eventually initiates surging of the engine. These studies indicate that one must include rotating stall dynamics in any study of active control of axial compressors.
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Distributed nonlinear modeling and stability analysis of axial compressor stall and surge Conference Paper · August 1994 DOI: 10.1109/ACC.1994.752492 · Source: IEEE Xplore
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DISTRIBUTED NONLINEAR MODELING AND STABILITY ANALYSIS OF AXIAL COMPRESSOR STALL AND SURGE Catherine A. Mansoux, Daniel L. Gysling, Joga D. Setiawan, James D. Paduano Massachusetts Institute of Technology Cambridge, MA 02139 Abstract This paper presents a nonlinear formulation of the Moore-Greitzer rotating stall model that is suitable for control analysis and design. The model is validated by comparing stall inception experiments to simulated stall inception transients. The shape of the nonlinear compressor characteristic is shown to be a primary determinant of stall inception transient behavior. A Lyapunov stability analysis procedure is then presented. This procedure allows basins of attraction to be determined for various system operating points. Examples are given, and implications for design of rotating stall/surge control systems are discussed.
Nomenclature lc, B, m, µ, λ Compression system geometric parameters (see ) A Flowfield transformation operator, Eqs. (14), (15) E Flowfield transformation operator, Eqs. (14), (16) G Fourier transform matrix, Eq.. (17) M 2N+1 N Number of harmonics modeled in finite order approximation, Eq. (17) S (1xM) matrix to take mean of a distributed vector quantity, Eq. (20) T (Mx1) matrix to for zeroth mode of surge/stall model, Eq. (20) V Lyapunov function ƒ(⊇) Fourier transform, Eq. (13) n Harmonic number δ Perturbation quantity Laplacian ∆ Size of basin of attraction η Nondimensional axial distance along compressor duct, zero at compressor face θ Circumferential angle around compressor annulus τ Nondimensional time φ Flow coefficient, Eq. (1) φ (Mx1) vector containing flow coefficient values at distributed values of θ ψ Nondimensional instantaneous local pressure rise across compressor Θ Nondimensional velocity potential ΨC Pressure rise delivered across compressor ΨT Pressure drop imposed across throttle ΦT Inverse of ΨT KT Throttle constant, Eq. (2) Annulus average, Eq. (9) (⋅) (⊇)ac Non-axisymmetric component, Eq. (9)
1. Introduction Rotating stall and surge are violent limit cycle-type oscillations in axial compressors, which result when perturbations (in flow velocity, pressure, etc.) become unstable. Originally treated separately, these two phenomena are now recognized to be coupled oscillation modes of the compression system -- surge is the zeroth order or planar oscillation mode, while rotating stall is the limit cycle resulting from higher-order, rotating-wave disturbances. The importance of these phenomena to the safety and performance of gas turbine engines is widely recognized , and various efforts to either avoid or control both rotating stall and surge have been studied [2, 3, 4, 5, 6]. Central to the issue of control is reliable modeling that captures the essential physics of the process, without being too cumbersome for control law design and analysis. For example, CFD models of rotating stall exist [7, 8], but these are clearly not suitable for conventional control law design. On the other hand, various 1D models have been proposed for use in control law design [9, 10]. These models are useful for a limited class of compressors in which rotating stall is recoverable and nondebilitating (primarily centrifugal compressors). However, they ignore the essential physics of rotating stall, which is (at least) 2 dimensional. Several studies [11, 12, 13] have shown that, in high speed axial compressors common to gas turbine engines, rotating stall is the performance limiting instability, even when it eventually initiates surging of the engine. These studies indicate that one must include rotating stall dynamics in any study of active control of axial compressors. Thus a 2D model of limited (compared to CFD) dimensionality is desired. The Moore-Greitzer model has proved useful in this context for linear control [3, 18, 19, 20, 28]. But this model also incorporates various nonlinear features, which have not until recently been validated experimentally or exploited in a control context. For instance, the model captures