Design of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces

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1 Downloaded from orbit.dtu.dk on: Nov 9, 218 Design of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces Lauridsen, Jonas Skjødt; Santos, Ilmar Published in: Mechanical Engineering Journal Link to article, DOI: /mej Publication date: 217 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Lauridsen, J. S., & Santos, I. (217). Design of Robust AMB Controllers for Rotors Subjected to Varying and Uncertain Seal Forces. Mechanical Engineering Journal, 4(5), [618 ]. DOI: /mej General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Bulletin of the JSME Mechanical Engineering Journal Vol.4, No.5, 217 Design of robust AMB controllers for rotors subjected to varying and uncertain seal forces Dept. of Mechanical Eng., Technical University of Denmark Copenhagen, Denmark Abstract H µ Keywords Received: 16 November 216; Revised: 19 April 217; Accepted: 8 May Introduction H Paper No J-STAGE Advance Publication date: 17 May, 217 [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 1

3 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) H H µ H H µ KS T µ µ [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 2

4 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) Impeller seals Displacement sensors Axial AMB Radial AMB Balance piston hole-pattern seal A B µ H 2. Modelling of the Turboexpander A B x y 2.1. AMB Model s x i x f b (i x, s x ) = K i i x + K s s x K i K s 3 db ω c 1.5 khz G act [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 23

5 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) G act = K i 1 + s ω c 2.2. Model of Shaft x y x y G f ẋ f = A f x f + B f u, y = C f x f 1 khz x = T L T x f, A = T L T A f T R, B = T L T B f, C = C f T R 2.3. Seal Model - CFD vs Bulkflow f x = K k x + C c ẋ + M ẍ k K y c C ẏ M ÿ f y 2 3 Hz 3. Robust Control Design µ H 3.1. Control Design Objectives and Challenges ±4 % [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 24

6 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) Parameter Seal Length [mm] Rotor Diameter [mm] Inlet Clearance [mm] Exit Clearance [mm] Hole Depth [mm] Hole Diameter [mm] Hole Area Ratio Rotor Speed [rpm] Inlet Pressure [bar] Outlet Pressure [bar] Res. Temperature [C] Preswirl Fig. 2 Value Hole-pattern seal specification and parameters (left) & fluid structure (right). From (Nielsen et al., 212) Fig. 3 Hole-pattern seal direct and cross coupled stiffness coefficients obtained using CFD, Experiment and ISOTSEAL. Figures adapted from (Nielsen et al., 212) 8 coefficients are considered uncertain independent to each other, i.e. although, for example, the direct coupled stiffness K has the same nominal value in both vertical and horizontal directions they have independent uncertainties. Mass coefficients are neglected since the fluid is air (Nielsen et al., 212). The system should be robust against other unmodelled dynamics and against system changes over time due to wear and ageing. These robustness criteria are specified in ISO , which states that the closed loop sensitivity (disturbance to error) should be less than 3 for all frequencies in order to be classified as Zone A (ISO , 26). Unbalance response should be less than 1 µm for the complete operating range, assuming the shaft is balanced according to the G2.5 standard. The control currents should stay well within the actuation limits of ±5 A. Settling time should be less than 2 ms for step disturbances on input (force) for good force disturbance rejection Uncertainty Representation The nominal rotordynamic model consists of the reduced order shaft model, the negative stiffness from the AMBs and the nominal stiffness and damping from the seals. The perturbation model G f i is constructed using the nominal model and the uncertainty representation, which are combined and written in LFT form, illustrated in Fig. 5. Here is a 8 8 diagonal matrix representing the normalized uncertainties and satisfy 1. G f i can be written in state space form, as shown in Eq. (6), where A, B and C are the nominal system matrices. Here the input and output matrices are extended from the nominal model to include the input and output mapping B and C. Note that no extra system dynamics is added since the LFT only changes the [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 25

7 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) Fig. 4 Hole-pattern seal direct and cross coupled damping coefficients obtained using CFD, Experiment and ISOTSEAL. Figure adapted from (Nielsen et al., 212) w u Fig. 5 z Gfi y ( ) Uncertain plant representation using upper LFT, Gunc = Fu G f i, nominal system matrix A. A G f i = C C B B (6) B and C are constructed as follows and a thorough description of this process can be seen in (Lauridsen et al., 215). It can be shown that changes in stiffness (or damping) in a single direction at e.g. A x corresponds to a change in a single column of system matrix A, which corresponds to the node j where the stiffness (or damping) is altered.... a1, j a2, j... A f =.. (7) ai, j... The change of the system matrix in reduced form A is found using the same modal truncation matrices as used to reduce the nominal system, as shown in Eq. (8). It is noted that the applicability of using the same modal truncation matrices to reduce the matrix representing the change in system dynamics as were used for reducing the nominal system matrix is based on assumption rather than proof, however, this assumption has been shown to hold well in practice. A f in Eq. (7) can also be written as a column vector B f and a row vector C f and the change/uncertainty. The input mapping B and output mapping C of the uncertainties are thus given as shown in Eq. (1). Repeating this process 8 times (one for each stiffness and damping parameter) and assembling the coloums of B and rows of C and making an 8 8 diagonal matrix, yields the complete uncertainty representation. A = T L A f T R = T L B f C f T R = B C [DOI: /mej ] (8) (9) (1) 217 The Japan Society of Mechanical Engineers 26

