Department of Electrical Engineering Southern Taiwan University Robust Nonlinear Speed Control of PM Synchronous Motor Using Boundary Layer Integral Sliding Mode Control Technique Student: Jia-Je Tsai Adviser: Ming-Shyan Wang Date : 31th-Dec-2008 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 1, 47~54, JANUARY 2000 In-Cheol Baik, Kyeong-Hwa Kim, Associate Member, IEEE, and Myung-Joong Youn, Senior Member, IEEE

Department of Electrical Engineering Southern Taiwan University 2 Outline Abstract I. INTRODUCTION II. NONLINEAR SPEED CONTROL OF PMSM USING INPUT–OUTPUT LINEARIZATION Modeling of PMSM Nonlinear Speed Control of PMSM Using Input–Output Linearization Asymptotic Load Torque Observer III. QUASI-LINEARIZED AND DECOUPLED MODEL AND PROPOSED CONTROL STRATEGY Quasi-linearized and Decoupled Model Proposed Control Strategy for the Quasi-linearized and Decoupled Model IV. EXPERIMENTAL RESULTS V. CONCLUSION REFERENCES

Department of Electrical Engineering Southern Taiwan University 3 Abstract A digital signal processor (DSP)-based robust nonlinear speed control of a permanent magnet synchronous motor (PMSM) is presented. A quasi-linearized and decoupled model including the influence of parameter variations and speed measurement error on the input–output feedback linearization of a PMSM is derived. Based on this model, a boundary layer integral sliding mode controller is designed and compared to a feedback linearization-based controller that uses proportional plus derivative (PD) controller in the outer loop. To show the validity of the proposed control scheme, DSP-based experimental works are carried out and compared with the conventional control scheme.

Department of Electrical Engineering Southern Taiwan University 4 INTRODUCTION INTRODUCTION Permanent magnet synchronous motor (PMSM) drives are being increasingly used in a wide range of applications due to their high power density, large torque to inertia ratio, and high efficiency. This paper deals with the nonlinear speed control of a surface mounted permanent magnet synchronous motor with sinusoidal flux distribution. However, this approximate linearization leads to the lack of torque due to the incomplete current control during the speed transient and reduces the control performance in some applications such as industrial robots and machine tools.

Department of Electrical Engineering Southern Taiwan University 5 INTRODUCTION INTRODUCTION The nonlinear control method, called a feedback linearization technique, is applied to obtain a linearized and decoupled model and the linear design technique is employed to complete the control design. The feedback linearization deals with the technique of transforming the original system model into an equivalent model of a simpler form, and then employs the well-known and powerful linear design technique to complete the control design. In this paper, a quasi-linearized and decoupled model including the influence of parameter variations and speed measurement error on the nonlinear speed control of a PMSM is first derived and then the Robust control scheme employing a boundary layer integral sliding mode is designed to improve the control performance.

Department of Electrical Engineering Southern Taiwan University 6 Modeling of PMSM The machine considered is a surface mounted PMSM and the nonlinear state equation in the synchronous d-q reference frame can be represented as follows: where (1) (5) (4) (3)(2)

Department of Electrical Engineering Southern Taiwan University 7 Nonlinear Speed Control of PMSM Using Input–Output Linearization From (1) and the assumption that the load torque is constant, the relationship between the outputs and inputs of the model can be obtained as follows : Where The nonlinear control input which permits a linearized and decoupled behavior is deduced from this relationship as follows: (6) (7) (8)

Department of Electrical Engineering Southern Taiwan University 8 Nonlinear Speed Control of PMSM Using Input–Output Linearization Where V 1 and V 2 are the new control inputs. By substituting (8) into (6), the linearized and decoupled model can be given as As the control laws for the new control inputs, the linear controller employed by Le Pioufle becomes as follows: (9) (10) (11) (12)

Department of Electrical Engineering Southern Taiwan University 9 Nonlinear Speed Control of PMSM Using Input–Output Linearization The following error dynamics can be obtained as where (13) (14)

Department of Electrical Engineering Southern Taiwan University 10 Nonlinear Speed Control of PMSM Using Input–Output Linearization For the control schemes employed in this paper, an information on the acceleration (dΩ/dt) is needed for the state feedback and can be calculated from (1). The 0-observer is derived under the assumption that the time variation of the unknown and inaccessible input is zero.

