Trajectory optimization for soft landing of fast-switching electromagnetic valves

Transcription

1 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, 211 Trajectory optimization for soft laning of fast-switching electromagnetic valves T. Glück W. Kemmetmüller A. Kugi Automation an Control Institute, Vienna University of Technology, Gusshausstrasse 27 29, 1 Vienna, Austria ( {glueck, kemmetmueller, kugi}@acin.tuwien.ac.at) Abstract: The esign of a feeforwar controller that facilitates soft laning an time optimality of a fast-switching electromagnetic valve is presente. In particular, a mathematical moel of the consiere pneumatic switching valve is evelope an parametrize by means of nonlinear parameter ientification. Base on this moel, the input constraine point-topoint quasi-time-optimal control problem is formulate an the resulting two-point bounary value problem is numerically solve. Due to the input constraints, the quasi-time-optimal control trajectories show bang-bang behavior. The performance of the numerically etermine trajectories are emonstrate by means of measurement results on an experimental test bench. Keywors: electromagnetic valve, fast-switching valve, soft laning, feeforwar esign, optimal control 1. INTRODUCTION Fast-switching solenoi valves are use in various fiels of applications. For instance in the manufacturing inustry, fast sorting tasks are performe by means of short air pulses. In automation applications, pneumatic piston actuators are often controlle by means of pneumatic pulsewith moulation an in the automotive inustry, internal combustion engines are controlle by variable gas exchange timings. Common requirements of these applications are short switching times an minimal impact velocities of the plunger in orer to eliminate acoustic noise, to avoi amage of mechanical components or at least to reuce mechanical wear. Different approaches reporte in the literature aress the soft laning problem, covering cycle aaptive controllers such as repetitive learning control(tai et al., 21), iterative learning control (Hoffmann et al., 2) an extremum seeking(peterson an Stefanopoulou, 2). Feeback control strategies using state observers have been reporte by Eyabi an Washington (2); Peterson et al. (2). Koch et al. (22); Chung et al. (27); Chlany an Koch (28) enhance these concepts by feeforwar controllers, which are esigne by exploiting the ifferential flatness properties of the unerlying mathematical moel. In orer to minimize the transition time of the plunger from one valve seat to the other, the allowe input voltage range shoul be fully utilize. The flatness-base esign methoology, however, oes not allow to incorporate these constraints. One possibility of consiering the input constraints in the flatness-base esign is to assume a specific smooth trajectory shape, cf. Petit et al. (21); Chung et al. (27). In this work, a feeforwar controller for a fast-switching solenoi valve is esigne by point-to-point quasi-timeoptimal control, which enables the incorporation of input constraints in a irect way. The work is structure as follows: In Section 2, a mathematical moel of the consiere solenoi valve is evelope, which is parametrize by nonlinear ynamic least-squares ientification in Section. The optimal control problem is formulate in Section an numerical results are shown in Section 5. Measurement results from an experimental test bench are given in Section. 2. MATHEMATICAL MODEL The mathematical moel of the consiere fast-switching valve can be separate into three subsystems: the moel of the electromagnetic subsystem, the mechanical subsystem an the pneumatic subsystem. Since measurement results of the consiere fast-switching valve confirm that the valve is pressure-balance, the pressure forces acting on the plunger will be neglecte. In aition, it will be assume that the flow force is small in comparison to the magnetic force. Since no internal feeback from the pneumatic ynamics to the electromechanical subsystem is consiere, the optimal control problem can be formulate with the pneumatic subsystem being neglecte. 2.1 Electromagnetic subsystem In Figure 1, the equivalent magnetic circuit of the fastswitching valve is given. It comprises the flux-epenent effective core reluctance R fc (Φ fc ), the effective reluctance R fp oftheplunger,theeffectivereluctancer g (s)oftheair gap between the core an the plunger, an the reluctance R l which accounts for the leakage fluxes. The reluctances are moele as l fc µ µ fc (Φ fc )A fc, R fp = l fp µ µ fp A fp, R fc (Φ fc ) = R l = l l, µ A l R g (s) = 2s. µ A g (1) Here, l fc, l fp, an l l are the effective lengths of the core, the plunger, an the leakage flux lines, respectively. A fc, Copyright by the International Feeration of Automatic Control (IFAC) 1152

