Cross-coupled Model Predictive Control for Multi-axis Coordinated Motion

Cross-coupled Model Predictive Control for Multi-axis Coordinated MotionSystemsXu Zhang1, 2, Kuanyi Zhu2, Xueyou Yang11. College of Precision Instrument and Optoelectronics Engineering, Tianjin University,Tianjin 300072, ChinaE-mail: redheartzx@2. School of Electrical and Electronic Engineering, Nanyang Technological University,Singapore 639798, SingaporeAbstractA cross-coupled model predictive controller for multi-axis coordinated motion systems is presented by incorporating cross-coupled strategy into the model predictive control architecture. The control law is generated by minimizing a cost function for the augmented system model in which the synchronization errors are embedded. Simulation and experimental results conducted on a high-precision positioning system with two permanent magnet linear motors (PMLMs) demonstrate the effectiveness of the proposed approach.1.IntroductionHigh-precision multi-axis motion systems are essential elements in modern advanced manufacturing. The term “axis of motion” refers to one degree of freedom, or forward and backward motion along one direction, which may be linear or rotary motion. When two or more axes of motion are involved on a single machine, that machine is employing multi-axis motion. In precision motion control, besides good tracking performance of each individual axis is absolutely necessary, synchronization or coordination between multiple axes becomes another challenging problem needed to be solved [1].Demand for high-precision control of multi-axis motion systems calls for the development of high performance positioning devices and accompanying control approaches. With the attractive features such as high dynamic performance and most importantly high positioning precision, permanent magnet linear motors (PM LM s) are increasingly widely being applied in multi-axis motion systems [2]. To harness the potential of PM LM s, the main challenge lies in the control approach.Cross-coupled control is an appropriate strategy used for coordinated control of processes with multiple motion axes. This concept has been successfully applied to reducing contour error of multiple motion axes in machine tool control [3]–[6], minimizing differential velocity error of two driving wheels in a mobile vehicle [7], and synchronizing the multi-robot for assembly tasks respectively [8]. However, the principal shortcoming of the existing control techniques is that they cannot explicitly incorporate plant model uncertainties in the formulation of the high precision control system when using the traditional PID or fussy method.M odel predictive control (M PC) algorithm is an effective approach to improve motion tracking and coordination performance of multi-axis systems due to its control acumen to deal with the multivariable constrained control problems. In the past two decades, the theory of MPC has been well developed nearly in all aspects, such as stability, nonlinearity, and robustness [9]–[13]. Several methods based on M PC are presented for better positioning and tracking precision of a single axis or motor [14]–[17]. A cross-coupling framework of generalized predictive control is investigated by compensating both tracking error and synchronous error between the process and the reference model [18]. In this paper, design and implementation of a cross-coupled model predictive control algorithm are presented for multi-axis motion systems. This paper is organized as follows. In Section 2, the dynamics of the PM LM is presented. Process model and MPC is formulated in Section 3. In Section 4, synchronization strategy is proposed. The cross coupled M PC controller for multi-axis coordinated motion systems is presented in Section 5. Simulations and experiments are performed in Section 6, and finally, conclusions are given in Section 7.International Conference on