Robustness of the Moore-Greitzer Compressor Model s Surge Subsystem with New Dynamic Output Feedback Controllers

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1 Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 4-9, 14 Robustness of the Moore-Greiter Compressor Moel s Surge Subsystem with New Dynamic Output Feeback Controllers Alina Anersson Aners Robertsson Anton S. Shiriaev, Leoni B. Freiovich Rolf Johansson Department of Automatic Control LTH, Lun University, Sween ( alina@control.lth.se, anersro@control.lth.se, rolf.johansson@control.lth.se) Department of Applie Physics an Electronics Umeå University, Sween ( leoni.freiovich@umu.se) Department of Engineering Cybernetics Norwegian University of Science an Technology, Norway ( anton.shiriaev@itk.ntnu.no) Abstract: This work presents an etension of a esign proceure for ynamic output feeback esign for systems with nonlinearities satisfying quaratic constraints. In this work we use an aial gas compressor moel escribe by the -state Moore-Greiter compressor moel (MG) that has some challenges for output feeback control esign (Planovsky an Nikolaev 199), (Rubanova 1). The more general constraints for the investigation of the robustness with respect to parametric uncertainties an measurement noise are shown. Keywors: Dynamic output feeback, Compressors, Nonlinear control systems, Stability robustness, Constraints, Nonlinearity, Lyapunov stability 1. INTRODUCTION A gas compressor is a mechanical evice for compressing an supplying the air or other gas uner a certain pressure. An aial gas compressor is one of the main components of gas turbines, aircraft jet engines, high-spee ship engines an small-scale power stations (Greiter 1976). They are also wiely use in high-voltage installations in the blast furnaces, in the chemical an petroleum inustries (Veernikov 1974). In 1986 Moore an Greiter publishe a ifferential equations moel escribing the airflow through the compression system in turbomachines (such as gas turbines, fans, etc.) (Moore an Greiter 1986). The Moore-Greiter moel inclues the ifferential equations: t φ = + 1 (1 + φ) φ + R(1 + φ) t = 1 (φ u) β (1) t R = σr σr(φ + φ ), R() y = Here, both φ an represent the eviations of the average flow an pressure from their respective nominal values. The variable u is the control variable an y is the measurement (with only eviation of the average pressure available). This work was supporte by the Sweish Research Council through the project Active Control of Compressor Systems Base on New Methos of Nonlinear Dynamic Feeback Stabiliation The stall R is the square amplitue of oscillations of the rotary erangement. An inlet flow eformation may result in the rotating stall, surge (or ai-symmetric stall), or a combination of them (Greiter 1976). The flow oscillations an strong vibrations of the blaes in the machine may cause amage to the complete engine an even flow reversal is possible (Ng 7). The fully evelope surge is not viewe as an unstable state of the compressor but as a set of equilibria along which R is nonero Pauano et al. (1). Subsequently we thus want to control the compressor ynamics to the appropriate set-point (φ,, R) = (,, ). The challenges to stabilie the origin of the -state Moore- Greiter moel that were presente before in (Moore an Greiter 1986), (Shiriaev 9), (Shiriaev 1), (Rubanova 1) are: the linearie ynamics are not stabiliable; the fact that R cannot be measure or use for feeback esign; the presence of non-globally Lipschit (cubic) nonlinearity; the nonlinearity in φ-ynamics is known only approimately. In this work we iscuss the egree of robustness an present the metho that will simplify the controller choice base on a specific task. By this metho an with the help of the previous project results one can choose the optimal controller for the given moel an analye the quality of the controller esign. Copyright 14 IFAC 69

