A Control of an Electromagnetic Actuator Using Model Predictive Control

Transcription

1 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering A Control of an Electromagnetic Actuator Using Model Predictive Control Paolo Mercorelli Abstract In permanent magnetic machines the nonlinearity due to the quadratic terms of the current makes difficulties in the control system. In order to cancel the nonlinearity, a strategy based on a pre-compensation action, which is conceived through two input partition matrices is presented. After canceling the nonlinearity a Model Predictive Control is used to obtain a positioning of the actuator. Simulation results are reported to validate the proposed technique. Key-words: Electromagnetic actuators, Geometric approach, Invariant subspaces, Model predictive control I. INTRODUCTION AND MOTIVATIONS In the past three decades, research on the geometric approach to dynamic systems theory and control has allowed to develop instruments and to become a powerful and a thorough tool for the analysis and synthesis of dynamic systems,, 3. Over the same time period, mechanical systems used in industry and developed in research labs have also evolved rapidly. Interests in robust control by using geometric approach have been developed during the last years 4, 5, 6. Recently the interest in this topic has increased in theoretical aspects and applications as well, see for instance 7, 8, particulary on problems like non-interaction and noise localization, see 9. As an important mechatronic component electromagnetic actuators are used in many industrial applications, in particular, in automotive and production systems. In production systems they are used for movements and precise positioning. Mechanical or hydraulic-mechanical components have been replaced by electromagnetic actuators due to their high efficiency, excellent dynamic behaviour and small size. In many applications these mechatronic actuators are designed for movements. Generally, there is a large variety of different electromagnetic actuators for motions. For long strokes, AC linear motor concepts are often preferred while for micro and nano meter applications special designs based on piezoelectric or magnetostrictive principles have been frequently investigated. For moving distances between 5 and 5 mm, however, DC linear motors, especially using permanent magnets as excitation, have been proven to be advantageous. Particular applications using electromagnetic actuator are presented in,, and 3 in which different kinds of actuators are used to generate movement of valves for engine applications. This paper deals with a DC permanent magnet linear motor used Paolo Mercorelli is with the Institute of Product and Process Innovation, Leuphana University of Lueneburg, Volgershall, D-339 Lueneburg, Germany Tel.: +49-() , Fax: +49-() mercorelli@uni.leuphana.de. for industrial applications to move heavy mass (more than Kg) with high precision, see Fig.. The objective of this permanent magnets coil ferromagnetic material, e.g. iron Fig.. ferromagnetic material,e.g. iron permanent magnet coil Geometry of the actuator, front (top) and side section (bottom). paper is to show a position controller design for the linear motor given in Fig.. A controller based on a controlled invariant subspace cancels the current nonlinearity. The presented actuator can be described by using four state variables. The system is nonlinear and time varying. In particular, the nonlinearities consist of quadratic current terms i (t) i (t) and nonlinear induced voltages u q(t). Because of the special form of the current nonlinearity it is possible to eliminate, or at least to reduce, the nonlinearity by using a controller based on a controlled invariant subspace. In 4 the authors considered the same problem, and they proposed a solution based on the decoupling and thus considered two controlled invariant subspaces to be transformed into two constrained controlled invariant ones. Moreover, in 4 the authors realized a controller based on a decoupling idea which is realized by these two constrained controlled invariant subspaces. This idea appears now, in

