An optimal state feedback control method for 4 degrees of freedom - Rigid rotor active magnetic bearing system - Tran Xuan Minh

Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 155 AN OPTIMAL STATE FEEDBACK CONTROL METHOD FOR 4 DEGREES OF FREEDOM - RIGID ROTOR ACTIVE MAGNETIC BEARING SYSTEM Tran Xuan Minh* Thai Nguyen University of Technology SUMMARY Based on mechanical – electrical – magnetic principles, the paper presents detailed analyses to build a completed mathematical model for 4 degree of freedom - rigid rotor active magnetic bearing (AMB) system. Gyroscopic effect, one of significant reasons affecting to performances of system is mentioned in this research. By using the centralized approach, a state-space model for multi-input multi-output (MIMO) active magnetic bearing system is built. An optimal state feedback controller is then designed in order to directly formulate the performance objectives of the control system and provides the best possible control system for a given set of performance objectives. Zero steady-state error of system outputs is also given by the means of integrators which are added into the system. As a result, MIMO system’s responses achieve quick stabilization and good performances. Keywords: Active Magnetic Bearing (AMB); gyroscopic; MIMO; state-space; Linear Quadratic Regulator (LQR) INTRODUCTION* Active Magnetic Bearing (AMB) comprises a set of electromagnetic mechanisms to provide bearing forces which suspend rotor shaft freely in space. These systems utilize feedback control methods to stabilize the rotating motion of them. This advanced bearing technology offers many significant advantages, compared to conventional bearings, since mechanical non-contact between rotor shaft and static parts is generated by electromagnets. With a suitable active control approach, damping and bearing stiffness characteristics of AMB can be adjusted [1, 2]. Control methods contribute an important role in designing an AMB system. In many applications, however, the performance of a controller is highly influenced by the coupled impact in motion of the system which should not be neglected. Many different control methods have been applied successfully for AMB, with or without the mention of the gyroscopic effect [4-9]. These include conventional decentralized approaches such as PD, PID and nonlinear control methods such as * Tel: 0913 354975 feedback linearization, backstepping [4, 5], [7], [9]. A new trend for modern control methods is also attracted many interests. These centralized methods consisting of Pole- placement, LQR, LQG, H∞, μ-synthesis [6, 7], [9] increase quickly due to the rapid development of the sensor technology and digital signal processing recently. As a result, measurement and computation tasks of various physical signals can be implemented easily for the purpose of feedback control. In this research work, a fully mathematical model of 4-DOF AMB is described, in which the gyroscopic effect is also included in the system dynamics. A modern centralized control method is designed for a MIMO radial suspension system. By using this approach, the optimal controller is then proposed in order to yield high performance for the system in terms of control energy and control error. Obtained results show that an improvement in dynamic performance of the system can be achieved. This paper is structured in four parts. Part 2 dedicates to modeling of the system in terms of dynamics and electromagnetic issues. The control design is described in part 3. Part 4 is the computation and simulation results. Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 156 MODELING OF 4-DOF AMB SYSTEM Dynamic equations of rotor The object of this research work is a drive system including 2 active magnetic bearings arranged in both ends of the shaft. The AMB1 and AMB2 provide radial forces in the directions of xA, yA and xB, yB respectively. Figure 1. 