Compressor Surge Control Design Using Linear Matrix Inequality Approach

Transcription

1 Comressor Surge Control Design Using Linear Matrix Inequality Aroach Nur Uddin Det. of Electrical Engineering Universitas Pertamina Jakarta, Indonesia Jan Tommy Gravdahl Det. of Engineering Cybernetics Norwegian University of Science and Technology (NTNU) Trondheim, Norway Abstract A novel design for active comressor surge control system (ASCS) using linear matrix inequality (LMI) aroach is resented and including a case study on iston-actuated active comressor surge control system (PAASCS). The nonlinear system dynamics of the PAASCS is transformed into linear arameter varying (LPV) system dynamics. The system arameters are varying as a function of the comressor erformance curve sloe. A comressor surge stabilization roblem is then formulated as a LMI roblem. Solving the LMI roblem results in a feedback control gain for the comressor surge stabilization and stability roof of the closed loo system in the whole comressor oerating area. Simulation results show that the designed surge control system is able to stabilize comressor surge. Significant imrovement of the control system erformance is achieved by combining the LMI aroach and linear quadratic regulator (LQR). I. INTRODUCTION Comressor oeration at lower mass flows is limited by comressor surge. Comressor surge is an aerodynamic instability and results in axisymmetric oscillations of the comressor mass flow and the comressor ressure. This henomenon is indicated by fluctuations at the comressor mass flow, at the comressor discharged ressure, and at the comressor flow temerature, and followed by vibrations on the rotating arts. The vibrations reduce the reliability of the rotating arts and large amlitude vibrations lead to comressor damage, esecially to the comressor blades and bearing. A method for stabilizing comressor using a state feedback control and an active element (actuator) has been introduced by Estein et al. [1]. The method is known as active comressor surge control. Several studies on active comressor surge control using different control design methods by including linear and non-linear control methods for different actuators have been resented, and are summarized in [2] [4]. Most of the active surge control studies uses the Greitzer comression model to reresent the dynamics of comressor states (ressure and mass flow). The Greitzer comressor model is a model of comression system which is able to redict transient comressor states during comressor surge including comressor mass flow and comressor [5]. Physical observations show that comressor roduced ressure (comressor discharged ressure) has a non-linear relation to the comressor mass flow which is usually described in a comressor ma. A comressor ma is commonly rovided by the comressor manufacturer. However, the comressor ma may also be obtained through a comressor erformance test by following stes: oerate the comressor for several oerating oints, record the mass flow and ressure data, and do a curve fitting to aroximate the whole oerating oints. An aroximation of comressor erformance for a constant comressor seed has been introduced in [6], while for the varying comressor seeds has been resented in [7]. The main goal of an active comressor surge control system design is to make the closed loo comressor system to oerate stable in the surge area. Previously, the surge control was designed base on linear control aroach [8] [1]. Using the linear control aroach, the closed loo system achievies locally asymtotically stable such that the stabilized comressor surge area is limited. In order to make the closed loo comressor system stable in the whole surge region, the closed loo system must be globally asymtotically stable (GAS). A necessary condition for GAS is the existence of a Lyaunov function. Simon and Valavani alied Lyaunovbased control for an active surge control system using closecouled valve [11]. Krstic et al. introduced a non-linear control design method known as backsteing in [12]. The backsteing method rovides a systematic rocedure to find state feedback and Lyaunov function