2 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, 211 Φ fc c c,u c,u plunger R fc (Φ fc ) R l R g (s) s u s l c c,l c,l m v f c Θ Φ l R fp Φ fp cushione limit stops c v v s,w f m Fig. 1. Reluctance moel of the fast-switching valve. A fp, an A l are the corresponing effective areas. The effective length of the air gap is 2s, since there are two air gaps between the core an the plunger, cf. Figure 2. The corresponingareais enote by A g. Furthermore,µ enotes the permeability of air. The relative permeability µ fp of the plunger is assume to be constant. However, saturation of the core is phenomenologically moele as µ fc (Φ fc ) = ( k 1 ( Φfc A fc ) Φ fc ) 1 e k2 A fc +k, (2) with the constant parameters k i, i {1,2,}. From the magnetomotive force Θ = Ni, where i is the current an N is the number of winings, the flux Φ fc through the coil is given in the form Φ fc = Θ R. () Here, the equivalent reluctance R of the overall system reas as R(s,Φ fc ) = R fc (Φ fc )+ R l(r g (s)+r fp ). () R l +R g (s)+r fp Base on the flux linkage ψ = NΦ fc of the coil, Faraay s law yiels ψ = Ri+v, (5) t where R is the electric resistance an v is the voltage applie to the coil. The coil current i can be expresse in terms of the flux linkage an the air gap as i = R(s,ψ) N 2 ψ. () From the magnetic energy W m = = ψ ψ i(s, ψ) ψ R fc ( ψ) N 2 ψ ψ + 1 2N 2 R l (R g (s)+r fp ) R l +R g (s)+r fp ψ 2 the magnetic force f m (see Figure 2) can be euce as f m (s,ψ) = W m s = 1 R 2 l R g (s) 2N 2 (R l +R g (s)+r fp ) 2 ψ 2. s 2.2 Mechanical subsystem (7) (8) The overall mathematical moel is complete by the mass balance of the plunger. Figure 2 shows a schematic magnet Fig. 2. Schematic iagram of the valve. iagram of the fast-switching valve. Here, w is the plunger velocity. The mass of the plunger is enote by m v, the stiffness of the loa spring by c v, an the viscous amping coefficientuetothefrictionofthehousinganthesealing elements by v. The plunger is loae by the magnetic force f m (s,ψ) an a contact force f c (s,w). The latter moels the contact of the plunger with the lower an upper limit stops, which are reache at s = s l an s = s u, respectively. That is, f c (s,w) = s = [s l,s u ]. The mass balance of the plunger reas as t s = w (9a) t w = 1 ( f m (s,ψ)+f c (s,w) c v (s l c ) v w). m v (9b) Herein, c v l c enotes the preloaing force resulting from the loa spring. The limit stops of the plunger are cushione by viscoelastic sealings, which can be escribe by means of a linear Kelvin-Voigt moel c c,u (s s u ) c,u w for s > s u f c (s,w) = c c,l (s s l ) c,l w for s < s l else. i v (1) Strictly speaking, the limit stop woul entail a switche systemwithswitchingconitionsats = s l ans = s u.the switching structure, however, may inorinately complicate the optimal control problem. Alternatively, the contact characteristics is smoothe a by tanh-function, i.e., f c (s,w) =(α l β l w) 1 2 ( tanh(η l(s s l ))+1) +(α u +β u w) 1 2 ( tanh(η u(s s u )) 1). (11) The steepness of f c (s,w) can be ajuste by the parameters η l an η u. Moreover, β l an β u enote the viscous amping coefficients. In orer to approximately follow the linear spring characteristics at the contact points, the stipulation f c s = c c,i 1 s=si,w= 2 α iη i, i {l,u} () ismae.theparametersα l anα u eterminethemaximal spring forces of the contact moel. Figure shows the 115