4-DOF rigid rotor AMB system [5] Based on dynamic principles of a rotating object in literature [1, 2, 3, 4, 5, 6], equations of motion for rotating shaft can be recognized by Newton’s second law and Euler’s second law of motion. This arrangement is assigned 4 physical degrees of freedom, describing translations in the x- and y- directions and angular displacements about the xz (β) and yz (α) planes. This contributes to elements for structural matrices of the mechanical system which enable the Lagrange representation of the second order dynamical system. fMq Gq Bu F (1) where, T s sx yq Figure 2. The rigid rotor provided with magnetic bearings and sensors[2] 0 0 0 0 0 0 0 0 0 0 0 0 i i J m J m M ; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 k rm k rm J J G The presentation of the generalized force F: 1 2 3 4 f F F F F F Bu ; 0 0 1 0 0 1 1 0 0 1 0 0 ; 0 0 0 0 1 0 0 1 1 0 0 1 a b c d a b c d B C The mass matrix is symmetric and positive definite, T= 0M M ; y xJ J because the rotor is assumed to be symmetric; the gyroscopic matrix is skew-symmetric, T= -G G containing the rotor speed rm as a linear factor; Radial bearing forces are represented by four controlled forces, which act within the bearing planes in the x- and y- directions. T f xA xB yA yAf f f fu Each bearing force is be described as a linearized function of the rotor displacement in the rotor and the coil current. Hence, the following relationship results for the force vector fu can be shown as follows [2]: f s b iu K q K i (2) where, _ _ _ _( , , , );s s xA s xB s yA s yB T b bA bB bA bB diag K K K K x x y y K q sK is bearing stiffness vector; i : is vector of control current of four bearing coils; ( , , , ); T i iA iB iA iB xA xB yA yAdiag K K K K i i i iK i iK : is force/current vector; bq expresses the rotor displacements within magnetic bearings. For the purpose of simple differential equation description, the center of gravity (COG) coordinates, combined in q , have been introduced. However, rotor displacements involve the bearing coordinates bq . It therefore is necessary to transfer the bearing coordinates bq into the COG coordinates q through a linear transformation matrix b ST : Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 157 1 0 0 1 0 0 0 0 1 0 0 1 T b bA bB bA bB b s a b x x x y y a b y q q T q (3) By inserting of (2) and (3) into (1), the complete equation of motion can be obtained as follows: + + =s iSMq Gq K q BK i (4) where, s sS b SK BK T is the negative bearing stiffness matrix converted into COG coordinates. In general, displacement motions ,s sx y and angular motions , will be coupled due to the presence of the gyroscopic effect in matrix G if rotor speed 0rm . Electromagnetic equations of magnetic bearing model Assuming that in the air gaps and iron paths, the magnetic flux and flux density are constant. Moreover, the iron will be treated as operating below saturation then stored magnetic energy Wa, and magnetic flux in the paths Φa can be calculated by [4]: 2 2 a 0 1 W 2 2 2 fea iron a a a iron fe lB l B dV s A (4) where, Va = 2sAa; Aa is cross-section area of flux path. It is assumed that flux leakage is not existed, Φ = Φa = Φfe 0 0 0 0 2 1 2 a fe feiron iron iron fe a iron fe A NiNi l ll ls s A (5) When the magnetic flux in the air gaps is unity then making: B = Ba= Φ/Aa; If rotor is displaced by an amount of Δs then a magnetic force F is generated [1, 2], [4]: 2 2 a 0 2 W F 2 a feiron iron fe A N i s ll s (6) The relationship between magnetic force and current in (6) is represented by square term which indicates the nonlinearity. feiron iron fe ll is neglected since the equation above does not take into account magnetization effect of ferromagnetic materials. As a result, (6) becomes: 2 2 a 0 0 W F 4 aA N i s s s (7) Figure 3. An differential arrangement for electromagnets in x-axis Two functions for magnetic forces will be given by: 2 0 0 i i F K s s and 2 0 0 i i F K s s The total magnetic force F will be the difference between F+ and F-; 2 0 4 aA NK Under the assumption that 0s s and 0i i the linear function of F can be approximated by the first order of Taylor- expansion as follows [1, 2]: 2 0 0 3 2 0 0 4 4 s i Ki Ki F s i K s K i s s (8) State-space description The rotor dynamic equations and the linearized electromagnetic equations constitute a set of equations describing dynamic behaviour of the system. We introduce a state vector presenting for bearing displacements and their derivatives: T x y x yx the input vector: T xA xB yA yBi i i iu and, the output vector at the sensors of bearings: T c d c dx x y yy Then, the state-space description of the 4- DOF AMB system yields: Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 158 4 4 4 4 1 1 4 4 4 4 4 4 4 41 ; ; ; s i + b S 0 Ix Ax Bu A M BK T M