2 19th IFAC Worl Congress Cape Town, South Africa. August 4-9, 14. THE CONTROL SYNTHESIS METHOD (SHIRIAEV 1) We will apply this metho to the surge subsystem of the MG compressor moel (1). The surge subsystem is: t φ = + 1 (1 + φ) φ + t = 1 (φ u) β y = with the nonlinearity W {φ} (φ) = 1 (1 + φ) () We use the general form of a ynamic output feeback control law (Shiriaev 1): u = U(, y), ż = F(, y) (4) where U( ) an F( ) are smooth functions of appropriate imensions. The family of stabiliing output feeback controllers has the following structure: () u = Λ {u} + Λ{u} + ω u W (, ) t = Λ{} + (5) Λ{} + ω W (, ) with R, where Λ {u}, Λ {u}, Λ {}, Λ{}, ω u, ω are constants to be efine. The nonlinearities in the controller of Eq. (5) are static nonlinearities an efine as W (, ) = 1 (1 + t + t ) (6) where t, t are constants to be efine too. The close-loop system with the surge subsystem of Eq. () an the controller of Eq. (5) takes the form: 1 [ {u} φ φ = 1 ż β Λ β Λ{u} β Λ {} Λ {} }{{} + =A cl 1 ω u β ω }{{} B cl =[B cl1, B cl [ W {φ} (φ) W (, ) with R an the output matri C cl = [, t, t. (7) In (Rubanova 1) we presente some stabiliing controllers an new constraints for the corresponing parameters. The task now is to analye the quality of the set of the stabiliing controllers presente.. ROBUSTNESS OF THE CLOSED-LOOP SURGE SUBSYSTEM of stability of the close-loop system of Eq. (7) is base on the Circle criterion (Yakubovich 4), (Khalil ). As we alreay know, in the close-loop system of Eq. (7) there are two nonlinearities of Eqs. () an (6). We will simplify the notation where W {φ} (υ 1 ) = W {φ} (φ) W (υ ) = W (, ) υ 1 = C cl1 = [ 1 [ φ υ = C cl = [ t t One of the main parts of the constructive steps in esign is that nonlinearities W {φ} (υ 1 ) an W (υ ) have to satisfy the quaratic constraints C cl1 C cl [ W {φ} (υ 1 ) φ 4 W (υ ) 4 T T C T cl 1 C cl1 C T cl C cl (8) (9) (1) By using the simplifie notation of Eq. (8) we can rewrite the quaratic constraints of Eq. (1) W {φ} (υ 1 )υ 1 4 υ 1 W (υ )υ 4 υ (11) Since the static nonlinearity W (υ ) is assume to resemble the original nonlinearity W {φ} (υ 1 ) of the system of Eq. () we nee to have one aitional constraint that will be connecte to both nonlinearities (W {φ} (υ 1 ) W (υ ))(υ 1 υ ) (1) The three quaratic constraints of Eqs. (11-1) shoul be satisfie φ,,. In general we have [ ω1 ẋ = A cl + [B cl1, B cl ω (1) where = [φ T an ω 1, ω represent the nonlinearities of the form W {φ} (φ) an W (, ) an satisfy the given conitions of Eqs. (11-1). To check the stability of the close-loop system of Eq. (7) we will use the Circle criterion (CC) (Yakubovich 4), (Shiriaev 1). Stability conitions by using the Circle criterion Following the CC, to claim stability of the close-loop system of Eq. (7) it is enough to check the following two conitions: (1) There are constants τ 1, τ, τ such that τ 1 + τ + τ > an there are transfer functions G 11 (jω) = C cl1 (jωi n A cl ) 1 B cl1 The synthesis of stabiliing controllers an their application to the surge subsystem of the MG compressor moel was presente in (Rubanova 1). The alternative proof G 1 (jω) = C cl1 (jωi n A cl ) 1 B cl G 1 (jω) = C cl (jωi n A cl ) 1 B cl1 G (jω) = C cl (jωi n A cl ) 1 B cl (14) 691