the essential nonlinearities which determine rotating stall inception transient behavior . It also explicitly accounts for nonlinear coupling between rotating stall and surge . To begin to take advantage of the nonlinear form of the model, this paper presents several results. After a brief account of the fluid-dynamic PDEs, a nonlinear ODE form for use in control analysis and design is derived. This model is then verified against experimental data in which perturbations are large enough to cause nonlinearity to be important. This is an important step because various assumptions are made during modeling. A physically motivated Lyapunov analysis is then conducted for the rotating stall/surge problem. This analysis allows domains of attraction to be determined. Examples point out the important
physical aspects of rotating stall stabilization. A
2. Rotating Stall/Surge Model
ΚΤ depends on the degree of throttle closure. In a typical experiment, the throttle is slowly closed, the throttle characteristic becomes steeper (modeled by increasing ΚΤ in (2)), the intersection point between ΨC(φ) and ΨT(φ) changes, and the equilibrium operating point of the compressor moves from high flow to low flow (see Fig. 2). The stability of the equilibrium point represented by the intersection between ΨC(φ) and ΨT(φ) has been the topic of numerous studies, due to its importance in the safe, high performance operation of gas turbine engines. In our model, the system state under unsteady, possibly non-axisymmetric (i.e. circumferentially varying) conditions is characterized by three terms: the annulus averaged pressure difference across the compressor, ψ, the annulus averaged flow coefficient, φ, and the spatially distributed perturbation on φ, denoted δφ: φ (η, θ,τ) = φ + δφ(η,θ, τ),
where η is the nondimensional axial position in the compressor (the
stable equilibria unstable equilibria throttle characterstic
During quasi-steady operation, the total-to-static pressure rise delivered by the compressor is simply determined by its 'pressure rise characteristic,' denoted ΨC(φ) (Refer to Fig. 2). The pressure rise is balanced by a pressure loss across a throttling device, which can be either a simple flow restriction (used for testing compressors as components) or the combustor and turbine in a gas turbine engine. The balance between pressure rise across the compressor and pressure drop across the throttle is depicted as an intersection between the characteristics of the two devices, ΨC(φ) and ΨT(φ), where, for low pressure ratios, ΨT(φ) is usually taken to be a quadratic function of φ: ΨT = 1 K T⋅φ 2 2 . (2)
Fig. 1: Compression system components: A - inlet duct, B - compressor, C - downstream duct, D - throttle.
Pressure Rise (or Drop), ψ
(axial velocity) φ= (rotor speed)
Consider the schematic diagram of an axial compressor in Fig. 1. It consists of an upstream annular duct, a compressor modeled as an actuator disk, a downstream annular duct, and a throttle. During stable operation, flow through the compressor can be assumed to be circumferentially uniform (axisymmetric), and a single non-dimensional measure of flow through the compressor determines the system state. One such measure is the 'flow coefficient', which is simply the nondimensionalized value of the axial velocity:
Flow Coefficient, φ Fig. 2: Compressor and throttle behavior during a typical experimental test.
origin is chosen to be at the compressor face), θ is the circumferential position, and τ is non-dimensional time (in rotor revolutions). Note that evaluation of ΨC and ΨT must now be conducted more carefully due to the unsteady and non-axisymmetric character of the flow: ΨC is evaluated at η = 0 (the compressor face), and varies with both τ and θ − i.e. we evaluate ΨC(φ+ δφ(0,θ,τ)). Thus the compressor is viewed as a distributed memoryless nonlinearity operating on the local (in θ) flow coefficient. On the other hand, due to the nature of the downstream flowfield, ΨT can be evaluated for the annulus averaged flow; thus the pressure loss across the throttle is simply ΨT(φ). One additional variable must be introduced in order to set up the system of equations. The upstream flowfield, being two dimensional, admits both axial velocity perturbations (δφ) and circumferential velocity perturbations. Rather than introduce a circumferential velocity variable, we define the perturbation velocity potential δΘ, such that Ž(δΘ) =δφ Žη
Ž(δΘ) =δ(circ. vel.) Žθ .