Department of Electrical Engineering Southern Taiwan University 11 Asymptotic Load Torque Observer the inaccessible load torque can be assumed as an unknown constant. For a PMSM, the system equation for a disturbance torque observer can be expressed as follows: Where For this system,(D,C) is observable. The well-known asymptotic load torque observer can be designed as (15) (16)

Department of Electrical Engineering Southern Taiwan University 12 Quasi-linearized and Decoupled Model The actual nonlinear control input which employs the nominal parameter values and measured mechanical speed is expressed as follows: By substituting (17) into (6), a quasi-linearized and decoupled model can be obtained as follows: (17) (19) (18)

Department of Electrical Engineering Southern Taiwan University 13 Quasi-linearized and Decoupled Model The unwanted nonlinear terms, and, are not exactly known but can be estimated as and, and the estimation errors are bounded by some known functions, and The control input gain b is also unknown but its bound can be deduced. the feedback linearization technique is considered as a model- simplifying device for the robust control, and the control laws for the new control inputs and are derived using a boundary layer integral sliding mode control technique to overcome the drawbacks of the conventional nonlinear control scheme.

Department of Electrical Engineering Southern Taiwan University 14 Proposed Control Strategy for the Quasi-linearized and Decoupled Model Assume the bounds of parameter variations and speed measurement error as follows: Obtain the minimum value for by calculating the minimum value of each term of using (20) and summing up each term. Obtain the maximum value for by calculating the maximum value of each term of using (20) and summing up each term. (20)

Department of Electrical Engineering Southern Taiwan University 15 Proposed Control Strategy for the Quasi-linearized and Decoupled Model Obtain the estimate by calculating the average value of minimum and maximum values. Obtain the estimation error bound by calculating the absolute difference between minimum(or maximum) value and. Repeat Steps using (20) to obtain the estimate and the estimation error bound for. The bound on the control input gain is (21)

Department of Electrical Engineering Southern Taiwan University 16 Proposed Control Strategy for the Quasi-linearized and Decoupled Model The boundary layer integral sliding mode controller is considered to avoid the chattering phenomenon and the reaching phase problem The sliding surface is chosen for the input–output decoupled form of (18) which is second-order relative to as follows: The control law for is designed to guarantee as where (22) (23)

Department of Electrical Engineering Southern Taiwan University 17 Proposed Control Strategy for the Quasi-linearized and Decoupled Model In (23), sat(.) is the saturation function described as From the bound on the control input gain b of (21), the geometric mean b m can be defined as (24) (25)

Department of Electrical Engineering Southern Taiwan University 18 Proposed Control Strategy for the Quasi-linearized and Decoupled Model The bound on b can then be written as where (26)

Department of Electrical Engineering Southern Taiwan University 19 Proposed Control Strategy for the Quasi-linearized and Decoupled Model The sliding surface is chosen for the input-output decoupled form of (19) which is third-order relative to as follows: The control law for is designed to guarantee as where (27) (28)

Department of Electrical Engineering Southern Taiwan University 20 System Configuration TABLE I SPECIFICATIONS OF PMSM

Department of Electrical Engineering Southern Taiwan University 21 System Configuration Fig. 3. Configuration of the DSP-based experimental system

Department of Electrical Engineering Southern Taiwan University 22 Nonlinear Speed Control of PMSM Using Input–Output Linearization Fig. 1. Block diagram of the conventional nonlinear control scheme

Department of Electrical Engineering Southern Taiwan University 23 Fig. 2. Block diagram of the proposed robust nonlinear control scheme

