PRECISION CONTROL OF LINEAR MOTOR DRIVEN HIGH-SPEED/ACCELERATION ELECTRO-MECHANICAL SYSTEMS. Bin Yao
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1 PRECISION CONTROL OF LINEAR MOTOR DRIVEN HIGH-SPEED/ACCELERATION ELECTRO-MECHANICAL SYSTEMS Bin Yao Intelligent and Precision Control Laboratory School of Mechanical Engineering Purdue University West Lafayette, IN4797
2 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
3 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
4 LINEAR MOTOR VS ROTARY MOTOR Linear Motion Table Gear Leadscrew FLUX DENSITY (B) COIL COIL ASSEMBLY ASSEMBLY CURRENT (I) (I) FORCE (F=IxB f ) FORCE (F) Rotary Servo Motor MAGNET ASSEMBLY MAGNETIC ATTRACTION (Fa)
5 LINEAR MOTOR DRIVE SYSTEMS Mechanical simplicity (no mechanical transmission mechanisms), higher reliability, and longer lifetime No backlash and less friction, resulting in the potential of having high load positioning accuracy No mechanical limitations on achievable acceleration and velocity Bandwidth is only limited by encoder resolution, measurement noise, calculation time, and frame stiffness
6 CONTROL ISSUES OF LINEAR MOTOR Model Uncertainties Parametric uncertainties (e.g., load inertia) Discontinuous disturbances (e.g., Coulomb friction); external disturbances (e.g., cutting force) Drawback of without mechanical transmissions Gear reduction reduces the effect of model uncertainties and external disturbance Significant uncertain nonlinearities due to position dependent electro-magnetic force ripples (e.g. ironcore linear motors) Implementation issues (e.g., measurement noise)
7 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
8 MATHEMATICAL MODEL Linear Motor with Negligible Electrical Dynamics x 1 = x2 M x 2 = u Bx2 Ffn ( x2) Fr ( x1) + Fd ( x1, x2, t) y = x where x 1 x 2 1 : position M : mass of load/motor : velocity u : input voltage y : output B : viscous friction const F fn F d : nonlinear friction F : force ripple : lumped disturbance r
9 FRICTION MODELLING Coulomb Friction Discontinuous at Zero Velocity F fn A f F fn V(m/s) Continuous Friction Model Approximation F ( x ) = A S ( x ) fn 2 f f 2
10 FORCE RIPPLE MODELLING Cogging force is a magnetic force developed due to the attraction between the permanent magnets and the cores of the coil assembly. It depends only on the relative position of the motor coils with respect to the magnets, and is independent of the motor current Reluctance force is developed due to the variation of self inductance of the windings, which causes a positiondependent force in the direction of motion when current flows through the coils
11 MEASUREMENT OF FORCE RIPPLE P.6.4 Force ripple (v) Position (m) X-axis
12 FORCE RIPPLE MODELLING The identical and equally spaced permanent magnets of the linear motor at a pitch of P : F ( x1 + P ) F r ( x1 r = ) Approximation via first q harmonics function of position with period of P : where F ( x ) = S ( x ) A T r 1 r 1 r Fr ( x ) 1 A =, A, A ] r [ A r 1 s, A r 1 c, 2π 2π 2π q 2π q S ( 1) [sin( 1),cos( 1),,sin( 1),cos( 1)] T r x = x x x x P P P P rqs rqc T
13 Actual System Dynamics Parametrized Design Model where ~ ) ( x y d x S S x u x x x r T b f = + + = = θ θ θ θ θ ( ) ( ) T f f r r M x u Bx A S x S x A d = = ) ( ) ( r r fn fn F F F F d n n r b f d d d d A A B M = = = = = = ~,,,, θ θ θ θ θ ARC DESIGN MODEL ARC DESIGN MODEL where
14 ASSUMPTIONS AND OBJECTIVES Assumption: θ Ω θ = { θ : θmin < θ < θmax} d Ω = { d : d δ x,t } d ( ) Primary Objective: Synthesize a control input u such that the output y track the reference motion trajectory as closely as possible y r d y r (t) e(t) y t