3 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, 211 static contact force fc α l α u s l position s Fig.. Static contact force moel. smoothe contact characteristics (11) for the parameters β u = β l = an η u = η l = 1 1/m.. PARAMETER IDENTIFICATION AND MODEL VALIDATION Several parameters of the mathematical moel cannot be irectly etermine from ata sheets. Therefore, a parameter ientification of the overall moel is performe in orer to fit the erive moel to measurement results. First the electromagnetic subsystem is parametrize. For this, the reluctance () is reformulate in the structural equivalent form R(s,ψ) = p 1 ψ e p2 ψ +p + p (p 5 s+p ) p +p 5 s+p, (1) with the parameters p 1 = k 1l fc µ NA 2, p 2 = k 2, p = k l fc, fc NA fc µ A fc p = R l, p 5 = 2 (1), p = R fp. µ A g The unknown parameters θ {R,p 1,p 2,p,p,p 5,p }, with the electric resistance R from (5), are foun from the nonlinear ynamic least-squares ientification task 1 T min (i i m ) 2 t θ T s.t. t ψ =v Ri, ψ() = ψ (15) i = R(s m,ψ) N 2 ψ. Here, i m an s m are the measurements of the current an the plunger position, respectively. The measurements were carrie out uring the time interval t (,T) for ifferent steps in the input voltage v. Figure shows results of the parametrize moel for two ifferent voltage steps v. The results confirm that the reluctance moel () is capable of approximating the real electromagnetic behavior of the valve in an excellent way. The effect of saturation is negligible for small current values, cf. Figure (a). However, for large currents, Figure (b) clearly emonstrates the necessity of moeling the nonlinear saturation phenomena. In aition, the inuctance L = R(s,ψ)/N 2 is shown in Figure. The magnetic force f m is calculate from the ientifie reluctance moel accoring to (8). Due to temperature epenency an the presence of hysteresis of the viscoelastic material, the propose force moel (11) is only an s u approximation of the real viscoelastic behavior. Thus, the ientification of the mechanical parameters from the mass balance (9) by applying again nonlinear ynamic leastsquares ientification fails. In fact, the spring constant c v as well as the preloaing force c v l c were obtaine from measurements in stationary conitions. The parameters of the contact moel an the viscous amping coefficient v were ajuste in orer to approximately reprouce the contact behavior with the sealing. Base on the evelope moel, the optimal control problem is formulate in the next section.. OPTIMAL CONTROL PROBLEM The mathematical moel (5) an (9) with the state vector x = [s, w, ψ] T R with the constraine affine input u = v R can be written in the form t x = f(x)+bu, x() = x, (1) with the initial conition x R. The control objective is to fin an optimal control u U = [ u, u +] that guarantees a minimal transition time T f for a set-point change (u,x ) (u f,x f ), t (,T f ), (17) with u() = u, = f(x )+bu, u(t f ) = u f, = f(x f )+bu f (18) an the terminal conition x(t f ) = x f R. Therefore, the input-constraine point-to-point optimal control problem Tf min J(u) = ϕ(t f)+ l(u)t u U s.t. t x = f(x)+bu, x() = x (19), x(t f ) = x f, u U = [ u, u +] has to be solve. The summan ϕ(t f ) = T f of the cost functional J(u) represents the time optimality, whereas the integral term l(u) = ru 2 /2, with r >, serves as a regularization in orer to avoi singular arcs in the quasi-time-optimal control problem. Introucing the Hamiltonian (Bryson an Ho, 1975) H(x,u,λ) = l(u)+λ T (f(x)+bu), (2) with the ajoint states λ R, an applying a time transformation t = T f τ that maps the time interval t (,T f ) onto τ (,1), the optimal control problem can be reformulate by means of Pontryagin s maximum principle (Athans an Falb, 19) in form of a two-point bounary value problem, i.e. τ x = T f (f(x )+bu ) (21a) ( ) T τ λ = T f (x) x f λ (21b) u = argminh(x,u,λ ) u U x=x (21c) with bounary conitions x () = x an x (1) = x f (22) an the transversality conition H(x,u,λ ) τ=1 = 1 (2) 115