Gy Cx Du 0 B C C 0 D 0 M BK (9) Controller design Linear time-invariant state equations of proposed system can be expressed in generalized form: +x Ax Bu y Cx (10) with ; , ; ; ;n m n n n m m nR R R R Rx u y A B C This controlling design seek to minimize the quadratic performance index: 1 0 T T t t J dtx Qx u Ru (11) where, Q is state weighting matrix and positive semi-definite; R is control weighting matrix and positive definite; x is the state vector. Figure 4. LQR closed-loop block diagram In the LQR design approach, the only design parameters are the weighting matrices, Q and R. A LQR controller is constructed under the assumption that all states of the controlled system are available for feedback. Once the LQG controller is obtained, the dynamic behaviour of each controlled variable can be checked and the closed-loop poles can be evaluated [10]. Optimal control law denoting by optu can be represented as: -1 Topt = -u R B Px or opt = -u Kx (12) where, -1 T=K R B P is optimal feedback gain matrix. Should 1t in (11) be infinite, the matrix Riccati equations reduce to a set of simultaneous equations, where P is unique positive-definite solution to the following equation: T -1 T+ + - =PA A P Q PBR B P 0 (13) Therefore, if we want to modify the system behaviour, the weighting matrices Q and R can be adjusted to obtain different optimal feedback gain matrix K. A reasonable simple choice for the matrices Q and R is given by Bryson’s rule [9]. Then Q and R are selected diagonally with: 2 2 1 ; max acceptable value of 1 max acceptable value of 1,2,..., ; 1,2,..., ii i ii i Q x R u i j i l which corresponds to the following criterion: 2 2 1 10 ( ) ( ) l k ii i jj j i j J Q x t R u t dt (14) In this paper, we introduce an integral action based on the tracking error in order to achieve zero steady-state error under any changes in references. A number of integrators are added in the system model then the state-space model of the new augmented system cab be written as: 0 LQR Ix x r x y C A BK BK 0 C 0 I (15) where the additional auxiliary state variables ε act as “accumulated errors”. SIMULATION RESULTS In this section, AMB’s simulating parameters are taken from [7] in order to prove the effectiveness of the method: Rotor mass: m=12.4(kg); Distance between mass center and bearings/sensors: la= lb= 0.21(m); lc = ld = 0.215(m); Momen of inertia in i, j and k axes: Ji = Jj = 2.22x10-1(kg.m2) and Jk=6.88x10-3 (kg.m2). Speed of rotor ωrm = 15000(RPM). Ratio of magnetic force/current: Ki = 102.325 (N/A); Bearing Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 159 stiffness coefficient: Ks = -4.65x105(N/m); Earth’s gravity acceleration: g=9.81(kg.m/s2). The computation results of the optimal feedback gain matrix K in equation (12) can be obtained with given weighting matrices Q and R and Riccati equation solution P. 8 8 8 8 T 11 11 10 11 11 10 0.026 0.026 0.0089 0.0089 0.0055 0.0055 8.7 10 2.7 10 0.0089 0.0089 0.026 0.026 1.85 10 1.66 10 0.0055 0.0055 7.071 7.071 5.22 10 5.56 10 7.071 7.071 1.99 10 6.66 10 5.95 10 4.82 1 1.28 10 K 11 10 7.071 7.0710 7.071 7.0711.52 10 With the optimal feedback gain matrix K above, eigenvalues of closed-loop system can be achieved: -1242.1+517.7i; -1242.1+517.7i; -126.75+52.8i; -126.75-52.8i; -58.35+267.6i; -58.35-267.6i; -58.35+267.6i; -58.35-267.6i It is can be seen that all the closed-loop eigenvalues have negative real parts indicating that the closed-loop is asymptotically stable. We see in the Figure 5 and 6 that the LQR design exhibits excellent regulation performance. The zeros response for LQG design decays to zeros quickly with not much oscillation. The results indicate that the proposed optimal controller is superior in terms of both regulation performance and control efforts. The figure 7 shows that the performance of LQR method is much better than pole- placement method when the same step inputs are applied to. With the addition of integrators in the LQR model, steady-state errors are improved significantly. Moreover, in contrast to the pole-placement method, where the desired performance is indirectly achieved through location of closed-loop poles, the optimal control system directly addresses the desired performance objectives while minimizing the control energy. Figures 8 and 9 indicate the significant difference in terms of control efforts. Figure 