3 19th IFAC Worl Congress Cape Town, South Africa. August 4-9, 14 Ψ( ) 1 1 k (k k 1 ) k 1 (k 1 k 1 ) Ψ(, t) Fig. 1. The eample of the possibility to move the sector conition for the nonlinearity (loop-shaping). such τ 1 Re{ ω 1 υ υ 1 } τ Re{ ω υ + 4 υ } τ Re{( ω 1 ω ) ( υ 1 υ )} < (15) hols ω 1 C, ω C, ω R, where υ 1 = G 11 (jω) ω 1 + G 1 (jω) ω υ = G 1 (jω) ω 1 + G (jω) ω (16) () There are row matrices K 1 an K such that K 1 an K are linear relations satisfy all the quataric constraints an the matri is Hurwit. (A cl + B cl1 K 1 + B cl K ) (17) First, in orer to be able to use these two conitions we will choose the gains K 1 an K. In Fig. 1 the sector conition for some nonlinearity Ψ(, t) is illustrate as an eample. The given nonlinearity never leaves the sector area between two lines k Ψ(, t) k 1 (18) We can move the whole sector an the given nonlinearity clockwise on the same angle as the angle between the ero an the line k 1. As a result, we will have a new nonlinearity an a new sector conition for it to simplify the following calculations. We will rewrite the close-loop system of Eq. (7) an Eq. (1) as follows ω 1 + ẋ = A cl + [B cl1 B cl ω + (19) [B cl1 B cl That gives us where ẋ = A cl [B cl1 B cl [ ˆω1 + [B cl1 B cl ˆω = [A cl 4 B cl1 C cl 4 B cl C cl [ ˆω1 + [B cl1 B cl ˆω Â cl = [A cl 4 B cl1 C cl 4 B cl C cl () (1) is new state matri of a form of Eq. (17) an ˆω 1, ˆω are new nonlinearities such that ˆω 1 = ω 1 +, ˆω = ω + () an satisfy the same conitions of Eqs. (11-1). The new quaratic constraints are ˆω υ (ˆω 1 ˆω ) (υ 1 υ ) () Secon, as presente in (Shiriaev 1), by choosing τ 1, τ or τ equal to ero we can remove one of the terms of Eq. (5). For a start, we have to introuce the following notations for new transfer functions in orer to avoi the presence of long epressions: Ĝ 11 (jω) = C cl1 (jωi n Âcl) 1 B cl1 Ĝ 1 (jω) = C cl1 (jωi n Âcl) 1 B cl Ĝ 1 (jω) = C cl (jωi n Âcl) 1 B cl1 Ĝ (jω) = C cl (jωi n Âcl) 1 B cl G 1 (jω) = Ĝ(jω) + Ĝ1(jω) G (jω) = Ĝ11(jω) Ĝ1(jω) G (jω) = Ĝ1(jω) Ĝ(jω) G 4 (jω) = G (jω) + G (jω) (4) We will use τ 1 = an τ = 1 to be able to reuce the inequality of Eq.(5) with the new nonlinearities of Eq. () to Re{ˆω ˆυ } τ Re{(ˆω 1 ˆω ) (ˆυ 1 ˆυ )} < (5) with ˆυ 1 = Ĝ11(jω)ˆω 1 + Ĝ1(jω)ˆω ˆυ = Ĝ1(jω)ˆω 1 + Ĝ(jω)ˆω (6) that hols ˆω 1 C, ˆω C, ω R. To present the inequality of Eq. (5) as the matri form we have to change the variables as follows: (1) from the first part of the inequality of Eq. (5) we have Re{ˆω ˆυ } = = Re{ˆω (Ĝ1(jω)ˆω 1 + G 1 (jω)ˆω Ĝ1(jω)ˆω )} = 1 {ˆω Ĝ1(jω)(ˆω 1 ˆω )} + 1 {(ˆω 1 ˆω ) Ĝ 1(jω)ˆω } + Re{ˆω G 1 (jω)ˆω } (7) 69

4 19th IFAC Worl Congress Cape Town, South Africa. August 4-9, 14 () from the secon part of the inequality of Eq. (5) we have τ Re{(ˆω 1 ˆω ) (ˆυ 1 ˆυ )} = = τ Re{(ˆω 1 ˆω ) (ˆω 1 (Ĝ11(jω) Ĝ1(jω)))} + τ Re{(ˆω 1 ˆω ) (ˆω (Ĝ1(jω) Ĝ(jω)))} = τ Re{(ˆω 1 ˆω ) G (jω)(ˆω 1 ˆω )} + 1 (ˆω 1 ˆω ) τ G 4 (jω)ˆω + 1 ˆω τ G 4 (jω)(ˆω 1 ˆω ) (8) Now we are able to rewrite the inequality of Eq. (5) in the matri form [ [ ˆω 1 ˆω Π(jω) 1 > (9) ˆω 1 ˆω ˆω 1 ˆω with Π(jω) = [ Re{G1 (jω)}.5g 5 (jω).5g 5 (jω) τ Re{G (jω)} () with G 1 (jω) = Ĝ(jω) + Ĝ1(jω) G 5 (jω) = Ĝ1(jω) + τ G 4 (jω) (1) an it shoul be vali for some τ >, ω 1, ω C an ω R. The inequality of Eq. (9) is positive if the matri Π(jω) of Eq. () is positive efinite. The constraint of Eq. (9) inclues the both stability conitions of Eq. (15) an Eq. (17)..1 Eample To show the benefit of the alternative proof we choose the same numerical values for the linear part for the controller (5) as in (Rubanova 1): Λ {u} = 19, Λ {u} = 7, Λ {} = 7, Λ{} = 6 () an the corresponing parameters of the nonlinear part of the controller are: t = , t = 1.449; () ω u = 1, ω =.97; The controller is thus given by u = 19 7 W (, ) t = W (, ) (4) where W (, ) = 1 ( ) (5) Now we are able to check the conitions of Eq. (15), (1): (1) The transfer functions are.51s G 1 (jω) = s +.671s G 1 (jω) = G (jω) = G 4 (jω) =.58s s s + 9.9s s s +.56s s s +.56s (6) φ u Time t [s Fig.. Simulation results of the close-loop system with the controller of Eq. (5) an the system of Eq. (). We alreay know that the inequality of Eq. (9) is positive if the matri Π(jω) of Eq. () is positive efinite. In our case we have a matri an it will be sufficiently to show that its eterminant an iagonal elements are positive. The eterminant of the matri of Eq. () is positive for τ >.64 (7) The higher tolerance for state elimination or pole-ero cancellation of transfer functions of Eq. (6) forces aitional cancellations. For this eample we use high tolerance to be able to show the calculations with the lower orer polynomials. For higher orer polynomials the erivation of the eterminant of the matri Π(jω) will be complicate, but we will get approimately the same numerical value for τ. Hence, the rough approimation was chosen in orer to simplify the metho presentation in this work. () The eigenvalues of the matri of Eq. (1) are [.168;.875; 7.87, hence it is Hurwit. Running the simulation moel with the controller of Eq. (5) shows that the close-loop system of Eq. (7) is stable. The result of the simulation is shown in Fig.. 4. THE LYAPUNOV FUNCTION SEARCH METHOD FOR THE KNOWN SUBSYSTEM In the MG compressor moel of Eq. (1) the eviation of the average flow φ is not available for the measurements. The controller parameters are chosen separately for ifferent subsystems of the close-loop system or Eq. (7). We refer to (Shiriaev 1) an (Rubanova 1) for a more etaile eplanation. Thereby, to simplify the calculations, we will fin the quaratic function for the known subsystems from the system of Eq. (7) [ ż [ = A 1 First, we choose the quaratic function as + B cl W (, ) (8) 69

5 19th IFAC Worl Congress Cape Town, South Africa. August 4-9, 14 V (, ) = 1 [ T [ P (9) with a matri P T = P > such that the time-erivative of V (, ) is non positive. The quaratic constraint of Eq. (1) is vali for the static nonlinearity W (, ) from the close-loop system with the subsystem of Eq. (8) an the controller of Eq. (5). Then the time-erivative of the quaratic function of Eq. (9) is t V (, ) = = 1 [ T [ A T 1 P + P A 1 1 [ T [ [ A T 1 P + P A 1 [ C cl W (, ) [ T [ C 4 cl T C cl = 1 [ T [ A 1 P + P A 1 [ CT cl C cl [ [ T + P B clw (, ) [ T + P B clw (, ) [ T [ + P Bcl C cl T W (, ) (4) where matrices B cl, C cl are the same as in Eq. (7). A sufficient conition of the eistence of the negative time erivative of the quaratic function of Eq. (9) can be written as A T 1 P + P A 1 CT cl C cl < (41) P B cl = Ccl T To fin the the matri P for the epression of Eq. (41) we solve the conve optimiation problem by using CVX a software tool for conve optimiation by (Boy an Vanenberghe 4). The specification for CVX looks like: cv_begin sp variable P1(,) symmetric minimie() subject to P>; A1 *P+P*A1-/* C *C< P*B==C ; cv_en For the eample in the subsection.1 we get [ P = (4) In Fig. the time erivative of the quaratic function of Eq. (9) is presente. The matri P of Eq. (4) is an approimate solution an it is epening on the chosen accuracy for the calculations. For eample, the presence of the equality in the epression of Eq. (41) is making this mathematical problem complicate. In aition the CVX solvers will be not able to fin the solution for the similar conition as in Eq. (41) but for the close-loop surge subsystem of Eq. (7). That is why for the higher orer calculations we suggest to use V/t Time t [s Fig.. The time erivative of the quaratic function of Eq. (9). the conition of Eq. (9) to investigate the robustness of the controller of Eq. (5). 