simultaneously. Several works on active surge control using backsteing have been resented afterwards [13] [16]. However, the backsteing may result in a comlicated state feedback and can be difficult to be imlemented [17]. Two general state feedback control laws for comressor surge control have been derived using Lyaunov-based control method and guarantee the GAS of the closed loo system [18]. However, alying the Lyaunovbased control method is not straightforward and in general, it is not ossible to do erformance adjustment base on a cost function as in otimal control. A non-linear system can be aroximated by a linear system through a linearization around an oerating oint and therefore linear control methods are alicable. This aroximation is only valid for a limited region around the oerating oint as basis in the linearization. Therefore, linearization at several oerating oints is required to cover the whole oerating area. This will result in several linear systems and the arameters

2 Station A A i Inlet M T Motor M T c Comressor d w d V, w s Plenum V, L s t Piston w t Throttle u s Station B Fig. 1. COMPRESSION SYSTEM EQUIPPED PAASCS. are varying for each oerating oint. Linear matrix inequalities Station A Station B c A d (LMI)-based control is w B i wone d of the owerful control t design Inlet methods for such linear arameter varying system [], [2]. The LMI-based control is a convex otimization base on Throttle Comressor Plenum LyaunovMotor stability condition. A Lyaunov function candidate, which is a ositive definite function and the time derivative should be negative definite, is formulated as an LMI roblem []. The LMI roblem is solved to obtain a ositive definite matrix such that the time derivative of the Lyaunov candidate is negative definite. The LMI solution can be obtained using available comutational tools, for an examle the YALMIP Toolbox [22]. A iston-actuated active surge control system (PAASCS) is a method to stabilize comressor surge by dissiating the downstream comressor energy using a iston [17]. This aer is resenting an alication of LMI-based control method in an active comressor surge system with a case study on PAASCS. The goal is to obtain a state feedback control to stabilize the whole comressor oerating area and the control system has a erformance. The non-linear dynamics of PAASCS is transformed into a linear arameter varying system, and the comressor surge stabilization is formulated as an LMI roblem. II. PISTON ACTUATED ACTIVE SURGE CONTROL SYSTEM A iston actuated active surge control system (PAASCS) is an active surge control system utilizing a iston as an actuator to stabilize comressor surge. The iston generates mass flow to maniulate the comressor downstream ressure in order to stabilize comressor surge. PAASCS has been introduced in [17] and the model is shown in Figure 1. The model was develoed base on the Greitzer comressor model [5] and all assumtions in the Greitzer model are adoted. It is assumed that ressures at station A ( A ) and at station B ( B ) are equivalent to the ambient ressure. All ressures in the system are measured relative to the ambient ressure. Heat transfer in the system is neglected. Pressure dro along the inlet line is neglected such that the inlet ressure ( i ) is equal to A. There is no mass storage in the comressor such that the inlet mass flow ( ) is equal to the comressor discharge mass flow (w d ). t B A lenum is a model of downstream control volume. The lenum volume (V ) can be reresenting a volume of ieline and/or vessel. It is assumed that the ressure in the lenum ( ) is uniformly distributed in the lenum sace. A throttle is used to adjust the outlet mass flow (w t ). The throttle generates ressure dro ( t ) between lenum and the outlet. A iston is connected to the lenum for generating iston mass flow (w s ). The iston mass flos used to maniulate the lenum ressure to maintain the system to be stable. Dynamic equations of the system are given as follows [18]: ẇ i = A c [ c ] (1) ṗ = a2 V [ w t w s ] (2) where A c is the inlet cross section area, is the effective length of the inlet, a is the seed of sound, and V is