15 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
16 DIRECT ADAPTIVE ROBUST CONTROL Define a switching-function-like quantity as where p = e + k1 e = x2 x2, x2eq = y d e= y y () d t k eq 1 e Error dynamics where T Mp u ϕ θ d = + + T ϕ = [ x 2eq, x2, S f ( x2), Sr ( x1),1] x = y k e 2eq d 1
17 DIRECT ADAPTIVE ROBUST CONTROL ARC control law u = u + u, u T = ϕ ˆ θ us = us1+ us2, us1 = k2p Error dynamics T Mp + k p = u ϕθ + d Choose robust control such that ii pus2 where ε is a design parameter Example i a s a 2 s2 p ~ ~ ( + ) T u ϕ θ d s2 ε u 1 = ( θ θ ϕ + δd ) 4ε s2 max min 2 p
18 THEORETICAL PERFORMANCE OF DARC τ τ = ϕp If the adaptation function is chosen as, then the ARC control law guarantees that In general, all signal are bounded. Furthermore, the positive definite function defined by V s is bounded by where λ = 2 k θ 2 V s 1 max 1 V s = Mp 2 exp( λt) V s 2 ε () + [1 exp( λt)] λ p 2 (t) upper bound of p 2 (t) p 2 ( ) 2ε λ M p 2 (t) t
19 THEORETICAL PERFORMANCE OF DARC In addition, zero final tracking error is achieved in the presence of parametric uncertainties only ~ (i.e., d =, t t ) p 2 (t) upper bound of p 2 (t) p 2 ( ) = p 2 (t) t
20 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
21 IMPLEMENTATION ISSUES OF DARC ϕ Regressor has to be calculated on-line based on the actual measurement of the velocity x 2 The effect of velocity measurement noise may be severe Slow adaptation rate has to be used Model compensation depends on the actual feedback of the state Creates certain interactions between the model compensation ua u a and the robust control Complicates the controller gain tuning process us
22 DESIRED COMPENSATION ARC DCARC control law u = u + u, u = where Error dynamics τ = ϕ p T a s a ϕd d T ϕ = [ y, y, S ( y ), S ( y ),1] d d d f d r d T T Mp = us ϕd θ + ( θ1k1 θ2) e + θ3[ S f ( y d) S f ( x2)] + θ4b[ Sr( yd) Sr( x1)] + d Notice that (applying Mean Value Theorem) where g ( and are certain nonlinear functions f x2, t) gr ( x1, t) ˆ θ addition S ( y ) S ( x ) = g ( x, t) e, S ( x ) S ( y ) = g ( x, t) e terms f d f 2 f 2 r 1 r d r 2
23 DESIRED COMPENSATION ARC Robust control function u = u + u, u = k p k s1 where is a nonlinear gain such that the matrix A defined below is p.d. For example s s1 s2 s1 s ( T k s k θ k + θ + θ g f k1θ 2 + k1θ 3g f θ 4bg r ) 2 A = 1 T 1 3 ( k1θ 2 + k1θ 3g f θ 4bg r) Mk k k + θ k θ θ g + ( θ k + θ k g + θ g ) T s f f 4b r 2θ 1k1 u s2 Choose such that ( T ~ ~ i p u ϕ θ + d ) ii pu s2 s2 d ε 2
24 THEORETICAL PERFORMANCE OF DCARC If the DCARC law is applied, then In general, all signal are bounded. Furthermore, the positive definite function defined by is bounded by where V V s V s = Mp + s exp( λt) V λ = min{ 2k2 θ1max, k1} s In addition, zero final tracking error (i.e., asymptotic tracking) is achieved in the presence of parametric uncertainties only 1 2 Mk 2 1 e ε () + [1 exp( λt)] λ 2
25 Digital implementation of the adaptation law where Velocity-free implementation of adaptation law Θ = = = + otherwise < and ˆ ˆ if > and ˆ ˆ if ] 1) [( ˆ imin i max imax i min i i i i T j θ θ θ θ θ θ θ = Θ T j T j i d i i i dt e k e T j 1) ( 1, ) ( ) ( ˆ ϕ γ θ + + = Θ T j T j i d T j T j i d T j T j i d i i i edt t e edt k T j 1) (, 1) (, 1) (, 1 ) ( ) ( ˆ ϕ ϕ ϕ γ θ IMPLEMENTATION OF DCARC LAW
26 LINEAR MOTOR POSITIONING STAGE
27 EXPERIMENTAL SETUP Pentium II 4 DS113 Encoder Interface Encoder Singals 5X multiplication Encoder Signals X-axis Y-axis MATLAB/Simulink Real-Time Workshlp Main Processor A/D,I/O MLIB MTRACE Real-Time Interface Complier A/D, D/A, Digital I/O PWM Amplifiers ControlDesk Hall Sensors Host PC Servo Controller X-Y Stage (Driven by two linear motors)
