IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 5, SEPTEMBER 1999 567 Centrifugal Compressor Surge and Speed Control Jan Tommy Gravdahl, Member, IEEE, and Olav Egeland, Member, IEEE Abstract—Previous work on stabilization of compressor surge is extended to include control of the angular velocity of the compressor. First a low-order centrifugal compressor model is presented where the states are mass flow, pressure rise, and rotational speed of the spool. Energy transfer considerations are used to develop a compressor characteristic. In order to stabilize equilibria to the left of the surge line, a close coupled valve is used in series with the compressor. Controllers for the valve pressure drop and spool speed are derived. Semiglobal exponential stability is proved using a Lyapunov argument. Index Terms— Compressors, Lyapunov methods, modeling, surge control. I. INTRODUCTION C OMPRESSOR surge is an axisymmetric oscillation of the mass flow and pressure rise. Modeling and control of these oscillations is of considerable interest since surge limits the useful range of mass flows where the compressor operates stably. Large amplitude surge can also damage the compressor. Low-order models for surge in compression systems have been proposed by many authors, and a classical reference is [7]. However, the compression system model of [17] has been widely used for surge control design. It was derived for axial compression systems, but [19] showed that it is also applicable to centrifugal compressors. The model has two states, normalized mass flow and normalized pressure, and the compressor is treated as an actuator disc, with a third-order polynomial flow/pressure rise characteristic. Over the last decade many papers covering the area of surge control have been published. A review can be found in [16]. Of many possible actuation schemes, closed coupled valve (CCV) control is considered one of the most promising [6], [18], [21], [25], [27]. Experimental results of CCV control is reported in [6] and [21]. Surge in a compression system can be explained by the throttle line crossing the compressor characteristic in an area of positive compressor characteristic slope. A close coupled valve is placed immediately downstream of the compressor (hence close coupled), and active control of the valve pressure drop is utilized to make the slope of the equivalent compressor (compressor in series with the valve) negative and thereby stabilizing the system. This approach was used in [21], [25], and [27] for surge control of the model of [17]. Linear stability analysis was used in designing control laws resulting in local stability results. In [26] pressure disturbances were included in the analysis, and Manuscript received November 25, 1997; revised July 28, 1998. Recom- mended by Associate Editor, M. Jankovic. The authors are with the Department of Engineering Cybernetics, Norwe- gian University of Science and Technology, N-7034 Trondheim, Norway. Publisher Item Identifier S 1063-6536(99)06452-0. a nonlinear CCV control law was designed for the model of [17] using the method of Lyapunov. By applying backstepping, [13] developed nonlinear surge controllers for the same model, but included disturbances both in mass flow and pressure. Global stability results were presented. In [14] certain passivity properties of the model were utilized in designing a CCV control law. One drawback of CCV control is that the valve introduces a pressure drop in the compression system as discussed in [26]. When using the valve as a steady-state device, such as in [6], this loss may become unacceptably large. However, as pointed out in [25] and [26], a time varying valve will introduce considerably less pressure drop than a valve with constant pressure drop. Since compressors are variable speed machines, it is of interest to investigate the influence of speed transients on the surge dynamics. Models describing this interaction were developed in [8] and [12] for axial compressors, and in [10] and [15] (a preliminary version of this paper) for centrifugal compressors. As surge can occur during acceleration of the compressor speed, it is of major concern to develop controllers that simultaneously can control both surge and compressor speed. In this paper, a surge control law for variable speed cen- trifugal compressors is presented and analyzed. The speed is controlled with a PI-control law. Inspired by [9] and [29], we make a departure from the third order polynomial approximation of the compressor characteristic commonly used in the surge control literature. Fluid friction and incidence losses, as well as other losses, in the compressor stage are modeled, and a variable speed compressor characteristic is developed based on this. Both annular and vaned diffusers are studied. Semiglobal exponential stability results for the proposed controllers are given using Lyapunovs method, and the results are confirmed through simulations. II. MODEL The centrifugal compressor consists essentially of a sta- tionary inlet casing, a rotating impeller which imparts a high velocity to the gas, and a number of fixed diverging passages in which the gas is decelerated with a consequent rise in static pressure. The latter process is one of diffusion, and consequently, the part of the compressor containing the diverging passages is known as the diffuser, [2]. Fig. 1 is a diagrammatic sketch of the impeller and diffuser of a centrifugal compressor. The function of the inlet casing is to deliver gas to the impeller eye. A volute (also known as a scroll or a collector) may be fitted at the diffuser exit. Its 1063–6536/99$10.00  1999 IEEE