Although necessary for the derivation, δΘ will ultimately be eliminated from the equations, along with all of the partial derivatives with respect to space, leaving an operator-theoretic
ordinary-differential relationship. These are the relevant system states. Derivation of a model for the dynamics of the compression system is presented in other documents, and is beyond the scope of this paper. Here we will present the model in a format that is coherent and accessible from a control theoretic point of view, and discuss the dynamic implications of the model. Thus we present the fluid dynamics without further derivation, referring the interested reader to the relevant literature [11, 14, 15, 18, 27]. Note that time, τ, is nondimensionalized by the rotor revolution period. 1) For the upstream flow field we assume no incoming vorticity (clean inlet conditions) and thus the flow in this region is potential: 2δΘ
=0 η ≤ 0. (5) 2) The annulus averaged pressure rise across the compressor (indicated by an overbar) lags behind that imposed by the throttle characteristic, due to mass storage in the downstream ducting and plenum chamber: Žψ = 1 φ – ΦT ψ Žτ 4l B2 c
η = 0. (6)
3) The annulus averaged flow coefficient accelerates to balance the difference between the annulus averaged pressure rise delivered by the compressor and the annulus averaged pressure rise that actually exists across the compressor (in unsteady situations these can be different): Žφ 1 = Ψ φ + δφ – ψ Žτ l c C
η = 0. (7)
4) Finally, the non-axisymmetric (indicated by the subscript 'ac') part of the pressure rise delivered by the compressor acts to accelerate the flow through the rotor and stator passages, where part of this acceleration is due to the rotor moving through a nonuniform flow perturbation (this is reflected by the presence of a partial derivative with respect to θ): m
∂ δΘ ∂ δφ ∂ δφ +µ +λ = ΨC φ + δφ ∂τ ∂τ ∂θ
η = 0. (8)
n = –∞ n≠ 0
Θ n τ ⋅einθe
η ≤ 0.
This solution satisfies Laplace's equation, and the other boundary conditions of clean inlet flow experiments as described in . Using Equation (4), we can also write a solution for δφ in the upstream flowfield: ∞
n = –∞ n≠ 0
φn τ ⋅einθ e n η
η ≤ 0,
where φ n(τ) = n Θn τ . These relationships allow δΘ to be eliminated from the system, and Equation (8) to be written ∞
n = –∞ n≠0
m φ + µφ + inλφ ⋅einθ = Ψ φ + δφ n n C n n
where the partial derivatives have been eliminated, and φ n τ is the nth spatial Fourier coefficient of the axial velocity perturbation at the compressor face (η = 0): φ n(τ) = ƒ δφ(0,θ,τ) = 1 2π
δφ(0,θ,τ) e–inθ dθ
Introduction of the Fourier series solutions to eliminate spatial derivatives is a well-known method for handling distributed systems. The modal (or 'spectral') form of the equations allows linear control techniques to be applied. This approach has proven successful in laboratory stabilization of rotating stall [18, 19]. To study the effects of nonlinearity on the system, an operatortheoretic form is often more useful. By substituting (13) into (12), we can write the system (6,7,12) as follows:
0 = –1 lc
1 4lc B2
1 Φ (ψ) T 4l cB 2 1 Ψ (φ+δφ) lc C
ΨC ac = ΨC – ΨC
5) Definitions: lc, B, m, µ, λ are scalar geometric parameters defined in , Φ T(⊇) is the inverse of ΨT, and the definitions of 'annulus averaged' and 'non-axisymmetric' take the expected forms: ΨC = 1 2π
2.1 Control-Theoretic Formulation The system of Eqs. (5-8) is a coupled set of partial differential equations, in which two space dimensions and time are present. However the space dimensions can be eliminated by introducing the solution for the upstream velocity potential as follows:
E( ⋅ ) = ƒ–1
note that only through the nonlinearity of ΨC are the axisymmetric and the non-axisymmetric dynamics coupled. Thus if one assumes small perturbations, rotating stall and surge are decoupled phenomena and can be stabilized separately. One can also study 1D nonlinear perturbations alone [10, 23]. This approach carries with it the implicit assumption that the effect of rotating stall is already captured (at the 1D level) by an 'annulus averaged compressor characteristic' (different from the ΨC we have defined here). This is a useful approach in some centrifugal compressors and low-speed fans, if no abrupt drop in pressure rise is experienced when rotating stall occurs.
Σ n = –∞ n ≠0
m +µ ⋅ƒ( ⋅ ) n m +µ ⋅ 1 2π n
( ⋅ )⋅ein( θ – ζ ) dζ (15)
and A( ⋅ ) = ƒ–1 inλ⋅ƒ( ⋅ )
n = –∞ n≠0
inλ⋅ 1 2π
( ⋅ )⋅ein(θ – ζ ) dζ .
E and A are linear operators that represent spectrally the effect of the upstream flowfield, as well as allowing derivatives with respect to θ and η to be evaluated. The linear and nonlinear parts of Equation (14) have been separated for illustrative purposes, but the equations are clearly in the desired form, i.e. x = F(x).