2 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering the light of this investigation, as not necessary. Therefore the proposed method in 5 represents an enhancement of the proposal in 4 realizing a very simple controller structure which is based on the idea to transform a unique controlled invariant subspace into an invariant one. This paper considers the approach presented in 4 and write explicitly the precompensation matrix, after that consider a Linear Model Predictive Control for the positioning of the actuator. The paper is organized in the following way. Section II is devoted to the model description. Section III shows the problem statement. Section IV shows possible procedure to cancel the nonlinearity in the actuator. Section (IV-A) shows a Model Predictive Control (MPC) for the positioning problem. The simulation results close the paper. The main nomenclature u in (t): input voltage vector i(t): coil current vector ẋ(t) = v(t): velocity of the armature u q (t): induced voltage vector R c : coil resistance L c : coil inductance B g : magnetic flux density vector Θ M : magnetic voltage sources of the permanent magnets Θ Coil = Θ Coil + Θ Coil : magnetic voltage source of both involved coils F L (t): Lorentz force F (t): disturbance force mini(a, B) = n i= Ai imb: minimum A invariant subspace containing im(b) F: feedback gain matrix S i : input partition matrix I i : currents matrix I i : currents subspace R Ii : current controlled subspace candidate II. MODEL DESCRIPTION The geometry of the considered linear motor is given in Fig.. For geometry details see. The device consists of an outer and an inner iron part. Permanent magnets are mounted on top of the inner iron bar. Their polarization is indicated by the element arrows. The actuator s coils are attached to the outer iron. Each coil is equipped with a separate voltage input. For controller design purposes the dynamical model of the linear motor must be identified. A. The electrical system Figure demonstrates the equivalent electrical circuit diagram of the actuator s coils. The electrical system of the coils is generally given by: u in (t) = R c i(t) + L c i(t) t + u q (t), u in Fig.. R c < i L c Equivalent electrical circuit diagram. where u in (t) represents the input voltage vector, i represents the coil current vector and R c and L c the resistance and the inductance of the coil windings. The induced voltages u q (t) = lv(t)b g are generated due to the armature speed v(t). Parameter l represents the coil length and B g represents the magnetic flux density vector in the air gap. From Fig. it is easy to see that B g = B g B g T, where B g = B g. A vectorial expression of the induced voltage is not required since the geometry boundaries are quadratical, see Fig.. Finite-element simulations prove that considering a quarter of the geometry is sufficient for modeling the magnetic system of the actuator (see Fig. 3). Thus, taking into account all self and coupling inductances and reducing the geometry only to two coils (quarter of geometry), electrical system equation () can be derived: with i (t) t i (t) t = Rc L cl c5 R cl c R cl c4 Rc L c3l c5 L cl c5 Lc Lc4 L c3l c5 u q i (t) i (t) u in(t) u q (t) u in (t) u q (t) L c = L + L 4 ; L c = L + L 3 L c3 = L + L 3 ; L c4 = L + L 4 L c5 = LcLc4 L cl c3, + where L and L are the self inductances of coil and coil. The remaining inductances in equations () represent the coupled inductances among the coils., () ()

3 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering B. The magnetic system Figure 3 represents the reduced, quarter geometry as well as its equivalent magnetic circuit diagram (compare with geometry, Fig. ). The expressions Θ M and Θ Coil represent Fig. 3. Θ Coil R mcoil R mair R mm Θ M R mfeo Rmσ > Φ σ R mfei Φ Feo Φ Fei Quarter of the geometry and equivalent magnetic circuit diagram. the magnetic voltage sources of the permanent magnets and the coils. The magnetic resistances of the airgap, the iron parts, the coil and the permanent magnets are expressed by term R m. The magnetic flux through the equivalent elements is represented by Φ. The element Θ Coil = Θ Coil + Θ Coil represents the magnetic voltage source of both involved coils, where Θ Coil = i N and Θ Coil = i N, with N as the number of windings. Θ M represents the magnetic voltage source of the permanent magnets Θ M = H c h m, where H c is the coercive field and h m the thickness of the permanent magnets in direction of the magnetic