Advanced Computer Control2. Dynamics of the PMLMThe dynamics of the PM LM can be viewed as comprising of a dominantly linear model and a nonlinear remnant [19]. In the dominant linear model, the mechanical and electrical dynamics of a PM LM can be expressed as follows:d m MxDx F F ++= (1) ()ae a a a dI K xL R I u t dt++= (2)m f a F K I = (3)where x(t) denotes motor position; M ,D ,F m , and F d denote mechanical parameters, i.e., mass of the moving thrust block, viscosity constant, generated force, and system disturbance, respectively; u(t),I a ,R a , and L a denote electrical parameters, i.e., time-varying motor terminal voltage, armature current, armature resistance, and armature inductance, respectively; K f denotes an electrical–mechanical energy conversion constant; and K e is back electromotive force (EM F) voltage. The delay due to electrical transient response may be ignored since the electrical time constant is typically much smaller than the mechanical one. The following is the simplified model:()()()x axt bu t t ξ=++ (4) wheref e a K K R D a RM+=−fa Kb R M =()dF t M ξ=−.Note that Ӷ(t) includes extraneous nonlinear effectsthat may be present in the physical structure. Among them, the prominent nonlinear effect is due to ripple force arising from the magnetic structure of the PMLM and other physical imperfections, which can be viewed as a sinusoidal function of motor position with a period of Ԁ and amplitudes of A r and B r , i.e.,cos()sin()ripple r r F A x B x ωω=+ (5)where F ripple is ripple force. In particular, ripple force ismore complex in shape, e.g., due to variations in the magnet dimension.3. Process model and MPCA process with multiple PM LM s can be describedby the linear discrete-time CARIMA model:111()()()(1)()()/i i i i i i A q x t B q u t C q t ξ−−−=−+Δ(6)where t is the normalized discrete time, i denotes the corresponding motor of the process, x i (t) and u i (t) are the position and control input of the motor, respectively, Ӷi (k) represents system disturbance andunmodeled dynamics, A i (q -1)and B i (q -1) are the systempolynomials in the backward shift operator q -1,C i (q -1) is a polynomial used for robustness of the controller, and Ӕ is the differencing operator.In model predictive control, minimization of a cost function yields the control law. The cost function is:211{(()(|))Hpn c i i i k I w t k x t k t ===+−+¦¦,21((1|)}Hci i i k u t k t λ=+Δ+−¦ (7)where w i (t + k) is the reference signal, H p and H c are the prediction and control horizons, respectively, ӳi is a non-negative weighing factor for the control input. Based on MPC technique, the future output x i (t + k|t)can be expressed as(|)(1|)()i i ik ik x t k t G u t k t p t +=+−+,1,2,3,......,p k N = (8)where11k ik ik ik G G g q −+−=+ (9)()()(1)i ik ik ik i iiF H p t x t u t C C =+Δ− (10)G ik , F ik and H ik satisfy the Diophantine equationsk i i ik ik C A E q F −=Δ+ (11) k i ik i ik ik B E C G q H −=+ (12)with the corresponding degrees() () 1() max((), ())() max((), ())1ik ik iki i ik i i E G k F A C k H B C δδδδδδδδ==−=−=−­°®°¯(13) The matrix version of the predictor (8) is obtained asi i i i X G U P =+ (14)where[( 1|) ( 2|) ... ( |)]T i i i i p X x t t x t t x t N t =+++ (15) [( |) ( 1|) ... ( 1|)]T i i i i u U u t t u t t u t N t =ΔΔ+Δ+− (16)12 [() () ... ()]p N T i i i i P p t p t p t = (17)0100(1)(2)(1)(2)()000u u p p p u i i i i N i N i i N i N i N N i g g g G g g g g g g −−−−−…=…ªº«»«»«»«»«»«»«»«»¬¼%##%%%#%%# (18)4. Synchronization strategyIn multi-axis coordinated motion systems, both tracking performance and synchronization performance are specified. However, if the motors follow different trajectories, i.e., r i (t) r j (t), then a scalar of r i (t)/r j (t) should be introduced to describe the relationship between the ith motor and the jth motor accurately. So the synchronization error can be defined as() (() ()) ij ij i ij j t x t x t φαβ=− (19) where өij and Ӫij