Equations (14) can be further recast into a number of different forms, depending on the application, by proper application of the Fourier transform and its inverse. For instance, we have implemented an update equation for E(δφ) in our simulation, and then used the proper Fourier relationships to recover δφ. It is also possible, by choosing appropriate state variables, to reduce the system to one involving convolution with influence functions in the circumferential variable θ, and thus eliminate the summations over n. Finally, spatial discretization of δφ(θ) and substitution of matrix Fourier transformations in place of summations and integrals allows the system to be written as a finite-dimensional state-space system, with good numerical properties for simulation and control work . This is the form we will need in Section 4, so we briefly outline it here.
E = G–1DEG , where: lc
m+µ 0 1 m 0 +µ 1 m +µ 0 N m +µ 0 N
, and, A 0
If we define the following as the system's state vector:
φ = ψ
δφ(θ1 ) + φ δφ(θ2 ) + φ δφ(θ2N+1) + φ ψ
It will prove useful to define specially scaled, real-valued DFT matrices, which essentially decompose and synthesize Fourier series sine and cosine coefficients instead of exponential coefficients. These matrices are:
2 2N +1
1 1 2 2 cos(θ1 ) cos(θ2 )
1 2 cos(θ2N+1)
sin(θ1) sin(θ 2)
Note that the annulus averaged (surge) part of φ(θ) is now incorporated as part of the state, so the matrices E and A are slightly different than their continuous domain counterparts - the (1,1) entry of DE and DA now contain the surge dynamics. This approach simplifies the final form of the equations, and allows surge and rotating stall to be analyzed together. Equation (14) can now be rewritten as follows: φ + ΨC(φ φ) – T⋅ψ E⋅φ φ = –A⋅φ ψ =
1 S⋅ φ – Φ ψ T 4l cB2
where φ is the state vector representing the distributed flow coefficient state, S is a row vector that extracts φ from φ :
and T creates a distributed (vector) representation of a zeroth mode disturbance from its scalar equivalent: T = [1 1 ≡ 1]T . The purpose of this section was to introduce the Moore-Greitzer model in two forms that are useful to control theorists. Because of the assumptions on the upstream flowfield, the partial-differential nature of the original system of equations has been eliminated, and the dependence on axial position has been 'solved out' of the system, leaving a simpler set of equations.
S= [1/M 1/M ≡ 1/M]
0 λ –λ 0
, where θn = 2πn 2N + 1
Re φN Im φN
0 Nλ –Nλ 0
and N is a power of 2, then we can replace the Fourier transforms and inverses in (15) and (16) with DFT and inverse DFT matrices, if we assume that modes higher than the Nyquist do not appear in the compressor (more detailed models of unsteady compressor performance, coupled with experiments , justify this assumption). The result is a finite-order nonlinear state space model, which we proceed to derive.
G : φ → φ Re φ 1 Im φ1 Re φ2
= G–1DAG ,
G –1 ≡ G T
where the last equality holds because the DFT matrix is scaled by 2/(2N+1) instead of 2/(2N+1). Using these matrices, we can easily define matrix versions of the linear transformations E and A in Eqs. (14-16):
Refinement of the above model to include the effects of unsteadiness on the compressor characteristic has been conducted by Haynes et. al. , for the linearized case. Their refinement recognizes that the compressor characteristic is not in reality a memoryless nonlinearity. In fact, the lags inherent in the compressor pressure-rise response have a strong effect on both the damping and the rotation frequency of pre-stall waves. It is straightforward to append the dynamics introduced in  to the equations derived here; to avoid unnecessary complexity we will not present this extension. Simulations indicate that, although compressor pressure-rise lags change the effective inertias and stability characteristics of the system, they do not add significant extra dynamics -- in the parlance of the linearized model,
including compressor lags adds fast dynamics, but also alters the dominant root locations. Thus if we model the change in the dominant root locations and neglect the fast dynamics, the model will presumably be accurate at frequencies of interest. For a priori prediction of compressor behavior, incorporating compressor lags is vital, as discussed in  and .