flux Φ. The calculation of all magnetic resistances is based on the wellknown reluctance equation R m = l µs, (3) where l represents the length of the magnetic circuit, µ is its magnetic permeability and S represents the surface involved in the magnetic circuit. The reluctance of the iron parts, the coils and the air gaps are merged to single circuit elements (see Fig. 3). The leakage flux reluctance R mσ is derived from Finite-Element simulations. Solving the magnetic circuit leads to the magnetic fluxes through the circuit elements from which the magnetic flux density B g in the air gap can be derived. One possibility to obtain the solution is the use of superposition. Therefore, only one magnetic voltage source is active at a time in order to calculate the magnetic fluxes within the network (Fig. 3). It is known that the superposition is applicable to linear systems only. Since nonlinear saturation effects are supposed to be implemented, a model based on superposition delivers correct results only for the linear range of the reluctance. Nevertheless, in order to involve nonlinear effects we designed an iterative equation system which determines linearized reluctance data for each magnetic flux calculation sequence. C. The mechanical system Involving the coil current i and the flux density B g of the mechanical system of the actuator is identified. The mechanical system is given by: ẍ(t) = F L(t) F fric (t) + F (t), (4) m where x(t) is the position of the armature. F fric (t) = k d ẋ(t) is the friction force and F (t) is the disturbance force acting on the armature. The mass of the armature is given by m. The Lorentz force F L (t) = l(i (t)b g + i (t)b g ) is determined by the electrical and by the magnetic system. Assuming quadratically shaped geometry boundaries as shown in Fig., scalar Lorentz force expression is applicable. Using superposition, the Lorentz force expression can be formulated as: F L (t) = l ( N(i (t) i (t)) ) + g(t)(i (t) i (t)), A Coil R mres (5) where R mres is the magnetic resistance of the equivalent circuit (Fig. 3) with a short-cut voltage source Θ M. A Coil represents the coil area in flux direction. Expression g(t) substitutes ( ) R mσ Φ σ g(t) =, (6) R mair + R mcoil + R mfeo where Φ σ is the leakage flux generated by the permanent magnets. g(t) is time varying since R mfeo is subject to saturation effects. The presented system model was validated with finite-element programs. The modelling technique can be considered as a paradigm for similar actuator systems. III. PROBLEM STATEMENT The nonlinear influence of term N(i (t) i (t)) in equation (5) can be easily seen. It is obvious that this nonlinearity leads to problems of control. In this paper the following problem can be defined: To control the state variables i (t) and i (t) so that i (t) = i (t). In so doing, the nonlinear term N(i (t) i (t)) in equation (5) is canceled. The electrical model part of the system described in equation () is linear and the following problem can be formulated. Problem : (Invariance) Given the system represented by (), under the assumption u q (t) = determine, if possible, a state feedback u(t) = Fi(t) and an input partition matrix S i such that, for the state i(t) (i (t) and i (t)), and two reference identical input functions r(t), the following relationship i (t) = i (t) (7)

4 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering holds. Remark : Condition (7) allows to cancel the nonlinearity of the actuator as stated in (5). The idea is to realise condition (7) as a permanent working condition of the actuator. Given the dynamic system in equation () of a permanent magnetic machine, and let Rc R cl c A = L cl c5 L cl c5l c5 ; (8) B = R cl c4 Rc L c3l c L cl c5 Lc Lc4 L c3l c5, (9) then a controller based on a state feedback gain F and an input partition matrix S i which solve Problem. To be more precise, we look for an invariant and stabilizing state feedback matrix F, along with input partition matrices S i with i =,., such that, for the dynamic triples the requirements (I i, A + BF, BS i ), () R Ii = mini(a + BF, BS i ) imi i () can be achieved. In other words, we have to find the invariant subspace R Ii which depends on the actuator model parameters A and B. For the currents i (t) and i (t) for instance, this subspace is a subspace of