the lenum volume. The inlet mass flow dynamics is a function of the ressure difference between the comressor discharge and the lenum. Pressure at the comressor discharge is a result of energy conversion from mechanical into neumatic. It is a function of the comressor mass flow and the comressor seed as commonly shown in a comressor ma. The comressor ma usually consists of a lot of the comressor roduced ressure against the comressor mass flow for several comressor seeds. However, we consider only on a constant comressor seed in this study. A comressor erformance at a constant seed can be aroximated by a qubic function [6]: c = c + H 2 [ ( wi W 1 ) ( wi ) ] 3 W 1 where c is the shut-off value of the axisymmetric characteristic, W is the semi-width of the cubic axisymetric comressor characteristic and H is the semi-hight of the cubic axisymetric comressor characteristic, consults [6] for more detailed definition. III. LINEAR PARAMETER VARYING SYSTEM REPRESENTATION Define system states for the PAASCS as follows: (3) x 1 = w t (4) x 2 = c (5) and substituting into (1) and (2) results in: ẋ 1 = k 1 x 2 ẇ t (6) ẋ 2 = k m k 1 x 2 k 2 x 1 + k 2 w s (7) where k 1 = Ac, k 2 = a2 V and k m = dc d. The variable k m reresents the sloe of comressor erformance curve which is varying at each oerating oint. A state sace form of the PAASCS dynamics is given as follows: [ ] [ ẋ1 = ẋ 2 ] [ ] [ k 1 x1 + k 2 k m k 1 x 2 k2 ] [ ] 1 w s + ẇ t. (8)

3 ressure (kpa) % Aroximation 2 Pc=2.564; H=.; W=.11; wi=-.2:.1:.22; Pc=Pc+H*(1+1.5*(wi/W-1)-.5*(wi/W-1).^3); Comressor seed: rm G F %Arroximation 1 wi=.22:.1:.1; Pc = 1158*wi.^3-2534*wi.^ *wi +.97; Aroximation 1 Pd Aroximation mass flow (kg/s) Fig. 2. Comressor erformance curve obtained through a erformance test [23]. The equation (8) can be exressed by: ẋ = A(k m )x + Bw s + Dẇ t, (9) where x = [ [ ] ] T k x 1 x 2, A(km ) = 1, B = k 2 k m k [ ] 1 T [ ] T k2, and D = 1. The variable x is the system states, w s is the system inut, and w t is the system disturbance. The ẇ t is the rate change of comressor outlet flow and the value is assumed to be bounded as the outlet valve has usually slow dynamics. Eigenvalues of the oen loo system (8) are given by: s 1,2 = k mk 1 ± (k m k 1 ) 2 4k 1 k 2, (1) 2 where the real arts of eigenvalues indicate the system stability. Because the value of k 1 is constant and k 1 >, the comressor system stability deends on the value of k m, which is the comressor erformance curve sloe. Therefore, the comressor oerates stable at along comressor erformance curve at negative sloe (k m < ) and unstable at the ositive sloe (k m > ). The unstable comressor oerating along the ositive sloe is known as surge. For a comressor erformance curve described in (3), the curve sloe is given by: k m = 3 2W 3 ( wi ) 2 2 W 1, (11) which is a function of the comressor mass flow. The sloe is varying in a range of km k m k m, + where k m + is the maximum sloe and km is the minimum sloe. The maximum sloe is k m + = 3H 2W achieved at mass flow = W, while the minimum sloe is km = 2 w o [18]. Therefore, the system dynamics in (8) or (9) is a linear arameter varying (LPV) system as the system matrix A is a function of k m. From now we use notation A instead of A(k m ) in the interest of simlicity. E D C B A IV. LPV SYSTEM STABILIZATION A PAASCS alies a iston to generate mass flow for stabilizing comressor surge. The generated mass flos reresented by the system inut, w s, in (9). In order to design a controller, define a state feedback: w s = Kx (12) where K is a control gain matrix. The closed loo system of (9) is given as follows: ẋ = [A + BK] x + Dẇ t. (13) It is required to find K such that the closed loo system matrix is asymtotically stable. It is achieved iff [A + BK] is Hurwitz. The asymtotic stability of a system is guaranteed by the existence a Lyaunov function V (x), where: V (x) > and V (x) <. For (13), define a Lyaunov function candidate V (x) = x T P x (14) where P needs to satisfy the following conditions: P > (15) A T P + P A + K T B T P + P BK <. (16) Equation (16) is a bilinear matrix inequality (BMI) because it has multilication of two unknown