3. Comparison of Simulated and Experimental Stall Inception The nonlinear functions ΨC and Φ T govern both the stall inception and the fully developed stall behavior. Fully developed stall has been treated by other researchers [21, 22], and appears to be less relevant to the stabilization problem, for which we are primarily interested in the character and subsequent avoidance of stall inception. Thus we will concentrate here on the effect of ΨC and ΦT on nonlinear stall inception behavior, showing both experimental and theoretical results. Stall inception - that is, transition from axisymmetric flow to fully developed stall - is of interest because only during this transition process are two important criteria met by the flow field: 1) relatively small axial flow perturbations, and 2) strong influence of the nonlinearities. Meeting the first criterion is important to insure that the assumptions inherent in the model described above and in  are reasonable. Specifically, the "semi-actuator disk" assumption used to simplify the representation of the flow across the compressor is a more severe approximation during fully developed stall than during pre-stall and stall-inception conditions . Restricting the model validation to waves of small amplitude relative to fully developed stall does not pose a problem to control law development, because presumably any working controller would stabilize the system and avoid the fullydeveloped stall condition. The second condition above is important in the present study because understanding and accurately modeling nonlinear stall inception behavior is our goal. In some compressors, over a certain range of flow coefficients, linearization of the dynamics in (13) is a workable approach . However, to further extend the operating range of these compressors, and to control compressors in which nonlinearities are more severe, we must understand the nonlinear behavior. Our approach in the present study is to carefully model three experimental low speed compressors with different stall inception behavior. The parameters in the model are chosen on the basis of compression system geometry, modified by physical reasoning and comparison between simulation and experiment (i.e. a simplistic 'identification' procedure). Experimental stall inception data are then compared to nonlinear simulation, to verify that the model captures the important transient phenomena.
3.1 Compressor Models For convenience we will label the three experimental compressors used as compressors C1, C2, and C3. The details of the experimental setup for each compressor are described in references , , and  respectively. The geometric details of the compressors (those necessary to perform the simulation) are presented in Tables I and II, and the compressor and throttle characteristics are shown in Fig. 3. Note the following approximations in the choice of the model parameters: 1)
The 'B' parameter  is set to 0.1 in the experimental
compressors - this represents in all cases a 'worst case' maximum. Even using this maximum, the surge dynamics are very stable for the experiments we will present; this is most easily expressed by recognizing that the following approximation to Eq. (6) applies as B∅0: φ ≅ ΦT ψ
Thus the mean flow will follow the throttle characteristic very closely in these experiments, because the system has almost no mass storage capacity. 2) An 'unsteady loss parameter' is used to account for the compressor lags . A priori prediction of this parameter is difficult
C1 73 m/s 0.259 m 0.4202 10.8 8.0 0.1 2 0.65 0.18 0.44
C2 72 m/s 0.287 m 0.463 9.41 6.66 0.1 2 1.29 0.68 0.68
C3 36 m/s 0.686 m 0.431 6.3 4.75 0.1 2 0.42 0.25 0.4
Known part of characteristic Estimated part of characteristic Transient into rotating stall Throttle characteristic Equilibrium point
U (speed at mean wheel radius) R (wheel radius) φ at stall inception ΚΤ at stall inception lc B m µ λ λeff (adjusted to match frequencies) Measurement position, η Nondim. convection time (used for unsteady loss approximation) Discharge plenum pressure
choices are made for the shapes of the unknown portions of the characteristics. Details of this procedure
TABLE I - COMPRESSOR MODEL PARAMETERS
TABLE II - COMPRESSOR CHARACTERISTICS
C1: ψC(φ) = 1.975⋅φ – 0.0987⋅φ + 0.0512 ; φ ≤ 0.025 3 2 –12.776⋅φ + 6.394⋅φ – 0.295⋅φ + 0.0535 ; 0.025 < φ ≤ 0.30 –5.536⋅φ4 + 7.720⋅φ3 – 4.204⋅φ2 + 1.127⋅φ + 0.0719 ; φ > 0.30
C2: ψC(φ) =
12.117⋅φ2 – 2.423⋅φ + 0.221 ; –49.624⋅φ3 + 39.509⋅φ2 – 6.413⋅φ +0.395 ; –10.0695⋅φ 2 + 9.430⋅φ – 1.184 ;
φ ≤ 0.1 0.1 < φ ≤ 0.40 φ > 0.40
0.8 4⋅φ2 – 2⋅φ + 0.5 ; –143.14⋅φ3 + 143.04⋅φ2 – 44.683⋅φ +4.717 ; –13.365⋅φ2 + 11.574⋅φ – 1.920 ; –5.428⋅φ2 + 4.211⋅φ – 0.213 ;
φ ≤ 0.25 0.25 < φ ≤ 0.405 0.405 < φ ≤ 0.463 φ > 0.463
, so we have used physically reasonable values (~the convection time across one blade), adjusted to improve the model fit. 3) The simulations presented here also use an 'effective λ', which is adjusted from the geometrical value λ to match the perturbation rotation frequency. The final and most important model parameter is the axisymmetric compressor characteristic, ΨC. The characteristic shapes shown in Fig. 3 are determined in two steps. First, experimental data for each compressor is fit with polynomial segments. However, the experimental data extends only over the portion of the curves indicated by solid lines, because beyond this point the compressors are unstable and cannot be operated axisymmetrically. To extend the compressor characteristics beyond their measured portion, the nonlinear simulation is used. Simulated and actual stall transients are compared, and informed
C3: ψC(φ) =
Fig. 3: Compressor and throttle characteristics used to simulate rotating stall inception in compressor: (a) C1, (b) C2, and (c) C3. Transients of annulus average (a.a.) pressure vs. flow during stall inception are also shown.