controllability and it can be expressed by mini(a+bf, BS i ). This formulated criterion holds if the controlled subspace lies within the following subspace: I i R Ii = imi i. () In addition, the partition matrices S i satisfies the following relationships im(bs i ) = imb R Ii. (3) expression im of a matrix, e.g. imi i or imb, represents the image of that matrix which indicates the subspace (spam) created by the columns of this matrix. An invariant controllable subspace consists of state space vectors reachable through trajectories entirely lying in the subspace R Ii. Moreover, the trajectories lying in this subspace remain in this subspace. IV. ANALYSIS OF POSSIBLE PROCEDURES USING A GEOMETRIC APPROACH BASED ON A CONTROLLED INVARIANT SUBSPACE In 4 the authors proposed a decoupling control structure in order to achieve the cancelation of the nonlinearity stated by (5). Considering the following current subspaces: It is known that, mini(a, B) = P n i= Ai imb is the minimum A invariant subspace containing im(b). I (,) = im, I (,) = im. (4) In this approach, taking into account conditions () and (3), the input partition matrices S i(.,.) can be determined by: S i(,) = L cl c5 Lc Lc4 L c3l c5 S i(,) = L L c3l c5 c L cl c3l cl c4 L cl c3l c4l c5 L cl c3l cl c4, (5). (6) To complete the calculation of the input partition matrix it follows: S i(,) = L cl c5 Lc Lc4 L c3l c5. (7) Through symbolic calculation the following result is obtained: S i(,) = L cl cl c3l c5 L cl c3l cl c4 L L cl c5 c3 L cl c3l cl c4. (8) In 4, in order to satisfy condition (), matrix F must be designed such that: Rc R cl c L cl c5 L cl c5l c5 + R cl c4 Rc L c3l c L cl c5 Lc Lc4 L c3l c5 F = λ λ. (9) In order to obtain i (t) = i (t) in equation (9) the two eigenvalues of the dynamics of the currents must be the same and this can be calculated by using matrix F. Moreover, after that, the control system need proportional factor in one of the inputs to obtain the required condition i (t) = i (t). Parameter λ represents the eigenvalues of the desired electrical system. For stability, λ must lie in the negative real plane. Adjusting the values of λ we can obtain a desired dynamics of the electrical system of the actuator which influences the whole dynamics of the actuator. After the cancelation due to the currents compensation, considering i(t) = i (t) = i (t) the following state variables can be chosen x f (t) = i(t) y(t) v(t), () 3

5 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering where y(t) = x(t) and v(t) = ẋ(t), then the following matrices can represent the system: Aw = 6 4 ( Rc RcL c + L c L c5 L c L c3 L c5 B w = RcL c4 + Rc ) L c L c3 L c5 L c3 L c5 lbg 4lg A Coil k d m L cl c5 Lc Lc4 L c3l c , (), () in which g(t) is considered constant because of its small variability. A. Solving a linear position MPC optimization problem Considering the models described by the matrices in () and () in which Euler discretization is considered with k = nt s, n N, where T s is the sampling time, if y(t) = x(t) is the position of the actuator which is assumed to be the controlled output, and v(t) = ẋ(t), then the following system is obtained: where x f (k + ) = u (k) A wk x f (k) + B wk u (k) y(k) = H k x f (k), (3) x f (k) = i(k) y(k) v(k) (4) and matrix H k = is the output matrix which determines the position of the valve according to the whole system represented by matrix A wk. In the MPC approach just two samples are considered: y(k + /k) = H k A wk x f (k) + H k B wk u mpc (k) (5) y(k + /k) = H k A wk x f(k) + H k A wk B wk u mpc (k) + H k B wk u mpc (k + ). (6) Equations (5) and (6) can be vectorially expressed as where and F p = Y(k) = G p x f (k) + F p (k)u mpc (k), (7) U mpc (k) = Hk Bwk H k Awk Bwk H k Bwk u mpc (k) u mpc (k + ), G p =, (8) Hk Awk H k A wk If the following performance criterion is assumed, J = N j= ( ) TQp ( y d (k + j) y(k + j) y d (k + j)) ) y(k + j) + N ( TRp u mpc (k + j)) u mpc (k + j), j=. (9) where y d (k + j), j =,,..., N is the position reference trajectory (desired trajectory) and N the number of samples of the prediction horizon, and Q p and R p are non-negative definite matrices, then the solution minimizing performance index (9) may be then obtained by solving J u mpc =. (3) A direct off-line computation may be obtained explicitly as u mpc = (F T p Q pf p + R p ) ( F