variables, P and K. Solving a BMI roblem is difficult, and it is recommended to convert a BMI roblem into a linear matrix inequalities (LMI) roblem by the following stes []: a. Define Y = P 1 and do re- and ost- multilication to (16) such that it results in: Y A T + AY + Y K T B T + BKY <. (17) Inequality (17) is still a BMI. b. Define L = KY and substitute into (17) such that it results in: Y A T + AY + L T B T + BL <. (18) Inequality (18) is a LMI. Because the comressor ma sloe is varying in a certain range, we need to define two LMIs which reresents the extreme oerating region: Y A T 1 + A 1 Y + L T B T + BL < () Y A T 2 + A 2 Y + L T B T + BL < (2) where A 1 = A(k m) for the minimum sloe and A 2 = A(k + m) for the maximum sloe, resectively. Both LMIs are then solved simultaneously to find a ositive definite matrix Y and a matrix L such that the both LMIs in () and (2) are satisfied. The YALMIP Toolbox together with Matlab is one of the available software to solve the roblem. Commands for solving the roblem are given as follows: Y = sdvar(2,2); L = sdvar(1,2, full ); F = [Y > ]; F = [F, [Y*A1 +A1*Y+Bs*L+L *Bs ] < ]; F = [F, [Y*A2 +A2*Y+Bs*L+L *Bs ] < ]; solvesd(f,-trace(y))

4 TABLE I PAASCS PARAMETERS [23] Parameter Value Unit Parameter Value Unit a 34 m/s V.12 m 3.8 m A c.38 m 2 ρ kg/m 3 A s.314 m 2 m s 2 kg wi [kg/s] ref wi [kg/s] ref [kpa] [kpa] Fig. 3. COMPRESSION SYSTEM STATES WITHOUT SURGE CONTROL. ws [kg/s] Fig. 4. CLOSED LOOP TIME RESPONSE. While the matrices Y and L are found, the matrices P and K are obtained by P = Y 1 and K = LY 1, resectively. A simulation is done to evaluate the closed loo system erformance using arameters data in Table I and a comressor erformance curve shown in Figure 2. The simulation scenario is given as follows. A comressor is initially oerating steady at mass flow.6 kg/s and then the oerating oint is changed at t = 5 seconds by reducing the mass flow to.15 kg/s which is crossing the comressor surge line. The simulation results are shown in Figure 3. It is shown that the oen loo comressor system exeriences oscillation in lenum ressure and inlet mass flow, and is known as comressor surge. Moreover, the comressor surge is known as a dee surge as the comressor mass flos reversed (negative mass flow). While Figure 4 shows simulation results of the closed loo system using the designed control law, the system oerates stable in the desired oerating oint, which means that the PAASCS is able to stabilize comressor surge. However, the closed loo system has a long settling time and the iston mass flos very high comare to the comressor mass flow which is not ractical. A iston with a large diameter and fast movement is required to generate the high iston mass flow. The erformance of the closed loo system is therefore needed to be imroved. V. LMI-LQR METHOD Considering the simulation results in the revious section, the closed loo system erformance needs to be imroved. Using the LMI formulation in (16), we can not do any erformance adjustment as the matrices A and B are given from the lant, and the matrices P and K are obtained through a comutational rocess using the YALMIP Toolbox. Performance adjustment bases on a cost function is commonly done in otimal control theory, for examle linear quadratic regulator (LQR). Fortunately, combination of LQR method and LMI method for control system design (LMI-LQR) has been resented in [], [], [24], [25] and summarized as follows. The LMI-LQR gives an oortunity to adjust the erformance of a closed loo system designed using LMI. The concet of LMI-LQR is described as follows. A LQR roblem for a system: ẋ = Ax + Bu () is basically to find a control gain K such that a states feedback control u = Kx minimizes a cost function: J = 1 2 ( x T Qx + u T Ru ) dt, (22) where Q and R >. Using u = Kx, the closed loo system of () is given by: ẋ = [A + BK] x (23)