are given in , and the results are shown in Figs. 4-6, which constitute the nonlinear model validation.
3.2 Discussion of Simulated and Experimental Stall Inception Transients The 'best fits' of the nonlinear simulation to the transient data are shown in Figs. 4, 5, and 6. In all cases the salient features are similar between the experiment and simulation, although the three compressors have very different stall inception behavior. Compressor C1 has a long, slow growth of pertur-bation waves into rotating stall. Compressor C2, on the other hand, has relatively small perturbation waves, followed by a sharp inception
wave that Experimental Stall Inception Data
δφ Probe #3
Probe #1 Simulated Stall Inception - Frequency Adjusted Using Effective λ
Probe #1 0
SCALE: δφ = 0.2
τ, Rotor revs
SENSOR LOCATIONS: η = -0.5
Fig. 4: Comparison of experimental and simulated stall inception for compressor C1. δφ
Experimental Stall Inception Data
Probe #3 Probe #2 Probe #1 δφ Simulated Stall Inception - Frequency Adjusted Using Effective λ Probe #3 Probe #2 Probe #1 0
5 SCALE: δφ = 0.2
10 15 τ, Rotor revs SENSOR LOCATIONS: η ≅ -0.2
Experimental Stall Inception Data
25 2 1 3
Fig. 5: Comparison of experimental and simulated stall inception for compressor C2.
leads quickly into fully developed rotating stall. Compressor C3 is even more severe in this regard: nonlinear influences are seen while the waves are still quite small, and the stall wave grows quickly into a violent nonlinear event. Despite these differences in behavior, the proper choice of the shape of ΨC allows experimental observations to be closely matched by the nonlinear model. There is a larger discrepancy between the simulated and the measured stall inception behavior for compressor C3 than for C1 and C2. Several factors contribute to this discrepancy. Most important is the effect of non-axial flow on the measurements. Hot-wire anemometers are used to measure the flow velocity in the experiments; these devices effectively measure the absolute
Simulated Stall Inception - Frequency Adjusted Using Effective λ
2 SCALE: δφ = 0.2
6 8 τ, Rotor revs
SENSOR LOCATIONS: η≅0
Fig. 6: Comparison of experimental and simulated stall inception for compressor C3.