Department of Electrical Engineering Southern Taiwan University 24 EXPERIMENTAL RESULTS The design parameters used for the conventional nonlinear control scheme are selected as K 11 =2700, K 21 =900, and K 22 = For the proposed robust nonlinear control scheme, the design parameters are selected as λ 1 =2700, λ 2 =900, η 1 = 1, η 2 = 10, ψ 1 = 0.005, and ψ 2 = The observer gains are selected as l 1 = and l 2 = to locate the double observer poles at -400 when there are no parameter variations

Department of Electrical Engineering Southern Taiwan University 25 EXPERIMENTAL RESULTS Fig. 4. Speed response and q-axis current under no inertia variation (a) Conventional control scheme (b) Proposed control scheme

Department of Electrical Engineering Southern Taiwan University 26 EXPERIMENTAL RESULTS Fig. 5. Speed response and q-axis current under +200% inertia variation (a) Conventional control scheme (b) Proposed control scheme

Department of Electrical Engineering Southern Taiwan University 27 EXPERIMENTAL RESULTS Fig. 6. Speed response and q-axis current under +200% inertia and +20% flux variations (a) Conventional control scheme (b) Proposed control scheme

Department of Electrical Engineering Southern Taiwan University 28 EXPERIMENTAL RESULTS Fig. 7. Values of sliding surfaces s 1 and s 2 during speed transient under +200% inertia and +20% flux variations.

Department of Electrical Engineering Southern Taiwan University 29 CONCLUSION Based on a quasi-linearized and decoupled model, the design methods for the proposed control scheme have been given using the boundary layer integral sliding mode control technique Compared with the conventional nonlinear control scheme, the proposed robust nonlinear control scheme provides good transient responses under the inertia and flux variations. For the proposed control scheme, the chattering phenomenon and the reaching phase problem can be avoided by introducing the boundary layer integral sliding mode control technique. It can be said that the proposed control scheme has the robustness against the unknown disturbances. Therefore, it can be expected that the proposed control scheme can be applied to the high-performance applications such as the machine tools and industrial robots.

Department of Electrical Engineering Southern Taiwan University 30 REFERENCES [1] G. Champenois, P. Mollard, and J. P. Rognon, “Synchronous servo drive: A special application,” in IEEE-IAS Conf. Rec., 1986, pp. 182–189. [2] M. Fadel and B. De Fornel, “Control laws of a synchronous machine fed by a PWMvoltage source inverter,” presented at the EPE, Aachen, RFA, Oct [3] W. Leonhard, Control of Electrical Drives. Berlin, Germany: Springer-Verlag, 1985, pp. 240–246. [4] T. Rekioua, F. M. Tabar, J. P. Caron, and R. Le Doeuff, “Study and comparison of two different methods of current control of a permanent magnet synchronous motor,” in Conf. Rec. IMACS-TC1, vol. 1, Nancy, France, 1990, pp. 157–163. [5] B. Le Pioufle and J. P. Louis, “Influence of the dynamics of the mechanical speed of a synchronous servomotor on its torque regulation, proposal of a robust solution,” in EPE, vol. 3, Florence, Italy, 1991, pp. 412–417. [6] J. J. Carroll, Jr., and D. M. Dawson, “Integrator backstepping techniques for the tracking control of permanent magnet brush DC motors,” IEEE Trans. Ind. Applicat., vol. 31, no. 2, pp. 248–255, Mar./Apr [7] B. Le Pioufle, “Comparison of speed nonlinear control strategies for the synchronous servomotor,” in Electric Machines and Power Systems (EMPS). Philadelphia, PA: Taylor & Francis, 1993, vol. 21, pp. 151–169. [8] A. Isidori, Nonlinear Control Systems: An Introduction. Berlin, Germany: Springer- Verlag, 1985, pp. 156–163. [9] J. J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice- Hall, 1991, pp. 207–208, 277–291.

Department of Electrical Engineering Southern Taiwan University 31 Thanks for your attention