4 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, 211 ψ in mvs ψ in mvs i in A L in H v in V normalize time (a) Set-point change of v = 2.5V. i in A L in H v in v Simulation Measurement normalize time (b) Set-point change of v = V. Fig.. Ientification results for ifferent voltage set-point changes. Measure air gap s m, simulate flux linkage ψ, measure an simulate current i m an i, an simulate inuctance L. s in µm w in m/s ψ in mvs normalize time τ λ 1 λ 2 λ normalize time τ v in V i in A f in N normalize time τ (a) Optimal state trajectories. (b) Optimal ajoint state trajectories. (c) Optimal voltage v, current i, magnetic force fm ( ), contact force fc ( ) an sum force fsum ( ). Fig. 5. Numerical results of the time optimal control problem for the opening motion (s u s l ). resulting from the free en time T f. The superscript refers to optimal variables. Owing to the input affine system representation (1), the minimization problem (21c) can be explicitly solve resulting in an optimal control function u for u u u = ξ(λ) = u for u ( u,u +) (2) u + for u u +, with u = 1 r (λ ) T b. (25) Note that λ T b = λ. In the limit case for r each time λ crosses zero, u switches between the limits u an u +. Then, the solution of the optimal control problem (19) is a bang-bang control, except for λ =. Also note that for r = the optimal control problem (19) is singular (Bryson, 1999) since u can neither be erive from the minimization problem (21c) nor from the relate first-orer necessary conition H/ u =. Then, the optimal control input may be euce from some total time erivatives of H/ u =, cf. Kelley et al. (197). 1155

5 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, 211 s in µm w in m/s ψ in mvs normalize time τ λ 1 λ 2 λ normalize time τ v in V i in A f in N normalize time τ (a) Optimal state trajectories. (b) Optimal ajoint state trajectories. (c) Optimal voltage v, current i, magnetic force fm ( ), contact force fc ( ) an sum force fsum ( ). Fig.. Numerical results of the time optimal control problem for the closing motion (s l s u ). 5. NUMERICAL RESULTS OF THE OPTIMAL CONTROL PROBLEM Numerical solutions of the two-point bounary value problem (21)-(2) can be obtaine using the Matlab function bvpc(), cf. Shampine et al. (2). Figures 5 an show numerical results of the time-optimal point-to-point transition for the opening (u =,x = [s u,, ] T ) (u f = v l,x f = [s l,, ψ l ] T ) an for the closing (u = v l,x = [s l,, ψ l ] T ) (u f = v u,x f = [s u,, ψ u ] T ) of the fast-switching valve within the normalize transition time τ (,1). Here, v l an ψ l enote the resulting set-point voltage an flux linkage, respectively, cf. (18), at the upper limit, an v u an ψ u the set-point voltage an flux linkage at the lower limit. Note that for all numerical solutions outsie the vertical ashe lines the initial an final values are hel constant for illustration purposes only. The optimal state trajectories for opening the valve are given in Figure 5(a). Figure 5(c) shows the corresponing optimal voltage v, which is nearly bangbang because r in the penalty term is chosen very small r = 1 8 1/V 2. Moreover,Figure 5(c) contains the optimal current i, the magnetic force fm, the contact force fc an the sum of the forces fsum acting on the plunger. The optimal ajoint states are shown in Figure 5(b). They exhibit oscillations at the beginning of the consiere time interval. Although the incorporation of the contact moel (11) results in a locally numerical stiff moel, the oscillation is not a numerical artifact but inherent to the moel. Numerical results for ifferent contact stiffnesses c ci, i {u,l},ofthecontactmoelconfirmthisstatement. Figure shows analogous numerical results for closing the valve with r = 1 8 1/V 2. It is worth noting that in this case the solution of the optimal control problem is not purely bang-bang. Whenever λ vanishes (approximately for τ [.,.]), u oes so as well. This happens if fm vanishes, implying that only the spring force accelerates the plunger. For confirming the numerical results of the two-point bounary value problem, the optimal control problem was aitionally solve by full iscretization with SNOPT, see Gill et al. (2), resulting in the same optimal control trajectories.. IMPLEMENTATION AND EXPERIMENTAL RESULTS The obtaine numerical results were verifie on an experimental test bench. The power electronics use for riving the magnetic valve allows the accurate measurement of the current an the voltage. The position an velocity of the plunger were obtaine from a laser vibrometer (Polytec) an the measuring an control system SPACE 15 was