axial plus radial flow perturbations. In the C3 experiment, the hot wires are mounted very close to the compressor face (η = 0), where significant non-axial and reverse flow perturbations exist. In the C1 and C2 experiments, on the other hand, the hot wires are mounted further upstream of the compressor (η = -0.5 and η = 0.2 respectively) where the measured perturbations are primarily axial; a 'fluid dynamic filter' exists (Eq. (11)) that smoothes out the internal flow details, allowing the more global influences (those modeled by Moore-Greitzer) to be observed. To understand the influence of the nonlinearity in ΨC on stall inception, consider a sinusoidal velocity perturbation being mapped through a compressor characteristic, shown in Fig. 7. From the figure it is clear that at the peak of the characteristic, a linear representation of ΨC is insufficient; the slope of the characteristic is near zero, so higher order derivatives become important. The high velocity portion of the wave experiences attenuating pressure forces, because it accesses the stable side of the characteristic, while the low flow side experiences destabilizing pressure forces. Because of the interaction caused by partials with respect to θ and τ (Eq. 8), the pressure rise does not act alone to accelerate the flow. Instead there is an 'integrating action' of, for instance, the first sinusoidal harmonic. If the integral effect of the positive and negative parts of the characteristic causes a net attenuating effect, then the wave will die away (or converge to a small amplitude limit cycle). If the integral effect is amplifying, the wave will continue to grow, with a speed that depends at each instant on the wave shape and its mapping through ΨC and ΨT. The two extremes of this general behavior are: 1) gradual characteristics, whose behavior can be adequately characterized using a linearized analysis (with the proper choice of slope), and 2) characteristics with abrupt changes in slope. In the latter case (when the unstable side is steeper as in Fig. 7), a sharp drop in pressure rise is experienced by the low-flow part of the wave. This pressure drop overrides other influences and causes a quick, localized deceleration of the flow (as in Fig. 6). This drop in flow
further reduces the overall pressure rise delivered by the compressor, ψ
0.8 0.6 0.4 0.2 0
0 0.2 0.4 0.6 0.8
φ Ψc (φ(θ))
Note drop in Ψc
Fig. 7: Effect of nonlinear characteristic on pressure forces which act to accelerate a velocity perturbation. Pressure forces (on right) below the mean value act to decelerate the flow. Note the significant crossfeed between the first harmonic and the zeroth and second harmonics.
often manifest themselves as high spatial frequencies in the stall transients. In both C2 and C3, the rotating stall precursor wave starts as a sinusoidal wave (first spatial harmonic). During stall inception, however, significant spatial harmonic content above the first harmonic exists in the wave shape as it transitions into rotating stall. Figure 7 is an example of how this comes about - it shows a first harmonic perturbation δφ being transformed into a 2nd harmonic acceleration term δψ. Figure 8 shows the result during stall inception, what began as a 1st harmonic perturbation becomes a multi-harmonic perturbation, eventually transitioning again to a 1st-harmonic dominated fully developed stall cell. Thus approximations which model only the first spatial harmonic (Galerkin approximation, ) may not capture the important nonlinear effects at stall inception accurately enough to allow realistic control law design. It should be noted in closing this section that the philosophy of the Moore-Greitzer model is to ignore blade-to-blade flow details when studying transient rotating stall behavior. This philosophy, although validated in a wide variety of both high and low speed compressors  is not universally applicable. Day  has measured stall inception behavior that must be observed at higher resolution to be fully understood. In such cases, the MooreGreitzer model requires further refinement; 3-dimensional effects need to be at least partially accounted for to properly model these stall phenomena.
4. Lyapunov Analysis of Rotating Stall which in turn moves the mean operating point towards lower values (in order to satisfy the throttle characteristic, Eq. 21). The system thus evolves into rotating stall at a rate that is premature when viewed from the linearized analysis. In fact, with the proper initial condition, a compressor can go into stall while at an annulus-averaged operating point that is still stable in the linearized sense. In these cases, the domain of attraction of the operating point has become very small, because of the existence of a nearby abrupt change in the nonlinear mapping ΨC. Section 4 makes this observation mathematically rigorous. Also important to the stall inception behavior is the slope of the throttle characteristic, ΨT. If ΨT is steep at the nominal operating point (∂ψT/∂φ large), the annulus-averaged flow coefficient is insensitive to changes in the pressure rise delivered by ΨC. This is the case in C1 (Fig. 3a), in which the throttle discharges to a plenum below atmospheric pressure, making the throttle line steep during stall initiation. This steep slope, combined with the shape of ΨC, accounts for the slow transient into rotating stall shown in Fig. 4.
