known. In particular, valve and compressor characteristics are assumed to be available. The main aim of the present work is to cope with the case in which both these characteristics and the Greitzer parameter are unknown. We consider a typical compressor controller configuration in which the controlled input is the throttle valve position while the measured output is the total pressure at the compressor inlet. The advantages of this choice are discussed in Giannattasio, Micheli, and Pinamonti (2000). For this system we propose a very simple non-identifier-based high-gain adaptive control, often referred to as λ-tracker (Blanchini & Ryan, 1999; Ilchmann, 1993; Ilchmann & Ryan, 1994), the implementation of which does not require any information on the system characteristics. We show that, under very mild assumptions, this control assures boundedness and global convergence of both the system state and the adapted feedback gain. To prove this result we cannot rely on strict minimum-phase conditions and this will enforce us to show convergence with a properly tailored geometric proof. This proof is based on the construction of an invariant set, the boundaries of which are formed by arcs of the (uncontrolled) system trajectory and arcs of the (unknown) compressor and valve characteristics.

We also deal with the problem of control bounds (which are not considered in the convergence proof). It is well known that input saturation may strongly reduce the domain of attraction. As a consequence, under too strict bounds, the controller can fail to remove the system from surge cycles. Nevertheless, it will be shown that this problem is an intrinsic limit of the plant. Precisely, if the proposed setup fails in suppressing limit cycles, no control with the same actuator will succeed, no matter how it is generated and which outputs are considered.

We finally provide both numerical and experimental results to validate the approach.

Section snippets

Model description and assumptions

Consider the compressor-plenum plant depicted in Fig. 1, represented by the following second-order model (Greitzer, 1976a) (see also Arnulfi et al., 1999b; Badmus, Chowdhury, & Nett, 1996)ẋ1(t)=−B[x2(t)−Ψs(x1(t))],ẋ2(t)=1B[x1(t)−u(t)Γs(x2(t))],where x1 is the dimensionless flow rate in the compressor duct, x2 represents the dimensionless pressure in the plenum, B>0 is the Greitzer parameter, Ψs(x1) and Γs(x2) are the static characteristics of the compressor and the throttle valve,

Main result

We introduce now an adaptive λ-tracker control law (Ilchmann, 1993; Ilchmann & Ryan, 1994), which drives the output to zero within a prescribed tolerance level. Such a tolerance is introduced together with a dead-zone characterized by the following inequality|y(t)|=|x2(t)−Ψs(x1(t))|⩽ε,where no parameter adaptation occurs. Define function σλ:R+R+σλ(ξ)=0if0⩽ξ⩽λ,ξ−λifξ>λ..Note that σλ(|y|) is the distance of y from interval Iλ=[−λ,+λ]. Rewrite Eqs. (1) asẋ1(t)=−B[x2(t)−Ψ(x1(t))],ẋ2(t)=1B[x1(t)−u

Control saturation

In the presence of control saturation, convergence is assured only for initial conditions which are sufficiently close to the equilibrium point so that our basic goal of suppressing developed surge cannot be met. The previous results do not take into account this problem. It is very easy to show, by numerical simulation, that under too strict conditions (i.e., too small a value of At,max, defined in Section 2) the plant cannot be removed from surge cycles. In this section we show that,

Experimental results and concluding discussions

Theorem 3.1 shows that, for any tolerance ε, the system output is asymptotically driven to the tolerance interval, Iε. Although, in principle, we would like to meet condition y=0, imposing a dead-zone of tolerance is fundamental in practice, due to the presence of measurement noise. In this sense, an efficient rule to choose ε is to set it equal to the maximum of the disturbance absolute value. Indeed, if ε is greater we loose precision, conversely, if ε is smaller the gain may increase without

Franco Blanchini was born on 29 December 1959, in Legnano (Italy). He received the Laurea degree in Electrical Engineering from the University of Trieste, Italy, in 1984. In 1985 he was Lecturer of Numerical Analysis at the Faculty of Science at the University of Udine, Italy. He was Research Associate of System Theory from 1986 to 1991. In 1992 he became Associate Professor of Automatic Control at the Engineering Faculty of the University of Udine. He is currently Full Professor of Automatic

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    Franco Blanchini was born on 29 December 1959, in Legnano (Italy). He received the Laurea degree in Electrical Engineering from the University of Trieste, Italy, in 1984. In 1985 he was Lecturer of Numerical Analysis at the Faculty of Science at the University of Udine, Italy. He was Research Associate of System Theory from 1986 to 1991. In 1992 he became Associate Professor of Automatic Control at the Engineering Faculty of the University of Udine. He is currently Full Professor of Automatic Control since November 2000. He is affiliated to the Department of Mathematics and Computer Science in Udine and he is Director of the Laboratory of System Dynamics of the Department. He is Associate Editor of Automatica. He was member of the program committee of the Conference on Decision and Control in 1997, 1999 and 2001. At the beginning of his research activity he was interested in numerical methods for analysis and synthesis of linear systems and in the theory of generalized linear systems. Presently, his main research activity is in the field of robust control, especially Lyapunov methods and L1 control theory. His research interests include also the control of constrained systems, mechanical systems and the control of distribution networks.

    Pietro Giannattasio was born on 12 September 1958, in Bari (Italy). He received the Laurea degree in Mechanical Engineering from the University of Bari, Italy, in 1985. From 1985 to 1989 he worked with a R&D Company (Tecnars, Bari) where he was involved in research activities in the fields of combustion, fluid dynamics and heat transfer. From 1989 to 2000 he was Research Associate of Mechanical Engineering at the Università of Udine, Italy. He is currently Associate Professor of Mechanical Engineering at the Università of Udine since September 2000. He is affiliated with the Department of “Energetica e Macchine” in Udine. At the beginning of his research activity in the University, he was interested in the computational fluid dynamics, with special reference to the numerical solution of the incompressible Navier–Stokes equations and to the simulation of the unsteady compressible flow in the pipe systems of internal combustion engines. He was also involved in the phenomenological modelling of engine combustion. Presently, his main research activities are in the fields of pulsating combustion and of the surge instability of compression systems.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Jan Willem Polderman under the direction of Editor Robert R. Bitmead. Supported by the Italian National Council of Research (C.N.R.).

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