utilize for ata processing. The experimental results reveale small moel inaccuracies, which may be attribute to temperature epenency of the coil resistance an of the contact behavior of the limit stops an to stick-slip effects. However, by slightly moving the switching-points of the input voltage, it is possible to open an close the valve in minimal time an with almost zero velocity at the limit positions. Figure 7 shows the measurement results for the opening an the closing of the magnetic valve. For the opening scenario in Figure 7(a), in contrast to the numerical results, the normalize opening time of τ m =.87τ was neee. It is evient, that by applying the structural equivalent input trajectory it is possible to minimize the impact velocity at the seals. The same applies to the valve closing in Figure 7(b). A normalize time ratio of τ m =.8τ was neee for the set-point change within minimal time. 7. CONCLUSION AND FUTURE WORK In this work, time-optimal feeforwar trajectories for fast-switching valves are presente. Nonlinear ynamic least-squares ientification was performe to parametrize 115

6 Preprints of the 18th IFAC Worl Congress Milano (Italy) August 28 - September 2, wm in m/s.1.2 wm in m/s.2.1 im in A.8. im in A.8. v in V normalize time τ m v in V normalize time τ m (a) Opening motion (s u s l ). (b) Closing motion (s l s u). Fig. 7. Measurement results from an experimental test bench. the evelope mathematical moel. The point-to-point quasi-time-optimal control problem was reformulate by means of Pontryagin s maximum principle an numerically solve by a irect approach. In the last part, the applicability of the time-optimal feeforwar trajectories is emonstrate by means of measurement results. Future work aresses the cycle-base aaption of the time-optimal feeforwar trajectory in orer to account for time-varying parameters an moel mismatches. ACKNOWLEDGEMENTS We thank Prof. Dr.-Ing Knut Graichen for his helpful iscussion on formulating an solving the optimal control problem. REFERENCES Athans, M. an Falb, P. (19). Optimal Control: An Introuction to the Theory an Its Applications. McGraw- Hill, New York. Bryson, A. (1999). Dynamic optimization. Aison- Wesley. Bryson, A. an Ho, Y. (1975). Applie optimal control. John Wiley & Sons, New York. Chlany, R.R. an Koch, C.R. (28). Flatness-base tracking of an electromechanical variable valve timing actuator with isturbance observer feeforwar compensation. IEEE Transactions on Control System Technology, 1(), 52. Chung, S.K., Koch, C.R., an Lynch, A.F. (27). Flatness-base feeback control of an automotive solenoi valve. IEEE Transactions on Control System Technology, 15(2), 9 1. Eyabi, P. an Washington, G. (2). Moeling an sensorless control of an electromagnetic valve actuator. Mechatronics, 1, Gill, P.E., Murray, W., an Sauers, M.A. (2). User s Guie for SNOPT Version 7: Software for Large- Scale Nonlinear Programming. URL sbsi-sol-optimize.com. Hoffmann, W., Peterson, K., an Stefanopoulou, A.G. (2). Iterative learning control for soft laning of electromechanical valve actuator in camless engines. IEEE Transactions on Control System Technology, 11(2), Kelley, H., Kopp, R., an Moyer, H. (197). Singular extremals, chapter, 11. Acaemic Press. Koch, C.R., Lynch, A., an Chlany, R. (22). Moeling an control of solenoi valves for internal combustion engines. In Proceeings of the IFAC Symposium on Mechatronic Systems, California, USA. Peterson, K.S., Grizzle, J.W., an Stefanopoulou, A.G. (2). Nonlinear control for magnetic levitation of automotive engine valves. IEEE Transactions on Control System Technology, 1(2), 5. Peterson, K.S. an Stefanopoulou, A.G. (2). Extremum seeking control for soft laning of an electromechanical valve actuator. Automatica,, Petit, N., Milam, M.B., an Murray, R.M. (21). Inversion base constraine trajectory optimization. In Proceeings of the 5th IFAC Symposium on Nonlinear Control Systems. St. Petersburg, Russia. Shampine, L., Glawell, I., an Thompson, S. (2). Solving ODEs with MATLAB. Cambrige University Press. Tai, C., Stubbs, A., an Tsao, T. (21). Control of an electromechanical actuator for camless engines. In Proceeings of the American Control Conference,