are the coupling and synchronizationfactors between the ith motor and the jth motor, respectively. M oreover, even if Ĩij (t) = 0 may be achieved at an instant time, the overall system may not be synchronized perfectly due to ӔĨij (t + 1) 0,where ӔĨij (t + 1) = Ĩij (t + 1) íĨij (t). In this case, the difference between the consecutive synchronization errors should also be considered as1() (1 )()ij ij t q t ψφ−=− (20)denoting[( 1|) ( 2|) ... ( |)]T ij ij ij ij p t t t t t N t φφφΦ=+++ (21) [( 1|) ( 2|) ... ( |)]T ij ij ij ij p t t t t t N t ψψψΨ=+++ (22)For the synchronization error and its difference, bysetting ӿij(t + 1|t) = 0,it can be obtained thatij ij L Ψ=Φ (23)where00001100011100011L −=−−−ªº«»«»«»«»«»«»¬¼""%##%%" (24)The cost function is introduced as1{()()}i i i i i i i nT T c i W X W X U U I λ=−−+=¦111n nij ij i j i T −==++ΦΦ¦¦ȁ (25)where[( 1) ( 2) ... ( )]T i i i i p W w t w t w t N =+++ (26)TI L L υη=+ȁ (27)and I is the unit matrix.5. Cross-coupled MPC controllerBy minimizing the cost function (25), the Cross-coupled MPC control law can be obtained:()()21111122122222312n m m m m n m m m m n n n W P P P U U W P P P U W P αβαβ=−==ΩΓ§ªº−−Λ−ªº¨«»«»¨«»−−Λ−«»¨«»«»¨«»«»¨«»¨¬¼−¬¼©¦¦##()()12222211210m m m m m n mn mn mmn n m P P P P αββαββ=−=·ªº¸«»Λ−¸«»+¸«»¸«»¸«»¸Λ−¬¼¹¦¦# (28)where Ө,ӓ,Өij are expressed as follows1111n n nn ΩΩΩ=ΩΩªº«»«»«»¬¼"#%#" (29)100T Tn G G Γ=ªº«»«»«»¬¼"#%#" (30)2, ,Ti i ij Tij ij j i i G G I if i jG G if i j λαρΟ+=ΩΛ<­°=®−°¯ (31) 122211()i nmi mi im m m i I αβα−==+Ο=+Λ+¦¦ (32)The block diagram shown in Figure 1 illustrates thestructure of cross-coupled MPC controller.Figure 1. Structure of cross-coupled MPCcontroller6. Simulation and experimentFigure 2. High-precision positioning controlsystemThe effectiveness of the proposed control approach will be illustrated by computer simulations and experiments on a high-precision positioning system with two PMLMs, as shown in Figure 2. The dynamics of the positioning system can be described by11120.653()11.006()x xt u t =−+ ()11.418sin 20.074x π+ ()10.313cos 20.074x π− (33)2225.917() 5.048()x xt u t =−+ ()20.676sin 20.051x π+ ()2cos 0.19120.051x π− (34)With a sampling interval T s = 0.0008 seconds, the corresponding linear discrete-time system models are given by111()() ()(1) ()/(1),i i i i i i A q x t B q u t C t q ξ−−−=−+− 1, 2i = (35)where11211122111112112() 1.000 1.9840.984() 1.000 1.9950.995() 3.502 3.483() 1.613 1.610 10.8,,,,.A q q q A q q qB q q B q qC C q −−−−−−−−−−−=−+=−+=−=−==−In this study, two PM LM s are desired to track thesame square wave. The parameters of the cross-coupled M PC controller are set as: N p = 10; N c = 4; ӳ=50,ӵ = 20,ӯ = 10,ө12 = 1,Ӫ12 = 1. Figure 3-5 show the desired tracking trajectories of two PMLMs, the simulation and experimental results for the cross-coupled MPC controller, respectively.Figure 3. Desired tracking trajectoriesFigure 4. Simulation results for the cross-coupled MPC controllerFigure 5. Experimental results for the cross-coupled MPC controller7. ConclusionTo achieve better synchronization performance while maintaining good tracking response, a cross-coupled M PC controller for multi-axis coordinated motion systems has been presented. Computer simulations and experiments on a high-precision positioning system with two PMLMs have shown that the improved performance of the control system can be obtained with the control parameters to be set as follows: first, N p ,N c and ӳare selected appropriately so that the control system provides a stable andsatisfactory tracking response and then, the weighting factorsӵandӯare introduced to reduce the synchronization errors caused by the mismatched dynamics, system disturbances, and so on. Although the same desired trajectory is considered in this study, different trajectories can be realized easily by setting the synchronization factorsөand Ӫʳappropriately.8. 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