8 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) ^ - w 2 + G act w z Gfi y u K w W P e W u z P z u w u P P K z e G act G f i W P W u ˆ P 3.3. Robust Control Design Interconnection and Weight Functions W p W 1 W 2 e W p W p = s M + w B s + w B A W p A 1 1 M 9.5 db w B 2 ms W u 1.4 khz W u 3.4. Robust Control Synthesis P H ˆ γ = F l (P, K) γ γ 1 γ H H γ Reduce Conservatism by D-scaling H ˆ D w z D D 1 D µ 3.5. Results S i S o ±4 % H [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 27

9 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) 2 Sensitivity using H controller 2 Sensitivity using µ controller Singular Values (db) Singular Values (db) S o S i γ -8 S o S i γ Frequency (rad/s) wp Frequency (rad/s) wp H µ 5 Control Sensitivity using H controller 4 Control Sensitivity using µ controller Singular Values (db) -5-1 KS o KS i γ Frequency (rad/s) wu Singular Values (db) KS o KS i γ Frequency (rad/s) wu H µ µ 1 db H KS i KS o ±4 % H µ A x A y B x B y 1 N 2 ms A x ±4 % µ H x y Worst Case Unbalance Response G f ( ) G f ( ) = F u ( G f i, ) S i F u (Ω) [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 28

10 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) H controller µ controller 2 2 Displacement (m) 1-1 Displacement (m) A x A y B x B y Time (s) -2-3 A x A y B x B y Time (s) H µ.6 H controller.6 µ controller.4.4 Current (A).2 Current (A) A x A y B x B y Time (s) A x A y B x B y Time (s) H µ y max y max = σ(g f ( ))F u (Ω) Ω Ω F u (Ω) G2.5 Ω 2 3 Hz µ 4 µm ±4 % 4. LPV Control Design G f i 2 3 Hz 2 3 Hz Results of Spin-up Test - LPV vs µ Controller µ 2 3 Hz 1 s µ 14 Hz ±4 % 5 ms [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 29

11 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) 1 6 Spinup - µ vs. LPV controller LP V Ax LP V Ay LP V Bx LP V By µax µay µbx µby 2 Displacement (m) Spinup - µ vs. LPV controller 1 6 LP V Ax LP V Ay LP V Bx LP V By µax µay µbx µby 2 Displacement (m) Time (s) Fig Time (s) Spin-up test time response comparison, using LPV and µ controllers. A force input disturbance of 1 N enters after s and.8 s and a displacement output disturbance of 2 µm enters after.5 s and.85 s, both at A x. The µ controller is designed for operation at 14 Hz rotational speed. alternating between acting on the input and output signals. A force input disturbance of 1 N enters after s and a displacement output disturbance of 2 µm enters after 5 ms, both at A x. The µ controller performs, not surprisingly, best near its design operational point at 14 Hz and worse when operating away from it. Worst case performance of the µ controller is illustrated in Fig. 11 (left) for low operational speeds in the beginning of the spinup and in Fig. 11 (right) for high operational speeds. It is observed, that for these operational ranges, the LPV controller has a faster settling time, below 2 ms as required, where the µ controller takes longer time. Also, the response is more oscillatory in the case of the µ synthesized controller compared to the LPV controller. Fig. 12 shows S o using LPV and µ controllers for plant variations due to plant changes in the operational speed range of 2-3 Hz. The plot confirms worse performance of the µ synthesized controller, i.e: I) Oscillatory behaviour due to higher peak of S o. II) Slower disturbance rejection due to lower crossover frequency. 11 Sensitivity - µ vs. LPV controller Singular Values (-) So LP V So µ Limit Frequency (rad/s) Fig. 12 Closed loop output sensitivity using LPV and µ controllers for plant variations due to change in operational speed in the range 2-3 Hz. The yellow line indicates the ISO upper limit. 5. Conclusion Robust control design is suggested for handling uncertain seal forces in AMB systems. Significant performance improvement is shown for robust control, incorporating model uncertainty, compared to nominal model based control. This clearly demonstrates the need for incorporating uncertainties into the model based con[doi: /mej ] 217 The Japan Society of Mechanical Engineers 1 2

12 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) µ µ References [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 11 2

13 Lauridsen and Santos, Mechanical Engineering Journal, Vol.4, No.5 (217) [DOI: /mej ] 217 The Japan Society of Mechanical Engineers 12 2