5 and the cost function (22) can be exressed as: J = 1 2 x T ( Q + K T RK ) xdt. (24) Assuming (23) is asymtotically stable by existing a Lyaunov function: V (x) = x T P x > (25) then time derivative of V (x) along the system trajectories (23) is given by: V (x) = x(a T P + P A + K T B T P + P BK)x < (26) where P >. The negative definiteness of (26) can be reinforced by defining: V (x) < x T ( Q + K T RK ) x <. (27) A time integration from to of (27) will result in: or V ( ) V () < x T ( )P x( ) x T ()P x() < x T ( Q + K T RK ) xdt (28) x T ( Q + K T RK ) xdt. (29) Since (23) is asymtotically stable then x( ) is equal to zero such that (29) becomes: x T ()P x() > x T ( Q + K T RK ) xdt. (3) Equation (3) shows uer bound of the cost function (24), such that the cost function will be minimum by minimizing the matrix P : min P,K xt ()P x() s.t V (x) < x T ( Q + K T RK ) x. (31) Exressing the minimization constraint along the trajectories of system (23), the (31) becomes: min P,K xt ()P x() s.t (A + BK) T P + P (A + BK) + (Q + K T RK) <. (32) Since the initial condition x() is given, it can be eliminated from (32) and the minimizing roblem becomes: min tr(p ) P,K s.t (A + BK) T P + P (A + BK) + Q + K T RK <. (33) Equation (33) is a non-convex otimization roblem where the constraint is BMI. A transformation is required to transform the BMI into LMI as done in the revious section. Define Y = P 1 and do re- and ost- multilications to the otimization constraint of (33) such that the constraint becomes: Y A T + AY + Y K T B T + BKY + Y QY + Y K T RKY <. (34) Define L = KY and substituting it into (34) results in: Y A T + AY + L T B T + BL + Y QY + L T RL <. (35) By using Schur comlement, (35) can be exressed as: (AY + BL)T + (AY + BL) Y L T Y Q 1 <. L R 1 (36) which is in a LMI. A detail exlanation of Schur comlement is given in the Aendix. Equation (33) is therefore exressed as a convex otimization as follows: max tr(y ) Y,L s.t (AY + BL)T + (AY + BL) Y L T Y Q 1 <. L R 1 (37) Alying the LMI-LQR method in PAASCS design is resented [ as follows. ] Define weighting cost function matrices 1 Q =, and R = 1. YALMIP Toolbox is used to 1 solve the LMI by the following commands: Q=eye(2); R=1; Y = sdvar(2,2); L = sdvar(1,2, full ); F = [Y >= ]; F = [F, [-A1*Y-Bs*L + (-A1*Y-Bs*L) Y L ;... Y inv(q) zeros(2,1);... L zeros(1,2) inv(r)] > ]; F = [F, [-A2*Y-Bs*L + (-A2*Y-Bs*L) Y L ;... Y inv(q) zeros(2,1);... L zeros(1,2) inv(r)] > ]; solvesd(f,-trace(y)) K = double(l)*inv(double(y)); and running the code results in a new control gain K. Simulation of PAASCS using the new control gain for the same system arameters and simulation scenario results in an imrovement of the closed loo system erformance as shown in Figure 5. Simulation of the PAASCS designed using LMI- LQR results in stabilized comressor surge where the control system requires much less iston mass flow and much faster system resonse than the PAASCS designed using LMI only. VI. CONCLUSION A control design of an active comressor surge control system using LMI method has been resented. The LMI method rovides an analytic solution to obtain a control gain and a stability roof. A study case of alying the LMI method for the PAASCS resulted in a linear state feedback control which is able to stabilize the comressor surge. The control system erformance is imroved significantly by combining the LMI method and LQR method. This method is very systematically for nonlinear control system design by aroaching a nonlinear system as a linear arameter varying system and the control system design roblem is exressed as a LMI roblem. The LMI solution is obtained directly using the available software. ACKNOWLEDGMENT The authors acknowledge the financial suort of Siemens Oil and Gas Solutions Offshore through the Siemens-NTNU collaboration roject in a eriod of 29-3.