28 PERFORMANCE INDEXES Transient Performance: e M = max t { e( t) } Final Tracking Accuracy: e F = max t [ T 2, T ]{ e ( t ) } f f Average Tracking Performance: L 2 [ e ] = 1 T f T f e 2 dt Average Control Input: L 2 1 T T f [] u = u () t f 2 dt Degree of Control Chattering: L [ u] [ ] N 2 c u =, L2 u = u j T u ( j 1) T L2 u N j = 1 1 [ ] ( ) ( ) 2
29 COMPARATIVE EXPERIMENTS (Y-AXIS) PID with model compensation: ARC: DRC: u = ˆ θ () ( ) ˆ () ( ) ˆ 1 yd t + θ 2 y d t + θ 3() S f ( y ) K pe Ki edt K S f 2 ( y ) = arctan(9 y ), K π k 1 p = , K i = = 4, k2 = 32, ˆ() θ = [.5,.25,.1,] Γ = diag[5,,2,1] T 5, d e K d = 18 Same controller law with ARC but without parameter adaptation DCARC: k1 = 4, k2 = 32, ˆ() θ Γ = diag[25,,5,1] = [.5,.25,.1,] T
30 SINUSOIDAL TRAJECTORY TRACKING Set 1: - To test the nominal tracking performance, the motors are run without payload Set 2: - To test the performance robustness of the algorithms to parameter variations, a 2lb payload is mounted on the motor Set 3: - A large step disturbance (a simulated.5v electrical signal) is added at about t=2.5s and removed at t=7.5s to test the performance robustness of each controller to disturbance
31 TRACKING ERRORS (UNLOADED) (µ m) Tracking Error 6 4 PID DRC ARC DCARC Time (sec) y d =.5sin(4t)
32 TRACKING ERRORS (LOADED) 5-5 PID (µ m) Tracking Error -5 DRC ARC DCARC Time (sec) y d =.5sin(4t) 2lb Load
33 PARAMETER ESTIMATES (LOADED) Estimate of θ 1 Estimate of θ 3 Estimate of θ Time (sec) Time (sec) Time (sec) ARC y d =.5sin(4t) Estimate of θ 3 Estimate of θ 1 Estimate of θ Time (sec) Time (sec) lb Load Time (sec) DCARC
34 CONTROL INPUTS (LOADED) 2 1 PID (V) Control input DRC ARC DCARC Time (sec)
35 TRACKING ERRORS (DISTURBANCE) 5-5 PID (µ m) -5 DRC Tracking Error 5-5 ARC DCARC Time (sec) y d =.5sin(4t) 2lb Load Step disturbance between 2.4 and 7.4 sec
36 TRANSIENT PERFORMANCE e M PID DRC ARC DCARC Set1 Set2 Set3
37 FINAL TRACKING ACCURACY e F Set1 Set2 Set3 5 PID DRC ARC DCARC
38 CONTROL EFFORT L2[u] PID DRC ARC DCARC Set1 Set2 Set3
39 CONTROL INPUT CHATTERING Cu PID DRC ARC DCARC Set1 Set2 Set3
40 POINT-TO-POINTMOTION TRAJECTORY Positsion (m) Velocity (m/s) Acceleration (m/s 2 ) Time (sec) Time (sec) Time (sec)
41 UNLOADED EXPERIMENTS
42 POINT-POINT MOTION TRACKING ERRORS 5 Tracking Error (µ m) PID DRC DCARC Time (sec)
43 DARC PARAMETER ESTIMATES Estimate of θ 3 Estimate of θ Time (sec) Estimate of θ Time (sec) Time (sec)
44 NONLINEAR VS CONSTANT GAIN Tracking Error (µ m) Tracking Error (µ m) Constant Gain Time (sec) Nonlinear Gain y d =.5sin(4t) Time (sec) 2lb Load Step disturbance between 2.4 and 7.4 sec
45 NONLINEAR VS CONSTANT GAIN Tracking Error (µ m) Tracking Error (µ m) Constant Gain Time (sec) Nonlinear Gain Time (sec) Point to Point Motion: acceleration 12m/sec 2 velocity 1 m/sec
46 LOADED EXPERIMENTS
47 TRACKING ERRORS WITH LOAD
48 SUMMARY OF DARC DESIGNS Unique Features of Direct Nonlinear Adaptive Robust Control Strategy: Nonlinear physical model based; easy to incorporate physical intuition into the controller design stage. Address nonlinearities associated with linear motor dynamics directly. Address parameter variations due to change of load, nominal value of disturbances, Effectively handle the effect of hard-to-model terms (e.g., uncompensated friction) Achieve high performance through the use of learning techniques such as on-line parameter estimation.