T p Q p ( ) Y dp (k) G p x f (k), (3) where Y dp (k) and Y p (k) are the desired output column vector and the measured or observed output vector. V. SIMULATION RESULTS The presented case considers a mass to be moved that equals Kg as a moving mass. The mass must be moved to reach a position of 8mm with respect to the initial one. In Fig. 4 the simulation results of the proposed approach Inputs (V) Input of the first coil Input of the second coil Fig. 4. Currents (A) Fig Inputs voltage u (t) and u (t) of the coils. Current of the first coil Current of the second coil Current i (t) and i (t) in the coils of the actuator. are shown. In particular, Fig. 4 shows input voltage coils and 4

6 Proceedings of the 3 International Conference on Applied Mathematics and Computational Methods in Engineering Velocity (m/sec.) Position (m) 9 x Desired position Obtained position Fig. 6. Position of the actuator Fig. 7. Velocity of the actuator. Fig. 5 shows compensated coil currents. In Fig. 6 the effects of these simulations on the position and in 7 on the velocity of the actuator are shown. VI. CONCLUSIONS The paper deals with a control for a permanent magnetic machine. The controller is designed in order to cancel the nonlinearity of the proposed machine. The proposed approach uses the invariant subspace theory. The method proposes a very simple controller structure which is based on the concept of the invariant subspaces to cancel the nonlinearity. After canceling the nonlinearity, a linear MPC strategy is used for the problem of the positioning. ACKNOWLEDGEMENTS Particular thanks are directed to Steffen Braune of the IAI (Institute for Automation and Informatics), Wernigerode (Germany) and to Kai Lehman for the work which he spent in IAI for technical support. This work would not have been accomplished without them. VII. REFERENCES G. Basile and G. Marro. Controlled and conditioned invariants in linear system theory. Prentice Hall, New Jersey-USA, 99. A. Isidori. Nonlinear Control Systems. Spring-Verlag, August W.M. Wonham. Linear multivariable control: a geometric approach. Springer Verlag, New York, S.P. Bhattacharyya. Generalized controllability, controlled invariant subspace and parameter invariant control. SIAM J. Alg. Disc. Meth., 4(4):59 533, December G. Marro and F. Barbagli. The algebraic output feedback in the light of dual-lattice structures. Kybernetika, 35(6):693 76, D. Prattichizzo and P. Mercorelli. On some geometric control properties of active suspension systems. Kybernetika, 36(5):549 57,. 7 P. Mercorelli. Robust decoupling through algebraic output feedback in manipulation systems. Kybernetika, 46(5):85 869,. 8 P. Mercorelli. Geometric structures for the parameterization of non-interacting dynamics for multi-body mechanisms. International Journal of Pure and Applied Mathematics - IJPAM, 59(3):57 73,. 9 P. Mercorelli. Decoupling dynamic regulator with a minimum variance error estimator for online parameters indentification of permanent magnet three-phase synchronous motors. In Proceedings of the 6th IFAC Symposium on System Identification, pages , Brussels,. D. Howe and Z.Q. Zhu. Status of linear permanent magnet and reluctance motor drives in europe. nd International Symposium on Linear Drives for Industry applications LDIA 98, pages 7, 998. P. Mercorelli. A two-stage augmented extended kalman filter as an observer for sensorless valve control in camless internal combustion engines. IEEE Transactions on Industrial Electronics, 59(): ,. P. Mercorelli. A hysteresis hybrid extended kalman filter as an observer for sensorless valve control in camless internal combustion engines. IEEE Transactions on Industry Applications, 48(6):94 949,. 3 A. Fabbrini, A. Garull, and P. Mercorelli. A trajectory generation algorithm for optimal consumption in electromagnetic actuators. IEEE Transactions on Control Systems Technology, (4):5 3,. 4 P. Mercorelli, K. Lehmann, and Steven Liu. On robustness properties in permanent magnet machine control using decoupling controller. In Proc. of the 4th IFAC International Symposium on Robust Control Design, ROCOND 3, Milan, (Italy), 3. 5 P. Mercorelli. On the nonlinearity compensation in permanent magnet machine using a controller based on a controlled invariant subspace. Procedia Engineering a journal of Elsevier Science, 5:59 6,. 5