When ΨT is shallow (∂ψT/∂φ small), on the other hand, the nonlinearity of ΨC couples more immediately into changes in the mean flow as follows: ΨC maps energy from higher harmonics into the zeroth harmonic (this effect is represented by the function ΨC φ + δφ , and can be seen in Fig. 7). This causes a loss in annulus-averaged pressure rise, which must be accompanied by a relatively large drop in φ if ΨT is shallow (Eq. (21)). Thus the rate at which the system evolves into rotating stall depends in part on the slope of ΨT. Compressor C3 is the best example of this type of behavior - the transient from stall inception to fully developed stall is under one rotor revolution, partially because of the shape of ΨT. Another important conclusion one can draw from the results presented here is that nonlinear effects, when they are important,
Having qualitatively validated the model in a nonlinear sense, we now turn to quantitative techniques for control law analysis. Lyapunov methods are the foundation for various nonlinear control design procedures (such as feedback linearization, sliding mode control, and Lyapunov control), and thus a Lyapunov stability analysis of rotating stall is a logical first step. Furthermore, we present herein a complete method, based on Lyapunov concepts, for assessing the large-amplitude disturbance rejection characteristics (domains of attraction) of axial compression systems. To conduct a Lyapunov analysis and subsequently deduce stability regions, we first translate the origin to the equilibrium point, as in Fig. 9. We then find a suitable 'incremental energy' or Lyapunov function to characterize the system state. Although any positive
follows: Using Eq. (19),
DFT coefficient magnitude (64 points)
φ = Gφ φ ⋅DA⋅ Gφ φ =0 φ T⋅A⋅φ
5th Harmonic 0
Time, rotor revolutions
Fig. 8: Participation of 1st 5 Fourier harmonics in stall inception. Note that harmonics 2 and 3 temporarily become larger than the 1st harmonic during stall inception. Simulation results are for compressor C3 (same as Fig. 6). ψ ΨT(φ)
which is the two-dimensional analog of the "incremental power production and dissipation" equation motivated and developed by Simon . Taken to the 2D continuum limit (i.e. M∅ ), (25) can be expressed as follows: V = 1 2π
φ(θ) ΨC φ(θ) dθ – ψ⋅ΦT ψ
Ψc(φ) Fig. 9: Coordinate system for Lyapunov stability calculation.
definite function based on the system state is a candidate Lyapunov function, careful choice of the form leads to a more elegant and physically meaningful formulation. Consider the function 2 1 φT ⋅E⋅φ φ + 2B2l c⋅ψ 2M
which is the annulus averaged incremental energy production of the compressor, minus the incremental energy dissipation of the throttle. This measure of compression system stability was originally proposed by Gysling . Since ψ⋅ΦT ψ is a positive definite function of ψ (see Fig. 9: ΦT ψ lies strictly in the first and third quadrant), the 'worst case' value of V will occur when ψ is zero. Thus we can consider the nonlinear stabilityT of the rotating stall system by simply analyzing the function φ ⋅Ψc φ , (or, equivalently, the integral term in Eq. (26) ). 4.1 Basins of Attraction Using the Lyapunov analysis presented above, it is possible to determine a class of perturbations that are guaranteed to be stable. This 'basin of attraction' can be defined by finding the largest hypersphere: T S ∆ = φ : 1 φ ⋅ E⋅φ φ ≤ ∆2 2M
The last equality holds due to the form of DA, which appears in (19). Equation (24) reflects the fact that Aφ is simply φ rotated by π/2, and is thus orthogonal to it φ . Using (24), Eq. (23) simplifies to V = 1 φ T⋅ΨC φ – ψ⋅ΦT ψ M , (25)
inside which V is always less than zero. An estimate of ∆ can be motivated as follows. For a given operating point, we can compute the function φ⊇ΨC(φ) as a function of φ as shown in Fig. 10. This function 'maps' a perturbation φ(θ) into a value of V, as shown in the figure. The shape of φ⊇ΨC(φ) shown in Fig. 10 is quite general near the local maximum of ΨC for stable equilibria. φ⊇ΨC(φ) will invariably have a local maximum that goes through the origin, be negative for values of φ greater than zero, and become positive for values of φ below some value that we call -d.
where M=2N+1 and, from here on, all states and functions are evaluated about the new origin. It is straightforward to show that E is positive definite based on the definitions (18), so V is positive definite. This form of the Lyapunov function clearly accounts for all perturbations that might exist in the compressor, and thus suitably characterizes the system’s 'incremental energy'. More important, however, is the way this choice for V effects the form of V. By taking the derivative of (22), substituting Eqs. (20), and simplifying, we have φ + 1 φT⋅ΨC φ – ψ⋅ΦT ψ V = – 1 φ T⋅A⋅φ M M . (23) We can eliminate the first term in this equation, because A is a rotation matrix whose eigenvalues are all on the jω-axis, as
Based on this discussion we see that, at stable equilibria and for perturbations φ(θ) whose minima are greater than -d, V in Eq. (26) will always be negative. Thus these perturbations will become smaller in magnitude, where magnitude is measured by V in Eq. (22). Stated mathematically, we have
min φ(θ) ≥ –d