5. CONCLUSIONS In control theory it is very important to investigate the robustness of the stabiliing controllers. We presente an etension of a previous research base on a proceure for ynamic output feeback esign for systems with nonlinearities satisfying quaratic constraints (Shiriaev 9), (Shiriaev 1), (Rubanova 1). The new constraint for the robustness investigation were presente as an inequality of Eq. (9). The coefficient τ belongs to the conition for the nonlinearities of Eq. (1) as shown in the inequality of Eq. (15). It is possible to fin some positive τ for the controllers of Eq. (5). In other wors, the controllers have a certain egree of robustness an there is a possibility to resist some of the parametric uncertainty an measurement noise. The alternative proof of stability of the close-loop system of Eq. (7) uses less number of conitions, then in (Rubanova 1). The results are applicable for ynamic output feeback esign for systems with nonlinearities satisfying quaratic constraints. Also, there is a possibility to inclue a new quaratic constraint in the analysis. The inequality of Eq. (9) is a more general stability conition than presente in the previous part of the research in (Rubanova 1). The matri Π(jω) of Eq. (9) is shown. ACKNOWLEDGEMENTS This work was supporte by the Sweish Research Council through the project The project Active Control of Compressor Systems Base on New Methos of Nonlinear Dynamic Feeback Stabiliation eals with a number of facts relate to the output feeback stabiliation of the Moore-Greiter compressor moel. The authors from Lun University are members of the LCCC Linnaeus Center an the elliit Ecellence Center at Lun University. 694

6 19th IFAC Worl Congress Cape Town, South Africa. August 4-9, 14 REFERENCES Boy, S. an Vanenberghe, L. (4). Conve Optimiation. Cambrige University Press, UK. Greiter, E. (1976). Surge an rotating stall in aial flow compressors, part I/II. ASME J. Engineering for Gas Turbines an Power, 98, Khalil, H. (). Nonlinear Systems (r e.). Prentice Hall, Upper Sale River, New Jersey Moore, F. an Greiter, E. (1986). A theory of poststall transients in aial compressor systems: Part I - Development of equations. ASME J. Engineering for Gas Turbines an Power, 18, Ng, E., Ng, Y., an Liu, N. (7). Compressor Instability with Integral Methos. Springer, Berlin Heielberg. Pauano, J., Greiter, E., an Epstein, A. (1). Compression system stability an active control. Annual Review of Flui Mechanics,, oi: /annurev.flui Planovsky, A. an Nikolaev, P. (199). Unit Operations an Equipment of Chemical Engineering (1st e.). Mir Publishers, Moscow. (in English). Rubanova, A., Robertsson, A., Shiriaev, A., Freiovich, L., an Johansson, R. (1). Analytic parameteriation of stabiliing controllers for the surge subsystem of the Moore-Greiter compressor moel. In American Control Conference (ACC1), Washington, D.C., USA. Shiriaev, A., Freiovich, L., Johansson, R., an Robertsson, A. (1). Global stabiliation for a class of couple nonlinear systems with application to active surge control. Dynamics of Continuous, Discrete an Impulsive Systems, Series B, Applications an Algorithms, 17(6b), Shiriaev, A., Johansson, R., Robertsson, A., an Freiovich (9). Criteria for global stability of couple systems with application to robust output feeback esign for active surge control. In IEEE Multi-Conference on Control Applications (MSC9), Saint Petersburg, Russia. Veernikov, M. (1974). Compressor an Pump Installations in Chemical, Petrochemical an Refining Inustries (r e.). Vysshaya Shkola, Moscow. (in Russian). Yakubovich, V., Leonov, G., an Gelig, A. (4). Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. Worl Scientific Publishing Co, Singapore. 695