49 SUMMARY OF DARC DESIGNS Comparative Experimental Results Illustrate the above claims The proposed schemes outperform a PID controller with model compensation significantly in terms of output tracking accuracy Tracking errors for high-speed/high-acceleration movements are very small, even during transient period. Final tracking errors are mostly within measurement resolution level of 1 µm
50 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
51 OUTLINE Motivation and Issues Problem Formulation Direct Adaptive Robust Control (DARC) Desired Compensation DARC Indirect Adaptive Robust Control (IARC) Integrated Direct/Indirect ARC (DIARC) Conclusions
52 COMPARATIVE EXPERIMENTS Off-line estimates (no load): Off-line estimates (with 2lb load): Bounds θ =.27, θ =.273, θ = θ = ( V / m / s ) θ θ min max = [.2,.22,.2, 1] = [.12,.35,.2, T 1] Initial parameter estimates ˆ θ = [.5,.24,.5, ] T T
53 POINT-TO-POINT MOTION TRAJECTORY Positsion (m) Velocity (m/s) Acceleration (m/s 2 ) Time (sec) Time (sec) Time (sec)
54 TRACKING ERRORS (NO LOAD) 2 Tracking error without load in µm (a) DARC (b) IARC (c) DIARC time (s)
55 PARAMETER ESTIMATES (NO LOAD) solid: DARC.45 dotted: IARC θ 1.4 solid(thick): DIARC time (s)
56 PARAMETER ESTIMATES (NO LOAD) θ 2.3 solid: DARC.28 dotted: IARC.26 solid(thick): DIARC time (s)
57 PARAMETER ESTIMATES (NO LOAD).2.18 solid: DARC dotted: IARC.16 solid(thick): DIARC θ time (s)
58 TRACKING ERRORS (LOADED) 2 Tracking error with load in µm (a) DARC (b) IARC (c) DIARC time (s)
59 PARAMETER ESTIMATES (LOADED) θ 1.8 solid: DARC.7 dotted: IARC solid(thick): DIARC time (s)
60 PARAMETER ESTIMATES (LOADED) θ solid: DARC dotted: IARC solid(thick): DIARC time (s)
61 PARAMETER ESTIMATES (LOADED).2.18 dotted: IARC solid(thick): DIARC θ solid: DARC time (s)
62 CONCLUSIONS Unique Features of Integrated Direct/Indirect Adaptive Robust Control Strategy: Nonlinear physical model based; easy to incorporate physical intuition into the controller design stage and estimator designs. Effectively deal with nonlinearities, parameter variations, and hard-to-model terms through both non-linear high-gain feedback and fast adaptation Separate estimation process from control law design to achieve accurate parameter estimates Achieve best tracking performance through the use of both estimation based parameter adaptation and the fast dynamic compensation based parameter adaptation.
63 ACKNOWLEDGEMENTS The project is supported in part by National Science Foundation -- CAREER grant CMS Purdue Research Foundation
64 More Information can be downloaded from:
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