6 wi [kg/s] [kpa] ref ws [kg/s] Fig. 5. CLOSED LOOP TIME RESPONSE LQR. REFERENCES [1] A. H. Estein, J. E. F. Williams, and E. M. Greitzer, Active suression of comressor instabilities, in Proc. of AIAA 1th Aeroacoustic Conference, Seattle, 86. [2] F. Willems and B. de Jager, Modeling and control of comressor flow instabilities, Control Systems, IEEE, vol., no. 5,. 8 18, oct 99. [3] N. Uddin and J. T. Gravdahl, Bond grah modeling of centrifugal comression systems, SIMULATION, vol. 91, no. 11, , 5. [4] J. T. Gravdahl and O. Egeland, Comressor surge and rotating stall: Model and control. London: Sringer Verlag, 99. [5] E. M. Greitzer, Surge and rotating stall in axial flow comressor, art I: Theoritical comression system model, J. Engineering for Power, vol. 98,. 8, 76. [6] F. K. Moore and E. M. Greitzer, A theory of ost stall transients in an axial comressors system: Part I-Develoment of equation, J. Engineering for Gas Turbine and Power, vol. 18, , 86. [7] J. T. Gravdahl, O. Egeland, and S. O. Vatland, Drive torque actuation in active surge control of centrifugal comressor, Automatica, vol. 38, , 22. [8] J. E. F. Williams and X. Y. Huang, Active stabilization for comressor surge, J. Fluid Mechanics, vol. 24, , 89. [9] D. Gysling, D. Dugundji, E. M. Greitzer, and A. H. Estein, Dynamic control of centrifugal comressor surge using tailored structures, ASME J. Turbomachinery, vol. 113, , 91. [1] J. Pinsley, G. Guenette, A. H. Estein, and E. M. Greitzer, Active stabilization of centrifugal comressor surge, ASME J. Turbomachinery, vol. 113, , 91. [11] J. S. Simon and L. Valavani, A lyaunov based nonlinear control scheme for stabilizing a basic comression system using a close-couled control valve, in Proc. of the American Control Conference, 91, [12] M. Krstic, I. Kanellakooulos, P. V. Kokotovic et al., Nonlinear and adative control design. John Wiley & Sons New York, 95, vol. 8. [13] M. Krstic, J. Protz, J. Paduano, and P. Kokotovic, Backsteing designs for jet engine stall and surge control, in Decision and Control, 95., Proceedings of the 34th IEEE Conference on, vol. 3. IEEE, 95, [14] J. T. Gravdahl and O. Egeland, Comressor surge control using a close-couled valve and backsteing, in American Control Conference, 97. Proceedings of the 97, vol. 2. IEEE, 97, [15], Control of the three state moore-greitzer comressor model using a close-couled valve, in Proc. 97 Euroean Control Conference, 97. [16] A. Banaszuk and A. J. Krener, Design of controllers for mg3 comressor models with general characteristics using grah backsteing, in American Control Conference, 97. Proceedings of the 97, vol. 2. IEEE, 97, [17] N. Uddin and J. T. Gravdahl, Active comressor surge control using iston actuation, in Proc. of the ASME Dynamics System and Control Conference, Virginia, 1. [18], Two general state feedback control laws for comressor surge stabilization, in 24th Mediterranean Conference on Control and Automation (MED), Athens, Greece, June 6, [] C. Olalla, R. Leyva, A. El Aroudi, and I. Queinnec, Robust lqr control for wm converters: an lmi aroach, Industrial Electronics, IEEE Transactions on, vol. 56, no. 7, , 29. [2] J. Mohammadour and C. W. Scherer, Control of linear arameter varying systems with alications. Sringer, 2. [] S. P. Boyd, Linear matrix inequalities in system and control theory. Siam, 94, vol. 15. [22] J. Löfberg, Yalmi: A toolbox for modeling and otimization in matlab, in Comuter Aided Control Systems Design, 24 IEEE International Symosium on. IEEE, 24, [23] N. Uddin and J. T. Gravdahl, Active comressor surge control system by using iston actuation: Imlementation and exerimental results, in Proc. of the 11th IFAC Symosium on Dynamics and Control of Process Systems, including Biosystems (DYCOPS-CAB 6), Trondheim, Norway, June 6. [24] J. Löfberg, Modeling and solving uncertain otimization roblems in yalmi, in Proceedings of the 17th IFAC World Congress, 28, [25] V. Balakrishnan and L. Vandenberghe, Connections between duality in control theory and convex otimization, in American Control Conference, 95. Proceedings of the, vol. 6. IEEE, 95, APPENDIX APPENDIX: LMI-LQR DERIVATION (A + BK) T P + P (A + BK) < (Q + K T RK) A T P + K T B T P + P A + P BK + Q + K T RK < X(A T P + K T B T P + P A + P BK + Q + K T RK)X < XA T + XK T B T + AX + BKX + XQX + XK T RKX < XA T + L T B T + AX + BL + XQX + XK T RKX < XA T + AX + L T B T + BL + XQX + L T RL < XA T + AX + L T B T + BL + [ X L ] [ ] [ T Q X R L By using Schur comlement it can be reresented as: XA T + AX + L T B T [ + BL + ] [ ] [ ] X L T Q X }{{}}{{} R L S 11 S 12 and then } {{ } S 22 XAT + AX + L T B T + BL X L T X Q 1 L R 1 ] < < }{{} S <. Schur comlement of s 22 in S: [ ] s11 s S = 12 < s s s 11 s 12 (s 22 ) 1 s < 22