5. State responses of 4 DOF AMB using LQR + integrators Figure 6. Output responses of 4 DOF AMB using LQR + integrators Figure 8. Control efforts of 4 DOF AMB using Pole-placement method Figure 9. Control efforts of 4 DOF AMB using LQ controller + integrators Figure 7. Comparisons of output responses between different methods Trần Xuân Minh Tạp chí KHOA HỌC & CÔNG NGHỆ 122(08): 155 - 160 160 CONCLUSION In this paper, a detailed mathematical model of a 4 DOF AMB system has been constituted in which the gyroscopic effect is also mentioned. The proposed LQ regulator has performs excellent qualities of both regulation performance and control effort. This centralized control approach is successfully constructed under the assumption that all states can be measured. Moreover, as integrator parts have been added to LQR model, steady-state errors are reduced significantly in dynamic output responses of the system. Therefore, this design can be a good reference for an alternative realistic designs, such as optimal output feedback design and observer design, since the issues of unmeasurable physical quantities are considered. Those unsolved problems are wished to be investigated in other research works. REFERENCES 1. Akira Chiba, Tadashi Fukao, Osamu Ichikawa, Masahide Oshima, Masatsugu Takemoto and David G. Dorrell, Magnetic Bearings and Bearingless Drives. Newnes, 2005. 2. Gerhard Schweitzer and Eric H. Maslen, Magnetic Bearings: Theory, Design, and Application to Rotating Machinery. Springer- Verlag, 2009. 3. R. D. Smith and W. F. Weldon, “Nonlinear control of a rigid rotor magnetic bearing system: Modeling and simulation with full-state feedback”, IEEE Transactions on Magnetics, Vol. 31, No. 2, 1995. 4. M. S. de Queiroz and D. M. Dawson, “Nonlinear control of Active Magnetic Bearing: A backstepping approach”, IEEE Transactions on Control Systems Technology, Vol. 4, No. 5, March 1996. 5. Abdul R. Husain, Mohamad N. Ahmad and Abdul H. M. Yatim, “Deterministic models of a Active Magnetic Bearing System”, Journal of Computers, Vol. 2, No. 8, October 2007. 6. Tian Ye, Sun Yanhua, Yu Lie, “LQG Control of Hybrid Foil-Magnetic Bearing”, 12th International Symposium on Magnetic Bearings, 2010. 7. Chunsheng Wei, Dirk Soffker, “MIMO-control of a Flexible Rotor with Active Magnetic Bearing”, 12th International Symposium on Magnetic Bearings, 2010. 8. Quang Dich Nguyen, Nobukazu Shimai, Satoshi Ueno, “Control of 6 Degrees of Freedom Salient Axial-Gap Self-Bearing Motor”, 12th International Symposium on Magnetic Bearings, August, 2010. 9. Luc Quan Tran, Xuan Minh Tran, “Design a state feedback controller with Luenberger observer for 4 degree of freedom - rigid rotor active magnetic bearing system”, Proceedings of the First Vietnam Conference on Control and Automation, November, 2011 (in Vietnamese). 10. Robert L. Williams II, Douglas A. Lawrence, Linear State-Space Control Systems, John Wiley & Sons, 2007 TÓM TẮT MỘT PHƯƠNG PHÁP ĐIỀU KHIỂN TỐI ƯU PHẢN HỒI TRẠNG THÁI CHO HỆ THỐNG Ổ ĐỠ TỪ CHỦ ĐỘNG 4 BẬC TỰ DO ROTOR CỨNG Trần Xuân Minh* Trường Đại học Kỹ thuật Công nghiệp – ĐH Thái Nguyên Dựa trên các nguyên lý về cơ- điện- từ, bài báo trình bày những phân tích chi tiết để xây dựng nên một mô hình toán học đầy đủ cho hệ thống ổ đỡ từ chủ động (Active Magnetic Bearing-AMB), rotor cứng, 4 bậc tự do. Ảnh hưởng hồi chuyển là một trong những nguyên nhân chính ảnh hưởng đến chất lượng làm việc của hệ thống cũng được đề cập đến trong nghiên cứu này. Bằng cách sử dụng giải pháp điều khiển tập trung, một mô hình không gian trạng thái cho hệ nhiều đầu vào, nhiều đầu ra (MIMO) được xây dựng cho hệ thống ổ đỡ từ chủ động. Một bộ điều khiển phản hồi trạng thái được thiết kế nhằm trực tiếp đưa ra các tiêu chí chất lượng của hệ thống điều khiển và tạo thành một hệ điều khiển tốt nhất có thể ứng với các tiêu chí chất lượng làm việc cho trước. Sai lệch tĩnh của tín hiệu đầu ra cũng được cải thiện thông qua các bộ tích phân được bổ sung vào mô hình điều khiển của hệ thống. Các kết quả mô phỏng cho ứng của hệ MIMO đạt được độ ổn định hóa nhanh chóng và chất lượng làm việc tốt. Từ khóa: Ổ đỡ từ chủ động (AMB); ảnh hưởng hồi chuyển; MIMO;không gian trạng thái; LQR. Ngày nhận bài:09/6/2014; Ngày phản biện:11/8/2014; Ngày duyệt đăng: 25/8/2014 Phản biện khoa học: PGS.TS Lại Khắc Lãi – Đại học Thái Nguyên * Tel: 0913 354975